Reviews in Computational Chemistry, Volume 15

Keviews in Computational Chemistry Volume 15

Keviews in Computational Chemistry Volume 15 Edited by

Kenny B. Lipkowitz and Donald B. Boyd

8W I LEY-VCH N E W YORK

CHICHESTER

WElNHElM * BRlSBANE

SINGAPORE

TORONTO

Kenny B. Lipkowitz Department of Chemistry Indiana University-Purdue University at Indianapolis 402 North Blackford Street Indianapolis, Indiana 46202-3274, U.S.A. [email protected]

Donald B. Boyd Department of Chemistry Indiana University-Purdue University at Indianapolis 402 North Blackford Street Indianapolis, Indiana 46202-3274, U.S.A. [email protected]

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Preface A Tribute to the Halcyon Days of QCPE Computational chemists deal with information. We use computers to calculate numbers and compare our numbers to experimental data, when available. To perform our calculations we need computer programs. In the realm of programs, one of the great boons to theoretical and computational chemists, especially in the earlier days of the field, was the Quantum Chemistry Program Exchange (QCPE). It is an appropriate time to pay tribute to QCPE because the man who managed it for the last 33 years, Mr. Richard W. Counts, retired at the end of July 1999. Not many months earlier, his long time coworker, Dr. Margaret (Peggy) Edwards, also retired from QCPE. We do not have space here-nor have we been able to obtain all the historical documents-to recount the full history of QCPE, but it is worthwhile to give a brief summary of an institution that had such a significant impact on the field as it exists today. The older readers of this book series know well QCPE, but some of our younger readers may not have a full appreciation of this organization. QCPE was founded at the inspiration of Professor Harrison Shull, a theoretician at Indiana University, Bloomington (IUB). His vision was to have a central, international repository of software used by quantum chemists, particularly ab initioists. At meetings and elsewhere, he convinced his fellow theoreticians of the advantage of exchanging computer programs. He pointed out that it was wasteful of the time of graduate students at every university to have to write a program to do the same quantum mechanical integral calculation that had already been programmed elsewhere. To avoid “reinvention of the wheel,” it made sense to have these widely needed, standard programs available. A second motivation for setting up a library of shared software was to create a more or less permanent repository. So if a graduate student finished a thesis and left a university or if a professor changed research interests, the fruits of their labors in terms of software created would not be lost or lie unused on some forgotten shelf. A third motivation for a central repository was to create an intermediary between the code writerdowners and users. Quantum chemistry professors whose students had created useful programs often shared copies with V

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other groups. However, the users in the other groups might not understand the requirements of operation or the limitations for getting useful results. Hence these users would constantly be asking the developers for help. For very widely used programs, such requests for help could eat up a significant amount of time and distract the original developer from other work. So, someone at a central depository could field at least some of these questions from the users, thereby freeing the developers from being constantly on call for technical support. There was general agreement among the theoreticians that an organization like QCPE would be useful, but it took the hard work and dedication of Dr. Shull and his colleagues at IUB to bring QCPE to reality. QCPE was launched in April 1962 with 23 pieces of software, mostly quantum chemistry subroutines, ready for distribution. Among the individuals who helped in the early days of operation were Dr. Keith Howell, Shull’s postdoctoral associate from England, and later Dr. Franklin Prosser. (Frank Prosser, incidentally, retired in 1999 from the IUB computer science department; his Ph.D. was in physical chemistry.) Dr. Stanley A. Hagstrom, another theoretician at IUB who became emeritus professor in 1994, also played a vital role in the birth of QCPE, especially at the technical level. He developed initial submission procedures, documentation guidelines, and distribution procedures. In late 1964, a significant event occurred in the life of QCPE when funding was secured from the Directorate of Chemical Sciences of the Air Force Office of Scientific Research, which was part of the U.S. Office of Aerospace Research. QCPE served as a conduit through which individual researchers could donate their programs. The programs were checked to make sure that they compiled, performed as claimed, and contained at least a minimal amount of documentation in the form of “comment cards” or separate write-up. Then the availability of the programs was announced through QCPE’s catalog, and the software was sent to individuals who paid the modest distribution and handling costs. In the 1960s, the software was distributed on computer cards and magnetic tape. Generally the programs were written in the then current versions of FORTRAN, and they ran on mainframe computers, such as the behemoths of International Business Machines (IBM) and Control Data Corporation (CDC). A few of the programs ran on the machines of other manufacturers such as Burroughs, Honeywell, Univac, and a sprinkling of smaller companies that have long since disappeared from the scene. In the early years, QCPE regularly published a list of its members, which was several pages long. QCPE also published a quarterly newsletter with news, announcements of new members, and progress reports from individual theoretical chemistry research groups around the world. Looking through the early issues of the newsletters, one sees familiar names like Ruedenberg, Simonetta, Moskowitz, Frost, Hall, Bishop, Kaufman, Michels, Rein, Calais, Trindle, Kestner, Simpson, Davidson, Coulson, Fischer-Hjalmars, ROOS, Ballhausen, Dahl, Bloor, Krauss, Lykos, Allen, Sutcliffe, Schaad, Csizmadia, Lehn, Momany, Pople, Berry, Debye, Freed, Hoffmann, Scheraga, Roothaan, Wahl,

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Bader, McWeeny, Cruickshank, Morokuma, Hermann, Snyder, Fukui, Ohno, Kutzelnigg, and many others who shaped the field in the 1960s. These reports increased awareness of what each group was currently working on, again to help avoid duplication of effort (and perhaps accelerate competitive races). By 1965, QCPE membership had grown to 425 individuals, it had a library of 71 programs, and 500 copies had been distributed. Back then, the most frequently requested programs were for running extended Hiickel molecular orbital calculations and for evaluating two-electron integrals as needed for what we now call ab initio calculations. In 1967 Dick Counts, with a physics background and a master’s degree, was hired from IUB’s Aerospace Research Applications Center to become administrator of QCPE. All through the 1970s QCPE continued growing and provided exemplary service to the community of theoretical chemists with an ever expanding library of programs. Some of the deposited programs ran without problems, but others were written very specifically for one machine or for just one machine configuration. Dr. Hagstrom and other colleagues at IUB provided assistance to QCPE by getting such programs operational on other machines. We do not have space here to list all the programs in the library, but among the ones then popular was CNDOANDO program (QCPE 141) from Paul Dobosh in John Pople’s group at Carnegie-Mellon University. Also, Pople’s group released Gaussian 76 to QCPE in 1978 (QCPE 368). (For a complete list of QCPE software, see QCPE‘s new website at http://qcpe.chem.indiana.edu/.) In 1971 QCPE received a grant from the National Science Foundation, which put the organization on a solid financial footing. Starting in 1973, QCPE became self-supporting. A very modest annual membership fee was charged members. Users purchasing software at the modest distribution cost was another source of revenue. In 1981 the Q C P E Newsletter was formalized as the Q C P E Bulletin. It was published quarterly and included short citable articles, editorials, announcements of newly deposited software, and news of interest to the community. Richard Counts was editor and Peggy Edwards, a former secretary at Eli Lilly and Company with a Ph.D. in English, was assistant editor. To make sure QCPE‘s role inside and outside the university was proceeding in the proper direction and staying current with changing events, a QCPE Advisory Board was appointed by Mr. Counts with approval of the chairman of the IUB chemistry department. Harry Shull was the chairman and the members were Norman L . Allinger, Harry F. King, Max M. Marsh, Horace Martin, David Pensak, and Michael Zerner. The industrial representatives were Mr. Marsh (a physical chemist and research advisor at Eli Lilly and Company, who was one of the first chemists to foresee the usefulness of computer-aided drug design) and Dr. Pensak (the group leader at DuPont, who was one of the first, if not the first, computational chemist interviewed for an article in the N e w York Times). Also in 1981 when Prof. Shull moved to a new position in California, Prof. Hagstrom was named director of QCPE, while Mr. Counts and Dr.

viii Preface Edwards continued to manage the day-to-day operations. In effect, Stan Hagstrom served as faculty advisor to QCPE. From April 1980 to April 1981, 451 programs were distributed to the United States, 212 to West Germany, 138 to Great Britain, 106 to Japan, and 77 to Switzerland. Also in 1981, the shortlived U.S. National Resource for Computational Chemistry (Lawrence Berkeley Laboratory, Berkeley, California) ceased operations and turned its software collection over to QCPE. Besides serving as a repository of software and producing the Bulletin, QCPE performed another valuable service in the 1980s. Mr. Counts organized annual summer workshops on Practical Applications of Quantum Chemistry. Most of these intense week-long courses were held at IUB, but one was held in Oxford, England, and another in Marlboro, Massachusetts. The workshops were taught by practicing computational chemists and offered hands-on experience in running important programs in QCPE’s holdings. Back in the early 1980s input data was still prepared on IBM punch cards, and the jobs were run on the mainframes at IUB. The workshops exposed 20-25 individuals each year to computational chemistry tools. Not all the individuals taking the courses were newcomers to the field; many were experienced users who had come to learn about the latest programs and the advantages and pitfalls of each method. Don Boyd was on the faculty of four of these workshops, and Ken Lipkowitz taught at three of them. The QCPE workshops were so effective at training users and generating revenues that other universities and organizations emulated them and captured most of the market for such courses after late 1980s. Many popular programs such as the molecular mechanics program MM2 from Allinger’s group and the semiempirical molecular orbital programs (MIND0 and MNDO) appeared in QCPE’s catalog. However, a significant milestone occurred in May 1983 when the MOPAC program (A General Molecular Orbital Package) was deposited by Dr. James J. P. Stewart. He was on extended leave from the University of Strathclyde, Scotland, and was working as a postdoctoral associate in Michael J. S. Dewar’s group in Austin, Texas. MOPAC (QCPE 455) became by far the most popular and influential program in QCPE’s offerings. The appearance of MOPAC coincided with the manufacture of the hugely successful VAX 11/780 superminicomputers from Digital Equipment Corporation (DEC). These machines changed significantly the way computational chemistry was being done at that time and expanded the horizon of computing for many chemistry departments. The late 1980s thus saw an increasing number of QCPE holdings that ran on departmental computers. Many programs that were originally developed for large mainframes were ported to these less expensive machines and eventually to personal computers. Another step in QCPE’s history occurred in 1984 when Professor Ernest R. Davidson was invited to move his group to IUB from the University of Washington, Seattle. Ernie Davidson replaced Stan Hagstrom as faculty advisor. Mr. Counts and Dr. Edwards continued to manage the day-to-day operations. To provide new viewpoints, the membership of the QCPE Advisory Board was

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rotated. Drs. Enrico Clementi and Isaiah Shavitt were enlisted in 1986, Drs. Donald B. Boyd and Gilda H. Loew were enlisted in 1987, Drs. Charles Bender and Herschel J. R. Weintraub were enlisted in 1989, Dr. Hare1 Weinstein was enlisted in 1991, and Dr. James J. P. Stewart was enlisted in 1992. In its heyday, QCPE distributed about 2500 programs per year. Mr. Counts hired students to help him and Dr. Edwards with the heavy workload. The software catalog became so thick that it was broken into subcategories. A standardized format for citing QCPE software was published in the QCPE Bulletin, and indeed QCPE programs have been widely cited in the scientific literature. The name of the organization and the bulletin was shortened from Quantum Chemistry Program Exchange to simply QCPE, indicating that the software library had evolved from being just about quantum chemistry to computational chemistry in general. The IUB chemistry department faculty-and especially the chairmen of the department-viewed QCPE as departmental ‘Lproperty.” Nevertheless, Richard Counts ran the operation essentially independently, and many QCPE members were unaware of the departmental ties. The revenues QCPE generated at its zenith tempted some chemists at IUB to dream of QCPE as a potential source of research funds or as a way to leverage donations of computer hardware from major manufacturers. However, QCPE was primarily a service to the community; it did not become a big revenue generator. But the organization held its own financially and contributed to the international visibility of the IUB chemistry department. One of the reasons for the popularity of QCPE was that most of the programs distributed were in the form of source code. By obtaining source code, other researchers could extend, modify, and perhaps improve a piece of software. In contrast, few of the software companies that sprang into existence in the 1980s to serve the growing computational chemistry market distributed their source codes. Nevertheless, a number of factors undermined the important role QCPE was playing. The 1980s and 1990s witnessed the commercialization of software by increasingly large companies in the computational chemistry business. Customers had to buy commercial versions of MOPAC, AMPAC, MM3, Gaussian, and other popular programs to obtain the latest versions with the most features and most bugs fixed. The QCPE software holdings became less relevant to the present-day mode of computing with graphical user interfaces (GUIs).By 1990, the QCPE library did have some programs with GUIs, as well as some elaborate programs qualifying for the name LLsystem”or “package.” Another major trend in the 1990s was the emergence of the Internet, which afforded individuals easy, independent ways of distributing software they produced. The healthy flow of new programs being deposited in QCPE gradually diminished. The number of programs being requested also dropped in the 1990s, although exact figures are unavailable. Software had been deposited by American chemists as well as researchers in many countries besides the United

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States. However, distribution of programs in the last 10 or so years was largely out of the United States. Interestingly, the Japanese remained some of the main customers of QCPE’s holdings. Another trend impacting QCPE was the fact that users wanted and expected technically supported software: that is, they wanted to be able to call a toll-free telephone number and ask questions about the operations of a program. Mr. Counts provided some support to QCPE “customers” on an ad hoc basis. His efforts at keeping QCPE running are to be applauded, but it was hard for one person to compete with the large software companies. The last major tribute to QCPE was at a symposium organized “in honor of R. W. Counts for service to the field of computational chemistry” held by the Computers in Chemistry Division (COMP) of the American Chemical Society at the 207th National Meeting, March 13-17, 1994, San Diego, California. At the half-day symposium, Harry Shull (then provost at the Naval Postgraduate School, Monterey, California, and now retired) reminisced about “The QCPE Experiment.” Professor N. L. (Lou) Allinger (University of Georgia) spoke on “Funding Computational Chemistry in the 80’s and 90’s’’ and pointed out the need for professors to sell software in order to make up for the increasing difficulty in obtaining government grants. Professor M. C. (Mike) Zerner (University of Florida) explained the “Whys and Why-Nots of Commercially Distributed Software” and used the opportunity to answer a few critics of his handling of his ZINDO semiempirical MO program. Finally, Richard Counts spoke on “Thirty Years of the Software Support Problem.” He explained his philosophy that the ownership of the programs in the QCPE library remains in hands of the developers who submit them. He saw the role of QCPE was to distribute source code faithfully. A theme often sounded by the QCPE Advisory Board was that QCPE needed to adapt to new technologies and to bolster itself against both the free, independent exchange of software between scientists and the commercialization of computational chemistry software. After about 1991 the QCPE Advisory Board did not meet and was eventually dropped. Some of the recommended changes in QCPE’s goals and strategies were impossible to implement because of restrictions imposed by university and departmental policies. QCPE did respond to changes in technology, but at the same time Richard Counts realized that he had to tailor QCPE‘s services to meet the needs of clientele in some parts of the world without e-mail and/or web browsers. In 1989, QCPE became reachable via e-mail over BITNET. Starting in 1993, QCPE made its catalog available by file transfer protocol (ftp) and began distributing software that way. Also in 1993, QCPE acquired an e-mail address on the Internet. Whereas at one time 2000 members were receiving the QCPE Bulletin, the membership slipped toward 1000 in the last few years. The issues of the bulletin became thinner. Fewer programs were being deposited. No doubt the declining revenues created problems for the organization. The people with power to control QCPE’s destiny in the 1980s could not or would not make the

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changes necessary for the changing technological and market environments. QCPE could be a case of the old saw that if you fail to keep moving ahead, you will slide backward. At present, some members of the IUB chemistry department feel that QCPE should continue, whereas others think that it has served its useful lifespan and should be allowed to rest in peace. Likewise in the broader community of computational chemists, some think that QCPE has fulfilled its role (very admirably) but is no longer needed. Other people believe that QCPE’s mission to serve as a stable repository of computational chemistry software remains. For now, QCPE continues at IUB, but its exact role and structure are still being sorted out. As we write this, the chair of the IUB chemistry department has appointed a temporary director to handle QCPE’s operations. In 1999 software continued to be deposited, and programs were distributed at a rate of only about 15 per month. Deposits (and hence distributions) may pick up when the user community is reassured about the organization’s viability. The QCPE library is presently approaching 775 programs for mainframes and workstations, plus about 200 additional programs for desktop computing. This collection represents hundreds of thousands of line of source code, much hard work, and immense creativity. Although the QCPE Bulletin has been suspended, plans call for resuming it on the QCPE home page. Other exciting developments are also planned to make QCPE more web-based in 2000 and beyond. We wish the stewards of QCPE well. We will miss the interaction with Richard Counts, Peggy Edwards, and the others who have retired. Many messages have poured into QCPE thanking these individuals for their contributions and wishing them well. All the leaders who were involved in the organization and running of QCPE, plus all the individuals who deposited programs, participated in the workshops, or otherwise served the organization, deserve our profound thanks. They and QCPE helped bring about the birth of computational chemistry.

Information Resources for Chemists Information is in abundance these days. Being able to retrieve information from the Internet and from the numerous databases of chemical information is, of course, one of the great uses of computers. A relatively recent, very powerful software tool for literature retrieval is SciFinder from Chemical Abstracts Service (CAS, Columbus, Ohio), the world’s leading provider of chemical information, SciFinder lets people use natural language queries to search CAS’s largest database. In 1997 CAS introduced an inexpensive, watered-down version of their search engine software: SciFinder Scholar, designed for the academic market, enables chemistry students and faculty to explore abstracts and chemical structures, although not with all the “bells and whistles” available in the full version.

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SciFinder and SciFinder Scholar access two databases, CAPlus and the Registry File. Both databases are maintained and marketed by STN International, of which CAS is a partner along with FIZ Karlsruhe in Germany and the Japan Science and Technology Corporation. STN produces about 200 separate databases related to chemical information. CAPlus contains some of the items traditionally found in the old hard-bound Chemical Abstracts (CA), plus all articles from more than 1350 important chemical journals since October 1994, as well as citations for document types not covered in CA such as biographical items, book reviews, editorials, errata, letters to the editor, news announcements, product reviews, meeting abstracts, and miscellaneous items. CAPlus currently contains almost 16 million abstracts of journal articles (from 8,000 journals), patents, and other documents. The Registry File is the world’s largest chemical substance database, currently with more than 21 million records. SciFinder Scholar lets students and faculty search the CAPlus and Registry databases by authors’ names, concepts, chemical names, molecular formulas, CAS Registry Numbers, chemical structures, and specific references such as a patent number. SciFinder allows these options, plus other ones for handling the queries and hits. The latest version of SciFinder also can search Medline. Although SciFinder and SciFinder Scholar are very user friendly and are great software products, the user may not realize that the searches are not retrieving all the information stored in the 200 databases at STN. To illustrate that other STN databases contain pertinent information, we present in Table 1 the results of an STN search we did about 5 years ago. We searched for common terms obviously relevant to our audience, such as “computational chemistry,” “molecular orbital,” and “force fields.” These terms are listed across the columns of the table. In most of the searches we qualified the search inquiry so that only hits relevant to computational chemistry would be retrieved. The table shows clearly that the CAPlus file had a great many hits, as expected. However, it is also obvious that most of the other STN databases also contain pertinent hits. We present these results just to illustrate the sort of information that is accessible with the full-blown STN search tools. Being able to modify or extend software tools is desirable. As mentioned, QCPE distributed almost all its software as source code, so development and customization was possible with their software. In contrast, commercial software is almost never distributed as source code and can be very expensive unless mass-produced. On the subject of software and its monetary value, it is interesting to look at the touchy subject of using illegally copied software. The 1999 Annual Report on Global Software Piracy from the Software and Information Industry Association estimates that 25% of the software in use in the United States is pirated, thereby depriving the developers/manufacturers of $2.8 x 109. In the People’s Republic of China, a whopping 95% of the software in use is thought to be pirated, amounting to $1.2 x 109 in unrealized revenue. In third place by revenue lost is Japan, where 31% of the software is thought to be pirated, totaling $0.6 x 109. As anyone who has spent months or years creating tens of thousands of lines of programming can appreciate, it is frustrating when

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users benefit from the software but the developers do not get full return on their investment. On a per capita basis, software piracy amounts to $1-$15 per person per year in most nations ($11 in the U.S.). Typically, what happens is that one copy of (mostly American) software is purchased legally; it is copied and installed on many machines. Not all the piracy is by individuals at home or school. The Business Software Alliance, an international group that opposes counterfeiting of computer programs, estimates that in the U.S., 25% of the software used by individuals in business is obtained illegally; in Israel, it is 48%. Information is in abundance and free on the World Wide Web (WWW). Every computational chemist, as well as most other computer-literate people, use the free search engines to locate information on the Internet. However, the spectacular growth of the Web has made it difficult for the search engines to keep up. A recent study determined that at best only 16% of the current websites are catalogued in the databases of the search engines. Altavista rated highest at 15.5%. Surprisingly, some of the other well-known search engines had as little as 2%. Thus at present, it is difficult or impossible for Web surfers to locate all the information actually available on the Internet. The number of websites is an estimated 6.5 million and still growing rapidly. If the growth becomes less torrid, which is not likely to happen soon, the search engine databases may eventually be able to catch up, although there is so much catalogable information in the world, it will be difficult to ever catch up completely. Finally, we offer a few words about words. Although this may be an arcane topic of interest only to editors who have nothing more important to do with their time, it is nevertheless fascinating to watch language-as well as the databases-attempt to keep up with high technology. It is reported that the word “Internet” was first used in 1974, but it was only in the last 15 years that this word became widespread. Five years ago, the phrase “home page” came into being to describe a presentation on the Internet compatible with WWW browser software. With the increasing elaboration and layering of home pages, this phrase appears to be giving way to “web site” as the preferred terminology. It is still too early to know if the standard spelling will settle on “website” and whether the “w” will be capitalized. The hybrid term “web page” is also encountered. Over the last few decades, we have witnessed an evolution in what we call the software. In the 1960s and 1970s, the terminology was simply “programs” or “code” or “software.” However, with the commercialization of software in the 1980s to meet the needs of a dramatically growing molecular modeling software market, plus the increasing elaborateness of the programs with graphical user interfaces, the software started to be called “systems” or “packages”. In the 1990s, the word “solutions” came into vogue. Solutions is a glitzier word that marketing people can easily love. It would be wonderful if software did solve all our problems, but we know this is not true; at best the programs help us attack a research problem . . . whether they provide a “solution” to our research problems is more problematic.

ANABSTRb APILIT2c CABAd CAPluse CBNBf CEABAg CENh CERABI CINi CJACSk CJAOACl CJELSEVIERm CJRSC. CJVCHo CJWILEYP COMPENDEXq CONFSCP DISSABSs INSPECt INVESTEXT" IPAv JICST-Ew JPNEWSx KKFy METADEX= NAPRALERT"" NTISbb PAPERCHEM2cc PROMTdd RAPRAee

Database 6 268 40 10,025 0 54 20 31 0 7,987 0 178 1,255 72 60 747 63 654 6,156 1 32 919 1 19 179 3 516 23 14 137

6 0 3 104 6 6 72 0 0 0 0 4 28 7 9 43

11 24 109 96 3 20 1 3 1 0 119 1 23 1 2

Molecular Orbitala

Computational Chemistry

Molecular Modeling" 1 13 6 804 0 12 75 0 19 2,397 0 33 4 44 82 169 5 113 133 95 25 28 4 48 2 0 68 2 356 0

Molecular Mechanicsa 4 19 22 1,419 0 14 25 2 1 3,518 0 128 616 26 135 249 21 3 19 809 6 14 127 0 9 3 1 90 7 31 63

Force Fielda 4 39 3 1,870 0 13 11 14 0 3,864 1 130 445 90 150 400 15 249 2,801 4 2 83 0 14 57 1 145 11 11 75

Search Term Molecular Graphicsa 0 8 3 97 0 19 1 0 1 467 0 9 74 8 25 80 3 7 73 3 6 39 0 0 0 0 15 0 16 2

Molecular Dynarnicsa 4 88 13 1,955 0 59 41 69 1 3,798 0 54 416 55 236 1,518 38 48 1 5,834 28 3 694 0 77 534 0 752 2 66 345

Table 1 Number of Hits in S T N Databases for Various Common Terms Used in the Field of Commtational Chemistry

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T h i s search term was qualified to obtain hits only within the context of computers, computations, or calculations. bANABSTR (Analytical Abstracts) is a bibliographic database covering the worldwide literature on analytical chemistry. It is produced by the Royal Society of Chemistry and covers 1980 to date. cAPILIT2 is a bibliographic database containing citations for nonpatent literature pertaining to the petroleum and petrochemical industries, including information on alternate energy sources and environmental effects. It is produced by the American Institute of Petroleum and covers 1964 to date. dCABA (CAB Abstracts) database is a bibliographic database covering worldwide literature from all areas of agriculture and related sciences. It produced by CAB International and covers 1973 to date. eCAPlus is the most current and most comprehensive chemistry bibliographic database available from the Chemical Abstracts Service (CAS) and covers from 1967 to date. CAplus coverage includes international journals, patents, patent families, technical reports, books, conference proceedings, and dissertations from all areas of chemistry, biochemistry, chemical engineering, and related sciences. fCBNB (Chemical Business NewsBase) is a bibliographic database covering news on the chemical industry worldwide. It is produced by the Royal Society of Chemistry and covers 1984 to date. gCEABA (Chemical Engineering And Biotechnology Abstracts) is a bibliographic database covering the international literature on chemical engineering and biotechnology. It is produced by a group of European chemical societies and covers 1975 to date. T E N (Chemical & Engineering News Online) is a full-text database containing the entire text from each issue of the American Chemical Society’s printed Chemical & Engineering News weekly magazine. It covers 1991 to date. CERAB (Ceramic Abstracts) is a bibliographic database covering worldwide literature on all aspects of ceramics. It is produced by the American Ceramic Society and covers 1976 to date. rCIN (Chemical Industry Notes) is a bibliographic database covering worldwide business events in the chemical industry. It is produced by the American Chemical Society and covers 1974 to date. CJACS (Current Journals of the American Chemical Society) was a database of full-text articles published in 23 ACS journals during the period from 1982 through 1993. It is no longer available via STN. ‘CJAOAC was a database of full-text articles published in the Journal of AOAC International on analytical chemistry during the period from 1987 through 1993. It is no longer available via STN. mCJELSEVIER (Current Journals of Elsevier) was a database of full-text articles published in four journals published by Elsevier Science during the period from 1990 through 1993. It is no longer available via STN. *CJRSC (Current Journals of the Royal Society of Chemistry) was a database of full-text articles published in 15 journals published by the U.K. society during the period from 1987 through 1993. It is no longer available via STN. oCJVCH was a database of full-text articles published in Angewandte Chemie by VCH Publishers during the period from 1988 through 1993. It is no longer available via STN. PCJWILEY (Current Journals of Wiley) was a database of full-text articles published in five polymer journals published by John Wiley & Sons during the period from 1987 to through 1993. It is no longer available via STN. 4COMPENDEX (COMPuterized ENgineering InDEX) is a bibliographic database covering the worldwide literature in engineering and technology. It is produced by Engineering Information and covers 1970 to date. CONFSCI (Conference Papers Index) is a bibliographic database of international research papers and findings presented at scientific and technical conferences and meetings throughout the world. It is produced by Cambridge Scientific Abstracts and covers 1973 to date. notes continued on next page

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sDISSABS was a database of Ph.D. dissertation abstracts. It was produced by University Microfilms and covered 1861 to date. It is no longer available via STN. fINSPEC (Physics, Electronics, and Computing) is a bibliographic database providing access to the worldwide literature on physics, electrical engineering and electronics, control theory and technology, and computers and computing. It is produced by the Institution of Electrical Engineers (U.K.) and covers 1969 to date. uINVESTEXT is the world's largest full-text database of investment research reports. It contains business information on publicly traded companies in a wide variety of industries. It is produced by Thomson Financial Services and covers 1982 to date. UIPA(International Pharmaceutical Abstracts) is a bibliographic database containing international coverage of pharmacy and health-related literature in information, the practice of pharmacy, pharmaceutical education, and the legal aspects of pharmacy and drugs. It is produced by the American Society of Health-System Pharmacists and covers 1970 to date. 'uJICST-E is a bibliographic database with English citations and abstracts covering the literature published in Japan on all fields of science, technology, and medicine. It is produced by the Japan Information Center of Science and Technology and covers 1985 to date. XJPNEWScovered news about Japanese industry, products. and technology. It was produced by COMLINE Business Data and covered 1986 through 1993. It is no longer available through STN. YKKF(Kunststoffe Kautschuk Fasern) is a bibliographic database containing references to the international scientific and technical literature on production, processing, applications, and technological properties of plastics, rubbers, and fibers, and the fundamental physical and chemical properties of polymers. It is produced by the Deutsches Kunststoff-Institut and covers 1973 to date. zMETADEX (METals Abstracts/Alloy InDEX) is a bibliographic database covering the worldwide literature on metallurgy and materials. It is produced by Materials Information and covers 1966 to date. aaNAPRALERT (NAtural PRoducts ALERT) contains bibliographic and factual data on natural products, including information on the pharmacology, biological activity, taxonomic distribution, ethnomedicine and chemistry of plant, microbial, and animal (including marine) extracts, and data on the chemistry and pharmacology of secondary metabolites that are derived from natural sources. It is produced by the University of Illinois School of Pharmacy and covers 1650 to date. bbNTIS is a multidisciplinary bibliographic database of publications, especially unrestricted reports, on research, development, and engineering projects, sponsored by U.S. and non-US. governments. It is produced by the National Technical Information Service and covers 1964 to date. ccPAPERCHEM2 is a bibliographic database that contains international patent and journal literature pertaining to pulp and paper technology. It is produced by the Institute of Paper and Science Technology and covers 1967 to date. d.IPR0M-I (Predicasts Overview of Markets and Technology) is a database that provides international news coverage of companies, products, markets, and applied technology for all industries. It is produced by the Information Access Company and covers 1978 to date. eeRAPRA is a bibliographic database about rubber, plastics, adhesives, and polymeric composites. It covers the worldwide scientific, technical, and trade literature, as well as patent documents pertinent to the rubber and plastics industries. It is produced by RAPRA Technology Ltd. and covers 1972 to date. ffSCISEARCH (Science Citation Index Expanded) contains records published in the Science Citation Index and additional records from about a thousand journals covered in the Current Contents series of publications. It is produced by the Institute of Scientific Information and covers 1974 to date. ggVtB (Verfahrenstechnische Berichte) is a bibliographic database covering worldwide literature in the field of chemical and process engineering and related fields. It is produced by BASF AG and covers 1966 to date.

Preface xvii

This Volume Providing ideas and knowledge in the area of computational chemistry and molecular modeling remains the mission of Reviews in Computational Chemistry. We endeavor to present high quality tutorials and reviews at a relatively low cost in comparison to the potential of using that knowledge in your studies and careers. Many of our chapters are designed to be used in conjunction with classroom teaching. These chapters can also be used as supplementary reading material. We do not, however, present problem sets because these have to be tailored to the available software and hardware as well as the nature of the classes being taught. Professional computational chemists come from a variety of backgrounds including theoretical chemistry, organic chemistry, crystallography, biophysics, etc. However, it is probably safe to say that many of today’s practicing computational chemists received their introduction to the field by graduate studies in quantum chemistry. As pointed out about 45 years ago by Henry Eyring, John Walter, and George E. Kimball,” “In so far as quantum mechanics is correct, chemical questions are problems in applied mathematics. In spite of this, chemistry, because of its complexity, will not cease to be in large measure an experimental science. . . .No chemist, however, can afford to be uninformed of a theory which systematizes all of chemistry even though mathematical complexity often puts exact numerical results beyond his immediate reach.” Thus, this fifteenth volume focuses on quantum chemistry, an area that many consider to be the central core of computational chemistry. However, as theoretical chemists quickly learn if they are hired into the pharmaceutical industry, computational chemistry is much more than quantum chemistry. Accordingly, our next volume (Volume 16) will focus on some modern computational tools and concepts used in molecular design. Chapter 1 of Volume 15 deals with density functional theory (DFT). As with many quantum mechanical calculations, it is easy to become wrapped up in the theory and lose sight of the chemical phenomena we are trying to explain with the calculations. Equally important to bow the numerical calculations are done is bow the results can be interpreted to gain chemical insight. Dr. F. Matthias Bickelhaupt and Professor Evert Jan Baerends show how the results of DFT calculations can be analyzed to open up chemical understanding. This chapter illustrates that the plain numbers from a quantum mechanical calculation can be interpreted to be conceptually useful to chemists. In many ways, this chapter evokes memories of the famous way Professor Roald Hoffmann has extracted information from extended Huckel molecular orbital calculations. In Chapter 2, Professor Michael A. Robb and colleagues Drs. Marco Garavelli, Massimo Olivucci, and Fernando Bernardi describe strategies for ”H. Eyring, J. Walter, and G. E. Kimball, Quantum Chemistry,Wiley, New York, 1944, p. iii (Preface).

xviii

Preface

modeling photochemical reactions. Although many practicing computational chemistry in the pharmaceutical industry are concerned only with ground states of molecules, many light-induced transformations of synthetic interest exist. Also many natural phenomena relevant to living systems, atmospheric conditions, and interstellar chemistry involve excited states, so the ability to predict the outcome of photochemical events is significant. Excited states offer special challenges for computational experiments, especially for a reaction where the various energy surfaces are intersecting. These conical intersections must be mapped out to determine the mechanism of a photochemical reaction. Drs. Larry A. Curtiss, Paul C. Redfern, and David J. Frurip present a tutorial on how to compute enthalpies of formation in Chapter 3. Often a computational chemist will want to know how stable a molecule is. The techniques described in this chapter can answer this question. The authors, who have studied what has been called computational thermochemistry, describe ab initio molecular orbital methods (including the highly accurate and popular Gn methods), density functional methods, semiempirical molecular orbital methods, and empirical methods (such as based on bond energies). These methods are richly illustrated with detailed, worked out examples. In prior volumes, we have had essays describing the history of computational chemistry in the United States, Great Britain, and France. We think these chapters serve not only to record the important trends and shaping events but also to acquaint newcomers with the evolution of the field in various parts of the globe. Continuing with this series, we are delighted in this volume to present a chapter about the growth of computational chemistry in Canada by Professor Russell J. Boyd.+ More so than in the United States, computational chemistry in Canada has a quantum mechanical flavor. In fact, Canada has become known for contributing many advances in density functional theory. More recently, the seeds of computational chemistry are germinating there in the form of nascent companies related to pharmaceutical research. To make the information in our books easily accessible and retrievable, we have always tried to provide thorough author and subject indexes. In addition, a website is maintained for Reviews in Computational Chemistry at http://chem.iupui.edu/rcc/rcc.html.It includes our author and subject indexes, plus color graphics, errata, and other material as adjuncts to the chapters. When we began this book series, we did not know how long the series would be. Many of our colleagues have commented to us that they are surprised at the quantity of chapters we have published. Our readers have told us that they find the books useful and that the series should continue as it is. We +This chapter’s mention of the Second Canadian Symposium on Theoretical Chemistry and the hugely popular world’s exposition, Expo 67, which were held contemporaneously in Montreal in the summer of 1967, brings back fond memories for one editor. The Canadian symposium was the first scientific conference that DBB ever attended. Although proudly bearing the same Scottish last name. the editor and author share no known familial relationship.

Preface xix appreciate their trust. We are grateful to our authors for their superb chapters. We thank Mrs. Joanne Hequembourg Boyd for indispensable editorial assistance. We hope these books will have enduring value to our readers and authors in their learning, teaching, and research. Donald B. Boyd and Kenny B. Lipkowitz Indianapolis October 1999

Contents 1.

Kohn-Sham Density Functional Theory: Predicting and Understanding Chemistry E Matthias Bickelhaupt and Evert J a n Baerends Introduction Scope Historical Overview Outline The Kohn-Sham Molecular Orbital Model MO-Theoretical Analysis of Chemical Bonding: Beyond a Qualitative M O Theory Introduction Electrostatic Interaction and Steric Repulsion Attractive Orbital Interactions Interplay of Steric Repulsion and Orbital Interaction The Electron Pair Bond and Pauli Repulsion Introduction The Potential Energy Surfaces of CN and CP Dimers Bonding in C N and CP Dimers: Qualitative Considerations Bonding in C N and CP Dimers: Quantitative Analysis Summary The Three-Electron Bond and One-Electron Bonding Introduction The Fragment Approach to the Three-Electron Bond Summary The Role of Steric Repulsion in Bonding Models Introduction Structure and Inversion Barrier in AH; Radicals Interhydrogen Steric Repulsion Versus A-H Electronic Interaction in AH; Radicals Summary Strongly Polar Electron Pair Bonding Introduction

1

11 11 14 23 28 34 34 36

40 42 48 49 49 50 54 55 55 57 59 63 65

65

xxi

xxii

Contents The Polar C-Li Electron Pair Bond in Monomeric CH,Li The Polar C-Li Electron Pair Bond in Tetrameric CH,Li Analysis of the Charge Distributions in CH,Li Oligomers Summary Conclusions and Outlook Acknowledgments References A Computational Strategy for Organic Photochemistry Michael A. Robb, Marco Garavelli, Massimo Olivucci, and Fernando Bernardi Introduction Modeling Photochemical Reactions Aims and Objectives Characterization of Conical Intersections “Noncrossing Rule” and Conical Intersections Conical Intersection Structure An Example: The S, /So Conical Intersection of Benzene Practical Computation of Photochemical Reaction Paths Quantum Chemical Methods and Software for Excited State Energy and Gradient Computations Conical Intersection Optimization Locating Decay Paths from a Conical Intersection Semiclassical Trajectories Mechanistic Organic Photochemistry: Some Case Studies Three-Electron Conical Intersections of Conjugated Hydrocarbons Conical Intersections of n-n* Excited States The S, /So Conical Intersection of Protonated Schiff Bases Competitive Ground State Relaxation Paths from Conical Intersection Competitive Excited State Photoisomerization Paths Conclusions Acknowledgments References

66 71 73 75 75 76 76

87

87 87 95 96 96 100 105 108 108 110 112 118 121 122 123 129 133 137 139 140 141

Theoretical Methods for Computing Enthalpies of Formation of Gaseous Compounds Larry A. Curtis, Paul C. Redfern, and David]. Frurip

147

Introduction Enthalpies of Formation Overview of Theoretical Methods Test Sets for Assessments of Predictive Methods

147 149 152 154

Contents

4.

xxiii

Quantum Chemical Methods Ab Initio Molecular Orbital Methods Extrapolation Methods Density Functional Methods Semiempirical Molecular Orbital Methods Illustrative Examples of Quantum Chemical Methods Empirical Methods Bond Energy Approach Benson’s Method Correcting from the Condensed Phase to the Gas Phase Concluding Remarks Acknowledgments References

155 155 156 180 181 185 189 190 193 199 201 202 202

The Development of Computational Chemistry in Canada Russell J. Boyd

213

Introduction In the Beginning There Was Quantum Chemistry and Spectroscopy Expo 67 and Fullerenes Canadian Association of Theoretical Chemists Demographic Facts Toward a Steady-State Population Family Trees and Trends Departmental Histories University of Montreal (1954) University of British Columbia (1957) University of Alberta (1959) University of Ottawa (1959) University of Saskatchewan (1959) Lava1 University ( 196 1) University of Toronto (1961) University of Waterloo (1961) McGill University (1962) Queen’s University ( 1962) University of New Brunswick (1962) McMaster University (1963) University of Calgary (1964) University of Western Ontario (1965) York University (1965) Simon Fraser University (1966) University of Manitoba (1966) Carleton University (1970) Dalhousie University (1970)

213 216 218 222 223 234 235 236 236 238 241 244 245 246 246 250 253 255 260 261 263 265 267 269 271 272 273

xxiv

Contents University of Guelph (1970) University of Sherbrooke (1970) Computational Chemistry in Canadian Industry Hypercube, Inc. Ayerst Laboratories Merck Frosst Canada Inc. Xerox Research Centre of Canada ORTECH, Inc. BioChem Therapeutic Advanced Chemistry Development, Inc. SynPhar Labs, Inc. Bio-Mkga Astra Other Examples History of Theoretical Chemistry at the National Research Council of Canada High-Performance Computing in Canada Major Conferences Fifth International Conference on Quantum Chemistry Second World Congress of Theoretical Organic Chemists Canadian Computational Chemistry Conference Spreading Their Wings Acknowledgments References

274 2 76 276 276 277 277 278 279 279 279 280 280 280 28 1 281 282 283 283 283 284 284 286 286

Author Index

301

Subject Index

313

Contributors Evert Jan Baerends, Scheikundig Laboratorium der Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands (Electronic mail: [email protected]) Fernando Bernardi, Dipartimento di Chimica “G. Ciamician” dell’universith di Bologna, Via Selmi 2, 40126 Bologna, Italy (Electronic mail: [email protected])

F. Matthias Bickelhaupt, Afdeling Theoretische Chemie, Scheikundig Laboratorium der Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam,

The Netherlands (Electronic mail: [email protected])

Russell J. Boyd, Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada B3H 453 (Electronic mail: boyd@cheml .chem.dal.ca) Larry A. Curtiss, Materials Science and Chemistry Divisions, Argonne National Laboratory, Argonne, Illinois 605 15, U.S.A. (Electronic mail: [email protected]) David J. Frurip, Analytical Sciences Laboratory, Dow Chemical Company, Midland, Michigan 48667, U.S.A. (Electronic mail: [email protected]) Marco Garavelli, Department of Chemistry, King’s College London, Strand, London WC2R 2LS, United Kingdom (Electronic mail: [email protected]) Massimo Olivucci, Istituto di Chimica Organica, Universitd degli Studi di Siena, Via Aldo Moro, 1-53100 Siena, Italy (Electronic mail: [email protected]) Paul C. Redfern, Materials Science and Chemistry Divisions, Argonne National Laboratory, Argonne, Illinois 60515, U.S.A. (Electronic mail: [email protected]) Michael A. Robb, Department of Chemistry, King’s College London, Strand, London WC2R 2LS, United Kingdom (Electronic mail: [email protected]) xxv

Contributors to Previous Volumes‘ Volume 1 David Feller and Ernest R.Davidson, Basis Sets for Ab Initio Molecular Orbital Calculations and Intermolecular Interactions. James J. P. Stewart,t Semiempirical Molecular Orbital Methods. Clifford E. Dykstra,* Joseph D. Augspurger, Bernard Kirtman, and David J. Malik, Properties of Molecules by Direct Calculation. Ernest L. Plummer, The Application of Quantitative Design Strategies in Pesticide Design. Peter C. Jurs, Chemometrics and Multivariate Analysis in Analytical Chemistry. Yvonne C. Martin, Mark G . Bures, and Peter Willett, Searching Databases of Three-Dimensional Structures. Paul G . Mezey, Molecular Surfaces. Terry I?. Lybrand,¶ Computer Simulation of Biomolecular Systems Using Molecular Dynamics and Free Energy Perturbation Methods. “When no author of a chapter can be reached at the addresses shown in the original volume, the current affiliation of the senior or corresponding author is given here as a convenience to our readers. +Current address: 15210 Paddington Circle, Colorado Springs, CO 80921-2512 (Electronic mail: [email protected]). *Current address: Department of Chemistry, Indiana University-Purdue University at Indianapolis, Indianapolis, IN 46202 (Electronic mail: [email protected]). ¶Current address: University of Washington, Seattle, WA 98195 (Electronic mail: [email protected]).

xxvii

xxviii Contributors to Previous Volumes Donald B. Boyd, Aspects of Molecular Modeling. Donald B. Boyd, Successes of Computer-Assisted Molecular Design. Ernest R. Davidson, Perspectives on Ab Initio Calculations.

Volume 2 Andrew R. Leach,* A Survey of Methods for Searching the Conformational Space of Small and Medium-Sized Molecules. John M. Troyer and Fred E. Cohen, Simplified Models for Understanding and Predicting Protein Structure. J. Phillip Bowen and Norman L. Allinger, Molecular Mechanics: The Art and Science of Parameterization. Uri Dinur and Arnold T. Hagler, New Approaches to Empirical Force Fields. Steve Scheiner, Calculating the Properties of Hydrogen Bonds by Ab Initio Methods. Donald E. Williams, Net Atomic Charge and Multipole Models for the Ab Initio Molecular Electric Potential. Peter Politzer and Jane S. Murray, Molecular Electrostatic Potentials and Chemical Reactivity. Michael C. Zerner, Semiempirical Molecular Orbital Methods. Lowell H. Hall and Lemont B. Kier, The Molecular Connectivity Chi Indexes and Kappa Shape Indexes in Structure-Property Modeling. I. B. Bersukert and A. S. Dimoglo, The Electron-Topological Approach to the QSAR Problem.

Donald B. Boyd, The Computational Chemistry Literature. *Current address: Glaxo Wellcome, Greenford, Middlesex, UB6 OHE, U.K. (Electronic mail: ad2295 [email protected]). +Current address: College of Pharmacy, The University of Texas, Austin, TX 78712 (Electronic mail: [email protected]).

Contributors to Previous Volumes

xxix

Volume 3 Tamar Schlick, Optimization Methods in Computational Chemistry. Harold A. Scheraga, Predicting Three-Dimensional Structures of Oligopeptides. Andrew E. Torda and Wilfred F. van Gunsteren, Molecular Modeling Using NMR Data. David F. V. Lewis, Computer-Assisted Methods in the Evaluation of Chemical Toxicity.

Volume 4 Jerzy Cioslowski, Ab Initio Calculations on Large Molecules: Methodology and Applications. Michael L. McKee and Michael Page, Computing Reaction Pathways on Molecular Potential Energy Surfaces. Robert M. Whitnell and Kent R. Wilson, Computational Molecular Dynamics of Chemical Reactions in Solution. Roger L. DeKock, Jeffry D. Madura, Frank Rioux, and Joseph Casanova, Computational Chemistry in the Undergraduate Curriculum.

Volume 5 John D. Bolcer and Robert B. Hermann, The Development of Computational Chemistry in the United States. Rodney J. Bartlett and John F. Stanton, Applications of Post-Hartree-Fock Methods: A Tutorial. Steven M. Bachrach,” Population Analysis and Electron Densities from Quantum Mechanics.

-

‘Current address: Department of Chemistry, Trinity University, San Antonio, TX 78212 (Electronic mail: [email protected]).

xxx

Contributors to Previous Volumes

Jeffry D. Madura," Malcolm E. Davis, Michael K. Gilson, Rebecca C. Wade, Brock A. Luty, and J. Andrew McCammon, Biological Applications of Electrostatic Calculations and Brownian Dynamics Simulations.

K. V. Damodaran and Kenneth M. Merz Jr., Computer Simulation of Lipid Systems. Jeffrey M. Blaneyt and J. Scott Dixon, Distance Geometry in Molecular Modeling. Lisa M. Balbes, S. Wayne Mascarella, and Donald B. Boyd, A Perspective of Modern Methods in Computer-Aided Drug Design.

Volume 6 Christopher J. Cramer and Donald G. Truhlar, Continuum Solvation Models: Classical and Quantum Mechanical Implementations. Clark R. Landis, Daniel M. Root, and Thomas Cleveland, Molecular Mechanics Force Fields for Modeling Inorganic and Organometallic Compounds. Vassilios Galiatsatos, Computational Methods for Modeling Polymers: An Introduction. Rick A. Kendal1,t Robert J. Harrison, Rik J. Littlefield, and Martyn E Guest, High Performance Computing in Computational Chemistry: Methods and Machines. Donald B. Boyd, Molecular Modeling Software in Use: Publication Trends. Eiji Osawa and Kenny B. Lipkowitz, Appendix: Published Force Field Parameters.

-

"Current address: Department of Chemistry and Biochemistry, Duquesne University, Pittsburgh, PA 15282-1530 (Electronic mail: [email protected]). +Current address: Metaphorics, 130 Aka Avenue, Piedmont, CA 9461 1 (Electronic mail: [email protected]). *Current address: Scalable Computing Laboratory, Ames Laboratory, Wilhelm Hall, Ames, IA 5001 1 (Electronic mail: [email protected])

Contributors to Previous Volumes

xxxi

Volume 7 Geoffrey M. Downs and Peter Willett, Similarity Searching in Databases of Chemical Structures. Andrew C. Good" and Jonathan S. Mason, Three-Dimensional Structure Database Searches. Jiali Gao,t Methods and Applications of Combined Quantum Mechanical and Molecular Mechanical Potentials. Libero J. Bartolotti and Ken Flurchick, An Introduction to Density Functional Theory. Alain St-Amant, Density Functional Methods in Biomolecular Modeling. Danya Yang and Arvi Rauk, The A Priori Calculation of Vibrational Circular Dichroism Intensities. Donald B. Boyd, Appendix: Compendium of Software for Molecular Modeling.

Volume 8 ZdenZk Slanina, Shyi-Long Lee, and Chin-hui Yu, Computations in Treating Fullerenes and Carbon Aggregates. Gernot Frenking, Iris Antes, Marks Bohme, Stefan Dapprich, Andreas W. Ehlers, Volker Jonas, Arndt Neuhaus, Michael Otto, Ralf Stegmann, Achim Veldkamp, and Sergei F. Vyboishchikov, Pseudopotential Calculations of Transition Metal Compounds: Scope and Limitations. Thomas R. Cundari, Michael T. Benson, M. Leigh Lutz, and Sham 0. Sommerer, Effective Core Potential Approaches to the Chemistry of the Heavier Elements.

-

'Current address: Bristol-Myers Squibb, 5 Research Parkway, P.O. Box 5100, Wallingford, CT 06492-7660 (Electronic mail: [email protected]). +Current address: Department of Chemistry, University of Minnesota, Minneapolis, MN 55455 (Electronic mail: [email protected]).

xxxii

Contributors to Previous Volumes

Jan Almlof and Odd Gropen,t Relativistic Effects in Chemistry. Donald B. Chesnut, The Ab Initio Computation of Nuclear Magnetic Resonance Chemical Shielding.

Volume 9 James R. Damewood Jr., Peptide Mimetic Design with the Aid of Computational Chemistry.

T. P. Straatsma, Free Energy by Molecular Simulation. Robert J. Woods, The Application of Molecular Modeling Techniques to the Determination of Oligosaccharide Solution Conformations. Ingrid Pettersson and Tommy Liljefors, Molecular Mechanics Calculated Conformational Energies of Organic Molecules: A Comparison of Force Fields. Gustavo A. Arteca, Molecular Shape Descriptors.

Volume 10 Richard Judson," Genetic Algorithms and Their Use in Chemistry. Eric C. Martin, David C. Spellmeyer, Roger E. Critchlow Jr., and Jeffrey M. Blaney, Does Combinatorial Chemistry Obviate Computer-Aided Drug Design? Robert Q. Topper, Visualizing Molecular Phase Space: Nonstatistical Effects in Reaction Dynamics. Raima Larter and Kenneth Showalter, Computational Studies in Nonlinear Dynamics. Stephen J. Smith and Brian T. Sutcliffe, The Development of Computational Chemistry in the United Kingdom. +Address: Institute of Mathematical and Physical Sciences, University of Tromsa, N-9037 Tromsa, Norway (Electronic mail: [email protected]). "Current address: CuraGen Corporation, 322 East Main Street, Branford, CT 06405 (Electronic mail: [email protected]).

Contributors to Previous Volumes xxxiii

Volume 11 Mark A. Murcko, Recent Advances in Ligand Design Methods. David E. Clark, Christopher W. Murray, and Jin Li, Current Issues in De Novo Molecular Design. Tudor I. Oprea and Chris L. Waller, Theoretical and Practical Aspects of ThreeDimensional Quantitative Structure-Activity Relationships. Giovanni Greco, Ettore Novellino, and Yvonne Connolly Martin, Approaches to Three-Dimensional Quantitative Structure-Activity Relationships. Pierre-Alain Carrupt, Bernard Testa, and Patrick Gaillard, Computational Approaches to Lipophilicity: Methods and Applications. Ganesan Ravishanker, Pascal Auffinger, David R. Langley, Bhyravabhotla Jayaram, Matthew A. Young, and David L. Beveridge, Treatment of Counterions in Computer Simulations of DNA. Donald B. Boyd, Appendix: Compendium of Software and Internet Tools for Computational Chemistry.

Volume 12 Hagai Meirovitch, Calculation of the Free Energy and the Entropy of Macromolecular Systems by Computer Simulation. Ramzi Kutteh and T. .?I Straatsma, Molecular Dynamics with General Holonomic Constraints and Application to Internal Coordinate Constraints. John C. Shelley and Daniel R. BCrard, Computer Simulation of Water Physisorption at Metal-Water Interfaces. Donald W. Brenner, Olga A. Shenderova, and Denis A. Areshkin, QuantumBased Analytic Interatomic Forces and Materials Simulation. Henry A. Kurtz and Douglas S. Dudis, Quantum Mechanical Methods for Predicting Nonlinear Optical Properties. Chung F. Wong, Tom Thacher, and Herschel Rabitz, Sensitivity Analysis in Biomolecular Simulation.

xxxiv

Contributors to Previous Volumes

Paul Verwer and Frank J. J. Leusen, Computer Simulation to Predict Possible Crystal Polymorphs. Jean-Louis Rivail and Bernard Maigret, Computational Chemistry in France: A Historical Survey.

Volume 13 Thomas Bally and Weston Thatcher Borden, Calculations on Open-Shell Molecules: A Beginner’s Guide. Neil R. Kestner and Jaime E. Combariza, Basis Set Superposition Errors: Theory and Practice. James B. Anderson, Quantum Monte Carlo: Atoms, Molecules, Clusters, Liquids, and Solids. Anders Wallqvist and Raymond D. Mountain, Molecular Models of Water: Derivation and Description. James M. Briggs and Jan Antosiewicz, Simulation of pH-Dependent Properties of Proteins Using Mesoscopic Models.

Harold E. Helson, Structure Diagram Generation.

Volume 14 Michelle Miller Franc1 and Lisa Emily Chirlian, The Pluses and Minuses of Mapping Atomic Charges to Electrostatic Potentials.

T. Daniel Crawford and Henry E Schaefer In, An Introduction to Coupled Cluster Theory for Computational Chemists.

Bastiaan van de Graaf, Swie Lan Njo, and Konstantin S. Smirnov, Introduction to Zeolite Modeling. Sarah L. Price, Toward More Accurate Model Intermolecular Potentials for Organic Molecules.

Contributors to Previous Volumes xxxu Christopher J. Mundy, Sundaram Balasubramanian, Ken Bagchi, Mark E. Tuckerman, Glenn J. Martyna, and Michael L. Klein, Nonequilibrium Molecular Dynamics. Donald B. Boyd and Kenny B. Lipkowitz, History of the Gordon Research Conferences on Computational Chemistry. Mehran Jalaie and Kenny B. Lipkowitz, Appendix: Published Force Field Parameters for Molecular Mechanics, Molecular Dynamics, and Monte Carlo Simulations.

Reviews in Computational Chemistry Volume 15

CHAPTER 1

Kohn-Sham Density Functional Theory: Predicting and Understanding Chemistry F. Matthias Bickelhaupt” and Evert Jan Baerendst *Fachbereich Chemie, Philipps-Universitat Marburg, HansMeerwein-Strape, 0 - 3 5 0 3 2 Marburg, Germany, (present address): Afdeling Theoretische Chemie, Scheikundig Laboratorium der Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands, and tscheikundig Laboratorium der Vrije Universiteit, De Boelelaan 1083, NL-1081 H V Amsterdam, The Netherlands

INTRODUCTION Scope Over the past decade, Kohn-Sham density functional theory (DFT) has evolved into what is now one of the major approaches in quantum chemistry.1-20 It is routinely applied to various problems concerning, among other matters, chemical structure and reactivity in such diverse fields as organic, organometallic, and inorganic chemistry, covering the gas and condensed phases as well as the solid state. What is it that makes Kohn-Sham DFT so attractive? Certainly, an important reason is that it represents a first-principles Reviews in Computational Chemistry, Volume 15 Kenny B. Lipkowitz and Donald B. Boyd, Editors Wiley-VCH, John Wiley and Sons, Inc., New York, 0 2000

1

2

Kohn-Sham Density Functional Theory

method whose practical implementation combines a high efficiency (a favorable order-N3 or better scaling of the computational cost, where N is a size parameter, e.g., the number of electrons)4 with a relatively high accuracy (often agreeing with experiment within a few kcal/mo1).1J,13-19 In the present chapter, however, we wish to focus on a different but, in our opinion, equally important aspect of Kohn-Sham DFT: its status as a physically meaningful one-electron or molecular orbital (MO) model. Our purpose is to demonstrate the potential of such a method, which integrates efficiency, accuracy, and a transparent physical model: not only can one accurately predict and model the behavior and properties of realistic systems, but these can also be understood, within the same quantum chemical approach, in terms of familiar concepts from MO theory. Striving for accurate calculation of particular chemical phenomena and endeavoring to understand these calculations have not always gone hand in hand. We will argue that, in fact, Kohn-Sham DFT not only offers a road to accurate calculation and prediction, but also allows interpretation and understanding of chemical bonding phenomena using elementary physical concepts. Quantum chemists have at times expressed their regret that often the more involved and accurate calculations become, the more elusive the understanding of the results proved to be. The simple MO model that is the very essence of the Kohn-Sham (KS) approach to DFT makes it possible to reconcile the search for accuracy with the search for understanding.

Historical Overview The history of density functional theory begins in the 1920s with the notion by Thomas21 and Ferrniz2J3 that the ground state energy of a system of electrons moving in the external potential v(v) of a nuclear frame may be expressed directly and alone in terms of the electron density. That such an approach would greatly reduce the effort of solving the many-body problem is clear: the complicated N-electron wavefunction Y(1,2, . . . , N) with its 4N degrees of freedom (three space coordinates and one spin coordinate per electron) is replaced by the much simpler electron (spin) density p(r, s ) , which has only four degrees of freedom. At that time, however, it was not at all clear whether such a density functional model was really legitimate in the sense that it could be proven to be an exact reformulation of traditional wave mechanics. The original idea, implemented through a statistical approximation of the electron kinetic energy in an atom, was further developed and extended by Dirac and von Weizsacker during the 1930s.9324 But it was not until 1964 that Hohenberg and Kohn6 provided a solid foundation with their fundamental first theorem, expressing that indeed the ground state energy E is uniquely determined by the corresponding electron density p(v), that is, E = E[p(r)]. The second Hohenberg-Kohn theorem, furthermore, provides the energy variational principle for DFT. It states that for a well-behaved trial density ptrial,the energy functional yields an energy that is higher than or equal to that belonging to the

Introduction

3

exact ground state density p, that is, E[pt,,al] 2 E[p]. Thus, the exact ground state electronic energy and density can be calculated without recourse to the Schrodinger equation, at least in principle. That is, the first Hohenberg-Kohn theorem merely proves the existence of such a functional relationship between E and p, but it does not give any prescription how to systematically construct or find E[p]. The introduction in 1965 by Kohn and Sham7 of a practical computational scheme may, therefore, be considered to be the next major milestone in the development of formal DFT. The essential ingredient in this approach is the postulation of a reference system of N noninteructing electrons, moving in an effective external potential u,(Y), the so-called Kohn-Sham potential, instead of the electrostatic potential ~ ( rof) the nuclei:

The first term in brackets is the usual kinetic energy operator. The noninteracting reference system has the property that its one-determinantal wavefunction of the lowest N orbitals yields the exact density of the interacting system with external potential v(r)as a sum over densities of the occupied orbitals, that is, p(r) = Cl1$~12, and the corresponding exact energy E[p(r)]. The Kohn-Sham potential should account for all effects stemming from the electron-nuclear and electron-electron interactions. Not only does the Kohn-Sham potential contain the attractive potential ~ ( rof) the nuclei and the classical Coulomb repulsion VcOul(r)within the electron density p(r), but it also accounts for all exchange and correlation effects, which have so to say been “folded into” a local potential v,,(r):

This leaves us with an effective one-electron formulation of the quantum manybody problem, which is used in essence by all current implementations of DFT. Note, however, that the exact exchange-correlation potential vx,(r) is not known as some analytical expression in the density p(r).Thus, approximations to v,,(r) have been developed: for example, the local density approximation (LDA) or the generalized gradient approximations (GGA or nonlocal DFT), whose quality determines the level of density functional theory applied. Interestingly, the Kohn-Sham approach formally validates the X a method, already formulated in 1951 by Slater, as a first (local-exchange-only) approximation of the exchange-correlation potential vxc(r) in Kohn-Sham theory, using Slater’s famous p*/3 potential, uXa(r ) ,which derives, by averaging, from the nonlocal Hartree-Fock exchange potential.

4

Kohn-Sham Density Functional Theory

Outline This chapter is organized as follows. First we discuss the status of the molecular orbital model of Kohn-Sham theory, comparing it to the HartreeFock molecular orbital method. It will prove possible to understand the similarities and the differences between the two types of molecular orbitals from an analysis of the components that make up the Kohn-Sham potential v,(r).Then we analyze the chemical bond within an MO-theoretical framework. The Kohn-Sham MOs are convenient because of their connection with exact densities (for isolated fragments and for the total system); the good approximation of the exact kinetic energy by the Kohn-Sham electron system is an advantage. A very helpful bond energy decomposition scheme is introduced and discussed. The sections thereafter apply the bond energy decomposition and are devoted to various aspects of the electron-pair and three-electron bond as well as electrostatic interaction and Pauli repulsion in the Kohn-Sham MO model. Finally, we summarize our main conclusions and discuss future prospects.

THE KOHN-SHAM MOLECULAR ORBITAL MODEL The M O concept is directly related to an approximate wavefunction consisting of a Slater determinant of occupied one-particle wavefunctions, or molecular orbitals. The Hartree-Fock orbitals are by definition the ones that minimize the expectation value of the Hamiltonian for this Slater determinant. They are usually considered to be the best orbitals, although it should not be forgotten that they are only optimal in the sense of energy minimization. The basic assumption of qualitative MO theory (QMO)is that, in spite of the well-known shortcomings of the single-determinantal wavefunction, the changes in energy of a system due to various perturbations [geometry deformations, interaction between (molecular) fragments] can be described qualitatively at the single particle level, that is, within the M O model. One primarily focuses on changes of the one-electron energies of the orbitals in response to the perturbations, as an indication of the trend in total energy. The successes of this approach are documented in a series of excellent textbooks and papers,25-30 which include references to the abundant literature on the subject. The Q M O approach has been mostly applied in the context of semiempirical calculations, although as a matter of fact many ab initio calculations, even those that use sophisticated techniques for the inclusion of correlation, very often cast their explanations somewhat illogically in simple MO language. Nevertheless, although M O theory has become a workhorse for everyday explanatory activity of chemists, it still suffers from the double odium of inaccuracy and frequent semiempiricism.

The Kohn-Sham Molecular Orbital Model

5

Following the Hohenberg-Kohn theorem, which says that the energy of a system of electrons in an external local potential u(r) (in our case, the nuclear potentials) is a functional of the electron density p(r), Kohn and Sham introduced one year later an orbital model into density functional theory. Their basic ansatz (as yet unproven!) is that there exists for every system of interacting electrons moving in an external local potential v(r) a local potential u,(r) such that a system of noninteracting electrons will obtain precisely the same density as the (exact) density of the interacting electron system. The noninteracting electrons have one-electron wavefunctions (MOs) that follow from the KS equations:

h&(r,

5)

= [-%V

+ us(r)]+i(r,s) = &(r,

5)

[31

Kohn and Sham thus introduced an independent particle (orbital) model in rather pure form. For some time a physical meaning of these KS orbitals has been denied. Their only purpose in the theory was considered to be the building of the exact density by solving (conveniently!) one-electron equations. This density could then be substituted into some good approximation to the Hohenberg-Kohn density functional for the energy to obtain an accurate total energy. However, we wish to argue (see Refs. 31-33) that, on the contrary, the KS orbitals are perfectly suited for use in the orbital theories of chemistry. The KS orbitals even have advantages over the Hartree-Fock orbitals, as will be discussed below. When forming the one-determinantal wavefunction with Kohn-Sham orbitals instead of Hartree-Fock orbitals, and taking the expectation value of the Hamiltonian, one obtains

=

T, + V -I-Wcoul + W,

The factors of lh in the electron-electron interaction terms, the Coulomb and exchange energies, Wcoul and W,, prevent double counting. It is to be noted that the electron-nuclear attraction energy V = jpv dr and the electron-electron Coulomb repulsion Wcoul = (?h)jp(r1)p(r2)/yl2 drldr2 = (?h)jp(v, )Vcoul(rl)dr, are equal to the exact quantities, since the KS density p is equal to p a c t . Then T, is the kinetic energy of the KS orbitals, which is different from the exact kinetic energy. W, is the exchange energy calculated with KS orbitals instead of

6

Kohn-Sham Density Fmctional Theoy

HF orbitals. Of course, EKS is not the exact energy. It does not play a significant role in DFT calculations, where one focuses on approximations of the exact energy E , which are obtained by approximations to the exchange-correlation functional Ex&],

Note that this equation defines Exc[p], which is simply introduced as the unknown difference between the exact energy and the known quantities (from a Kohn-Sham calculation): T,, Jpvdr, and (l/z)!pVcou,dr. Since E is a functional of the density (the Hohenberg-Kohn theorem), as well as T, (the HohenbergKohn theorem applied to noninteracting electrons), jpv dr, and ( %)JpVCou,dr, Ex,will also be a functional of the density. It is usually obtained as an integral over space of the exchange-correlation energy density E,,(Y), which itself is a functional of the density,

Increasingly accurate approximations of the spatial function E ~r)have ~ ( become available over the years. The simplest approximation is the local density approximation (LDA), where at point r the E,,(T) value is taken to be that of a homogeneous electron gas that would uniformly have the same density as the actual system has at point t: Already this approximation, which would be hard to justify physically, works rather well. This fact was already apparent from the success in the 1960s and 1970s of the X a method, which involves use of an exchange-only LDA. Recently, improved functionals have been devised that use besides the value of the density at point r also derivatives of the density, IVpl, or higher derivatives such as the Laplacian, V2p. We will take these developments for granted and focus here on the difference between the Hartree-Fock and the Kohn-Sham one-electron models. We first consider the correlation energy E, of DFT, which is defined in Eq. [6] as the difference between the exact energy and the energy of the KS determinantal wavefunction, E , = E - EKS, and also, as can be seen from Eqs. [5] and [6], as the nonexchange part of Exc: E , = Ex, - WX' This DFT definition differs from the standard definition of correlation energy in quantum chemistry, which is the difference between the exact energy and the Hartree-Fock determinantal energy EHF. The differences between the KS and HF models can further be highlighted with the help of the correlation corrections to the various eiiergy components defined above. The correlation correction is always defined as the difference between the exact quantity and either the KS or the HF one. In Eq. [8] below, the exact quantities are unsubscripted, the correlation correction is denoted by a subscript c if the difference is with respect to the KS quantity, and a subscript c plus superscript HF if the difference is with respect to Hartree-Fock (note that T K S = Ts):

The Kohn-Sham Molecular Orbital Model

7

According to Table 1, where these correlation corrections are given for the N, molecule at three internuclear distances,34 the KS kinetic energy deviates much less from the exact one than does the H F kinetic energy. Both T K S and F F are lower than the exact kinetic energy, but the former has only half the error of the latter at equilibrium distance. The KS error is not sensitive to the N-N distance, but the (restricted)H F error increases strongly. As a result, at 3.5 bohr, the HF error is four times as large as the KS error. These errors can be put in perspective by noting that the bond energy of N, is 10 eV, so the errors which are about 9 eV for KS and 17-33 eV for HF are quite significant (see Table 1). Since the kinetic energy becomes lower when orbitals are more diffuse, the results suggest that the HF orbitals are more diffuse than the KS orbitals, and therefore the HF electron density should also be more diffuse. This is corroborated by the results for the electron-nuclear energy, V; Because the KS density is equal to the exact density, there is actually no error in the KS V, so V, = 0. However, the H F density gives a rather large error. The error is negative (i.e., the VHF is not negative enough) so again the density seems to be too diffuse and must contract toward the nuclei to make the electron-nuclear energy more negative. The magnitude

Table 1 Correlation Corrections (eV) for Various Energy Components with Respect to Kohn-Sham and Hartree-Fock One-Determinantal Wavefunctions, for N, at

Three Internuclear Distancesa

~

Internuclear Distances (bohr) R(N- N )

T

Tc

TFF V

vc

VFF WCOUl WCO"1,C W%,C EC E? ~~

2.074 (= R e )

3.0

3.5

2976.66 8.95 17.01 -8261.48 0.0 -15.18 2042.54 0.0 7.46 -12.92 -12.76

2950.10 8.92 27.75 -7843.33 0.0 -36.19 1846.36 0.0 19.48 -17.44 -16.41

2953.18 8.52 33.09 -7721.43 0.0 -47.86 1786.72 0.0 26.67 -20.41 -18.69

~

aThe terms are defined in the text. 1 bohr = 0.529 A.

8

Kohn-Sham Density Functional Theory

of the error increases again strongly with internuclear distance. The electronelectron repulsion term Wcoul depends only on the electron density, and therefore has no error in the KS case. There is, however, a significant error in the HF case. The too-low electron-electron repulsion energy of HF again is in agreement with a too diffuse density. The exchange energies calculated with the KS orbitals ( W,) or with the HF orbitals ( W?”) are rather close. In spite of the large differences in the individual energy terms, the total correlation energies of the HF and the KS determinantal wavefunctions are rather close, indicating that considerable cancellation occurs between the errors of opposite signs in the various energy components in the HF case. How should we interpret this difference between the HF and KS orbital models? The HF determinant is often denoted as the “best” one-determinantal wavefunction (and therefore the HF orbitals as the “best” orbitals) because it is the determinantal wavefunction yielding the lowest energy (i.e., expectation value for the exact Hamiltonian). However, we have noted that the HF wavefunction gives quite large errors in important energy terms such as the kinetic energy and the electron-nuclear and electron-electron Coulomb energies. In N2 the electron-nuclear energy is not negative enough by 15 eV at R e , and by almost 50 eV at 3.5 bohr (to be compared to a bond energy of 10 eV and to a zero error in this term for the KS determinant). HF makes this error because it can lower the kinetic energy and the electron-electron repulsion energy by making the orbitals (hence the density) too diffuse, without being “punished” too much in terms of the electron-nuclear attraction energy. However, this imbalance increases the error in all these energy terms. In short, HF is only trying to minimize the total energy, and it will make large errors in individual energy components if, in doing so, it lowers the total energy, even if only barely. It has been noticed35736 that this “freedom” of HF to distort the density and the orbitals, if only the energy decreases, may lead to a distorted picture of chemical bonding-for instance, to localized orbitals (ionic bonds)-whereas more accurate wavefunctions (CASSCF) yield a more covalent picture. One can turn around the foregoing argument about the “distortion” effected by HF and note that the KS determinant manages to improve significantly the kinetic energy and various Coulomb energy terms with respect to HF, with only a small rise of the total energy, and therefore the (total) correlation energy. If the criterion for “best determinantal wavefunction” were based not only on the correlation error in the total energy but also on the correlation errors in the physically important energy components discussed above, the KS determinantal wavefunction would clearly be “better.” It may not be useful to argue about whether Hartree-Fock or Kohn-Sham orbitals would be “better.” But it is important to note that the KS orbitals are in no way “unphysical,” nor are they unsuitable for use in MO theoretical considerations. The properties of the KS orbitals we have noted above are a direct consequence of the form of the local potential v,(r) in which the KS electrons move:

The Kohn-Sham Molecular Orbital Model

9

The leading terms in vs are the attractive nuclear field u(r) and the repulsive electronic Coulomb potential vcoul(r).They determine the large-scale features of the spectrum of orbital energies and the shape of the MOs. The electronic Coulomb potential vcoul(r) and the nuclear attraction u(r)occur also in exactly the same form in the effective one-electron Hamiltonian for the HF orbitals (the Fock operator). The next important term is the potential of the exchange or Fermi hole, up1e(r),which has the following origin. Evidently electrons should not move in the field vcoul of the total electronic density, including their own density. An electron at a certain position creates a hole around itself in the total electronic density, so that it sees a density p + phole. The hole density should integrate to -1 electron, so that the electron moves properly in the field of N - 1 other electrons. This is called the self-interaction correction, which is the largest part of the exchange “field” in both the HF and KS Hamiltonians. There is a difference, though. The HF exchange potential is orbital dependent and contains for each orbital-apart from other contributions-a self-interaction correction hole that is minus that orbital density. The KS Hamiltonian is the same for all orbitals, being an average of the exchange holes of all occupied orbitals. This difference is usually not very important, although occasionally it leads to noticeable differences in the orbital energy spectrum. An important difference between the Fock operator and the KS effective Hamiltonian is that there are no more terms in the Fock operator, but the KS Hamiltonian contains three more terms, of which @Ie is particularly important (we will not discuss the less important potentials vc,kin and vresp; see Ref. 33 and takes into account that references therein). The Coulomb hole potential @Ie the hole around an electron should not only account for the self-interaction correction and other “genuine” exchange effects, but should also reflect the Coulomb repulsion between the electrons [see Ref. 37 for an extensive account and examples of exchange (Fermi)holes and Coulomb holes]. When an electron travels through a molecule, it “pushes away” the other electrons. This is the phenomenon of electron correlation, which is not taken into account in the Hartree-Fock model but enters the KS model through the $“Ie potential. Since other electrons are “kept out of the way,” the electron sees the nucleus less screened, and the KS orbitals tend to be less diffuse than the HF orbitals and are able to build the exact density. This explains the differences we observed above between the Hartree-Fock and Kohn-Sham orbitals and energy terms. The effect of becomes particularly striking in the case of a dissociating electron pair bond, like in H, at large distance.33.37 The existence of left-right correlation implies that an electron close to the left nucleus (A say) will see the other electron at the nucleus B to the right. The hole in the electron density should thus be completely around the reference electron at the left nucleus. HartreeFock does not have this property; its hole is symmetrical and takes away half an electronic charge density around each nucleus. Hartree-Fock has too much

10

Kohn-Sham Density Functional Theory

weight for “ionic configurations,” with both electrons of the electron pair bond on the same site, in particular at long bond distances. So our reference electron sees a partly screened nucleus and responds by staying away from the nucleus; that is, Hartree-Fock builds a too diffuse density. The potential @’le(r), however, corrects for this error. When added to v,hole, it builds a proper localized exchange-correlation hole containing minus one electronic charge density around the reference electron at nucleus A. The HF error is thus annihilated by $“le(r). This prevents the orbitals and density from becoming distorted (e.g., too diffuse, or unduly localized at one end of a bond35936) and results in the “advantages” of the KS orbitals noted above. For a more detailed account of the nature of the potentials in the KS Hamiltonian, see Ref. 3 3 . For now, the important conclusion is that the KS orbitals are perfectly suited for use in MO theoretical considerations. There is no need to have any reservations against their use for this purpose. They do not deviate in ways we do not understand from the orbitals we are used to. Similar nodal patterns occur (bonding and antibonding character) and similar orbital energy spectra (see also Stowasser and Hoffmann38). “Similar” does not mean identical: there are minor but not unimportant differences-for instance, in transition metal (TM) complexes. It is well known that orbital energies from Hartree-Fock and approximate methods such as extended Hiickel (EH) differ notably in cases exhibiting great variation in the spatial extent of the orbitals. This occurs in most TM complexes, where the d orbitals are much tighter than the ligand orbitals, which are also often (symmetry-)delocalized over several in the Fock operator for an orbital ‘pi ligands. The exchange potential z&’~ with predominantly d character will have, as the self-interaction part, the potential of a localized hole that is very much like the negative of a d-orbital density. It will be strongly attractive and pull the d orbitals down in the orbital spectrum. Even if in a photoelectron spectroscopy experiment the d electrons are often the first to be ionized, their orbitals may in an HF calculation still be found below the energies of ligand orbitals. There is nothing wrong with this, but it has made HF orbitals less convenient for easy qualitative interpretation purposes than EH orbitals. The KS orbitals, with their averaged exchange hole, usually exhibit an occupied orbital energy spectrum more like EH, and thus conform better to the naive expectations. We conclude that KS orbitals seem to be just as suitable, if not better, for qualitative MO theoretical considerations than other orbitals, e.g., HF orbitals. The KS orbitals offer the advantage, in particular over semiempirical orbitals, but also over HF, that they are connected in an interesting way with the exact wavefunction and with exact energetics. So the MO-theoretical analysis put forward in the next section deals with energetic contributions that sum up to the exact or, with the present state of the art in density functionals, at least accurate interaction energy. The KS model offers an MO-theoretical “universe of discourse” in which molecular energetics can be interpreted in terms of considerations that until now were necessarily inaccurate and qualitative. Is this MO-

MO-Theoretical Analysis of Chemical Bonding

11

theoretical analysis therefore revealing “true” phenomena? This question is as hard to answer as another question: Is electron correlation (not defined in a statistical sense, but according to either the quantum chemical or the DFT definition) a true physical phenomenon? It is a man-made concept, related to the introduction of a convenient trial wavefunction, that is useful for our communication and understanding. So are the KS molecular orbitals, and the considerations based on them (see next section), that together constitute the KS MO model of chemical bonding.

MO-THEORETICAL ANALYSIS OF CHEMICAL BONDING: BEYOND A QUALITATIVE MO THEORY Introduction MO theoretical analyses of chemical bonding and reactivity have often been kept at a qualitative level. The guiding principle in judging the influence of various perturbations is usually simply their effect on the (sum of the) orbital energies. We wish to emphasize and illustrate in this section that it is possible to carry out the analyses in more detail, to obtain a considerably more complete view of the physics of chemical bond formation. Such analyses have been carried out in the context of Hartree-Fock wavefunctions, starting with the work of Morokuma39.40 and since pursued by many working with Hartree-Fock, Hartree-Fock-Slater, and semiempirical wavefunctions.41-46 It is an asset of the Kohn-Sham model that it has a direct connection with the exact wavefunction and, via the exchange-correlation functional E,,[p], the exact energetics. Of course, in the original Hohenberg-Kohn-Sham theory E,,[p] was only defined for ground state densities. Its domain can be extended to arbitrary proper densities (integrating to N electrons, positive everywhere) by the constraint search definition of Levy,47 but it is not clear what exactly the presently available approximate functionals (LDA or GGA) are giving when densities are inserted that arise for trial wavefunctions that play a role in the analysis of bonding to be discussed below. However, the energetic effects we shall be discussing are mostly determined by other energy terms, and E,,[p] does not play an important role for the interpretation. It is a small part of the total potential energy (electron-electron plus electron-nuclear Coulomb interaction energies), which together with the kinetic energy, will be the primary tool for the interpretations. We may be confident that E,,[p] is giving a reasonable representation of the exchange and correlation energies not only in the isolated fragments and in the complete systems after the bond formation, but also in the intermediate steps we will be considering. In the same way, it has been found that excited Kohn-Sham determinants usually give quite good approximations

12

Kohn-Sham Density Functional Theoy

for excitation energies, in spite of the application of E,,[p] to excited state densities. Again, the excitation energy is primarily determined by other energy components (kinetic and Coulombic), and apparently E,,[p] gives, not unexpectedly, a reasonable description of exchange and correlation effects for the excited state density as well as for the ground state density. Similar considerations apply to the application of E,,[p] to the density po that will feature in the steric repulsion between two systems as discussed below. We will consider two interacting systems A and B, which in the simplest case are both atoms but usually are larger molecular fragments. It is no more complicated to carry out the analysis in terms of fragment MOs (FMOs)than in terms of atomic orbitals (AOs),and, of course, chemists always have recognized natural divisions of a system in interacting subunits (metal and ligands in TM complexes, donor and acceptor molecule in molecular complexes, solvent molecules and solute, etc.). Sometimes the choice of fragments is not unambiguous. Rather than hampering the analysis, one can usually gain additional insight by investigating alternative choices for the fragmentation (see the section below on Strongly Polar Electron Pair Bonding). We distinguish basically three steps in the interaction. The first one consists of bringing the unperturbed fragment charge distributions from infinity to the positions they will have in the final, interacting situation, giving rise to a superposition of fragment densities pA + pe . The accompanying energy change, AVelstat,is simply the classical electrostatic interaction between the fragment charge distributions, which is in general attractive (provided the fragments are not too close). In the second step, we go from pA + pB to a wavefunction of the composite molecule in which we do not allow for any relaxation other than obeying the antisymmetry requirement. The associated energy change, AEPauli, is designated Pauli repulsion and, as explained below, is responsible for any steric repulsion (exchange repulsion, kinetic repulsion, Born repulsion, and overlap repulsion are other names for this repulsive contribution originating from the Pauli antisymmetry principle). The trial wavefunction associated with this stage is denoted Y O , which we will represent by a determinantal wavefunction Y: that can be generated from the KS determinants of the (overlapping) systems A and B by antisymmetrization and renormalization (see below). Ziegler,42943 who was the first to apply this type of analysis within the context of DFT (Xct or Hartree-Fock-Slater), called the combined steps 1 and 2 the steric interaction step. In case of neutral fragments, it can be useful indeed to combine AVelstatand AEPauliin the steric interaction HAEO. However, we will reserve the term “Pauli or steric repulsion” for AEpauliand the term “electrostatic interaction” for AVelstat.The third step consists of the “relaxation” of the system to its , correfinal ground state energy corresponding to the wavefunction, ’I?and sponding KS determinant Ys.The relaxation is effected by mixing of virtual orbitals into the occupied orbitals and is appropriately termed the orbital interaction step.

MO-Theoretical Analysis of Chemical Bonding

13

a I

HOMO

t SOMO

4

+I

I

A

B

A

B

Figure 1 (a) Orbital interaction diagram for an electron pair bond plus lower lying occupied levels. (b)Orbital interaction diagram for a donor-acceptor interaction plus lower lying occupied levels.

Before discussing these steps in turn, we show in Figure 1 simple MO schemes of two common bonding situations. In Figure la, an electron pair bond is formed between two singly occupied frontier orbitals (SOMOs).At the same time, however, lower lying occupied orbitals will overlap and exert repulsion by destabilizing two-orbital four-electron interactions. We depict in Figure la the repulsive interaction of the frontier (singly occupied) orbitals with a subvalence or upper core orbital. At lower energies, there will be core orbitals, which at typical chemical bond lengths will have only small overlap with each other and will therefore have little mutual repulsion. Their repulsion with valence orbitals can still be significant. Although the orbital interactions leading to bond formation usually receive much more attention, these repulsive interactions are equally important for a complete understanding of chemical bond lengths and strengths and will therefore be highlighted in our discussion. Figure l b depicts a donor-acceptor interaction between an occupied frontier orbital (HOMO)on

14

Kohn-Sham Density Functional Theory

fragment B and an empty one (LUMO)on A. This occurs, of course, again with the simultaneous presence of the repulsive interactions among lower orbitals, and of the occupied frontier orbital (the donor orbital) with the lower lying occupied orbitals of the other fragment. We discuss below the physics of the classical electrostatic attraction AVelstat and the steric or Pauli repulsion AEPauli.Thereafter, we turn to the stabilizing or bonding interactions, both electron-pair bond formation and donor-acceptor interactions, as well as stabilization coming from admixing of (higher) virtual orbitals on one fragment due to the potential field of the other fragment (polarization). Finally, in the last part of this section we discuss some aspects of the mutual influence between the various interactions.

Electrostatic Interaction and Steric Repulsion When the two systems A and B are brought from infinity to their equilibrium postions, the wavefunctions Y A and Y B of the subsystems will be overlapping. The Pauli principle is obeyed by explicitly antisymmetrizing (operator and renormalizing (factor N) the product wavefunction:

a)

The interaction energy AEO is defined unambiguously as AEO = EO - E A - EB. As mentioned before, it contains a term that can be conceived as the classical electrostatic attraction between the unperturbed charge distributions of A and B and a repulsive term that originates from the Pauli antisymmetry principle and corresponds, as we will see, to the intuitive concept of steric repulsion that is widely used in chemistry. To better understand the repulsive character of AEO, we first consider the electrostatic interactions. It is important to recognize that the density po corresponding to YO differs from a simple superposition of the densities P A and pB. It is however useful to first define the electrostatic interaction between unmodified electronic charge densities p* and pB and nuclear charges 2, and Z , when put at their final positions:

where

-

15

MO-Theoretical Analysis of Chemical Bonding

is the attractive potential of the nuclei of system A (and V, similarly for the nuclei of B). We note that the first and last terms in Eq. [l11 are repulsive, whereas the second and third are attractive. At large distance (when p* and pB do not overlap), these terms cancel. When the systems approach each other, and p* and pB start to overlap, the last, repulsive, term becomes smaller than the other terms. It is well known from elementary electrostatics that two interpenetrating charge clouds have a repulsion that is smaller than the one for point charges at the centers of charge. So it is easily seen that for two approaching atoms with spherical charge densities, the total electrostatic energy between the unmodified charge distributions becomes attractive for neutral systems at the distance range of interest. Only at very short distances, too short to be of chemical interest, the nuclear repulsion (first term), which becomes singular for R +0, dominates all other terms and causes AVelstatto become repulsive. These effects usually occur in the same way for larger fragments. An illustration is provided in Figure 2, which displays AV,,,,,, as a function of the C-C distance for two C W fragments approaching each other in the linear NC-CN system (equilibrium bond length 1.39 A). AVelstat is negative (attractive) due to the mentioned penetration effect up to a very short C-C distance, where the singular nuclear repulsion starts to overtake. So we note first that contrary to rather widely held belief, the steric repulsion is not an electrostatic effect (“charge clouds repelling each other”). In the second place we note that the idea that the inner repulsive wall of the E-versus-R curve of a chemical bond is determined by the internuclear repulsion is not correct, this repulsion 20

15 10

$

5

E

O

a> C

I

0

I

0

-5

-10

-15 -20

1

0

2

3

4

NC-CN A Figure 2 Plot of AV,,, f , AVPauli,A P , AEPauli,and AEo (eV) as functions of carboncarbon distance R (in for NC-CN.

&,

16

Kohn-Sham Density Functional Theory

typically being effective at too short a range. There are other repulsive terms in the interaction, to be discussed shortly, that are responsible for the repulsive wall when the internuclear distance becomes a little shorter than the equilibrium distance. We note in Figure 2 that the AEO term is indeed repulsive, in spite of the attractive AVclstat. To consider the steric repulsion embodied in AEO more closely (see accounts in Refs. 48 and 49), we take as an example just singly occupied orbitals, qA and qB, respectively, on the fragments A and B. They should have a nonzero overlap S = (qA Iqe), hence being spinless orbitals or spinorbitals of the same spin (precisely the same argument is used for closed shells of doubly occupied orbitals qAand qB).We obtain, with x1 denoting both the space and spin variables, x1 = ( y l , s1 ),

The d operator in this case is just (1- PI, ), with P12the permutation operator. Equation [12] demonstrates that po is different from P A + pB= IqAP+ IqB12.The first two terms show that the density at A and B is actually the original density enhanced by a factor 1/(1- S2) > 1. The last term in the expression for pO on the last line of Eq. [12] shows that this enhancement is effected by a depletion of density from the overlap region, where both qA and vB, hence the product qAqB,have sizable values. In Figure 3 we show a plot of the difference density ApO = po - p A - pB in the NC-CN molecule (A and B are the CN. monomers). This dimer, which is discussed more fully in the next section, is an example of the situation for which a simple MO representation is depicted in Figure la: an electron pair bond can be formed by two singly occupied orbitals (the CN 5 0 SOMOs in the case of the symmetrical NC-CN), and repulsive interactions arise from lower lying, overlapping, doubly occupied orbitals, notably the 40.(The SOMOs have spin up and down and do not overlap; they only contribute to the ApO by their overlap with the same-spin lower orbitals on the other fragment, see below.) The depletion in the overlap region and the enhanced density at the atoms are rather striking. The shape of the ApO type of electron depletion in the overlap region is rather typical (see other such plots for Ag-048 and K' in a W, cluster49). We note that this charge rearrangement is different from the buildup of electron density in the overlap region normally associated with bond formation. It is the relaxation from the wavefunction YO to the fully converged wavefunction Y, which is the second step in our analysis of the chemical bond formation, that will bring charge back into the overlap region. The change in the density from the superposition of the fragment densities to p0 may be

MO-Theoretical Analysis of Chemical Bonding

17

6.5

6

5.5

5

4.5

4

3.5 3 2.5 2

1.5

1

0.5

n

0

1

2

3

4

Figure 3 Contour plot of the difference Apo between the electron density of NCCN and the densities of the CN. radicals (contour values: kO.001, k0.002, f0.005, 50.01, k0.02, k0.05, kO.1, k0.2, 50.5, 0.0 e bohr-3). Asterisks indicate the positions of the

nuclei.

viewed as a manifestation of the Pauli principle. According to this principle, which follows from the antisymmetry requirement for fermion wavefunctions, electrons are not allowed to be at the same place with the same spin. The antisymmetrization we had to carry out actually reduces the probability density in the overlap regionso from what it would be if the necessary antisymmetry of the wavefunction had not been taken into account.

18

Kohn-Sham Density Functional Theory

The energy change from E A + E B to EO can be written as the electrostatic interaction defined above, plus all other effects lumped together into the Pauli repulsion term AEPauli:

It is always possible to write the energy corresponding to a wavefunction as the sum of the kinetic energy (the expectation value of the kinetic energy operator) and of the potential energy, which is the expectation value of all Coulombic operators, so we write AEO as the sum of kinetic and potential energy differences, AEO = AVO + AT0 = AVelstat + AVpauli + A P

~ 4 1

AEPauliis broken up into a potential energy (AVpauli)and a kinetic energy ( A P ) part. The potential energy part AVPaulirepresents the change in the potential energy due to the change in density from P A + pB to PO. This change represents the charge flow out of the overlap region into the atomic regions, which is depicted in Figure 3. This is the opposite of the buildup of a “bond density,” which is known to occur in most chemical bonds. However, it is again a common misconception to consider the buildup of a bond density as favorable for (i.e., lowering) the potential energy and, therefore, as an important energetic factor in the bond formation. On the contrary, in the overlap region the potential in which the electrons move is higher (less negative) than close to the nuclei, even when the screening effect of the other electrons on the nuclear attractive potential is taken into account. The Coulombic wells of the nuclear potential are so deep around the nuclei that flow of electronic density out of the overlap region closer to the nuclei, as is happening in the steric repulsion step of the bond formation, is actually lowering the potential energy considerably. Therefore, AVPauli is often a more negative term than the electrostatic interaction AV,,,,,,, as is very clear in Figure 2. On the other hand, the flowback of charge in the relaxation step to the final wavefunction and charge density, to create the “bond density,” will be unfavorable for the potential energy. The name steric or Pauli repulsion for AEpaUlialready suggests it is repulsive (positive, antibonding), in spite of the negative contribution AVPauli. The repulsive character is due to the strongly positive AT0 (see Figure 2). Steric repulsion is evidently a kinetic energy effect and may also be appropriately called kinetic repulsion. Let us consider the simple case that qA and q B are symmetry equivalent (e.g., the 1s orbitals of He, or triplet H2).The well-known MOs 1

1

MO-Theoretical Analysis of Chemical Bonding

19

can be formed as a linear transformation of the (qA,qB) set: w = cpT, where the underlining indicates a row vector, T is the transformation matrix, and S is the overlap integral of qA and cpB. We recall that forming a Slater determinantal wavefunction with orbitals that are obtained by a linear transformation yields just the Slater determinant of the original orbitals multiplied by the determinant of the transformation matrix: IyI = lcpl det T. (Here the determinantal bars indicate, as usual, formation of theslate; determinantal wavefunction, including a normalization factor.) So the Slater determinantal wavefunction for triplet H2 on the basis of the orthogonal set (yig,yiI1)is just the antisymmetrized and renormalized product wavefunction Yo = NA{qAqB}:

l/m

Yo = lyig~wuctl= IcpAaq,aIdetT = NA{cpAacp,a}

[I61

Note that the additional normalization factor det T appears because the qAand cpB in the Slater determinant are not orthogonal; therefore, Iq,acp,al is not is not equal normalized. For the same reason, the electron density of IcpAacpBal to simply the sum of the orbital densities. Since in the Slater determinant Ivgaiy,al the orbitals wg and yi, are orthogonal, the density is the sum IwgP + Iwu12, which is easily seen to be just the po of Eq. [12]. When the overlap between qA and cpB is appreciable, the rise in kinetic energy AT0 will be considerable. This may intuitively be seen from the nodal plane in the antibonding orbital w,, when one remembers that the kinetic energy, which can be written in different forms by a partial integration,

will become high when there are large regions in space with high gradients lVwJ for some orbitals. This is evidently the case for tyu around the nodal plane. Of course this argument is not restricted to a symmetrical situation. When one carries out the transformation from the set qA,cpBto an orthogonal set yil ,w2 by a Gram-Schmidt orthogonalization of qBonto the unmodified cpA = v1:

then the kinetic energy is invariant to further orthogonal transformations of the yil ,yi2 set (for instance to yilg,yiu in case of symmetry-equivalent orbitals (PA,(PB). The density po is, of course, also invariant under such a transformation. If the orbital qAis a core orbital and cpB a valence orbital, the transformation to w1 ,w2 will be closer to the one that would be obtained by a diagonalization (in the

20

Kohn-Sham Density Functional Theory

qA,qBspace) of the KS Hamiltonian 6: = %V2 + v,[pO](r).Diagonalization of fi: leads to “canonical” orbitals for the steric repulsion situation to which meaningful orbital energies can be assigned. We have been discussing the case of two overlapping fragment orbitals of the same spin, but the whole treatment carries over without change to doubly occupied orbitals qAand q B ,with only insertion of an occupation number 2 in appropriate places (notably in Eq. [12] for the density po in front of each of the three terms). Equation [18] shows that the kinetic energy increase that determines the repulsive character of the steric repulsion step will be significant in two cases. First, even if the kinetic energies TAand TBare themselves small, as is the case when qAand q B are valence orbitals (e.g., occupied bond orbitals, or lone pair orbitals), the rise in kinetic energy will still become significant when the overlap becomes large, by virtue of the S2/(1 - S2) prefactor. This will occur when the relatively diffuse valence orbitals are pushed too close. This is precisely the phenomenon of steric repulsion, well known in chemistry. We have therefore traced its origin to a purely quantum mechanical effect, namely, the rise in kinetic energy due to the necessary antisymmetrization of the overlapping wavefunctions in accordance with the Pauli principle. The effect can be conveniently described in terms of the required mutual orthogonalization of the fragment orbitals. We emphasize again that the steric repulsion is not an electrostatic effect of Coulomb repulsion between overlapping charge clouds. There is a second important instance in which a large increase of the kinetic energy may occur according to Eq. [18]. When qAis a core orbital, its kinetic energy will be much larger than that of valence orbitals. On the other hand, the overlap with valence orbitals will usually be small. Deep core orbitals would bring in a huge amount of kinetic energy if they were mixed, in the orthogonalization process, into valence orbitals. However, since the overlaps between diffuse valence orbitals and the very tight deep core orbitals are very small, the S2/(1 - S2) prefactor will be small, and the effect of the deep core will actually not be large. The opposing effects of diminishing overlap and increasing kinetic energy going from (sub)valence to (upper) core and deep core orbitals, leads to a maximum effect for upper core orbitals that still have sufficient size to overlap with valence orbitals, yet have already such a large kinetic energy that orthogonalizing the valence orbitals onto them gives rise to a large kinetic repulsion. Transition metal complexes comprise a typical example. The lone pair orbitals of ligands like CO, H20, C1-, and 0 2 - experience significant repulsion from the upper core shells 3s and 3p (first transition series). The Pauli repulsion with these shells determines the repulsive wall in the metal-ligand E-versus-R curve. Of course, there is also overlap with deeper core orbitals, and so their effect is less important. An illustration is provided elsewhere (Figures 2 and 3 of Ref. 35), where the behavior of the Pauli repulsion is demonstrated along the Mn-0 bond distance in MnO;. The fragments are Mn2+, which has 5 electrons with spin up in the 3d orbitals, and the 0:- cage, which has 5 electrons

MO-Theoretical Analysis of Chemical Bonding

21

with spin down in the oxygen 2p-based e and t, orbitals. The latter will form bonds with the 3d orbitals that also have e and t2 symmetry. Near the equilibrium Mn-0 distance, there is little overlap between one combination oft, 0 2p and Mn 3d orbitals, and the repulsion is completely due to the overlap of the 0 2p with the Mn 3s and 3p shells (see Ref. 35 for a more complete discussion of this example). On top of the attractive AVelstat and the sterically repulsive AEPauli, we will have attractive orbital interaction energies, as discussed later. Thus, it is the Pauli repulsion between the atoms (fragments) that determines the inner repulsive wall of the E-versus-R curve. Another simple and clear example is the Li, molecule, which has a remarkably longer bond length and smaller bond dissociation energy than its for Re, 1.1 eV vs. 4.8 eV for D e ) . first-period congener, H, (2.67 vs. 0.74 i% There are two important differences. First, the lithium 2s and 2p valence AOs are very diffuse and begin to build up a bond overlap at larger internuclear distances than the hydrogen 1s valence AO (see the section below on Strongly Polar Electron Pair Bonding and also see Bickelhaupt et al.51). Second, lithium atoms contain a I s core shell. Even though these cores do not overlap at the equilibrium bond distance, the Li valence 2s orbital overlaps with the 1s core on the other atom. The orthogonality requirement of the 2s orbital onto the Is orbital on the other center leads to a strong increase in the kinetic energy (see AT0 in Figure 4a). In H, (see Figure 4b), there is no orthogonalization, and therefore A P is identically zero. The increasing A P in Li, builds a repulsive wall that forces the Li-Li bond length to stay long. The absence of A P in H, allows the bond in H, to become short, although the fact that AVelstat in H, is building a repulsive wall at considerably shorter distance than in Li, also contributes to making the very short bond length in H, possible. It is nevertheless clear that A T is the determining factor for the repulsive wall that, together with the early buildup of bond overlap between the valence AOs, causes the long Li-Li bond length. (Parenthetically we remark that we should have AEO = AVelstat in H,, since both AVPauli= 0 and A 7 0 = 0, but this is not exactly given by most density functionals owing to deficiencies in the self-interaction cancellation; this is ignored in these plots.) It is well known that the full A T value in H, (molecular kinetic energy minus sum of atomic kinetic energies) is by no means zero; at the equilibrium distance it is positive and equal to the dissociation energy. We refer to the analysis by Ruedenbergs2.53 of the role of kinetic energy and potential energy in the bond formation in H; and H, . So the Pauli repulsion of the Li 2s electrons with the I s core together with the early buildup of bond overlap due to the diffuse nature of the 2s and 2p valence AOs cause the Li-Li bond to stay long and weak. This is yet another illustration of our earlier assertion that it is not the nucleus-nucleus Coulombic repulsion that determines bond lengths, but the Pauli principle. The Pauli principle gives the atoms their spatial extent, by forbidding the electrons to all huddle together in the Is shell, but forcing them to occupy the higher nl shells.

Kohn-Sham Density Functional Theory

22

10

8

2

0

-2

0

a

1

2

Li-Li

3

4

3

4

A

10

8

v)

6

c)

0

>

co4

-8 a ! Q3.

2 0

-2

0

1

2

b

H-H A Figure 4 (a) Li,: the energy terms (eV) AVelstat, AVpaulirA P , AEPauli, and AEO as functions of R (A). (b) H,: the energy terms (eV) AVelsrat, AVpaul,, A P , AEPauli, and AEO as functions of R (note that here AEO = AVelStat; AEPaul, = AVpauli, = A T = 0).

This is one factor determining the spatial extention of matter. In much the same way, the Pauli principle gives chemical bonds their length. The valence electrons of one atom are not allowed (at least not readily) to penetrate the occupied shells of the neighboring atom, and thus atoms cannot be pressed too close. This is the second way in which the Pauli principle acts to give matter its

MO-Theoretical Analysis of Chemical Bonding

23

spatial extent. The bonding in H,, even though this molecule is often considered the prototype system for an electron pair bond, is in fact rather atypical in the sense that the electron pair bond can be formed without interference of any lower lying core shells or occupied valence orbitals. Usually such other occupied orbitals are present and play an important role in determining the bond length and strength. In the next section, we will discuss in some detail the example of bonding between two CN. radicals, where the bond between the singly occupied 5 0 orbitals is heavily influenced by the presence of the fully occupied 40 N lone pair orbital. In this case there is, in contrast to the case of Li, ,not only overlap between the singly occupied unpaired electron orbital (2s and 50, respectively) and the opposite closed shell (1s and 40, respectively), there is also considerable overlap between the 40 orbitals, contributing strongly to the Pauli repulsion. This Pauli repulsion, as well as the Pauli repulsion with the 5 0 electrons, will of course be different for the different bonding modes (NC-CN, CN-CN, and CN-NC). A detailed analysis is provided in the next section. Before leaving the subject of Pauli repulsion, it is interesting to note that Pauli repulsion becomes more important in situations of steric crowding. If we expand the 1/(1- S 2 ) factor in To,we see that the rise in kinetic energy for two overlapping occupied shells can be expanded in S2 as follows:

P - T A - TB = S’(TA + T B ) + S4(TA + T B ) + S 6 ( T A + T B ) + . . . [19a] However, when one is dealing with a shell vBoverlapping simultaneously with two shells qA,with equal overlaps S, the increase of the kinetic energy would be

The Pauli repulsion cannot be obtained as a sum of pairwise interactions, but there is a strong three-body effect in the S4 and higher terms.49

Attractive Orbital Interactions The interactions we have been discussing are typically between fully occupied shells, or between a singly occupied valence orbital with fully occupied valence and lower levels on the other fragment. These repulsive interactions are always present (except in H, ). Nevertheless, most interest in theories of chemical bonding has centered on the attractive interactions that cause atoms (and larger fragments) to stick together, i.e., to form bonds. In our decomposition scheme, these interactions arise at the orbital interaction step when the system “relaxes” from the wavefunction YO to the final wavefunction Y . In the KS model, the KS orbitals of the determinantal wavefunction Y f relax, by mixing in virtual orbitals in a simple self-consistent field calculation, to the final KS orbitals that build the exact density and form the KS determinant Y s. If the two

24

Kohn-Sham Density Functional Theoy

interacting fragments are closed-shell systems (see Figure lb), the orbital interactions will consist of charge transfer or donor-acceptor interactions between occupied orbitals on one fragment and virtual orbitals on the other. At the same time, polarization will occur, consisting of occupied-virtual interactions on one fragment. Charge transfer and polarization cannot be strictly separated, and we will not attempt to do so. If there are singly occupied orbitals, usually one on each fragment, the orbital interaction will primarily involve formation of an electron pair bond by the pairing up of the unpaired electrons in a bonding orbital (see later section for three-electron bonds). We refer to Bickelhaupt et al.54 and to the next section for a detailed discussion of examples of pair bond formation in the presence of occupied-occupied and occupied-virtual interactions. We will consider the distinction between electron pair bond formation energy and other interaction energies below, but we first derive a decomposition of the total orbital interaction into components that can be labeled by the irreducible representations of the point group of the system, which was originally introduced by Ziegler and Rauk.55 Such a decomposition may be particularly helpful in distinguishing, for example, (r and n interactions, at least when they occur in different irreducible representations. We are interested in the energy change when going from YO with density po to the final system with wavefunction Y and KS determinant Ysand density p = pexact. We will denote this energy change arising from orbital interactions as AE,, . One can express the densities in terms of the basis functions {x,) with the help of the corresponding density matrices PO and P,

It was shown by van Leeuwen and Baerendss6 that the energy change, when going from an initial density p i to a final density pf, can be obtained from a path integral along a path in the space of densities that connects the initial and final densities. The path is arbitrary, since the initial and final energies depend only on the respective densities. In the present case, we may take the simple linear path measured by the parameter t, from po = p ( t = 0 ) to p = p e x a c t = p ( t = l), p ( t ) = po

+ tAp;

P,,(t)

= PEv

+ tAPPv

Here, we have used the fact that the derivative of the energy with respect to a density matrix element is the corresponding one-electron Hamiltonian matrix

MO-Theoretical Analysis of Chemical Bonding

25

element, F,, ( t )= dE/dP,,( t).We continue to write the matrix elements of the KS one-electron Hamiltonian hs as the Fock matrix; incidentally, the relation F,, = dE/dP,, holds for the Hartree-Fock case as well. We also define the “transition state” matrix element F: as the integral fiF,,( t)dt. Approximate so-called transition state expressions for energy differences were derived earlier by Slater for ionization and excitation energies57 and by Ziegler and Raukss for energy differences related to general density differences, by means of Taylor expansions of the energy in small changes of the density. Slater performed the Taylor expansion around the midpoint or “transition state” density pTS = I%( p + PO), obtaining with F;fys = F,,(t = %) an expression correct to second order in AP,,,,. Ziegler included expansion at the beginning (PO) and end (p) points, obtaining with FF: = l/sF,,(po) + %F,,(pTS) + %F,,(p) an expression for AE correct to fourth order in APPV.Equation [21] generalizes these results to an exact expression for AE. In practice the t integral can be done very accurately by some Gauss numerical integration method (see, e.g., Chapter 20.5 of Ref. 58) over the [0,1] interval, but it is rarely necessary to go beyond the Simpson rule,58 which is identical to Ziegler’s expression for Ps. Y It is possible to use as the basis functions (x,) symmetry-adapted combinations of primitive basis functions. This affords a decomposition of the orbital interaction energy of Eq. [21] according to irreducible representations of the point group

since there are no matrix elements of either P or F,, between functions p and ? v belonging to different irreducible representations r. Many examples are availand are given in following sections. Here we able in the literat~re42,43,54~59-67 show the decomposition of the orbital interaction terms for the cases of the tetrahedral complex35 MnO, (Figure 5) and the octahedral c0mplex5~360 Cr(CO), (Figure 6). Five electron pair bonds are formed between the d5 Mn2+ ion, with unpaired electrons in both the two e-type and the three tz-type 3d orbitals, and the 0:- cage with unpaired electrons in matching orbitals of e- and t,-types. It can be seen from Figure 5 that the t2 bonds are stronger and start to form at longer distance than the e-type bonds. This can be related to the predominantly CY character of the t, bonds, and TC character of the e bonds. It is interesting to note that these bonds are virtually homopolar in the KS calculations, whereas the e and t2 orbitals are rather unsymmetrical in the HartreeFock model. The HF e orbitals localize on the oxygens, having only 1.1electron in the 3d rather than the expected 2; the t, orbitals on the other hand have 4.2 e in the 3d rather than 3. Only at the level of the complete active space selfconsistent field (CASSCF)method do the 3d occupations come close to 2 and 3 in e and t2 symmetry, respectively.35 The failure of the KS orbitals to exhibit the distortion of the electron distribution suffered by the Hartree-Fock model can

26

Kohn-Sham Density Functional Theory 40

20

Bc:

(II

0

4 +I

8

a,

d

-20

-40

-60

1

Mn-0 8,

2

3

Figure 5 Attractive orbital interaction terms (eV) between Mn2+ and 0;- in MnO, in various irreducible representations as functions of internuclear separation R (A).

be related to the existence of the Coulomb hole potential in the KS potential v, . The importance of left-right correlation in this case (between metal and 0, cage), as well as how its absence in the HF model distorts the HF charge density, is discussed extensively elsewhere.35 Figure 6 displays the orbital interaction energies as a function of distance for the interaction between Cr in its (t2J configuration and the (CO), cage. The tZgsymmetry represents the 71: backbonding out of the dt2e(d,,, dxy, dyz)orbitals into the CO 71:" orbitals. Clearly this interaction sets in at larger bond length and is much stronger than the o donaEven at tion in eg symmetry out of the So lone pairs into the dCg(d,2-y2and dZ2).

MO-Theoretical Analysis of Chemical Bonding

2o 18

I -

16

-

I

14 -

10 -

I

27

I

AE

12

g

0

-----

5

-2-

l u

-4-

0

AEO

4

-6

-

-8

-

Cr-C

A

Figure 6 Attractive orbital interaction terms (eV) between Cr and (CO), in Cr(CO), in various irreducible representations as functions of R (A).

the equilibrium distance, the 7c bonding is still much stronger. All other interactions are much less important. We refer to earlier work59 for a discussion of these other interactions and of the mutual strengthening (synergistic effect) of 7c back-donation and CT donation. It is also possible to perform a basis set transformation from primitive basis functions to symmetry combinations of the KS MOs of the atoms or larger fragments that constitute a system. In that case the population matrix elements P,, become more meaningful, because they reflect the involvement of the fragment MOs in the orbitals of the total system. A Mulliken population analysis i n

28

Kohn-Sham Density Functional Theory

this symmetrized fragment orbital (SFO) basis, for instance, yields gross populations of SFOs, which for a virtual orbital give an indication of the population that the virtual orbital acquires as a result of admixing in the occupied orbital space. For an occupied orbital, the populations give an indication of the charge donation out of the orbital. Combined with the decomposition of the interaction energies between fragments we have been discussing in this section, this whole fragment-based approach affords an excellent analysis method for interactions between chemically meaningful moieties.

Interplay of Steric Repulsion and Orbital Interaction The two components into which we have decomposed the interaction energy act almost always simultaneously and will influence each other. We end this section by giving an account of this mutual influence and of the distinction between donor-acceptor and electron pair bond formation energies. These issues will reappear in the discussions of specific cases in the sections that follow. The total interaction energy can be written as follows:

The preparation energy51J9-61>133AE,,,, is the energy required to prepare the fragments for the interactions described by the other energy components. Usually the preparation energy will contain the energy required to deform the geometries of the fragments from their shapes as isolated substances to the geometries they have in the interacting system. Sometimes this is a minor geometric and energetic effect, such as the lengthening of a CO bond from its value in free CO (1.128 A) to the value in a carbonyl complex (ca. 1.15 A); sometimes it is a significant perturbation, such as changing the planar CH; radical to the umbrella shape of the CH, fragment in ethane. The preparation energy may also include an electronic excitation energy when the ground state electronic configuration is not the most suitable for an energy decomposition. For instance, the Fe2(CO)8fragment of Fe,(CO), and Fe3(CO),, has a ground state configuration suitable for o-donor and r-acceptor interaction with Fe(CO), [to form Fe,(CO),,], but it must be prepared by an electronic excitation to an electronic configuration suitable for n-donor and o-acceptor interaction with CO [to form Fe,(CO),] (see Ref. 61). The simplest interpretation of AEoi occurs when two closed-shell systems are interacting by a donor-acceptor interaction, as depicted in Figure lb. This charge transfer will usually coexist with other effects such as polarization (occupiedhirtual mixing on one fragment due to the presence of another fragment) and relief of Pauli repulsion. This is illustrated in Figure 7, where we

MO-Theoretical Analysis of Chemical Bonding

29

out-of-phase

polarizationand charge transfer

2n'

in-phase

AI

lx:

AI

I 7

AI

T

out-of-phase polarization and charge transfer

%

in-phase

CN

CN

Figure 7 n-Orbital interaction diagram for the CN dimers. Note that the n-electronic where the subscript pb denotes pair bond [see structure is the same for YO and Eq. WI).

"Pob,

show the situation in n symmetry for the linear interaction between two CN* radicals to be discussed in the next section (closed-shell holds in this case for the n symmetry only, but for now we consider the interactions in just the n symmetry). One can picture the n-bond formation between the two fragments as donative bonding (charge transfer) from an occupied l n A to the empty 2nB and vice versa. At the same time, however, the Pauli repulsion that exhibits itself in the formation of the occupied antibonding combination, lnA - l n B , is relieved by admixture of 2nA - 2xB, which similarly leads to occupation of the 2n and electron depletion from 1 n. We cannot decompose these simultaneous processes; instead, we denote them collectively as the AEn term in the decomposition of the orbital interaction energy AEoi according to symmetry. Interactions may occur not only between fully occupied orbitals on one side and empty orbitals on the other side, but also between singly occupied orbitals that form an electron pair bond. It is possible in this case to simply stick to the symmetry decomposition of the orbital interactions, noting only that the irreducible representation r where the electron pair bond is formed will have a large AEr for that reason. It is also possible, at least when the electron pair bond is between two identical fragments, to carry the analysis somewhat further and

30

Kohn-Sham Density Functional Theoy

make an estimate of the energy of electron-pair bond formation versus the donor-acceptor interactions that will also be present (see Ref. 54).Although we will not go into so much detail in the examples to be discussed in the following sections, we present for completeness this last step in the energy decomposition in this section. As an example, we take the electron pair bonds of NC-CN and CNNC in 0 symmetry between the 5 0 singly occupied highest occupied orbitals (SOMOs) of the two CN- radicals, two systems treated in the next section. Below the 5 0 orbitals, there are fully occupied orbitals, the most important one orbital, see next section). The being the 40 N lone pair orbital (the oHOMO wavefunction YO is written in this case as follows: Yo = N /(closed shells)A (closed shells)B5oAa(l) SoBp(2)l

[24]

Since the fragment orbitals in Y O are overlapping, the determinantal wavefunction will not be normalized, and we have added a normalization factor. For Y O , only one of the two valence bond (VB) structures is taken, for reasons to become clear presently, so its symmetry is C, rather than Dmh, and the C, symmetry must be used in the analysis. As we have seen, one way to evaluate the energy of Y O is to first orthogonalize the A and B orbitals onto each other. Once the orbitals are in an orthogonal set, the density p0 can be written as a sum of orbital densities, and the energy can be obtained from the Slater-Condon rules for evaluating matrix elements between determinantal wavefunctions. The occupied 40 lone pair orbitals will overlap with each other, which will induce steric repulsion. Such steric repulsion can conveniently be represented in an elementary MO diagram as Figure l b as a four-electron two-orbital destabilizing interaction. As for 5 0 , it is to be realized that the 5 0 A a orbital is orthogonal to 5 0 B p on account of the spin orthogonality, so there is only Pauli repulsion coming from the orthogonality requirement of the 5 0 of one side on the samespin occupied orbitals of the opposite side, and vice versa. We may let the wavefunction Y O of Eq. [24] relax to the SCF solution by admitting interactions with the virtual orbitals. This yields the total orbital interaction energy AEoi. It will contain, in addition to donor-acceptor interactions, also the energy lowering connected with the formation of an electron pair bond. This will in our example be part of the energy in 0 symmetry. To get an estimate of the pair bond energy, one might consider as an intermediate step the pair bond (pb) wavefunction54 Y$,: Y$, = N [(closedshells), (closed shells), (50,

+ 50,)21

[251

The electrons in the 5 0 orbitals have now been allowed to pair up in the 5 0 A + 5uB bonding orbital. The energy of the electron pair bond is now estimated from (defined as) AEPb = AEib - AEO. Figure 8, which displays the various

MO-Theoretical Analysis of Chemical Bonding

31

I

IEpb A E ~

AEo'l

0

YDb

d CN fragments

%CF

Figure 8 Diagram of the relation between the various energy changes used in the interaction energy analysis.

energy components we have been distinguishing, also indicates AEpb,Note that to the left of Figure 8, the familiar terms AEO and AE,, are indicated, and we are now dealing with a further decomposition of the AEoi step, as depicted to the right in Figure 8. We use YEb to define a pair bond energy, but an alternative definition would of course correspond to the valence bond wavefunction in which Y O is combined with the determinantal wavefunction in which the electron spins have been interchanged: I(c1osed shells) SOAP( 1)5oEa(2)1. The wavefunction with these two VB configurations does give a somewhat lower energy in the case of H, at equilibrium distance, compared to the MO-LCAO wavefunction, but it is well known that the interpretation of the bonding is not really different in the VB and MO cases.68 The energy lowering upon bond formation in either the VB or the MO description is caused by the resonance integral (hopping integral, interaction matrix element) (SOAlheff150B). The MO representation of Eq. [25] fits in naturally with the MO-based presentation we are giving. Note that in the wavefunction YEb the 50, + 50, orbital will overlap with lower lying orbitals of the same symmetry, notably the 4 0 +~ 40,. This embodies the Pauli repulsion of the 50 SOMO orbitals with the 40 HOMO

32

Kohn-Sham Densitv Functional Theorv

orbitals (note that the 50, - 50, is empty; there is no Pauli repulsion between SOMO and HOMO in the C, symmetry of D,, ,in accordance with the single occupation of the 50). Finally, the wavefunction Yo is allowed to relax to pb. the SCF Kohn-Sham solution Ys. The admixture of virtual orbitals yields the relaxation energy AErelax(see Figure 8). This relaxation step contains charge transfer and polarization contributions, which partly serve to relieve the steric repulsion. The various steps are illustrated with the orbital interaction diagram of Figure 9, where only 40 and 50 orbitals are used. The steric interaction energy, corresponding to YO, consists for a large part of the four-electron two-orbital destabilizing interaction between the two occupied 40 orbitals. We have indicated this by a sizable splitting between the stabilized bonding and destabilized antibonding orbitals. In the case of Li,, the occupied 1s core orbitals would be

out-of-phase

polarization and

41

t

41 v

7 Pauli repulsion

41 t

4lr

YtJ

Y O

CN

CN

(CN) *

I

'1

(CN),

(CN),

in-phase

%F

(CN),

Figure 9 o-Orbital interaction diagram for the CN dimers, representing the interaction between the CN 40 and So fragment orbitals, The first step, formation of YO, corresponds to the steric interaction (AEO). The next step, drawn with lighter lines, corresponds to the formation of the "pure" pair bond, that is, the fictitious situation of forming 50 + 50' without the Pauli repulsion with the 40 + 40' (and 3 0 + 3o', etc.) in-phase combinations. Going from Y O to YE, represents the formation of the pair bond (AEpp) including this Pauli repulsion. In the final step, the wavefunction Y& is allowed to relax to the SCF solution YScF by the admixture of virtual orbitals, yielding AErelax.

MO-Theoretical Analysis of Chemical Bonding

33

the analogs of the present 40. In that case, the splitting between the bonding and antibonding combinations of the 1s core orbitals would not be large, and little repulsion would come from core-core overlap. The singly occupied SOMO orbitals will be destabilized by overlap with the fully occupied closed shells on the opposite fragment. So both the (CN), dimer and Li, will have Pauli repulsion by the singly occupied orbital ( 5 0 and 2s, respectively) with the next lower doubly occupied shell on the opposite fragment (40 and Is, respectively). This is indicated in the diagram of Figure 9 by a destabilization of the SOMOs in the Y O wavefunction. The second step in the interaction consists of the formation of YEb, containing the doubly occupied bonding orbital, which yields the energy lowering AEPh in Figure 8. Conceptually, we may consider the change from the YO to Y$, to occur via the formation of the strongly stabilized 50, + 50, (or 2sA + 2sB in case of Liz); see the lighter levels in Figure 9. The intermediate step denoted by the lighter levels in Figure 9 is not associated with a trial wavefunction; in Yo this pair-bonding orbital will already be pb destabilized again by repulsive interaction with the occupied 40, + 40, orbital (Is, + 1s.B in Liz). Although one cannot associate a wavefunction with the situation depicted with the lighter lines in Figure 9, it will nevertheless be useful to keep in mind that the total AEpb contains not only the “pure” pair bond formation energy but also the above mentioned repulsive effect. In the last step, ,,’ is formed by allowing virtual orbitals to mix in. It is to the wavefunction Y be noted that in the virtual orbital space, there is the antibonding 5 0 , - SOB, which will mix with the 4a, - 40,. Here the 5 0 acts effectively as an acceptor orbital and relieves the Pauli repulsion of the lone pair orbitals 4 0 by stabilizing the antibonding partner in the bondindantibonding set of 40-derived levels. This is completely analogous to the interaction of the 2n, - 2nB with the In, In, that we met in the n: symmetry. However, it is to be noted that the 5 0 , being much lower in the orbital energy spectrum than the genuine virtual orbitals like the 271, can be quite effective in this acceptor function. More detailed discussion, in particular for the differences in this relaxation term between the N-N and the C-C bonded (CN), dimers, is provided in the next section. The relaxation energy obtained in this last step, of course, must be combined with the pair bond energy AEph to obtain the total orbital interaction energy AE,, (see Figure 8). This total orbital interaction energy may always be obtained straightforwardly; it is not necessary to go through the further decomposition steps we have been discussing here. The case of a homopolar electron pair bond is relatively straightforward, since the mixing of the two singly occupied orbitals is determined by symmetry. Decomposition in the case of a heteropolar electron pair bond requires a choice to be made for the mixing of the SOMOs in the pair bond wavefunction-for instance, according to the mixing in the final SCF wavefunction. In the sections that follow, we discuss several examples of the energy decomposition analyses we have presented thus far.

34

Kohn-Sham Density Functional Theow

THE ELECTRON PAIR BOND AND PAUL1 REPULSION Introduction In this section and the following one, we discuss how the concepts of the one-, two-, and three-electron bonds-originally introduced by Pauling69-71 in the context of valence bond theory-are defined and how they can be analyzed within the framework of Kohn-Sham MO theory. The fragment approach, as pointed out in the preceding section, is of eminent importance here because it reintroduces the concept of a local chemical bond in the otherwise delocalized picture of molecular orbital theory. Thus, in MO theory a two-center, oneelectron (2c-le) bond A . B arises from the interaction between a singly occupied molecular orbital (SOMO) on fragment A with an unoccupied one (e.g., the LUMO) on fragment B as shown schematically in 1. Likewise, a two-center, two-electron (2c-2e) or electron pair bond A-B is formed when two SOMOs with antiparallel spins interact (2), and a two-center, three-electron (2c-3e) bond A :. B emerges from the interaction of a SOMO and an occupied MO, typically the HOMO (3). 2 0 1e

2c-2e

2c-3e

,-.

A

A*B 1

B

A

A-B 2

6

A

A:.B

B

3

These level splittings are the essential features that determine the type of chemical interaction, i.e., 2c-le, 2c-2e or 2c-3e bond. However, as discussed in the preceding section, there are nor many examples in which these bonds occur in such a "pure" state. Archetypal representatives of these bonding modes (e.g., Hi, H2 ,HeH) are in fact rather atypical for chemical bonds in general. In nearly all other cases, we have to deal with mono- or polyatomic fragments, carrying additional electrons in other orbitals. These electrons will interfere with and affect the nature of the primary frontier orbital interactions. Here we focus on the nature of the electron pair bond and how this bond can be influenced by Pauli repulsion effects due to other fragment orbitals such as lone pairs.S4,72

The Electron Pair Bond and Pauli Repulsion

35

The nature of the three-electron (2c-3e) bond as well as the relation between 2c-3e and 2c-le bonding will be the subject of the following section. One class of systems in which the effect of such an interplay of primary electron pair bonding and “secondary” frontier orbital interactions is quite pronounced consists of the CN dimers 4. All three C2N2 isomers have been experimentally characterized. The first one, cyanogen (formally 1,4diazabutadiyne, NCCN, 4a), was prepared as early as 1815 by none other than Gay-Lussac.73 It is stable under ambient conditions and has been studied intensively since.74>75In contrast, its positional linear isomers isocyanogen (CNCN, 4b)54,76-103 and especially diisocyanogen (CNNC, 4 ~ ) are ~ 6rather unstable. It is therefore not surprising that their discovery was reported nearly two centuries later, in 1988 and 1992, respectively. We will show that the differences in stability along 4a-4c would be even larger if the trends in primary 2c-2e bonding between the cyanide SOMOs were not damped, to some extent, by secondary interactions with the CN lone pair orbitals. We will elaborate on this shortly.

4a

4b

4b

5a

5b

5b

An interesting comparison can be made with the corresponding phosphorus compounds, the CP dimers 5, about which much less is known. Only three reports have appeared, and they all concern the mass spectrometric (MS) or photoelectron spectroscopic (PES) detection in the gas phase of 1,4-diphosphabutadiyne (PCCP, 5a),104-106 the phosphorus analog of the stable cyanogen (4a). To our knowledge, no reports have appeared concerning the other two isomers CPCP (5b) and CPPC (5c). One of the interesting aspects of the CP dimers (5) is that the central electron pair bond is in a sense more pure than that in 4: that is, it is much less affected by interferences with other electrons. What is the reason for this? In the following, we examine the nature of the central bond between the CX. radicals in 4 and 5 (X = N, P). To what extent can this bond be considered to be a simple 0 electron pair bond? Does 7c bonding make a significant contribution? What exactly determines the degree to which the 2c-2e bond interferes with other electrons, and how does this affect molecular structure and bond strength? The discussion of these questions is based on our DFT investiga-

36

Kohn-Sham Density Functional Theory

tions, which were carried out at the BP86/TZ2P leve157~107-109unless stated otherwise.54~110For computational details, see Ref. 110.

The Potential Energy Surfaces of CN and CP Dimers CN Dimers Before we turn to bonding theoretical considerations, let us first get a more precise idea of the potential energy surfaces of 4 and 5. All three CN dimers have stable minimum energy structures of linear symmetry: D,, for NCCN and CNNC, C,, for CNCN (see Figure 10). Interestingly, the central bond d2 ( 6 ) becomes both shorter and weaker as one couples the two CN radicals via C-C (4a),N-C (4b), or N-N (4c):in this order, d2 contracts from 1.373 to 1.305 to 1.274 A, whereas the corresponding bond dissociation enthalpy (BDE = -AHat 298.15 K ) decreases from 136.6 to 113.5 to 68.2 kcal/ mol, respectively. Our valuesl*O for the internal bond lengths d2 of 4a and 4b

dl

x-c-c-x

d2

d3

(X = N, P)

C-X-C-X

c-x-x-c 6

are generally in good agreement with those7*,91~92~9~~9*~102,106,111-113 obtained by other theoretical calculations and by a variety of experimental methods (Figure 10 and Table 2), and the experimental NC-CN bond strength amounts to 134.7 k 4.2 kcal/mol,l14 only 1.9 kcal/mol less than our value. To our knowledge, no experimental geometry or BDE values have been reported for 4c. The contraction of d2 can be attributed to the combined effect of the smaller effective size of nitrogen versus carbon and to nitrogen's higher electronegativity, which causes the weakly C-N antibonding cyanide SOMO to have a lower amplitude on nitrogen. These effects lead to an onset of both repulsive and bonding interactions at shorter bond distance if N instead of C gets involved in the central bond. We come back to this later. There is yet another linear isomer: CCNN (4d).This can be conceived as a codimer of C, and N, monomers, held together by a donor-acceptor bond between the 30, LUMO of C, and the doubly occupied 30, of N, . The central

The Electron Pair Bond and Pauli ReDulsion 1.165 1.373 1.165 N-C-C-N

4a D,h -140.5 (0) [-136.61

1.19 1305 1.167 C-h-C-N

4b

1.587 1.336 P-c-c-P

1.189 1.274 1.189 C-N-N-C

1.587

5a D.h -1 54.7 (0) [-152.21

1.639 c-P-c-P

Cv ,

-1 17.2 (0) [-113.51

37

1.709

5b

1.580

c

C,

5b'

C-v

-68.9 (2) [-68.7]

1.629 c-P-P-c

134.7"

2.216

-69.7 (0) [-68.7]

1.629 113.2"

4C 0-h -71.4 (0) [-68.21

5C

5c'

D.h

-5.8 (4) [-7.51

czv

-78.6 (0) [-77.91

1.704

1.277 1.267 1.140

C-C-N-N

1.282 1.645 C-C-P-

1.929

P

6% .418 yN$ 123.1"

4d

c,

-68.5 (0) [-65.0]

5d

c,

-87.6 (0) [-85.8]

4c' C2" -2.7 (1) [-2.81

Figure 10 Selected C2N2 (4) and C2P2 isomers (5): geometries (A, degrees) and electronic energies (kcaYmol) relative to two CX radicals (X = N, P); number of imaginary frequencies in parentheses, and in square brackets enthalpies at 298.15 K from BP86KZ2P computations.

C-N bond distance in 4d is 1.267 A. A more detailed discussion of this bond has been given by Scheller, Cederbaum, and Tarantelli.115 CCNN is rather high in energy, in fact even higher than the least stable CN dimer 4c (although by only 3.2 kcal/mol). Apparently, the CN dimers, especially NCCN, are thermodynamically stable with respect to dissociation. Another point concerns the inertness or kinetic stability with respect to unimolecular rearrangements. We have investigated the transformations of 4a to 4b, and of 4b to 4c via transition states in which the respective CN groups

38

Kohn-Sham Density Functional Theoy

Table 2 Central Bond Length d,

(A) for Linear CN (4) and CP (5)Dimers (6)

NC-CN

CN-CN

CN-NC

PC-CP

CP-CP

CP-PC

(44

(4b)

(44

(54

(5b)

(54

BP86KZ2Pa

1.373

1.305

1.274

1.709

2.216

B3LYPe

1.381 1.395 1.375

1.318 1.322 1.307

1.279 1.294 1.274

1.336 1.36

Values

Theoretical MNDOb MP2c CEPAd

Experimental HR-IRf HR-Ramang

MWh

ED'

X-ray'

1.389 1.380 1.3881

1.314 1.312k 1.300

aNonlocal DFT calculation with Becke-88-Perdew-86 functional; Bickelhaupt and Bickelhaupt, Ref. 110. bSemiempirica1 calculation with Modified Neglect of Diatomic Overlap; Bock and Bankmann, Ref. 106. <Second-order Mdler-Plesset perturbation theory; Nguyen, Ref. 95. dCoupled Electron Pair Approximation; Botschwina and Sebald, Ref. 98. eHybrid DFT with Becke-3-Lee-Yang-Parrfunctional; Ding et al., Ref. 102. f)ligh-resolution infrared spectroscopy; Maki, Ref. 111. gMorino et al., Ref. 112. hStroh and Winnewisser, Ref. 78. $Electron diffraction. ~ M ~ l land e r Stoicheff, Ref. 113. &Weiset al., Ref, 91. 'Boese et al., Ref. 92.

undergo cyanide/isocyanide rearrangements. The 298.15 K activation enthalpies AH* for these reactions are 57.6 and 80.4 kcal/mol, respectively. The reverse barriers are 34.5 (4b-+4a)and 35.1 kcal/mol(4c+4b). Sunil, Yates, and Jordan9' reported a low energy transition state of D,, symmetry for direct conversion of 4c into 4a; it is only 16.8 kcal/mol above 4c (at the MP4//MP2 level, i.e., fourth-order Maller-Plesset calculations were done on a strbcture optimized with second order MP perturbation theory; for an overview of ab initio methods; see, e.g., example, Ref. 116).We have found a similar structure, 4c', which is 23.2 kcal/mol above 4c (Figure 10).A vibrational analysis reveals, however, that this is actually a transition state for the automerization of 4a and not for the interconversion 4a+4c. Ding et a1.102 located a cyclic transition state for the isomerization of 4c to 4a. But this TS again is rather high in energy, being 39.4 kcaYmol above 4c [at the CCSD(T)//B3LYPlevel; coupled cluster calculations with single, double, and (through perturbation theory) triple excitations were done on a structure optimized with Becke-3-Lee-Yang-Parr hybrid DFT; see, e.g., Refs. 116 and 1171. Thus, caught in potential energy wells of at least 34 kcaYmol, all three CN dimers are kinetically stable toward unimolecular isomerization at room temperature.

The Electron Pair Bond and Pauli Repulsion

39

CP Dimers The potential energy surface of the CP dimers looks quite different from that of the CN dimers. Only the PCCP molecule (5a) resembles its nitrogen analog (4a)to some extent in that it is a stable linear species with CP units that are bound by AH = -152.2 kcal/mol via a carbon-carbon bond of 1.336 A (Figure 10). Our BP86RZ2P bond strength fits nicely in between the experimental values of 148.9 f 7.4104 and 157.4 k 9.2 kcaYmollo5 obtained through mass spectrometric equilibrium measurements. The PC-CP bond is even somewhat stronger (16 kcal/mol) and shorter (0.04 A) than the NC-CN bond. This is indicative of (partial) multiple C-C bond character. Thermodynamically, PCCP should therefore be just as stable a molecule as the wellknown cyanogen (4a).However, in the design of strategies toward its synthesis, one must take into account that, probably, 5a has a low kinetic stability. This is suggested by the rather small HOMO-LUMO gap of only 2.5 eV in the n: system; for comparison, the HOMO-LUMO gap in NCCN is 5.6 eV.110 The differences between 4 and 5 become larger when the CP radicals are connected via phosphorus. The weakening of the central bond d2, for example, is much more pronounced along the series 5a-5c than for the nitrogen analogs 4a-4c. The bond dissociation enthalpy decreases from 152.2 to 68.7 to 7.5 kcaYmol for PC-CP, CP-CP, and CP-PC, respectively (compare the structures in Figures 10).The bond d, elongates from 5a to 5b to 5c (from 1.336 to 1.709 to 2.216 A, respectively) instead of decreasing as in 4a - 4c, of course, simply because the phosphorus atom is larger than carbon or nitrogen. More importantly, the linear CPCP (5b) and CPPC (5c) are no longer minimum energy structures. Instead, structure 5b represents a second-order saddle point connecting two bent, C, symmetric species 5b' (Figure 10). The latter is the actual equilibrium structure of CP-CP with a C-P-C angle of 134.7'. Note that the preference for the nonlinear structure is marginal: on the zero Kelvin potential energy surface, 5b' is only 0.8 kcal/mol more stable than 5b, and the 298 K bond dissociation enthalpies for both are equal (-AH = 68.7 kcaymol). It is therefore conceivable that another quantum chemical method may yield the linear 5b as the equilibrium structure. This would, however, not affect our concept of the essential physics: CPCP (either 5b or Sb') is much less prone to adapt a linear geometry than PCCP. Apparently, the CPCP structure is highly flexible. In addition, its barrier for isomerization to PCCP (5a) via a cyclic intermediate (5ab) is extremely low-effectively zero (see Eq. [26]; the drawings schematically indicate geometries, not valence structures). P-C-P,

5b'

C

-

, / y FJ-c-c 5ab

--+

p-c-c-P

[261

5a

The tendency to distort, away from linearity, increases even further when both CP radicals bind via phosphorus as in CPPC (5c),which is a fourth-order

40

Kohn-Sham Density Functional Theoy

saddle point. [A fourth-order saddle point has four normal modes, each of which is associated with imaginary frequencies (i.e., the force constants are negative). A structure tends to deform along such a normal mode toward a geometry of lower energy. For comparison, a “normal” transition state is a “first-order” saddle point on the potential energy surface.] Our purpose here is not to give a full account of the potential energy surface, but we want to mention the most stable minimum energy structure involving P-P bonding (see Ref. 110 for other C2P2 species): the C,, symmetric 5c’ at -77.9 kcal/mol (AH)relative to two CP. The P-P bond in 5c’ is 2.362 8, (0.146 A longer than in the linear 5c), and the P, unit is symmetrically bridged by each of the two carbon atoms with C-P bond distances of 1.802 A and a dihedral C-P-P-C angle of 113.2’ (Figure 10). There is no C-C bond (dcc = 2.272 A). Finally, not a CP dimer but still an interesting C2P2 isomer, is the linear CCPP (Sd), which can be viewed-analogously to its nitrogen counterpart CCNN (4d)-as a donor-acceptor bound codimer of C2 and P, with a C-P bond length of 1.645 A.It is the second most stable linear C,P, isomer, at 66.4 kcal/mol above PCCP (5a).

Bonding in CN and CP Dimers: Qualitative Considerations Electronic Structure of the Monomers Figure 11 shows the valence levels of the CN and CP fragments, together with a schematic representation of the corresponding orbitals. At the lower end (i.e., 30 for CN and 5 0 for CP), of the orbital spectrum, we have the tsHOMO-l which is given by the bonding 2s(C) + 2s(N)or 2s(C) + 3s(P)combination. This low energy orbital as well as the high energy unoccupied 60(CN) and 80(CP) are of minor importance for the central bond in the CN and CP dimers 4a-4c and 5a-5c. What matters, instead, are the frontier orbitals in the middle of the orbital spectrum. They determine the bonding capabilities of the CN. and CP. radicals: in 0 symmetry, they are the oHOMO and the oSOMO (i.e., 40 and 5 0 for CN; 6 0 and 70 for CP) and, in n symmetry, the zHOMO and the nLUMO (i.e., In and 2.n for CN; 2n and 3n for CP). Both oHOMO and cSOMO are essentially nonbonding orbitals. The former provides the axial N or P lone pair, whereas the latter carries the unpaired electron. The doubly degenerate nHOMOconstitutes the two n bonds in CN and CP; its antibonding counterpart is the unoccupied TcLUMO. Orbital Interaction How do these fragment orbitals interact in the dimers? We recall from our earlier discussion that in 0 symmetry, the SOMOs on the two monomers + provide the two-center electron-pair (2c-2e) bond by forming the (oSOMO OsOMO ) 2 configuration (7).An important feature inherent to the SOMO is its rather low energy compared to completely unoccupied orbitals such as the

The Electron Pair Bond and Pauli Repulsion

41

Figure 11 Valence orbitals of CN and CP radicals.

LUMO. Thus, not being separated from the occupied orbitals by a large gap such as is typically the case for a LUMO, the SOMO is predestined to enter into a subtle interplay of stabilizing and destabilizing interactions with the closedof the other fragment: the SOMO may act either as an unoccupied shell cHOMO which leads to stabilization, or it orbital accepting charge from the oHOMO, may act as an occupied orbital whose electron experiences Pauli repulsion with the same-spin electron in the oHOMO. These are the “secondary” interactions interfering with the plain 2c-2e bond that have been discussed at the end of the preceding section and in the introduction to this one. Another important Pauli

42

Kohn-Sham Density Functional Theorv

repulsive component in o symmetry stems from the destabilizing four-electron interaction. two-orbital o H O M O k oHOMO RHOMO-LUMO

. . . ,

7

8

In x symmetry, there is a stabilizing donor-acceptor interaction involving four causing two "parelectrons between the doubly degenerate xHOMO and xLUMO tial x bonds" (8).They are opposed by the Pauli repulsive nHOMOk nHOMO two-center four-electron (2c-4e) interaction. Note the difference in nature between o and the x bonds: the former is an electron pair bond between singly occupied orbitals, whereas the latter evolves from a donor-acceptor or charge transfer interaction between occupied and unoccupied orbitals. Before we discuss the actual energetic effects of the various orbital interactions, let us take a more detailed look at the shape of the CN and CP frontier orbitals (see the contour plots in Figure 12). An important feature is their delocalized nature. In particular the SOMO, carrying the unpaired electron, has significant amplitude at both ends of the diatomic. Thus, CN and CP are clearly ambident radicals. In terms of simple valence bond structures, they are best represented as resonances 9 and 10.

9

10

Owing to their ambident character, CX. radicals may, in principle, form electron pair bonds either via carbon or via the heteroatom, leading to 4a-4c and 5a-Sc. The SOMO is however not evenly distributed, having a more extended, higher amplitude lobe at the carbon side. The strength of the electron pair bond 7 and the stability of the dimer should therefore decrease in the order XC-CX > cx-cx > cx-xc.

Bonding in CN and CP Dimers: Quantitative Analysis Trends in Bonding: Linear CN Versus CP Dimers The qualitative considerations above nicely match the trends in bond strength computed at the BP86A'Z2P level of DFT: along both series 4a-4c and

The Electron Pair Bond and Pauli Repulsion

...........

43

. . . I

40

60

(-12.3 eV)

(-1 1.6 eV) . . ........ ...

,

I

50

";

70

(-8.1 eV)

(-9.9 eV)

.................

....................... ._.. ...... -. , . , ............ .,.. . . . . . . ... . ,. . ;

17c (-9.7 eV)

. I

I I

..

.................. , .

2n; (-7.5 eV)

. . _._._ ...................... . ..

,

,I

'._................ ./..

__ ......... ....

27c (-1.7 eV)

I

'

37c (-2.9 eV)

Figure 12 Contour plots of CN and CP frontier orbitals (orbital energies in eV, contour values: 0.0, k0.02, k0.05,&0.1, k0.2, t0.5 bohr-3'2). Asterisks indicate the positions of the nuclei. 5a-5c the stability of the central bond decreases (Figure 10). However, a more quantitative analysis shows that these similar trends have quite different origins (see Table 3 and Figure 13).The first step of forming the dimer is the adaptation of the monomer's geometry to the situation in the composite molecule; this corresponds to a slight reduction or increase of the C-X bond length (X = N, P). The associated preparation energy AE,,,, is rather small in all cases and has no influence on the overall trend (Table 3). The actual interaction energy between the monomers, including the preparation energy, decreases in both the

OSOMO)

I HOMO)

I OSOMO)

0.46 0.26 0.35 0.09

-154.7

Overlap Integral Between Fragment Orbitals (CX I CX)b 0.21 0.43 0.31 0.37 0.22 0.31 0.41/0.24~ 0.28 0.32 0.14 0.13 0.12

-71.4

"Carried out at the BP86rTZ2PIIBP86rTZ2P Level. bcrHOMO, crSOMO, rrHOMO are 40,50, l n for CN, and 80,95, 371 for CP (Figures 12 and 13). cIn case of CX-CX (X = P, N): (+ left CX I Cp right CX) I ($ right CX I Cp left CX).

HOMO 1 XHOMO)

(OHOM0

(OHOMO

(aSOMO

-117.2

-140.5

AE

-75.2 3.8

0.27 0.27 0.41/0.29~ 0.10

-68.9

-75.4 6.5

-120.8 3.6

-143.9 3.4

AEint

AE,,,,

-159.4 4.7

-225.0 149.6

-320.9 161.5

-446.5 371.3

-386.3 265.5

-288.2 144.3

AEoi AEO

AEx

-168.3 -56.7

(5b)

CP-CP

PC-CP (54

Bond Energy Decomposition (kcaVmol) -360.7 -243.3 -306.9 -85.8 -77.6 -79.4

CN-NC (44

-230.0 -58.2

CN-CN (4b)

AEcr

NC-CN (44

Table 3 Analysis of the Central Bond in Linear CN and CP Dimersa

0.16 0.29 0.22 0.07

-5.8

-13.1 7.3

-104.1 91.0

-71.5 -32.6

CP-PC (54

The Electron Pair Bond and Pauli Repulsion

5a

45

5c :I'

#

-19.1

CNA CNE,

%A+%,

NCCN

f

IhAdUB]

-20.7

CNA CNB

CNNC

Figure 13 Schematic CJ orbital interaction diagrams for PC-CP (Sa), CP-PC (Sc), NC-CN (4a), and CN-NC (4c). MO and FMO energies in electron volts. Left panels: FMOs; central panels: primary or "first-order'' interaction ( i s . , no ~ S O M O / ~ H O Mmixing); O right panels: final situation including all interactions.

C N dimers (from -140.5 kcal/mol t o -71.4 kcal/mol) and in the linear CP dimers (from -154.7 kcal/mol down t o only -5.8 kcaYmol) as we go from C-C to C-X to X-X coupling. In the CN dimers, the reduction in bond strength is caused by an enormous increase in AEO repulsion (from 144.3 kcal/mol to 371.3 kcal/mol) which is counteracted but not compensated for by a sizable increase in AE,; (from -288.2 kcal/mol to -446.5 kcal/mol). In contrast, in the CP dirners the calculated bond strength decreases because of a weakening of the bonding orbital interactions ALEoi (from -320.9 kcal/mol to -104.1 kcallmol), especially those associated with the c bond (AE,) in spite of an opposite trend of the repulsion AEO, which actually decreases (from 161.5 kcal/mol to 91.0 kcal/ rnol). The increase in Pauli repulsion along the C N dimers 4a-4c is caused by the increase in overlap between the closed-shell oHOMO orbitals and between the nHOMOorbitals (Table 3), which have higher amplitudes on nitrogen (Figure 12). In NCCN (4a), the repulsive overlap between the C N cHOMO orbitals,

46

Kohn-Sham Density Functional Theoty

for example, is relatively small (0.26),leading to a correspondingly weak interaction. This is illustrated by the relatively small separation between the primary or “first-order” bonding CTHOMO+ CYHOMOand antibonding (JHOMO - oHOMO combination belonging to the fictitious situation in which only oHOMO/OHOMO and o ~ ~ M ~ / ( J S O Minteraction O has occurred, but not yet mixing between oSOMO and cHOMO (see orbital interaction diagram for 4a in Figure 13, central panel). This can be compared to the schematic situation with lighter lines in Figure 9, but note that we have indicated the relatively strong interaction beorbitals (overlap 0.46) by a strong stabilization/ tween the oSOMO orbitals destabilization. In CNNC (4c), the overlap between the oSOMO decreases, but that of the CTHOMOorbitals rises to 0.37 and, likewise, the gap between “first-order” bonding and antibonding combinations decreases for the oSOMO orbitals and increases for the oHOMO orbitals (4c in Figure 1 3 ) . Interference of the Primary Electron Pair Bond with Other Orbitals What about the increase in stabilizing orbital interactions along the series 4a-4c? This does not correlate with the bond overlap (osoMoI osoM0)which, as qualitatively predicted, decreases from 0.46 to 0.21. Here, the “secondary” o ~ ~ M ~ / ~ H O interactions M O that interfere with the primary electron pair bond come into play. Indeed, to first order, the ~~~~~/~~~~~ interaction decreases along 4a-4c, as indicated by the reduced splitting between the occupied oSOMO + oSOMO and the unoccupied oSOMO - oSOMO in the first-order panels for 4a and 4c (see Figure 13). Thus, to first order, the oSOMO + oSOMO descends strongly and comes out close to (in fact, in 4a even below) the oHOMO + oHOMO. This causes strong mutual repulsion that pushes the o H O M O + oHOMO upward; this is shown in the right panel for 4a in Figure 13. As a result, the electron pair bonding component of AE, [the energy lowering that is associated electron pair configuration but with the formation of the (osoM0+ oSOMO)2 with (oHOMo + also contains the repulsive interaction of (osoM0+ oSOMO)2 oHOMO )2, as discussed earlier] and therefore also AEoi become less bonding in 4a. (For a detailed account of the quantitative decomposition of the AE, term, see Bickelhaupt et a1.54) In contrast, in 4c it is the occupied o H O M O - oHOMO orbital that approaches in first order the unoccupied o~~~~ - ‘JSOMOfrom below (see 4c in Figure 13, central panel). The resulting donor-acceptor interaction is about twice54 as strong as that in 4a and causes AE, and AE,; in 4c to be significantly more bonding (Table 3). This is the possible role of the oSOMO to act as an electron acceptor orbital mentioned earlier. At the same time, the + oSOMO)2 with ( ( 3 ~ 0+~o 0 H O M 0 ) ’ is much repulsive interaction of (oSOMO smaller because of the larger energy gap. The overall effect of the combined “secondary” interactions is the increase of BE, and AEoi along the series 4a-4c. Note that without these effects, the difference in bond strength between NCCN and CNNC would have been much larger. The next point concerns the question of why the repulsive and bonding

The Electron Pair Bond and Pauli Repulsion

47

orbital interactions in the three linear CP dimers follow the opposite trend with respect to those in the CN dimers. That is, why do both AEO and BE,, decrease along the series 5a, Sb, and 5c? In the first place, as we mentioned, the CP orbitals are more extended and diffuse at the P side. This causes overlaps to be smaller, and AEoi and AEO to be of smaller magnitude, as soon as phosphorus becomes involved in the central bond (see Table 3). Thus, the Pauli repulsion contained in AEO and, even more so, the bonding orbital interactions AEOi decrease along the series 5a-5c instead of increasing as they do along 4a-4c. The other important point is that the primary or “first-order’’ electron pair bond is much less affected by secondary interactions with other orbitals: the decrease in oSOMO + oSOMO electron pair bonding, if we go from 5a to Sc, is no longer compensated by a strongly stabilizing oHOMO/OS~MOinteraction. This - crHOMO orbital not coming close enough in energy to arises from the oHOMO - crSOMO (compare 5a and 5c in Figure 13) because of the the empty oSOMO smaller CTHOMOk CTHOMOsplitting in the CP dimer and because of the larger o ~ ~ ~ gap in~the CP / monomer ( s (3.5 ~ eV~in CP ~ vs. ~ 2.4 eV in CN; see Figures 11 and 13). The carbon-carbon bonds in 5a and 4a, the most stable CP and CN dimer, respectively, are of comparable strength (AE = -154.7 and -140.5 kcal/mol, respectively) and the differences in bonding mechanisms are subtle (Table 3). As mentioned above, the oS~MO/(ZHOMOrepulsion is less pronounced in PCCP. This leads to a somewhat stronger orbital (AE,. ) and overall interaction ( A E ) ,a slight reduction of the bond length d, (1.336 in 5a vs. 1.373 A in 4a), and a k nnOMO repulsion contained in AEO (Table 3). The small increase of zHOMO oSOMO + c S O M O of 5a does not drop below the oHOMO + G ~ but becomes ~ ~ the highest occupied cr orbital at -9.1 eV, unlike the situation in NCCN, where it ends up at -19.1 eV (compare 5a and 4a in Figure 13).

T h e Role of n Bonding There is also an important n-bonding contribution BE, to the central bond of the CX dimers. Although much smaller than AE-, as can be seen in Table 3, AE, is substantial in the sense that it is on the order of half to even somewhat more than the overall bond energy AE. In terms of simple VB structures, the nature of the CX dimers is therefore best represented by resonances 11:

00

Ic=x-c=xI

0000

I C=X-X=CI

- /00

,c-XSC-x,

\

0, - ,000 ,c-xsx-c, 11

O

48

Kohn-Sham Density Functional Theory

Linear versus Nonlinear Geometry In the preceding discussion, we have compared linear CP and CN dimers. However, as mentioned in the beginning of this section, 5b and 5c are actually higher order saddle points. In contrast to their nitrogen analogs, they tend to adapt nonlinear structures, which leads to Sb' and 5c' (see Figure 10).This is so for reasons best described as subtle. One of the factors that plays a role again is + the secondary four-electron interaction between the "first-order " oSOMO oSOMO and oHOMO + oHOMO contained in the AEoi term (see Ref. 110 for the numerical results of a detailed analysis). In principle, this interaction always favors a bent, nonlinear structure because bending (e.g., as shown in Eq. [27]) reduces the repulsive overlap between the lobes of the oSOMO and cHOMO that are pointing toward each other in the linear species. "271

The question of bending or not bending is determined by the trend in Pauli C and ~ the ~ repulsion, which correlates with the overlap between the T oSOMO (12).By symmetry, the overlap (S) between these orbitals is zero in the moves out of the nodal linear species. But on bending, the lobe of the oSOMO plane of the T C ~ and ~ overlap ~ ~ begins , ~ to, build up (12, right).

The increase in overlap and the resistance against bending is stronger for C-C and weaker for X-X coupling because the oSOMO has a larger lobe on carbon than on the heteroatom for both CN and CP (Figure 12). Resistance against and distortion is weakest in the case of P-P coupling because both the oSOMO ~cHOMOof CP have a lower weight on the heteroatom than the corresponding frontier orbitals of the more polar CN. Thus, CPPC has the strongest bias toward nonlinearity and eventually adopts structure 5c'.

Summary In this section of the chapter, we have shown how to analyze the 2c-2e or electron pair bond both quantitatively and qualitatively within the theoretical

~

~

The Three-Electron Bond and One-Electron Bonding

49

framework of the Kohn-Sham MO model. In this context, “quantitative” refers not merely to an accurate computation of, say, the bond strength or distance. The more important point is that also the qualitative picture benefits from an accurate description of the sometimes subtle features in the bonding mechanism. This is important for a true understanding. A point in case is our example of the CN dimers in which the primary electron pair bonding between the CN SOMOs is delicately influenced by “secondary” Pauli repulsive interactions with closed shells that are close in energy. We have shown how this reduces the difference in bond strength between the most stable (NCCN) and the least stable (CNNC) isomers. In the isoelectronic linear CP dimers, the effect of this interference is much less severe because in CP the energy gap between the SOMO and the HOMO in CJ symmetry is larger than in CN. The electron pair bond in the linear CP dimers is, so to say, more “pure” and more in conformity to the common conception based on standard textbook examples such as the H, molecule. But even here, the interference of closed shells with the electron pair bond plays a role in the tendency of CPCP and CPPC to adopt nonlinear equilibrium structures.

THE THREE-ELECTRON BOND AND ONE-ELECTRON BONDING Introduction In the preceding section, we discussed the electron pair (2c-2e) bond and how it can be influenced by Pauli repulsion of the SOMOs with other electrons. In the three-electron (2c-3e) bond, Pauli repulsion plays an even more fundamental role, as we will see.72 The idea of the three-electron bond was introduced in the early 1930s by Pauling in the context of the valence bond (VB) model of the chemical bond.70.71 Since then, it has been further developed both in VB and in MO theory and has become a standard concept in chemistry.118-129 In VB theory,70>71>118-123the two-center, three-electron (2c-3e) bond between two fragments A and B is viewed as arising from a stabilizing resonance between two valence bond structures in which an electron pair is on fragment A and an unpaired electron on B (13a),or the other way around (13b):

A:

-B 13a

t )

A*

:B

13b

A sizable resonance stabilization is achieved only if the energies associated with

configurations 13a and 13b are “similar.” As mentioned before, the picture in

50

Kohn-Sham Density Functional Theoy

MO theorylls-125 is that of a closed-shell orbital, typically a lone pair, of one fragment A interacting with the singly occupied molecular orbital (SOMO) of the other fragment B, resulting in a doubly occupied bonding MO ((3) and a singly occupied antibonding MO ( 0 " )of the composite molecule A:.B (14):

+

: cJ* ',,

0'

A:

A:.B

*B

14

It follows from 14 that the 2c-3e bond may be viewed as being composed of an electron pair bond (cr)2 counteracted by a destabilizing component due to the , formally to a bond order of 1/2 or less. antibonding electron [i.e., ( c J " ) ~ ] leading To arrive at a stable 2c-3e bond, the interacting fragment molecular orbitals (FMOs)must be close in energy-similar to the requirement for 13a and 13b in the VB model-and they should have sufficient overlap S because both these factors stabilize the electron pair bonding configuration (012. Note, however, that the antibonding 0'' MO is generally more destabilized than the bonding cr MO is stabilized, and this excess destabilization aggravates with increasing overlap. Thus, when S exceeds a certain critical value, the net 2c-3e interaction becomes nonbonding or even repulsive. In the simple Huckel model with overlap, for example, the net 2c-3e interaction associated with two initially degenerate FMOs on fragments A and B, respectively, is optimal for S = 0.17 and becomes antibonding for S > 0.33.120,123

The Fra ment Approach to the Three-E ectron Bond

f

The preceding MO-theoretical analysis of the 2c-3e bond is done from the point of view of the composite molecule A :. B and the properties of its MOs. Here, we follow a different-although equivalent-approach, in which we choose to analyze the 2c-3e bond from the perspective of the interacting fragments. This allows us to describe the bonding mechanism not only qualitatively but also quantitatively, just as we have already done for the electron pair bond. As an example, we take the CZhsymmetric sulfur-sulfur bound dimer radical cation of hydrogen sulfide, H2S:.SH; (15), which was found by Gill and Radom120 to be the only minimum on the MP2/6-31G" potential energy surface of (H,S)$'.

The Three-Electron Bond and One-Electron Bonding

51

1

15 The simplicity of this particular model system has the advantage of allowing us to concentrate on the main features of its electronic structure and three-electron bonding mechanism. But 15 is certainly also of experimental relevance. It is the archetype of the dialkyl sulfide dimer radical cations, R2S :. SR;, which have been intensively studied in the gas phase with mass spectrometric techniques.126,127As to the nature of the sulfur-sulfur bond in H2S:. SHS, we must also clarify to what extent this bond is really provided by orbital interactions. After all, we are dealing with a cationic model system and, therefore, one might expect electrostatic forces to be quite important. Of more general interest, however, is the attempt to identify, both qualitatively and quantitatively, the repulsive component in the three-electron orbital interactions that causes the 2c-3e bond order to be less than 1/2. Table 4 shows the results of our sulfur-sulfur bond energy decomposition into the classical electrostatic interactions (AV,,,,,,) between the unperturbed charge distributions of the H,S+* and H2S fragments, the Pauli repulsive (AEpauli)and the bonding (AEOi)orbital interactions (see the above section on MO-Theoretical Analysis of Chemical Bonding). There is a discrepancy of some 10 kcaUmo1 between our best ab initio value [-31.9 kcaYmol at CCSD(T)]and the nonlocal DFT value (-42.8 kcal/mol at BP86) for the overall bond energy AE. This relatively large error for the nonlocal functionals was attributed recently130 to a well-known deficiency of the existing exchange functionals to properly cancel the self-interaction part of the Coulomb energy in the case of delocalized ionization out of symmetry-equivalent weakly (or non)overlapping orbitals.131 Although this particular deficiency of the functionals should be kept in mind, it does not impede qualitative analysis. To assess possible effects of varying the computational level on our physical model of the 2c-3e bond, this analysis has been done at various levels of DFT, local (Xa-VWN57.109) as well as nonlocal (BP86107J08 and PW9119,132).The Xa-VWN value of the overall bond energy (AE = -50.8 kcal/mol) is 8 kcal/mol more bonding than that of BP86 (AE = -42.8 kcal/mol), in line with the general tendency of the local density approximation (LDA) to overbind.l.2 But, although numerically different, the relative proportions of the different physical terms (AVelstat, AEPauli,AEo, ) in the S-S interaction are very similar for all three levels of DFT, yielding the same physical picture. The subsequent discussion is based on the BP86/TZ2P computations, which give the least overbinding.

52

Kohn-Sham Density Functional Theoy

Table 4 Analysis of the Sulfur-Sulfur

Three-Electron Bond in H,S BP86c

XCt-VWNb

Optimum d,,

Bond Distance (A) 2.778

2.886

Bond Energy Decomposition (kcal/mol) -23.8 -18.4 30.2 25.4 -58.1 -51.3 -51.7 0.9

-44.3 1.5

-50.8

-42.8

Fragment Orbital Overlaps (H,S 0.20 0.04 0.04

I H,S+*) 0.18 0.04 0.03

:. SHga PW91d 2.859 -19.4

26.0

-52.5 -45.9 1.5 -44.4 0.18 0.04 0.03

"Carried out using the TZ2P basis set, see Bickelhaupt et al., Ref. 72. hLocal density approximation (LDA) with Slater's Xu functional for exchange (Ref. 57) and the functional of Vosko, Wilk, and Nusair (Ref. 109) for correlation. CNonlocal DFT with the Becke-88 functional for exchange (Ref. 107) and the Perdew-86 functional for correlation (Ref. 108). Wonlocal DFT with the Perdew-Wang-91 exchange-correlation functional (Refs. 19 and 1321.

The sulfur-sulfur bond in H2S:.SH; is mainly provided by the threeelectron orbital interactions between the 1b, orbitals of the two fragments, that is, the 3px lone pair of H,S and the 3px SOMO of H2S" (see Figure 14). Combining the repulsive and bonding orbital interactions together into one term, we obtain

which gives a three-electron interaction AE,,.,, of -25.9 kcal/mol. This is about 60% of the net interaction AEint (Table 4).However, the electrostatic interaction AVelstat, although clearly smaller than AE2c-3e, is an important component too; at -18.4 kcal/mol it still contributes somewhat more than 40% to the net sulfur-sulfur interaction. The energy AE,,,, , required to bring the fragments from their equilibrium geometry to the geometry they acquire in the composite system, does not play an important role; it is very small, about 1 kcal/mol, because the two H,S moieties in H,S :. SHt are hardly deformed with respect to free H2S and H,S+.. Next, we examine the three-electron orbital interactions in more detail. The two 3px orbitals participating in the three-electron bond are pointing toward each other, leading to a considerable overlap integral of 0.1 8 at a relatively

The Three-Electron Bond and One-Electron Bonding

S3

Figure 14 Fragment orbital interactions in H,S :. SH; (MO energies in eV).

large equilibrium bond distance of 2.886 A (Table 4). This shows up in a large splitting of 2.4 eV between the bonding and the antibonding combinations (Figure 14). For comparison, the overlap and interaction between other H2S and H,S+- fragment MOs is much smaller: the 2a, lone pairs, for example, have a small mutual overlap of 0.03 and a splitting between bonding and antibonding combinations of only 0.2 eV. Note the close agreement, which is probably fortuitous, between our (0.18) and the optimal Hiickel 2c-3e bond overlap (0.17). In Figure 15, we illustrate how the three-electron bond (see 16 in Figure 15) can be interpreted in the fragment approach. The repulsive term AEPauli contained in AE2,-,, is mainly due to the destabilizing interaction 17 between the unpaired 3p,a electron on H2S+. and the same-spin 3p,a electron of the lone pair on H2S (the excess spin is arbitrarily chosen a; see Figure 15). Note that this two-electron, two-spin orbital repulsion for ci spin is analogous to the well-known four-electron, two-orbital destabilizing interaction between closed shells. The latter type of Pauli repulsive orbital interactions also occurs in H2S:. SH;. In principle, it may therefore contribute to AEPauli,but it is less important because of relatively small overlaps (e.g., between the H2S and H2S+2a, lone pairs). The bonding orbital interaction AEoi is simply provided by a

54

Kohn-Sham Density Functional Theoy

:+..

3Px

'-1..

*',

+:

3px a '*.,s

"'.st "

I

:+a

''-+" ,/" 3p,

--

AEpauli

-:

3p,

.

.+-

p"...,

..*+

,/'

3px p

"

3Px P + 3Px P

3Px a + 3Px a

3Px + 3Px

- 3Px P

. -<

. ,%+

,:3px+

AE2c-3e

3Px P

3Px a - 3Px a

3Px

+: 3px

18

17

16

+

E2c-1e

Figure 15 Qualitative and quantitative decomposition of the three-electron bond (16) into a Pauli repulsive component (17) plus a one-electron bond (18) between the interacting fragments. one-electron bond 18 between the 3p,p electron of the lone pair on H2Sand the empty 3p,p orbital on H2S+*,i.e., AE,, = AE2,-,, (see Figure 15). In this way, the three-electron bond is naturally linked to the one-electron bond plus a repulsive term. And we can easily quantify the different terms of this qualitative picture (Table 4). The one-electron bonding component in H2S:. SH; (AE2,.le = AE,, = -51.3 kcal/mol) is, in absolute terms, about twice as strong as the repulsive term (AEPauli= 25.4 kcal/mol) and the resulting three-electron bond (AE2c.3e= -25.9 kcal/mol).

Summary In this section, we have evaluated and probed the very useful concept of the 2c-3e bond within modern Kohn-Sham M O theory. This enables us to separate and quantitatively describe the different physical terms such as the classical electrostatic attraction and the three-electron orbital interactions. The former is not unimportant in our cationic H2S:.SH$ model, but the latter interactions still dominate with a contribution of about 60% to the overall bond strength. More importantly, our quantitative fragment approach sheds new light on the nature of the 2c-3e bond itself. It shows that this bond can be thought of as consisting of two physically different components that can also be quantified: (1)a Pauli repulsive term, arising from the unpaired electron on H,S+* interacting with a same-spin electron of the lone pair on H,S, and (2) a one-electron bonding term. This analysis creates a more quantitative link between one- and three-electron bonding. Further, the analysis shows that Pauli repulsion is inherently connected with 2c-3e bonding, contrasting the situation

The Rote of Steric Repulsion in Bonding Models

55

of the 2c-2e bond in which secondary Pauli repulsion effects may be very important (e.g., CN dimers) but not necessarily (e.g., H, or CP dimers). And it makes clear that 2c-3e bonds are in principle always weaker than the corresponding 2c-1 e bonds.

THE ROLE OF STERIC REPULSION IN BONDING MODELS Introduction So far, we have looked at different modes of bonding and how Pauli repulsive orbital interactions may either influence them (2c-2e bond) or be an essential part (2c-3e bond). In this section, we examine a different role of Pauli repulsion, namely, the one it plays in the absence of bonding interactions between groups. Here, it is responsible for the fact that such groups, A and B say, repel each other. Or, to put it in another way, steric repulsion between A and B is a pure quantum effect, caused by the Pauli repulsion between same-spin electrons of the different fragments, such as the well-known two-center, fourelectron (2c-4e) repulsion (19) or the two-center, two-same-spin-electron (2c-2sse) repulsive component (20) of the three-electron bond. 2c-4e :

A

' I

',

A---B 19

2c-2sse

B

A

A---B

B

20

As pointed out earlier, the physical basis of this repulsion is the increase in kinetic energy of the electrons due to the Pauli exclusion principle, which is most easily seen from the large gradients induced in the wavefunctions by the orthogonality requirement. A correct description of Pauli repulsive interactions between valence, subvalence, and core electrons as well as of electrostatic interactions is an essential requirement for accurate quantum chemical predictions. We now show that a repulsion in nonbonded proper analysis of steric repulsion-Pauli

56

Kohn-Sham Density Functional Theorv

interactions-is also essential for arriving at a correct physical model of a system, i.e., for understanding its chemistry and structure.133 We demonstrate this for the AH; radicals, where A is one of the group 1 4 atoms C, Si, Ge, and Sn (21). AH; radicals and the corresponding cations play an important role in many areas of chemistry. They occur as reactive intermediatesl34-137 and are (for A = Si, Ge) involved in processes [e.g., chemical vapor deposition (CVD)] which are important for the production of high technology electronic devices (see, e.g., Refs. 138-145). Furthermore, they appear naturally-as building blocks-in theoretical analyses of AH3X molecules and the corresponding AX bond (see citations 2-6 in Ref. 133). The A-X bond, in turn, is involved in many standard organic and organometallic reactions.134-137

AH3-planar

AH3-pyramidal C3"

D3h

21

A true understanding the nature of the AH', radicals and their structural trends due to varying A is thus interesting both from a practical and a bonding theoretical point of view. The structural trends we are referring to are the striking and systematic increase in the degree of pyramidalization and the height of the inversion barrier when the central atom A is running down in group 14, starting with the flat D3h symmetric methyl radica1.26~28.133~'46-169 Similar trends are known for the closed-shell group 15 (AH,) and group 16 hydrides (AH, )2633,164-177 as well as for the allylic CH,=CH-AH; anions, where A is a group 14 atom.178 The trend in pyramidalization is generally explained in qualitative M O theoretical terms through the operation of a second-order Jahn-Teller effect (see, e.g., Chapters 7 and 9 of Ref. 28) as shown in 22 for AH', : (1)the mixing between the nonbonding np, SOMO and the AH antibonding LUMO stabilizes and pyramidalizes AH!; (2) this effect becomes stronger for the heavier (more electropositive and diffuse) central atoms A, because the SOMO-LUMO gap becomes smaller owing to the higher energy of the np, SOMO and the less A-H antibonding nature of the LUMO; (3)

The Role of Steric Repulsion in Bonding Models

57

the Jahn-Teller effect is opposed by the rising energy of the le, orbitals which is ascribed to the loss of M-H bonding overlap; (4) thus, only CH; remains planar because the Jahn-Teller effect is not strong enough in this case to outweigh the le, destabilization.

22 This qualitative model, based on semiempirical MO theory, focuses entirely on the so-called electronic effects, as the A-H bonding orbital interactions are often called. However, steric repulsion (i.e., the destabilizing orbital interactions) between the hydrogen substituents in AH; is just as important in the interplay of mechanisms that determine whether the molecule adopts a planar or a pyramidal shape. In fact, as will become clear from the following discussion, which is based on a Kohn-Sham DFT study at the BP86/TZ2P 1eve1,107~108steric repulsion turns out to be the decisive factor in determining the pucker of our example.133

Structure and Inversion Barrier in

AH; Radicals

The structural data and inversion barriers of the AH; radicals, collected in Tables 5 and 6, respectively, confirm the increasing degree of pyramidalization as A becomes a heavier group 14 atom: the H-A-H bond angle p (= 120.00", 112.66", 112.44", 110.56")decreases and the inversion barrier corrected for

58

Kohn-Sham Density Functional Theory

Table 5 Geometries of AH; Radicals Computed at BP86/TZ2Pa

d*H

(A)

Planar AH;c CH;-planar SiH;-planar GeH;-planar SnH;-planar

1.088 1.470 1.505 1.733

Pyramidal AH;@ CH;-pyramidalf SiHj-pyramidal GeH;-pyramidal SnH;-pyramidal

1.094f 1.484 1.524 1.755

a

(d%) 90 90 90 90 106.06f 106.06 106.31 108.36

P

(deg)

NIMAGb

120 120 120 120

0 1 (i 610.9)d 1 (i 554.8)d 1 (i 436.5)d

112.66f 112.66 112.44 110.56

-f

0 0 0

aNonlocal DFT calculation with Becke-88-Perdew-86 functional and doubly polarized triplezeta STO basis set; see Ref. 133; see 2 1 for definition of geometrical parameters. bNumber of imaginary frequencies (vibrational analysis carried out at Xa-VWNITZ2P). coptimized in D 3 h symmetry. dImaginary frequency (in cm-1) corresponding to A," symmetric inversion of AH;. eoptimized in C,, symmetry. fd,, optimized in C,, symmetry with fixed a from SiH3-pyramidal C,, optimization.

Table 6 Calculated Inversion Barriers AE,,, of AH; Radicals (kcal/mol) BP86iTZ2Pa

CH; SiH; GeHj SnH;

Remaining Literature

AEinv

AEinv+ AZPEa

UHF

0.0 4.4 4.3 5.8

0.0 3.7 3.8 7.0

0.0 7.6< 7.9 10.2c

Post-HF 4.4d, 5.8e, 4.4f

4.6h, 4.5'

aRef. 133, AZPE from Xa-VWNTTZ2P frequencies. bIR, Ref. 146. 4JHF/3-21G", Ref. 152. dMP4/6-3 1G*//HF/6-3 lG"+AZPE, Ref. 179. eCISDICGF-TZ2P, Ref. 163. fCISDIST0-DZP+TZP//CISD/STO-DZP, Ref. 168. d R (inferred using two assumed forms of potential function), Ref. 147. hCASSCF/MRSDCI, Ref. 161. WMP4SDTQfBAS4//UMP2/BAS2,Ref. 157. rREMPI, Ref. 150.

Experimental 0.Ob

5.3q 5 . 0 ~ 4.41

The Role of Steric Repulsion in Bonding Models

59

zero point vibrational energy effects AEi,,, + AZPE (= 0.0,3.7,3.8,7.0kcal/mol) increases monotonically along CH; , SiH; , GeH; , and SnH; . Note, however, that SiH; and GeH; have essentially the same degree of pyramidalization and that the inversion barrier of GeHj is slightly higher only after correction for AZPE. The equilibrium A-H bond length increases from 1.088 A in CH;planar to 1.755 8, in SnH;-pyramidal. The transition states for inversion (AH;planar) are characterized by one imaginary frequency (= i 610.9, i 554.8 and i 436.5 cm-I), which decreases along SiH;, GeH; ,and SnH; .The planar transition states display a slight A-H contraction of 0.01-0.02 8, with respect to the pyramidal equilibrium structures. Our BP86/TZ2P results133 agree well with most of the available literature data, as can be seen in Table 6.146-16*,179For a detailed comparison with the various experimental and theoretical studies, see Ref. 133.

Interhydrogen Steric Repulsion Versus A-H Electronic Interaction in AH; Radicals To understand the trends in pyramidalization and inversion barrier of the four AH; radicals, we divide the overall interaction energy AE between the central atom A and the three hydrogen atoms into three components as shown in Eq. [29]. A + 3W

+ AH;

AE = AE(A-sp3) + AEjnt(A-H3)

+ AEin,(H3) [29]

The promotion (rehybridization) energy AE(A-sp3) is the amount of energy required to bring the group 14 atom A from its s2p2 ground state to its valence sp3 configuration (Eq. [30]).The H . . . H interaction energy AEi,,(H3) corretriangle in its quartet valence configuration sponds to the formation of the (H.)3 and in the geometry it acquires in the overall molecule (Eq. [31]).The interaction energy AEint(A-H, ) corresponds to the actual energy change when the prepared A-sp3 and (He)3fragments are combined to form the final A-H bonds (Eq. [32]). A + A-sp3

AE(A-sp3)

A - s ~ ’+ (W)3+ AH;

AEi,,(A-H3)

= AEO(A-H3)

+ AE,i(A-H,)

[32]

How are the various energy terms related to the electronic structure and the orbital interactions? First, we consider the formation of the quartet (Ha),

60

Kohn-Sham Density Functional Theory

23

fragment in 23 (see also Chapter 5.2 of Ref. 28 and Chapter 2 of Ref. 26): the three same-spin, singly occupied hydrogen 1s AOs enter into a 3c-3e interaction, which yields a bonding la; and a degenerate pair of antibonding l e i orbitals, each occupied by one p electron. This primarily gives rise to steric repulsion AEO(H,) contained in AEint(H3). Next, we inspect the orbital interactions between A-sp3 and (H.)3.In planar AH;, three (polar) electron pair bonds are formed through the ns + la; and the doubly degenerate npx,y+ l e i orbital interactions. The A-np, A 0 turns, essentially unchanged, into the AH; la; SOMO because it has no overlap with valence orbitals of the hydrogen fragment (Figure 16, left). The combined orbital interactions AEOi(A-H3 ) dominate the net A-H interaction energy AEint(A-H3 ) because the two fragments have opposite spin, and Pauli repulsion can thus only occur through core-valence overlap. In pyramidal AH;, the npz orbital of the A atom does have overlap and mixes in a bonding fashion with (H*),-la; (Figure 16, right). This yields an additional stabilizing contribution to AEint(A-H3). We have analyzed the H . . . H and A-H interactions for three geometries of each AH; radical (Eq. [33]): (1)AH;-planar, the optimized planar structure; (2)AH;-pyramidal", in which dAHis kept fixed to its value in the planar radical, whereas the H-A-H angle p is bent to its value in the optimized pyramidal structure; (3) AHj-pyramidal, the optimized pyramidal structure in which dAH is allowed to elongate to its equilibrium value. Note that for both CH;pyramidal" and CHj-pyramidal the optimum H-A-H angle p of

The Role of Steric Repulsion in Bonding Models

61

Figure 16 Schematic orbital interaction diagram for planar ( D 3 h )and pyramidal (Ch)AH:. SiHj-pyramidal was used, because there is no stationary point corresponding to a pyramidal methyl radical (Table 5). H-A,

/H bending H (dAHfixed)

AH:-planar

dAHelongation A--' H ~

?

AH;-pyramidal'

H H AH;-pyramidal

[331

It appears that the geometry of AH; is primarily determined by the subtle balance between the H . . . H steric repulsion AEO(H3) and the A-H bonding orbital interactions AEoi(A-H3). These two terms set the trends in the net H . . . H and A-H interactions AEint(H3) and AEi,,(A-H3 ), respectively (Eqs. [31] and [ 3 2 ] ) The . promotion energy AE(A-sp3) has no influence at all on the geometry because, for a given central atom A, it leads to a constant (endothermic) contribution. It is therefore not discussed further. In Table 7, we have collected the changes in H . . . H and A-H interactions on bending AH;planar to AH;-pyramidal", ie., AAEi,,(H, ) and AAEint(A-H3 ), respectively, in terms of which we discuss the trend in AH; pyramidalization. Note that for the heavier AH; radicals, the sum of AAEint(H3)and AAEin,(A-H3), that is, the energy released upon pyramidalization (AEpyr,Table 7), is very similar but not equal to the inversion barrier (AEinv,Table 6). This is because AEinvalso

62

Kohn-Sham Density Functional Theory

Table 7 Changes in H . . . H and A-H Interaction

Term

Interaction on Pyramidalizing AH; Radicalsa

CH;b

SiH;

Sum

Change in Bond Energy Terms (kcal/mol) AAEi"t(H3) 4.0 ( 5.4)~ 1.1 AAEint(A-H3) 0.4 (-1.9)' -5.3 AEPP 4.4 -4.2

H...H: A-H, :

(Change in) Fragment Orbital Overlaps (1s I 1s). 0.328 0.16e m p , I 14) 0.21 0.24

H...H

A-H,

GeH;

SnH;

1.o -4.7 -3.7

0.6 -6.3 -5.7

0.19 0.24

0.10e 0.28

aAH; radicals are deformed from the planar D,, optimum to a C,, structure in which dAHis from the D,, and a from the C,, optimization; see Eq. [19] and Table 5 . ba from SiH; C,, optimization. CAAEO(H,),that is, the sum of changes in electrostatic and Pauli repulsive H . . . H interactions. dAA.E,,(C-H, ), that is, the change in C-H, bonding orbital interactions. eA(ls I 1 s ) = 0.02 on pyramidalization in all four cases.

accounts for the relaxation of the A-H bond which in AE,,, has been kept frozen at the equilibrium value of the planar radical. There is a striking difference between CH; and the heavier homologs. In CH; , the H . . . H steric repulsion AEO(H3) and thus AE,,,(H, ) is significantly stronger and increases much more upon pyramidalization. This is seen most clearly from a comparison of AH;-planar and AH;-pyramidal": the H . . . H interaction becomes more repulsive by 4.0 kcab'mol for CH; and by only 1.1 kcal/mol or less for SiH; ,GeH; ,and SnH; (Table 7). This trend is also reflected by the decreasing lak-le; energy gap shown in Figure 17. The reason is a shorter A-H bond length and, as a result of this, shorter H . . . H interatomic distance in CH;, which leads to a larger (H I s I H' Is) overlap. The H . . . H repulsion is slightly relieved and thus partly hidden after the A-H bond is allowed to elongate in AH;-pyramidal (not shown in Table 7).133 The short C-H bonds are related to the compact nature of the carbon 2s and 2p AOs (Figure 18), which causes optimal bond overlaps and AE,, at shorter bond lengths (Table 5).The valence ns and np AOs become significantly more extended and diffuse (i.e., the effective size of A increases) and A-H bonds thus elongate, along the series C, Si, Ge, and Sn (Figure 18).The origin of this phenomenon is the increasing number of core shells with respect to which the valence ns and np orbitals must be orthogonal (Figure 18). This has also been termed intraatomic Pauli repulsion,62.180 because the valence electrons must occupy the more diffuse and higher energy atomic orbitals of higher quantum number instead of going, together with all other subvalence and core electrons, into the lowest I s AO. This arrangement is a direct consequence of the Pauli principle. It is well known18O that for each I the first nl shell (2p, 3d, etc.) is particularly contracted, since there is no orthogonality condition on a deeper shell of the same 1. Accordingly, the distinct step in A-H bond lengths,

The Role of Steric Repulsion in Bonding Models

Si

"

-1 5

Ge

63

Sn

2s

Figure 17 Orbital energies of C, Si, Ge, Sn, and corresponding AH; radical.

in the geometry of the

hence in AAE,,,(H3), is from the 2p of carbon (no p core at all) to the 3p of silicon, the first atom A with a p core. The overall A-H bonding interaction AEin,(A-H3 ) is largest for CH; (not shown in Table 7), but the additional stabilization upon pyramidalization is the weakest. This is again most clearly demonstrated by a comparison of AH;planar and AHj-pyramidal": AAEi,,(A-H3 ) is 0.4, -5.3, -4.7, and -6.3 kcal/ mol along the series C H j ,SiHj ,GeH; ,and SnH; .Note that the change of -1.9 ) of the methyl radical is also weakly kcal/mol in orbital interaction AE,,(C-H, stabilizing. The trend in AAE,,,(A-H3) follows that of the increasing gain in A-H, overlap A(np, I la;) (Table 7) and the associated stabilization caused through np, + la; orbital interaction. The npx,y + l e i orbital interactions, although significantly stronger, have less influence on the trend in pyramidalization because they do not undergo such a drastic change as does the np, + la; interaction. They weakly favor a pyramidal geometry, through a delicate interplay of different factors, in spite of a slight loss of (np, I lei.,) overlap.133 [In this notation, the subscript "1-X"on the orbital of e symmetry (there is only one in this particular case) indicates the x-component, i.e., the component overlapping with the px A 0 of the central atom.]

Summary The preceding section on the bonding in AH; radicals illustrates how a quantitatively accurate description of the various physically different interactions is not only important for making the right prediction (e.g., accurate geom-

64

Kohn-Sham Density Functional Theory

I Carbon 2p

Tin

5p

Tin

5s

_./.__......____ -.__

--

-. ....

I

Carbon 2s

Silicon

3s

Figure 18 Contour plots of the ns and np atomic orbitals of C, Si, and Sn (contour values: 0.0, f0.02, k0.05, kO.10, k0.2, kO.5 bohr-3'2; nodal surfaces dash-dotted). Dots indicate the positions of the nuclei in the corresponding AH; radical. Ge 4s and 4p are not shown; they are only slightly larger than Si 3s and 3p.

etry and inversion barrier of AH;) but especially also for finding the correct qualitative MO model that adequately accounts for the sometimes delicate balance of cooperating and counteracting mechanisms behind the observed trends. Here, we have focused on the role of steric repulsion between nonbonded groups, that is, the hydrogen substituents in the AH; radicals. Our Kohn-Sham MO approach yields the following picture. The CH; radical is planar because of the steric repulsion between the hydrogen ligands. The steric H . . . H repulsion is much weaker for the heavier central atom homologs in which the ligands are farther removed from each other. Electronic effects (i.e., electron pair bonding between central atom and hydrogen ligands) always favor a pyramidal structure (although only slightly so for the methyl radical) through the additional stabilization of the unpaired electron in A-np, (Figure 16).Thus, the diminishing steric repulsion allows for an increasing degree of pyramidalization along the series SiHj ,GeH; , and SnH; . This explanation differs from the

Strongly Polar Electron Pair Bonding

65

classical explanation for the trend in AH; geometry and inversion barrier as sketched in the introduction of this section. The difference is that the main opposing factor to pyramidalization is the increase in repulsive H * . . H (1s I 1s) overlap and not the loss in (npx,,I lei) bonding overlap. Note however that the qualitative Walsh diagram 22 is in principle still valid; only the reason for the rise in energy of the le, MOs of AH; upon pyramidalization has changed.

STRONGLY POLAR ELECTRON PAIR BONDING Introduction In this section, we elaborate on the character of the electron pair bond in strongly polar molecules. A well-known representative that serves here as an example is the carbon-lithium bond in organolithium chemistry. In the recent past, various high level ab initio investigations have yielded a predominantly ionic picture of the C-Li bond (24a).181-183 This picture is mainly based on the results of a number of advanced schemes for the analysis of the electron density distribution, for example, the natural population analysis (NPA) developed by Reed, Weinhold, and others,l84-186 as well as topological methods like Collins and Streitwieser's integrated projected population (IPP),187 or the atoms in molecules (AIM) approach of BaderlssJ89 (see also Ref. 190). These schemes yield strongly positive lithium atomic charges that range from +0.75 through +0.90 electron.191-195 This high charge suggests that organolithium oligomers (RLi), can be conceived as saltlike aggregates of lithium cations and carbanions, bound by electrostatic forces (24a). ti 24a

24b

However, a detailed reexamination within the framework of Kohn-Sham MO theory leads us to a view that differs in a number of ways.51 In the following, we show that the C-Li bond in CH,Li (25)may very well be envisaged as an electron pair bond (24b), although a rather polar one, of course. But our point is not just the shift of the bonding picture back to the more covalent side of the 24a-24b spectrum. In particular, if we go to the oligomers (e.g., the methyllithium tetramer 27), a fundamentally new phenomenon occurs in the C-Li bonding mechanism. This phenomenon emphasizes the presence of dis-

66

Kohn-Sham Density Functional Theoy

CH3Li

(CH3Li)2

C3”

C2h

25

26

(CH3L04 Td

27

Crete covalent components and may modify the conception of polar bonds in general.51 Complementary to the electronic structure analyses, we present a new scheme for analyzing the charge density-termed “Voronoi deformation density” or VDD charges (see below; see also Ref. 51)-and explain why previous charge analyses do not justify the current concept of the C-Li bond as being largely ionic. Our discussion is based on nonlocal DFT computations at the BP86/TZ2P leve1107J08 (see Ref. 51 for computational details).

The Polar C-Li Electron Pair Bond in Monomeric CH,Li We begin with the C-Li bonding mechanism in monomeric methyllithium 25. This can be analyzed in two ways: homolytically, as an interaction between CH; and Li‘, and heterolytically, as an interaction between CHg and Li+.It is instructive to compare the two approaches. First, the homolytic view is considered. The valence electronic structures of CH; and Li’ are schematically shown in Figure 19. Lithium has a singly occupied 2s orbital and a set of empty 2p AOs, only 2 eV higher in energy. The orbital spectrum of CH; consists of the doubly occupied la, bonding, involving carbon 2s) and le, orbitals (CJ,--~ bonding, involving carbon 2p, and 2p,) and their antibonding counterparts, the 3a, LUMO and 2e, “LUMO+l”. The 2a, SOMO (essentially nonbonding) is located in between. In CH,Li, the methyl 2a1 and the 3 eV higher energy lithium 2s enter into a strongly polar electron pair bond (see Figure 20) as reflected by the increased population196 of the methyl 2a,: P(2a1) = P(SOM0-a,) = 1.40 e (Table 8). A significant overlap of 0.33 leads to a substantial 2a, k 2s mixing and is responsible for the covalent character of the C-Li bond, together with a sizable contribution of lithium 2p,, which acquires a population of 0.19 e (not shown in

Strongly Polar Electron Pair Bonding

Figure 19 Valence MO scheme for Lie and CH;

67

.

Table 8). The same 2p, function is unoccupied in the calculation of the methyl radical in the presence of a ghost lithium atom using the geometry of CH,Li. This shows that the lithium 2p, orbital acts like a “normal” valence orbital in the description of the C-Li bond and not, as suggested previously,l97~198as a superposition function. The strong charge donation from Li to C is in line with the difference in electronegativity between these atoms, and with the modern picture of a strongly polar carbon-lithium bond.181-183

Figure 20 Orbital interaction diagram for CH,Li.

68

Kohn-Sham Density Functional Theoy

Table 8 Analysis of C-Li

Bonding in (CH,Li)z

CH; + Li'

CH; + Li'

Bond Energy Decomposition (kcal/rnol)c -15.0 -62.9

(CH,Li),b

-5.9

-1.0

-85.8 -387.0 -18.7

-20.9 45.0 -198.9

-63.9 40.1 -32.1

-491.5 520.8 -401.3

-174.8 0.6 0.0

-55.9 10.4 0.0

-372.0 66.7 -3.3

-174.2

-45.5

-308.6 -126.6

Fragment Orbital Overlape

(CH,), + (Li), (SOMO-a, I SOMO-

CH, + CH,'

Li

+ Li'

(CH,

(Li),:

L:

0.33

ad (SOMO-t, I SOMOt2)

0.55 0.29 0.09 0.65 0.23

(2a, I2a,) (2s 12s) (2Pz I2PJ

Fragment Orbital Population (e1ectrons)f

P(SOM0-a, ) P(SOMO-t,) P(SOMO-a, ) P(SOMO-t, 1

1.40

0.50

1.02 1.43g 0.91 0.69

~BP86/TZ2P//MP2(full)/6-31+G*; see Ref. 51 for computational details. b(CH;)4 + (Li*)4. = AEint+ AE,,,, = AEOi+ BE,,,,, + AV,,,,,, + AE,,,, (Eqs. [34]-[39]);BE = overall energy

change for formation of (CH,Li), from CH, and Li ions or radicals; AEint = interaction between (CH,), and (Li), fragments; AE,,, = preparation energy required to form the (CH,), and (Li), fragments from the corresponding CH, and Li ions or radicals; AVelsrat= classical electrostatic interaction between the unperturbed charge distributions of the (CH, )n and (Li), fragments; AEpauii.= Pauli repulsion between occupied fragment orbitals; AEOi= AE,, + AEt2 + AE,,,, = orbital interaction, composed of the electron pair bond of the SOMOs in A , and T2 symmetry of the (CH,), and (Li), fragments plus a rest term. dAEo,i = oligomerization energy of CH,Li. coverraps between orbitals of the indicated fragments. fP(cp) is the gross Mulliken population (Ref. 196) that fragment orbital cp carries in the overall molecule. gPopulation of one member of the triply degenerate T, set.

Furthermore we have quantitatively analyzed the C-Li bonding mechanism in the CH,Li monomer 25, dimer 26 (not discussed here), and tetramer 27 (see later) through a decomposition of the overall bond energy AE. The latter corresponds to the formation of (CH,Li), from the corresponding methyl and lithium radicals and is made up of two major components (Eq. [34]).

Strongly Polar Electron Pair Bonding nCH;

+ nLio -+ (CH,-Li),

AE = AEi,, + AE,,,,

69

WI

The preparation energy AEpKep is the amount of energy required to assemble the fragments between which there is a carbon-lithium bond and to bring them into the geometry that they acquire in the overall molecule: the CH; radicals must be pyramidalized and, in the case of the oligomers, brought together in an outer (CH; ,) “cage” (Eq. [35]), whereas the lithium atoms must form the inner (Lie), cluster (Eq. [36]). The interaction energy AE,,, corresponds to the actual energy change when the “prepared” (CH; ), and (Lie), fragments are combined to form the C-Li bond (Eq. [37]).

(CH;), + (Lie), -+ (CH,-Li),

AEi,,

[371

The interaction energy is further split up into three physically meaningful components as discussed earlier: (1) the classical electrostatic interaction AVelstat between the unperturbed charge distributions of the prepared fragments which is usually attractive, (2) the Pauli repulsive orbital interactions AEpauli,and ( 3 )the stabilizing orbital interaction AE,, (Eq. [38]):

The orbital interactions can be further split up into the contributions from each irreducible representation r of a (CH,Li), system (Eq. [39]): AE,, = E A E= ~ AE,,

r

+ AE,, + AE,,,,

[391

We anticipate that in the case of the oligomers two dominant contributions can be recognized and, accordingly, we have partitioned the orbital interactions as follows: ( 1)AEal, the contribution from A, symmetry in which a lower energy SOMO (i.e., SOMO-a,, see Figure 21) on (CH;), and another one on (Lie), interact, (2) AE,,, the contribution from T, symmetry in which a higher energy SOMO (i.e,, SOMO-t,) on (CH;), and another one on (LP), interact, and ( 3 ) AE,,,,, a term containing the rest of the contributions from the remaining symmetries. In CH,Li, there is only one SOMO on each fragment (CH; and Lie): the two SOMO-a, orbitals 2a, and 2s, respectively (Figure 20). The CH,-Li orbital interaction AE,, of -63.9 kcal/mol is almost exclusively provided by the polar electron pair bond AEa, between these two SOMOs (Table 8); a small contribution of -1 .0 kcal/mol stems from a n-type interaction between C--H bonding le, orbitals and lithium 2p. The orbital interactions are opposed by a

70

Kohn-Sham Density Functional The0y

Figure 21 Orbital interaction diagram for (CH,Li),.

net repulsion AEpaUli+ AVelstat = 8.0 kcal/mol (mainly due to mutual corevalence overlap) and a preparation energy AE,,,, = 10.4 kcal/mol (caused by the pyramidalization of CH; ), leading to the overall C-Li bond energy AE = -45.5 kcal/mol. Correction for zero point energy effects leads to a bond dissociation energy Do(CH,-Li) of 43.5 kcal/mol (AZPE = -2.0, BP86/TZ2P), in good agreement with previous theoretical Do values of 42.15 and 43.7 kcal/ m 0 1 . I ~ ~The corresponding 298.15 K bond dissociation enthalpy, AHdiss,29s(CH3-Li) = 43.7 kcal/mol, leads to a heat of formation AH,(CH,Li(g)) = 29.2 kcal/mol, using the experimental heats of formation114 of 34.8 k 0.3 and 38.1 kcal/mol for CH; and Li., respectively. The basis set superposition error (BSSE)200 is very small, only 0.2 kcaYmol, and is therefore neglected. In the preceding paragraph, we took a homolytic or covalent approach, dividing the system into radicals. But of course, one might also choose to analyze the interaction from an ionic or heterolytic point of view. Two striking changes occur if the heterolytic approach is chosen (Table 8): the electrostatic interaction AVelstat increases from -32.1 kcal/mol to -198.9 kcal/mol, and the interaction AEa, between the methyl 2a1 orbital (lone pair on CH,) and the lithium 2s (LUMO of Li+) decreases from -62.9 kcallmol to -15.0 kcal/mol. The increase of the electrostatic attraction is not unexpected as one goes from neutral to oppositely charged fragments: for example, two point charges of +1 and -1 electron separated by 2.005 8, yield an electrostatic attraction of -166

Strongly Polar Electron Pair Bonding

71

kcal/mol (our value of about -200 kcal/mol is even larger, since the CHJ lone pair is oriented toward Li+).The sizable 2a, + 2s interaction of -15 kcal/mol (associated with a charge transfer of half an electron) is a quantitative measure for the tendency of methyllithium to deviate from the purely ionic structure 24a. Overall, owing to the charge separation, the heterolytic dissociation of CH,Li (-AE = 174.2 kcal/mol) is about 4 times more endothermic than the homolytic dissociation (-AE = 45.5 kcal/mol). In this respect, it seems more natural to conceive the carbon-lithium bond as a polar electron pair bond. This is also in line with the dipole moment of 5.6 D; a complete electron transfer from Li to C would lead to a dipole moment of 9.5 D, as pointed out before by Schiffer and Ahlrichs.199 Furthermore, the heterolytic approach provides an unbalanced starting point for the oligomers: excessive and unphysical electrostatic repulsion, especially between lithium cations (e.g., +843 kcal/mol for Lit+), must be compensated by even larger donor-acceptor and electrostatic interactions between the methyl anion cage and the lithium cation cluster. Such a picture does not serve our understanding of C-Li bonding and the cohesion within the central lithium cluster. We conclude that the Kohn-Sham orbital electronic structure and our quantitative bond energy analysis suggest that a homolytic approach is in many respects the more natural one. Therefore, we discuss the C-Li bond in the methyllithium tetramer solely in terms of the homolytic approach.

The Polar C-Li Electron Pair Bond in Tetrameric CH,Li The carbon-lithium bond in (CH,-Li), is formed between the (CH, )4 and (Li), fragments. These have quartet electronic valence states with one electron in a C-C (or Li-Li) bonding orbital (SOMO-a, ) and three electrons in a set of triply degenerate C-C (or Li-Li) antibonding orbitals (SOMO-t,, Figure 21). The preparation energy of the (CH;), “cage” (66.7 kcallmol) is again mainly due to methyl pyramidalization. This fragment has a relatively small energy gap of 1.2 eV between SOMO-a, (2al, C-C bonding) and SOMO-t, ( 3t2, C-C antibonding) because the four methyl 2a, SOMOs from which they originate are quite far away from each other (dCc = 3.6 A). Interestingly, the formation of the tetrahedral quartet is slightly exothermic (-3.3 kcal/mol) and not endothermic, as might be expected for four strongly overlapping ((2s I 2s) = 0.65) same-spin 2s orbitals. The reason is a significant stabilization of the 2p orbitals in the lithium cluster and a strong 2s-2p rehybridization (effectively an s+p transfer), which heavily stabilizes both l a , and It,. Besides a smaller separation in space compared to the methyl groups (dLiLi = 2.4 A), the main reason for the strong overlap between the lithium AOs is the very extended nature of these orbitals, as illustrated by Figure 22, which compares the 2s and 2p orbitals of lithium and carbon. This turns out to have an important consequence (see below).

72

Kohn-Sham Density Functional Theoy

c 2s

Li

2s

Li

2p

*

Figure 22 Contour plots of Li and C 2s and 2p AOs (contour values: 0.0, k0.02, k0.05, f O . 10, k0.2, k0.5 bohr-3’2; radial nodes dash-dotted). Asterisks indicate the Li-Li separation of 2.147 A in (CH,Li), (26).

In contrast to the monomer, the carbon-lithium bond in the methyllithium tetramer 27 is provided by two distinct orbital interactions: the SOMO-a, interaction 2a, l a , and the triply degenerate SOMO-t, interaction 3t25 It, (Figure 21). Strikingly, the former gives an essentially covalent electron pair bond of -85.8 kcal/mol; the SOMO-a, populations are 1.02 and 0.91 e for the tetramethyl (2a, ) and the tetralithium ( l a , ) fragment, respectively (Table 8)! The extremely low polarity of this C-Li electron pair bond is due to the very low energy of the l a , orbital: 3 eV below lithium 2s and only 0.8 eV above (CH;), 2a, (Figure 21). As mentioned just before, this is the important +_

Strongly Polar Electron Pair Bonding

73

consequence of the very strong stabilization of the strongly overlapping lithium 2s and 2p AOs in the (Lio)4cluster. The triply degenerate SOMO-t, interactions provide a strong bond of -387.0 kcal/mol (i.e., -129.0 kcal/mol per electron) because of the stabilization It, associated with the sizable charge donation from the higher energy (-3.5 eV) to the (CH; )4 3t2 orbitals, which overrules the unfavorable effect of the relatively small overlap of 0.29 (Table 8). This T, component of the C-Li bond is highly polar and significantly increases the tetramethyl 3t, population (P(SOM0-t,) = 1.43 e). Note, that the specific depopulation of the (weakly) Li-Li antibonding It, orbitals, together with the remaining population of the Li-Li bonding la,, also improves the cohesion within the positively charged cluster. When all interactions are taken together, we arrive at an overall bond energy AE of -308.6 kcal/mol (Table 8). The additional stabilization of -126.6 kcal/mol with respect to four isolated methyllithium monomers corresponds to the oligomerization energy for reaction [40]: 4 CH,Li

-+ (CH,Li),

[401

Analysis of the Charge Distributions in CH,Li Oligomers Complementary to the orbital electronic structure analysis, we carried out analyses of the charge density distribution using the following methods: the Hirshfeld scheme,201 the Voronoi deformation density (VDD) method,51 and the natural population analysis (NPA); the latter was carried out using MP4SDQ ab initio theory instead of DFT. In the Hirshfeld method, a hypothetical “promolecule” with electron density Cp, is constructed by the superposition of spherically symmetrized charge densities pB of the isolated atoms B. The electron density p of the real molecule at each point in space is then distributed over the atoms A in the same ratio wA = (pA / CpB) as they contribute charge density to that point in the promolecule. The Hirshfeld atomic charge QF is obtained by subtracting the resulting partial electron density associated with atom A from the corresponding nuclear charge Z , (Eq. [41]).

The Voronoi deformation density approach, is based on the partitioning of space into the Voronoi cells of each atom A, that is, the region of space that is closer to that atom than to any other atom (cf. Wigner-Seitz cells in crystals; see Chapter 1of Ref. 202). The VDD charge of an atom A is then calculated as the difference between the (numerical) integral of the electron density p of the real molecule and the superposition of atomic densities CpB of the promolecule in its Voronoi cell (Eq. [42]):

74

Kohn-Sham Density Functional Theory

cell A

(We restrict ourselves here to overall neutral molecules.) Thus, the VDD atomic charges offer a way of quantifying the deformation density p - XpB on an atomic basis by means of a simple geometric partitioning of space. They merely monitor if charge "flows" away from or toward the space around a certain nucleus upon the formation of the molecule from its atoms. Therefore, the physical interpretation is rather simple and straightforward: a positive or negative corresponds to the loss or gain of electrons in the Voronoi atomic charge cell of atom A. For further development of the VDD method, see Refs. 203 and 204. The results of the charge density analyses are collected in Table 9 in the form of the lithium atomic charges for the methyllithium monomer and tetramer; we have also added the dimer, which was left out in the discussion of electronic structure. The VDD lithium charges are relatively small, and they decrease from +0.38 to +0.26 to +0.13 e along the series CH,Li (25), (CH,Li), (26),and (CH,Li), (27).This clearly shows that the shift of electron density from lithium to methyl decreases upon oligomerization. This is also confirmed by the Hirshfeld lithium charges, which decrease from +0.49 to +0.42 to +0.30 e along the same series. This trend agrees with the electronic structure analysis, where it shows up in the increasing population of the (Lie), fragment orbitals SOMO-a, and SOMO-t, . The trend is indicative of the increasing importance of a covalent component in the carbon-lithium bond (see above). Note from Table 9 that the carbon-lithium bond is significantly more ionic according to NPA (ca. 90%) than according to VDD (40-10%) or Hirshfeld (50-30%). Furthermore, the NPA charges do not monitor the reduction of Li+CH, charge transfer upon oligomerization. We consider all three charge analysis schemes to be satisfactory approaches for the definition of basis set independent and chemically meaningful atomic charges (see Bickelhaupt et al.,51 Wiberg and Rablen,205 and Meister and Schwarz206).The fact that there

exDD

Table 9 Lithium Charges (electrons)in Methyllithium Oligomers as Obtained by Three Methods of Electron Density Analysisa Voronoi deformation density (VDD)b Hirshfeldb Natural population analysis (NPA).

+0.38 +0.49 +0.85

+0.26 +0.42 +0.88

+0.13 +0.30 +0.86

Wsing MP2(fu11)/6-31+G*geometries (second-order Meller-Plesset perturbation theory with core electrons included in the perturbation treatment). bBP86/TZ2P (nonlocal DFT with the Becke-88-Perdew-86 functional). cMP4SDQ/6-31+G" (fourth-order Merller-Plesset perturbation theory with single, double, and quadruple excitations).

Conclusions and Outlook

75

is nevertheless a significant discrepancy between the methods demonstrates, in our opinion, that the degree of ionicity of a bond obtained on the basis of atomic charges should not be regarded as an absolute quantity. Therefore, the designation of the C-Li bond as ionic on the basis of, for example, NPA or Bader atomic charges of roughly +0.9 electron for lithium is unsupported.

Summary The preceding discussion illustrates the power of Kohn-Sham MO theory

to disclose subtle and yet fundamental features of a polar chemical bond. The

C-Li bond that we have chosen as our example may very well be conceived as an electron pair bond, although it is of course very polar, as we have shown for CH,Li. But of more significance is the change in character of the polar C-Li bond when the CH,Li molecules form oligomers. As we have seen, the C-Li orbital interactions in the methyllithium tetramer consist of two discrete components: (1)a virtually covalent electron pair bond in A, symmetry between C-C and Li-Li bonding (CH;), and (Li1)4 fragment orbitals, and (2) a strongly polar interaction between the corresponding C-C and Li-Li antibonding fragment orbitals. These results do not deny the polar nature of the C-Li bond, but they do emphasize its dual character. The charge density analyses confirm this picture from Kohn-Sham MO theory. They also show that atomic charges are not absolute quantities and should not be overrated. The label “ionic” for the C-Li bond based on such quantities is therefore questionable.

CONCLUSIONS AND OUTLOOK The Kohn-Sham molecular orbital model constitutes an attractive and simple one-electron model for the discussion of features of chemical bonding. It is certainly an advantage that the electronic charge density provided by this MO method is in principle correct for all systems involved, both before the interactions are taken into account and when the final system has been formed. The Kohn-Sham potential shares a number of properties with the Fock operator of Hartree-Fock theory, but also has interesting additional structure (notably the potential of the Coulomb correlation hole charge density). Thus, the KS potential provides the one-electron model with a firm physical basis. Its connection (in principle) with exact energies via the exchange-correlation functional Ex,is a distinct advantage to the theoretical chemist seeking to understand chemistry. The most important purpose of this tutorial is, however, to show that it is possible to extend the traditional qualitative MO theory to arrive at a more detailed analysis of molecular interactions. Explicit calculation of physically well-defined energies enables one to obtain a more complete view of the physics

76

Kohn-Sham Density Functional Theorv

of chemical bond formation. The electrostatic interactions are of course important; indeed, it is an asset of Kohn-Sham calculations that they provide in principle exact electronic charge densities. In addition, however, we have highlighted the importance of kinetic energy effects, in particular when it comes to the understanding of repulsive energy contributions. We have emphasized the importance of Pauli repulsion, both between valence electrons (as an explanation of the steric repulsion) and between valence electrons and (upper) core electrons as an explanation for the inner repulsive wall in the chemical bond. We stress that we view this extended MO theory not as a contender against the well-known qualitative MO theory, but as a next step in refinement and interpretive power.

ACKNOWLEDGMENTS We thank the Deutsche Forschungsgemeinschaft (DFG), the Fonds der Chemischen Industrie (FCI), and the Nederlandse Organisatie voor Wetenschappelijk Onderzoek ( N W O )for financial support. F.M.B. gratefully acknowledges a DFG Habilitation Fellowship.

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CHAPTER 2

A Computational Strategy for Organic Photochemistry Michael A. Robb, * Marco Garavelli,‘kMassimo Olivucci,t and Fernando Bernard8 “Department of Chemistry, King’s College London, Strand, London WC2R2LS, United Kingdom, tIstituto di Chimica Organica, Universita degli Studi di Siena, Via Aldo Moro, I-53 100 Siena, Italy, and SDipartimento di Chimica “G. Ciarnician” dell’llniversita di Bologna, Via Selmi 2, 401 26 Bologna, Italy

INTRODUCTION Modeling Photochemical Reactions The aim of this chapter is to provide an introduction to the practical computational investigation of photochemical reaction mechanisms. During the last decade or so, the speed of computers has grown considerably, and now the computational investigation of realistic models of organic compounds is becoming a standard practice. Current applications range from the investigation of the mechanism of synthetically useful reactions to the study of shortlived organic intermediates detected in the interstellar medium. For thermal reactions, standard state-of-the-art ab initio quantum chemical methods are already capable of providing a complete description of what happens at the Reviews in Computational Chemistry, Volume 15 Kenny B. Lipkowitz and Donald B. Boyd, Editors Wiley-VCH, John Wiley and Sons, Inc., New York, 0 2000

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A Computational Strategy for Organic Photochemistry

molecular level during bond-breaking and bond-forming processes. In particular, it is possible to compute the transition structure that connects a reactant to a product and the associated energy barrier with almost chemical accuracy (ca. 1 kcal/mol error). Furthermore the reaction path (i.e., the progression of the molecular structure from the reactants toward the transition state and the product) can be determined, in a completely unbiased way, by computing the minimum energy path (MEP)1connecting the reactant to the product on the ( 3 N - 6)-dimensional potential energy surface of the system. A detailed understanding of the reaction pathway in the excited state manifold will increase our ability to design new and to control known photochemical reactions. As an example, the conversion of light into chemical energy in plants and animals involves extended conjugated moleculescarotenoids and retinals-bound in protein complexes. The use of such extended systems in optical data storage and processing technology is now being investigated. Photobiological systems exploit the ability of these chromophores to undergo cis-trans isomerization and to transduce radiative energy into thermal energy on picosecond or shorter time scales. Recent advances in timeresolved spectroscopy2 (e.g., the use of ultrafast laser pulses) have provided a powerful tool to monitor reaction dynamics on the femtosecond time scale and have made direct observation of these processes possible, increasing our understanding of the excited state structures and dynamics for model systems. UItrafast (femtosecond) radiationless decay has been observed, for example, for simple dienes,3 cyclohexadienes,4~5and hexatrienes,6 and in both free7 and opsin-bound8 retinal protonated Schiff bases. However, a complete understanding of molecular dynamics on multiple electronic states is required to interpret these laser experiments with confidence and to understand the principles involved in the design of optical devices. Until recently, reaction path computations were mainly limited to the investigation of thermal reactions and thus to reactions occurring on a single potential energy surface. Photochemical processes, where the reactant resides on an excited state potential energy surface and the products accumulate on the ground state, could not be easily investigated. In photochemistry, the reaction path must have at least two branches: one located on the excited state and the other located on the ground state energy surface. The main difficulty associated with such computations lies in the correct definition and practical computation of the “funnel” region, where the excited state reactant or intermediate is delivered to the ground state. Thus, while the progression on excited state energy surface (i.e., the excited state “branch” of the reaction path) may be investigated with the same methods used for thermal reactions, there was no general way of defining the “locus” where the excited state branch of the reaction path was connected to the ground state branch. During the last decade, computational tools have been developed and strategies discovered to explore electronically excited state reaction paths.9-17 The goal of such computational approaches, in the study of photochemical mechanisms, is the complete descrip-

Introduction

89

tion of what happens at the molecular level from energy absorption to product formation. This review focuses on some of these new theoretical tools, with case studies to show how they can be applied. Modern textbooks on photochemistry with a good theoretical treatment include Refs. 9-11. We shall begin with some simple ideas that are the main focus of the theoretical study of excited state processes. A very schematic view of the course of a photochemical reaction is given in Figure 1. Following light absorption, the system is promoted to an excited state (R +hv, R”).Photoproduct formation can then occur by adiabatic reaction (R“+ P’*) on the excited state (a photochemical process) followed by emission (P’* -hv, P’) or by internal conversion to the ground state (a photophysical process). However, the most common mechanism involves a nonadiabatic radiationless decay process, which either regenerates the reactant R (a photophysical process) or generates new products P (a photochemical process). Whether a photochemical reaction occurs thus depends on the relative rates of photochemical processes that generate new molecular structures versus competing photophysical processes that convert between electronic states at the same nuclear geometry. The competition between photochemical and photophysical processes may occur via two different mechanisms: branching at a same decay channel (see left side of Figure 1) or competition between two different decay channels (see right side of Figure 1). Understanding the mechanism of this nonadiabatic radiationless decay is central to explaining excited state processes. There are two possible mechanisms (see nonadiabatic reactions in Figure 1).When real surface crossings exist (conical intersection, see left side of Figure 1)and are accessible, the Landau-

Landau-Zener/Massey parameter. (ps to fs)

TS Theory (1 kcal mol-‘ - 5 ps)

1

Fermi Golden-Rule

(ps to ns)

-f

Non Adiabatic Events

Figure 1 Schematic view of a course of a photochemical reaction. Processes can occur on a picosecond (ps) or femtosecond (fs) time scale.

90

A Computational Strategy for Ovganic Photochemistry

Zener modelllJ3 (discussed in more detail later) provides a semiclassical model for fast radiationless decay. In this chapter, we use the word “real” to indicate an actual occurrence of something (e.g., a crossing) as distinguished from a near miss. By “accessible” we mean that there is a reaction coordinate with a sufficiently low energy barrier that leads from the initial excited state geometry to the crossing region. In this case, internal conversion can take place within a single vibrational oscillation through, or near, the surface crossing (R“+ P or R* -+ R), and radiationless decay occurs on a scale of picoseconds to femtoseconds. If surface crossings are not present, or are present but not easily accessible, the process of radiationless decay is better described as the transformation of electronic energy into a manifold of vibronic states associated with the lower electronic state (see right side of Figure 1).This process is governed by the density of vibrational states and Franck-Condon factors (overlap of ground and excited state vibrational wavefunctions) according to the Fermi golden rule formalism (Ref. 11, pp. 257). Decay of the Fermi golden rule type occurs at a local minimum on the excited state (which might be either an avoided crossing, as shown in Figure 1, or the minima R” or I“*).This process is much slower than decay at a surface crossing, and typical internal conversion rates for aromatic hydrocarbons are found to be 105-106 s-1. Recently it has been shown that certain photochemical radiationless reactions are extremely fast. For example, cis-trans isomerization of the retinal chromophore in the vision process occurs in about 200 fs. This demonstrates that these reactions are likely to occur via real surface crossings, since these can be faster than processes such as fluorescence and internal conversion via avoided crossings or at local minima, which take place on the time scale of a microseconds to nanoseconds. Recent complementary theoretical computations have shown that low-lying intersections (real crossings) between the photochemically relevant excited state and the ground state occur with a previously unsuspected frequency.14215 Such crossings-that is, conical intersections in the case of two singlet (or two triplet) states, or singlet-triplet intersectionsprovide a very efficient “funnel” for radiationless deactivation9~11~13-17 (internal conversion and intersystem crossing) and, in turn, prompt photoproduct formation. To summarize, the decay probability (i.e., the internal conversion rate) at an avoided crossing or a local excited state minimum is controlled by the interaction between the vibrational energy levels of the ground and excited state potential energy surfaces by using the Fermi golden rule, and thus radiationless decay and fluorescence can occur on competitive time scales. On the other hand, radiationless decay at a conical intersection implies that the internal conversion process can approach 100% efficiency13 so that any observed retardation in the internal conversion or reaction rate (i.e., the competition with fluorescence) must reflect the presence of some excited state energy barrier that separates R* from the intersection structure. Finally, in the case of radiationless

Introduction

91

decay that leads to a chemical reaction, the molecular structure at the intersection must be related to the structure of the photoproducts. The traditional view of deactivation of an electronically excited intermediate by internal conversion, mainly due to the work of Van der Lugt and Oosteroffls on the ring opening of cyclobutene, was formulated in terms of avoided crossing funnels. Thus internal conversion rates were predicted to be slow as in aromatic hydrocarbons because of the finite gap at the avoided crossing (energy gap law). Because it was impossible to compute methodologically correct reaction paths in the 1960s, the true “real” nature of the funnel was missed. In Figure 2 we show the relation between an avoided crossing (a section of a cone along R-P) and the double-cone topology of a real conical intersection. In two dimensions, the Van der Lugt and Oosteroff model is refined by replacing the “avoided crossing” with an “unavoided crossing” (i.e., a conical intersection as shown in Figure 2).15 The Van der Lugt and Oosteroff model reaction avoided crossing path R + P is replaced by a path involving a real surface crossing R + CI -+ P.In the subsequent discussion, CI (in bold) refers to conical intersections, while CI (plain text) stands for configuration interaction. Teller19 was the first to point out that in a polyatomic molecule the noncrossing rule, which is rigorously valid for diatomics, fails. Rather, two electronic states, even if they have the same symmetry, are allowed to cross at a conical intersection. Accordingly, radiationless decay from the upper to the lower intersecting state can occur within a single vibrational period when the

I

I

I

I

R Figure 2 Topological relation between an avoided crossing and a conical intersection (CI).The avoided crossing is the cross section through the cone along R-P. The minimum energy path is R-CI-P and passes through the apex of the cone.

92

A Computational Strategy for Organic Photochemistry

system travels in the vicinity of such intersection points. In the field of photochemistry, Zimmerman20 and Michl21 were the first to suggest, independently, that certain photoproducts originate from internal conversion at a conical intersection. Zimmerman and Michl used the term “funnel” for this feature. We can now suggest a theoretical basis for the computational modeling of photochemical reactions. We shall call this method the pathway approach, following the suggestion of Fuss et a1.22 According to this approach, the excited state motion is determined by the structure of the relevant excited and ground state potential energy surfaces. In simple terms, information on the excited state lifetime and on the type of photoproducts generated is obtained by following the detailed relaxation and reaction paths of the molecule along the potential energy surfaces from the Franck-Condon (FC) point (i.e., vertically excited geometry), or excited state intermediate, to the ground state. This approach is part of a more general way of considering photochemistry, which was employed in a book 11 published in 1994: it follows the pathway on the potential energy surfaces and pays attention to local details such as slopes, barriers, saddle points, and collecting funnels. Key elements of the pathway approach are the existence of low-lying real crossings or excited state products that are preceded by a transition state controlling the rate of transformation or decay of the excited state reactant. This feature has been suggested by experimental observations. Experiments on isolated molecules in cold matrices, in expanding jets, and in solution have revealed the presence of thermally activated fast radiationless decay channels in polyenes, such as hexatrienes and octatetraenes,23J-4 and in aromatic compounds, such as benzene, azulene, and azoalkanes.25J6 In particular, recent low temperature spectroscopic investigations of isolated polyene molecules have provided evidence that photoinduced double-bond trans + cis isomerization may occur via a nonadiabatic reaction path where the excited state intermediate decays to the ground electronic state but at a highly twisted molecular geometry. The original suggestions of Teller,l9 Zimmerman,20 and Michl21 have now been fully verified by computational results,25-40 which provide a particularly clear illustration of the application of the pathway approach to the study of organic photochemistry. In Figure 3 we illustrate the results of two different experiments on all-tr~ns-octa-1,3,5,7-tetraene (all-trans OT). The first experiment (Figure 3a) is due to Kohler and coworkers,23 who recorded the fluorescence lifetime of S, (2A,) all-trans OT as a function of the temperature. In this experiment, the all-trans OT molecules are isolated in a molecular cavity of frozen n-hexane and do not interact with each other. From Figure 3a, one can see that at temperatures above 200 K, the fluorescence lifetime drops dramatically, indicating fast decay of the excited state molecules to the ground state. This event was assigned to the opening of a thermally activated efficient radiationless decay channel The second with a barrier height of about 1500 cm-1 (4.3 kcal/mol~).~l~14 experiment (Figure 3b) is due to Petek and coworkers,24 who reported the fluorescence decay rate of S, all-trans O T molecules measured in free jet expan-

Introduction

0

100

93

200

Temperature I K (a) ----

Radiationless

Decay

\

150

i

-

/

100

-50

-0

0

I

f

I

I

1000

2000

3000

4Ooo

Energy Above the Origin I cm’l

J

X

4

s \

@?d

(b) Figure 3 “Opening” of a fast radiationless decay channel in all-trans octatetraene in (a) matrix-isolated conditions and (b) expanding cool jet. (From Ref. 14.)

sion as a function of the excitation energy. These authors proposed that (under isolated conditions in a cool jet) trans -+ cis motion in all-trans OT is responsible for the observed radiationless decay channel on S, (2A,), which opens up at about 2100 cm-1 (ca. 6 kcal/mol) excess energy. These experimental results can be rationalized by means of the potential energy profiles shown in Figure 4. In both experiments referred to above, the fluorescence lifetime decreases slowly and almost linearly as the S, excess vibrational energy is increased until an energy threshold is reached and a dramatic decrease in excited state lifetime is observed. Quantum chemical computations

94

A Computational Strategy for Organic Photochemistry Photochemistryor emission depending on barrier

0 Ultrafast

M*-+TS+CI

R

0

Photochemistry

FC+CI

R

Photophysicsonly

M*+CI

*

R

Figure 4 “Opening” of a fast radiationless decay channel via conical intersection for (a) a barrier controlled reaction, (b) a barrierless path, and ( c ) an uphill path without transition state (sloped conical intersection). M‘ is an excited state intermediate and FC is a Franck-Condon point.

of the S, reaction path of all-trans OT28 have revealed that the energy threshold corresponds to a transition state that connects an excited state intermediate to a conical intersection funnel. This result is schematically illustrated in Figure 4a. The computation predicts a 7.5 kcal/mol barrier, in good agreement with the experiment. In the absence of a barrier, the reaction becomes ultrafast (see Figure 4b), and no fluorescence can be observed. Features such as excited state intermediates and funnels are conveniently optimized25-40 by means of gradient optimization methods as critical points on the potential energy surface associated with the photochemically relevant excited state (usually the first singlet or triplet excited state). Standard quantum chemical methods can be used for the calculation of the excited state potential energy surface per se. However, in excited state chemistry, the knowledge of the molecular structure of the funnel clearly appears to be of vital importance for the rationalization and prediction of the observed photoproduct distribution. We expect the photoproduct molecular structure to be related to the molecular structure of the decay channel in more or less the same way in which the structure of a thermal product is related to that of the corresponding transition state. Similarly, detailed knowledge of the energetic stability of the decay channel relative, for instance, to the excited state equilibrium structure of the reactant-is expected to be related to the excited state lifetime. In other words, excited state energy barriers may control the time taken by the system to reach the decay channel. Thus, the new feature that occurs in the quantum chemistry of photochemical reactions is the characterization of the conical intersection

Introduction

95

and its relation to other features on the ground and excited state potential energy surface.

Aims and Objectives The remainder of this chapter is divided into three parts. The first two parts are theoretical, while the third illustrates the application of these theoretical concepts via some case studies drawn mainly from our own work. Our objective is to outline a computational approach to photochemistry. We aim for a complete description of what happens along the reaction coordinate from absorption to photoproduct formation. We shall show how this can be achieved by mapping of the photochemical reaction path computed by following the MEP from the excited state intermediate (or from the Franck-Condon structure) to the ground state photoproduct through a conical intersection. This method (the pathway approach22), discussed previously, pays attention to local details and properties like slopes, saddle points, barriers, and funnels (such as conical intersections) and has an intimate connection to the approach that describes ultrafast photochemical processes by means of the motion of wavepackets or semiclassical trajectories on potential surfaces. This description is becoming increasingly important because of recent advances in femtosecond spectroscopy and ultrafast laser techniques.2-8J3.24 The first theoretical section is conceptually oriented. We shall discuss thc special “features” occurring when more than one energy surface is involved in the chemical reaction (e.g., avoided and real crossings) as well as the general structure of the “photochemical reaction path” that connects the excited state reactant and the photoproducts through the funnel. The second theoretical section is more practically oriented. Here we shall illustrate the computational tools and strategies that are being used to compute a photochemical reaction path. This discussion includes the computation of the excited state branch of the reaction path, explains how to study the branching at a conical intersection and compute “competing” ground state relaxation paths, and briefly discusses semiclassical trajectories. We focus on the special techniques that are required for the study of excited state processes. In principle these techniques can be implemented in a variety of computational algorithms as long as they can produce analytical gradients and Hessians and give a balanced description of the energetics. It is not our purpose to discuss the details of particular algorithms that have been implemented. Rather we focus on the conceptual background that is necessary to use such methods effectively. Thus we limit ourselves mainly to some general remarks about current quantum chemical methods and algorithms that are suitable for computational organic photochemistry. Finally, in the last section, we illustrate via case studies a few general results obtained in the field of mechanistic organic photochemistry and com-

96

A Computational Strategy for Organic Photochemistry

pare them with modern (time-resolved)3-*.14.’5,23.24 and traditional41 experimental data. We shall concentrate on the following points: the funnel structure, the existence of chemically or stereochemically distinct competing paths (i.e., funnels, on the excited state energy surface), and the reaction path branching (i.e., the competing ground state relaxation paths) at a specific funnel. Our objective is to illustrate how the observed photoproduct stereochemistry and distribution in a photochemical organic reaction must depend on radiationless decay via a conical intersection (i.e., a real surface crossing).

CHARACTERIZATION OF CONICAL INTERSECTIONS The way the energy of a molecular system varies with small changes in its structure is specified by its potential energy surface (PES). A potential energy surface is a mathematical relationship linking molecular structure and molecular energy. The concepts of energy surfaces for molecular motion, equilibrium geometries, transition structures, and reaction paths depend on the BornOppenheimer approximation to treat the motion of the nuclei separately from the motion of the electrons. Minima on the potential energy surface for the nuclei can then be identified with the classical picture of equilibrium structures of molecules (i.e., reactant, product, and intermediates); saddle points can be related to transition states and reaction rates (see Refs. 42 and 43 and references cited therein). Minima, maxima, and saddle points can be characterized by their first (i.e., the gradient) and second (i.e., the Hessian) derivatives of the energy.42.43 If the Born-Oppenheimer approximation is not valid-for example, in the vicinity of surface crossings-nonadiabatic coupling effects (that couple nuclear and electronic motion) need to be taken in account to correctly describe the motion of the molecular system. This is done, for instance, when one needs to describe a jump between two different PESs. In this case, one uses semiclassical theories and the surface-hopping method, which we discuss subsequently. We now discuss in some detail how the region in which nonadiabatic effects become important can be characterized topologically.

“Noncrossing Rule” and Conical Intersections In photochemistry one must deal with a new type of potential surface feature (surface crossings and conical intersections), and we now introduce this subject. In diatomic molecules the PES of two states (e.g., the ground state and the first excited state) will intersect only if the states have a different (spatial or spin) symmetry. However, this statement is not true in polyatomic sys-

Characterization of Conical Intersections

97

terns.19344745 Rather, the correct statement is: two PESs of a polyatomic molecule can in principle intersect even if they belong to states of the same symmetry and spin multiplicity. The preceding sentence leaves open the question of whether such intersections actually occur in polyatomic systems. We now give a quantitative analysis of this situation.45 If we suppose we have all but two of the solutions for the electronic part of the Schrodinger equation, and +1 and 42 are any two functions that, together with the known solutions, constitute a complete orthonormal set (the two missing solutions correspond to the two states whose energies are El and E,, respectively, and whose crossings we are interested in), then it must be possible to express each of the two remaining electronic eigenfunctions (which describe the states we want to examine) in the form

The resulting secular equation is then

H22 -

and we can write down the expressions for the energies E l and E, of the two states as follows:

where the matrix elements are defined

Now, to have degenerate solutions (i.e., an unavoided crossing), the discriminant must vanish, and it is necessary to satisfy two independent conditions:

H,, = H,,

H,, (= H21) = 0

PI

This requires the existence of at least two independently variable nuclear coordinates. Since in a diatomic molecule there is only one variable coordinate-the interatomic distance-so the noncrossing rule can be stated as follows: For states of different (spatial or spin) symmetry, HI, is always zero, and the two surfaces cross when H,, = H22. This is possible for a suitable

98

A Computational Strategy for Organic Photochemistry value of the single variable coordinate. Otherwise, if the two states have the same symmetry, they will not intersect.

However in a system of three or more atoms (N is the number of atoms), there are enough degrees of freedom for the rule to break down: the two conditions (Eq. [ S ] ) can be simultaneously satisfied by choosing suitable values for two independent variables, while the other n - 2 degrees of freedom ( n= 3 N - 6 ) are free to be varied without leaving the crossing region. If we denote these two independent coordinates by x1 and x2, and take the origin at the point where HI, = H,, = W and H,, (= H21) = 0, in the hypothesis of the first-order (i.e., linear)46approximation, the secular equations may be cast in the form 1x2

(i.e., fill=

w + h l x l , etc.)

or

where m = %(h, + h,), k = %(h, - h,). The eigenvalues are

E = W + mx, f .\ik2x:

+ l2x?

PI

Equation [8] is the equation of an elliptic double cone (i.e., with different axes) with vertex at the origin (it will be a circular cone only for the case k = 1). Thus, such crossing points are called conical intersections. Indeed, if we plot the energies of the two intersecting states against the two internal coordinates x 1 and x2 [whose values at the origin satisfy the two conditions and H,, = H22 and H,, (= H21)= 01, we obtain a typical double-cone shape (see Figure 5 ) . Now let us inquire into the physical meaning of the two conditions HI = H2,and H,, (= H,,) = 0. If we consider the basis and 4), of the secular equation (Eq. [2]) as the diabatic components of the adiabatic electronic eigenfunction (a diabatic function describes the energy of a particular spin-coupling or atomic orbital occupancies,l4 while the adiabatic function represents the surface of the real state), the crossing condition (real or avoided) is fulfilled when the two diabatic components 4, and 42 cross each other, and this happens that is, when the energy of the two diabatic potentials (Hll is when H,, = H2,,

Characterization of Conical Intersections

2 dimensional

99

n-2 dimensional intersection space

branching space

(4

(b)

Figure 5 (a) Typical double-cone topology for a conical intersection. (b)Relation between the “branching space” (xl,x2)and the “intersection space” (spanning the remainder of the ( n - 2)-dimensionalspace of internal geometric variables.

and H,, is the energy for the diabatic the energy for the diabatic function function 4,) is the same. At the crossing of the diabatic functions (Hll = IT2,), the expressions for the energies of the two real states become (from Eq. [ 3 ] )

El

= H , , = H12

E,

= H,, = H,,

191

and the energy gap between the two real states is

Thus, if the off-diagonal (resonance) term is not zero, the crossing will be avoided, and the potential surfaces of the two real states will “split.” The energy separation at an avoided crossing thus depends on the magnitude of H,,. HI, will be zero (and the crossings will be unavoided) when the two electronic states have a different (spatial or spin) symmetry. However, in general, H,, is generally not zero for states of the same symmetry (and thus will generate avoided crossings). This is the noncrossing rule. Anyway, as we have seen, this rule is true only for diatomic molecules,l9 and in a polyatomic system one can always have unavoided (i.e., real) crossings for suitable values of a pair of independent coordinates (xland xz),which will simultaneously satisfy Eqs. [5]. In conclusion, the most general statement for unavoided crossings’9 is: for a polyatomic system, two states (evenwith the same symmetry)will intersect along an (n - 2)-

100 A Computational Strategy for Omanic Photochemistry

dimensional hyperline (i.e., a line in more than three dimensions) when the energy is plotted against the n internal nuclear coordinates [the two dimensions referred to above are the two independent variables (i.e., x1 and x2) defined previously]. Theoretical investigations of surface crossings have required new theoretical techniques based on the mathematical description of conical intersections, and we now briefly review the central theoretical aspects. For two geometric variables, two surfaces of the same multiplicity intersect as a double cone (Figure 5). If one moves in the plane spanned by the two directions x1 and x2 (the so-called brunching space46), the degeneracy is lifted. In n dimensions, the degeneracy persists along an ( n- 2)-dimensional hyperline (called intersection space): if we move from the apex of the cone along any of the remaining n - 2 internal coordinates defining the intersection space, the degeneracy is not lifted. This ( n- 2)-dimensional space is a hyperline consisting of an infinite number of conical intersection points (see Figure Sb). It can be demonstrated47 that these two directions are given as the gradient difference vector

and the gradient of the interstate coupling vector

where C, and C2 are the configuration interaction (CI) eigenvectors in a C1 problem, H is the CI Hamiltonian, and Q represents the nuclear configuration vector of the system. The vector x2 is parallel to the nonadiabatic coupling vector g(Q).

The vector g(Q)is the coupling term that gives the magnitude of the coupling between the Born-Oppenheimer states described by C, and C2 as a function of the nuclear motion along Q.

Conical Intersection Structure To understand the relationship between the surface crossing and photochemical reactivity, it is useful to draw a parallel between the role of a tran-

Characterization of Conical Intersections 2 01

TS

R

Figure 6 Comparison of the role of (a) a transition state (TS)in thermal reactivity and (b) a conical intersection (CI) in photochemical reactivity.

sition state in thermal reactivity and that of a conical intersection in photochemical reactivity.14 In a thermal reaction, the transition state (TS)forms a bottleneck through which the reaction must pass, on its way from reactants (R) to products (P) (Figure 6a). The motion through the TS is described by a single vector, the transition vector x1 (i,e., the eigenvector related to the imaginary vibrational frequency). A transition state separates the reactant and product energy wells along the reaction path. An accessible conical intersection (CI) (Figure 6b) also forms a bottleneck that separates the excited state branch of the reaction path from the ground state branch. The crucial difference between conical intersections and transition states is that whereas the transition state must connect the reactant energy well to a single product well via a single reaction path, an intersection is a “spike” on the ground state energy surface (see Figure 6b). The CI may connect the excited state reactant to two or more products (PI and P2) on the ground state via a branching of the excited reaction path (in the plane x1 and x2) into different ground state relaxation valleys. The branching is possible even in the first-order approximation because of the elliptic nature of the double cone; however in the case of an elliptic cone the branching will occur at most along two directions (see PI and P2 in Figure 6b).The nature of the products generated following decay at a surface crossing will depend on the ground state valleys (relaxation paths) that can be accessed from that particular structure. Different topological situations are possible for unavoided crossings between surfaces. One can have intersections between states of different spin multiplicity [an ( n - 1)-dimensional intersection space in this case, since the interstate coupling vector vanishes by symmetry], or between two singlet surfaces or two triplets [and one has an n - 2)-dimensional conical intersection hyperline in this case]. We have encountered situations in which both types of

102 A Computational Strategy for Organic Photochemistry

singlet

Energy

(n-n*) 3

Wn*)

&q

3

(n-n*)

(c)

(dl

Figure 7 Topological possibilities for the crossing of two states: (a) typical ( n - 2)dimensional conical intersection between two states of the same spin multiplicity (i.e., two singlets or two triplets); (b) ( n - 1)-dimensionalintersection between two states of different spin multiplicity (i.e., singlet and triplet); ( c ) singlet and triplet conical intersection occurring at the same ge0metry3~;and (d) Renner-Teller-like degeneracy (a “touching” rather than a crossing).4* Whereas for examples a-c the gradient of the surfaces at the crossing is different from zero, in d it is zero.

intersection occur at the same geometry.32 These three various topological possibilities are summarized in Figure 7a-c; notice that for these examples, the gradient of both surfaces at the crossing is not zero. A conical intersection is just a Jahn-Teller-like degeneracy48>49that occurs without symmetry. One might ask if a Renner-Teller-like degeneracy48 (i.e., the gradients of both surfaces go to zero at the degeneracy) can occur without having its origin in the symmetry of the states. This situation (shown in Figure 7d) is characterized by the fact that the gradients of both states are zero (i.e., they are true minima). Thus this situation is a “touching” rather than a crossing. The topography of the potential energy surfaces in the vicinity of a conical intersection can also be characterized by the relative orientation of the two potential surfaces, as discussed by Ruedenberg et al.46 In this review we use

kX7 &:

Characterization of Conical Intersections 10.3

x2

Peaked Conical Intersection

(4

Sloped Conical Intersection (b)

x2 Intermediate Conical Intersection (c)

Figure 8 Topological possibilities for conical intersections (characterized according to Ref. 46): (a) peaked, (b) sloped, and (c) intermediate conical intersections.

Ruedenberg’s terminology: peaked, sloped, and intermediate, as shown in Figure 8. Often the chemically relevant conical intersection point is located along a valley on the excited state potential energy surface (i.e., a peaked intersection). Figure 9 illustrates a two-dimensional model example that occurs in the photochemical trans -+ cis isomerization of octatetraene.28 Here two potential energy surfaces are connected via a conical intersection. This intersection

Figure 9 Two-dimensional model surface for the photochemical cis -+ trans isomerization of octatetraene.

104 A Computational Strategy for Organic Photochemistry appears as a single point (CI)because the surfaces are plotted along the branching space (xi,xz).The intermediate M" is reached by relaxation from the Franck-Condon region (FC),and it is separated from the intersection point by a transition state (TS).In this case (a peaked type of intersection), the molecular structure of the intersection and the reaction pathway leading to it can be studied by computing the MEP connecting FC to M" and M" to CI using the standard intrinsic reaction coordinate (IRC) method.50 However, in certain situations (sloped intersections, see Figures 4c and 8b) there is no transition state connecting M" to the intersection point. In such situations, mechanistic information must be obtained by locating the lowest lying intersection point along the n - 2 intersection space of the molecule. The practical computation of the molecular structure of a conical intersection energy minirnum47.51 is illustrated in the next subsection. This technique provides information on the structure and accessibility of the intersection point that controls the locus and efficiency of internal conversion. Nonadiabatic events (transition from the excited state to the ground state at the conical intersection) pose a serious challenge because the nonadiabatic transition is rigorously quantum mechanical without a well-defined classical analog. At a simple level of theory13 (we return to a better treatment subsequently), the probability of a surface hop is given as follows: P = exp[-(x/4)5]

~ 4 1

where the Massey parameter is

Thus this simple theory predicts that radiationless transitions will occur when the energy gap AE(Q)is small and the scalar product between the velocity vector and the nonadiabatic coupling Q . g(Q)is large. Here Q is the nuclear coordinate vector in Eq. [ l l ]and g(Q) is defined in Eq. [13]. The motion to ground state photoproducts following decay via internal conversion at a conical intersection channel requires a study of the possible ground state relaxation processes. The initial relaxation direction (IRD) method, which locates and characterizes all the relaxation directions that originate at the lower vertex of the conical intersection cone, has been implemented52 and is illustrated in detail in the next section. The MEP starting along these relaxation directions defines the ground state valleys, which determine the possible relaxation paths and ultimately the photoproducts that can be generated by decay. Although this information is structural (i.e., nondynamicd),it provides insight into the mechanism of photoproduct formation from vibrationally cold, excited state reactants such as those encountered in many experi-

Characterization of Conical Intersections 105 ments in which slow excited state motion or/and thermal equilibration is possible (in cool jets, in cold matrices, and in solution). When such structural or static information is not sufficient (i.e., the excited state may not decay at the minimum of the conical intersection line, or the momentum developed on the excited state branch of the reaction coordinate may be sufficient to drive the ground state reactive trajectory along paths that are far from the ground state valleys), a dynamics treatment of the excited state/ ground state motion is required.53.54 These techniques also are illustrated in the next subsection.

An Example: The S,/So Conical Intersection of Benzene We now give an example of the way in which the topological features just discussed occur in practical problems. In S, benzene there is a threshold of about 3000 cm-1 for the disappearance of S, fluorescence (see Refs. 25 and 54 and references cited therein). This observation is assigned to the opening of a very efficient, radiationless decay channel (termed “channel 3 ” ) leading to the production of fulvene and benzvalene via a prefulvene intermediate (see Figure 10a). Ab initio CASSCF12 (complete active space self-consistent field) calculations show25 that the general surface topology of the excited state energy surface (Figure lob) is consistent with that shown in Figure 5a (and illustrated in three dimensions in Figure 9). Thus the observed energy threshold, which is reproduced by multireference MP2 (Mdler-Plesset (second-order) computations,Ss corresponds to the energy barrier that separates S, benzene from an S, / So conical intersection point. A molecular species has only a transient existence in the region of a conical intersection. Thus, the molecular structure at such a point can be derived only from theoretical computations. The optimized conical intersection structure for S, benzene is shown in Figure 11. The structure contains a triangular arrangement of three carbon centers corresponding to a -(CH)3kink of the carbon skeleton. The electronic structure corresponds to three weakly interacting electrons in a triangular arrangement; these electrons are loosely coupled to an isolated radical that is delocalized on an ally1 fragment. This type of conical intersection structure appears to be a general feature in conjugated systems and has been documented in a series of polyene and polyene radicals.27 We will illustrate this point in the last subsection. The electronic origin of this feature can be understood by comparison with H3, where any equilateral triangle configuration corresponds to a point on the Do/D, conical intersection (this is an example of Jahn-Teller degeneracy) in which the three H electrons have identical pairwise interactions. We now discuss the -(CH)3kink electronic structure and coupling involved in photoproduct formation after decay back to the ground state. In the conical intersection surface topology, illustrated in Figure lob, there are two

106 A Computational Strategy for Organic Photochemisty

Figure 10 (a) Photochemical transformation of benzene to fulvene and benzvalene. (b) The change in spin coupling for a circuit around the apex of the CI in the plane xl,x2. Bold lines between atoms represent bonding interaction, whereas up and down arrows are used to designate +1/2 and - 1/2 electron spin, respectively.

geometrical distortions (Figure 12) that lift the degeneracy (xland x2). By examining the molecular structure of the conical intersection, its electronic distribution, and the directions indicated by the x1 and x2 vectors, we can derive information on the photochemical reaction path and on the possible coupling patterns involving the three unpaired electrons of the kink and leading to the final photoproducts. Three different pairings (1-6,5-6, and 1-5; see Figure 11 for numbering) of two of the three electrons of the kink are possible (see Figure lob). The gradient difference vector xl, which is almost parallel to the S, transition vector, predicts a relaxation toward the prefulvene diradical in the positive direction (1-5 pairing) and a planar ground state benzene in the negative direction (i.e., reversing the arrows in Figure 12).The other two couplings are described by the nonadiabatic coupling vector x2 (shown in Figure 12), which is the other geometrical coordinate shown in Figure lob. It describes the simultaneous double bond reconstruction that occurs upon relaxation. This direction corresponds to localizing the .n bonds in either the 1-6 or 5-6 posi-

Characterization of Conical Intersections 107

n

Figure 11 Computed S,/So conical intersection structure for benzene. The relevant geometrical parameters are in angstrom units. The -(CH)3- kink is framed. (From Ref. 25).

tions (1-6 and 5-6 pairings in the positive and negative direction of x2, respectively), Thus from Figure lob, a “circuit” of the conical intersection changes the

coupling in the triangular arrangement of the three carbon centers corresponding to the -(CH)3- kink of the carbon skeleton. This discussion is more general than it appears. As we will show in the next section (see Figure 15, below), for the conical intersection geometry that has been optimized on the S, surface for the problem of cyclohexadiene/ hexatriene photochemical interconversion, there are again three different pairings of two out of three electrons. The recoupling of the three electrons of the -(CH)3- kink appears to be a general feature in the decay and ground state relaxation valleys departing from conical intersections in polyenes.

Figure 12 Computed branching space vectors (gradient difference vector x1 and nonadiabatic coupling vector x2 ) for S, /So conical intersection of benzene.

108 A Computational Strategy for Organic Photochemistry

PRACTICAL COMPUTATION OF PHOTOCHEMICAL REACTION PATHS In this section, we describe the computational strategies and techniques needed to determine a photochemical reaction path from the Franck-Condon point (FC) to the ground state photoproduct (P). The techniques include methods for computing the excited state electronic energy, methods to determine the molecular structure of stationary points [intermediates (M”), transition states (TS)], low-lying crossings [either a conical intersection (CI) or a singlet-triplet crossing], and methods to construct excited state reaction paths and ground state relaxation paths in term of MEP (see Figure 9 for a model surface). We also give some discussion of dynamics and application of semiclassical trajectories.

Quantum Chemical Methods and Software for Excited State Energy and Gradient Computations The study of photochemical mechanisms presents a considerable challenge for computational chemistry. The objective is a complete description of the reaction path from the Franck-Condon region to the ground state product. Thus the details of the excited state reaction path and the region where a nonadiabatic event takes place at a surface crossing must be treated in an accurate and balanced way. One needs both analytical gradients and second derivatives at all points on the potential energy surface. Dynamic electron correlation (i.e., correction for the incorrect instantaneous repulsion of electrons in doubly occupied orbitals) is crucial for accurate energies. There are two fundamentally different approaches to the computation of excited states, which we shall refer to as wavefunction methods and response methods. The oldest approach, which is currently undergoing a revival, features response theories in which the excited state is computed as the first-order response of the ground state wavefunction. While the derivation of response methods is complex, and beyond the scope of this tutorial, at the simplest level, these response methods reduce to the configuration interaction singles (CIS)”6 approach. The CIS method can yield a good representation of the excited states that are dominated by single excitations from the ground state and can often provide a good starting point for computations. This method also forms the basis of most semiempirical approaches. However, methods such as CIS neglect the effects of dynamic electron correlation. At higher levels, response theories include the various equation-of-motion (EOM) approaches, which have seen much recent development in a coupled cluster implernentation.57J8 The most recent developments of response methods are referred to as time-dependent density functional theory (DFT).S9360 However, in both EOM and time-

Practical Computation of Photochemical Reaction Paths 109 dependent DFT, one ends up with an “effective Hamiltonian” in the space of a single excitation CI from the ground state (i.e., on the space of CI singles), where the effects of dynamic correlation are included. Response methods are limited to problems that can be described in terms of the “linear response” of the ground state. What this means in practice is that the excited state must have a nonzero projection on the space of single excitations. However, the excited states involved in photochemical processes often involve (after decay from the optical state) essentially doubly excited states and curve crossings with the ground state. Whereas response methods seem to be capable of reproducing the energy difference between ground and excited states (i.e., vertical excitation energies), they are completely untested for the type of problem addressed in this chapter. However, this problem is currently a field of intense research activity. Standard wavefunction methods (i.e., other than DFT), which have been extensively applied both to the computation of vertical (i.e., at ground state equilibrium geometry) excitation energies and excited state reaction paths are the current preferred method for applications in this field. Wavefunction methods that are used in studying photochemical mechanisms are limited to those that can describe excited states correctly. Unfortunately, standard methods for the evaluation of the ground state PES such as SCF and DFT cannot describe excited states because they are restricted to the aufbau principle. The ab initio CASSCF method12 is the main wavefunction method used for geometry optimization because it permits the gradient and second derivatives to be computed analytically,6l and this method has been implemented in widely distributed programs such as Gaussian 94,62 MOLCAS,63 COLUMBUS,64 and GAMESS.65 (See Ref. 66 for a quantum chemical software compendium). However, when accurate energetics are required, a treatment of dynamic correlation with the MP2 method has been implemented (CASSCF/MP2),67.68and this gives reliable results at low computational cost. Another method, which gives the results similar to those obtained with CASSCF/MP2, is the multireference configuration interaction method (MR-CI)@;however this method is at present limited to small systems, owing to its high computational cost. Analytical gradients and Hessians are available for CASSCF, and it is expected that this technology will be extended to the MR-CI and MP2 methods soon. Further, by virtue of the multireference approach, a balanced description of ground and excited states is achieved. Unfortunately, unlike “black boxes” such as first-order response methods (e.g., time-dependent DFT), CAS-based methods require considerable skill and experience to use effectively. In the last section of this chapter, we will present some case studies that serve to illustrate the main conceptual issues related to computation of excited state potential surfaces. The reader who is contemplating performing computations is urged to study some of the cited papers to appreciate the practical issues. There is also a large literature of excited state computations carried out using semiempirical methods with CI such as ZIND070 and MND071; these methods have been applied to photochemical problems by Klessinger and

1 10 A Computational Strategy for Organic Photochemisty

coworkers,72 Momicchioli and coworkers,73 and Orlandi and coworkers;74 however, gradient-based methods for surface crossings have not yet been implemented in these methods. In recent years, there has been considerable interest in modeling large molecular systems by combining quantum mechanics with force field methods such as molecular mechanics (MM).75 In general, such methods are based on SCF theory, which cannot be applied to excited states. Recently, we developed76,77 an approach based upon valence bond (VB) theory (which uses a parameterized Heisenberg Hamiltonian78.79 to represent the quantum mechanical part in a VB space) together with an MM force field. This approach yields a modeling method, called MM-VB (i.e., molecular mechanics valence bond),75,76 that reproduces the results of CASSCF computations for ground and excited states, yet is fast enough to allow dynamics simulations. A recent benchmark80 gives a good indication of the accuracy of such techniques in styrene photophysics. The ab initio CASSCF method (together with CASSCF/MP2) is currently the preferred choice for computing the PES of excited states. This method can treat up to 12 active electrons in routine application. When the size or the complexity of the molecular system under investigation does not allow the use of ab initio methods, information on the general structure of the potential energy surface can still be obtained by using hybrid methods such as MM-VB. Because this tutorial does not discuss general methods for excited state energy computation (vertical excitation energies, etc.), we concentrate, in the following, on the special problems associated with the computation of reaction paths that span one or more potential energy surfaces.

Conical Intersection Optimization The process of locating and computing the structure of an energy minimum or saddle point on the potential energy surface is usually referred to as geometry optimization. At both minima and saddle points, the gradient must be zero. Usually a geometry optimizer starts its search at a suitable initial molecular structure. It evaluates the energy and gradient (the gradient corresponds to the direction in which the energy decreases most rapidly: the steepest descent direction) and determines the direction and magnitude of the next structure that should be closer to the target point. The Hessian is also used to specify the curvature of the surface and to provide a quadratic representation of the PES in the vicinity of the point. In this subsection, we discuss the special problem of optimizing the surface crossing and finding relaxation paths from it. As reported in the introduction, we have indicated that in some situations there is no transition state connecting an excited state intermediate (M")to the conical intersection point (sloped conical intersections, see Figures 4c and 8b). In such situations, mechanistic information associated with surface crossings must be obtained by locating the lowest lying intersection point along the n - 2 intersection space of the molecule.

Practical Computation of Photochemical Reaction Paths 1 1 1 The practical computation of the molecular structure of a conical intersection energy minimum can be illustrated by making an analogy with the optimization of a transition structure (see Figure 6 ) . A transition structure is the highest energy point along the path (MEPI) joining reactants to products and the lowest energy point along all the other n - 1directions orthogonal to it. One can optimize such a structure by minimizing the energy in n - 1 orthogonal directions and maximizing the energy in the remaining direction corresponding to the reaction path (i.e., the transition vector x1 in Figure 6a). The technique for locating the lowest energy intersection point47J1 exploits the fact that the branching space directions x1 and x2 play a role analogous to the reaction path at the transition state. Accordingly, the lowest energy point on a conical intersection is obtained by minimizing the energy in the ( n- 2) dimensional intersection space (x3, x4, . . . , XJ, which preserves the degeneracy (see Figure 5b) and minimizes the energy gap in the branching space (xl and x2).In practice, to properly locate these low energy stationary points at which two potential energy surfaces have the same energy, one must carry out constrained geometry optimizations that optimize the geometry in directions orthogonal to the two directions x1 and x2. It is important to appreciate that the gradient on the excited state PES will not be zero at an optimized conical intersection point (see Figure 7a,b), because it looks like the vertex of an inverted cone (see Figures 5 and 6b).Rather, it is the projection of the gradient of the excited state PES onto the orthogonal complement of x1 and x2 [i.e., the (n - 2)-dimensional hyperline] that goes to zero when the geometry of the conical intersection is optimized. This situation is distinguished from an “avoided crossing minimum” or Renner-Teller degeneracy of two surfaces (see Figure 7d; indeed this is a rather than an intersection point); both the “avoided crossing minimum” and Renner-Teller degeneracy of two surfaces correspond to real minima at which the gradient on the excited state PES would go to zero. Thus, in an optimized conical intersection point, both conditions in Eq. [16] must be fulfilled:

E2-E,=0

The practical algorithm,47.51 which was first implemented in the Gaussian 94 program package,62 can be described as follows. For minimization of E , - El in the x1 and x2 plane, we have:

a aQ

-(E2 - El)2 = -2(E2 - E,)xl

where x1 is the gradient difference vector (see Eq. [ll]). The length of x1 has no significance-only its direction. lxll will be large if the potential energy surfaces have opposite slope but very small if they have nearly the same slope. This

-

1 1 2 A Computational Strategy for Otganic Photochemistry means that the size of the step should depend only on E , - E, and suggests that we should take the gradient along the step to the minimum of E, - El to be

f = -21E,

-

El\-

X1

Clearly f will go to zero when E , = E l , independently of the magnitude of x l . Note, however, that the gradient will also go to zero if El is different from E, but the two surfaces are parallel (i.e., xl,the gradient difference vector, has zero length). In this case the method would fail. This situation will occur for a Renner-Teller-like degeneracy, for example. Of course, in this case, the geometry can be found by normal unconstrained geometry optimization. If we now define the projection of the gradient of E , onto the n - 2 orthogonal complement to the plane xl,x2 as g = P - aE2

aQ

where P is the projection operator, then the gradient to be used in the optimization becomes g=g+f

[201

The general behavior of such an optimization procedure is illustrated in Figure 13, where we report two PES intersecting along the x1 and “intersection space” coordinates (x3,. . ., xn). Although this example is not realistic because we started very far from the optimized structure, notice that starting from this initial structure (initial point), the method first finds the closest intersection point. After that, it moves downhill along the intersection space.

Locating Decay Paths from a Conical Intersection Since the minima and saddle point are well defined points on the PES, it is possible to define a unique reaction path. The minimum energy pathway1 can be defined as the path traced by a classical particle sliding with infinitesimal velocity from a saddle point down to each of the minima. The MEP [which can be computed by using the intrinsic reaction coordinate (IRC) method*l] is a geometrical or mathematical feature of the PES, like minima, maxima, and saddle points. Since molecules have more than infinitesimal kinetic energy, a classical trajectory will not follow the MEP and may in fact deviate quite widely from it, as in the case of a “hot” system (i.e., one with an high excess vibrational energy), which would require a dynamic treatment of the motion on the PES.

Practical Computation of Photochemical Reaction Paths 1 13

Iterations Figure 13 General behavior of a conical intersection optimization procedure. This contrived example was started from an almost planar geometry (much further from the optimum geometry than normal practice). The curve shows the rapid approach to the degenerate situation followed by minimization (retaining the degeneracy).

However, the MEP may be a convenient measure of the progress of a molecule in a reaction, because in general a molecule will move, on average, along the MEP in a well-defined valley, and it is a good approximation of the motion of vibrationally cold systems (e.g., for photochemical reactions in which the excited state reactant has a smallkontrolled amount of vibrational excess energy). We have shown that an accessible conical intersection forms a bottleneck that separates the excited state branch of a nonadiabatic photochemical reaction path from the ground state branch, thus connecting the excited state reactant to two or more products on the ground state surface via a branching of the

1 1 4 A Computational Strategy for Organic Photochemistry excited reaction path into several ground state relaxation channels (see Figure 6b). The nature of the products generated following decay a t a surface crossing will depend on the ground state valleys (reaction paths) that can be accessed from that particular structure. We implemented a gradient-driven algorithm to locate and characterize all the relaxation directions departing from a single conical intersection point.52 The MEP starting along these relaxation directions defines the ground state valleys, which determine the possible relaxation processes and the photoproducts. This information is structural (and thus excludes dynamical effects such as lifetimes and quantum yields) and provides insight into the mechanism of photoproduct formation from vibrationally cold excited state reactants such as those encountered in many experiments in which slow excited state motion or/and thermal equilibration is possible (in cool jets, in cold matrices, and in solution).Under these conditions of low vibrational excess energy, semiclassical dynamics yields the same mechanistic information as derived from topological investigation of the PES; this is because the surface structure is expected to play the dominant role in determining the initial molecular motion in the decay region. The MEP connecting the reactant to the product of a thermal reaction is uniquely defined by the associated transition structure. The direction of the transition vector (i.e., the normal coordinate corresponding to the imaginary frequency of the TS) is used to start an MEP computation. One takes a small step along this vector x1 (shown in Figure 6a) to points R or P and then follows the steepest descent paths connecting this point to the product or reactant well. The small step vector defines the initial relaxation direction (IRD) toward the product or reactant. It is obvious from Figure 6b that this procedure cannot be used to find the IRD for a photochemical reaction since, as discussed above, the downhill direction lies in the plane xl,x2,The general situation is illustrated in Figure 14a. In the linear approximation, since the cone is elliptic (see discussion in the preceding section) two steep sides (see Figure 14b) exist in the immediate vicinity of the apex of the cone. As one moves away from the apex along these steep directions, real reaction valleys (as in Figure 14a rather than approximate ones) develop, leading to final photoproduct minima. Thus in reality the firstorder approximation will break down at larger distances, and there will be more complicated cross sections and more than two relaxation channels. Also there are symmetric cases (such as H 3 ) in which the tip of the cone can never possibly be described by Eq. [8] because one has three equivalent relaxation channels from the very beginning of the tip of the cone. In Figure 14b we show the potential energy surface for a “model” elliptic conical intersection46 plotted in the branching plane (xl,x2).Because, as stated earlier, the cone is elliptic in the linear approximation (i.e., the base of the cone is an ellipse rather than a circle), there are two “steep” sides of the ground state cone surface and two “ridges”. There are two preferred directions for downhill motion located on the steep sides of the ground state cone surface. A simple

Practical Computation of Photochemical Reaction Paths 115

Figure 14 Illustration of the general procedure used to locate the initial relaxation direction (IRD) toward the possible decay products. (a) General photochemical relaxation path leading (via conical intersection decay) to three different final structures. (b) Potential energy surface for a “model” elliptic conical intersection plotted in the branching plane. (c) Corresponding energy profile (as a function of the angle a)along a circular cross section centered on the conical intersection point and with radius d.

procedure for defining these directions involves the computation of the energy profile along a circular cross section of the branching plane centered on the vertex of the cone (0,O) as illustrated in Figure 14b, c. This energy profile is given in Figure 14c as a function of the angle a and for a suitable choice of the radius d. It can be seen that the profile contains two different energy minima. These minima (M, and M, in Figure 14b) uniquely define the IRD from the vertex of the cone. The two steepest descent lines starting at M, and M, define two MEPs describing the relaxation processes in the same way in which the transition vector x1 (see Figure 6a) defines the MEP connecting reactant to products. Notice that there are also two energy maxima TS,, and TS,, in Figure 14b, c. These maxima can be interpreted as the “transition structures” connecting M, and M, along the chosen circular cross section. It can be seen in Figure 14b, c that these transition structures locate the energy ridges that separate the IRD “valleys” located by M, and M, Thus, while there is no analog of the transition vector for a conical intersection, the simple case of an elliptic cone shows that the IRDs are still uniquely defined in terms of MI and M, .Whereas the IRD from a TS connects the reactant to the product, there are two IRDs

.

1 16 A C,‘omputationalStrategy for Organic Photochemistry -

-

from an elliptic conical intersection leading to two different photoproduct Valleys (and one of these photoproducts may correspond to the original reactant). For the elliptic (i.e., first-order46) cone model, discussed above, there can be at most two minima (M, and M2) defining two distinct IRDs (excluding the case of the cone that becomes circular, when there are an infinite number of equivalent directions of relaxation). These minima are located on the branching plane (xl,x,). However, this model of the potential energy surfaces at a conical intersection point is not general enough to give a correct description of all relaxation paths for a real system. First, there may be more than two possible IRDs originating from the same conical intersection. Second, some IRDs may lie out of the branching plane because the real (xl,x2)space is, in general, curved. However, the ideas introduced above can be easily extended to search for IRDs in the full n-dimensional space surrounding a conical intersection point by replacing the circular cross section with a (hyper)spherical cross section centered at the vertex of the cone. Locating stationary points on the hypersphere involves constrained geometry optimization, in mass-weighted coordinates, with a “frozen” variable d. Although all results (IRD vectors and MEP coordinates) are generally given in mass-weighted Cartesian coordinates, the actual computations are carried out using mass-weighted internal coordinates.81 We have presented the full mathematical details elsewhere.52 We must emphasize that the procedure outlined above is designed to locate the points at which the relaxation paths begin (i.e., these points define the IRD). Once the points have been found for some small value of d, one must compute the associated MEP that defines the relaxation paths leading to a ground state energy minimum (as stated before). Thus the approach outlined above provides a systematic way to find the MEP connecting the vertex of the cone to the various ground state photoproduct wells. Since more than one MEP originates from the same conical intersection point, this procedure also describes the branching of the excited state reaction path occurring at the intersection point. A simple example serves to illustrate the foregoing approach. In Figure 15 we show the conical intersection geometry that has been optimized on the S, surface for the problem of cyclohexadiene (CHD) / cZc-hexatriene (cZc-HT) photochemical interconversion.82 This species has a characteristic -( CH)3kink (this geometric triangular arrangement is typical of polyene conical intersections,27 as we will see in the next section), and three different ground state recoupling paths are possible, as shown in Figure 16a. Thus one expects to find three ground state reaction pathways (leading to R, PI,and P,, as illustrated schematically in Figure 16b) corresponding to the three different recoupling patterns involving the triangular kink (see the analogy with the three recoupling patterns in benzene discussed in the preceding section and shown in Figure 10). The real (computed) surface topology is illustrated in Figure 17. Close to the apex of the cone, there are only the two reaction valleys directed toward R and

Practical Comiiutation of Photochemical Reaction Paths 11 7 Ring Closure

Ring Opening

Figure 15 Computed S,/S, conical intersection structure for the problem of cyclohexadiene/hexatrienephotochemical interconversion. The relevant geometrical parameters are in angstrom units.

Figure 16 (a) The three different electron recoupling patterns from the conical intersection shown in Figure 15. (b) The branching of the photochemical reaction path through a conical intersection.

1 18 A Computational Strategy for Organic Photochemistvy

Figure 17 Computed relaxation paths from the conical intersection of Figure 15. Although two similar valleys develop close to the crossing point, the third one (initially a ridge) starts far and is energetically unfavored.

PI, whereas the direction leading to P, (the more unstable diradical intermediate) is a ridge. The valley to Pz develops only at a larger distance from the apex of the cone (see Figure 17). Thus calculations show36>82and experiments confirm4JJ33-87 that there are two almost equivalent relaxation paths, which will be populated after decay from the conical intersection: one leading to cyclohexadiene (R) and the other to hexatriene (PI), with very similar quantum yields (i.e., product ratio). We shall return to discuss this problem in a little more detail in the next section.

Semiclassical Trajectories The techniques outlined above provide information on the structure and accessibility of the photochemical reaction paths. As mentioned, this inforrnation is structural (i.e., nondynamical) and provides insight into the mechanism of photoproduct formation from vibrationally “cold” excited state reactants such as those encountered in many experiments where slow excited state motion or/and thermal equilibration is possible (in cool jets, in cold matrices, and in solution).

Practical Computation of Photochemical Reaction Paths 119 In many cases, such structural or static information is not sufficient. The excited state may not decay at the point where the excited state path (MEP) intersects the n - 2 hyperline. Alternatively, the momentum developed on the excited state branch of the reaction coordinate may be sufficient to drive the ground state reactive trajectory along paths that are far from the ground state valleys. In such cases, a dynamics treatment of the excited state/ground state motion is required for mechanistic investigations. Furthermore a dynamics treatment is required to gain information of the time scales and quantum yields of the reaction. In principle, the current implementation of Car-Parrinello,ss which is based on a local DFT method, provides a method for ab initio molecular dynamics. This method uses semi-classical equations of motion to propagate the nuclei classically in concert with wavefunction propagation. But since this method is based on DFT, it cannot describe an excited state PES. Further, the wavefunction is propagated by means of using classical dynamics and is thus inapplicable to curve-crossing problems. For small systems, a parameterized potential can be developed and full quantum dynamical treatment is possible.”3 In our own work,S4 we have used classical dynamics with a hybrid quantum mechanicallforce field (MM-VB).75,76 This method employs a “direct” procedure for solving the equations of motion (i.e., the gradient that drives the dynamics is evaluated “on the fly”), and thus one avoids the tedious, and often unfeasible, parameterization of an analytical expression of a multidimensional energy surface.53 The trajectory-surface-hopping algorithm of Tully and Presis used to propagate excited state trajectories onto the ground state in the region of a conical intersection. We now briefly describe the way in which the nonadiabatic event (surface hop) can be described in “on the fly” dynamics methods. We can represent the time-dependent wavefunction in the CI space as a vector:

where C, is a complex coefficient giving the contribution of state K. If the time step is sufficiently small, the solution of the time-dependent Schrodinger equation can be propagated in concert with the nuclei in the dynamics simulation as a ( t )= exp[iH(t)t]a ( 0 )

1221

where H is the matrix representation of the Hamiltonian in the CI basis. The projection (( a(t)lYk))of the a ( t )on the adiabatic basis states (i.e., Y k , eigenvectors of H ) gives the “occupancy” of these states, as illustrated in Figure 18. In the surface hop method, the gradient is always computed from an adiabatic

120 A CompHtationaE Strategy for Organic Photochemistry

I

so

Figure 18 ‘cOccupancy’’of the states (i.e., projection on the adiabatic basis states of a(t) given in Eq. [22])for a molecular system wavefunction on the S, excited (Yl ) or So ground (Yo)states.

basis state (i.e., an eigenvector of H ) and the projection of the a ( t ) on the adiabatic basis states is used to decide which adiabatic basis state is used for the gradient. The reader is referred to Refs. 26 and 92 for examples of the application of the surface hop method using quantum chemistry methods. Surface hop approaches are not very well defined for the case of the trajectory that recrosses the region of strong coupling many times.93,94 This problem occurs in the case of a sloped intersection (see Figure 4c or 8b). One possible solution is the “multiple spawning” method of Levine and Martinez,95796 where trajectories are spawned each time the trajectory passes through the region of strong interaction and a frozen Gaussian (representing a wavepacket) is associated with each spawned trajectory so that the population transfer can be computed quantum mechanically. In 1998 we implemented97 a third approach that uses mixed state dynamics (first used with a diatomics-inmolecules model by Gadea and coworkers98). Here the nuclear dynamics are controlled by the Ehrenfest force, that is, the gradient is computed directly from the a ( t ) of Eq. [22]. Thus the trajectory “feels” both potential surfaces and the nonadiabatic couplings all the time. The most important aspect of mixed state approaches is the integration of the time-dependent Schrodinger equation for the electronic wavefunction in concert with nuclear propagation, so that the method is closer to exact quantum methods than the surface hop approach. This method has been used successfully for investigation of the excited state lifetimes and decay process;97 however, the accurate determination of quantum yields remains an outstanding problem. The problem of combining dynamics with quantum chemistry is at the forefront of current research. The best method to be used will emerge as a compromise between quantum mechanical rigor and computational feasibility.99

Mechanistic Organic Photochemistry: Some Case Studies 12 1

MECHANISTIC ORGANIC PHOTOCHEMISTRY: SOME CASE STUDIES As reported in the introductory section, "real" crossings (such as conical intersections and singlet-triplet crossings) are now known to occur with a previously unsuspected frequency in organic systems.13.14 Thus decay at real crossings provides a common mechanism for excited state radiationless transitions and for the generation of photoproducts. Although there are, in principle, at least three different ways of generating a photoproduct-namely, via an adiabatic reaction, via decay at an avoided crossing, and via decay at a real crossing (see Figure 1)-here we deal exclusively with the third process. Thus, the investigation of the mechanism of a photochemical reaction requires, as a primary step, the investigation of the structure and energetics of the low-lying real crossings for the system under investigation and the study of the ground state reaction paths, which originate at this decay funnel. The methodology described in the section on computational methods can be used. Here we provide some background (by way of examples) for initiating theoretical work on photochemical problems. Thus, within the theoretical framework described in the preceding sections, we hope that the case studies we now discuss give the reader some insight into the types of problem that may be encountered in some very different photochemical systems. Recent computational results do indicate some general features of surface crossings and of the related photochemical reaction paths. We attempt to classify these features from a conceptual and chemical point of view. These case studies will also serve to indicate that present theoretical methods and computational technology can go a long way toward completing our understanding of photochemical reaction mechanisms. Through the analysis of some selected case studies, we hope to establish the following ideas, which appear to be general: 1. Photochemistry of conjugated hydrocarbons can be rationalized by the cornmon electronic and molecular structure of the surface crossing between a covalent excited state and the ground state. 2. Photochemistry of azoalkanes and enones is dominated by intersections of ~l-n"and 7c-n" states with the ground state as well as by the intervention of the triplet manifold. 3 . Photochemistry of protonated Schiff bases is also based on conical intersections; however, the excited state is ionic and corresponds to an intramolecular charge transfer state; thus the theoretical aspects of the problem are distinct from polyenes. 4. The quantum yield in the radiationless decay and competitive photoproduct formation process in cyclohexadienelcZc-hexatriene(1)system is controlled

122 A Computational Strategy for Organic Photochemistry by competitive ground state relaxation paths, which originate from a single conical intersection channel. 5. The all-trans-hepta-2,4,6-trieniminiumcation (2), a retinal protonated Schiff base model, may undergo trans + cis isomerization of the double bond at either position 2 or 4.Thus, the photochemistry is dominated by the structure of the competitive excited state reaction paths leading to distinct conical intersection structures.

1

2

Three-Electron Conical Intersections of Conjugated Hydrocarbons The application of different spectroscopic techniques to low temperature samples of “isolated” conjugated molecules100-102 has begun to provide very detailed information on the excited state dynamics of these organic systems and indicates that there is a small threshold energy to ultrafast radiationless decay. Use of the ab initio methods described earlier suggests that the general mechanism that “triggers” the decay is a displacement of the electronically excited equilibrium structure toward a “critical” configuration in which the excited and ground states cross at a conical intersection? The molecular geometry at the point of decay shows, in many examples, a “kink” located at a -(CH)3- segment. In Figure 19 we compare the structure of this -( CH)3- segment for all-trans-octatetraene (all-trans-OT), S, benzene, and S, cyclohexadiene intersections. Comparison of these structures reveals common structural and electronic features. Each structure contains a triangular arrangement of three carbon centers corresponding to a -( CH),kink of the carbon skeleton in all-trans-OT27~28and benzene25 and to a triangular arrangement of the -CH, and -CH-CH2 terminal fragments in cyclohexadiene.36 As mentioned previously, the electronic structure in each case corresponds to three weakly interacting electrons in a triangular arrangement (like H3)103 which are loosely coupled to an isolated radical center (this is delocalized on an ally1 fragment in all-trans-OT, benzene, and cyclohexadiene). This type of conical intersection appears to be a general feature of linear polyenes and polyene radicals. These intersections are located at the end of the excited state reaction path, that is, the MEP on the S, potential energy surface, which has the general structure illustrated in Figure 4a (for short polyenes, such as butadiene, it has the general structure illustrated in Figure 4b). The structure and energetics of the excited state decay path for six ( n = 3, . . . ,8) all-trans conjugated hydrocarbons, three polyenes (butadiene, hexatriene, and octa-

Mechanistic Owanic Photochemistry: Some Case Studies 123

r\

Octatetraene

and Short Polyenes

Hexatrime/

Cyclohexadiene

Figure 19 Structures of S, /So conical intersections in conjugated hydrocarbons showing the -(CH)3- kink structure in all-trans-octatetraene,benzene, and cyclohexadiene. Also the typical triangular D, /Do conical intersection for H, is

illustrated.

tetraene), and three polyene radicals (allyl, pentadienyl, and heptatrienyl) have been documented by means of the ab initio CAS-SCF method with the DZ+d basis set and an active space including all n and n* orbitals and electrons.27

Conical Intersections of n--7t" Excited States In other classes of organic molecules, the electronic structure of the lowest lying intersection changes. We have published detailed results on photorearrangements of the olefin-carbonyl Paterno-Buchi systern,30 a,@-enones,31 P,~-enones,32azocornpounds (diazomethane33 and cyclic diazoalkenesl04), and acylcyclopropenes.34 Whereas hydrocarbon photochemistry typically involves a low energy 7c-n" doubly excited covalent state, the singlet photochemistry of carbonyl and azo compounds is dominated by wto-7c" excitations.

3

As an example of this class of system, we discuss some results on the photochemistry of 2,3-diazabicyclo[2.2.l]hept-2-ene (DBH) (3). The photochemistry of this system is remarkably complex because it involves a study of the

124 A Computational Strategy for Otganic Photochemistry

ground state So surface as well as l(n-ni‘),3(n-n*), and 3(n-n“) surfaces and their crossings. All the topological features suggested in Figure 6 actually occur in practice, and a complete understanding of the photochemistry of this species involves a study of the evolution of singlet and triplet photoexcited DBH along a network of 18 ground and excited state intermediates, 17 transition structures, and 10 “funnels,” where internal conversion (IC)or intersystem crossing (ISC) occurs.104 Whereas the mechanistic photochemist specifies a mechanism in terms of a sequence of molecular structures that occur along various reaction paths, one can use theoretical computations as a complementary and powerful tool to investigate reaction mechanism. One can use theoretical computation to actually find a viable mechanism, determine the nature of the structures involved (intermediates, transition states, surface crossings, etc.), and discover whether they lie on the excited or ground state branch of the reaction path and whether they are on the triplet manifold (accessible via triplet sensitization or intersystem crossing) or lie on the singlet manifold (via direct irradiation). This information is essential to generate some understanding, also allowing us to generalize the results in a qualitative way. Thus, our purpose in discussing DBH is to illustrate how these goals can be achieved in practice. The reaction pathways for the photochemistry of DBH (3) are illustrated schematically in Figure 20 as a sequence of structures along a network of reaction paths. DBH and its derivatives denitrogenate photochemically (and thermally) through an a C-N cleavage to yield cyclopentanediyl 1,3biradicals (7and 8). These biradicals usually cyclize to housanes, or they may undergo rearrangement to form cyclopentenes via 1,2 hydrogen shift. Photochemical transformations of DBH derivatives other than denitrogenation have been observed in a few cases. When certain prerequisites (e.g., increased ring stiffness and strain effects) are met, p C-C cleavage occurs. Increased ring stiffness due to the additional etheno bridge and simultaneous allylic stabilization of the resulting biradical results in the concurrent formation of azirane (9) and the usual housane. The companion schematic representation of the diazenyl region of the computed potential surface is shown in Figure 21, where the intermediates are labeled in the same way as in Figure 20. The accuracy of the methodology used in this study has been tested against other azoalkane spectroscopic data.105 For instance, the computed 0-0 singlet excitation energies are 84.0 kcal/mol [82.7 including the zero-point energy (ZPE)correction] for pyrazoline (experimental value 82 kcaYmol) and 73.7 kcal/mol for 2,3-diazabicyclo[2.2.2]oct-2-ene (DBO) (12)(experimental value 76 kcal/mol). The energy barriers for a C-N bond cleavage of singlet-excited pyrazoline and DBO are 7.6 (6.4including ZPE correction) and 11.4 kcal/mol, respectfully, which are comparable to experimental values of 6-9 k/cal/mol for a pyrazoline derivative and 8.6-10.2 kcal/ mol for DBO derivatives.

.

exo-endo

triplet diazenyl biradcals

K

d

cydopentene

D30

6

Jr "[I

biradical

triplet hydrazonyl

a

.

-

tnplet

singlet

endo

N*

10

* ~ 2

, Q

housane

cyclopentene

Figure 20 Sequence of structures along the reaction paths network computed in the photochemistry of 2,3diazabicyclo[2.2.l]hept-2-ene (DBH) species (3). (Adapted from Ref. 104.)

housane

biradcal

singlet hydrazonyl

exo-en+

226 A Computational Strategy for Orgunic Photochemistry

Figure 21 Companion schematic representation for the computed diazenyl region potential surface (the intermediates are labeled in the same way as Figure 20). (Adapted from Ref. 104.)

12

Three cyclic excited state species are reached following evolution from the Franck-Condon region: two metastable singlet (n-no) and triplet (n-n") species (4 and 6) and a stable excited state 3(n-n")-3(n-n*) intermediate (5). Structure 4 can decay directly to So (via the conical intersection labeled 10,ll) or undergo ISC to generate 5 andlor 6 . Structure 6 can decay directly to the T, diazenyl biradical or undergo IC to generate the mixed 3(n-n*)-3(n-n*) intermediate 5. Finally, the much more stable species 5 cannot be converted to the other excited state intermediates but can only react via either a C-N or p CC cleavage (to generate 11 or 9, respectively).

Mechanistic Organic Photochemistry: Some Case Studies 127 It is clear that the photochemistry of azo compounds is complex because of the delicate intertwining of the 3(n-x"), 3(n-n"), and l(n-n') states. The surface crossing region of the diazenyl biradicals is quite novel in that a fourfold crossing occurs: that is, the *(n-n")/S, crossing occurs at the same geometry as the 3(n-x")P(n-n") crossing (see structures 10,ll in Figure 21). The bent forms or in a C-N-N linear form. diazenyl biradical exists in C-N-N The bent forms correspond to real biradical intermediates with three possible conformations (exo, endo 11,or exo-endo 10; see Figure 20) and are generated by decay at the conical intersection corresponding to the linear form (see Figure 21). Indeed the linear form corresponding to structure 1 0 , l l corresponds to a highly unstable configuration, where the So, 3(n-n"), 3(n-n"), and l(n-7c") states are degenerate and form a multiple funnel. This funnel is entered via a cleavage by overcoming the S, or T, 1/3(n-n") transition structures shown in Figure 21. In Figure 22 we show the molecular and electronic structure of DBH at the conical intersection 1O,11. The origin of the four-fold crossing can easily be

Figure 22 DBH ground state equilibrium structure 3 and molecular and electronic structure for the computed low-lying real crossing 10,11. In this system the S, (n-n"), So, T2(n-n")/T1(n-n*)conical intersections and the T,(n-n")/So and T2(n-n")/So triplet/singlet crossings occur at the same molecular structure. The relevant geometrical parameters are in angstrom units.

128 A Computational Strategy for Organic Photochemistry rationalized from the character of the two unpaired electrons in these structures (see dashed line in Figure 22). These two electrons can be considered to be almost uncoupked from each other, and since the coupling between the two radical centers is so small, the triplet and singlet states must be degenerate. Furthermore, one nitrogen atom is left with a singly occupied p orbital and a lone pair located along orthogonal axes in space. The l(n-n*) and 3(n-n*) states can be derived from the So and 3(n-n“) states by swapping the relative occupancies of the singly occupied p orbital and lone pair. However, this difference will not affect the energy, and therefore all four states [l(n-n”), So, 3(n-n*), and 3(n-n“)] will be degenerate. This behavior is consistent with the directions defined by the gradient difference and derivative coupling vectors at the multiple funnel (corresponding to “superimposed” S,/S, and TI /T2conical intersections); the vectors indicate the type of molecular distortion required to lift the degeneracy. These vectors correspond to two “orthogonal” bendings of the N-N-C angle, which would split the degeneracy by increasing the coupling between the two radical centers. The decay channel for the triplet-sensitized photolysis also has an unusual surface topology, which is indicated in Figure 21. In the Franck-Condon region, T, has (n-n”)character and lies about 34 kcaUmol above TI, which has (n-n,) character. As the structure of the molecule gets distorted from the Franck-Condon geometry, the 3(n-n*) and 3(n-n”) states become strongly mixed so that the two triplet states can be distinguished only with difficulty, especially at structures where the C-N=N-C bridge is twisted. In other words, the electronic structures of the TI and T2 states become a combination of (n-n”) and (n-n”) configurations. Near the triplet (n-n”) energy minimum (6)there is a conical intersection between the (n-n”)and (n-n”) triplet states. The electronic structure of the TI and T2 states in the vicinity of the intersection can be understood on the basis of the derivative coupling and gradient difference vectors. The derivative coupling vector involves an N-N-C bending, and the gradient difference vector involves twisting of the C-N=N-C bridge (see Figure 21). The N-N-C bending distortion causes the “pure” (n-n”) and (n-n*)states to cross. Thus on one side of the upper (T,) cone is the (n-n*) state and on the other side is the (n-n”)structure (this structure corresponds to a steep T2/TT,intersection point). No stable (n-n”)minimum has been located on T,. The orthogonal C-N=N-C distortion has the effect of producing “mixed” (n-n*) and (n-n”)states, that is, states that can be represented by a mixture of (n-n”) and (n-n“)character. Thus decaying from the tip of the conical intersection in the direction of the gradient difference vector generates the mixed (n-n”)-(n-n*) minimum (5) on the TI state. (In 5 , the n and n orbitals are strongly mixed so that the location of the two radical centers is ambiguous). These effects are typical of the presence of a conical intersection: as one moves in a circle centered on the conical intersection point, the wavefunction changes in a continuous fashion from (n-n“) to (n-n”) + (n-n“)to (n-d) to (n-n*)- (n-n*) to -(n-n*).

Mechanistic Ovganic Photochemisty: Some Case Studies 129 The preceding discussion reveals a few of the complexities of the photochemistry that is typical of carbonyl and azo compounds. Results on the olefin-carbonyl Paterno-Buchi system,30 and on the photorearrangements of a, p-enones,31 p, yenones,32 azo compounds (diazomethane33 and cyclic diazoalkenesl04), and acylcyclopropenes34 show similar features. In these examples, one encounters fourfold intersections as well as conical intersections and singlet-triplet crossings. Thus the potential surfaces are more complex than in hydrocarbon photochemistry.

The S,/S, Conical Intersection of Protonated Schiff Bases The photoisomerization of protonated Schiff bases (protonated imines) occurs along a fully barrierless reaction path37-40 (see model surface in Figure 4b). Because of the different electronic nature of the S, state, however, the molecular structure and electronic distribution of the lowest lying conical intersection is completely different from that of polyene hydrocarbons. The s, state of cis-C,H,NHt, a short protonated Schiff base analog of tZt-hexa-1,3,5triene (13),is ionic 1B,-like,37 whereas the S, energy surface of the corresponding polyenel06 is the covalent 2A, state. In what follows, we will see that relaxation in this ionic state leads to an intramolecular charge transfer state.

The evolution of the cis-C,H,NH; isomer along the interstate MEP connecting the FC structure of the cis isomer to the So trans and cis product wells is illustrated in Figure 23. Along this path, the energy difference between the S, and S, (covalent 2A,-like) states is large (>25kcal/mol), and thus it appears that covalent S2 is not involved in the reaction. The S, relaxation path ends a t a point where the S, (1B,-like) and So potential energy surfaces cross at a conical intersection. The intersection point has a central double bond twisted by about 80", which provides a route for fully efficient nonadiabatic cis-trans isomerization. Starting from this point, we have located, via computation, two So relaxation paths. The first path is a continuation of the excited state path and ends at the all-trans-C,H,NH$ energy well. The second path describes the backformation of the reactant. Whereas both the doubly excited 7c-7c." state of polyenes and singly excited n-n" state of azoalkanes and carbonyl compounds are diradical and do not involve charge transfer from one region of the molecule to another, the singly excited n-nah state of protonated Schiff bases does. Figure 24 shows the evolution of the Mulliken charges (with hydrogens summed into neighboring heavy atoms) along the reaction coordinate. The cis-S,+S, FC excitation

130 A Computational Strategy for Organic Photochemistry

MEP co-ordinate

(a.u.)

Figure 23 Energy profiles along the three minimum energy paths (MEPs) describing the relaxation from the Franck-Condon (FC) and conical intersection (CI)points Curves with open squares and solid squares define the excited (1B,-like) and ground state branches of the cis + trans photoisomerization path. Solid triangles define the ground state cis back-formation path. Open circles show the dark (2A,-like) state energy along the excited state branch of the photoisomerization path. The structures (geometerical parameters in angstrom units and degrees) document the geometrical progression along the photoisornerization path. (From Ref. 37.)

results in a partial single electron transfer toward the NH, end of the molecule, which is consistent with the charge transfer associated with the HOMO-LUMO singly excited 1B,-like nature of S, .*079108 Accordingly, the positive charge migrates toward the -CH, molecular end. The most striking feature is the large, but regular, increase (from ca. 0.0 to +0.39) of the charge a t the y carbon center along the S, path. This center is adjacent to the rotating bond, a n d a stabilization of its positive charge must have a n important effect o n the stability

Mechanistic Organic Photochemistry: Some Case Studies 13 1

Figure 24 Evolution of the Mulliken charge distribution along the excited (1BJike) and ground state branches of the cis -+ trans photoisomerization MEP (see Figure 23) connecting cis-C,H,NH; (FC) to traas-C,H,NH;. The charges are given in atomic units, and the value of the central torsional angle is given in degrees. (Adapted from Ref. 37.)

132 A Computational Strategy for Organic Photochemistry of the twisted configuration. Along the isomerization coordinate, the excited

state charge distribution is smoothly changed, and this change continues into the ground state branch of the reaction (i.e., after decay), where the positive charge is shifted back toward the -NH, molecular end. As shown in Figure 24, in the vicinity of the conical intersection point about 70% of the positive charge is localized on the allyl fragment due to depopulation of its singly occupied molecular orbital (SOMO). Note that in the cis-C,H,NH; isomer, the increase in polarization along the computed MEP is gradual and a polarization corresponding to the migration of roughly 0.5 electron toward the -CHCH=NH; is already present in the untwisted FC region. This situation is described by the four resonance formulas reported in Figure 25. The existence of a conical intersection point and the motion of charge observed along the computed isomerization coordinate can be rationalized by means of the “two-electron, two-orbital model” of Michl, Bonacic-Koutecky, et al.103109 According to this theory, the twisted conical intersection structure corresponds to a “critically heterosymmetric biradicaloid. ” A heterosymmetric biradicaloid is a structure in which two localized orbitals have different energies but do not interact. This is the situation found in the conical intersection structure presented above, where the SOMO 7c orbital of the allyl fragment and the SOMO 7c orbital of the --CHCHNH; fragment are not overlapping. In this condition, the energy separation of the ionic S, and covalent So states depends on the difference between the ionization potential of the allyl SOMO and the electron affinity of the -CHCHNHi SOMO. These quantities can be changed as a function of the fragment structure. Thus along the last part of the S, reaction coordinate, the geometry of the two fragments is such that these

Initial relaxation

Figure 25 Resonance formulas describing the gradual increase in polarization along the photoisomerization MEP of the cis-C,H,NH; isomer. (From Ref. 37.)

Mechanistic Organic Photochemistry: Some Case Studies 133

Figure 26 Plot of the x-electron density for the degenerate So and S, states at the conical intersection structure (CI). The arrows indicate the number of electrons migrated from the CH2CHCH- allyl fragment to the -CHCHNH; fragment. (From Ref. 37.)

energies become equal. Consequently, the S, energy is lowered, and ultimately the S, surface crosses with the So surface. This interpretation is strongly supported by the x-electron densities for the degenerate So and S, states reported in Figure 26. In the So -+ S, transition, electron occupation of the two SOMO x orbitals is shown to differ by one electron shift from the allyl fragment to the -CHCH,NHi fragment, according to the previous model.

Competitive Ground State Relaxation Paths from Conical Intersection We now return to the problem discussed briefly in the preceding section: branching at a conical intersection. For a photochemical reaction involving decay at a conical intersection, the reaction coordinate will have two branches. The first (excited state) branch describes the evolution of the molecular structure of the excited state intermediate until a decay point is reached. At this point, the second (ground state) branch of the reaction coordinate begins, which describes the relaxation process ultimately leading to product formation. As we have discussed, the ground state relaxation paths departing from a single conical intersection point can be unambiguously defined and computed with a gradient-driven method.52 We now illustrate in more detail an application of

134 A Computational Strategy for Organic Photochemistry such methods to the description of the radiationless decay and competitive photoproduct formation process in the cyclohexadiene (CHD)/cZc-hexatriene (cZc-HT) (1) system.36382,110 (14) Irradiation at 254 nm transforms 2,5-di-tert-butylhexa-1,3,5-triene (a hexatriene with a dominant cZc equilibrium conformation) into the corresponding cyclohexadiene (15) with a 0.54 quantum yield. The reverse reaction transforms 1,4-di-tert-butylcyclohexa-1,3-diene(15) into the corresponding hexatriene (14) with a 0.46 quantum yield.83-85 Consistently, the computed

14

15

structure of the low-lying part of the S, (2A, ) potential energy surface of these molecules shows that both the direct (CHD -+ cZc-HT) and reverse (cZc-HT -+ CHD) photochemical reactions involve the formation and decay of a common excited state intermediate110 (see Figure 27). This intermediate corresponds to excited state CZC-HT(cZc-HT”),and it is predicted to decay to the ground ( l A l ) state via a conical intersection (CICHD),which has been located about 1 kcal/mol above cZc-HT“ (see Figure 27a). The detailed structure of CICHDwas given in Figure 15. This is a polyene type conical intersection showing an interchain -( CH)3- kink. Decay through a conical intersection and the subsequent evolution on the ground state surface can be studied by means of quantum or semiclassical dynamics.s3~99~lll For a “cold” or thermalized excited state of a sizable organic molecule, the structure of the potential energy surface is expected to play the dominant role in determining the initial molecular motion in the decay region. Thus, one expects that excited state stationary points and MEP will provide the important mechanistic information. Given the tetraradical electronic nature of this structure, relaxation from CICHD may occur along three different routes, as illustrated in Figures 16 and Figure 2%. Each route is associated with a different bond formation mode which is, in turn, driven by the recoupling of four weakly interacting electrons. Accordingly, the route labeled PI leads to relaxation toward CZC-HT,the route labeled R leads to CHD, and the route labeled P, leads to a methylenecyclopentene diradical (MCPD).The mechanism of product formation in CHD photochemistry, in the limit of a “cold” excited state, has been investigated via a systematic search for the ground state relaxation paths departing in the region of CICHD and defining the accessible product valleys. We assume that the photoproducts originate from an excited state intermediate so “cold” that the ground state trajectories lie very close to the computed MEP and that the surface hop occurs in the vicinity of the optimized conical intersection point. The results of these computations yield a description

Mechanistic Organic Photochemistry: Some Case Studies 135

czc-HT

Y 2 Q6

CZC-HT'

'

+\

GroundState Relaxation Path

"f

CHD

,-i 5

--_. C' cno

MCPD

(a) (b) Figure 27 (a) The cyclohexadiene (CHD)/cZc-hexatriene (cZc-HT) photoconversion problem involves the formation of a common excited state intermediate (cZc-HTF*)) and its decay via a conical intersection point (CICHD).(b) Because CI,,, has a tetraradical electronic nature, relaxation on So may occur along three different routes (a, b, and c) associated with different bond formation modes and different recouplings. (From Ref. 82.)

of the ground state relaxation paths departing in the vicinity of the CI,,, structure. The results were illustrated schematically in Figure 17 (see also the corresponding discussion). The excited state reaction path connecting the intermediate cZc-HT" to CICHD leads to an intermediate or slightly sloped conical intersection (see Figures 8c and 8b). The excited state path controls the motion leading to decay at CI,,, , and the ground state relaxation paths control the evolution following the decay and the photoproduct formation process. With the hypothesis that significant picosecond vibrational relaxation takes place in solution at room temperature, the initial relaxation direction (IRD) computed in the region of the conical intersection provide insight into the mechanism of the CHDIcZc-HT photochemical interconversion. The cZc-HT" intermediate, which is produced by either CHD or cZc-HT irradiation,'*o decays via a facile vibrational displacement leading toward the CICH, decay point. After decay, the reaction path bifurcates along two ground state relaxation valleys leading to CHD and cZc-HT. A third relaxation valley leading to MCPD is not directly connected to the conical intersection point CICHD and originates after the energy ridge

1.36 A Computational Strategy for Organic Photochemistry

RDG,,,, has split into two new ridges (comprising the MCPD “valley”) around 1.0 amul’2bohr from the decay point (see Figure 17). Distances are measured in mass-weighted Cartesian coordinates as defined in Ref. 81. The distances of the initial part of the CHD, cZc-HT, and MCPD valleys from the decay point and the steepness of the slope provide qualitative information on the extent of the “catchment region” associated with a specific photoproduct (see Figure 28). A path (i.e., a MEP determined via IRD computations) that develops close to the conical intersection point and is lower in energy will be associated with a larger “catchment region,” and therefore there will be a higher probability of populating the associated valley upon decay from the conical intersection. Figure 28 (see also the model surface in Figure 17)suggests that the CHD and CZC-HT“catchment regions” must be similar in size. Thus we expect that the decay of cZc-HT“ will generate the products CHD and CZCHT with similar yields. Hence, the photolysis of CHD is predicted to generate cZc-HT with a quantum yield (DcZc-HT < 1 because of the competitive CHD back-formation. On the other hand, the computational results suggest that MCPD can give only low quantum yield in the photolysis of CHD, since the MCPD product formation path has a higher valley with respect to the other paths. Furthermore, the MCPD product formation path is topologically inhibited because, in the immediate vicinity of the conical intersection, it corresponds to a ridge (i.e., RDGMMCPD), and ridgelike paths will be populated only at a high kinetic energy. Because the ridge pathway is not populated, the photolysis of either CHD or cZc-HT must have a similar outcome. In fact, since both these reactants yield the same cZc-HT* 2A, intermediate,llO cZc-HT is predicted to yield CHD with a quantum yield (DcZc.HT+CHD that is related to the quan-

valleys located

the conical intersection I

s1/s0

~~~(a.u.1 CHD

Figure 28 Energy profiles of the three valleys located around the conical intersection clCHD

-

Mechanistic Organic Photochemistry: Some Case Studies 13 7 turn yield of cZc-HT produced via CHD photolysis (@‘CHD+cZc.HT); namely, @ ‘ c ~ c - 0~ (1 ~ - +@‘cHL)+~z~-HT ~ ~ ~ 1The interpretation of the CHD/cZc-HT photolysis just presented is compatible with available experimental data. The excited state energy barrier of roughly 1 kcal/mol computed via multireference MP2 theory36 is consistent with the observed picosecond lifetime of the 2A, state following CHD direct irradiation.4.5 Although there are no measurements of @DCHD+CHD(i.e., CHD back-formation during photolysis of CHD) or @cZc-HT+CHD (i.e., CHD formation during photolysis of cZc-HT), @‘CHD+cZc.HT is 0.414,s (i.e., suggesting efficient CHD back-formation), On the other hand, irradiation of 2,5-di-tert(15) butylhexa-l,3,5-triene (14)produces 1,4-di-tert-butylcyclohexa-1,3-diene with a 0.54 quantum yield, The reverse reaction occurs with a 0.46 quantum yield,*3-85 in agreement with the predicted relationship @‘CHD+cZc-HT N (1QCZc.HT+CHD ). In other substituted and polycyclic molecules,86~~7 steric and strain effects may greatly differentiate the slopes of the CHD and CZC-HT valleys, leading to values of mCHD and @cZc.HT far from 0.5.

Competitive Excited State Photoisomerization Paths The all-truns-hepta-2,4,6-trieniminium cation (a “longer” protonated Schiff base with four conjugated double bonds) has an S, MEP that is similar to the MEP of Figure 23. However, in contrast to the short protonated Schiff base discussed above, this molecule may undergo trans -+ cis isomerization at the double bond in positions 2 and 4:

H

H

2 C2-C3

z

c4-c5

The presence in the excited state of competing channels that are barrierless or nearly barrierless (see model surface in Figure 4b) is an interesting feature of longer protonated Schiff bases.39.112 Here we characterize the S, MEPs of both these isomerization processes. In Figure 29 we report the two MEPs. Again the structure of such paths is similar to the one of Figure 23 but with a longer energy plateau (from 1to 3 a.u. (bohr) and from 1 to 5 a.u. for the C,-C, and C4-C, isomerization, respectively). The two paths are nearly barrierless with only a small (4kcal/mol)

I-

138 A Computational Strategy for Organic Photochemistry

10

F

-

h

8 w

I

,I

0' -10

Y

v

4

-20

93"

-30 -40

2

4

'89" 1.47

_T_I

14

Figure 29 Energy profiles along the MEPs describing the competing excited state isomerization paths from the Franck-Condon point (FC) to the decay points S , / S , CIcz-c3 and S, /So CI,,-c5 of all-trans-C,H,NH; (the relaxed planar stationary point is labeled SP). Open squares define the S, (1B,-like) energy. Open circles indicate the position of the conical intersections. The structures (geometerical parameters in angstrom units and degrees) document the geometrical progression along the path. (From Ref. 112.)

energy difference along the long plateau region, in favor of the isomerization at position C,-C, . However, the initial part of the MEPs, which is dominated by double bond expansion, is similar. Thus, in contrast with 2-cis-C,H6NH;, the MEP coordinate of this longer protonated Schiff base shows that evolution along the torsional coordinate (and therefore either of the two competing paths) begins only after relaxation in the vicinity of the planar stationary point SP. Initial relaxation along a totally symmetric mode followed by motion along nontotally symmetric (twisting) modes seems to be a general feature of the excited state behavior of these and closely related compounds (see Ref. 112). The results of analytical frequency computations112 confirm that in a similar fashion t o the short protonated Schiff base, the FC structure evolves along a coordinate dominated by totally symmetric stretching modes, and the structure of region I is that of a valley (see Figure 30). This is also confirmed by the direction of the gradient at the FC point and by observation, at 0.73 a.u. distance, of real (although extremely small) frequencies (8 and 59 cm-1) along the two relevant torsional modes. As seen in the case of the shorter molecule 2-czs-C,H6NH;, there is one imaginary frequency at the S, stationary point. However, the magnitude of this imaginary frequency and that of the frequency

Conclusions 139

Figure 30 Structure of the S, energy surface of all-truns-C,H,NH;. The initial part of the MEPs (see framed region in Figure 29) are indicated by heavy lines. The frequencies are computed at the points indicated with an open square and at the relaxed planar stationary point SP (solid square). The frequencies along the modes leading to decay are given on the surface. The frequency along the SP mode, which correlates with the initial gradient is given in parenthesis. The direction of the initial gradient is represented by arrows on the top structure. Torsional coordinates are indicated by curved arrows on the right and left structures. (From Ref. 112.) corresponding t o the alternative isomerization mode are only 45 i (C,-C, torsional mode) and 62 cm-’ ((2,-C, torsional mode), thus reflecting the existence of the energy plateau and the general flatness of region 11, as well as competition between the two isomerization processes.

CONCLUSIONS We hope the reader has been convinced that it is technically feasible to describe a photochemical reaction coordinate, from energy absorption to photoproduct formation, by means of methods that are available in standard quantum chemistry packages such as Gaussian (e.g., OPT = Conical). The conceptual problems that need to be understood in order to apply quantum chemistry to photochemistry problems relate mainly to the characterization of the conical intersection funnel. We hope that the theoretical discussion of these problems and the examples given in the last section can provide the information necessary for the reader to attempt such computations.

140 A Computational Strategy for Organic Photochemist y

Any mechanistic study undertaken using quantum chemistry methods requires considerable physical and chemical insight. Thus for a thermal reaction, there is no method that will generate automatically all the possible mechanistic pathways that might be relevant. Rather, one still needs to apply skills of chemical intuition, and it is necessary to make sensible hypotheses that can then be explored computationally. In excited state chemistry, these problems are even more difficult, and we hope the examples given in the last section provide a bit of this required insight. However, the DBH example shows just how complex these problems can become when many electronic excited states are involved. In addition to the conceptual and physical problems associated with mechanistic studies in photochemistry, the actual computations are technically difficult to carry out. Whereas single reference quantum chemistry methods such as SCF, DFT, and CIS require no special physical insight to use, multireference methods such as CASSCF require physical insight and technical skill to use effectively. One needs to select a reference space from the outset. Thus, one needs a good chemical model or picture of the nature of the excited states involved, and then one must choose a reference space that is compatible with this initial insight. Obviously, one’s initial choice can be refined as a computational study progresses, but an initial insight is a fundamental prerequisite. It may be that time-dependent DFT or EOM57-60 methods will prove to be useful in excited state mechanistic problems. These methods can be used as a black box. But whether they can be applied to problems of the type we have discussed remains an open question. For the future, it is clear that dynamics methods are almost essential if one is going to examine the interesting results that are coming from femtosecond spectroscopy and to study quantum yields. These methods are just beginning to be exploited, and this is an exciting new direction for quantum chemistry. We have not commented on the role of the solvent or the role of the environment provided by a biochemical system. There are no special problems related to excited state chemistry for the former, and one can look forward to applications to biochemical systems to appear in the near future.

ACKNOWLEDGMENTS This chapter is based (in part) on lecture courses given by M. Olivucci at the EPA Summer School in Noorwick, Holland, June 16-20,1998, and by M. A. Robb at the Jyvaskyla Summer School, Jyvaskyla, Finland, August 3-21, 1998. We are grateful to the organizers of these schools for the opportunity to present this material and especially to the participants for their searching questions. We are also pleased to acknowledge coworkers Michael Bearpark, Paolo Celani, Stephane Klein, Naoko Yamamoto, Barry Smith, and Thom Vreven, whose work we have used in this tutorial and who have commented on various parts of the manuscript. This research has been supported (in part) by an EU TMR network grant (ERB 4061 PL95 1290, Quantum Chemistry for the Excited State). We are also grateful to NATO for a travel grant

References 141 (CRG 950748). The continuous support of the EPSRC (U.K.) under grants H58070 H94177 J/25123 and K 04811 is also gratefully acknowledged.

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CHAPTER 3

Theoretical Methods for Computing Enthalpies of Formation of Gaseous Compounds Larry A. Curtiss,’kPaul C. Redfern,:kand David J. Fruript “Materials Science and Chemistry Divisions, Argonne National Laboratory, Argonne, Illinois 60525, and +AnalyticalSciences Laboratory, Dow Chemical Company, Midland, Michigan 48667

INTRODUCTION Knowledge of the thermochemistry of molecules is of major importance in the chemical sciences and is essential to many technologies. Thermochemical data provide information on stabilities and reactivities of molecules that are used, for example, in modeling reactions occurring in combustion, the atmosphere, and chemical vapor deposition. Thermochemical data are a key factor in the safe and successful scale-up of chemical processes in the chemical industry. Despite compilations of experimental thermochemical data of many compounds, there are numerous molecular species for which data are lacking. In addition, the data in the compilations are sometimes incorrect. Experimental measurements of thermochemical data are often expensive and difficult, so it Reviews in Computational Chemistry, Volume 15 Kenny 8. Lipkowitz and Donald B. Boyd, Editors Wiley-VCH, John Wiley and Sons, Inc., New York, 0 2000

147

148 Theoretical Methods for Computing Enthalpies of Gaseous Compounds is highly desirable to have computational methods that can make reliable predictions. Before discussing the techniques for making predictions, we present some fundamental thermochemical concepts. For the typical reaction aA + bB +-cC + dD

PI

the reaction enthalpy AH; is determined from standard enthalpies of formation, AH;, of the species in the reaction according to AH; = cAH7 + dAHF - aAH7 - bAH?

PI

The standard enthalpy of formation AH? is the enthalpy change upon formation of the material in its standard state from the elements in their standard states. For example, the enthalpy of formation for liquid CH,Cl, at 25°C is the enthalpy change accompanying the reaction C(graphite) +H,(g) + Cl,(g)

-+ CH,Cl,(liq)

[31

Enthalpies of formation are determined experimentally in a number of ways. Perhaps the most common way is through precise oxygen combustion measurements. Where data do not exist in tabulations, estimation techniques-either empirical or based on quantum chemical calculations-are available. The development of faster computers coupled with the advances in theoretical methodologies during the last 10-15 years has resulted in tremendous progress in computational capabilities for the calculation of accurate thermochemical data, as well as many other applications. Progress in computer technologies has followed two tracks in recent years.1 First, there is the increase in the speed of the individual workstations. It is now possible to purchase a desktop workstation with gigaflop speed at a price that is affordable to most research groups. Thus, it is becoming commonplace for desktop computers to have the power of a supercomputer from 5-10 years ago. The second track of computer development is the massively parallel computer systems. At present these systems generally range in speed from 100 gigaflops to 1 teraflop and are usually located in large computer centers. These systems provide computational resources several orders of magnitude greater than those of the desktop computers. Also personal computers are being linked in inexpensive Beowolf clusters. Advances in computer power are likely to continue in the near future, thus increasing the size of systems for which accurate thermochemical data can be calculated. The concomitant advances in theoretical methodologies and algorithms have also played a vital role in increasing computational capabilities for theoretical thermochemistry. These advances include ( 1)new methods for accurate treatment of electron correlation in molecules and atoms such as coupled cluster and quadratic configuration interaction methods, (2)new basis sets such as the correlation consistent basis sets, and ( 3 )development of “model chemistry”

Introduction 149 methods such as Gaussian-2 (G2) theory, Gaussian-3 (G3) theory, and the complete basis set (CBS) methods. In addition, faster methods for locating equilibrium structures and transition states and calculating vibrational energies have enhanced the capabilities of the computational chemists. Likely new advances in methodology, as well as increases in computer resources, will make theoretical thermochemistry increasingly accurate and more feasible in the future. In this chapter we discuss various computational methods that are available for the calculation of enthalpies of formation. The focus is largely on gas phase molecules and ions, although the condensed state is treated to some extent in the section on empirical methods. Most, but not all, of the discussion pertains to species containing first- and second-row elements (Li-Ar) plus hydrogen. There has been much less work done on the development of methods for higher 2 elements. In the remainder of the Introduction we describe how an enthalpy of formation is calculated, give an overview of the computational approaches that are used, and describe the G2 test set of data used for assessing quantum chemical methods. The next section covers quantum chemical methods, including ab initio molecular orbital theory, density functional theory, and semiempirical molecular orbital theory, The third section deals with empirical methods. Both the third and fourth sections present some worked examples of calculating enthalpies of formation for selected methods.

Enthalpies of Formation The standard enthalpy of formation of a compound, AH?, is defined as the increment in enthalpy associated with the reaction of forming a given compound from its elements, with each substance in its thermodynamic standard state at the given temperature.2 The thermodynamic cycle for the enthalpy of formation of methane (CH, ) from the standard states of carbon and hydrogen (graphite and hydrogen molecules) is shown in Figure 1. The enthalpy of formation of a molecule cannot be calculated directly because the enthalpies of the elements in their standard states (e.g., graphite) cannot be determined with standard quantum mechanical methods. Instead, the enthalpy of formation is calculated indirectly3 by means of the combination of experimental and theoretical data and coupled thermodynamic cycles shown in Figure 1.First, atomic enthalpies of formation of the elements in their standard states at 0 K, AH? (atom, 0 K) are obtained from experimental data in standard thermodynamic tables.2.4 Values for hydrogen through chlorine are given in Table 1. The experimental enthalpies of formation of Si, Be, and A1 have large uncertainties (up to 1.9, 1.2, and 1.0 kcal/mol, respectively). This means that any calculated enthalpies of formation containing these atoms may have uncertainties due to the use of these atomic enthalpies. There have been some attempts to calculate these atomic enthalpies by theory.5-7 The atomic enthalpies for other first- and second-row elements are quite accurate (+0.2 kcal/mol or better). The atoms are then combined to form the desired molecule. The energy

150 Theoretical Methods for Computing Enthalpies of Gaseous Compounds

Figure 1 Thermodynamic cycle for the enthalpy of formation of methane (CH?) from the standard states of carbon and hydrogen (graphite and H2). The quantities in italics are calculated in typical therrnochemical quantum chemical predictions.

Table 1 Enthalpies of Formation at 0 K for Gaseous Atoms and (H298- HO) Values for Elements in Their Standard States from Experimenta Atoms

H Li Be 0 C N 0

F Na Mg A1 Si P S C1

UFrom Ref. 1 except where noted. "Ref. 4.

AH? (0 K) 51.63 rt 0.001 37.69 rt 0.2 76.48 k 1.2 136.2 k 0.2h 169.98 rt 0.1 112.53 rt 0.02 58.99 rt 0.02 18.47 k 0.07 25.69 k 0.17 34.87 ? 0.2 78.23 rt 1.0 106.6 rt 1.9 75.42 k 0.2 65.66 k 0.06 28.59 k 0.001

H298

- HO

1.01 1.10 0.46 0.29 0.25 1.04 1.04 1.05 1.54 1.19 1.08 0.76 1.28 1.05 1.10

Introduction 151

for this step is the negative of the atomization (dissociation) energy CD, and can be calculated by simply subtracting the quantum mechanical energies of the atoms from the quantum mechanical energy of the molecule (electronic energy + zero-point energy). An enthalpy of formation at 0 K is then evaluated by subtracting the calculated atomization energy CD, from the experimental enthalpies of formation of the isolated atoms. For any molecule, such as A,B,H,, the enthalpy of formation at 0 K is given by AH?(A,B,H,,OK) =zAHP(A,OK)+yAHF(B,OK)+zAH<(H,OK)-CD,

[4]

Temperature corrections for the atomic enthalpies are also needed in the thermodynamic cycle in Figure 1 to obtain the enthalpy of formation at 298 K or some other temperature. Theoretical enthalpies of formation at 298 K are calculated by correction to AH? (0 K) as follows: AH~(A,B,H,, 298 K) =AH?(A,B,H, ,O K) + [Ho(A,B,H,, 298 K ) -Ho(A,B,H,, OK)]-~[Ho(A,298K)-Ho(A, OK)],, -y[Ho(B,298 K)-Ho(B, 0K)],,-~[Ho(H,298K) -Ho(H,OK)l,,

[51

The enthalpy corrections (in square brackets) are treated differently for compounds and elements. The correction for the A,B,H, molecule is made with calculated frequencies for the vibrations in the harmonic approximation for vibrational energy8 scaled by an appropriate factor to get agreement with experiment,9 the classical approximation for translation (3RT/2) and rotation (3RT/2 for nonlinear molecules, RTfor linear molecules), and the PVterm. The harmonic approximation may not be appropriate for some low frequency torsional modes, although the error should be small in most cases.10 The elemental corrections are for the standard states of the elements (denoted “st” in Eq. [5]) and are taken directly from experimental compilations (seeTable 1).The resulting values of AH? (298 K) are often discussed as theoretical numbers, although they are based on experimental data for monatomic and standard species. We also note that it is possible to calculate free energies of formation of gaseous compounds by means of calculated vibrational frequencies and the classical approximation for rotational and translational contributions to the entropy.8 In this chapter we focus largely on enthalpies of formation of neutral molecules. However, the enthalpies of formation, AH?, of ions are also of interest. They can be obtained by combining calculated ionization potentials, electron affinities, or proton affinities: ABH, + ABH; + eABH, + H+ -+ ABH;,, ABH, + e- -+ ABH;

ionization potential proton affinity electron affinity

with calculated enthalpies of formation of the corresponding neutrakll

152 Theoretical Methods for Cornbutinn Enthalbies o f Gaseous Combounds

Overview of Theoretical Methods The methods available for computing enthalpies of formation fall into two general groups: those based on purely empirical schemes and those founded on quantum chemistry. The quantum chemical methods can be further divided into three types: ab initio molecular orbital theory, density functional theory, and semiempirical molecular orbital theory. A summary of the types of method used to calculate enthalpies of formation is given in Table 2, along with some specific examples. This is not meant to be a comprehensive tabulation, but rather a list of some of the more popular approaches in use today. Table 3 names some of the commercially available computer programs having capabilities to calculate thermochemical data. The ab initio molecular orbital methodslz-17 are based on approximate solution of the Schrodinger equation. The ab initio methods that are used for predicting thermochemical data can be divided roughly into three categories: (1)very high level quantum chemical calculations with no experimental input and extrapolation to the complete basis set limit, (2) techniques that combine an intermediate level of ab initio quantum chemical calculation with some form of molecule-independent empirical parameters, and ( 3 )techniques that use molecule-dependent empirical parameters obtained from accurate experimental data in combination with intermediate level ab initio quantum chemical calculations. In the ab initio approaches of types 2 and 3 , the experimental input is through external corrections, whereas in the semiempirical methods (see below) the experimental input is through internal parameters. The ab initio molecular orbital methods are the most accurate of all the possible approaches to computing thermochemical data, but require the most computer resources. Table 2 Some Methods for Calculating Enthalpies of Formation Tvues

Examples Benson group additivity Molecular mechanics (MM3, etc.)

Quantum chemical Semiempirical molecular orbital Density functional theory

MIND0/3, MNDO, AM1, PM3 B3LYP, BLYP

Ab initio molecular orbital

BAC-MP4 Isodesmic reaction schemes 52 PCI-x CBS-Q G1, G2, G3

w1,w2

CCSD(T)with basis set extrapolation

AI, DF, SE AI, DF DF AI, DF, SE, FF A1 AI, DF A1 A1 AI, SE, FF SE, FF FF AI, DF, SE DF SE SE

Gaussian 98 Q-Chem Jaguar Spartan GAMESS NWCHEM MOLPRO ACES I1 Hyperchem Chem 3D PCMODEL CADPAC ADF MOPAC AMPAC

www.emsl.pnl.gov:208O/docs/nwchem/nwchem. html www.tc. bham.ac.uk/molpro/ www.qtp.ufl.edu/Aces2/ www. hyper.com www.camsci.com www.serenasoft.com www.oxmol.co.uWsofare/unichem/spedcadpac.shtml tc.chem.vu.nV-visser/SCM/ home.att.net/-mrmopac/, www.schrodinger.com/mopac.html www.semichem.com/ampac. html

www.msg.ameslab.gov/GAMESS/GAMESS.html

www.gaussian.com www.q-chem.com www.schrodinger.com/jaguar2.html www.wavefun.com

Web Site

eAI, ab initio molecular orbital methods; DF, density functional methods; SE, semiempirical molecular orbital methods; FF, force field methods.

Capabilities"

Program

Table 3 Some Quantum Chemical Programs That Can Be Used for Thermochemical Calculations

154 Theoretical Methods for Computing Enthalpies of Gaseous Compounds Density functional methodsl s-22 have in the last decade become widely used by chemists to calculate properties of molecules. Physicists have used these methods for many years to study the electronic structure of solids. Density functional theory is based on the work of Kohn,20 who showed that the ground state energy of a quantum chemical system is uniquely determined by its electron density. This relationship is more easily handled than the complicated Schrodinger equation used in ab initio molecular orbital theory. The advantage of density functional theory is that because of its simplicity it is relatively fast and can be applied to fairly large molecules. However, the functional that gives the exact energy is not known, and there is no systematic way in which the exact energy of a system can be approached, whereas ab initio molecular orbital theory can be systematically (but tediously) improved to approach the exact energy. The density functional models currently being used by quantum chemists may be divided into nonempirical and empirical types. The empirical types such as the B3LYP method of Becke20 includes some parameterization of the functionals based on accurate experimental data for a training set of molecules. Density functional methods require less computer time than ab initio methods, but are not yet as accurate. Semiempirical molecular orbital method+-25 incorporate parameters derived from experimental data into molecular orbital theory to reduce the time-consuming calculation of two-electron integrals and correlation effects. Examples of semiempirical molecular orbital methods include Dewar’s AM1, MNDO, and MIND0/3. Of the three quantum chemical types, the semiempirical molecular orbital methods are the least sophisticated and thus require the least amount of computational resources. However, these methods can be reasonably accurate for molecules with standard bond types. Empirical methods26 use known experimental enthalpy data to estimate enthalpies and bond energies for unknown compounds. Among the methods in this group are the bond energy approach and Benson’s rules. The empirical methodologies still hold an important place in the tool box of the scientist simply because these methods are so easy to use and are of proven reliability. The empirical methods require little in the way of computer resources and can handle very large molecules. These methods can be reasonably accurate for molecules that have standard bond types.

Test Sets for Assessments of Predictive Methods Critical documentation and evaluation of theoretical models of electronic structure are essential. If such methods are to become proper tools for chemical investigation, their predictions must be presented together with convincing evidence of reliability. This problem can be approached by assembling a large set of good, credible experimental data and systematically comparing them with corresponding results from theoretical models. The mean differences between

Quantum Chemical Methods 155 the two sets of numbers are then measures of their combined error. If the experimental data set is limited to measurements of very high accuracy, these mean differences document the overall accuracy of the theory. In this manner, reasonable error bars can be placed on theoretical predictions in situations where experimental results are either unavailable or suspect. A widely used test set of experimental data is the G2 test set27-29 consisting of 125 reaction energies, chosen because they have well-established experimental values. Almost all molecules in this test set contain only one or two nonhydrogen atoms (the two exceptions are CO, and SO,). The G2 test set includes 55 atomization energies, 37 ionization energies, 25 electron affinities, and 7 proton affinities. In 1997, the test set was expanded to include larger, more diverse molecules and was named the G2/97 test set.3330 It includes 148 enthalpies of formation of neutrals (at 298 K), 88 ionization potentials, 58 electron affinities, and 8 proton affinities for a total of 302 experimental energies. In contrast to the original G2 test set, enthalpies of formation at 298 K, instead of atomization energies, were used for comparison between experiment and theory. The 148 enthalpies in G2/97 correspond to 29 radicals, 35 nonhydrogen systems, 22 hydrocarbons, 47 substituted hydrocarbons, and 15 inorganic hydrides. The set includes molecules containing up to six nonhydrogen atoms from first and second rows of the periodic table. For assessment of different methods reported in this chapter, we used the G2/97 test set less three ionization potentials (C,H,CH, -+ C,H,CH;, C,H,NH, + C,H,NH;, C,H,OH -+ C,H,OH+), resulting in a total of 299 energies. These three ionization potentials were excluded to make comparisons on an equal basis, since for some methods these species are computationally difficult. The enthalpies of formation at 298 K, ionization potentials, electron affinities, and proton affinities were calculated. In some of the assessments in this chapter, we only include results for the 148 neutral enthalpies of formation from the G2/97 test set. Figure 2 gives the relative CPU times for several of the quantum methods applied to benzene, one of the largest molecules in the set.

QUANTUM CHEMICAL METHODS Ab Initio Molecular Orbital Methods Since the early 1970s when ab initio molecular orbital calculations became routine, one of the major goals of modern quantum chemistry has been the prediction of molecular thermochemical data to so-called chemical accuracy (+1kcal/mol). One of the problems was that the Hartree-Fock calculations done in the 1970s gave large errors (up to 100 kcal/mol) in bond energies. Prediction of accurate thermochemical data required going beyond Hartree-

15G Theoretical Methods for Computing Enthalpies o f Gaseous Compounds

Figure 2 Relative times of different methods for calculating the enthalpy of formation of benzene.

Fock theory to include a sophisticated treatment of electron correlation, and this made the calculations very difficult. After two decades of work, considerable progress was made in attaining the goal of a k1 kcaUmol accuracy through advances in theoretical methodology, development of computer algorithms, and increases in computer power. It is now possible to calculate reliable thermochemical properties for a fairly wide variety of molecules. At the ab initio molecular orbital level, the methods currently used for computing thermochemical data range from very high levels of theory to those that combine moderate levels of theory with some form of empirical input. The former type is limited to smaller molecules and can attain accuracies of k0.5 kcaYmol, whereas the latter methods can be applied to larger molecules with somewhat less accuracy. As described earlier, ab initio molecular orbital methodsl2-17 for prediction of thermochemical data can be divided into three categories: (1)those using very high level calculations with extrapolation, (2)those implementing moderate level calculations with molecule-independent parameters, and ( 3 )those of moderate level calculations with molecule-dependent parameters. In this section we discuss some aspects of these different approaches,

Extrapolation Methods In principle it is known how to compute the thermochemical properties of most molecules to very high accuracy (uncertainty of k0.5 kcal/mol). This can be achieved by accounting for electron correlation, such as can be obtained by means of coupled clustered theory with single, double, and triple excita-

Qualztum Chemical Methods 157 tionsl5,16,31-33 [CCSD(T)]or quadratic configuration interaction with single, double, and triple excitation34 [QCISD(T)]methods, and with very large basis sets. The results of these calculations are then extrapolated to the complete basis set limit and corrected for some smaller effects not included in the calculations such as core-valence effects, relativistic effects, and atomic spin-orbit effects.35-37 Unfortunately, this approach is limited to small molecules because of the -n7 time scaling (with respect to the number of basis functions) of the correlation methods and the large basis sets used. This methodology has been used by Dunning, Feller, and coworkers35J8-43 at Pacific Northwest National Laboratories to study systematically a large number of small molecules having one and two nonhydrogen atoms. Their work was based on CCSD(T)calculations with correlation consistent basis set~.4~-46They have applied these calculations to a diverse enough set of molecules to demonstrate convincingly that it does perform to a very high level of accuracy. Martin has also developed methodologies for very accurate calculations based on extrapolation procedures.7J7~47-50Recently, Martin and de Oliveira set forth the W1 and W2 quantum chemical methods51 that use an extrapolation based on a series of high level CCSD(T)calculations with correlation consistent basis sets. The W1 method uses a small empirical correction, whereas the more computationally intensive (and more accurate) W2 method does not use an empirical correction. Other groups have also used this type of approach to computational thermochemistry.5J6.52-56 Grev and Schaefers used coupled clustered methods [CCSD(T)]with large basis sets to study the thermochemistry of CH, and SiH, hydrides and some of their cations. The researchers achieved bond energies accurate to 0.5 kcaUmol (mean absolute deviation) without any empirical corrections for these small molecules. Petersson and coworkers53 have used QCISD(T) with very large basis sets to achieve a mean absolute deviation of 0.53 for a subset of the G2 test set of reaction energies. Bauschlicher, Langhoff, Taylor, and coworkers36>54-56used an approach based on converging to the one-particle limit through the use of atomic natural orbitals at a moderate level of correlation treatment. The correlation treatment is calibrated against full configuration interaction calculations on smaller systems, or against accurate experimental data in some cases.

Methods with Molecule-Independent Empirical Parameters Because the very high level calculations described in the preceding section are difficult to extend to larger molecules, an alternative approach is to use a series of high level correlation calculations [e.g., fourth-order perturbation theory (MP4),QCISD(T),or CCSD(T)]with moderate-sized basis sets to estimate the result of a more expensive calculation. The Gaussian-n27-29.57-60 and CBS series6.61-63 described in the following sections exploit this idea to predict thermochemical data. In addition, molecule-independent empirical parameters are used in these methods to estimate the remaining deficiencies in the calcula-

158 Theoretical Methods for Computing Enthalpies of Gaseous Compounds tion. This will work if the remaining deficiencies are systematic and scale as the number of pairs of electrons. Such an approach has been quite successful in the Gaussian-n series and the latest version, Gaussian-3 theory,64 achieves an overall accuracy (mean absolute deviation) of one kcal/mol. Gaussian-2 (G2) theory Ideally, a sucessful method for computation of thermochemical data has several features: (1)it should be applicable to any molecular system in an unambiguous manner, (2) it must be computationally efficient so that it can be widely applied, (3) it should be able to reproduce known experimental data to a prescribed accuracy, and (4) it should give similar accuracy for species for which the data are not available or for which experimental uncertainties are large. The Gaussian-n methods were developed with these objectives in mind. Gaussian-1 ( G l ) theory was the first in this series.27.28 We will not cover G1 theory in this chapter because it was replaced by G2 theory, which eliminated several deficiencies in G1, and because G2 is currently the most widely used method of this series. Gaussian-2 theory29 is a composite technique in which a sequence of welldefined ab initio molecular orbital calculations is performed to arrive at a total energy of a given molecular species. Geometries are determined using secondorder Moller-Plesset perturbation theory.65 Correlation level calculations are done using Moller-Plesset perturbation theory up to fourth order66 and with quadratic configuration interaction.34 Large basis sets including multiple sets of polarization functions are used in the correlation calculations. A series of additivity approximations makes the technique fairly widely applicable.67 Unlike many other approaches, G2 theory is not dependent on calibration with experimental data for related species either through isodesmic reactions or in some other manner. It does have one molecule-independent semiempirical parameter that is chosen by fitting to a set of accurate experimental data. The principal steps in G2 theory are summarized here. 1. An initial equilibrium structure is obtained by geometry optimization at the Hartree-Fock (HF) level with the 6-31G(d) basis.68.69 Spin-restricted Hartree-Fock (RHF) theory is used for singlet states and spin-unrestricted Hartree-Fock theory (UHF) for others. 2. The HF/6-31G(d) equilibrium structure is used to calculate harmonic frequencies mi,which are then scaled empirically by a factor of 0.8929 to take account of known deficiencies at this level.70 These frequencies give the zeropoint energy, AE(ZPE)

used to obtain E , in step 7. 3. The equilibrium geometry is refined at the MP2/6-31G(d) level [Moller-Plesset perturbation theory to second order with the 6-31G(d) basis

Quantum Chemical Methods 159

set], using all electrons for the calculation of correlation energies. This is the final equilibrium geometry in the theory and is used for all single-point calculations at higher levels of theory in step 4. All these subsequent calculations include only valence electrons in the assessment of electron correlation. 4. The first higher level calculation is at full fourth-order Maller-Plesset perturbation theory66 with the 6-311G(d,p) basis set71J2 [i.e., MP4/6311G(d,p)].This energy is then modified by a series of corrections from additional calculations including (a) a correction for diffuse functions,73,74 AE( +); (b) a correction for higher polarization functions73 on nonhydrogen atoms, AE(2df ); (c) a correction for correlation effects beyond fourth-order perturbation theory using the method of quadratic configuration intera~tion,3~ AE(QC1); and (d) a correction for larger basis set effects and for nonadditivity caused by the assumption of separate basis set extensions for diffuse functions and higher polarization functions, AE(+3df,2p). The single-point energy calculations required for these corrections are described in Table 4. 5. The MP4/6-311G(d,p) energy and the four corrections from step 4 are combined in an additive manner: E(combined) = E[MP4/6-311G(d,p)]+ AE(+) + AE(2df) + AE(QC1) 171 + AE(+3df,2p) 6. A “higher level correction” (HLC) is added to give the total electronic energy, E,,

E,(G2) = E(combined) + AE(HLC)

PI

The HLC is equal to -An, -Bn, where the n, and n, are the number of p and a valence electrons, respectively, with n, 2 np. For G2 theory,29 A = 4.81 millihartrees and B = 0.19 millihartree (equivalent to 5.00 millihartrees per electron pair). The B value was chosen so that E , is exact for the hydrogen atom. The A value was determined to give a zero mean deviation from experiment for the atomization energies of 55 molecules having well-established experimental values. The higher level correction takes into account remaining deficiencies and makes G2 theory slightly “empirical,” though only a single moleculeindependent parameter is used. 7. Finally, the total energy at 0 K is obtained by adding the zero-point correction, obtained from the frequencies of step 2, to the total energy: E,(G2)=Ee(G2)+ AE(ZPE)

t91

The energy E , is referred to as the “G2 energy.” The final total energy is effectively at the QCISD(T)/6-311+G(3df,2p) level if the different additivity approximations work well. The validity of these

4(a) 4( b) 4(~) 4(d) 4'(d) 4"(C) 4"(d)

G2

G2 (MP2) G2(MP2,SVP)

Step

Method

AE(+)= E[MP4/6-31 l+G(d,p)]-E[MP4/6-311G(d,p)] AE(2df) = E[MP4/6-311G(2df,p)] -E[MP4/6-31 lG(d,p)] AE(QC1)= E[QCISD(T)/6-311G(d,p)J-E[MP4/6-31 lG(d,p)] AE(+3df,2p)= E[MP2/6-311+G(3df,2p)] - E[MP2/6-31 lG(Zdf,p)]- E[MP2/6-31 l+G(d,p)]+ E[MP2/6-31 lG(d , ~ ) ] - [E(MP2/6-311G(d,p)J AE' (+3df,2p) = [E(MP2/6-311+G(3df,2~)] AE"(QCI) = E[QCISD(T)/6-31G(d)]- E[MP4/6-3 l G (d)] AE"(+3df,2p) = [E(MP2/6-311+G(3df,2~)] - [E(MP2/6-31G(d)]

Corrections

Table 4 Energy Corrections (step 4) for G2, G2(MP2), and G2(MP2,SVP) Theories

Quantum Chemical Methods 161 additivity approximations was investigated by performing complete QCISD(T)/6-311+G(3df,2p)calculations on the set of 125 test reactions, and in most cases, the additivity approximations were found to work well.67 All the calculations required for G2 theory can be done with the quantum chemical computer programs Gaussian75 and MOLPRO (see Table 3). In the Gaussian code there is a keyword “G2” that does a G2 calculation automatically. G2 theory has an average absolute deviation from experiment of 1.48 kcallmol for the G2/97 test set.3930 It has an average absolute deviation of 1.56 kcal/mol for the 148 enthalpies of formation of neutrals in this test set. Table 5 gives a complete summary of the average absolute deviations, including a breakdown into different molecule types. Variations of G2 Theory Variants of G2 theory have been proposed. The purpose of some of these has been to reduce the computational expense of the calculations, whereas others have been aimed at improving the accuracy. G2(MP2) theory is a variation of G2 theory that uses reduced orders of Marller-Plesset perturbation theory.76 In this theory the basis set extension corrections of G2 theory in steps 4a, 4b, and 4d are replaced by a single correction obtained at the MP2 level with the 6-311+G(3df,2p) basis set, AEf(+3df,2p),as given by step 4’(d) in Table 4. The total G2(MP2) energy is thus given by E,[G2(MP2)] = E[MP4/6-311G(d,p)] + AE(QC1)+AE’(+3df,2p) + AE(HLC)+ AE(ZPE)

POI

where the AE(QC1) and AE(HLC) terms are the same as in G2 theory. The G2(MP2) energy requires only two single-point energy calculations, QCISD(T)/6-311G(d,p) and MP2/6-311+G(3df,2p), since the sum of the E[MP4/6-311G(d,p)] and AE(QC1) terms in Eq. [lo] is equivalent to the QCISD(T)/6-311G(d,p) energy, and the QCISD(T)/6-311G(d,p) calculation provides the MP2/6-31 lG(d,p) energy needed to evaluate AEi(+3df,2p). The absence of the MP4/6-311G(2df,p) calculation in G2(MP2) theory provides significant savings in computational time (approximately a quarter) and disk storage such that larger systems can be handled than in G2 theory. G2(MP2) theory is somewhat less accurate than G2 theory, having an average absolute deviation of 1.89 kcallmol for the G2/97 test set (see Table 5). A variation of G2 theory that uses reduced orders of Marller-Plesset perturbation theory in combination with a smaller basis set for the quadratic configuration correction is G2(MP2,SVP) theory.77,7* The “SVP” refers to the split-valence plus polarization basis, 6-31G(d),used in this QCISD(T) correction. In this theory the final energy is given by E,[G2(MP2,SVP)] = E[MP4/6-31G(d)] + AE”(QC1)+ AE”(+3df,2p) [I11 + AE(HLC) + AE(ZPE)

1.41 1.46 1.02 1.18 2.12 0.70 0.74 1.03 1.23 1.30

1.13 0.98 1.34 0.94 1.72 0.68 0.56 0.87 0.84 1.01

1.41 1.41 1.08 1.56 2.44 1.29 1.48 0.95 1.16 1.48

G2 1.72 1.94 0.77 2.03 3.24 1.83 1.89 1.20 1.36 1.89

G2(MP2)

“The numbers in parentheses refer to the number of pieces of data used for the comparison.

Ionization energies (85) Electron affinities (58) Proton affinities (8) Enthalpies of formation (148) Nonhydrogen compounds (35) Hydrocarbons (22) Substituted hydrocarbons (47) Inorganic hydrides (15) Radicals (29) All (299)

G3(MP2)

G3 1.77 2.11 0.91 1.92 3.53 0.78 2.04 0.91 1.19 1.89

G2(MP2,SVP)

1.55 1.12 1.41 1.54 2.68 1.49 1.17 1.18 1.01 1.46

CBS-Q

2.11 3.13 1.99 1.78 1.68 1.72

CBS-q

3.04 4.12 4.47 2.61 1.68 2.04

cb5-4

Table 5 Average Absolute Deviations (in kcal/mol) from Experiment of Various Composite Ab Initio Molecular Orbital Methods for Ionization Energies, Electron Affinities, Proton Affinities, and Enthalpies of Formation in the G2/97 Test Seta

Quantum Chemical Methods 163 where the AE”(QC1) and AEf’(+3df,2p) correction terms are relative to the 6-31G(d)basis set (see Table 4). The HLC for G2(MP2,SVP) theory is A = 5.13 millihartrees and B = 0.19 millihartree. The G2(MP2,SVP) energy requires only two single-point energy calculations, QCISD(T)/6-31G(d) and MP2/6311+G(3df,2p),since the sum of the E[MP4/6-31G(d)]and AE”(QC1) terms in Eq. [ll]is equivalent to the QCISD(T)/6-3lG(d)energy, and the QCISD(T)/63 1G(d) calculation provides the MP2/6-3 1G(d) energy needed to evaluate AEf’(+3df,2p).The use of the 6-31G(d) basis set in the quadratic configuration interaction calculation instead of the 6-311G(d) basis, as in the G2 and G2(MP2) theories, reduces computational time and disk space. G2(MP2,SVP) theory has an average absolute deviation similar to that of G2(MP2)theory (see Table 5), but the method requires about half the computer resources. Other modifications of G2 theory have been investigated that use high levels of theory for correlation effects, geometries, and zero-point energies.79 A high level of correlation treatment was examined using Brueckner doubles [BD(T)]*O>*’and coupled cluster [CCSD(T)]31-33methods rather than quadratic configuration interaction [QCISD(T)]. These methods are referred to as G2(BD) and G2(CCSD), respectively. The use of geometries optimized at the QCISD level rather than the second-order Maller-Plesset level (MP2) and the use of scaled MP2 zero-point energies rather than scaled Hartree-Fock (HF) zero-point energies have also been examined. These methods are referred to as G2NQCI and G2(ZPE=MP2), respectively. (By convention, “//” means “at the geometry of,” so the energy is computed by the method named before the double slash using atomic coordinates obtained in energy optimization at the level named after the double slash.) Inclusion of high levels of correlation treatment has little effect except in the cases of multiply bonded systems. In these cases, better agreement is obtained in some cases and poorer agreement in others, so that there is no improvement in overall performance. The use of QCISD geometries yields significantly better agreement with experiment for several cases. The use of MP2 zero-point energies gives no overall improvement. These methods may be useful for specific systems. Bauschlicher and Partridge82 have proposed a modification of G2 theory that uses geometries and vibrational frequencies from density functional theory (DFT) methods. In this method, referred to as G2(B3LYP/MP2/CC), the QCISD(T) step in G2(MP2) theory is replaced by a CCSD(T) calculation. In addition, the HF/6-3 1G(d) zero-point energies and MP2/6-3 1G(d) geometries are replaced by zero-point energies and geometries obtained from density functional theory [B3LYP/6-31G(d)]. This modification does not improve the average absolute deviation, but it does decrease the maximum errors compared with the G2(MP2)approach. Morokuma and coworkers*3-85 have proposed a family of modified G2 schemes based on geometry optimization and vibrational frequency calculations using density functional theory tB3LYP16-311G(d,p)] and electron correlation using coupled cluster methods. These schemes use spin-projected Mdler-Plesset theory for radicals*G and triplets and may be

164 Theoretical Methods for Computing Enthalpies of Gaseous Compounds more reliable for radicals with large spin contamination. Durant and Rohlfing have assessed G2 theory for the calculation of transition states and have proposed a modification that is based on QCISD geometries (G2Q).87 Extension of G2 Theory to Third-Row Non-Transition-Metal Elements Gaussian-2 theory has been extended to include molecules containing thirdrow non-transition-metal elements Ga-Kr.88 Basis sets compatible with those used in G2 theory for molecules containing first- and second-row atoms were derived for this extension. G2 theory for the third row incorporates the following modifications: 1. The MP2 geometry optimizations and the HF vibrational frequency calculations use the 641(d) basis set of Binning and C~rtiss899~0 for Ga-Kr along with 6-31G(d) for first- and second-row atoms, referred to overall for simplicity as “6-31G(d).” The same scale factor (0.8929) is used for the zeropoint energies. 2. The MP2, MP4, and QCISD(T) calculations (step 4 in section on G2 theory above) use the 6-31 1G basis and appropriate supplementary functions for first- and second-row atoms and corresponding sets that were developed88 for Ga-Kr, referred to overall as “6-311G”, again for simplicity 3. The splitting factor of the d-polarization functions for the 3df basis set extension is 3 rather than the factor of 4 used for first- and second-row atoms. The 3d core orbitals and 1s virtual orbitals are frozen in the single-point correlation calculations. 4. First-order spin-orbit energy corrections A E ( S 0 ) are included in the G2 energies for the third row including 2P and 3P atoms and 2n molecules. Values for the these corrections are obtained from spin-orbit configuration interaction calculations.*8+’1 The average absolute deviation from experiment of 40 test cases containing one or two nonhydrogen atoms (atomization energies, ionization energies, electron affinities, proton affinities) involving species containing Ga-Kr atoms is 1.37 kcal/mol. This is only slightly greater than for the G2 treatment of firstand second-row molecules for the 125 reaction energies of small molecules, for which the average absolute deviation is 1.21 kcal/mol. The inclusion of firstorder atomic and molecular spin-orbit corrections is important for attaining good agreement with experiment. When the spin-orbit correction is not included, the deviation increases to 2.36 kcaUmol. In several cases lacking a good distinction between the core orbital and valence orbital energy levels, such as GaF,, this can cause a problem with the frozen core approximation in G2 theory when applied to elements in the third rowa92,93Remedies for this problem have been suggested.Q93 G2 theory has also been extended to include molecules containing K and Ca94 in a manner similar to that used for Ga-Kr. The G2(MP2)and G2(MP2,SVP) theories can be used for the third r0w.~5 The formulation is analogous to that for first- and second-row elements with

Quantum Chemical Methods 165 the exception of the additional spin-orbit correction term. The average absolute deviation for the third-row test set using G2(MP2) and G2(MP2,SVP) theories is 1.92 kcal/mol for both methods. Gaussian-3 (G3) Theory Gaussian-3 theory,64 like Gaussian-2 theory,29 is a composite technique in which a sequence of well-defined ab initio molecular orbital calculations is performed to arrive at a total energy of a given molecular species. In G3 theory, steps 1-3 are the same as in G2 theory (see above). Steps 4-7 are modified as follows: 4. A series of single-point energies calculations is carried out at higher levels of theory. The first higher level calculation is complete fourth-order Merller-Plesset perturbation theory with the 6-31G(d) basis set [i.e., MP4/631G(d)].This energy is then modified by a series of corrections from additional calculations: (a) a correction for diffuse functions, AE(+); (b) a correction for higher polarization functions on nonhydrogen atoms and p functions on hydrogens, AE(2df,p); (c) a correction for correlation effects beyond fourth-order perturbation theory using the method of quadratic configuration interaction, AE(QC1); and (d) a correction for larger basis set effects and for the nonadditivity caused by the assumption of separate basis set extensions for diffuse functions and higher polarization functions, AE(G3Large). The single-point energy calculations required for these corrections are listed in Table 6. 5. The MP4/6-31G(d) energy and the four corrections from step 4 are combined in an additive manner along with a spin-orbit correction AE(S0)for atoms only (hydrogen through chlorine): E(combined) = E[MP4/6-31G(d)] + AE(+)+ AE(2df,p) + AE(QC1) [121 + AE(G3Large) + AE(S0) The atomic spin-orbit correction is taken from experiment96 where available and accurate theoretical calculations in other cases. The values are listed in Table 7. 6. A “higher level correction” (HLC) is added to take into account remaining deficiencies in the energy calculations: E,(G3) = E(combined) + E(HLC)

~131

The HLC is -An, - B ( n , - n p )for molecules and - Cn, - D(n, -n,) for atoms (including atomic ions), where n , and n, are the number of p and a valence electrons, respectively, with n, 2 n,. The number of valence electron pairs corresponds to np Thus, A is the correction for pairs of valence electrons in molecules, B is the correction for unpaired electrons in molecules, C is the correction for pairs of valence electrons in atoms, and D is the correction for

4(4 4(b) 4(c) 4(4 4'(4

G3

~

AE(+) = E[MP4/6-31+G(d)]- E[MP4/6-31G(d)] AE(2df) = E[MP4/6-3 lG(2df,p)] - E[MP4/6-31G(d)] AE(QCI) = E[QCISD(T)/6-31G(d)] - E[MP4/6-31 G(d)] AE(MP2) = E[MP2(full)/G3Large]- E[MP2/6-31G(2df,p)] - E[MP2/6-31+G(d)]+ E[MP2/6-31G(d)] AE' (MP2) = E[MP2/G3MP2Large] - E[MP2/6-31G(d)l

Corrections

aHigher level correction in G3 has A = 6.386, B = 2.977, C = 6.219, D = 1.185, whereas G3(MP2) has A = 9.279, B = 4.471, C = 9.345, D = 2.021. bThe density functional based G3 methods use B3LYP/6-31G(d)geometries and frequencies (scaled by 0.96). The other steps are the same as in the G3 methods. Higher level corrections in G3//B3LYP has A = 6.760, B = 3.233, C = 6.786, D = 1.269, whereas G3(MP2)//B3LYPhas A = 10.041, B = 4.995, C = 10.188, D = 2.323.

G3(MP2)

SteD

Method

Table 6 Energy Corrections (Step 4 in Text) for G3 and G3(MP2) The0riesa.b

Quantum Chemical Methods 167 Table 7 Atomic Spin-Orbit Corrections (millihartrees) Atomic Species

H (2s) He (IS) Li (2s) Be ( I S ) B (2p)

c (3P)

N (4s) 0 (3P) F (2p) Ne ( I S ) Na (2s) Mg ('S) Al (2P) Si (3P) p (4s)

s (")

CI (2P)

Ar (IS) He+ (2s) Li+ (IS) Be+ ( 2 s ) B+ (IS) C+ (2P)

AE(SO)a 0.0 0.0 0.0 0.0 -0.05 -0.14 0.0 -0.36 -0.61 0.0 0.0 0.0 -0.34 -0.68 0.0 -0.89 -1.34 0.0 0.0 0.0 0.0 0.0 -0.2

Atomic Species

N+ (3P)

o+(4s)

F+ (3P) Ne+ (2P) Na+ (IS) Mg+ (2s) Al+ (1s) si+(2P) P+ (3P)

s+ (4s) c1+(3P)

Ar+ (2P) Li- (IS)

B-

(3P)

c- (4s)

0- (2P) F- (IS) Na- (IS) Al- (3P)

si- (4s) P-

s-

(3P)

(2P) C1- (IS)

AE (SO)"

-0.43 0.0 -0.67 -1.19 0.0 0.0 0.0 -0.93 -1.43 0.0 -1.68 -2.18 0.0 -0.03' 0.0 -0.26' 0.0 0.0 -0.28' 0.0 -0.45' -0.88' 0.0

aspin-orbit corrections are from Ref. 96 except where noted. bcalculated value, Ref. 30.

unpaired electrons in atoms. For G3 theory, A = 6.386, B = 2.977, C = 6.219, and D = 1.185 millihartrees. The A, B, C, D values are chosen to give the smallest average absolute deviation from experiment for the G2/97 test set? 7. Finally, the total energy at 0 K is obtained by adding the zero-point energy, obtained from scaled (0.8929) HF/6-31G(d) vibrational frequencies to the total energy: E0(G3) = E,(G3) + E(ZPE)

~ 4 1

The energy E , is referred to as the "G3 energy." The final total energy is effectively at the QCISD(T,FULL)/G3Large level if the different additivity approximations work well. The average absolute deviation from experiment of G3 theory for the G2/97 test set is 1.01 kcal/mol (see Table 5). For the subset of 148 neutral enthalpies of formation the average absolute deviation is 0.93 kcal/mol. The corresponding deviations for G2 theory are 1.49 and 1.56 kcal/mol, respectively. The improvement over G2 theory is shown in Figure 3 for different types of molecule in the G2/97 test set.

168 Theoretical Methods for Computing Enthalpies of Gaseous Combounds

2-

All

Figure 3 Comparison of the accuracy of G2 and G3 methods for calculating enthalpies of formation (based on the 148 enthalpies of formation in the G2/97 test set). The goal is to get below 1 kcal/mol. The G3 deviations for all of the enthalpies of formation in the G2/97 test set are listed in Table 8. Also listed in this table are the deviations for G2 and G2(MP2) theories, as well as several other methods described in this tutorial. Many of the deficiencies in G2 theory for the G2/97 test set have been eliminated in G3 theory. Of particular importance is the improvement for 35 nonhydrogen systems, such as SiF, and CF,, for which the average absolute deviation decreases from 2.54 kcaVmol (G2 theory) to 1.72 kcaYmol (G3 theory). Another significant improvement is found for the 47 substituted hydrocarbons in the test set, for which the average absolute deviation decreases from 1.48 kcal/mol to 0.56 kcal/mol. The deviations from experiment for a series of hydrocarbons (Figure 4)show excellent agreement with experiment. Variations of G3 Theory At least two variations of G3 theory have been proposed. The first does the basis set extensions at the second-order MarllerPlesset level. This method, referred to as G3(MP2) theory,97 has an average absolute deviation from experiment of 1.30 kcaVmol for the G2/97 test set and 1.18 kcal/mol for the subset of 148 neutral enthalpies (see Table 5 ) . This is a significant improvement over the related G2(MP2) theory. The new method provides significant savings in computational time compared to G3 theory (see Figure 2). The modification to step 4 in G3 theory is shown in Table 6, along with the new higher level correction parameters. A second variation of Gaussian-3 (G3) theory uses geometries and zeropoint energies from B3LYP density functional theory [B3LYP/6-31G(d)]instead of geometries from second-order perturbation theory [MP2/6-31G(d)] and zero-point energies from Hartree-Fock theory [HF/6-31G(d)].98 This varia-

Quantum Chemical Methods 169 tion, referred to as G3//B3LYP, has an average absolute deviation from experiment for the G2/97 test set of 0.99 kcaYmo1 compared to 1.01 kcal/mol for G3 theory. A summary of the performance of G3//B3LYP for neutral enthalpies of formation is given in Table 9, Generally, the results from the G3 and G3//B3LYP methods are similar, but there are some exceptions. G3//B3LYP theory gives significantly improved results for several cases for which MP2 theory is deficient for optimized geometries, such as C N and 0;.However, G3//B3LYP does poorly for ionization potentials that involve a Jahn-Teller distortion in the cation (CH,', BF;, BCl; ) because the B3LYP/6-31G(d) method, used for geometries, does poorly for these Jahn-Teller states.98 The G3(MP2)method is also modified to use B3LYP/6-31G(d)geometries and zero-point energies. This variation, referred to as G3(MP2)//B3LYP,has an average absolute deviation of 1.25 kcal/mol compared to 1.30 kcal/mol for G3(MP2)theory. The use of density functional geometries and zero-point energies in G3 and G3(MP2)theories is a useful alternative to MP2 geometries and HF zero-point energies. Some of the Gaussian-n methods are available as keywords in the Gaussian program. The G1, G2, and G2(MP2) methods are incorporated in Gaussian 9475 and later versions. The G3, G3(MP2), G3//B3LYP, and G3(MP2)// B3LYP methods are available in Gaussian 98 (A.7 and later versions)99 and can be run automatically with keywords. Compilations of Gaussian-n energies are available on the Internet.100 Complete Basis Set Methods Petersson et al.61-63 developed a series of methods, referred to as complete basis set (CBS)methods, for the evaluation of accurate energies of molecular systems. The central idea in the CBS methods is an extrapolation procedure to determine the projected second-order (MP2) energy in the limit of a complete basis set. This extrapolation is performed pair by pair for all the valence electrons and is based on the asymptotic convergence properties of pair correlation energies for two-electron systems in a natural orbital expansion. As in G2 theory, the higher order correlation contributions are evaluated by a sequence of calculations with a variety of basis sets. Ochterski, Petersson, and Montgomery63 proposed three CBS models, referred to as CBS-4, CBS-q, and CBS-Q. Among the three, the CBS-4 method is the most widely applicable, whereas the CBS-Q method is the most accurate. The CBS-Q method involves a series of calculations with the QCISD(T), MP4(SDQ),MP2 (with CBS extrapolation), and HF levels of theory with progressively larger basis sets. The steps in the CBS-Q method are listed in Table 10. The largest basis set used in the CBS-Q method is 6-311+G(3d2f,2df,2p), where the values in parentheses indicate the multiple sets of polarization functions used for the second-row elements, first-row elements, and hydrogen, respectively. The CBS-q method is similar but uses smaller basis sets, extending its range of applicability to larger molecules. The CBS-4 method does not have a QCISD(T) step at all, and the highest order component is obtained from an MP4(SDQ) calculation.

Species

G3 0.3 -0.5 1.4 1.3 0.9 1.0 0.3 0.9 0.6 -0.8 1.0 -0.3 0.2 2.1 1.3 1.0 0.9 0.5 -1.8 -0.4 -0.1 2.2 0.7 -0.7 0.2 0.3 -1.8 0.2 0.3 0.3

G3 33.0 82.2 141.1 92.4 101.8 34.0 -18.2 84.3 44.5 -10.2 8.4 -57.5 -65.4 63.1 84.9 46.9 7.3 32.6 3.1 -4.5 -21.9 49.4 -80.8 54.9 12.3 -20.4 106.7 31.3 -26.7 9.7

Expth

33.3 81.7 142.5 93.7 102.8 35.0 -17.9 85.2 45.1 -11.0 9.4 -57.8 -65.1 65.2 86.2 47.9 8.2 33.1 1.3 -4.9 -22.1 51.6 -80.1 54.2 12.5 -20.1 104.9 31.5 -26.4 10.0

AH? (298 K) (kcal/mol)

0.01 -2.78 1.80 1.44 1.09 0.82 -0.05 1.25 0.59 -0.96 1.09 -0.39 0.29 2.34 2.84 1.90 0.99 1.25 -1.21 0.63 0.31 2.88 0.05 -0.09 0.68 0.03 -1.51 0.30 0.95 0.54

G3(MP2)

Table 8 Deviations of Calculated Enthalpies of Formation from Experimenta

0.6 -1.5 0.6 -1.0 1.4 -0.1 0.7 -1.1 0.1 -0.2 0.3 0.3 1.o 2.9 0.5 1.2 2.2 0.2 -0.7 -0.1 0.4 2.0 1.3 -1.6 -0.2 0.5 -2.4 0.3 1.8 0.7

G2 0.21 -2.36 0.27 -1.50 1.06 -0.59 0.17 -1.35 -0.07 -0.06 0.56 1.10 1.60 2.72 0.34 1.09 2.02 -0.21 -0.99 0.75 1.23 2.41 1.69 -2.11 -0.72 -0.21 -2.61 0.12 2.87 1.27

G2(MP2) -0.35 -0.76 -0.12 -1.01 -0.07 -0.28 -0.22 -1.30 -0.36 -0.98 0.41 0.14 0.93 2.34 1.54 2.15 2.73 1.35 0.03 0.66 0.91 0.01 0.38 -1.97 -0.97 -0.56 -2.16 -0.71 0.64 0.49

CBS-Q

Deviation (kcaUmo1)

-0.31 -0.81 -0.06 -1.02 0.16 -0.57 -0.18 -1.13 0.03 -0.54 0.32 0.25 0.51 2.20 1.31 2.00 2.63 1.73 0.70 0.51 0.19 0.14 -0.22 -1.78 -0.84 -0.50 -1.74 -0.23 0.31 0.16

CBS-QB3

-2.74 -2.57 -0.33 0.09 -1.79 0.42 0.60 -1.11 -0.54 -1.18 -0.16 -0.36 0.54 1.28 3.43 3.91 3.65 2.40 0.49 0.91 1.17 -1.99 -3.58 -3.20 0.62 1.98 -6.40 -3.47 -0.23 -0.16

CBS-4

cci,

AIF, AICI, CF,

G2-2 test set BF, BCI,

so,

FC1 Si2H, CH,CI CH,SH HOCl

sc so c10

NaCl SiO

c12

s2

0, HOOH F2 CO, Na, Si, p,

NO

H,CO CH,OH N2 H2"H,

-271.4 -96.3 -289.0 -139.7 -223.0 -22.9

-26.0 -48.0 0.0 22.8 21.6 0.0 -32.5 0.0 -94.1 34.0 139.9 34.3 30.7 0.0 -43.6 -24.6 66.9 1.2 24.2 -13.2 19.1 -19.6 -5.5 -17.8 -71 .O -270.9 -96.3 -290.1 -143.0 -223.9 -24.6

-26.6 -48.1 2.1 24.9 21.8 1.1 -3 1.3 0.7 -95.3 30.0 139.8 35.5 31.6 1.1 -44.8 -23.9 65.8 1.7 25.9 -12.5 17.7 -19.5 -5.1 -1 7.4 -67.1 -0.5 0.0 1.1 3.3 0.9 1.7

0.6 0.1 -2.1 -2.1 -0.2 -1.1 -1.2 -0.7 1.2 4.0 0.1 -1.2 -0.9 -1.1 1.3 -0.7 1.1 -0.5 -1.7 -0.7 1.4 -0.1 -0.4 -0.4 -3.8 -1.81 -0.56 0.36 2.98 0.00 2.32

0.56 -0.34 -1.99 -2.52 -0.33 -2.02 -1.70 -1.30 0.83 3.33 2.80 -0.17 0.54 -0.42 1.55 0.64 3.21 -1.10 -2.18 -1.42 1.20 0.07 0.38 -0.66 -3.94 0.0 2.0 -1.4 2.8 5.5 2.8

2.0 1.4 -1.3 -0.9 0.6 -2.4 -0.2 -0.3 2.7 2.4 -0.4 -1.3 -3.2 -1.4 1.2 -1.7 1.o -2.6 -2.2 0.7 2.9 0.9 -0.2 0.5 -5.0 0.39 4.21 -0.51 5.3 1 7.00 6.50

2.60 1.85 -1.12 -0.54 1.44 -2.07 1.20 0.67 4.22 2.77 0.03 -1.12 -1.34 0.73 1.70 1.07 2.85 -1.94 -1.42 1.54 2.48 1.50 0.55 2.03 -1.25 -1.06 4.02 2.45 9.32 3.58 10.01

0.96 0.33 -2.01 -2.08 0.48 0.20 0.20 0.60 2.08 0.37 -0.10 -0.45 1.54 1.73 0.63 1.22 1.25 0.73 -0.34 0.54 3.43 1.09 0.53 1.40 -0.41 -2.35 1.13 1.25 6.48 2.03 6.34

1.oo 0.54 -0.80 -1.02 0.75 0.35 0.57 -0.02 1.40 0.22 -0.98 0.11 0.38 -0.11 2.14 0.32 0.15 0.46 -1.82 -0.99 3.14 0.39 0.28 0.54 -0.25

continued

-2.68 7.54 -2.03 7.26 0.80 11.46

0.72 0.27 -4.87 -1.42 -4.93 -0.95 4.12 -4.53 -0.64 -0.74 -0.14 -2.99 1.51 -0.87 -0.57 1.33 2.15 1.52 -8.66 -3.07 4.91 1.83 1.98 -1.72 -1.40

CF,CN CH,CCH (propyne) CH2=C=CH, (allene) C3H4 (cyclopropene) CH3CH=CH2 (propylene) C3H6 (cyclopropane) C3H8 (propane) CH2CHCHCH, (butadiene) C4H, (2-butyne) C4H6 (methylene cyclopropane) C4H6 (bicyclobutane) C4H, (cyclobutene) C,H, (cyclobutane) C,H, (isobutene) C,H,, (trans-butane) C4H,, (isobutane) C,H, (spiropentane)

c2c14

F2O CIF, C2F4

0 3

PF,

CS2 COF, SiF, SiCl, N2O ClNO NF,

cos

Species

Table 8 Continued G3 -35.9 24.7 -145.7 -384.9 -158.4 21.4 13.4 -31.6 -224.2 34.9 6.5 -36.0 -162.3 -6.4 -120.2 44.4 45.0 68.4 4.7 13.4 -25.3 26.7 35.2 46.4 54.5 39.5 6.8 -4.0 -30.4 -32.3 44.7

Exptb

-33.1 28.0 -149.1' -386.0 -158.4 19.6 12.4 -31.6 -229.1 34.1 5.9 -38.0 -157.4 -3.0 -118.4 44.2 45.5 66.2 4.8 12.7 -25.0 26.3 34.8 47.9 51.9 37.4 6.8 -4.0 -30.0 -32.1 44.3

AH: (298 K) (kcaVmol)

2.8 3.3 -3.4 -1.1 0.0 -1.7 -1 .o 0.1 4.8 -0.8 -0.6 -1.9 4.9 3.4 1.8 -0.2 0.5 -2.2 0.0 -0.7 0.3 -0.4 -0.4' 1.5 -2.6 -2.1 0.0 0.0 0.4 0.2 -0.4

G3 3.38 5.31 -4.02 -2.14 1.05 -2.45 -0.49 -0.84 -5.21 -2.31 -1.72 -4.32 4.30 4.98 0.96 0.28 1.45 -1.62 0.50 -0.75 0.08 0.77 -0.06 2.01 -2.28 -1.41 0.00 0.37 0.18 0.03 -0.49

G3(MP2) 2.7 2.1 -0.5 -7.1 3.8 -0.6 0.8 3.7 -5.4 1.1 0.5 0.4 8.2 4.6 4.8 -1.5 -0.9 -2.9 -0.5 -0.9 0.4 -1.7 -2.1 0.3 -3.0 -2.9 -0.2 -0.6 0.4 0.3 -1.4

G2 4.79 4.92 0.76 -5.26 3.27 0.69 2.82 5.37 -3.69 2.78 2.05 2.69 10.06 8.54 5.85 -2.20 -1.52 -3.47 -1.30 -1.52 -0.52 -2.56 -3.06 -0.47 -3.69 -3.76 -1.14 -1.64 -0.78 -0.82 -2.39

G2(MP2) 3.92 5.53 -1.65 2.94 8.87 -0.17 1.51 2.78 -1.20 0.45 0.29 -0.05 7.09 11.19 3.23 -0.48 1.26 -3.60 -1.40 -1.70 -0.89 -2.19 1.96 0.21 -3.69 -3.42 -1.5 1 -1.81 -0.81 -1.08 -1.51

CBS-Q

Deviation (kcal/mol)

2.71 3.77 -2.63 2.08 4.70 1.16 -0.24 1.91 -2.35 -0.41 -0.25 -4.02 5.51 7.68 2.51 -0.09 1.81 -3.22 -1.17 -1 .so -0.77 -1.88 -0.56 0.65 -3.22 -2.95 -1.40 -1.44 -0.62 -0.89 -0.98

CBS-QB3

2.75 4.57 -3.74 -3.75 8.14 -5.11 -0.95 -4.91 -1.75 -12.08 -9.73 -10.87 4.30 9.45 -1.24 2.02 2.14 1.47 1.96 5.95 3.28 1.91 14.43 7.88 5.82 3.09 5.40 3.33 4.67 4.66 11.01

CBS-4

C,H, (benzene) CH2F2 CHF, CH,CI, CHCI, CH,NH, (methylamine) CH,CN (methyl cyanide) CH,NO, (nitromethane) CH,ONO (methyl nitrite) CH,SiH, (methyl silane) HCOOH (formic acid) HCOOCH, (methyl formate) CH,CONH, (acetamide) C,H,NH (aziridine) NCCN (cyanogen) (CH, ),NH (dimethylamine) CH,CH,NH, (trans-ethylamine) CH,CO (ketene) C,H,O (oxirane) CH,CHO (acetaldehyde) HCOCOH (glyoxal) CH,CH,OH (ethanol) CH,OCH, (dimethylether) C,H,S (thiirane) (CH, ),SO (dimethyl sulfoxide) C,H,SH (ethanethiol) CH,SCH, (dimethyl sulfide) CH,=CHF (vinyl fluoride) C,H,Cl (ethyl chloride) CH,=CHCI (vinyl chloride) CH2=CHCN (acrylonitrile) CH,COCH, (acetone) CH,COOH (acetic acid) CHiCOF (acetyl fluoride)

19.7 -107.7 -166.6 -22.8 -24.7 -5.5 18.0 -17.8 -15.9 -7.0 -90.5 -85.0 -57.0 30.2 73.3 -4.4 -11.3 -11.4 -12.6 -39.7 -50.7 -56.2 -44.0 19.6 -36.2 -11.1 -8.9 -33.2 -26.8 8.9 43.2 -51.9 -103.4 -105.7 20.4 -108.4 -167.1 -22.3 -24.6 -4.5 17.8 -17.8 -15.7 -6.8 -90.6 -86.6 -55.9 31.4 73.6 -3.5 -11.3 -12.1 -12.6 -39.8 -51.6 -56.3 -44.4 18.8 -34.7 -10.7 -8.9 -34.4 -26.7 5.3 44.8 -52.0 -103.3 -105.8

-0.6 0.7 0.5 -0.5 0.0 -1.0 0.2 0.0 -0.2 -0.2 0.1 1.6 -1.1 -1.2 -0.3 -0.9 0.0 0.8 0.0 0.1 0.9 0.1 0.4 0.8 -1.5 -0.4 0.0 1.2 -0.1 3.6 -1.6 0.0 -0.1 0.1

1.13 0.19 -0.07 0.01 0.67 -1.39 0.11 -1.64 -1.19 -0.35 -0.50 0.67 -2.12 -1.41 -0.58 -1.45 -0.41 0.83 -0.45 -0.17 0.53 -0.46 -0.30 1.50 -1.41 0.31 0.58 1.43 0.05 4.39 -1.20 -0.45 -0.95 -0.54 -3.9 3.1 4.3 0.6 1.0 0.0 -0.1 2.7 2.7 0.4 2.0 3.8 0.2 -0.3 -1.5 0.3 0.8 0.8 1.3 1.3 2.9 1.o 2.0 0.7 -1.4 -0.4 0.2 1.7 0.8 3.7 -2.7 1.1 1.5 2.0 __

-5.05 3.88 5.19 2.30 3.81 -0.21 -0.51 3.73 4.12 0.07 3.21 4.79 0.41 -0.51 -1.70 -0.18 0.34 1.16 1.69 1.46 3.92 1.14 2.24 1.59 0.49 0.14 0.73 1.80 1.15 4.29 -3.29 0.95 2.37 2.58 -1.48 1.73 2.38 2.45 5.35 -1.41 -0.72 1.54 1.70 0.77 1.11 2.38 -0.89 -1.49 -1.43 -1.47 -0.82 0.64 0.22 0.04 1.49 -0.15 0.47 1.49 -0.11 0.38 0.61 0.73 0.87 4.10 -3.03 0.02 0.55 0.77 11.03 -1.39 1.24 0.13 1.36 -0.02 3.20 0.87 5.65 2.74 0.09 -0.87 -0.08 -1.15 0.06 1.55 1.70 -1.90 2.81 0.60 0.96 -0.42 2.30 0.74 -0.63 0.97 -0.85 3.06 -1.03 4.69 -0.84 1.21 -0.30 2.25 0.33 1.09 0.51 2.41 0.11 1.57 1.47 1.45 0.03 1.28 0.85 0.38 1.21 7.35 -0.502.82 0.12 3.35 0.37 3.46 0.51 0.96 0.08 3.04 3.36 6.26 -2.38 -3.74 0.68 3.09 0.81 0.42 0.21 1.02 continued

G3(MP2) 0.03 0.46 -0.15 0.27 -0.46 0.00 1.98 -0.48 0.84 1.14 1.16 -0.68 1.28 0.03 -0.46 -1.16 -1.68 1.21 0.03 -0.18 -0.97 -1.21

G3 0.2 0.4 0.5 1.1 0.2 -0.5 -0.2 -1.2 -0.1 0.5 0.5 -1.2 1.1 0.1 -0.1 -0.8 -1.2 0.8 0.2 0.0 -0.7 -0.2

G3 -58.2 -31.9 -65.7 -52.8 -5.9 -7.8 27.7 27.1 33.7 -0.5 33.7 136.3 70.5 -2.5 -3.9 4.9 -2.5 29.0 28.7 21.5 13.0 8.1

Exptb

-58.0 -31.5 -65.2 -51.7 -5.7 -8.3 27.5 25.9 33.6 0.0 34.2 135.1 71.6 -2.4 -4.1 4.1 -3.7 29.8 28.9 21.5 12.3 7.9

AH? (298 K) (kcaYmo1)

CBS-QB3 0.12 0.80 0.00 1.06 0.14 -0.53 0.88 -1.03 0.23 1.13 -0.02 -1.95 -0.28 -0.21 -0.26 -0.54 -1.82 0.69 -1.11 -1.10 -1.95 1.34

CBS-Q 1.56 1.54 -0.17 0.73 -0.53 -0.75 1.19 -1.48 -0.14 1.13 0.39 -2.08 -0.23 0.08 -0.33 -1.08 -1.88 0.89 -1.18 -1.47 -2.38 2.26

G2(MP2) 2.97 1.29 1.07 2.28 0.75 -0.83 -1.34 -2.67 -2.91 1.08 0.01 -4.28 -1.80 0.60 0.14 -0.85 -1.76 0.06 -1.80 -2.32 -3.30 2.02

G2 1.8 1.1 1.2 2.3 1.4 -1.0 -2.4 -2.2 -2.2 1.1 -0.3 -3.6 -1.1 0.4 -0.3 -0.7 -1.4 -0.1 -1.0 -1.3 -2.0 0.7

Deviation (kcaVmo1) 3.09 4.86 2.97 1.98 3.23 3.33 8.21 6.06 8.44 0.37 0.33 -1.34 0.95 1.29 -0.70 -3.32 -1.87 1.71 1.40 2.97 3.67 0.74

CBS-4

"Deviation = experiment - theory. The 148 enthalpies of formation in this table are from the G2/97 test set (Refs. 3 and 30). The G2-1 subset molecules are ones from the original G2 test set (Ref. 29), and the (22-2 subset are new ones added to the G2/97 test set. bSee Ref. 3 for experimental references.

CH,COCl (acetyl chloride) CH,CH,CH,CI (propyl chloride) (CH, ),CHOH (isopropanol) C,H,OCH, (methyl ethyl ether) (CH, ),N (trimethylamine) C,H,O (furan) C,H,S (thiophene) C,H,N (pyrrole) C,H,N (pyridine) H2 HS CCH CZH3 CH,CO (2A') H2COH (2A) CH30 CS (2A') CH,CH20 (,A") CH,S (2A') C,H, (2A') (CH3),CH (2A') (CH,),C (t-butyl radical) NO,

Species

Table 8 Continued

Quantum Chemical Methods 175

Figure 4 Relative CPU times (for a single processor workstation) and accuracy for G3 energy calculations on molecules containing up to 1 0 carbons.

Several empirical corrections are added to the resulting energies in the CBS methods to remove the systematic errors in the calculations (see Table 10).The CBS-Q method contains a two-electron correction term similar in spirit to the higher level correction used in G2 theory, a spin correction term to account for errors resulting from spin contamination in UHF wavefunctions for open-shell systems, and a correction to the sodium atom to account for core-valence correlation effects. The CBS-4 and CBS-q methods also contain a one-electron

Table 9 Average Absolute Deviations (kcal/mol) from Experiment of Various Composite Ab Initio Molecular Orbital Methods for Ionization Energies, Electron Affinities, Proton Affinities, and Enthalpies of Formation in the G2/97 Test SetU Ionization energies (85) Electron affinities ( 5 8 ) Proton affinities (8) Enthalpies of formation (148) Nonhydrogen compounds (35) Hydrocarbons (22) Substituted hydrocarbons (47) Inorganic hydrides (15) Radicals (29) All (299)

G3IB3LYP

G3(MP2)//B3LYP

CBS-QB3

1.10 0.95 1.22 0.93 1.65 0.57 0.70 0.78 0.76 0.99

1.37 1.44 0.89 1.13 1.99 0.75 0.70 0.93 1.1 8 1.25

1.35 1.09 1.26 1.19 1.96 1.27 0.82 0.95 0.93 1.22

&Thenumbers in parentheses refer to the number of pieces of data used for the comparisons.

1 76 Theoretical Methods for Computing Enthalpies of Gaseous Compounds Table 10 Enerav Corrections for CBS Methods Method

Correctionsa3b

E(SCF) = HFICBSB3 AE(MP2) = MP2/CBSB3 - HF/CBSB3 AE(QC1) = QCISD(T)/6-31+G(d‘) - MP4(SDQ)/6-3l+G(d’) AE(MP3,4) = MP4(SDQ)/CBSB4- MP2/CBSB4 AE(CBS) = E2(CBS)/CBSB3- AE(MP2) AE(1NT) = CBS-INT/CBSB3 - E2(CBS)/CBSB3 AE(EMP) = -O.O0579(0Iii) AE(SP1N) = -O.O0954(A(S’)) ZPE = HF/6-31G(d’) frequencies scaled by 0.91844 E, = E(SCF) + AE(MP2) + AE(QC1) + AE(MP3,4) + AE(CBS) + AE(1NT)+ AE(EMP) + AE(SP1N) + ZPE

CBS-Q

E(SCF) = HF/CBSBl AE(MP2) = MP2/CBSB2 - HF/CBSB2 AE(MP3,4) = MP4(SDQ)/6-31G - MP2/6-31G AE(CBS) = E2(CBS)/CBSB2- AE(MP2) AE(1NT) = CBS-INT/CBSBZ - E2(CBS)/CBSB2 AE(EMP) = -O.O0552(OIii) One-electron empirical correction = -0.00455( n, + np) AE(SP1N) = -O.O3843(A(S2)) ZPE = HF/3-21G( ”) frequencies scaled by 0.91671 E, = E(SCF) + AE(MP2) + AE(MP3,4) + AE(CBS) + AE(1NT) + AE(EMP) + one-electron correction + AE(SP1N) + ZPE CBS-4

E(SCF) = HF/CBSB3 AE(MP2) = MP2/CBSB3 - HFlCBSB3 AE(CC) = CCSD(T)/6-31+G(df)- MP4(SDQ)/6-31+G(df) AE(MP3,4) = MP4(SDQ)/CBSB4- MP2/CBSB4 AE(CBS) = E2(CBS)/CBSB3- AE(MP2) AE(1NT) = CBS-INTICBSB3 - E2(CBS)/CBSB3 AE(EMP) = -O.O0579(OIii) AE(SP1N) = -0.00954(A(S2)) ZPE = B3LYP/CBSB7 frequencies scaled by 0.99 E, = E(SCF) + AE(MP2) + AE(CC) + AE(MP3,4) + AE(CBS)+ AE(1NT) + AE(EMP) + AE(SP1N) + ZPE

CBS-QB3

E2 = second order perturbation correction CBS-INT = difference between the total CBS correction to the MP4 or QCISD(T) energy and the total CBS correction to the second-order correlation energy OIii = IS1,,21L,,where I,, = ( X J 2 is the intraorbital interference factor and ISI,? = jlaQf,P+,ld~ is the absolute overlap integral A(S2) = ((S2) from HF/CBSB3 calculation - ( S 2 ) of the pure spin state) for CBS-Q and CBS-QB3 and ((S2) from HF/CBSBl calculation - (S2) from pure spin state) for CBS-4 ZPE = zero-point vibrational energy aThe CBSBI, CBSB2, CBSB3, CBSB4, and CBSB7 basis sets defined in Refs. 62 and 101. bThe 6-31GcGt basis set from the literature is listed as “6-31+G(d‘)” in computer programs such as Gaussian; the notations are synonymous.

Quantum Chemical Methods 177 correction term to improve the computed ionization energies and electron affinities. The CBS-Q method has an average absolute deviation from experiment of 1.46 kcal/mol for the G2/97 test set and an average absolute deviation of 1.54 kcal/mol for the 148 enthalpies of formation of neutrals in this test set. A complete summary of the average absolute deviations is given in Table 5 , including a breakdown into different types of molecules. The CBS-q and CBS-4 methods are less accurate, with average absolute deviations of 3.13 and 4.12 kcaYmol, respectively, for the 148 neutral enthalpies of formation in the G2/97 test set. CBS-QB3 is a modification of the CBS-Q method that uses B3LYP/63 1l G (2d,d,p) geometries and zero-point energies.101 The zero-point energies are scaled by 0.99. This differs from CBS-Q in which frequencies are calculated at the HF/6-31Gt level and the geometries are calculated at the MP2/6-31Gt level. In addition, the QCISD(T)/6-31+Gtenergies in the CBS-Q method are replaced with CCSD(T)/6-31+Gt energies, and spin-orbit interactions are included in the atomic calculations, as with G3 theory. The empirical coefficient of the overlap-interference term is changed to -0.00579 hartree, and the spin correction parameter is changed to -0.00954 hartree. A summary of the steps in CBS-QB3 is given in Table 10. This method is an improvement over the CBS-Q method. A summary of the average absolute deviations is given in Table 9, including a breakdown into different types of molecule. The CBS-QB3 method has an average absolute deviation from experiment of 1.22 kcal/mol for the G2/97 test set and an average absolute deviation of 1.19 kcaYmol for the 148 enthalpies of formation of neutrals in this test set. CBS-RAD is a modification of the CBS-Q method that is specifically designed to calculate the enthalpies of formation of radicals accurately.102 Optimized geometries and scaled zero-point energies (empirical scale factor = 0.9776) are obtained at the QCISD/6-3 1G(d) level. The QCISD(T)/6-31+Gt energies in the CBS-Q method are replaced with CCSD(T)/6-31+Gtvalues. For large radicals, RMP2/6-31G(d)geometries are a good alternative. Other modifications of this method include the use of B3LYP/6-31G(d)geometries. Because these geometries are sensitive to spin contamination, they cannot be used in systems with large spin contamination. B3LYP/6-31G(d) zero-point energies are not as sensitive to spin contamination and can be used for large radicals. The CBS methods are available as keywords in the Gaussian program. The CBS-Q, CBS-q, and CBS-4 methods are incorporated in Gaussian 9475 and later versions. The CBS-QB3 method is available in Gaussian 98.99 Scaling Methods Another approach based on molecule-independent parameters is a multiplicative scaling of the calculated energy using parameters determined by fitting to experimental data. In some of the first work in this area Truhlar, Gordon, and coworkers103-106 suggested scaling the correlation energy, referred to as the “scaling all correlation energy” (SAC) method, to improve reaction energies and barriers. The parameters were optimized for different bond-breaking reactions and bond types, and the authors also pre-

1 78 Theoretical Methods for Computing Entbalpies of Gaseous Compounds sented “standard” values based on averaging over several different bonds. Subsequently, Siegbahn et al.107-110 presented further work based on the principle that in a balanced treatment roughly the same percentage of the correlation energy is obtained for every system (atomic and molecular). This method, referred to as the parameterized correlation method (PCI-X), has a single adjustable parameter X that scales the total correlation energy obtained with a modest basis set. A single common parameter is used for all systems and is obtained from fitting to a set of well-known experimental data. Recently, Truhlar and coworkers111-114 presented more elaborate schemes for scaling based on molecule-independent parameters fitted to a set of experimental data. The motivation in this work was to find accurate methods for calculating continuous potential energy surfaces. These methods are based on multicoefficient (MC) fits in which the total energy is written as a linear combination of energy terms (basis set contributions from different levels of theory). The investigators used total energies based on correlation consistent basis sets, referred to as the multicoefficient correlation method (MCCM),112 the G2 energy equation (MCG2),114 and the G3 energy equation (MCG3).l13 Up to 10 coefficients are used in the fits. The test set of experimental data included 49 molecules for the MCCM method, 3 1 for the MCG2 method, and 49 for the MCG3 method. The coefficients were obtained by least-squares fitting to the training set. The resulting average absolute deviations are very good. For example, a nine-parameter fit for MCG3 gives an average absolute deviation of 0.89 kcal/mol for the 49 molecules. For comparison, G3 theory gives an average absolute deviation of 1.01 kcal/mol for the G2/97 test set of 299 energies. Although the MCG3 results are based on a much smaller set and use experimental zero-point energies, the results are intriguing and suggest that these methods are promising. Methods with Molecule-Dependent Empirical Parameters Finally we describe several methods that combine molecule-dependent empirical parameters with a moderate level ab initio molecular orbital method. The BAC-MP4 method of Melius and coworkers115-118 combines a computationally inexpensive molecular orbital method with a bond additivity correction. This procedure uses a set of accurate experimental data to obtain a correction for bonds of different types that is then used to adjust calculated thermochemical data such as enthalpies of formation. Quite accurate results can be obtained if suitable reference molecules are available and if the errors in the calculation are systematic. The computational methodology is based on an MP4/6-3 l G (d,p)//HF/6-3 l G (d) calculation. A pairwise additive empirical bond correction is derived for different bonds from fitting to experimental enthalpies of formation or in some cases to high quality ab initio computations. In addition, for open-shell molecules an additional correction is needed to compensate for spin contamination of the wavefunction from higher spin states in the unrestricted Hartree-Fock (UHF) method.

Quantum Chemical Methods 179 The use of isodesmic reactions is another example of this approach, which can be quite accurate for molecules having no unusual bonding.119-123 In the isodesmic approach a reaction is chosen with the same number of chemical bonds of each formal type (e.g., C-C, C=C, C-N, C-H) on both sides of the reaction, and all the species, except the species of interest, have accurate experimental thermochemical data available.119 A moderate level of theory is used to calculate the reaction energy, and the enthalpy of formation of the unknown species is then extracted. Use of specially chosen reactions can give quite accurate enthalpies of formation because of cancellation of correlation effects and also because of use of accurate experimental data. However, suitable reference molecules are often not available, making this method inapplicable in many cases. This type of approach has been used since the 1970s to improve thermochemical data.119 Recently, the bond additivity approach has been combined with more accurate ab initio methods to improve the accuracy of these methods.120-123 For example, when the isodesmic approach is applied to a subset of the G2/97 test set, the average absolute deviation of G2 theory improves from 1.49 to 0.54 kcal/mol.l21 A summary of some results for the isodesmic approach, when applied to a subset of 37 molecules in the G2/97 test set, is given in Table 11. Peterson and coworkers123 presented a slightly different formulation of the isodesmic approach in which isodesmic bond additivity corrections for bonds of several types are determined by least-squares fits to the enthalpies of formation of 76 organic species. This type of approach also reduces the average absolute deviations of standard methods such as G2 or CBS-Q to around 0.5-0.8 kcal/mol. Recently, Friesner et al.124 proposed a method referred to as 52 theory to predict accurate thermochemical data. This approach is based on the generalized valence bond-localized M~ller-Plesset method (GVB-LMP2) and includes parameters that depend on the number of electron pairs and whether the pairs are 0 or 71: types. Thus, the parameterization in the 52 method is molecule dependent. The GVB-LMP2 method scales as n3 as opposed to n6 or n7 for the MP4, QCISD, or CCSD methods, so 52 is much faster than G2. The J2 method

Table 11 Comparison of Average Absolute Deviations for Standard and Isodesmic

Methods.

Average Absolute Deviation (kcal/mol)

Method

Standard

Isodesmic

0.67 0.71

0.61 0.75 0.54 0.64

~

G3 G3 (MP2) G2 G2(MP2) B3LYP BLYP ~~

1.46

1.95

2.34 7.26 ~~~

~

USubset of 37 molecules from the G2/97 test set.

1.29

1.88

180 Theoretical Methods for Computing Enthalpies of Gaseous Compounds

(with five parameters) performs much better than G2 theory (0.87 vs. 1.63 kcal/ mol for a set of 67 molecules), although it does not do as well as G3 theory (0.87 vs 0.74 kcallmol). It has been developed only for first-row molecules.

Density Functional Methods Over the past 30 years, density functional theory has been widely used by physicists to study the electronic structure of solids. More recently chemists have been using the Kohn-Sham version of density functional theory (DFT)125 as a cost-effective method to study properties of molecular systems. The density functional models currently being used by quantum chemists may be divided into nonempirical and empirical types. The simplest nonempirical type is the local spin density functional, which treats the electronic environment of a given position in a molecule as if it were a uniform gas of the same density at that point. One of these is the SVWN functional that uses the Slater (S) functional126 for exchange and the uniform gas approximate correlation functional of Vosko, Wilk, and Nusair (VWN).127The more sophisticated functional BPW91 combines the 1988 exchange functional of Beckel28 with the 1992 correlation functional of Perdew and Wang.129 Both components involve local density gradients as well as densities. The Becke part involves a single parameter that fits the exchange functional to accurate computed atomic data. The BP86 functional is similar but uses the correlation functional of Perdew.130 The BLYP131 functional also uses the Becke 1988 part for exchange, together with the correlation part of Lee, Yang, and Parr.132 This LYP functional is based on a treatment of the helium atom and hence really only treats correlation between electrons of opposite spin. A number of other functionals use parameters that are fitted to experimental energies in the original G2 test set. These give a functional that is a linear combination of Hartree-Fock exchange, 1988 Becke exchange, and various correlation parts. This idea was introduced by Becke,133 who obtained parameters by fitting to the molecular data. This is the basis of the B3PW91 functional. The others (B3P86 and B3LYP) are constructed in a similar manner, although the parameters are the same as in B3PW91.134 In several validation studies for molecular geometries and frequencies, DFT has given results of quality similar to that of MP2 theory.135>136 DFT has also been examined for use in calculation of thermochemical data.3,30,133,136-139 For example, Beckel33 found that the B3PW91 functional with a numerical basis set gave an average absolute deviation of 2.4 kcal/mol for the 55 atomization energies in the original G2 test set, about twice as large as from G2 theory. He also found similar performance for ionization potentials and proton affinities. Bauschlicher138 examined several DFT methods (BLYP, B3LYP, BP86, B3P86, BP) for the 55 atomization energies in the original G2 test set using the same 6-311+G(3df,2p) basis set. He found that B3LYP gave the best agreement with experiment (average absolute deviation of 2.20 kcal/mol).

Quantum Chemical Methods 181 The average absolute deviations for the 148 enthalpies of formation in the G2/97 test set for seven density functionals (SVWN, BLYP, BPW91, BP86, B3LYP, B3PW91, B3P86) are listed in Table 12.3 The results are based on the 6-31 1+G(3df,2p)basis set and most use MP2(FULL)/6-31G" geometries and scaled HF/6-3 1G" zero-point energies. The local density approximation has a very large average absolute deviation (91 kcal/mol) owing to systematic prediction of overbinding. The gradient-corrected functionals have average absolute deviations of 20 kcaYmol or less for enthalpies of formation. The B3LYP functional performs the best, with an average absolute deviation of 3.05 kcal/mol. The B3PW91 functional is only slightly worse, with an average absolute deviation of 3.49 kcaYmo1. The methods containing the Becke three-parameter functional perform better than the Becke 1988 functional128 for atomization energies, ionization energies, and proton affinities. There has been considerable effort devoted to finding improved functionals.140-147 Becke proposed a new density functional within the generalized gradient approximation that includes second-order gradients and noninteracting kinetic energy density.140.141 This method has more empirical parameters than the B3LYP method (10 vs. 3) but gives a significant improvement in accuracy for the G2/97 test set. A number of studies have been published on the Perdew-Burke-Ernzerhof exchange-correlation functiona1.142-144 This functional is found not to perform as well as the empirical B3LYP functional, but a hybrid based on it does nearly as we1L143 Kafafi and El-Gharka~y1~8.1~9 proposed a simple coupling scheme between HF, gradient-corrected local spindensity exchange, and Vosko, Wilk, and Nusairl27 correlation functionals. Combined with an empirical atom equivalent scheme that corrects the atomic energies, the authors found improved results for 150 species compared to the B3LYP method.

Semiempirical Molecular Orbital Methods Semiempirical molecular orbital methods23-25 were developed several decades ago when computers were much slower and had less memory. These methods treat only valence electrons and neglect certain interaction terms or replace them with empirical terms. These approximations greatly reduce the computational cost, but make the outcome dependent on having a good set of experiment data to use in the parameterization. The basic assumption of semiempirical methods is the zero-differential-overlap approximation (ZDO).This approximation neglects all products of basis functions depending on the same electron coordinates that are located on different atoms. There are three general types of integral approximation.14~150The best level is neglect of diatomic differential overlap approximation (NDDO),which keeps higher multipoles of charge distributions in the two-center interactions. The second approach is intermediate neglect of differential overlap approximation (INDO), which neglects all two-center, two-electron integrals that are not of the Coulomb type.

4.14 3.03 1.48 3.05 5.24 2.76 2.10 1.84 2.80 3.31

(4.17) (2.82) (1.32) (3.09) (5.14) (2.92) (2.22) (1.74) (2.85) (3.29)

B3LYPb 4.35 3.35 1.08 3.49 5.04 3.92 2.77 1.99 3.21 3.64

B3PW91 13.24 13.73 1.04 17.99 7.90 30.81 25.49 7.86 13.53 15.38

B3P86 5.98 2.62 1.96 7.11 10.41 8.09 6.10 3.13 6.09 5.78

BLYP

BP86 4.62 4.45 1.44 20.22 16.71 25.82 26.80 8.16 15.76 12.27

BPW91 5.09 2.79 1.35 7.88 12.35 4.85 7.99 4.21 6.48 5.93

13.61 16.08 5.61 90.91 73.68 133.71 124.41 33.65 54.55 52.40

SVWN

acalculated using MP2(fu11)/6-31G' geometries and 0.8929 scaled HF16-31G" frequencies. bResults in parentheses use B3LYP/6-31Ga geometries and 0.96 scaled frequencies. As can be seen, the predicted ionization potentials, electron affinities, proton affinities, and enthalpies are not highly sensitive to the molecular geometries and vibrational frequencies adopted.

IP (83) EA (58) PA (8) Enthalpies (148) Nonhydrogen compounds (35) Hydrocarbons (22) Substituted hydrocarbons (47) Inorganic hydrides (15) Radicals (29) All (297)

TvDe

Table 12 Average Absolute Deviations (kcaYmo1)from Experiment of Various Density Functional Methods for Some Experimental Observables, Including 148 Neutral Enthalpies of Formation, in the G2/97 Test Seta

Quantum Chemical Methods 183 The third approach is the complete neglect of differential overlap approximation (CNDO), in which only the one- and two-center, two-electron integrals remain. The direct application of these methods (NDDO, INDO, or CNDO) is not useful because of the approximations, so it is necessary to include parameters in place of all or some of the integrals. These parameters are based on atomic or molecular experimental data. Several versions of modified INDO (MINDO)that employ such a parameterization have been proposed. These include MINDOll, MINDOR, MIND0/2’, and MIND0/3, only the last of which resulted in a quantum chemical program that was widely used. MIND0/3151 is parameterized for H, B, C, N, 0,F, Si, P, S, and C1, although certain combinations of these atoms are not parameterized. The MIND0/3 method is no longer heavily used because the parameterized NDDO methods are generally more accurate. Various parameterizations of NDDO have been proposed. Among these are modified neglect of diatomic overlap (MND0),152 Austin Model 1 (AM1),153 and parametric method number 3 (PM3),154 all of which often perform better than those based on INDO. The parameterizations in these methods are based on atomic and molecular data. All three methods include only valence s and p functions, which are taken as Slater-type orbitals. The difference in the methods is in how the core-core repulsions are treated. These methods involve at least 1 2 parameters per atom, of which some are obtained from experimental data and others by fitting to experimental data. The AM1, MNDO, and PM3 methods have been focused on ground state properties such as enthalpies of formation and geometries. One of the limitations of these methods is that they can be used only with molecules that have s and p valence electrons, although MNDO has been extended to d electrons, as mentioned below. The performance of the semiempirical methods for the calculation of thermochemical data depends on the extent to which the physics is included in the model and how well the neglected features can be accounted for by the parameterization. These methods can be assessed by validation against accurate experimental data or high level ab initio predictions. A summary of results for four semiempirical methods (MIND0/3, MNDO, AM1, and PM3) for the neutral enthalpies of formation in the G2/97 test set is given in Table 13. Overall, the newest method, PM3, does the best with an average absolute deviation of 7.02 kcal/mol. It has average absolute deviations of 3.91 and 4.27 kcal/mol for the subgroups of hydrocarbons and substituted hydrocarbons, respectively. Numerous other semiempirical methods have been proposed. The MNDO method has been extended to d functions by Theil and coworkers and is referred to as MNDO/d.ls5~1S6For second-row and heavier elements, this method does significantly better than other methods. The semi-ab initio is based on the NDDO approximation and calculates method 1 (SAM1)157>15* some one- and two-center two-electron integrals directly from atomic orbitals.

184 Theoretical Methods for Computing Enthalpies of Gaseous Compounds Table 13 Summary of Average Absolute Deviations (kcal/mol)Between Experimental Heats of Formational and Semiempirical Results on the G2/97 Test Set0

Nonhydrogen compounds Hydrocarbons Substituted hydrocarbons Inorganic hydrides Radicals All

MINDOI3b

MNDOc

AMld

PM3e

11.63 (20) 7.14 (22) 6.64 (45) 3.62 (11) 13.53 (24) 8.63 (122)

15.45 (33) 4.77 (22) 5.36 (47) 4.81 (15) 13.83 (28) 9.15 (145)

16.00 (31) 4.80 (22) 5.86 (47) 4.84 (15) 14.11 (28) 9.48 (143)

10.12 (29) 4.04 (22) 3.82 (47) 5.58 (14) 12.25 (28) 7.02 (140)

"The number of species used in the comparisons is given in parentheses. bExcludes Na,, Si,, NaCI. .Excludes LiH, BeH, SiH, (3B, ), SiH,, PH,, Liz, LiF, Na,, Si,, P,, NaC1, SiO, CIO, FCl, Si,H,, HOCI, BCI,, AIF,, AlCI,, SiF,, SiCI,, CINO, PF,, CIF,, CH,SiH,, CH,COCI. dExcludes Liz, LiF, Na,, Si,, NaCI. ?Excludes LiH, Liz, LiF, Na,, Si,, NaCI, BF,, BCI,.

For a large set of neutral, closed-shell first-row molecules, the average absolute deviation is 4.44 kcal/mol in SAM1, compared to 7.24 and 4.85 kcal/mol in AM1 and PM3, respectively.157~15* The electronic energy calculated by the MIND0/3, MNDO, AM1, and PM3 methods is normally converted automatically in the computer program (Table 2) to an enthalpy of formation by subtracting the electronic energy of the isolated atoms and adding the experimental atomic enthalpies of formation. The zero-point energies and temperature corrections (0 to 298 K) are assumed to be included implicitly by the parameterization. For a molecule ABH,, the AHf is defined in these methods as

Although this procedure works quite well,1s9 it is not very satisfactory theoretically. Problems could arise if these methods are applied to transition states where there is one less vibrational frequency. Problems could also arise if zeropoint energies of thermal corrections were to change during the course of a reaction. In summary, semiempirical methods such as AM1 or PM3 have the advantage of being computationally very fast and allowing large molecules to be computed with minimal computer resources. The semiempirical methods are not nearly as accurate as the ab initio methods or even density functional methods. Some disadvantages of these methods include the following: (1)they can be applied only to molecules containing elements for which they have been parameterized, (2) the errors are less systematic than at an ab initio level of calculation, and (3) semiempirical methods depend on the availability of accurate experimental data (or reliable ab initio data).

Quantum Chemical Methods 185

B

Illustrative Exam les of Quantum Chemical Metho s Example 1. Calculation of the G2 Enthalpy of Formation at 298 K of CH, The geometry of CH4 is first optimized at the HF/6-31G(d)level, and the HF/6-31G(d)vibrational frequencies are calculated. A scale factor of 0.8929 is applied to the vibrational frequencies that are used to calculate the zero-point energies and the thermal correction to 298 K [keyword of freq(readisotopes)in the Gaussian 98 computer program]. The geometry is then optimized at the MP2(FULL)/6-31G(d)level and used to obtain the single-point energies listed below; note that all these energies can be obtained from four single-point energy calculations [QCISD(T,E4T)/6-3llG(d,p), MP4/6-31 l+G(d,p), MP4/631 1G(2df,p),and MP2/6-31 l+G(3df,2p)]. Energies (hartree) ~

QCISD(T)/6-31lG(d,p) MP2/6-311G(d,p) MP4/6-31 lG(d,p) MP2/6-3 11+G(d,p) MP4/6-31 l+G(d,p) MP2/&-311G(2df,p) MP4/6-311G(2df,p) MP2/6-31 l+G(3df,2p) ZPE, scaled

-40.40589 -40.3 7923 -40.40502 -40.37952 -40.40533 -40.39765 -40.42466 -40.40567 0.04266

A(QC1)= QCISD(T)/6-311G(d,p)- MP4/6-311G(d,p) = -0.00086 A(+) = MP4/6-311+G(d,p)- MP4/6-311G(d,p) = -0.00030 A(2df,p) = MP4/6-311G(2df,p) - MP4/6-311G(d,p) = -0.01964 A(MP2) = MP2/6-311+G(3df,2~) - MP2/6-311+G(d,p) MP2/6-311G(2df,p) + MP2/6-311G(d,p) = -0.00772 HLC = -A(valence e- pairs) -B[(valence e- pairs) + (unpaired e-)] = -0.00481(4) -0.00019(4 + 0) = -0.02000 E,[G2] = MP4/6-311G(d,p) + A(QC1) + A(+) + A(2df,p) + A(MP2) + HLC = -40.45355 E,[G2] = E , + ZPE = -40.41089 Using the atomic energies for carbon atom and hydrogen atom from Ref. 25, we calculate the enthalpy of formation of CH4 at 298 K as follows:

CD,(CH,) = E(C) + 4E(H)- E(CH4) = [(-37.78432) + (4)(-0.5) (-40.41088)] = 0.62657 hartree = 393.17 kcaVmol

AH? (0 K, CH4) = AH? (C, 0 K) + 4AH7 (H, 0 K) - ZDo(CH4) = (169.98) + (4)(51.63)- (393.17) = -16.67 kcal/mol

186 Theoretical Methods for Computing Enthalpies of Gaseous Compounds AH? (298 K, CH,) = AH? (0 K, CH4) + [Ho(CH,, 298 K) - Ho(CH4,O K)] - [Ho(C,298 K) - HO(C, 0 K)] - 4[Ho(H, 298 K) - Ho(H, 0 K)] = -16.67 + (2.39) - (0.25) + (4)(1.01)= -18.57 kcal/mol

The atomic thermal corrections are from Table 1, and [Ho(CH,, 298 K) Ho(CH4, 0 K)] = E,,(298 K) + Etran(298K) + Ero,(298 K) + APV(298 K) = 0.0233 + 0.889 + 0.889 + 0.593 = 2.3943 kcal/mol. The latter quantities are calculated from calculated vibrational frequencies and classical equations in Ref. 8. The calculation of the enthalpies of formation from other total energies such as from B3LYP, G2(MP2), G3, or CBS-Q is done in a similar manner except that different total energies for the molecule and atoms are used. In subsequent examples, only calculations of the total energies are described.

Example 2. Calculation of the G2(MP2) Energy for CH, The same procedure is used for geometries and frequencies as was done for the G2 energy in Example 1. Note that all the energies listed below can be obtained from two single-point energy calculations: QCISD(T)/6-311G(d,p) and MP2/6-311+G(3df,2p). Energies (hartree) QCISD(T)/6-311G(d,p) MP2/6-311 G(d,p) MP2/6-311+G(3df,2p) ZPE, scaled

-40.40589 -40.37923 -40.40567 0.042 66

A(MP2) = MP2/6-311+G(3df,2p) - MP2/6-311G(d,p) = -0.02643 HLC = -A(valence e- pairs) - B[(valence e- pairs) + (unpaired e-)] = -0.00481(4) -0.00019(4+0) = -0.02000 Ee[G2(MP2)]= QCISD(T)/6-311G(d,p)+ A(MP2) + HLC = -40.45232 E,[G2(MP2)] = E, + ZPE = -40.40966

Example 3 . Calculation of G3 Energy for CH, The same procedure as applied for the G2 energy is used for geometries and frequencies. Note that all the energies listed below for G-3 can be obtained from four single-point energy calculations: QCISD(T,E4T)/6-31G(d),MP4/63 1+G(d), MP4/6-3 1G(2df,p),and MP2(FULL)/G3Large. Energies (hartree) QCISD(T)/6-31G(d) MP2/6-31G(d) MP4/6-31G(d) MP2/6-31+G(d)

-40.35595 -40.33255 -40.35479 -40.33408

Quantum Chemical Methods 187 MP4/6-3 1+G(d) MP2/6-31G(2df,p) MP4/6-3 1G(2df,p) MP2(full)/G3Large ZPE, scaled

-40.35651 -40.38492 -40.41 114 -40.44716 0.04266

A (QCI) = QCISD(T)/6-31G(d)- MP4/6-31G(d) = -0.00116 A ( + ) = MP4/6-31+G(d)- MP4/6-31G(d) = -0.00173 A(2df,p) = MP4/6-31G(2df,p) - MP4/6-31G(d) = -0.05635 A(MP2) = MP2/G3Large - MP2/6-31+G(d)- MP2/6-31G(2df,p) + MP2/6-31G(d) = -0.06071 HLC = -A(valence e- pairs) - B(unpaired e-) = -0.006386(4) - 0.002977(0) = -0.02554 E,[G3] = MP4/6-31G(d) + A(QC1) + A(+) + A(2df,p) + A(MP2) + HLC =

-40.50028 E,[G3] = E, + ZPE = -40.45762

Example 4. Calculation of CBS-Q Energy for CH, The geometry is first optimized at the HF/6-31G(df)level and the HF/63 lG(d’) vibrational frequencies are calculated. The 6-31G(d’) basis set combines the sp functions of 6-31G with the polarization exponents of 6-311G(d,p).A scale factor of 0.91844 is applied to the vibrational frequencies that are used to calculate the zero-point energies and the thermal correction to 298 K. Next the MP2(FC)/6-31G(d’)optimization is performed and this geometry is used in all subsequent single-point energy calculations. In a frozen-core (FC) calculation, only valence electrons are correlated. Energies (hartree) QCISD(T)/G-31+G(dr) MP4(SDQ)/6-31+G(d’) MP4(SDQ)/CBSB4 MP2/CBSB4 HF/CBSB3 MP2/CBSB3 E2(CBS)/CBSB3 CBS-INT/CBSB3 OIii (S2) from CBSB3 ZPE, scaled

-40.359954 -40.356209 -40.389690 -40.367470 -40.212014 -40.404 123 -0.211511 -0.2041 72 2.07691 5 0 0.043655

E(SCF) = HF/CBSB3 = -40.212014 AE(MP2) = MP2/CBSB3 - HF/CBSB3 = -40.404123 - (-40.212014) = -0.1 92 109 - MP4(SDQ)/6-3l+G(d’)= -40.359954 + AE(QC1) = QCISD(T)/6-31+G(df) 40.356209 = -0.003745

1 88 Theoretical Methods for Computing Eiithalpies of Gaseous Compounds

AE(MP3,4) = MP4(SDQ)/CBSB4- MP2/CBSB4 = -40.389690 + 40.367470 = -0.02222 AE(CBS) = E2(CBS)/CBSB3- AE(MP2) = -0.211511 + 0.192109 = -0.01 9402 AE(1NT) = CBS-INT/CBSB3 - E2(CBS)/CBSB3= -0.204172 + 0.211511 = 0.00733 9 AE(EMP) = -O.O0533(OIii) = -0.00533(2.076915) = -0.011070 AE(SP1N)= -O.O092((S2) from CBSB3 - (S2) from pure spin state) = 0 - 0 = 0 E,[CBS-Q] = E(SCF) + AE(MP2) + AAE(QC1) + AE(MP3,4) + AE(CBS) + AE(1NT) + AE(EMP) + AE(SP1N)= -40.453221 E,[CBS-Q] = E, + scaled ZPE = -40.409566 The CBSB3 refers to the 6-311++G(3d2f,2df,2p) basis set with 3d2f on second-row atoms, 2df on first-row atoms, and 2p on hydrogen. The CBSB4 refers to the 6-31+G(d(f),p)basis set with a d function on first- and second-row atoms, an f function on some second-row atoms, and a p function on hydrogen. The E2(CBS)/CBSB3,CBS-INTKBSB3, OIii, and (S2) terms are in an archive at the end of an MP2/CBSB3 CBSextrap=(nmin=lO,pop)calculation from the Gaussian 98 computer program. This is part of a calculation using the keyword CBS-Q. The (S2)refers to S(S + l),where S is the total spin of the chemical species [e.g., for a molecule with one unpaired electron (doublet), S = 0.5 and S(S + 1)= 0.751. The EZ(CBS)is the complete basis set correction to the MP2 energy, and CBS-INT is the complete basis set correction to the QCISD(T) energy. The term with OIii is a size-consistent empirical correction involving overlap integrals between the most similar a and p orbitals and the square of the trace of the first-order wavefunction. The parameter -0.00533 with this term was obtained by minimizing the root mean square error (experiment - theory) for 55 accurately known dissociation energies. Example 5. Calculation of AH; of C,H, Using an lsodesmic Scheme In this example we illustrate how to calculate the G3 AH? of propane using the isodesmic scheme. The illustrative reaction is: CH3CH,CH3 + CH,

+ 2 CH3CH3

First, calculate the G3 enthalpy of reaction using the G3 enthalpies of formation of the species involved in this reaction. Species

G3 AHy(298 K) (kcal/mol)

Expt. AHF(298 K) (kcaVmo1)

CH, C2H6 C,H*

-18.15 -20.39 -25.33

-17.9 -20.1 -25.0

Empirical Methods 189 AHr (G3) = 2AHF(C,H6) - AH?(C,H,) - AH?(CH,) = 2(-20.39) - (-25.33) - (-18.15) = 2.70 kcaUmol

Then write an expression for the enthalpy of reaction using experimental enthalpies of formation of the reference species (CH, and CH,CH, ), set it equal to AHr (G3), and solve for the enthalpy of formation, x, of the desired species (CH,CH,CH, ). 2AH?(C2H6, expt) - x - AH?(CH, ,expt) = AH,.(G3) = 2AH?(C&,eXpt) - AH?(CH,,expt) - AH,(G3) = W20.1) - (-17.9) - (2.70) = -25.0 kcal/mol x = AHF(C,H,,isodesmic) = -25.0 kcal/mol X

EMPIRICAL METHODS For the pragmatic chemist or engineer, empirical methods have been and continue to be useful in the prediction of thermodynamic and other properties. In the chemical industry, where new molecules are conceptualized and brought to large-scale production rapidly, the determination of accurate thermodynamic data is sometimes crucial to a successful scale-up. Not only are these data needed for process optimization by means of such widely used process simulators such as ASPEN,160 but also for important early stage safety information. For example, a common calculation performed for a proposed new chemical process is the maximum temperature rise that might be expected in a reactor “pot” under so-called worst-case conditions. Most typically, the worst-case condition is assumed to be zero heat losses to the vessel or to the environment (i.e., a completely adiabatic system). The calculated final elevated temperature is then evaluated to determine whether these conditions might cause a pressure increase due to a change in vapor pressure of the solvent or a change in mechanism favoring some undesired chemistry (e.g., a gassy decomposition). This pressure rise might, ultimately, rupture the vessel if adequate relief is not available. For this adiabatic temperature rise calculation, the crucial parameter is the reaction enthalpy, AHr, since for most materials the heat capacities C, are relatively invariant and known (or easily estimated), the masses of the vessel m, and contents m, are known, and the degree of conversion (Conv) most often is assumed to be one:

ATadiabatic

AH,Conv + m,C,(contents)

= m,C,(vessel)

[I61

With rapidly changing competitive conditions in the global chemical industry, it is more important than ever to be able to determine confidently and quickly

I YO Theoretical Methods for Computing Enthalpies of Gaseous Comporinds

physical and thermodynamic properties of chemical species by nonexperimental methods. This philosophy is described in more detail elsewhere.161 The rapidly developing field of quantum mechanics (and the even more rapid improvements in computational hardware and software) are very promising for those in need of better estimation techniques, and an increasing number of chemists and engineers are turning to these techniques for routine use. Yet the empirical methodologies still hold an important place in the toolbox of the scientist simply because these methods are so easy to use and are of proven reliability.

Bond Energy Approach The bond energy approach is typically not directly aimed at the determination of standard enthalpies of formation for individual molecules although there are several exceptions.162 The bond energy technique is still worth discussing in the context of this chapter, however, since many researchers use it routinely to estimate reaction enthalpies. The energy required to break a chemical bond and separate the fragments to infinite distance in the gas phase at 0 K is commonly referred to as a bond energy. Bond energy tables using this definition cannot be used to predict enthalpies of formation, but they may be used to predict enthalpies of reaction (at 0 K). This may be done by summing the bond energies for bonds broken and subtracting the sum of bond energies for bonds formed, being careful to account for any additional energy effects such as ring strain:161 AH, = C (energy of bonds broken) - C (energy of bonds formed) [ 171

h more useful approach that deals with partial bond contributions to the gas phase enthalpy of formation at 298.15 K is that of Benson.162 Basically the bond contributions can be added together to predict either the enthalpy of formation of a molecule or the reaction enthalpy directly. A simple example of how the bond energy approach may be used is illustrated in the following trivial example. Suppose one needed to determine the reaction enthalpy for the following hypothetical reaction:

Empirical Methods 191 For the purposes of an engineering design at an early stage, the reaction enthalpy may be needed to an accuracy of only, say, k8 kcal/mol. Without resorting to a single bond energy table, one can predict that this reaction will be close to thermoneutral because the bonds broken and formed are so similar: Bonds Broken

Bonds Formed

c-0

c-0

c-Cl

c-CI

One may legitimately argue that the bond strengths of aliphatic and aromatic carbon bonds are different, but these differences are smaller than the desired accuracy. Bond energy tables may be found in a number of references.163-167 Because the methods used to extract these energies sometimes differ, it is important not to mix energies from different tables unless one is certain that the conventions are equivalent. Also, some bond energy tables distinguish between different types of bond, e.g., aliphatic vs. aromatic.165 Another example of this approach is given in Eq. [19]. Suppose the reaction enthalpy is desired for the gas phase reaction:

1191

0

From the bond energy table in Wall,163 we construct the following table: Bonds Broken c-CI N-H

Bond Energy (kcal/mol) 78

92.2

Bonds Formed

C-N H-CI

Bond Energy (kcal/mol) 66

102

The reaction enthalpy is then calculated as follows: I: (energy of bonds broken) - I: (energy of bonds formed) = (78 + 92.2) - (66 + 102) = +2 kcal/mol. For comparison, the calculated enthalpy for a simple analog reaction: EtCl + EtNH,

+ Et,NH + HC1

1201

(all species in the gas phase) is -1.3 kcal/mol using data from standard sources,168,169 which is reasonably close to the predicted enthalpy.

192 Theoretical Methods for Computing Enthalpies of Gaseous Compounds This example highlights an important concern whenever one is estimating a reaction enthalpy, regardless of the technique chosen. The bond energy approach (like most group contribution techniques) is based on gas phase thermodynamic data. However, very little industrial chemistry takes place in the gas phase (excluding, of course, the high temperature cracking processes which are mainly endothermic). Reaction enthalpies can be highly dependent on the state of the material. If the HC1 product in the example above were to end up in an aqueous state (i.e., if the reaction took place in water), the reaction enthalpy could be as much as 18 kcal/mol more exothermic because gaseous HC1 has a very high enthalpy of solution in water. Techniques to correct gas phase enthalpies to condensed phase values are discussed in Ref. 161 and group contribution techniques for estimating vaporization enthalpies may be found in Ref. 170. The following is an example of a reduction reaction using hydrogen:

a+4H2

Bonds Broken

c=cx2

c=o

H-H

x

4

- Q/

Bond Energy (kcal/mol) 145 x 2 173 103.2 x 4

Bonds Formed

c-c

C-H 0-H

x2 x6

x2

+

H20

[21]

Bond Energy (kcaYmol) 80 x 2 98.2 x 6 109.4 x 2

As in the preceding example, the reaction enthalpy is then calculated as follows: C (energy of bonds broken) - C (energy of bonds formed) with the result of -92.2 kcal/mol. One might consider subtracting 1 kcaYmol from this enthalpy to compensate for the probable difference in ring strain energy between cyclohexene and cyclohexane. For comparison, we may calculate (using experimental enthalpy data) the reaction enthalpy for the following analog reaction:

Using gas phase data from standard sources,168~169the calculated enthalpy is -90 kcal/mol, which is reasonably close to the bond energy estimate. The bond energy method does have its place in the toolbox of chemists and engineers who need a number quickly and are typically not concerned with having a value that is more accurate than, say +5-10 kcal/mol. The limitation with this method is that no account is typically taken of the differing electronic environment of the bonds in various molecules, which is known to affect the

Empirical Methods 193 bond energies. When one begins to take these effects into account, the next logical step is second-order group contribution methods, as discussed next.

Benson’s Method A natural extension of the bond energy approach is to account for interactions close to the chemical bond in question (which certainly affect the stability, and hence the thermodynamic properties). Based on this concept, a number of “group contribution” methods have been developed over the years, and many of these methods have been reviewed in Ref. 171. Benson’s second-order group contribution method, probably the most successful and widely embraced method, was developed some 30 years ago as an improvement to bond energy (or bond contribution) methods for the prediction of thermochemical properties.167 This improvement was accomplished by accounting for: 0 0 0

the effect of the bonded environment around a group ring strain effects isomeric effects (cidtrans, ortho/meta/para) nonbonded interactions (gauche effects)

As in any so-called group contribution technique, the molecule is presumed to be comprised of substructural units or groups. Each group contributes a constant amount to some thermodynamic property (or other property) of the molecule. The end result is a method that is more accurate than bond energy methods (typical uncertainties of 2-3 kcaYmol for materials containing C, H, N, and 0). However, a group contribution method is more complex to use than a bond energy method and requires more data for parameterization (because the number of groups is larger). A group consists of a group center (always and only a polyvalent moiety) plus any attached (bonded) monovalent and polyvalent ligands. Examples of possible monovalent moieties include H, F, C1, Br, I, CN, NO, and NO,. Examples of polyvalent moieties include 0, CO, C, and N. In the standard notation, the central polyvalent atom is listed first, followed by a dash, followed by all the moieties bonded to that center. For example, in the hypothetical structural fragment below, the notation for the group associated with the tetravalent center, X, is X-(A),(B)(D). Since the attachment D is monovalent, there is no associated group. Since A and B are polyvalent, there are corresponding groups with these as centers to complete the molecule. A ‘’ I

D-X-B I \ A ‘ / \

194 Theoretical Methods for Computirtg Ertthalpies of Gaseous Compounds

It is important to note that Benson’s method may require the use of correction terms for molecules having rings, cidtrans or ortho/para isomers, 1,5repulsions, or gauche interactions. The user is referred to Benson’s book162 for a complete description of the method. Some pertinent notation used in this method is as follows: Notation C d‘

Cb CP

ct co 0

Valence

Description

4 2 1 3 1 2 2

Tetravalent carbon (alkanes) Double-bonded carbon (alkenes) Benzene-type carbon (aromatic) Aromatic carbon at ring junction (polyaromatics) Triple-bonded carbon (alkynes) Carbonyl group (aldehydes, ketones, esters, carboxylic acids) Oxygen (noncarbonyl oxygen atom in ethers, esters, acids, alcohols) Trivalent nitrogen (amines) Aromatic nitrogen (pyridine, pyrazine, and pyrimidine, but not pyridazine) Divalent sulfur (sulfides)

Nb

3 0

S

2

N

Examples of Benson’s Method As an example of the group contribution concept, consider the Benson methylene groups shown below, where C represents a tetrahedral carbon, and Ch represents an aromatic carbon, C, represents a doubly bonded carbon. Each of these values was determined from experimental thermodynamic data. Note that the contribution of the C-(H),(X)(Y)group to the enthalpy of formation is different in each bonded environment. Benson Group

Contribution to AH; (kcallmol, 298.15

C-(H),(C), C-(H)2(Cb ) 2 C-(H),(C)(O) C-(H),(C)(N) C,-(H),

-4.93 -6.50 -8.10 -6.60 +6.26

Four worked examples of Benson’s method are now given. Example 1 . 4-Amino-2-pentanol (g)

K)173

Empirical Methods 195 In general, each multivalent atom must be counted as the central atom in a unique Benson group. Thus in the case of 4-amino-2-pentanol (g), there are seven multivalent atoms (five carbons, one nitrogen, and one oxygen), and thus we must end up with seven Benson groups. In addition, there may be corrections such as ring strain or other steric effects. Arbitrarily starting from the left, the first carbon is bonded to three hydrogens and one carbon. The Benson group notation is C-(H),(C). Note that at the opposite end of the molecule, there is an identical C-(H),(C) group. In Benson’s original notation, the central, multivalent atom is always listed first, followed by a dash, then everything bonded to that atom is listed to the right of the dash. Continuing on, left to right, the next multivalent atom is a carbon, bonded to one hydrogen, two carbons, and a nitrogen. In Benson’s notation, this group is then C-(H)(C),(N). The next group is a methylene bonded to two carbons. In standard notation, this is C-(H),(C),. The next carbon is bonded to one hydrogen, two carbons, and one oxygen: C-(H)(C),(O). The amine group is thus N-(H),(C), and the remaining hydroxyl group is then 0-(H)(C). In this example, there are no corrections such as ring strain or other steric effects to consider.

Note: There is no experimental value.

Example 2. Chlorocyclopropane (g)

A

The chlorocyclopropane molecule contains three groups and a ring correction. The two methylene groups are identical: C-(H),(C), . The third structural group, which includes the chlorine, is C-(C),(H)(Cl). The correction for ring strain would be close to that for cyclopropane.

196 Theoretical Methods for Combutina Enthalbies of Gaseous Combounds Benson Group

Contribution to AH,“(kcaumol, 298.15 -4.93 -14.80 27.60 2.94

C-(H),(C), x 2 C-(C),(H)(Cl) Cyclopropane ring

Total

x

K)173

2

Note: There is no experimental value.

Example 3 . Acrylic acid (g)

Starting at the left, we have a doubly bonded carbon (C, in Benson notation) bonded to two hydrogens. Since, in Benson’s scheme, the other doubly bonded carbon is implied, it is omitted from the notation, and we have Cd(H)2.The next group also involves a doubly bonded carbon, connected to one hydrogen and a carbonyl moiety. As discussed above, the carbonyl atomic grouping is always kept together as a divalent pseudoatom. Thus, this group is C,-(H)(CO). The next group is then CO-(C,)(O) followed by 0-(CO)(H). Benson Group

Contribution to AH,“(kcal/mol, 298.15 K)173

C,-(H), C,-(H)(CO) CO-(C,)(O) O-(CO)(H) Total

6.26 5.00 -32.00 -58.10 -78.84

Note: The experimental value is -77.0 kcaVrnol (Ref. 172).

Example 4 . Cblorobenzene (g)

In Benson notation, an aromatic carbon, Cb, is presumed to be attached to two other aromatic atoms. Thus, for example, benzene can be entirely described by six Cb-(H) groups. In the present example, the molecule is divided into five Cb-(H) and one C,-(CI) groups. The ring strain (in this case a stabilization) is automatically accounted for in the contribution from each group containing a “Cb,”and no additional groups or contributions are necessary.

Empirical Methods 197 ~~~~~

Benson Group

Contribution to AH: (kcaumol, 298.1.5 K)173

3.3 x 5. -3.80 12.70 Note. The experimental value is 12.4 kcallmol (Ref. 168).

Resources and Tools for Benson’s Method Because of its accuracy and relative ease of use, Benson’s second-order group contribution method has been incorporated into many computer programs. Among these are the CHETAH programI73 (Chemical Thermodynamic and Energy Release Program-see also a description of an earlier version of this program in Ref. 174);the LOADER prograrn,l75 which is part of the Physical Property Data Service (PPDS) system of programs; the NIST Standard Reference Database 18 THERM/EST p r ~ g r a m ; and l ~ ~the NIST Standard Reference Database 25 Structures and Properties Database and Estimation program.177 Overall, both CHETAH and Structures and Properties are most useful for reaction enthalpy estimations. The CHETAH parameter set contains contributions for over 600 groups, about double the number available in other sets. The recent set of Benson parameters developed by Domalski et al.’7* and incorporated in the THERWEST program is also recommended. The distinguishing feature of the latter parameter set is that it includes parameters for solids and liquids as well as gases. Although the set of Reid and Hearing171 has been widely used in the chemical engineering community, we do not recommend it, because oxygen compounds are poorly treated. A more comprehensive description and comparison of tools available for thermodynamic data estimation may be found in Ref. 161. One limitation of Benson’s method is the lack of adequate data for many chemical species, resulting in missing data for important building blocks in many compounds of industrial relevance. The following is a summary of some of the techniques by which one is able to accurately estimate (predict) such missing Benson group data and, thus, allow for an easy and accurate completion of a calculation or simulation. The philosophy here is similar to that often used in regard to missing force field parameters in a molecular mechanics calculation. Besides the methods described below, one may use a quantum chemical approach (using methods described in other sections of this chapter and also in Ref. 179)to determine the enthalpy of formation of a suitable small molecule containing the missing group. Simple Substitution On occasion, one may substitute one Benson group for another one of “similar” nature. For example, replacement of a doubly bonded carbon, C,, in the group environment, for an aromatic carbon generally leads to only small errors (+2 kcal/mol). Note that the central/main part of the group is the central atom and all univalent attachments. The environment of the group consists of all polyvalent attachments. Substitutions in the “environment” are allowable but the environmental changes should be for as similar an

198 Theoretical Methods for Computing Enthalpies of Gaseous Compounds environment as possible. Substitutions in the central group are not allowed in this technique. Consider the following groups and their values from CHETAH 7.2.173 Contribution to AH: (kcal/mol)

GrouD

Note that the substitution of C for either Cd or C, (in the environment) has a similar, minor effect. Note that the substitutions are in the environment part of the group. Many such substitutions can be made without significant loss of accuracy. As shown above, one may tabulate related groups in order to gauge the accuracy of potential substitutions. Interpolation Between and Extrapolation from Known Benson Group Parameters Interpolation and extrapolation of known group values are convenient methods to estimate data for a missing group. It is important to emphasize again that one should attempt to perform these operations in the "environment" (whenever possible), keeping the central group constant. Failing this, one can interpolate or extrapolate in the central group while keeping the environment constant but with a usually lower accuracy. Central group changes lead to much larger changes in the thermodynamic parameters, and so the potential for error is also larger. What is the proper interpolation and extrapolation technique? Experience has shown that differences in families are never quite constant, implying that linear interpolation and extrapolation is only a zeroth-order approximation. In some of the examples to follow, enough information is available to permit the use of a quadratic scheme, which may be the best approximation. For most systems, the differences between the linear and quadratic methods will be small.

Example 5. Determine AH;and 5 (Entropy) for C-(C)2(Cb)2 by Interpolation in the Environment AH,"(kcal/mol)

S (cal/mol deg)

0.50 2.81

-35.10 -35.18 [-35.11 Not available Not available

~~~

C-(C), C-(C),(C,) c-(c)2(cb)2 13

c-(c)(cb C-(C,), a

4.2 = 1/3(7.00 - 2.81) + 2.81.

[4.2]"

Not available 7.00

Empirical Methods 199 Example 5 illustrates determination of the unknown C-(C)2(Cb)2group by interpolation in the series C-(C), to C-(C,), . This represents interpolation in the environment. In this and all subsequent examples, data are taken from Ref. 173. The AH; in square brackets is from linear interpolation between C-(C),(C,) and C-(Cb),. It is probably more accurate to use this value than the value of AH: = 2.81 kcal/mol, which would result from simple substitution of C-(C),(C,) for C-(C)2(Cb)2;see preceding discussion.

Example 6. Determine AH; and S for C-(H)(Br),(C) Using Interpolation in the Central Group C-(H),(C) C-(H),(Br)(C) C-(H)(Br),(C)

AH: (kcal/mol)

S (cal/mol deg)

-10.08 -5.40 [1.80]

30.41 40.80

[49.70]

[1.01 C-(Br)3( C) C-(Br),

9.00 20.1

58.60

Comments

Linear fit between C(H),(Br)(C) and C-(Br),(C) Graphical interpolation, nonlinear

8O.P

#Values for the complete molecule C-(Br), were obtained from literature data contained in the NIST Structures and Properties program (Ref. 177).

This example shows how to determine the correct group value for AGand

S for C-(H)(Br),(C) by considering known values in the sequence C-(H),(C) to

C-(Br), . This example represents changes in the maidcentral group. Note that the entropy values for Benson groups such as C-(Br), have the symmetry contribution removed. The user must add in any symmetry contribution after the molecule is “built” with the complete set of Benson groups. For consistency in any interpolation scheme, one must remove the symmetry contribution from the entropy for the whole molecule. In this case, owing to the tetrahedral symmetry, an amount R In 12 (where R is the gas constant) was subtracted from the literature entropy value for C-( Br), .

Correcting from the Condensed Phase to the Gas Phase Most experimentally determined thermochemistry data are for the condensed phase. Yet Benson’s method is typically parameterized for the gas phase. Therefore, one usually must convert results from gas to condensed phase and vice versa.

200 Theoretical Methods for Computing Enthalpies of Gaseous Compounds In some cases, a literature source of thermodynamic data may exist, allowing one to perform the conversion. Fortunately, standard references (such as Refs. 168 and 180) frequently tabulate both the condensed and gas phase thermochemistry values. When that is not the case, the following relations may be used, where the enthalpies and entropies of vaporization and fusion must be

~ 3 1 ~ 4 1 1251 [261 where AS,, = AHv,,/298.1S and AS,,, = AHf,,/298.15 Note that we need to know only the enthalpies of fusion or vaporization (at 298.15 K) to convert enthalpies of formation, but we also need to know the vapor pressure of the liquid to convert the entropy. In CHETAH,173 the reference state pressure (Pref) is 1 atmosphere. Measured enthalpies of fusion are usually at the melting point, but given the heat capacities of the liquid and the crystal, one can convert the enthalpy of fusion at the melting point to that at 298.15 K. That difference may be quite small for materials melting within 50-100 K of 298.15 K. If an experimental vaporization enthalpy cannot be found, one may use the enthalpy of vaporization of a “similar” molecule from one of the above sources. Characteristics that should be as similar as possible include molecular weight, heavy atom content (halogens, sulfur, etc.), and hydrogen-bonding characteristics. Barring using data for a similar molecule, one may use an enthalpy of vaporization, in kcal/mol, that is one-tenth of the molecular weight (this is equivalent to saying the enthalpy of vaporization is 100 cal/g). Care must be exercised in using this approximation, because hydrogen-bonding systems (alcohols, amines, acids, etc.) have larger enthalpies of vaporization (150-250 caYg), whereas highly halogenated systems have smaller values (SO-SO caYg). For enthalpies of fusion at the melting point, one may use Walden’s rule:*81 AH,

(cal/mol) = 0.13 T,,,(K)

~ 7 1

and then correct to 298.15 K. Walden’s rule will give values that are too low for long flexible molecules and values that are too high for very small spherical or cyclic molecules. Also, we note again that Liebman and coworkers170 have published accurate group methodologies for determining these transition properties.

Concluding Remarks 201

CONCLUDING REMARKS The thermochemistry of molecules is of major importance in the chemical sciences and is essential to many technologies. Recent advances in theoretical methodology, computer algorithms, and computer power now allow us to compute reliable thermochemical data for a fairly wide variety of molecules. In this chapter we have reviewed the current state of the art in computational thermochemistry. The most accurate methods are based on ab initio molecular orbital theory. The ab initio methods that are used for predicting thermochemical data can be grouped into three types. The most accurate of these are very high level quantum chemical calculations that are based on an extrapolation of large basis set correlated calculations with correction for relativistic effects. The proper use of these calculations generally requires a significant expertise. The second type of ab initio approach combines an intermediate level of ab initio quantum chemical calculation with some form of molecule-independent empirical parameters. These approaches are generally reliable and easy for the nonexpert to use. The principal methods of this type are the Gn and CBS methods, both of which are available in the recent Gaussian programs. The latest in the Gn series, G3 theory, gives thermochemical data accurate to f 1 kcal/mol in most cases. The third type of approach includes techniques such as BAC-MP4 that use molecule-dependent empirical parameters obtained from accurate experimental data in combination with intermediate level ab initio quantum chemical calculations. These methods are computationally efficient, but they depend on the availability of accurate experimental data. They are also more difficult for the nonexpert to use because of the various empirical parameters. Currently density functional theory is not as accurate as the ab initio methods for thermochemistry, but considerable effort is being put into development in this area. Because of its computational efficiency, density functional theory is a promising area for the future of computational thermochemistry. Semiempirical molecular orbital methods such as AM1 or PM3 have the advantages of being computationally fast, easy to use, available in many quantum chemistry codes, and not highly demanding of computer resources. The methods can give useful thermochemical data even for large molecules such as encountered outside the realm of chemical physics. However, the semiempirical methods are not nearly as accurate as the ab initio methods when assessed on common test sets. Empirical methods use known experimental enthalpy data to estimate enthalpies and bond energies for unknown compounds. The empirical methodologies remain important simply because these methods are easy to use, fast, and of known reliability. These methods can be reasonably accurate for molecules that have standard bond types. Despite the successes, much remains to be done in the future to further develop capabilities for accurate prediction of thermochemical data. Among the challenges will be extension of the accurate methods to larger molecules, ob-

202 Theoretical Methods for Computing Enthalpies of Gaseous Compounds taining increased accuracy in predictions, and extension to heavier elements. The increase in power obtainable from new generations of computers, such as massively parallel architecture,l will play an important role in meeting these challenges.

ACKNOWLEDGMENTS The work at Argonne National Laboratory was supported by the Division of Materials Sciences, Office of Basic Energy Sciences, Department of Energy, under Contract W-31-109ENG-38. We thank Dr. Anwar Baboul for helpful discussions. One of the authors (DJF) gratefully acknowledges the assistance of the following Dow Chemical colleagues for their contributions in the preparation of this chapter: Dr. Joseph R. Downey, Dr. Marabeth S. LaBarge, Dr. Alan N. Syverud, and Dr. Gary Buske.

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2 10 Theoretical Methods for Computing Enthalpies of Gaseous (:ompounds 156. W. Thiel, Adv. Chem. Phys., 93, 703 (1996). Perspectives on Semiempirical Molecular Orbital Theory. 157. M. J. S. Dewar, C. Jie, and J. Yu, Tetrahedron, 49, 5003 (1993). SAM1; the First of a New Series of General Purpose Quantum Mechanical Molecular Models. 158. A. J. Holder, R. D. Dennington, and C. Jie, Tetrahedron, 50, 627 (1994). Addendum to SAM1 Results Previously Published. 159. W. J. Thiel and M. G. Hicks, J . Comput. Chem., 7, 213 (1986). Reference Energies in Semiempirical Parametrizations. 160. ASPEN PLUS User's Guide (1994), Aspen Technology Inc., 251 Vassor St., Cambridge, MA 02139 (1994). 161. D. J. Frurip, A. Chakrabarti, J. R. Downey, H. D. Ferguson, S. K. Gupta, T. C. Hofelich, M. S. LaBarge, A. J. Pasztor Jr., L. M. Peerey, S. E. Eveland, and R. A. Suckow, in International Symposiumon Runaway Reactions and Pressure Relief Design, CCPS/AIChE, Boston, MA, August 2-4,1995. Determination of Chemical Process Heats by Experiment and Prediction. 162. S. W. Benson, Tbermocbemical Kinetics, 2nd ed., Wiley-Interscience, New York, 1976. 163. F. T. Wall, Chemical Thermodynamics, 2nd ed., W. H. Freeman, San Francisco, CA, 1965. 164. J. A. Kerr, Chem. Rev., 66, 465 (1966). Bond Dissociation Energies by Kinetic Methods. 165. J. D. Cox, Tetrahedron, 18, 1337 (1962). A Bond Energy Scheme for Aliphatic and Benzenoid Compounds. 166. R. Walsh, Acc. Chem. Res., 14, 246 (1981). Bond Dissociation Energy Values in SiliconContaining Compounds and Some of Their Implications. 167. S. W. Benson, J. Cbem. Educ., 42, 502 (1965). Bond Energies. 168. J. B. Pedley, R. D. Naylor, and S. P. Kirby, Thermochemical Data of Organic Compounds, 2nd ed., Chapman & Hall, New York, 1986. 169. D. D. Wagman, W. H. Evans, V. B. Parker, R. H. Schumm, I. Halow, S. M. Bailey, K. L. Churney, and R. L. Nuttall, 1.Phys. Chem. Ref. Data, 11,Suppl. 2 (1982). The NBS Tables of Chemical Thermodynamic Properties. Selected Values for Inorganic and C1 and C2 Organic Substances in S1 Units. [Errata: Ibid., 19, 1042 (1990).] 170. J. S. Chikos, W. E. Acree Jr., and J. F. Liebman, in Computational Thermochemistry, K. K. Irikura and D. J. Frurip, Eds., ACS Symposium Series 677, American Chemical Society, Washington, DC, 1998, pp. 63-91. Estimating Phase-Change Enthalpies and Entropies. 171. R. C. Reid, J. M. Prausnitz, and B. E. Poling, The Properties of Gases and Liquids, 4th ed., McGraw-Hill, New York, 1987. 172. R. Vilcu and S. Perisanu, Rev. Roum. Chim., 25,619 (1980). The Ideal Gas State Enthalpies of Formation of Some Monomers. 173. B. K. Harrison, CHETAH 7.2, The ASTM Computer Program for Chemical Thermodynamic and Energy Release Evaluation (also available as NIST Special Database 16);available from American Society for Testing and Materials, 100 Barr Harbor Drive, West Conshohocken, PA 19428-2959. See also http://www.chetah.usouthal.edu/. 174. D. J. Frurip, E. Freedman, and G. R. Hertel, in Proceedings oftbe International Symposium on Runaway Reactions, Cambridge, MA, March 7-9, 1989, pp. 39-51. A New Release of the ASTM CHETAH Program for Hazard Evaluation: Versions for Mainframe and Personal Computer. 175. LOADER2 User Manual, PPDS Support Desk, National Engineering Laboratory, East Kilbridge, Glasgow, G75 OQU, Scotland, U.K. LOADER is part of the PPDS suite of programs. 176. NlST Standard Reference Database 28, NIST THERWEST Program version 5.0 (1993). Estimation of Chemical Thermodynamic Properties of Organic Compounds. See also, Ref. 178. 177. NIST Standard Reference Database 25, NIST Structures and Properties Database and Estimation Program, Version 2.0, 1992. 178. E. S. Domalski and E. D. Hearing,J. Phys. Chem. Ref. Data, 22, 805 (1993). Estimation of the Thermodynamic Properties of Carbon-Hydrogen-Nitrogen-Oxygen-Sulfur-Halogen Compounds at 298.15 K.

References 2 1 1 179. D. J. Frurip, N. G. Rondan, and J . W. Storer, in Computational Thermochemistry, K. K. Irikura and D. J. Frurip, Eds., ACS Symposium Series 677, American Chemical Society, Washington, DC, 1998, pp. 319-340. Implementation and Application of Computational Thermochemistry to Industrial Process Design at the Dow Chemical Company. 180. D. R. Stull, E. F. Westrum Jr., and G. C. Sinke, The Chemical Thermodynamics of Organic Compounds, Wiley, New York, 1969. (Out of print; largely superseded by Ref. 168.) 181. D. J. W. Grant and T. Higuchi, Solubility Behavior of Organic Compounds, Techniques of Chemistry, Vol. 21, Wiley, New York, 1990, p. 22.

CHAPTER 4

The Development of Computational Chemistry in Canada Russell J. Boyd Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada B3H 413

INTRODUCTION The development of computational chemistry in Canada is inextricably linked with the emergence of theoretical chemistry as one of Canada’s strongest scientific disciplines. For this reason, the present account chronicles the history of both computational and theoretical chemistry in Canada. The first theoretical chemists were appointed to Canadian universities in the mid- to late 1950s, long after strong traditions for theoretical chemistry had been established in Europe, the United Kingdom and the United States. Furthermore, many of the first Canadian theoreticians often carried out experiments or maintained experimental laboratories in departments of chemistry in which the dominant figures were typically experimental physical chemists or organic chemists with a penchant for natural products and synthesis. The late development of theoretical chemistry in Canada was a consequence of the relatively weak research base in Canadian universities prior to 1950. In fact, research in Canadian universities during the first half of the twentieth century lagged seriously behind that of the leading universities of the major industrialized nations. Only McGill University and the University of Reviews in Computational Chemistry,Volume 15 Kenny B. Lipkowitz and Donald B. Boyd, Editors Wiley-VCH, John Wiley and Sons, Inc., New York, 0 2000

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21 4 The Development of Computational Chemistry in Canada

Toronto had gained international reputations as major research institutions before the Second World War. Although some research was carried out prior to 1950 at about 20 universities across Canada, the majority of the significant research was done in the laboratories of the National Research Council of Canada (NRC) in Ottawa. From its humble beginnings in 1916, the NRC developed into a research organization which by the 1960s operated some of the premier research laboratories in the world. Long before Canada developed a reputation in theoretical and computational chemistry, the one Canadian scientist who needed no introduction was Gerhard Herzberg. His book on atomic structure1 was known to many generations of chemistry students, and his book on diatomic molecules2 and the later one on polyatomic molecules” became standard references for theoretical chemists worldwide. Herzberg was born in Hamburg, Germany, on December 25, 1904 and died in Ottawa on March 3, 1999. After earning his doctorate in engineering physics from the Technical University of Darmstadt in 1928, he did postdoctoral research at the universities of Gottingen and Bristol. In 1930 he returned to Darmstadt and during the next five years his research in atomic and molecular spectroscopy flourished. As his scientific reputation was growing, he received an invitation from Dr. J. W. J. Spinks to join the Department of Physics at the University of Saskatchewan. During his 10 years in Saskatoon he made major applications of spectroscopy to physics and chemistry. In 1945 he was appointed professor of spectroscopy at the Yerkes Observatory of the University of Chicago. After three years in the United States, Herzberg returned to Canada to take up a position at the National Research Council laboratories in Ottawa. When he was awarded the Nobel Prize in chemistry in 1971, the citation stated, Dr. Herzberg is generally considered to be the world’s foremost molecular spectroscopist, and his large institute in Ottawa is the undisputed centre for such research. In 1994 on the occasion of Herzberg’s ninetieth birthday, the Canadian Journal of Physics published a special issue. The festschrift, edited by Donald Betts and John Coxon of Dalhousie University, contains 72 papers by many of the world’s leading spectroscopists and theoretical chemists. In his acknowledgment, Nobel laureate Dudley Herschbach summarized Herzberg’s inestimable impact with the following tribute? With pleasure we dedicate this paper to Gerhard Herzberg. For us, as for a host of others, he has through his books served as a superb, inspiring mentor. As well as showing us the exhilarating scope of spectroscopic methods, he has taught us to orient our work toward basic questions and

Introduction 2 15 to strive for treatments that take account of complications as simply as possible. An article entitled simply “G. H.” by Boris Stoicheff is recommended to those who are interested in learning more about Herzberg’s remarkable career and scientific achievements. Stoicheff’s article appeared in the April 1972 special issue of Physics in Canada, published in honor of Gerhard Herzberg on the occasion of his receipt of the Nobel Prize. Another landmark in the history of theoretical chemistry in Canada pertained to funding. To stimulate research in Canadian universities, the NRC established a grants program and awarded research grants on a competitive basis to university researchers. During his presidency (1952-1962), E. W. R. Steacie established the principle that NRC’s extramural budgets for grants and fellowships should rise to match the intramural budget ($215 million in 19621963). Ultimately the NRC was in the difficult position of having to choose between funding its own laboratories and the peer-reviewed applications from external (university) researchers. To alleviate this conflict of interest, the Canadian government transferred in 1978 the research grants and scholarships programs of the NRC to a new federal granting agency, the Natural Sciences and Engineering Research Council of Canada (NSERC). Research in Canadian universities expanded rapidly in the 1960s for several reasons. The launching of Sputnik in 1957 by the Soviet Union shocked the Western nations and led to a rapid expansion of the existing universities and the creation of many more. Consequently, funding for research grew dramatically, many young people (including the author) were attracted to science, and universities started to put more emphasis on research. The number of universities offering the Ph.D. degree in science increased from 2 in 1940 to more than 25 in 1970. With this brief introduction, it is useful to reflect on the state of the development of theoretical chemistry in 1970, little more than a decade after its establishment in Canadian universities. In 1970 on the occasion of its twenty-fifth anniversary, The Chemical Institute of Canada published a 290-page book entitled Chemical Canada. The authors intended the book “to present a general account of the impact of things chemical in Canada up to, and including, the present time.” The book consists of three parts, the third part being a 50-page summary of research and development in Canada. The R&D section is organized by topics and devotes only two short paragraphs to theoretical chemistry. These two paragraphs are reproduced here for their interesting historical perspective.5 The very rapid increase in the number of theoretical chemists in Canada since 1962 does not reflect the true growth rate, or one which can be expected to continue; the sudden increase in numbers was caused by many

21 6 The Development of Computational Chemistry in Canada universities hiring their first theoretician. Although there has been a marked reluctance in many chemistry departments to consider theoretical chemistry as a discipline in itself, worthy of being pursued for its own sake, about eight departments having one theoretician have plans to hire another in the near future, and two departments have initiated plans to form centers of theoretical research consisting of about six staff members. The solution to the problems involved in the theoretical study of the electronic structure of large molecules and reaction rate theory demands very large and very fast computers, larger and faster than those now available in Canada. If these studies are to proceed, greatly expanded computing facilities must be made available. The type of computer envisaged and necessary for these calculations would be of such a large size that Canada could afford only one or two such installations. It is a general feeling that by the establishment of such a central computational facility, Canada could, for a nominal amount, be in the forefront of a number of critical areas in scientific research and development. The data collected for the 1970 chapter support the authors’ statements and their prophetic predictions. Moreover, their comments about computing facilities are still relevant, three decades later.

IN THE BEGINNING THERE WAS QUANTUM CHEMISTRY AND SPECTROSCOPY The early efforts in quantum chemistry in Canada, as in other countries, were closely connected with electronic spectroscopy. Later the scope expanded to include all forms of spectroscopy. One of the Canadian pioneers whose interests epitomize the history of quantum chemistry is Camille Sandorfy. After graduating from the University of Szeged in his native Hungary and completing a D.Sc. under the supervision of Profs. Louis de Broglie and R. Daudel at the Sorbonne in Paris, Sandorfy took up a postdoctoral fellowship with Dr. R. Norman Jones, a spectroscopist at the National Research Council of Canada laboratories in Ottawa. By the end of 1953, Sandorfy had settled in Montreal, where he began a highly distinguished career, initially as a theoretical chemist and ultimately as a full-time spectroscopist, specializing in the far-ultraviolet and infrared regions. He developed the basis for a very strong group in quantum chemistry that continued to flourish under the leadership of Dennis Salahub and his younger colleagues at the University of Montreal. In the years 1945-1955 theoretical chemists as well as chemical spectroscopists concentrated their efforts on systems containing n: electrons. Such

In The Beginning There Was Quantum Chemistvy and Spectroscopy 21 7 molecules can be easily treated by simple molecular orbital methods. It was considered a normal procedure in those times to treat the n electrons as a separate quantum chemical problem to which the (J electrons supply a supposedly uniform potential field. Sandorfy was struck by the artificial character of this approximation. His first target was the series of normal chain saturated hydrocarbons. A preliminary note on his results was published in the Comptes Rendus de I’Acade’mie des Sciences.6After taking up his position in Montreal, he greatly extended this work by introducing three different approximations, which he called the “C” “CH,” and “H” approximations.7 The first was based only on the atomic orbitals of the carbons, the second used the hybrid sp3 orbitals of the carbons and the I s orbitals of the hydrogens, and the third included all the valence atomic orbitals of the carbons and the hydrogens. These methods appear rather simple, yet they were the starting point of a long evolution. Gilles Klopman, whose research interests at Case-Western Reserve University later turned to modeling bioactive molecules, was the first to use Sandorfy’s methods. Kenichi Fukui made extensive use of them in his wellknown work on the structures and reactions of saturated hydrocarbons and their derivatives. Fukui added his frontier orbital considerations. Around 1959 the milieu of developments in quantum chemistry contributed to inspire William N. Lipscomb to conceive the extended Hiickel method, which was subsequently implemented by Lawrence L. Lohr and Roald Hoffmann.*a Soon thereafter, John Pople and his coworkers introduced self-consistent field methods based on the zero-differential overlap approximation.*b The arrival of ab initio programs based on Gaussian-type orbitals and more powerful computers reduced all these developments to historical souvenirs. It should be emphasized, however, that Sandorfy’s work constituted the first attempt to go beyond the n-electron approximation, and the first molecular orbital treatment of polyatomic molecules that took into account all valence electrons without using group orbitals. From among Sandorfy’s subsequent theoretical projects, two in particular should be mentioned: the first treatment of the excited states and electronic spectra of saturated hydrocarbons9 by a new n-electron method of the PariserParr type, and the first attempt to introduce Rydberg orbitals into molecular orbital calculations.10 In 1964 Sandorfy and his coworkers initiated an extensive experimental project to explore the electronic absorption spectra of saturated molecules in the far-ultraviolet region. Their first spectra of a series of paraffin hydrocarbons were published in 1967.11 This paper coincided with an article, also giving spectra of hydrocarbons, by J. Raymonda and W. T. Simpson.12 While there was general agreement between the recorded spectra of the two groups, the investigators’ interpretations were different; it was Sandorfy and his coworkers who put the spectroscopy of the paraffins on the right track by adopting an idea of R. S. Mullikenf3 that related to the Rydberg character of the bands of lowest

21 8 The Development of Computational Chernisty in Canada frequency found in their spectra. In this case too, a long evolution was forthcoming. The field has been summed up by M. B. Robin14 and, more concisely, by Sandorfy.15 Today, after many years of work in several laboratories the spectroscopy of cr-electron systems has advanced to a level previously attainable only for simple n-electron systems; it is possible, in a qualitative manner, to predict the bands that a given organic molecule, saturated or unsaturated, either heteroatomic or not, should possess in the near- or far-ultraviolet regions. Ultraviolet photoelectron spectroscopy helped greatly during this work of exploration. Camille Sandorfy is acknowledged as the first theoretical chemist to be appointed to the faculty of a department of chemistry in Canada. It is therefore fitting that he was the first theoretician to be honored by a special issue of the Canadian Journal of Chemistry. The 750-page special issue for July 1985 contained 117 papers spanning a broad range of topics relating to spectroscopy and theoretical chemistry. The Sandorfy issue is by far the largest issue the Canadian Journal of Chemistry has published. It was edited by two of his former Ph.D. students and younger colleagues at the University of Montreal, Gilles Durocher and Dennis Salahub, the former a spectroscopist and the latter a theoretical chemist.

EXPO 67 AND FULLERENES A key event in the emergence of theoretical chemistry and computational chemistry as strong disciplines in Canada took place in 1965 at the University of Alberta in Edmonton. Fraser Birss, a spectroscopist with a keen interest in theory, and Serafin Fraga, a recent arrival from the Chicago group of Mulliken and Roothaan, organized the first chemistry conference in Canada devoted entirely to theoretical chemistry. Participants in the Alberta Symposium on Quantum Chemistry recall that the main theme of the Edmonton meeting was whether the Hartree-Fock method was good enough. One notable lecture at the conference, often cited and reworded later, was the closing talk by Chris Wahl entitled, “Hartree-Fock Is Here. What’s Next?” While it is impossible to determine the full impact of the Edmonton conference, Ernest Davidson recalls that it was while he was driving back to Seattle through the Canadian Rockies that he got the idea for iterative natural orbitals.16 Also, Fritz Grein remembers a long discussion with Chris Wahl in the Faculty Club which gave him the idea for the work on MCSCF methods that is cited later in this chapter. Following a very successful conference in Edmonton, it was decided that a second theoretical chemistry conference would be organized in 1967 in Montreal by Camille Sandorfy (University of Montreal) and Tony Whitehead (McGill University). Montreal provided the perfect venue. Canada was cele-

Expo 67 and Fullererzes 219 brating its centennial, and the euphoria from coast to coast exceeded anything Canadians had known previously, or indeed would experience in the following three decades. Nowhere was the excitement greater than in Montreal, where 50 million people flocked to the Expo 67 site on islands in the historic St. Lawrence River. More than 100 pavilions were built to celebrate the theme, Man and His World, a title that within a generation would be deemed politically incorrect. The exhibition was testimony to the optimism and faith of Canadians and others in the future, and it celebrated progress and technology. There were pavilions with fantastic shapes, one of which made an indelible impression upon a young spectroscopist and is forever linked to one of the most exciting scientific discoveries of the 1980s. It is likely that virtually every participant in the Montreal conference visited the United States pavilion at Expo 67. But none could have foreseen that less than 30 years later the 1996 Nobel Prize in chemistry would be awarded for the 1985 discovery of fullerenes by Robert Curl, Harry Kroto, and Richard Smalley.17 Nor could anyone present have predicted that the name of the third ordered form of carbon was destined to have a Montreal connection. Harry Kroto suggested the name buckminsterfullerene for C,, because the giant molecule reminded him of Buckminster Fuller’s geodesic dome (Figure l),which he and his young family had visited during those memorable days in the summer of 1967, The U.S. pavilion, later dubbed Biosphere, became the unofficial symbol of the World’s Fair. Thirty years later it still attracts tourists and at a height of more than 200 feet is said to be the largest round structure in the world. 1967 was a remarkable year. The Vietnam War was being fought, Elvis married Priscilla, Che Guevara was assassinated, the hippie movement was beginning, the Six-Day War between Egypt and Israel forced the closing of the

Figure 1 The geodesic dome designed by Buckminster Fuller for the U.S. pavilion at Expo 67 in Montreal.

220 The Development of Computational Chemisty in Canada Suez Canal, Simon and Garfunkel were singing The Sound of Silence, the first heart transplant was done in South Africa, and Charles de Gaulle stood on a balcony in Old Montreal and uttered his famous “Vive le Qukbec libre.” At the Montreal symposium, it was decided to hold the third conference in the series in 1969 at the University of Toronto and to have the conference, like the Montreal meeting, organized by two cochairs from different institutions, a tradition that has been maintained by all subsequent symposia in the series. The Toronto symposium was organized by Richard Bader (McMaster University) and Imre Csizmadia (University of Toronto). Like for the first two symposia, the list of invited speakers included a large number of chemists from abroad. Despite its name, the Canadian Symposium on Theoretical Chemistry is an international conference. Typically about 70% of the invited speakers are from abroad and more than 60% of the participants are from 15 or more countries other than Canada. Attendance is generally in the range of 160-220 registrants. The fourth symposium, organized by Gulzari Malli [Simon Fraser University) and Bob Snider (University of British Columbia) was held in 1971 on UBC’s beautiful campus in Vancouver. Plans for the fifth symposium in the nation’s capital, Ottawa, were modified when the First International Congress of Quantum Chemistry was organized in 1973 in Menton, France. David Bishop (University of Ottawa) and Vedene Smith (Queen’s University) organized the fifth symposium in Ottawa in 1974. The shift from a biennial to a triennial conference established a natural rotation between the American Conference on Theoretical Chemistry,l* the Canadian Symposium on Theoretical Chemistry, and the International Congress of Quantum Chemistry. This rotation has continued without interruption for more than two decades. As indicated in Table 1, the Canadian Symposia on Theoretical Chemistry (CSTC) have continued to flourish more than 30 years after the original conference organized by Fraser Birss and Serafin Fraga. Amid the splendid scenery of the Rockies, it was decided at the Seventh Canadian Symposium on Theoretical Chemistry in 1980 in Banff, Alberta, that the two cochairs should be chosen 6 years in advance. This policy has helped to facilitate future planning and to ensure the continuation of the strong traditions associated with the CSTC. Thus, at the symposium in 1995 in Fredericton, New Brunswick, Fred McCourt (University of Waterloo) and Jim Wright (Carleton University) were chosen to organize the fourteenth symposium in 2001, an auspicious date for computational chemistry. Although our colleagues in other countries generally have warm memories of what they refer to as the “Canadian Conference,” there are very few written records of the CSTC. Indeed, the first published proceedings did not appear until 1984, when Andrk Bandrauk and Russell Boyd arranged to publish 16 of the 36 invited papers presented at the Eighth Canadian Symposium on Theoretical Chemistry in the Journal of Physical Chemistry.19 Similarly, there have been very few accounts written for a more general audience, the one exception being an account20 of the Halifax meeting. After extolling the plenary lectures

Expo 67 and Fullerenes 221 Table 1 Canadian Symposia on Theoretical Chemistry Number 1 2 3

4

5 6 7 8 9 10 11 12 13 14

Location

Year

Cochairs

Edmonton, AB Montreal, PQ Toronto, ON Vancouver, BC Ottawa, O N Fredericton, NB Banff, AB Halifax, NS Toronto, ON Banff, AB Montreal, PQ Fredericton, NB Vancouver, BC Ottawa, O N

1965 1967 1969 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001

F. W. Birss and S. Fraga C. Sandorfy and M. A. Whitehead R. F. W. Bader and I. G. Czismadia G. L. Malli and R. F. Snider D. M. Bishop and V. H. Smith Jr. F. Grein and W. Forst S. Huzinaga and W. G. Laidlaw A. D. Bandrauk and R. J. Boyd R. E. Kapral and J. Paldus B. L. Clarke and A. Rauk D. R. Salahub and B. C. Sanctuary W. J. Meath and A. J. Thakkar G. N. Patey and T. Ziegler F. R. W. McCourt and J. S. Wright

of Gerhard Herzberg and Roald Hoffmann (Figure 2), the 1971 and 1981 Nobel laureates in chemistry, this account noted: The technical program for the first day concluded with a late evening poster session. An assortment of liquid refreshments encouraged a high participation rate in the poster session, despite the late hour. Such inaugural benediction on the first day of the conference was, needless to say, the right kind of encouragement for stimulating the atmosphere which prevailed the rest of the week.

Figure 2 Andri. Bandrauk, Roald Hoffmann, Gerhard Herzberg and Russell Boyd at the Eighth Canadian Symposium on Theoretical Chemistry in Halifax.

222 The Development of Computational Chemistry in Canada

CANADIAN ASSOCIATION OF THEORETICAL CHEMISTS At the Seventh Canadian Symposium on Theoretical Chemistry, it was recommended by Bill Laidlaw, and subsequently approved by the approximately 40 persons in attendance at the business meeting, that the chairs of the seventh, eighth, and ninth symposia investigate the possibility of forming a federally incorporated association of theoretical chemists. On the basis of information gathered by Russell Boyd, it was decided at an impromptu meeting on June 1, 1982, in Toronto to apply for incorporation. The meeting was held during the 65th Chemical Conference and Exhibition of the Chemical Institute of Canada and attended by Russell Boyd, Fritz Grein, Ray Kapral, Bill Laidlaw, Joe Paldus, and Vedene Smith, all past or soon-to-be chairs of the Canadian Symposium on Theoretical Chemistry. On February 10, 1983, the minister of Consumer and Corporate Affairs issued Letters Patent creating a corporation under the name Canadian Association of Theoretical Chemists/Association Canadienne des Chimistes Thioriciens. The objects of the corporation are: (i) to organize the triennial Canadian Symposia on Theoretical Chemistry; (ii) to further the development of theoretical chemistry in Canada; (iii) and to engage in matters relating to theoretical chemistry as deemed appropriate by the Board of Directors.

The application was filed by Andrk Bandrauk, Russell J. Boyd, Sigeru Huzinaga, Raymond E. Kapral, William G. Laidlaw, and Josef Paldus. According to the by-laws of the corporation: (i) The Board of Directors shall accept applications for membership from any theoretical chemist holding a faculty position in a Canadian university or college who is eligible for grants in aid of research from the Natural Sciences and Engineering Research Council of Canada; (ii) The Board of Directors shall accept applications for membership from any theoretical chemist actively engaged in research at government, industrial and private institutes and laboratories in Canada; (iii) Membership shall be for life upon payment of a $5.00 fee. The $5 lifetime membership fee was levied to cover the roughly $300 cost of incorporation. At any one time, the board consists of the cochairs of the most recent Canadian Symposium on Theoretical Chemistry and the cochairs of the next two symposia.

Demoarabhic Facts 223

DEMOGRAPHIC FACTS One fascinating way to chart the development of computational chemistry in Canada is to consider the chronology of the appointments of theoretical and computational chemists in Canadian universities. Figure 3 plots the number of appointments in two-year intervals, beginning with 1954-1955 up to and including 1996-1997. The most evident feature of Figure 3 is the rapid increase in the early 1960s leading to a peak later in the decade, which was followed by a precipitous decrease in the early 1970s. A second peak, about half as high as the first, occurred about two decades later, in the late 1980s. The second peak appears to have been due to a combination of growth and retirements. From 1954, when Camille Sandorfy took up his position at the University of Montreal, to 1960, only 6 theoretical chemists were appointed. During the next five-year interval, 16 appointments were made. In the five-year interval 1966 to 1970, inclusive, a total of 35 theoretical and computational chemists were appointed to Canadian universities, a time of unprecedented opportunities. The growth rate was not sustainable. From personal experience, the author knows that appointments for theoretical and computational chemists were extremely rare in Canada for the decade beginning in 1971. Indeed, for the ten-year interval from 1971 to 1980, only 14 appointments were made: 6 in 1971-1975andSin1976-1980. While the first peak in Figure 3 is due to the rapid expansion of the Canadian university system in the 1960s, the second peak was a t least in part

0

1960

1970

1980

Two-Year interval

1990

Figure 3 Number of new appointments, in 2-year intervals, of theoretical and computational chemists to the faculties of the departments of chemistry of Canadian universities.

224 The Development of Computational Chemistry in Canada brought about by a Natural Sciences and Engineering Research Council (NSERC)program to maintain a pool of talented young researchers to take up those positions which were expected to be available as the first of the many professors recruited to Canadian universities in the early 1960s began to retire. Thus, at least 10 appointments were made between 1981 and 1989 under the terms of the NSERC University Research Fellowships (URF) program. Without the URF appointments, the situation in the 1980s would have been scarcely better than that of the 1970s. Here it may be remarked that the limited employment prospects noted above for the 1970s were not restricted to theoretical chemistry. In fact, the situation in Canada was much the same in all branches of chemistry and in many of the other sciences. Figure 3 is based on the information in Table 2, which lists 108 individuals, their year of appointment, and a few biographical details. The criteria for inclusion in Table 2 are: 1. The individual was actively involved in theoretical and computational chemistry at the time of his or her appointment; 2. The appointment must have been in a department of chemistry at the level of assistant professor or higher. 3 . The individual held one or more appointments per items 1 and 2 for a minimum of four years. Not included in Table 2 are the many experimentalists who became active in computational chemistry after their appointments. A few such individuals are discussed later in the chapter. The brief biographical sketches in Table 2 indicate some interesting facts. Of the 108 appointees, 58 (54%) obtained their first degree in Canada. Nine were undergraduates at the University of Toronto, whereas the Universities of British Columbia (UBC) and Western Ontario each count seven as alumni. McGill and Queen’s Universities follow with five each, while the University of Alberta graduated four with B.Sc. degrees. Another 16 Canadian universities graduated the remaining 19. Of the 50 (46%) appointees who obtained their first degrees abroad, 13 graduated from universities in the United States and 12 studied in the United Kingdom. Hungary and the former Czechoslovakia each account for three more. The other 19 received their first degrees from universities in 18 countries. Table 2 reveals some equally interesting facts on the sources of the Ph.D. degrees of the 108 individuals. Of the 48 (44%)Ph.D. degrees earned in Canada, the University of Toronto leads with 11,followed by McGill University with 6 and McMaster University with 5; UBC and Alberta account for 4 apiece, and Queen’s University and the University of Montreal each claim 3 . The remaining 12 individuals received their Ph.D. degrees from nine other Canadian universities. Of the 60 (56%)Ph.D. degrees earned abroad, 29 and 19 were received from universities in the United States and the United Kingdom, respectively.

B.Sc., 1943, Ph.D., 1946, Univ. of Szeged, Hungary (Prof. A. Kiss); D.Sc., 1949, Sorbonne, Paris (Profs. Louis de Broglie and R. Daudel); PDF, 1951-1953, NRC, Ottawa (Dr. R. N. Jones) B.A., 1950, M.Sc., 1953, Univ. of British Columbia; D.Phil., 1956, Oxford Univ. (Prof. C. A. Coulson); PDF, 1956, Univ. of British Columbia (Prof. C. Reid) B.Sc., 1953, Univ. of Alberta; Ph.D., 1958, Univ. of Wisconsin (Prof. C. F. Curtiss); PDF, 1958, NRC, Ottawa (Dr. W. G. Henry) B.Sc., 1953, M.Sc., 1955, McMaster Univ.; Ph.D., 1958, Massachusetts Inst. of Technology (Prof. C. G. Swain); NRC PDF, 1958-1959, Cambridge Univ. (Prof. H. C. Longuet-Higgins) B.A., 1953, M.A., 1954, Univ. of Saskatchewan; D.Phil., 1956, Oxford Univ. (Prof. C. N. Hinshelwood); NRC PDF, 1956-1958, Oxford Univ. (Prof. C. A. Coulson); PDF, 1958-1959, Univ. of Rochester (Prof. A. B. F. Duncan) B.A., 1951, M.Sc., 1953, Univ. of British Columbia; Ph.D., 1958, Cambridge Univ. (Prof. H. C. Longuet-Higgins); NRC PDF, 1957-1959, Univ. of Montreal (Prof. C. Sandorfy) B.Sc., 1948, Technical Institute of Prague; Ph.D., 1955, McGill Univ. (Prof. C. A. Winkler); PDF, 1958-1961, Univ. of North Carolina (Prof. 0. K. Rice) B.A., 1953, Ph.D., 1956, Univ. of Toronto (Prof. R. L. McIntosh); RS, DuPont, 1956-1961 B.A., 1954, M.A., 1955, Univ. of Toronto; Ph.D., 1958, Cambridge Univ. (Prof. H. C. Longuet-Higgins); PDF, 1958-1960, NRC, Ottawa B.Sc., 1956, M.Sc., 1959, Univ. of Sydney; Ph.D., 1962, Univ. of Illinois (Prof. H. S. Gutowsky) B.Sc., 1956, M.Sc., 1958, Univ. of Gottingen; Ph.D., 1960, Univ. of Frankfurt (Prof. H. Hartmann); NRC PDF, 1960-1962, Univ. of New Brunswick (Prof. R. Kaiser) B.Sc., 1956, Queen Mary College, London; Ph.D., 1960, Univ. of London (Prof. M. J. S. Dewar); PDF, 1960-1962, Univ. of Cincinnati (Prof. H. H. Jaffe) B.Sc., 1957, Ph.D., 1960, Univ. College, London (Prof. D. P. Craig); PDF, 19601962, Carnegie Inst. of Technology (Prof. R. G. Parr) continued

1954 1957

Camille Sandorfy Univ. of Montreal

John A. R. Coope Univ. of British Columbia Robert F. Snider Univ. of British Columbia Richard F. W. Bader Univ. of Ottawa, 1959-1963; McMaster Univ., 1963Fraser W. Birssb Univ. of Alberta, 1959-1987

1961

Wendell Forst Lava1 Univ., 1961-1986 John B. Moffat Univ. of Waterloo John P. Valleau Univ. of Toronto R. Julian C. Brown Queen's Univ., 1962-1966, 1969Friedrich Grein Univ. of New Brunswick Michael Anthony Whitehead McGill Univ. David M. Bishop Univ. of Ottawa 1963

1962

1962

1962

1961

1961

1959

K. Lenore McEwen Univ. of Saskatchewan, 1959-1977

1959

1959

1958

Biographical Sketch

Year

Name

Table 2 Chronology of Appointments" of Theoretical and Computational Chemists in Canadian Universities

L.Sc., 1954, Ph.D., 1957, Univ. of Madrid (Prof. A. Perez-Masia); PDF, 1958-1961, Univ. of Chicago (Prof. R. S. Mulliken); PDF, 1961-1962, Univ. of Alberta (Prof. F. W. Birss); Asst. Prof., 1962-1963, Royal Military College, St. Jean, Quebec DipLEng., 1956, Polytechnic Univ., Budapest; M.Sc., 1959, Ph.D., 1962, Univ. of British Columbia (Prof. L. D. Hayward); NATO PDF, 1962-1964, Massachusetts Institute of Technology (Prof. J. C. Slater) L.Sc., 1958, D.Sc., 1962, Univ. of Geneva (Prof. Th. Posternak); PDF, 1962-1964, Cyanamid European Research Inst. (Prof. R. F. Hudson) B.Sc., 1958, M.Sc., 1960, Univ. of British Columbia; D.Phil., 1963, Oxford Univ. (Prof. C. A. Coulson); PDF, 1963-1964, Univ. of Pennsylvania (Prof. H. F. Hameka) B.Sc., 1959, Univ. of Western Ontario; M.Sc., 1961, California Inst. of Technology (Prof. R. Mazo); Ph.D., 1963, Univ. of Alberta (Prof F. W. Birss); NRC PDF, Oxford Univ., 1963-1964 (Prof. C. A. Coulson) B.S., 1958, Univ. of California, Berkeley; A.M., 1959, Ph.D., 1963, Harvard Univ. (Prof. G. B. Kistiakowsky); NATO PDF, 1963-1964, Oxford Univ. (Prof. J. W. Linnett); RA, 1964-1965, Univ. of Wisconsin (Prof. W. Byers Brown) B.Sc., 1941, M.Sc., 1943, Univ. of Natal; Ph.D., 1951, Univ. of London (Prof. F. C. Tompkins); faculty member, 1951-1965, Imperial College, London B.Sc., 1960, Carleton Univ.; Ph.D., 1965, Univ. of Wisconsin (Prof. J. 0. Hirschfelder ) B.Sc., 1948, M.Sc., 1949, Ph.D., 1951, Univ. of Manchester (Prof. H. A. Skinner); faculty member, 1951-1965, Univ. of Manchester B.A., 1959, Willamette Univ.; Ph.D., 1964, Univ. of Washington; PDF, 1964-1966, Univ. of Wisconsin B.Sc., 1961, Univ. of Salford; M.Sc., 1962, Ph.D., 1964, Univ. of Manchester (Prof. H. 0. Pritchard); Welch PDF, 1964-1965, Rice Univ. (Prof. K. S. Pitzer); NRC PDF, 1965-1966, York Univ. (Prof. H. 0. Pritchard) B.Sc., 1958, Delhi Univ.; M.Sc., 1960, McMaster Univ. (Prof. G. W. King); Ph.D., 1964, Univ. of Chicago (Profs. R. S. Mulliken and C. C. J. Roothaan); PDF, 1964-1965, Yale Univ. (Prof. G. Breit); Asst. Prof., 1965-1966, Univ. of Alberta B.Sc., 1960, Ph.D., 1963, Univ. College London (Prof. D. P. Craig); Turner and Newall Fellowship, 1964-1966, Univ. of Manchester

1963 1964 1964

Serafin Fraga Univ. of Alberta

Imre G. Csizmadia Univ. of Toronto

Sandor Fliszar Univ. of Montreal D. A. Hutchinson Univ. of Victoria, 1964-1967; Queen’s Univ., 1967- 1986 William G. Laidlaw Univ. of Calgary

1965

Patrick W. M. Jacobs Univ. of Western Ontario William J. Meath Univ. of Western Ontario Huw 0. Pritchard York Univ. Margaret L. Benstonb Simon Fraser Univ., 1966-1981 Geoffrey Hunter York Univ. 1966 1966

Gulzari L. Malli Simon Fraser Univ.

Keith A. R. Mitchell Univ. of British Columbia

1966

1966

1965

1965

1965

Delano P. Chong Univ. of British Columbia

1964

1964

Biographical Sketch

Year

Name

Table 2 Continued

1966 1966 1966 1967 1967

Neil S. Snider Queen’s Univ. 1966-1995

John Walkley Simon Fraser Univ.

Robert Wallace Univ. of Manitoba

Thomas W. Dingle Univ. of Victoria

Byung Chan Eu McGill Univ. Che-Shung Linb Univ. of Windsor, 1967-1989

1967 1968

Vedene H. Smith Jr. Queen’s Univ.

N. Colin Baird Univ. of Western Ontario Jiri Cizek Univ. of Waterloo Sigeru Huzinaga Univ. of Alberta Josef Paldus Univ. of Waterloo 1968

1968

1968

1967

David P. Santry McMaster Univ.

1967

1966

Reginald Paul Univ. of Calgary

B.Sc., 1956, M.Sc., 1957, Univ. of Lucknow, India; M.Sc., 1962, Univ. of Alberta (Prof. F. W. Birss); Ph.D., 1966, Univ. of Durham (Prof. Lord Wynne-Jones and Prof. G. N. Fowler) B.Sc., 1959, Purdue Univ.; M.A., 1961, Ph.D., 1964, Princeton Univ. (Prof. D. Hornig); PDF, 1964-1965, Cornell Univ. (Prof. B. Widom); PDF, 1965-1966, Yale Univ. (Prof. M. Fixman) B.Sc., 1954, Ph.D., 1957, Univ. of Liverpool; PDF, 1957-1960, Univ. of California, Berkeley (Prof. J. H. Hildebrand); faculty member, Imperial College, London, 1960-1966 B.Sc., 1961, Ph.D., 1964, Univ. of Glasgow (Prof. A. L. Porte); NATO PDF, 19641965, Uppsala Univ. (Prof. P.-0. Lowdin); NATO PDF, 1965-1966, Univ. of Bristol (Prof. A. D. Buckingham) B.Sc., 1958, Ph.D., 1965, Univ. of Alberta (Prof. H. E. Gunning); PDF, 1965-1966, Oxford Univ. (Prof. C. A. Coulson); PDF, 1966-1967, Univ. of Ottawa (Prof. D. M. Bishop) B.S., 1959, Seoul National Univ.; Ph.D., 1965, RA, 1965-1966, Brown Univ. (Prof. J. Ross); RF, 1966-1967, Harvard Univ. (Prof. M. Karplus) B.Sc., 1956, M.Sc., 1960, National Taiwan Univ.; Ph.D., 1965, Univ. of Saskatchewan (Prof. K. L. McEwen); PDF, 1964-1966, Univ. of Alberta (Profs. F. W. Birss and S. Fraga); PDF, 1966-1967, Indiana Univ. (Prof. H. Shull) B.Sc., 1960, Ph.D., 1963, Univ. College, London (Prof. D. P. Craig); PDF, 19631964, National Physical Lab, England; PDF, 1964-1967, Carnegie-Mellon Univ. (Prof. J. A. Pople) B.A., 1955, Emory Univ.; M.Sc., 1957, Ph.D., 1960, Georgia Inst. of Technology (Prof. H. A. Gersch); FiLLic., FiLDr., Docent, 1967, Uppsala Univ. (Prof. P.-0. Lowdin) B.Sc., 1963, Ph.D., 1967, McGill Univ. (Prof. M. A. Whitehead); Robert A. Welch PDF, 1966-1968, Univ. of Texas (Prof. M. J. S. Dewar) M.Sc. (RNDr.), 1961, Charles Univ., Prague, Ph.D. (CSc.), 1965, Czechoslovak Academy of Sciences, Prague (Prof. J. Koutecky) B.Sc., 1948, Kyushu Univ., Ph.D., 1959, Kyoto Univ. (Prof. G. Araki), RA, 19591961, Univ. of Chicago (Prof. R. S. Mulliken) M.Sc. (RNDr.), 1959, Charles Univ., Prague; Ph.D. (CSc.), 1962, Czechoslovak Academy of Sciences, Prague (Prof. J. Koutecky); PDF, 1962-1964, NRC, Ottawa IDr. D. A. Ramsav) continued

1969 1969

Johannes P. Colpa Queen’s Univ., 1969-1991

Jack L. Ginsburg6 Saint Mary’s Univ., 1969-1998 Bryan R. Henry Univ. of Manitoba, 1969-1987; Univ. of Guelph, 1987Alan C. Hopkinson York Univ. Raymond E. Kapral Univ. of Toronto Andri. D. Bandrauk Univ. of Sherbrooke

Arvi Rauk Univ. of Calgary Stuart M. Rothstein Brock Univ.

Bruce L. Clarke Univ. of Alberta Frederick R. W. McCourt Univ. of Waterloo

1969

Alan R. Allnatt Univ. of Western Ontario

1970

1970

1970

1970

1970

1969

1969

B.Sc., 1963, Ph.D., 1967, Sheffield Univ. (Prof. P. A. H. Wyatt); PDF, 1967-1969, Univ. of Toronto (Profs. K. Yates and I. G. Csizmadia) B.S., 1964, King’s College, Wilkes-Barre, PA; Ph.D., 1967, Princeton Univ. (Prof. L. C. Allen); PDF, 1968, Massachusetts Inst. of Technology (Prof. J. Ross) B.Sc., 1961, Univ. of Montreal (Loyola College); S.M., 1963, Massachusetts Institute of Technology; Ph.D., 1968, McMaster Univ. (Prof. R. F. W. Bader); NATO PDF, 1968-1969, Oxford Univ. (Prof. C. A. Coulson) B.Sc., 1965, Univ. of Toronto; Ph.D., 1969, Univ. of Chicago (Prof. S. A. Rice); PDF, 1969 Univ. of California, Santa Cruz (Prof. T. L. Hill) B.Sc., 1963, Ph.D., 1966, Univ. of British Columbia (Prof. R. E Snider); NRC PDF, 1966-1967, Swiss Federal Institute of Technology (ETH),Zurich (Prof. H. Primas); NRC PDF/RA 1967-1969, Kamerlingh Onnes Lab, Leiden (Prof. J. Beenakker) B.Sc., 1965, Ph.D., 1968, Queen’s Univ. (Prof. S. Wolfe); NRC PDF, 1968-1970, Princeton Univ. (Profs. L. C. Allen and K. Mislow) B.S., 1960, Univ. of Illinois; M.S., 1966, Ph.D., 1968, Univ. of Michigan (Prof. S. M. Blinder); PDF, 1968-1969, Johns Hopkins Univ. (Prof. H. J. Silverstone); Asst. Prof., 1969-1970, Swarthmore College

B.S., 1953, Ph.D., 1957, California Inst. of Tech. (Prof. R. M. Badger); NSF PDF, 1956, Harvard Univ. (Prof. W. E. Moffitt); Asst. and Assoc. Prof., 1958-1968, Massachusetts Inst. of Technology B.Sc., 1956, Ph.D., 1959, Univ. of London (Prof. P. W. M. Jacobs); NATO PDF, 1959-1961, Univ. of Chicago (Profs. S. A. Rice and M. H. Cohen); faculty member, 1961-1969, Univ. of Manchester Candidaat, 1948, Doctoraal, 1953, Ph.D., 1957, Univ. of Amsterdam (Profs. J. A. A. Ketelaar and J. de Boer); PDF, 1961-1962, Cambridge Univ. (Prof. H. C. Longuet-Higgins) B.S., 1962, Temple Univ.; Ph.D., 1968, Rutgers Univ. (Prof. L. Goodman); PDF, 1968-1969, McMaster Univ. (Prof. R. F. W. Bader) B.Sc., 1963, Univ. of British Columbia; Ph.D., 1967, Florida State Univ. (Prof. M. Kasha), PDF, 1968-1969, NRC, Ottawa (Dr. W. Siebrand)

1968

Walter R. Thorson Univ. of Alberta, 1968-1994

1969

Biographical Sketch

Year

Name

Table 2 Continued

1970

1972

Saul Goldman Univ. of Guelph John M. Sichel Univ. of Moncton 1975 1975 1976

Russell J. Boyd Dalhousie Univ.

Paul W. Brumer Univ. of Toronto

Dennis R. Salahub Univ. of Montreal, 1976-1999; Steacie Institute of the National Research Council of Canada, 1999-

1972

1972

Robert J. Le Roy Univ. of Waterloo

1971

1970

1970

continued

B.S., 1962, Stanford Univ.; Ph.D., 1968, Univ. of California, Berkeley (Prof. A. Streitweiser Jr.); PDF, 1968-1970, Orsay, France (Prof. L. Salem) B.Sc., 1961, Carnegie-Mellon Univ.; A.M., 1962, Ph.D., 1966, Harvard Univ. (Prof. M. Gouterman); NIH PDF, 1968-1970, Uppsala Univ. (Prof. P.-0. L.owdin) B.Sc., 1964, Univ. of Western Ontario; MSc., 1966, Ph.D., 1969, Univ. of Toronto (Prof. I. G. Csizmadia); NRC PDF, 1969-1970, Univ. of York (Prof. B. T. Sutcliffe); NRC PDF, 1971, York Univ. (Prof. H. 0. Pritchard) B.Sc., 1965, M.Sc., 1967, Univ. of Toronto (Prof. G. Burns); Ph.D., 1971, Univ. of Wisconsin (Prof. R. B. Bernstein); NRC PDF, 1971-1972, Univ. of Toronto (Prof. J. van Kranendonk) B.Sc., 1964, Ph.D., 1969, McGill Univ. (Prof. G. C. B. Cave); NRC PDF, 19691972, Univ. of Florida (Prof. R. G. Bates) B.Sc., 1964, Ph.D., 1968, McGill Univ. (Prof. M. A. Whitehead); PDF, 1967-1969, Univ. of Bristol (Prof. A. D. Buckingham); RA, 1969-1970, Asst. Prof., 19701972, Lava1 Univ. B.Sc., 1967, Univ. of British Columbia; Ph.D., 1971, McGill Univ. (Prof. M. A. Whitehead); NRC PDF, 1971-1973, Oxford Univ. (Prof. C. A. Coulson); Killam PDF, 1973-1975, Univ. of British Columbia B.S., 1966, Brooklyn College; Ph.D., 1972, Harvard Univ. (Prof. M. Karplus); Institute Fellow, 1972-1973, Weizmann Inst. (Prof. R. D. Levine); RF, 19731974, Harvard College Observatory (Prof. A. Dalgarno) B.Sc., 1967, Univ. of Alberta; Ph.D., 1970, Univ. of Montreal (Prof. C. Sandorfy); NRC PDF, 1970-1972, Univ. of Sussex (Prof. J. N. Murrell); RA, 1972-1974, Univ. of Waterloo; RS, 1974, Johns Hopkins Univ.; RS, 1975-1976, General Electric, Schenectady

1970

David Rodney Truax Univ. of Calgary Charles H. Warren Dalhousie Univ. Stuart G. Whittington Univ. of Toronto James S. Wright Carleton Univ. Michael C. Zernerbac Univ. of Guelph, 1970-1982 Roy E. Kari Laurentian Univ. 1970

B.Sc., 1964, McGill Univ.; Ph.D., 1968, Columbia Univ. (Prof. M. Karplus); NRC PDF, 1968-1969, Univ. of Leiden (Prof. P. Mazur); NRC PDF, 1969-1970, Univ. of British Columbia (Prof. R. F. Snider) B.Sc., 1964, Ph.D., 1969, Univ. of Western Ontario (Prof. R. G. Kidd); PDF, 19691970, Univ. of Copenhagen (Prof. C. J. Ballhausen) B.Sc., 1964, Univ. of Western Ontario; Ph.D., 1968, McMaster Univ. (Prof. G. W. King); NRC PDF, 1968-1970, Uppsala Univ. (Prof. P.-0. Lowdin) B.A., 1963, M.A., 1967, Ph.D., 1972, Cambridge Univ. (Prof. D. Chapman)

1970

Bernard Shizgal Univ. of British Columbia

1976 1977

Bryan C. Sanctuary McGill Univ.

Paul G. Mezey Univ. of Saskatchewan Seamus O’Shea Univ. of Lethbridge

1982

Raymond A. Poirier Univ. of Toronto, 1982-1984; Memorial Univ., 1984John D. Goddard Univ. of Guelph 1983

1981

1981

1980

Tom Ziegler Univ. of Calgary

Ajit J. Thakkar Univ. of Waterloo, 1980-1984; Univ. of New Brunswick, 1984Ross W. Wetmore Univ. of Guelph, 1981-1985

1979

Glenn M. Torrie Royal Military College of Canada Grenfell N. Patey Univ. of British Columbia 1980

1979

Jean-Pierre Laplante Royal Military College of Canada

1977

Year

Name

Table 2 Continued B.Sc., 1967, Ph.D., 1971, Univ. of British Columbia (Prof. R. F. Snider); PDF, 19721974, Univ. of Leiden (Prof. J. Beenakker); Asst. Prof., 1974-1976, Univ. of Wisconsin, Madison M.Sc., 1967, Ph.D., 1970 (Prof. F. Torok), M.Sc. (Math) 1972, Univ. of Budapest; PDF, 1973-1976, Univ. of Toronto (Profs. I. G. Csizmadia and K. Yates) B.Sc., 1967, Univ. College Cork, Ireland; Ph.D., 1973, McMaster Univ. (Prof. D. P. Santry); PDF, 1973-1975, Univ. of Western Ontario (Prof. W. J. Meath); RA, 1975-1977, NRC, Ottawa (Dr. M. L. Klein) B.Sc., 1971, Ph.D., 1976, Univ. of Sherbrooke (Prof. A. D. Bandrauk); PDF, 19761977, Free Univ. of Brussels (Prof.G. Nicolis), PDF, 1977-1979, NRC, Ottawa (Dr. W. Siebrand) B.Sc., 1971, M.Sc., 1972, Ph.D., 1975. Univ. of Toronto (Prof. J. P. Valleau); NSERC PDF, 1975-1976, Univ. of Paris (Prof. D. Levesque) B.S., 1970, Memorial Univ.; M.Sc., 1972, Ph.D., 1975, Univ. of Toronto (Prof. J. P. Valleau); PDF, 1975-1977, Univ. of Paris (Prof. D. Levesque); RA, 1978-1980, NRC, Ottawa (Dr. M. L. Klein) B.Sc., 1973, Ph.D., 1976, Queen’s Univ. (Prof. V. H. Smith Jr.); NRC PDF, 19761978, Univ. of Waterloo (Profs. J. Paldus and I. Magee); RA, 1978-1980, Queen’s Univ. (Prof. V. H. Smith Jr.) B.Sc., 1972, Univ. of Ottawa; Ph.D., 1977, Univ. of Southern California (Prof. G. A. Segal); PDF, 1977-1979, Univ. of California, Berkeley (Prof.H. F. Schaefer 111); RS, 1979-1981, Smithsonian Astrophysical Observatory (Prof. A. Dalgarno) M.Sc., 1972, Univ. of Copenhagen; Ph.D., 1978, Univ. of Calgary (Prof. A. Rauk); PDF, 1978-1980, Free Univ., Amsterdam (Prof. E. J. Baerends); NSERC PDF, 1980-1981, McMaster Univ. (Prof. D. P. Santry) B.Sc., 1975, M.Sc., 1976, Laurentian Univ. (Prof. R. E. Kari); Ph.D., 1980, Univ. of Toronto (Prof. I. G. Csizmadia); NSERC PDF, 1980-1982, CNRS, Paris (Prof. R. Daudel) B.Sc., 1973, Univ. of Western Ontario; M.Sc., 1974, Ph.D., 1978, Univ. of Toronto (Prof. I. G. Csizmadia); NSERC PDF, 1978-1979, Univ. of California, Berkeley, and RA, 1979-1980, Univ. of Texas, Austin (Prof. H. F. Schaefer 111); RA, 19801982, NRC, Ottawa (Dr. W. Siebrand)

Biographical Sketch

Katherine V. Darvesh Mount Saint Vincent Univ.

Randall S. Dumont McMaster Univ. David M. Ronis McGill Univ.

Timothy A. Wildman McMaster Univ., 1986-1991 Gregory C. Corey Univ. of Montreal, 1987-1993, Saint Mary’s Univ., 1995-1999 Tucker Carrington Jr. Univ. of Montreal

Simon J. Fraser Univ. of Toronto Ian P. Hamilton Univ. of Ottawa, 1986-1992; Wilfred Laurier Univ., 1992Saba M. Mattar Univ. of New Brunswick

T. T. Nguyen-Dang Univ. of Sherbrooke, 1984-1987; Lava1 Univ., 1987David M. Wardlaw Queen’s Univ. John M. Cullen Univ. of Manitoba

Brian A. Pettitt Univ. of Winnipeg Axel D. Becke Queen’s Univ.

1989

1988

1988

1988

1987

1986

1986

1986

1986

1985

BSc., 1968, Alexandria Univ.; M.Sc., 1974, American Univ. in Cairo (Prof. G. Habashi); Ph.D., 1982, McGill Univ. (Prof. W. C. Galley); PDF and RA, 19821986, Univ. of Toronto (Prof. G. A. Ozin) B.Sc., 1977, M.Sc., 1978, Ph.D., 1982, Univ. of Manitoba (Prof. T. Schaefer); RA, 1982-1986, NRC, Ottawa (Dr. W. Siebrand) B.Sc., 1974, Univ. of New Brunswick, M.Sc., 1980, Univ. of Waterloo (Prof. R. J. Le Roy), Ph.D., 1984, Univ. of Waterloo (Profs. F. R. W. McCourt and W.-K. Liu); NATO PDF, 1984-1986, RA, 1986-1987, Univ. of Maryland B.Sc., 1981, Univ. of Toronto; Ph.D., 1985, Univ. of California, Berkeley (Prof. W. H. Miller); PDF, 1985-1987, Swiss Federal Institute of Technology (ETH), Zurich (Prof. M. Quack); RA, 1987-1988, NRC, Ottawa (Dr. P. R. Bunker) B.Sc., 1981, Univ. of Western Ontario; Ph.D., 1987, Univ. of Toronto (Prof. P. W. Brumer); NSERC PDF, 1987-1988, Columbia University (Prof. l? Pechukas) B.Sc., 1974, McGill Univ.; Ph.D., 1978, Massachusetts Inst. of Technology (Prof. I. Oppenheim); Miller Fellow, 1978-1980, Univ. of California, Berkeley; Asst. and Assoc. Prof., 1980-1988, Harvard Univ. B.Sc., 1980, Ph.D., 1984, Univ. of New Brunswick (Prof. F. Grein); Killam and NSERC PDF, 1984-1989, Dalhousie Univ. (Prof. R. J. Boyd) continued

B.Sc., 1975, Ph.D., 1982, Univ. of Toronto (Prof. P. W. Brumer); NSERC PDF, 1982-1984, California Inst. of Technology (Prof. R. A. Marcus) B.Sc., 1969, M.Sc., 1973, Univ. of Windsor (Prof. G. W. F. Drake); Ph.D., 1981, Univ. of Guelph (Prof. M. C. Zerner); PDF, 1983-1985, Harvard Univ. (Prof. W. N. Lipscomb) B.A., 1966, Oxford Univ.; Ph.D., 1970, Cambridge Univ.; Ciba-Geigy fellowship, 1970, Univ. of Paris; SRC fellowship, 1972, Univ. of Sussex B.Sc., 1976, Ph.D., 1982, Univ. of Toronto (Prof. P. W. Brumer); PDF and RA, 1982-1986, Univ. of Chicago (Prof. J. C. Light)

1984

1984

1984

B.Sc., 1968, M.Sc., 1971, Univ. of Manitoba (Prof. R. Wallace); Ph.D., 1985, Dalhousie Univ. (Profs. R. J. Boyd and R. E. Wasylishen) B.Sc., 1975, Queen’s Univ.; Ph.D., 1981, McMaster Univ. (Prof. D. W. L. Sprung); NSERC, Killam and Eastburn PDF, 1981-1984, Dalhousie Univ. (Prof. R. J. Boyd) B.Sc., 1975, Univ. of Lisge; Ph.D., 1980, PDF, 1980-1982, McMaster Univ. (Prof. R. F. W. Bader); PDF, 1982-1984, Univ. of Sherbrooke (Prof. A. D. Bandrauk)

1983

1992 1993 1994

Alain St-Amant Univ. of Ottawa

John R. Gunn Univ. of Montreal

1992

1991

1991

1991

1991

1990

1989

David Jack Concordia Univ.

Benoit Roux Univ. of Montreal Gustavo A. Arteca Laurentian Univ.

Donald D. Frantz Univ. of Lethbridge, 1991-1995 William J. Pietro York Univ,

Donald F. Weaver Queen’s Univ. Kathleen M. Gough Brock Univ., 1990-1995; Univ. of Manitoba, 1995Baltazar D. Aguda Laurentian Univ.

B.Sc., 1971, Ph.D., 1978, N. Copernicus Univ., (Prof. J. Karwowski); PDF and RA, 1979-1988, Univ. of Alberta (Prof. S. Huzinaga) B.Sc., 1981, Univ. of Lethbridge; M.Sc., 1984, Ph.D., 1987, Univ. of British Columbia (Prof. G. N. Patey); NSERC PDF, 1987-1989, Australian National Univ. (Prof. D. J. Evans) M.D., 1981, Ph.D., 1986, Queen’s Univ. (Prof. S. Wolfe); neurology residency, 1986-1 988, Dalhousie Univ. B.Sc., 1971, Loyola of Montreal; M.Sc., 1976, Ph.D., 1984, Univ. of Manitoba (Prof. B. R. Henry); RA, 1984-1986, NRC, Ottawa (Dr. W. Murphy); NSERC PDF, 1987-1989, McMaster Univ. (Prof. R. F. W. Bader) B.Sc., 1978, Univ. of the Philippines; Ph.D., 1986, Univ. of Alberta (Prof. B. L. Clarke); PDF, 1987, Univ. of Alberta; PDF, 1988-1990, Indiana Univ.-Purdue Univ. Indianapolis (Prof. R. Larter); PDF, 1990-1991, York Univ. (Prof. H. 0. Pritchard) B.Sc., 1980, Univ. of Alberta; Ph.D., 1989, Harvard Univ. (Prof. D. R. Herschbach); PDF, 1989-1990, Univ. of Rhode Island (Prof. D. L. Freeman) B.S., 1978, Polytechnic Inst. of New York; Ph.D., 1982, Univ. of California, Irvine (Prof. W. J. Hehre); PDF, 1982-1985, Northwestern Univ. (Profs. T. J. Marks and M. A. Ratner); Asst. Prof., 1985-1991, Univ. of Wisconsin, Madison B.Sc., 1981, M.Sc., 1984, Univ. of Montreal; Ph.D., 1990, Harvard Univ. (Prof. M. Karplus); PDF, 1990-1991, Atomic Energy Commission, France (Dr. J. Smith) B.Sc., 1980, M.Sc., 1981, Ph.D., 1985, National University of La Plata, Argentina (Profs. E. A. Castro and F. M. Fernandez); PDF and RA, 1986-1992, Univ. of Saskatchewan (Prof. P. G. Mezey) B.Sc., 1979, Univ. of Toronto; M.Sc., 1981, Ph.D., 1986, Univ. of Alberta (Prof. H. J. Kreuzer); NSERC Industrial PDF, 1986-1989, Alcan, Kingston; RA, 1990, Visiting Asst. Prof., 1991, Univ. of Toronto (Prof. J. C. Polanyi) B.Sc., 1986, Univ. of Winnipeg, M.Sc., 1988, Ph.D., 1992, Univ. of Montreal (Prof. D. R. Salahub), NSERC PDF, 1992-1993, Univ. of California, San Francisco (Prof. P. A. Kollman) B.Sc., 1987, Univ. of Calgary; Ph.D., 1992, Univ. of California, Berkeley (Prof. K. Dawson); PDF, 1992-1994, Columbia Univ. (Prof. R. A. Friesner)

1989

Mariusz Klobukowski Univ. of Alberta Peter G. Kusalik Dalhousie Univ. 1989

BiograDhical Sketch

Year

Name

Table 2 Continued

1995

Mark R. Roussel Univ. of Lethbridge Rent Fournier York Univ.

1997 1997 1997

Natalie M. Cann Queen's Univ.

Jeremy M. Schofield Univ. of Toronto

Igor M. Svishchev Trent Univ.

B.Sc., 1976, Acadia Univ.; M.Sc., 1980, Ph.D., 1991, Univ. of Toronto (Profs. J. E. Dove and P. G. Martin); PDF, 1991-1992, Dalhousie Univ. (Prof. P. G. Kusalik); RA, 1992-1994, Univ. of Toronto (Profs. J. R. Drummond and H.-R. Cho) B.Sc., 1988, Queen's Univ.; M.Sc., 1990, Ph.D., 1994, Univ. of Toronto (Prof. S. J. Fraser); NSERC PDF, 1994-1995, McGill Univ. (Prof. M. C. Mackey) B.Sc., 1984; Ph.D., 1989, Univ. of Montreal (Prof. D. R. Salahub); PDF, 19891991, Iowa State Univ. (Prof. A. E. DePristo); RA, 1991-1994, NRC, Ottawa (Dr. W. Siebrand); PDF, 1995-1996, Univ. of Nevada, Las Vegas (Prof. C. Chen) B.Sc., 1986, Univ. of Western Ontario; Ph.D., 1991, Univ. of Waterloo (Prof. F. R. W. McCourt); NSERC PDF, 1991-1993, Northwestern Univ. (Prof. G. C. Schatz); PDF, 1993-1996, Queen's University (Prof. D. M. Wardlaw) B.Sc., 1989, Univ. of New Brunswick; Ph.D., 1993, Dalhousie Univ. (Profs. R. J. Boyd and A. J. Thakkar); NSERC and Killam PDF, 1993-1997, Univ. of British Columbia (Profs. C. E. Brion and G. N. Patey) A.B., 1988, Amherst College; Ph.D., 1993, Massachusetts Inst. of Technology, (Prof. I. Oppenheim); PDF, 1993-1995, Univ. of Chicago (Prof. S. A. Rice); PDF, 19951997, Northwestern Univ. (Prof. M. A. Ratner) M.Sc., 1984, Moscow State Univ.; Ph.D., 1988, USSR Academy of Sciences (Prof. Buslaev); PDF and RA, 1991-1997, Dalhousie Univ. (Prof. P. G. Kusalik)

"Data through 1997 only. Among the abbreviations used in this table are: PDF, postdoctoral fellow; RF, research fellow; RA, research associate; RS, research scientist. bDeceased cNote added in proofs: Sadly, I note that Mike Zerner died on February 2, 2000.

1996

Mark Thachuk Univ. of British Columbia

1996

1994

Margot E. Mandy Univ. of Northern British Columbia

234 The Development of Computational Chemistry in Canada

The remaining 12 Ph.D. degrees were earned in 10 other countries. Of the 29 Ph.D. degrees earned in the United States, Harvard leads with 5 , followed by 3 each from Berkeley, MIT, and Wisconsin. The remaining 15 were received from 13 universities, including 2 each from Chicago and Princeton. The University of London tops the U.K. list with 6, followed by Cambridge with 4 and Oxford with 3 . The remaining 6 were awarded by five other universities in the U.K.

TOWARD A STEADY-STATE POPULATION One of the remarkable features of the data in Table 2 is that only five of the 108 individuals are deceased and only 10 others have severed their ties through resignation or retirement. Of the remaining 93, two have taken up fulltime administrative appointments and do not play an active part in their respective departments of chemistry. Another 13 have reached retirement age but continue to be associated with their respective departments. This leaves a total of 78 on the permanent teaching faculties of the universities. Since many of the retired professors are active in research and often continue to do some teaching, the number of theoretical and computational chemists active in teaching and/or research is estimated to have been about 85 in 1997. Figure 4 shows the number of active theoretical and computational chemists as a function of time. The nearly exponential growth in the 1960s, followed by slower linear growth in the

100

80 L

2 $ z -m Z

I-

60 40

20 0

1960

1970

Year

1980

1990

Figure 4 Number of active theoretical and computational chemists on the faculties of the departments of chemistry of Canadian universities as a function of time.

Family Trees and Trends 235 1970s and 1980s, and a leveling off in the 1990s is clearly evident. It must be emphasized that this figure underestimates the true level of activity in computational chemistry because many experimentalists have taken up computational chemistry during the past decade.

FAMILY TREES AND TRENDS Examination of the brief biographical sketches in Table 2 reveals some interesting trends, notably a shift from the United Kingdom as the preferred location for research experience in the 1950s and 1960s to the United States in the 1980s and 1990s. For example, 11 of the first 20 individuals on the list completed their doctorates or did postdoctoral research in Britian, five at Oxford, three at London, and three at Cambridge. In contrast, none of the last 20 individuals on the list completed any part of their education in the United Kingdom, whereas 9 received doctoral or postdoctoral training in the United States. Seven of the last 20 received all their degrees and postdoctoral in training in Canadian universities, whereas only one of the first 20 on the list received all his degrees in Canada. There has been a significant increase in the number and duration of postdoctoral positions before taking up permanent positions. The average time spent doing postdoctoral research for the first 20 appointees in Table 2 is just under 2 years, while for the last 20 on the list the corresponding figure is about 4 years. A number of interesting scientific family trees may be deduced from Table 2. For example, Alain St-Amant and Reni. Fournier can be traced back to Camille Sandorfy through Dennis Salahub. Mark Thachuk can be traced back to Bob Snider through Fred McCourt, and Peter Kusalik can be traced back to John Valleau through Gren Patey. A significant fraction of the appointees in Table 2 can be traced back to the legendary figures of C. A. Coulson, H. C. Longuet-Higgins, D. P. Craig, A. D. Buckingham, and others in the United Kingdom, M. Karplus, 0. K. Rice, R. S. Mulliken, C. C. J. Roothaan, J. C. Slater, J. 0. Hirschfelder, W. N. Lipscomb, and others in the United States, and P.-0. Lowdin in Sweden. In some cases it is difficult to complete the family trees in Table 2 because information is missing. Such omissions do not always indicate that the identity of the supervisor is unknown. John Valleau recounted to this author an amusing tale about his postdoctoral fellowship at the NRC in Ottawa.

I had somehow become interested in ultrasonics and wished to set up some experimental activity in the field. I explained this, as far as I remember, by mail to Bill Schneider (whom I knew quite well, having worked in his lab one summer), and the solution proposed was that I should be

236 The Development of Computational Chemistry in Canada ostensibly attached to Otto Maass. Now Otto, a dear sweet man, had just been retired from McGill and had been taken in by the NRC as a rescue. Of course, Otto had no interest in either theory or ultrasonics, so we had no scientific interaction; also, he never actually cottoned-on to the fact that I was doing experiments. But he wanted to do his social duty to me. So once a week he appeared at the door of my office, stuck his head in, and said, “Well . . , are they giving you enough pencils?” And each week I replied in an equally unvarying way, saying, “Yes; the problem is in finding enough erasers.”

DEPARTMENTAL HISTORIES The following sections outline the history of the development of theoretical and computational chemistry in 21 of the largest departments of chemistry in Canadian universities. The coverage is limited to universities with Ph.D. programs where at least one full-time theoretical or computational chemist was appointed in 1970 or earlier. The sequence is based on the order in which the first appointments were made in each department (see Table 2 ) . Where more than one university is listed in a given year, the order is alphabetical. Space limitations preclude more comprehensive coverage. Nevertheless, it is hoped that this account will give a general sense of the emphasis within each department and an overview of Canada’s strength in the various subtopics of computational and theoretical chemistry.

University of Montreal (1954) As indicated in an earlier section, the first appointment of a theoretical or computational chemist was made in 1954 when Camille Sandorfy became a faculty member at the University of Montreal. Ten years later, Sandor Fliszar joined the Department of Chemistry and developed a research program focused on electron distributions and a variety of charge-dependent properties. His many papers on correlations between atomic charges and NMR chemical shifts, the relationships between atomic charges and bond dissociation energies, and the partitioning of atomic charges into core and valence contributions have been summarized in his monograph.21 The University of Montreal showed much foresight with the appointment of Dennis Salahub in 1976. In the next two decades he made several key contributions to the development and applications of density functional (DF)methodology. His early work with the X a method22 helped to explain diverse complex phenomena in the area of transition metal clusters, their electronic and magnetic properties, and their use as models for chemisorption and catalysis. Explanations emanating from calculations on the reduction of surface mag-

Departmental Histories 23 7 netism following chemisorption and on the lack of thermal expansion in the FeNi Invar alloys are noteworthy from this early period. Salahub was among a handful of workers who recognized early the possibility of bringing DF methods into mainstream quantum chemistry. His group has made seminal contributions to incorporating quantum chemical techniques into the DF methods and software: efficient integral schemes, model core potentials, and analytic gradients have helped to extend the range of applications. He has shown the importance of incorporating nonlocal corrections to the density functionals for treating weaker interactions such as hydrogen bonds.23 Recently, accurate values for NMR chemical shifts,24 and even spin-spin coupling constants, have been calculated. Salahub’s research has helped to bring the formal and theoretical strengths of density functional theory (DFT) to bear on both qualitative and quantitative aspects of real chemical and physical problems. Recent efforts are aimed at describing chemical reactivity in complex environments; the efforts of the past years have advanced the tools to a state at which this is now a realistic, though ambitious, goal. Salahub’s group is making noteworthy contributions to a wide spectrum of frontier issues25 associated with DFT: new improved functionals have been proposed, tested, and implemented, along with others, in the code suite deMon, which was developed in Montreal and is in use at dozens of labs around the world. In the context of the deMon program, a fusion of DFT with other techniques (reaction fields, molecular dynamics, etc.) is under way. These advances in methodology and techniques have vastly extended the scope of DFT, and this is bearing fruit in Salahub’s laboratory in the area of transition metal chemistry26 and biomolecular modeling. Salahub moved to the Steacie Institute for Molecular Sciences of the National Research Council of Canada in July 1999. This will give him expanded opportunities to direct research with the aid of post doctoral fellows. The University of Montreal made a very significant commitment to theoretical and computational chemistry with the appointments of Greg Corey (1987), Tucker Carrington Jr. (1988), Benoit Roux (1991), and John Gunn (1994). Carrington’s research deals primarily with the development of new methods to calculate vibrational and rotational-vibrational energy levels of polyatomic molecules. He and his coworkers have developed a potential optimized discrete variable representation (PODVR) to choose DVR basis functions adapted to the potential.27 In a series of papers,28 Carrington’s group has shown that if matrix-vector products are evaluated efficiently, the Lanczos algorithm becomes an attractive alternative to standard Householder methods for calculating vibrational spectra. They have also compared carefully the advantages of using contracted or direct product basis sets with the Lanczos algorithm. Benoit ROUX’Sgroup is using molecular dynamics (MD) simulations to elucidate the fundamental principles governing the transport of ions with a special interest in constructing detailed atomic models of the gramicidin channel.29 They also use M D simulations and free energy methodologies to study

238 The Development of Computational Chemistry in Canada multiple occupancy effects on the conformation of the gramicidin channel in phospholipid bilayers. John Gunn is also interested in the statistical mechanics of complex systems. He is especially interested in the development of new algorithms and potential functions for protein structure prediction and refinement.30

University of British Columbia (1957) With Charles McDowell as the new head and a mandate to build a strong Department of Chemistry, the University of British Columbia (UBC) was the second Canadian university to make a serious commitment to theoretical and computational chemistry. John Coope and Bob Snider were appointed in 1957 and 1958, respectively. Coope had an early interest in semiempirical theories of the electronic structure of molecules and the solid state, and particularly in deeper theories of why simple semiempirical theories may work, including effective Hamiltonian, and spin wave interpretations of valence bond theory. He has been concerned with various mathematical aspects of molecular physics, including effective Hamiltonians, the formulation of resonance conditions, perturbation theory based on characteristic equations, and topological aspects of polycrystalline spectra. Much of his work has focused on the effects of electric and magnetic fields on molecules, including the effects of fields on molecular transport properties, and his findings were published in papers often coauthored with R. F. Snider. Their papers on irreducible Cartesian tensors are good examples31-33 of their research during the 1960s. Snider is best known for his paper reporting what is now referred to as the Waldmann-Snider equation.34 (L. Waldmann independently derived the same result via an alternative method.) The novelty of this equation is that it takes into account the consequences of the superposition of quantum wavefunctions. For example, while the usual Boltzmann equation describes the collisionally induced decay of the rotational state probability distribution of a spin system to equilibrium, the modifications allow the effects of magnetic field precession to be simultaneously taken into account. Snider has used this equation to explain a variety of effects including the Senftleben-Beenakker effect (i.e., is, the magnetic and electric field dependence of gas transport coefficients), gas phase NMR relaxation, and gas phase muon spin relaxation.35 Other novel work of Snider includes a paper36 on the kinetic theory of recombination and decay, which covers the whole range of concentrations from pure monomers to pure dimers. Thus it becomes possible to describe, in a uniform and consistent manner, the dynamical properties of a reacting gas of monomers and dimers. As well, the 1974 paper36 led to an understanding of the differences in predicted kinetic behavior if a dimer is treated as a pair of monomers. Some of Snider’s more recent papers37 question the validity of earlier approaches to the treatment of the density corrections to ideal gas kinetic

Departmental Histories 239 theory. Questions arose because the quantum formulation of the Boltzmann equation emphasizes, by its basic nature, aspects different from those stressed in the classical formulation, pointing to questions of consistency in the earlier treatments. On the occasion of the Canadian Centennial in 1967, Bob Snider wrote an interesting article on theoretical chemistry for Chemistry in Canada, which is no less valid today than it was a third of a century ago. Three paragraphs are reproduced here for their historical significance.38

In its broadest interpretation, theoretical chemistry includes any attempt to devise an explanation, however qualitative or quantitative, of any experimental chemical result. As such, a theory can serve two very different roles. Firstly, a theory provides a set of concepts and a language in which experiments can be discussed; secondly it may provide a tool for the quantitative prediction of experimental results. The first use of a theory is by now quite well developed in the sense that any new experimental discovery is fairly rapidly explained qualitatively, in fact, there are often several rival explanations. With such theories, it is sometimes possible to correlate different experimental results quantitatively with the use of a few, hopefully only one or two, empirical parameters. This is always beneficial to science as it organizes and summarizes large bodies of data. However, since we now believe that all of chemistry can be explained in terms of quantum mechanics, one must go further and deduce these empirical relations from basic principles. This either verifies the qualitative theory or replaces it by a more rigorous one and at the same time may give numerical values for the empirical parameters. It may be thought that this is merely a job for a computer and the theoretician is necessary only to write the computer program. In fact, computers are of great assistance in solving specific problems in a short time. But even if computers grow large enough to solve all the chemist’s problems, someone will be required to understand the answer so that the answer can be applied. This, of course, can be quite difficult if the result is expressed as a million numbers which could not possibly be memorized much less understood. Thus the result of a calculation must be in a form in which it is useful-which again means in terms of simple equations between experimental quantities. Here then is the real work for the theoretician-to start off with basic equations such as the Schrodinger equation and derive from them simple relations which may be compared with experiment. Theoretical chemistry as its own specialized field is growing very rapidly throughout the world. One prediction of its future39 is very easy to see, namely, that a much greater use will be made of abstract mathematics, not

240 The Development of Computational Chemistty in Canada only in the mathematical manipulations that are performed but also in the whole language in which theoretical problems are described. On the other hand, with the increasing elaborateness of experimental methods it will be increasingly important for a theoretician to understand the experimental methods in order to correctly transcribe the experimental problem into theoretical terms. In short, the theoretical chemists of the future must have the best theoretical and experimental background that they can obtain if they are to keep up with this challenging but fascinating field. Theoretical chemistry at UBC was further strengthened with the arrival of Delano Chong and Keith Mitchell in 1965 and 1966, respectively. Chong’s interests in quantum chemistry have spanned the full range from semiempirical to ab initio molecular orbital methods. His long-standing interests in perturbation methods and constrained variations have figured prominently in his publications. He is probably best known for his attempts to calculate the X-ray and UV photoelectron spectra of molecules, often by means of perturbation corrections to Koopmans’ the0rem.4~More recently he has shifted his focus to coupled pair functional methods and density functional methods, with a special interest in polarizabilities and hyperpolarizabilities.41 Although Keith Mitchell’s initial research was in quantum chemistry, in the early 1970s he became interested in the possibility of developing low energy electron diffraction (LEED) to learn about the details of structure at wellcharacterized single-crystal surfaces. The research of his group has given new insight into chemical bonding at surfaces. In particular, for chemisorption of electronegative atoms on transition metal surfaces, four types of situation have been recognized as being capable of giving ordered commensurate structures: (1)simple chemisorption, where metal atoms relax in the vicinity of the electronegative at0rn,~2(2) surface compound formation,43 ( 3 ) an independent reconstruction,44 and (4) underlayer formation.45 Although Mitchell and his coworkers note that some initial guidance for the choice of structural type can be obtained from Pauling-type bond length arguments, new theoretical approaches would be advantageous to give deeper insights into the relaxations and reconstructions identified experimentally in the surface structures. When Bernie Shizgal arrived at UBC in 1970, his research interests were in applications of kinetic theory to nonequilibrium effects in reactive systems. He subsequently applied kinetic theory methods to the study of electron relaxation in atomic and molecular moderators,46 hot atom chemistry, n~icleation~47 rarefied gas dynamics,48 gaseous electronics, and other physical systems. An important area of research has been the kinetic theory description of the high altitude portion of planetary atmospheres, and the escape of atmospheric species.49 An outgrowth of these kinetic theory applications was the development of a spectral method for the solution of differential and integral equations referred to as the quadrature discretization method (QDM), which has been used with considerable success in statistical, quantum, and fluid dynamics.50

Departmental Histories 241 UBC’s strong tradition for statistical mechanics was given another major boost when Gren Patey arrived in 1980. Within a few years he established a reputation as a leading researcher on the theory of liquids and solutions, and more recently interfacial phenomena.51 His group uses simulation techniques to investigate the equilibrium and dynamical properties of liquids,52 solutionss3 and molecular clusters.54 Their specific interests include the dynamics of ion solvation, the forces between immersed macroscopic objects, the coagulation of colloidal suspensions, and the closely related membrane fusion processes of importance in biological systems. In some of their simulations of the phase behavior of liquid crystals, they demonstrated that ferroelectric nematic liquid crystals can exist.55 Mark Thachuk joined the UBC Department of Chemistry in 1996. His research program focuses on the study of the dynamics and rates of chemical reactions and processes by mathematical and computational techniques. Typically, such investigations utilize classical, semiclassical, or quantum mechanics, and combine scattering theory with reaction rate and kinetic theories.

University of Alberta (1959) Theoretical and computational chemistry at the University of Alberta started in 1959 when Fraser Birss arrived in Edmonton. Within a few years he was joined by two excellent graduate students, Bill Laidlaw and Reg Paul, both of whom later started theoretical and computational chemistry at the University of Calgary. Serafin Fraga was a postdoctoral fellow with Birss in 1961-1962 and returned a year later to join the faculty. Fraga has suggested that being a theoretician in Alberta in the late 1950s was probably a difficult experience for Fraser Birss. Fortunately the head of the department was Harry Gunning, who had been brought from the Illinois Institute of Technology to transform the Department of Chemistry into a major research center. The fact that Gunning had been in the city of Chicago (which at the time was a preeminent center for theoretical chemistry) may have accounted for his support of work in that field. In the early 1960s all calculations had to be done with a Marchant electrical calculator. (Somewhere in the Department of Physics there was an LPG-30 computer, but it was not available to the theoretical chemists.) It was during Fraga’s postdoctoral year at Alberta that he and Birss developed a general formulation for performing self-consistent field calculations on open-shell configurations.56 In 1964 Birss and Fraga finally got a real computer, an IBM 1620, which they operated themselves, first with paper tape and finally with cards. As theoreticians they did not have much of an identity within the Department of Chemistry and therefore with Gunning’s approval they formed the Division of Theoretical Chemistry. To make the division better known, they organized a conference in 1965 called the Alberta Symposium on Theoretical Chemistry. Fraga recalls that the budget was really small: about $35 for accommodation

242 The Development of Computational Chemisty in Canada and meals in the university residences! He notes that “altogether it was a terrific experience, although 1 decided that I would never organize another meeting.” With that seminal conference, he and Fraser Birss had put Canada on the theoretical chemistry map by bringing strong contingents from the Chicago group and the Hirschfelder group to Alberta’s capital. By the time the Symposium on Theoretical Chemistry returned to Alberta in 1980, it had become the Canadian Symposium on Theoretical Chemistry, a triennial event with an excellent reputation as already mentioned. In the late 1960s, the University of Alberta, with Harry Gunning as head of the Department of Chemistry, decided to build a center of excellence in theoretical chemistry by hiring Sigeru Huzinaga, Walter Thorson, and Bruce Clarke in a two-year period. Sigeru Huzinaga, one of the pioneers in the development of Gaussian basis sets for molecular calculations, held the position of associate professor in physics at Kyushu University in the 1960s. During that time he was invited by R. S. Mulliken to join the research group at the Laboratory of Molecular Structure and Spectra at the Department of Physics of the University of Chicago, where, under Mulliken and C. C. J. Roothaan, he participated in the inception of computational quantum chemistry (1959-1961). He was appointed professor of chemistry at the University of Alberta in 1968. The scientific interests of Huzinaga are numerous. He initially worked in the area of solid-state theory. Soon, however, he became interested in the electronic structure of molecules. He studied the one-center expansion of the molecular wavefunction, developed a formalism for the evaluation of atomic and molecular electron repulsion integrals, expanded Roothaan’s self-consistent field theory for open-shell systems, and, building on his own work on the separability of many-electron systems, designed a valence electron method for computational studies on large molecules. Very early Huzinaga recognized the usefulness of Gaussian-type functions for the evaluation of molecular integrals. Subsequently, for 25 years he was engaged in the development of Gaussian basis sets for molecular calculations. His 1965 paper,57 which contained the first comprehensive compilation of Gaussian-type functions for the atoms H through Ne, was designated a Citation Classic by Current Contents in April 1980. The use of analytical basis sets in molecular calculations led Huzinaga to studies on the nature and manipulation of virtual space and on the role of the continuum in the superposition of configurations. He has written two books on the molecular orbital method and edited a compendium of Gaussian basis sets for molecular calculations.-58 In February 1992 the Canadian Journal of Chemistry honored Sigeru Huzinaga on the occasion of his sixty-fifth birthday with a special issue containing more than 50 papers by his coworkers, students, colleagues, and friends from many countries. The special issue, edited by Russell Boyd and Mariusz Klobukowski, is a fitting tribute to celebrate 40 years of achievement in theoretical chemistry and physics.

Depavtmental Histories 243

Huzinaga was the recipient of the 1994 John C. Polanyi Award of the Canadian Society for Chemistry. In his award lecture he described his model potential method, which deals only with the active electrons in molecular and solid state calculations. An invited review article,59 based on his 1994 Polanyi Award lecture, chronicles his efforts to develop a sound theoretical framework for the core-valence separation of electrons, a problem Van Vleck and Sherman60 once referred to as “the nightmare of the inner core.” In 1968, after 10 years on the faculty at MIT, Walter Thorson was appointed professor of chemistry at the University of Alberta. The research problems that attracted his attention and for which he developed insight and skill involved quantum dynamics of few-body systems or systems with a limited number of degrees of freedom, focused especially on cases of motions that were not separable or only poorly separable. The choice of few-body systems was always based on the idea that rigorous critical studies could provide important paradigms valid for application to more complex systems. Many such studies concerned the theory of atomic collisions and focused on interactions of electronic and heavy particle motions. This was the most fruitful area of research for Thorson and his coworkers; the concepts and insights they developed proved useful not only in collision theory itself, but also in applications to problems in the spectroscopy and properties of small molecules. A secondary topic was the vibrational dynamics of hydrogen-bonded systems, where proton motion is highly nonharmonic and strongly coupled to other vibrational motions of the system. His group made some exhaustive studies of the strongly H-bonded bifluoride ion [FHF-] as a paradigm that rigorously explored the dynamics and its consequences for the IR spectra of such a coupled system;61 a later classical dynamics study62 on the same system showed the relation between such coupled quantum states and classical resonances. Thorson’s interest in fundamental studies of atomic collision processes in the slow collision regime (up to 25 keV collision energies) arose from recognition that while the theoretical context of the problem of electronic states of stable molecules had become well defined by 1960, there was then no corresponding conceptual framework for describing what happens (for example) when a proton collides with a ground state hydrogen atom. The adiabatic or “Born-Oppenheimer” separation of electronic motions from heavy particle motions in a stable molecule is clearly relevant in some way to the physics of collision between two atomic or molecular systems. However, an electron bound to an atom in a colliding system has translational kinetic energy and momentum with respect to the system center of mass merely by being a passenger on the partner to which it is bound, and an explicit account of this is needed to rationalize the dynamics of excitation, charge transfer, or other processes, Thorson made fundamental contributions to the theory of such “electron translation factors” and their effects on collision processes in primary model systems, showing how the theory can be systematized to take such effects into account.63

244 The Development of Computational Chemisty in Canada

In reflecting on the Division of Theoretical Chemistry at the University of Alberta, Walter Thorson has written to this author: Until his untimely death in 1987, a vital ingredient in my own work and experience at Alberta was the presence and fellowship of a kindred spirit in Professor Fraser W. Birss, with whom I had many fruitful discussions and who shared with me a similar outlook on theoretical chemistry, the problems of computational chemistry and the aims of educating students in science. While Fraser and I never actually wrote a paper together, we influenced each other’s works and methods, and for me and my students the climate of theoretical and computational chemistry at Alberta was shaped in large measure by that collegiality. I would hazard that every one of the theoretical and computational chemists at Alberta during the period also shared this to some extent; Fraser Birss had the gift of stimulating a sense of value in ideas for their own sake, and it was partly his vision which imagined the possibilities in bringing such an unusual number of theoreticians together in one department. Nearly four decades after the arrival of Fraser Birss and the recent retirements from active teaching of Serafin Fraga, Sigeru Huzinaga, and Walter Thorson, it appears that the University of Alberta has abandoned its commitment to establishing one of the largest theoretical chemistry groups in Canada. Only Bruce Clarke and Mariusz Klobukowski continue to teach and do research in theoretical and computational chemistry. Some of the aura of the 1965 symposium has been diminished. Hopefully, additional appointments will restore Edmonton to its former glory as one of Canada’s major centers for theoretical and computational chemistry.

University of Ottawa (1959) Richard Bader joined the Department of Chemistry at the University of Ottawa in 1959. He left Ottawa four years later to return to his alma mater, McMaster University. He was immediately replaced in Ottawa by David Bishop. The key theme of Bishop’s theoretical and computational research has been its relevance to experiment and its reliability. In recognition of his relatively precise calculations of the properties of small atoms and molecules, Bishop is acknowledged to be the major player in theoretical nonlinear optics in Canada. His results not only have been cited as a benchmark for other calculations but also very often serve in the calibration of the experimental equipment itself. Initially the properties calculated were energetic in nature and related to IR spectroscopic measurements.64 Bishop’s work was the first serious attempt to calculate the rovibronic energies of the hydrogen molecule and molecular ion without using the Born-Oppenheimer approximation (i.e., three- and fourbody calculations). Many years later, this work is still cited and the relativistic

Departmental Histories 245 and radiative corrections are still used; in particular, his determination of the quadrupole moment of the deuteron remains the recommended value in the CERN handbook. Bishop’s attention turned to accurate calculations of electrical and magnetic properties, especially those of importance in nonlinear optics. Since most experiments in this field measure ratios, not absolute values, it is necessary to have a calculated value. Universally, Bishop’s helium nonlinear optical properties are used. In the same field, he was the first to seriously investigate the effects of electric fields on vibrational motions, with a much-quoted paper.65 His theory and formulation has now been added to two widely used computational packages: HONDO and SPECTROS. He has also derived a rigorous formula to account for the frequency dependence (dispersion) in nonlinear optical properties.66 He used this theory to demonstrate that the anomalous dispersion in neon, found experimentally, is an artifact of the measurements. Bishop has recently been studying the magnetic properties of small systems (including the Cotton-Mouton effect and the Faraday effect), once again providing accurate values with which the experiments can be judged. As well, his concerns with the effects of magnetic fields on vibrations have received widespread attention. Since joining the University of Ottawa in 1993, Alain St-Amant has focused on developing methods specifically designed to treat large systems, especially biological molecules.67 Whenever possible, he concentrates on methods that will be of use in both conventional Hartree-Fock and density functional programs. He is also involved in the development of linear scaling methods68 for electronic structure calculations, an intensely competitive topic in computational chemistry.

University of Saskatchewan (1959) When Lenore McEwen was appointed in 1959, she became the first woman named as a theoretical chemist on the faculty of a department of chemistry of a Canadian university. She maintained her interest in the electronic structures and spectra of nitrogen-oxygen compounds for several years after her arrival in Saskatoon, although most of her papers resulted from her postdoctoral research in Sandorfy’s group.69 Following McEwen’s resignation in June 1977, Paul Mezey took up the torch of computational chemistry in Saskatchewan. His research activities range from numerous new mathematical theorems on molecular stability, symmetry, chirality, reactivity, and molecular shape-complementarity to new computational methodologies that may be applied in many fields such as computeraided drug discovery, toxicological risk assessment, and molecular engineering. His numerous papers and books discuss his attempts to describe the shapes of molecules70 and to model chemical reactions at the molecular level.71 He is especially interested in using shape analysis72 of molecules in pharmaceutical compound design and in modeling large molecules such as proteins. He

246 The Development of Computational Chemistry in Canada developed macromolecular density matrix methods to image the electron density of proteins.73 Related methods show promise for the calculation of various molecular properties, such as the forces acting on the nuclei of macromolecules, and provide computational tools for the study of protein folding. In 1993, Mezey’s group reported74 their thorough analysis of all 20 geometric conformations of p-alanine by means of the shape group method that Mezey had pioneered 10 years earlier. Their shape-similarity analysis of the stable conformations of p-alanine provides an important test case of their methodology and represents a level of complexity approaching that of typical conformational problems in computer-aided drug design.

Laval University (1961) Laval University was one of the first Canadian universities to hire a theoretical chemist. Wendell Forst arrived in 1961 and developed a research program based on the theory of unimolecular reactions75 and quantum chemistry. He maintained ties with experimental physical chemistry through a strong interest in mass spectrometry and gas phase kinetics. In many of his papers he sought analytical solutions to fundamental problems.76 In 1986, after a quartercentury at Laval, he moved to the University of Nancy in France. T. T. Nguyen-Dang joined Laval University in 1987. His research has concentrated on the nonadiabatic dynamics of simple molecules. In this context, he has contributed to the generalization of the Numerov-Fox-Goodwin algorithm77 for multichannel Schrodinger equations involving nonadiabatic, kinetic couplings. He has also pioneered the development of high-order adiabatic representations using unitary transformations to resume first- or low-order nonadiabatic interactions.78 These methods are being used to study the dynamics of laser-driven molecules. The dynamics and structure of molecules in intense laser fields is the second main topic of Nguyen-Dang’s research, a theme he has maintained from his time at the University of Sherbrooke where, together with A. D. Bandrauk, he contributed seminal works on the adiabatic separation of dressed molecules within the Bloch-Nordsieck representation.79 At Laval, continuous efforts in the directions listed above led to the development of original nonperturbative wave packet methods for the simulation of the dynamics of the laser-driven molecules.*O Returning to the formal theory of dressed molecules treated completely within the a priori quantum electrodynamical representation of Coulomb systems, Nguyen-Dang and his group are seeking to develop symbolic tools for further analytical explorations of the high-order adiabatic representations they developed for these systems.

University of Toronto (1961) The origins of computational and theoretical chemistry at the University of Toronto can be traced to two key appointments made by Professor Donald J.

DepartmePttal Histories 247

Le Roy soon after his appointment in 1960 as head of the Department of Chemistry. In 1961 John Valleau became the first theoretical chemist to join the faculty at Toronto. The subsequent appointments of Ray Kapral(1969),Stuart Whittington (1970),and Paul Brumer (1975)led to the formation of the Chemical Physics Theory Group, designed to complement the excellent experimental chemical physics group of John Polanyi and his colleagues. The group of theoreticians soon became the strongest group of its kind in Canada and comparable to the best in the world. In the same year that John Valleau joined the faculty at Toronto, Le Roy also appointed Keith Yates, an organic chemist with a keen interest in the kinetics and mechanisms of organic reactions. During the following two decades, Yates and his colleagues built up Canada’s leading physical organic chemistry group. This achievement was, in part, due to the appointment of Imre Csizmadia to the faculty in 1964. Although Csizmadia and Yates both received their Ph.D. degrees in organic chemistry from the University of British Columbia, their paths to joining the faculty of the University of Toronto have little in common. Keith Yates grew up in England and after leaving school joined the Royal Navy. After several years of travel, he enrolled at the University of British Columbia (B.A., 1956, M.Sc., 1957, Ph.D., 1959). Following the completion of his doctorate in physical organic chemistry under the supervision of Prof. Ross Stewart, he went to Oxford and earned a D.Phi1. in 1961 as an NRC and NATO postdoctoral fellow. Imre Csizmadia completed his diploma in engineering in Budapest just prior to the Hungarian Uprising in 1956. Like many Hungarians he was granted refugee status in Canada and continued his studies at the University of British Columbia. Following the completion of his Ph.D. in organic chemistry, he joined Prof. John C. Slater’s group at MIT in 1962. During his first year at MIT he completed the first ab initio calculation with a Gaussian basis set on an organic molecule (HCOF). Given their complementary interests, it was natural that Csizmadia and Yates would collaborate on many computational projects. The arrival of Tom Tidwell in 1972 brought the number of organic chemistry faculty members with interests in computational chemistry to three, until Keith Yates retired in about 1990. John Valleau’s research has focused on Monte Carlo simulations for liquids and solutions. Many of his papers deal with the application of Monte Carlo methods to the structure of electrolyte solutions,gl the structure of double layers,*2 polyelectrolytes,s3 and the phase transition in Coulombic ~ysterns.8~ He made a major contribution with the invention of umbrella sampling.85 He noticed that one can sample from distributions other than the Boltzmann distribution, and then reweight. The original idea was to use this insight to estimate free energy differences at different temperatures, but it is also extremely helpful for solving, or helping with, quasi-ergodic problems. These come from slow convergence of the Markov chain along which the sampling is being carried out. Often, using umbrella sampling will help speed up con-

248 The Development of Computational Chemistry in Canada

vergence of the Markov chain to its limit distribution. It is now a widely used method. Valleau has extended the method so that it works simultaneously at several densities as well as at several temperatures (essentially a twodimensional umbrella). Imre Csizmadia is generally acknowledged to be the pioneer of ab initio theoretical organic chemistry in Canada. His interest in the optimization of Gaussian basis sets goes back to his postdoctoral research. The work of Csizmadia and his coworkers has been summarized in a book.86 Protonation and deprotonation in both ground and excited states have also constituted a major theme in Csizmadia’s work. His publications in this area reflect his strong ties to organic chemistry.87 Many of the papers from his group are associated with stereochemistry and his interest in the classification of the topology of potential energy hypersurfaces. This research has led to an ongoing interest in the development of peptide models by means of ab initio calculations.88 Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. The mechanisms of reactions that occur in condensed phases involve the participation of solvent degrees of freedom. In some cases, such as in certain ion association reactions involving solvent-separated ion pairs, even the very existence of reactant or product states depends on the presence of the solvent. Traditionally the solvent is described in a continuum approximation by reaction-diffusion equations. Kapral’s group is interested in microscopic theories that, by treating the solvent at a molecular level, allow one to investigate the origin and range of validity of conventional continuum theories and to understand in a detailed way how solvent motions influence reaction dynamics. While many aspects of the theory of classical condensed phase rate processes are well developed, no computationally tractable scheme based on a theory with controlled approximations exists for quantum systems, or for mixed quantum-classical systems, where some degrees of freedom, typically those of the solvent, may be treated classically. Kapral’s research in this area has

Departmental Histories 249

focused on the development of constrained MD schemes for classical and quantum reaction rates, path integral methods, and theories for mixed classicalquantum systems. These theoretical methods are used to study quantum reaction rates, such as proton transfer, in the condensed phase as well as in large molecular where new chemical mechanisms arise. The cluster environment differs radically from the bulk, and therefore Kapral is interested in exploring the distinctive chemistry that occurs in solvent-influenced reactions in these small model systems. This research forms part of a larger study of reaction rates and mechanisms in the mesoscale domain.93 Stu Whittington’s initial interests in statistical mechanics were focused on the use of Monte Carlo methods, especially for polymer problems.94 Minor early interests included solution dimensions and shapes of polysaccharides, as a function of linkage type, and so on. He was also concerned with how solution dimensions of copolymers (such as alginate) are affected by the comonomer sequence statistics. He then moved into percolation theory95 and its connection with gelation. In particular, his group showed that gels should have the same critical exponents as percolation, and not Flory-Stockmayer exponents. More recently he has worked on rigorous treatments of models of polymer adsorption, polymers in confined geometries, polymer collapse, knots and links in polymers, ribbon models of double-stranded polymers,96 lattice animals as models of branched polymers, and related topics.97 The research of Paul Brumer and his colleagues addresses several fundamental problems in theoretical chemical physics. These include studies of the control of molecular dynamics with lasers.98 In particular, the group has demonstrated that quantum interference effects can be used to control the motion of molecules, opening up a vast new area of research. For example, one can alter the rate and yield of production of desirable molecules in chemical reactions, alter the direction of motion of electrons in semiconductors, and change the refractive indices of materials etc. by creating and manipulating quantum interferences. In essence, this approach, called coherent control, provides a method for manipulating chemistry at its most fundamental leve1.99 Another interest of the Brumer group comes under the heading of classical and quantum chaos. The dynamics of most molecular systems is nonlinear and, if the nonlinearity is sufficient, the motion is chaotic (i.e., extremely sensitive to initial conditions). Brumer and his colleagues have explored the role of chaotic motion in chemical reactions and in isolated molecule dynamics as it relates to statistical theories of dissociation. Deep insights into isolated molecule dynamics have emerged. Their research program continues to focus on this area by exploring the role of chaos in quantum mechanics and in understanding quantum-classical correspondence in chaotic systems. This work also provides a general framework for the understanding of correspondence in all conservative systems.100 Theoretical computations on molecular dynamics require new developments in semiclassical mechanics to treat many degrees of freedom. Thus,

250 The Development of Computational Chemistry in Canada Brumer’s group is examining numerous methods, based on the semiclassical initial value representation, to develop a highly effective and general scheme for studying molecular motion. They are also interested in time- and frequencyresolved laser experiments101 that provide insight into the dynamics of atoms and molecules. Ongoing research in the group is dedicated toward developing a general computer-based approach to understanding these experiments and to learning what they tell us about molecular motion.

University of Waterloo (1961) Quantum chemistry at the University of Waterloo began with the appointment of John Moffat in 1961, only 2 years after the university received its charter. While Moffat’s primary research interests have centered on heterogeneous catalysis and surface chemistry, he had an interest in quantum chemistry since his time at DuPont (1956-1961). With the arrival of a mainframe computer (likely an IBM 650) at DuPont, he was asked if he would like to learn how to program it. He was thus initiated into basic machine language and began doing some simple molecular orbital calculations on nitriles. He continued this work at Waterloo. Among other topics, he worked on energy partitioning studies on nitriles.102 A few years later Moffat and Popkie showed that the sharing interference energies of various bonds correlated with the dissociation energies.103 The calculations were subsequently extended to surfaces. The Soviet Union’s military invasion of Czechoslovakia in August 1968 had a significant impact on the development of computational and theoretical chemistry in Canada. The troubles in Czechoslovakia led to Jiri Cizek and Joe Paldus accepting appointments at the University of Waterloo in 1968. Their many achievements include the first ab initio study of the coupled cluster method.104 Jiri Cizek’s research program centers on the quantum theory of molecular electronic structure and related developments in quantum chemical methodology, coupled-cluster approaches to many-electron correlation problems,’0” large-order perturbation theory,106 dynamical groups and exactly solvable models, lower bounds, and the use of symbolic computation language in physics and in chemistry. For the treatment of electron correlation, Cizek uses classical techniques as well as techniques based on mathematical methods of quantum field theory, namely, a coupled-cluster approach. A rapid development and deployment of these methods during the past decade was stimulated by the realization of the importance of size consistency or size extensivity in the studies of reactive chemical processes. Although truly remarkable accuracy and development have been achieved for ground states of closed-shell systems, an extension to quasidegenerate and general open-shell systems is most challenging. Cizek also works on the exploitation of these approaches to study the electronic structure of extended systems (molecular crystals, polymersl07). His many interests in-

Departmental Histories 252 clude the development of methods for the estimation of upper bounds and the use of symbolic computations in quantum chemical calculations.108 In the fall of 1958, Josef Paldus started a CSc. (Ph.D.) program a t the Institute of Physical Chemistry of the Czechoslovak Academy of Sciences in Prague, working under the supervision of Dr. J. Koutecky. He was Koutecky's first student to tackle the field of quantum chemistry and subsequently became one of the founders of the Prague theoretical school. In his CSc. thesis, Paldus succeeded in providing the first interpretation of electronic spectra of systems with nonbonding interactions, which were very puzzling to organic chemists at the time. His most significant contribution in this regard was a semiquantitative explanation of the optical spectra of [m, n]-paracyclophanes, barrelene, germacrol, and other systems involving so-called transannular interactions. In 1962 Paldus was awarded a National Research Council of Canada Postdoctoral Fellowship and joined the Division of Pure Physics at NRC to work in the Larger Molecules Section (LMS) under the supervision of Dr. D. A. Ramsay. This was just at the time when new, high resolution spectrographs were being obtained at LMS and computers were beginning to help with the analysis of complex molecular spectra. Aided by these new tools, Paldus embarked o n an analysis of the vibrational and rotational fine structure of glyoxal electronic spectra. Paldus resumed his position in Prague in August 1964 to continue his work in quantum chemistry and the analysis of glyoxal spectra. He subsequently returned to Ottawa twice (1966 and 1968) as an NRCC Visiting Scientist, to complete work on the 0-0 vibrational band of the first singlet n + 7c:' transition in glyoxal, as well as to collaborate on the rotational analysis of vibrationally excited bands. This work laid the foundation for extensive studies of glyoxal spectra by Ramsay's LMS group. During his 1968 stay at the NRCC in Ottawa, following the Soviet invasion of Czechoslovakia, Paldus decided to settle permanently in Canada. In December 1968, he accepted an offer of a visiting associate professorship in the Department of Applied Mathematics at the University of Waterloo, where he has remained to the present day; he obtained a regular appointment in September 1969 and tenure in July 1970. He was cross-appointed to the Department of Chemistry in June 1973. Since his appointment at the University of Waterloo, Paldus has fully devoted himself to theoretical and methodological aspects of atomic and molecular electronic structure, while keeping in close contact with actual applications of these methods in computational quantum chemistry. His contributions include the examination of stability conditions and symmetry breaking in the independent particle models,l09 many-body perturbation theory and Green's function approaches to the many-electron correlation problem,llO the development of graphical methods for the time-independent many-fermion problem,lll and the development of various algebraic approaches and an exploration of convergence properties of perturbative methods. His most important

2S2 The Development of Computational Chemistry in Canada

and influential work produced during the past two decades involves the development of the unitary group112 and coupled-cluster~~3 approaches to the many-electron correlation problem. These papers include several seminal contributions, which enabled the highly successful development of large-scale configuration interaction and both single-reference and multireference coupled cluster calculations; these works nowadays form a basis of many state-of-theart methods of post-Hartree-Fock quantum chemistry. Fred McCourt arrived at the University of Waterloo in 1970. His Ph.D. research had led to the first theoretical description of external magnetic field effects on the transport properties of gases of diamagnetic polyatomic molecules. These field effect phenomena were called Senftleben-Beenakker effects, after Herman Senftleben, who discovered such effects for gases of paramagnetic molecules, and Jan Beenakker, who first measured them for gases of diamagnetic molecules. The next few years involved a careful systematic extension of the perturbation expansion of the distribution functioddensity matrix for a polyatomic gas. This allowed the introduction of the now nearly universal “telephone number” notation for the identification of the many expansion terms involved.114 The methods of analysis were extended to include electric field effects in gases of polar molecules (mainly symmetric or near symmetric top molecules) and effects associated with second gradients of flow velocity and temperature (the so-called Burnett effects). An essentially complete understanding of the thermomagnetic torque in rarefied gases was achieved in this way. These developments led to a unified description of transport properties and relaxation phenomena along lines first suggested by Roy Gordon (Harvard). Key relaxation phenomena considered were the relaxation of the bulk magnetization of diatomic gases (as probed by NMR) and the scattering of light, especially depolarized Rayleigh scattering, by such gases. An ease of comparison between the various transport and relaxation phenomena was achieved by expressing all experimentally measurable quantities in terms of the “telephone number,” or effective cross sections, introduced in 1970.115 A second stage in the systematic development of transport and relaxation phenomena required an ability to evaluate the effective cross sections for a given potential energy surface describing the interaction between a pair of molecules. The extensive research of McCourt and his coworkers on these topics culminated in a two-volume monograph.116 Once a sound theoretical description of transport and relaxation phenomena was in place and methods for calculating the effective cross sections were available, McCourt turned his attention to the problem of obtaining accurate potential energy surfaces for the interactions between the atoms and molecules. In many cases a number of empirical potential surfaces were available from inversions of various sets of experimental data, none of them particularly definitive, with the exception of the interaction of molecular hydrogen with the rare gases, for which Bob Le Roy had obtained reliable results. Higher level ab initio calculations have been employed to give

Departmental Histories 253 reliable potential energy surfaces for a series of second-row diatomic species with He, Ne, and Ar.117 Bob Le Roy, son of D. J. Le Roy, came to Waterloo in 1972. His research focused on the properties and dynamical behavior of small molecules and molecular clusters, and on the intermolecular forces that govern them. The techniques used include analytic derivations, detailed quantum calculations for dynamics on multidimensional potential energy surfaces, classical Monte Carlo or molecular dynamics simulations, and nonlinear least-squares fits to experimental data of various types.1l8 Le Roy has a long-standing interest in the inversion of bound state or photodissociation data, and in the exploitation of the theoretically predicted characteristic near-dissociation behavior of simple molecules.1~9He uses the spectra of van der Waals dimers to obtain information about intermolecular forces, and in particular, about their anisotropy and dependence on internal coordinates. His group is developing accurate new methods for calculating the properties of these very floppy molecules and applying them to new spectroscopic data120 and other types of measurement to determine accurate multidimensional potential energy surfaces121 for systems such as He-HF and H,-CO. The structures and dynamical behavior of larger clusters consisting of a few to a few dozen atoms or molecules is of great interest because the properties of such aggregates are intermediate between bulk and isolated-molecule behavior, and their study promises insight into the molecular nature of solvation and phase transitions. Le Roy’s classical simulations of these systems have been combined with computer visualization techniques and quantum predictions of the spectral perturbation of a chromophore “solute” molecule in such clusters, to relate their microscopic dynamics to experimental observables.

McGill University (1962) Since his arrival at McGill University in 1962, Tony Whitehead has been known as one of the most colorful and entertaining members of the Canadian theoretical chemistry community. His appointment coincided with the establishment of Theoretica Chimica Acta, the first journal devoted specifically to the subjects of theoretical and computational chemistry. At the time, TCA, which was established in Germany, encouraged the publication of manuscripts in English, French, and German. The instructions to the authors indicated that Latin was also acceptable, presumably in keeping with the name of the journal. No articles written in Latin appeared in the first six volumes. However, the invitation to publish in Latin was a challenge to Whitehead and so with the help of a classicist he published a short note,122 Modus Computandi Ezgenvectores et Eigenaestimationes e Matrice Densitatis, in 1967. To assist readers lacking a knowledge of Latin, the abstract was also printed in the other three official languages of the journal. TCA was renamed Theoretical Chemistry Accounts:

254 The Development of Computational Chemistry in Canada Theory, Computation, and Modelling in 1997 and now publishes exclusively in English. During the early years at McGill University, Whitehead’s group concentrated on experimental nuclear quadrupole resonance spectroscopy123 and a variety of 7c- and all-valence electron semiempirical molecular orbital methods.124 His recent interests have included topics as diverse as density functional theoryl2s and related topics,l26 and molecular models of surfactants. The general themes of B. C. Eu’s research are theoretical studies in nonequilibrium statistical mechanics, extended irreversible thermodynamics for systems far removed from equilibrium, generalized hydrodynamics, and related topics in transport properties in both simple and complex fluids.127 Since Eu’s arrival at McGill in 1967, his group has developed and studied kinetic theories of irreversible processes in fluids (liquids and gases), the kinetic theory foundations of irreversible thermodynamics,‘28 and the kinetic theory of chemical reactions in solution, fluid dynamics, and rheology of complex liquids. In particular, chemical oscillations and waves in excitable media have been studied as part of a research program on nonlinear dynamics and generalized hydrodynamics. Mechanisms for pattern formations and selections and their irreversible thermodynamic bases have been investigated. Also, his group has studied extensively the equilibrium and nonequilibrium statistical mechanics of polymeric liquids. The Eu group apply their theories of nonlinear transport processes and related irreversible thermodynamics to transport properties129 of semiconductors, rheology of polymeric liquids and polyatomic liquids, and gas dynamics problems. Specifically, they calculate pair correlation functions and structure factors to correlate with experimental data on nonsimple liquids; various conductivities of charge carriers in semiconductors subjected to high external field gradients, and non-Newtonian viscosities of complex liquids subjected to high shear rates. In addition, kinetic theories of quantum gases and gases interacting with radiation are being investigated as part of the research program on irreversible thermodynamics and its kinetic foundations. Bryan Sanctuary’s research is based on nonequilibrium statistical mechanics, with emphasis on the applications. Before coming to McGill in 1976, he worked t o extend the Boltzmann equation to a quantum form valid for nondegenerate internal states. There are a few applications for this, such as pressure broadening, the effects of electric and magnetic fields on the transport properties of gases (the Senftleben-Beenakker effect), and NMR. Thus Sanctuary entered into a general study of NMR at a time when the experimental field was rapidly advancing. His present research is aimed at providing a theoretical basis for nuclear magnetic resonance and nuclear quadrupole resonance spectroscopies by means of general relaxation theory. His group has developed numerical methods for the assignment of multidimensional proton spectra of amino acid residues in proteins.130 The Sanctuary group studied the line shapes of the NMR spectra of solid H2, D, , and HD, and their dynamics. Analytical

Departmental Histories 255 solutions for a variety of spin systems were obtained131 as part of research on the theory and application of NMR spin dynamics. The development of pulse sequences in NMR to understand echo formation and structure has also been investigated. Research on multipole NMR was summarized in a 1990 review. 1 32 After eight years on the faculty at Harvard, David Ronis established in 1988 at McGill University a research program based on the nonequilibrium and equilibrium statistical mechanics in condensed complex systems. Specific examples include transport in membrane and zeolite channels, and the static and dynamic properties of suspensions of highly charged colloidal particles.133 In many respects, dilute colloidal suspensions mimic atomic systems and can be used to study static and dynamic processes such as solidification, rheology, shear-induced melting, and shear-induced order or pattern formation. Ronis’s group has developed several simple, but successful, theoretical models that explain many features of experiments on these systems.134 The interests of the Ronis group include correlations and conformations in polymer-coated ~olloids.13~ In particular, they are developing theories to describe the coupled intra- and interparticle interactions in systems comprising colloidal particles coated with charged or neutral, polymeric chains. This research is designed to answer several fundamental questions, including: What is the distribution of counterions in the polymer layer and in the space between the colloidal particles? They have developed a theory to explain the surface texturing observed on the surfaces of extruded plastics.136 Their current research includes a study of turbulent fluid flow and its effects on the thermodynamics and kinetics of phase transitions in binary liquid mixtures using renormalization group methods.

Queen’s University (1962) In one sense, research in theoretical chemistry at Queen’s University at Kingston originated outside the Department of Chemistry when A. John Coleman came in 1960 as head of the Department of Mathematics. Coleman took up Charles Coulson’s challenge150 to make the use of reduced density matrices (RDM) a viable approach to the N-electron problem. RDMs had been introduced earlier by Husimi (1940), Lowdin (1955), and McWeeny (1955). The great attraction was that their use could reduce the 4 N space-spin coordinates of the wavefunctions in the variational principle to only 16 such coordinates. But for the RDMs to be of value, one must first solve the celebrated N-representability problem formulated by Coleman, namely, that the RDMs employed must be derivable from an N-electron wavefunction.151 This constraint has since been a topic of much research at Queen’s University, in the Departments of Chemistry and Mathematics as well as elsewhere. A number of workshops and conferences about RDMs have been held, including one in honor of John Coleman in 1985.152 Two chemists, Hans Kummer [Ph.D. Swiss Federal Technical

2.56 The Development of Computational Chemistry in Canada

Institute (ETH), Zurich, with Prof. H. Primas] and Robert Erdahl (Ph.D. Princeton with Prof. L. C. Allen), arrived in 1965 as postdoctoral fellow’s with Coleman and stayed to become professors of mathematics. In 1962, when Prof. R. L. McIntosh came to Queen’s as the new head of the Department of Chemistry, one of the items on his agenda was to build up a strong group in theoretical chemistry. The first member was R. Julian C. Brown, who joined Queen’s University in September 1962 and began research and teaching in theoretical chemistry. He had just completed his Ph.D. at the University of Illinois under the supervision of H. S. Gutowsky; his thesis work was the theoretical analysis of spin-relaxation measurements in a liquid. His research at Queen’s continued in the areas of molecular dynamics in fluids and nuclear spin relaxation.153,154 He established a graduate course in quantum theory, in which, in those early days of NMR, the emphasis was placed on the quantum theory of spin systems in addition to the theory of electrons in atoms and molecules. At Queen’s it was natural for him to begin research in density matrix theory.155 In 1966, he returned to his native Australia; but he came back156 to Queen’s in 1969 and began experimental research on crystals, particularly disordered ammonium salts. Recently he has turned to computational studies of molecular dynamics in crystals.157 In 1966 Neil Snider joined Queen’s Department of Chemistry. He was followed by Douglas Hutchinson in 1964, Vedene Smith in 1967, and Hans Colpa in 1969 (see Table 2). Hutchinson left in 1984, while Colpa and Snider retired in 1991 and 1995, respectively. Meanwhile, Axel Becke and David Wardlaw arrived in 1984, Donald Weaver in 1989 and Natalie Cann in 1997. During his time at Queen’s University, Neil Snider was involved in theoretical research in two main areas: reaction dynamics and kinetics and equilibrium statistical mechanics of fluids. In the area of reaction kinetics, Snider’s work dealt in large part with understanding the effect of the details of the energy transfer probability on the kinetics of gas phase reactions wherein the rates are determined either in part or entirely by energy transfer.lS8 His work in the area of reaction dynamics was most notably directed toward understanding the dynamics of recrossing of reactant-product dividing surfaces by reactive and nonreactive classical trajectories.159 In the area of equilibrium statistical mechanics of fluids, Snider devoted effort to extending the scaling theory of critical phenomena. Most of his work in this area, however, dealt with the theory of classical simple dense fluids. In particular, he sought ways in which generalized van der Waals equations of state might be applicable to real liquids,l60 and he looked to the justification of such equations on the basis of statistical mechanical perturbation theory.161 Although Doug Hutchinson’s original interest was in semiempirical methods, some of his research at Queen’s was concerned with the theory of interaction of electromagnetic radiation with matter,162*163an interest aroused during his postdoctoral time with Hank Hameka at Penn and stimulated further

Departmental Histories 257 by a sabbatical year with C. Mavroyannis at NRC. As well, Hutchinson had an interesting and fruitful collaboration on various problems in electron spin resonance theory with his colleague at Queen’s, Jeff Wan.164 Vedene Smith came to Queen’s in 1967 for one of the Conferences on Reduced Density Matrices and Their Applications and returned later that year to join the Department of Chemistry. It should be no surprise that density matrices and electron density functions in position and momentum space have played a large role in the research program of Smith and his group. Although people had previously considered momentum densities and the related experimental quantity, the Compton profile, Smith was the first to point out and implement the use of the Fourier transform of the one-electron density matrix as the route to the momentum density, instead of the procedure of transforming the entire wavefunction and then reducing the N-electron momentum density to the one-electron momentum density. Benesch and Singh165 of the Smith group also showed the nature of the relationship between the momentum density and the charge density and introduced a function, today called B ( r ) ,which was the 3D-Fourier transform of the momentum density and the 1D-Fourier transform of the Compton profile. For these reasons, it has proven very useful in the analysis and interpretation of experimental Compton profile data. Schmider and Smith166 used B ( r ) to develop a very neat treatment of the modulation of the Compton profile by a weak laser field. Smith and his group have made a number of careful studies of the electron charge and momentum densities for atoms, molecules, and solids with respect to the role of electron correlation and bonding, topology, concentration regions, and shell structure. They have examined the intracule (relative electronelectron) and extracule (electron pair center-of-mass) distributions in both position and momentum space for ground and excited states at both SCF and CI levels. They were the first to do this at the CI level for more than two electrons. In addition to information about bonding and other properties, these pair densities are useful for developing exchange-correlation potentials with allowance for the kinetic energy contribution.167 These researchers have also calculated elastic, inelastic, and total scattering of X-rays and fast electrons, efforts that have been useful for the calibration of experiment, elucidation of bonding effects, and assessing the effects of basis set quality and electron correlation. Recently they have investigated the role of relativity.f68 The Smith group has also developed the methodology for making high precision calculations for small systems without invoking the BornOppenheimer approximation and have made calculations for two-electron atomic ions, small muonic molecules, and potentials of the screened Coulomb form. Their method for determining nonlinear parameters is now referred to as random tempering.169 Hans Colpa, known for his work on the pressure-induced rotational absorption spectra of hydrogen170 and on hyperfine coupling constants, came to Queen’s in 1969. The interpretation of Hund’s rules for energy differences in

258 The Development of Computational Chemistry in Canada ~~

atomic and molecular systems, and their common misinterpretation, became one of his major initial research interests. For more than 20 years, Colpa and his group had a very fruitful collaboration with the Max Planck Institute in Heidelberg and a successor research group in the Free University of Berlin. A primary interest of the work was in the mechanism of optical nuclear polarization (ONP),as observed in single molecular crystals at low temperatures. The particular importance of the level anticrossing (noncrossing) rule was stressed, and the impact of relaxation effects171 was considered in some detail. In addition, a formal theory for microwaveinduced ONP was derived.172 The role of tunneling in the formation and decay kinetics of photochemical proton transfer in aromatic single crystals was studied by means of time-resolved ONP and optically detected (spin resonance) methods.173 Returning to his alma mater, Queen’s University, Axel Becke has made significant advances in two research areas of quantum chemistry: the design of new computer algorithms174 for the computation of molecular energies and structures, and development of new density functional theories175 of the electronic structure of atoms, molecules, and condensed systems. Becke’s work on algorithms focuses on high numerical precision. Conventional quantum chemistry employs finite basis sets to represent electronic orbitals in molecules. Accuracy is limited by the nature and the size of the basis set, and systematic basis set extension is difficult. His approach, however, employs grids of several thousand points per atom to represent molecular functions.176 In addition to superior flexibility, the precision of this grid-based method is straightforwardly controlled by adjusting the number of grid points. Becke’s methodology is unique (though aspects of his procedure have been adopted by others) and is invaluable as a generator of benchmark results for the calibration of basis sets and theories. The product of this work is a full-featured molecular structure program called NUMOL, which he hopes to make available for general distribution. About 14 years ago, Becke discovered that a certain class of DFTs gives remarkably good molecular energies177 at significantly less cost than conventional non-DFT methods (i.e., correlated wavefunction methods). Indeed, density functional theory now holds great promise as the theory of choice for firstprinciples simulation of chemical reactions involving dozens or even hundreds of atoms, and of materials as well. Becke’s goal has been, and still is, to refine the accuracy and reliability of DFT as far as possible. He has shown the way to density functional theories with precision approaching that of the best available quantum chemical methods (typical thermochemical errors of a few kcal/mol). Data extremely difficult to obtain experimentally (e.g., accurate reaction energy profiles) now lie within the reach of a highly economical computational/ theoretical methodology.178 Becke’s recent theories have been incorporated into virtually every molecular structure program in the world, including major commercial packages.

Departmental Histories 259 David Wardlaw’s research in theoretical chemical physics focuses on the dynamics of chemical reactions and related molecular processes. His research program has three predominant themes: fundamental development, design and testing of simple models and interpretation and simulation of selected experiments. His postdoctoral research with Marcus led to flexible transition state theory (FTST),which predicts rate constants for barrierless association reactions (e.g., two radicals, or ion + molecule), and the reverse dissociation reacti0ns.17~FTST forms a “new” branch of transition state theory parallel to the famous RRKM theory for reactions with barriers developed by Rice, Ramsperger, Kassel, and Marcus (RRKM). FTST remains a topic of active research at Queen’s and elsewhere, and has been applied to numerous experimentally studied reactions. In the early 1990s Wardlaw and coworkers entered a relatively young but rapidly expanding field, the dynamics of chemical and physical processes under the influence of electric fields associated with very short, very intense laser pulses. Strong fields open up a domain of pathways to chemical products that cannot be explored by traditional chemical methods. This suggests the possibility of precise control over reactions, a long-pursued goal in chemistry. The Wardlaw group contributions to date180 in this field are twofold: development and refinement of simulation methods (both quantum and semiclassical), and creation of simplified interpretive models. Other topics that have been investigated include quantitative descriptions of classical and quantum chaotic dynamics,l81 extensive contributions to the theory of time delay in scattering systems,1*2 and developments in the modeling and interpretation of fall-off behavior (pressure dependence) of rate constants.1*3 The modeling and simulation components of research in the Wardlaw group are inextricably linked to various aspects of computational chemistry. At one end of the spectrum is the effort to render exact dynamics simulations feasible for systems with more than a few degrees of freedom. This requires fundamental developments in either the form or application of the equations of motion, yielding “new” or “improved” methodologies. A noteworthy example is their recently formulated “semiclassical” method (classical nuclei and quantum electrons) for treating the dynamics of molecules in intense laser pulses. At the other end of the spectrum is the development of computer programs or a suite of programs (software package). A pertinent example of the latter is the VARIFLEX software for the calculation of flexible transition state theory rate constants. Don Weaver joined the Department of Chemistry in 1989. He holds the distinction of being the only computational chemist in Canada who is also a practicing physician; he is a part-time clinical neurologist and professor of medicine at the Queen’s teaching hospitals. His research is at the chemistryneuroscience interface, and he has concentrated on macromolecular modeling of brain molecules using empirical force field methods.184 He has worked on new methods for searching conformational space, including genetic algorithms (GA)lgs and variable basis Monte Carlo (VBMC).

260 The Development of Computational Chemisty in Canada Weaver has developed force field parameterizations for phospholipids and various other biomolecules using ab initio molecular orbital methods. He has used semiempirical techniques to study amino acids, neurotransmitters, and various neuroactive drugs.186 From an applied perspective, he has performed computational quantitative structure-activity relationship studies (QSARs) on a wide range of anticonvulsant drug molecules. His work has led to the design and patenting of new chemical entities for treating epilepsy and Alzheimer’s disease. In conjunction with this design work, he cofounded Neurochem Inc. and Neuroceptor Inc. for the commercialization of his drug designs.187 Natalie Cann joined the Department of Chemistry at Queen’s in 1997. She is developing a research program focused on the study of separations of mixed systems. In particular, she is using simulations and analytic theories to model simple chiral mixtures.

University of New Brunswick (1962) Theoretical chemistry began at the University of New Brunswick in 1962 with the appointment of Fritz Grein. His early work focused on the development of methods to include a correlation factor (1 + arI2) in one-center wavefunctions in order to improve the ground state for two-electron molecules.137 Next he turned his attention to the development of configuration interaction (CI)programs for the calculation of the excited states of diatomic molecules.138 As noted earlier, he became interested in multiconfiguration self-consistent field methods (MCSCF) in 1965 as a consequence of his participation in the Edmonton conference. He developed the first MCSCF method with second-order convergence and later extended the method to excited states. One of his MCSCF papers139 was selected by Fritz Schaeferl40 as one of the landmark papers of quantum chemistry for the period 1928 to 1983. For more than two decades, Grein has carried out many careful studies of the excited states of small molecules by means of multireference CI calculations. Many of the computations for these studies were completed during his annual sojourns in Bonn. Some the best-known work of Grein and his coworkers involves the development of methods for the calculation of hyperfine coupling constants.141 More recently the focus has shifted to calculating magnetic g-tensors from highly correlated wavefunctions. Grein’s current interests include the study of stereoelectronic effects (such as the anomeric and reverse anomeric effects in acetal-like systems) in organic chemistry, a topic to which he has made important contributions.142 After four years at the University of Waterloo, Ajit Thakkar joined the Department of Chemistry at the University of New Brunswick in 1984, where he has maintained a very dynamic and productive research program centered around the quantum mechanical calculation of molecular properties and interactions. Some of his best known work is concerned with van der Waals forces and potential energy surfaces. His method for a generalized expansion for the

Departmental Histories 261 potential energy curves of diatomic molecules~43is cited by spectroscopists as the Thakkar expansion. Thakkar and his coworkers have also published important papers on polarizabilities, hyperpolarizabilities, and related properties.144 Thakkar and his group have completed many careful studies of intracules, extracules, Coulomb holes, and related topics.145 These position-space results are complemented by Thakkar’s many studies of momentum densities and related quantities. He also has a long-standing interest in electron and X-ray scattering.146 His current interests include the relationship147 between the aromaticity of heterocyclic compounds and their polarizabilities, and the prediction of “push-pull” molecules that have a large nonlinear optical response and are thus candidates for materials to be used in optical computers. Saba Mattar came to the University of New Brunswick in 1986. His research program is split between experimental and theoretical studies of the electronic structures and bonding in clusters and organometallic intermediates. Several experimental techniques are used to study rnatrix-isolated transient species, and the results are interpreted with the assistance of multireference CI calculations.*48 He also uses local density functional methods.149

McMaster University (1963) Richard Bader was among the earliest of workers to realize the importance of electron density in providing an understanding of chemistry. Early on he was led to formulate the first symmetry rule governing a chemical reaction in answer to the question of how the electron density changes in response to a motion of the nuclei. This rule, termed the pseudo- or second-order Jahn-Teller effect, provides the theoretical underpinnings of frontier molecular orbital theory and is still widely used in discussions of reaction mechanisms and molecular geometries. Beginning in the 1960s, Richard Bader initiated a systematic study of molecular electron density distributions and their relation to chemical bonding using the Hellmann-Feynman theorem.188 This work was made possible through a collaboration with the research group of Professors Mulliken and Roothaan at the University of Chicago, who made available their wavefunctions for diatomic molecules, functions that approached the Hartree-Fock limit and were of unsurpassed accuracy. The continuing study of molecular electron distributions led to the realization in 1972 that an atom, and the functional groups of chemistry, could be defined in terms of the fundamental topology exhibited by the electron density.189 This work culminated in the demonstration that the predictions of quantum mechanics could be extended to the topological atom, using the new formulation of quantum mechanics afforded by the work of Feynman and Schwinger. In particular, Schwinger’s principle of stationary action is used to identify the chemical atom with a proper open quantum system, one whose observables obey the correct equations of motion. Thus the functional groups

262 The Development of Computational Chemistly in Canada of chemistry are bounded space-filling objects whose properties are defined by the quantum mechanics of a proper open system, properties that faithfully recover the role of the functional group in chemistry. The topology of the electron density also leads to the identification of a chemical bond with a line linking neighboring nuclei along which the electron density is a maximurn.l90 This identification leads to a definition of molecular structure that is remarkable in its ability to recover all chemical structures. The dynamics of the density, as occasioned by nuclear displacements and analyzed by means of the mathematics of qualitative dynamics, leads to a complete theory of structural stability, one that clarifies the meaning of the making and breaking of a chemical bond. The Laplacian of the electron density plays a dominant role throughout the theory.191 In addition, Bader has shown that the topology of the Laplacian recovers the Lewis model of the electron pair, a model that is not evident in the topology of the electron density itself. The Laplacian of the density thus provides a physical valence-shell electron pair repulsion (VSEPR) basis for the model of molecular geometry and for the prediction of the reaction sites and their relative alignment in acid-base reactions. This work is closely tied to earlier studies by Bader of the electron pair density, demonstrating that the spatial localization of electrons is a result of a corresponding localization of the Fermi correlation hole. The theory of atoms in molecules~92recovers all the fundamental concepts of chemistry, of atoms and functional groups with characteristic properties, of bonds, of molecular structure and structural stability, and of electron pairs and their role in molecular geometry and reactivity. The atomic principle of stationary action extends the predictions of quantum mechanics to the atomic constituents of all matter, the proper open systems of quantum mechanics. All facets of the theory are predictive and, as a consequence, the theory can be employed in many fields of research at the atomic level, from the design and synthesis of new drugs and catalysts, to the understanding and prediction of the properties of alloys. Richard Bader’s seminal contributions to the field are beautifully documented in his most recent book193 and celebrated in the June 1996 issue of the Canadian Journal of Chemistry. The special issue of nearly 60 papers, edited by Russell Boyd and Nick Werstiuk, was published on the occasion of Bader’s sixty-fifth birthday. David Santry arrived at McMaster in 1967 following a postdoctoral fellowship with John Pople during the semiempirical, all-valence-electron period at Carnegie-Mellon University. He pursued a variety of research topics relating to electronic structures but is probably best known for his long-standing interest in infinite three-dimensional networks of hydrogen molecules194 and timedependent Hartree-Fock theory.195 Many of his papers are concerned with molecular polarizabilities and hyperpolarizabilities.’ 96

Departmental Histories 263

Since his arrival at McMaster in 1988, Randall Dumont has focused on statistical theories and their origin in quantum and classical mechanics. His interests include the development of Monte Carlo implementations of statistical theory wherein dynamical processes are simulated by random walks on potential energy surfaces. The breakdown of statistical theory and the appearance of nonexponential population decay are also topics of his ongoing investigations. Other questions of interest are the incorporation of quantum effects into statistical theory and the effects of collisions on reaction processes. He has a special interest in argon cluster evaporation in vacuum197 and in the description of simple isomerization reactions.198 His other interests include the semiclassical description of classically unallowed processes such as tunneling.199

University of Calgary (1964) Theoretical chemistry at the University of Calgary began with the arrival of Bill Laidlaw in 1964, two years before the university gained full autonomy from the University of Alberta. In his early work, Laidlaw focused on the applications of quantum chemistry to spectroscopic and structural problems. With teaching responsibilities in physical and theoretical chemistry, he wrote a textbook200 designed to introduce students to the world of quantum mechanics through spectroscopy. Later he became interested in symmetry breaking and molecular orbital instabilities in sulfur-nitrogen compounds.201 His continuing interest in the transition of open thermodynamic systems from a state with one symmetry to another state with a new symmetry is reflected in several of his papers. His many contributions include an important paper on light scattering.202 In recent years he has shifted the focus to applications of theoretical chemistry to a myriad of problems such as flow in porous media, enhanced recovery of trapped fluids in oil reservoirs, flow of heat in fruit, enzyme degradation in fruit, flight patterns of insects in forest environments, and mortality of insects in quarantine treatments. His work is truly a basket of delight, but nearly all of it linked to fluid mechanics modeling203 and simple models for kinetic processes. Laidlaw was joined by Reginald Paul, his former colleague from the University of Alberta, in 1966. Paul initially worked on the application of field theory to chemical physics. His many papers during the 1970s culminated in the publication of a book on the subject.204 Subsequently, he became interested in studying the interaction of electromagnetic fields with biological cells, and also in theoretical electrochemistry. The number of theoreticians at the University of Calgary doubled in 1970 with the arrival of Arvi Rauk and Rod Truax. Rauk has made many important contributions to theoretical organic chemistry. Perhaps his most significant contribution arises from his long-standing interest in the chiroptical properties of molecules, that is, properties connected with the handedness and optical

264 The Development of Computational Chemisty in Canada activity of molecules. These properties are important for many applications including asymmetric synthesis, drug design, and even optical recording devices. The work of the Rauk group has played a significant part in the analysis of vibrational circular dichroism spectroscopy,205 the only spectroscopic tool for the determination of absolute configurations of molecules. In the early 1970s Rauk and his coworkers derived the formalism to obtain optical rotatory strengths from CI wavefunctions and wrote the first ab initio computer program to calculate them. Subsequently the origins of optical activity in chiral carbonyl compounds and several other compounds were explained, and in the 1980s, Rauk’s interest turned to vibrational optical activity.206 He adapted the vibronic coupling theory of Nafie and Freedman for ab initio calculation and wrote a computer program (VCT9O)that has been widely used to calculate the vibrational rotatory strengths of compounds from first principles. His group derived sum rules for the phenomenon,207 and elucidated the role of electron correlation and the molecular force field in determining the signs and intensities of the circular dichroism of all vibrational transitions. Collaboration with experimentalists permitted comparison of theory and experiment, and also enabled the Rauk group to demonstrate the usefulness of theory for understanding conformational and compositional properties of synthetic polymers, proteins, and nucleic acids. Although all his research applications are in the realm of ab initio electronic structure theory and computations, Rauk has taught the theory of structure and bonding from a frontier orbital point of view. He is the author of a graduate-level textbook on the orbital interaction theory of organic chemistry.208 The research interests of Rod Truax fall under the general heading of symmetry and supersymmetry and their applications to problems of chemical and physical interest. He is especially interested in finding the symmetry associated with time-dependent models and exploiting this symmetry to compute solutions to the quantum mechanical equations of motion. The number of theoretical chemists at the University of Calgary grew to five with the appointment of Tom Ziegler in 1981. He was no stranger to the Department of Chemistry, having received his Ph.D. in 1978 for research on new computational methods based on density functional theory carried out under the supervision of Arvi Rauk. Ziegler quickly established a major research group in the general area of theoretical inorganic chemistry and the development of new computational methods based on DFT. Ziegler is best known as an early proponent of DFT209 and its practical applications. His 1994 Alcan Award lecture210 gives a masterful account of approximate density functional theory as a practical tool in studies on organometallic energetics and kinetics. He discusses electronic excitations and ionizations, electron capture, conformational changes, molecular vibrations, bond energies, and reaction profiles.

D@artmental Histories 265 His group implemented some of the first DFT methods for the calculation of bond energies, molecular structures, and reaction paths, and developed together with the group of Evert Jan Baerends (the Free University in Amsterdam) the Amsterdam density functional package (ADF). Ziegler’s group also implemented DFT methods for the calculations of NMR and ESR parameters.211 Early on Ziegler demonstrated the ability of nonlocal DFT to provide accurate structures and bond energies in transition metal complexes. He has applied DFT extensively to elementary reaction steps in homogeneous catalysis. More recently the Ziegler group has included steric bulk, solvation, and firstprinciples molecular dynamics212 in their study of elementary reaction steps.

University of Western Ontario (1965) The University of Western Ontario built up a strong theoretical chemistry group in the short span of four years with the appointments of Bill Meath and Patrick Jacobs in 1965, Colin Baird in 1968, and Alan Allnatt the following year. After 14 years on the faculty of Imperial College, Jacobs moved from London, England, to London, Ontario, where his research program focused on the optical and electrical properties of ionic crystals, as well as on the experimental and theoretical determination of thermodynamic and kinetic properties of crystal defects.213 Over the years his research interests have expanded to include several aspects of computer simulations of condensed matter.214 He has developed algorithms215 for molecular dynamics studies of non-ionic and ionic systems, and he has carried out simulations on systems as diverse as metals, solid ionic conductors, and ceramics. The simulation of the effects of radiation damage is a special interest. His recent interests include the study of perfect and imperfect crystals by means of quantum chemical methods. The corrosion of metals is being studied by both quantum chemical and molecular dynamics techniques. As noted elsewhere in this chapter, 1965 was a very significant year for the development of theoretical and computational chemistry in Canada. After completing his Ph.D. under the supervision of J. 0. Hirschfelder at the University of Wisconsin, Bill Meath immediately took up an appointment at the University of Western Ontario, where he developed a strong research program in three, often interconnected, areas. His early work on atomic and molecular properties and dispersion energies involved the development and application of ah initio pseudostate techniques for the reliable evaluation of atomic and molecular multipolar properties and dispersion energies for small species.216 This was followed by the development and application of practical constrained dipole oscillator strength (DOSD) techniques, based on a combination of experimental and theoretical input, for the reliable evaluation (errors < 1-2%) of the dominant dipolar

266 The Development of Computational Chemistry in Canada

properties and dispersion energies of a wide variety of large atoms and, in particular, molecules. These results, which have found application in a variety of areas from radiation research to the study of the additivity of molecular properties, furnish input for the construction of potential energy surfaces for atomic and molecular interactions. More recently the constrained DOSD methods have been extended to the reliable determination of anisotropic dipolar molecular properties and dispersion energies. A major contribution of Meath’s work on intermolecular forces and interaction energies has involved investigation of charge overlap effects in atomic and molecular interactions leading to representations of the correction terms (“damping functions”) needed to extrapolate the well-known results for longrange interaction energies to small values of the intermolecular distance. This, together with reliable results for the long-range energies, in particular the dipolar dispersion energies, has led to the development of successful models, including an exchange-Coulomb model, for both isotropic and anisotropic intermolecular potential energies. In his research on laser-molecule interactions, Meath has centered his efforts around the development of efficient numerical computational methods for dealing with the interaction of continuous wave and pulsed lasers with molecules.217 More recently the research has focused on the effects of molecular permanent dipoles and molecular structure, on laser-molecule interactions, and on the use of two-color laser-molecule interactions to control the population of molecular states.218 The derivation and use of analytical rotating wave approximation results for both one- and two-color laser-molecule couplings, which include the effects of permanent dipoles, has played an important role in this work.219 For example, for one-photon transitions the molecule-laser couplings actually decrease with increasing laser field strengths if the molecule has permanent dipole moments relative to the expected increase with increasing field strength for atoms. This leads to drastic molecular effects in the resonance profiles, and in the associated molecular state dynamics, for such transitions. For multiphoton transitions, both one- and two-color, the effects of permanent dipoles are also pronounced. After participating in the development of the MIND0 method in Michael Dewar’s group, Colin Baird initiated a research program at Western Ontario based on ab initio calculations on the ground and excited states of molecules.220J21 Over the years his interests shifted more to chemical education, with a special interest in developing concepts and materials for environmental chemistry. Alan Allnatt’s research interests at Western Ontario have been concerned with the statistical mechanics of the transport of matter through crystals. His earliest work centered on obtaining methods for calculating the equilibrium distributions and thermodynamic properties of the point defects (vacancies, interstitials, solutes) that make transport possible. He first studied dilute systems, so the methods could be largely analytical. The methods for ionic crystals,

Departmental Histories 267 Allnatt’s main interest, were based in part on extensions to lattice models of Mayer’s cluster methods for electrolyte solutions. For solids with low dielectric constants, these methods were best combined with a new exact formulation of the mass action equations for defect equilibria.222 At higher concentrations, purely numerical methods are required. Between 1980 and 1986 Allnatt’s group developed inexpensive methods for calculating equilibrium radial distribution functions for defects such as vacancies and solute ions in ionic crystals using lattice versions of the hypernetted chain equations.223 At the same time, they set up a general linear response theory for use with the particle-hopping models used to describe matter transport in crystals. Formal expressions for the Onsager phenomenological coefficients in this theory have been a fruitful starting point for both analytical and simulation studies, some of which are described in a monograph.224 Allnatt’s group has also developed a method for using Monte Carlo simulation to calculate the Onsager phenomenological coefficients. The method has been used to provide detailed numerical information on interacting lattice gas models of transport in concentrated alloys and highly defective systems. Furthermore, these workers have developed a second simulation method,225 which gives not only the transport coefficients but also the underlying time correlation functions.

York University (1965)

Many contributors to the field of computational chemistry were initially trained as experimentalists. Often their research interests were shaped by the circumstances in which they worked. Huw Pritchard is a case in point. After completing a Ph.D. in experimental thermochemistry and establishing a successful career as a theoretical chemist at the University of Manchester, he emigrated across the Atlantic Ocean in 1965 to then newly established York University in suburban Toronto. At York, he began to reestablish a program in computer applications in traditional theoretical chemistry. The promised computing facilities were a long time coming, and so he gave up wavefunctions for reactions and developed a research program with the emphasis on kinetics and relaxation phenomena, an appropriate choice given the emphasis on interdisciplinary topics at York university. In a long series of papers on the master equation, Pritchard and his coworkers elucidated for the first time the effects of rotational and vibrational disequilibrium on the dissociation and recombination of a dilute diatomic gas. Ultrasonic dispersion in a diatomic gas was analyzed by similar computational experiments, and the first example of the breakdown of the linear mixture rule in chemical kinetics was demonstrated. A major difficulty in these calculations is that the eigenvalue of the reaction matrix (corresponding to the rate constant) differs from the zero eigenvalue (required by species conservation) by less than

268 The Development of Computational Chemist? in Canada the rounding error of the machine, but this problem was subsequently resolved by a matrix perturbation approach.226 On a different front, one of the first successful modelings of a practical combustion system was achieved, that of the formation of NO in a jet engine combustor.227 The method proposed for nitric oxide abatement is being incorporated into the next generation of aircraft jet engines. Later, essentially complete agreement between experiment and numerical modeling was achieved for the thermal explosions of methyl isocyanide in spherical vessels.228 Recent work by Pritchard has concentrated on a state-to-state description of unimolecular reactions229 and an examination by classical trajectory methods of the effects of overall molecular rotation on the unimolecular rate. The latter calculations have revealed a most interesting aspect of computing in chaotic systems, namely, that the same algorithm gives different results on different machines for a trajectory with identical initial conditions, or even on the same machine with different releases of the same compiler. However, the ensemble average behavior, with an ensemble comprising 100 or more trajectories, is acceptably the same each time.230 Geoffrey Hunter, a former Ph.D. student of Pritchard’s at the University of Manchester, joined the faculty at York University in 1966. His research interests have often dealt with the enigmas of atomic and molecular quantum mechanics. This life-long interest can be traced back to his Ph.D. thesis, in which he set out to compute the ground state energy of the hydride ion using a wavefunction similar to the Born-Oppenheimer wavefunction of H;, except that the large and small masses were interchanged. However, he developed the theory in a general form that made it applicable to any three-particle system interacting by Coulomb forces. Computational results were published for several systems.231 A notable result of this work was a scaling of the electronic wavefunction to make the separated-atom limit of the adiabatic electronic energy exact, and in 1965, while in Texas, Hunter applied this scaling to a calculation of the hydrogen molecule, thereby showing that the published experimental dissociation energy of H, must be in error by about 4 cm-1. This result motivated Gerhard Herzberg to remeasure the dissociation energy by a more precise method. Hunter’s unique expertise and experience enabled him to realize that the square root of the electron density satisfies a one-electron Schrodinger equation and that the effective potential in this Schrodinger equation is, in principle, an exact representation of the motion of a single electron within a many-electron system.232 He subsequently showed that there is a surface enclosing a molecule outside of which the electron’s kinetic energy would be negative. He has proposed that this molecular envelope provides a nonarbitrary definition of molecular size and shape that is suitable for implementation in computer graphics. The arrival in 1969 of Alan Hopkinson from the University of Toronto, where he had worked with Yates and Csizmadia, broadened the theoretical interests at York to include theoretical organic chemistry. Hopkinson went to Toronto with a background in strong acid physical chemistry. Yates initially put

Departmental Histories 269 him on an extended Hiickel project involving amide hydrolysis, but Hopkinson quickly found that extended Hiickel theory could not satisfactorily answer their questions, so he quietly became more interested in the gas phase property of proton affinities. He persuaded Mike Robb and Roy Kari to allow him to use the Polyatom program that Csizmadia had brought from Slater’s group. He proceeded to show for the first time that as the size of the basis set is increased, the enthalpies of reaction, in this case the calculated proton affinities, converge on experimental values.233 Hopkinson was hired by York to teach theoretical organic chemistry (the Woodward-Hoffmann rules were then a hot topic) and to carry out experimental chemistry. Despite the limited computing capacity at York at the time, he managed to complete some work on the electrophilic addition to alkenes. He is probably best known, however, for his work on proton affinities, destabilized carbocations,234 organosilicon compounds, silyl anions and cations, and more recently, on the calculation of potential energy surfaces and thermodynamic properties. He has had a particularly fruitful collaboration with Diethard Bohme.235 Theoretical chemistry at York University was strengthened in the 1990s with the appointments of Bill Pietro in 1991 and Renk Fournier in 1996. Pietro wrote part of the Gaussian code as a graduate student and several modules of SPARTAN while an assistant professor at the University of Wisconsin. While he was in Madison he developed a research program based on molecular electronic devices.236 He expanded his interests to several facets of molecular electronics, including molecular electroluminescent materials, molecular electronic devices (diodes, switches, and sensors), and functionalized semiconductor nanoclusters.237 These new materials not only are scientifically very exciting, but they offer the possibility of revolutionary impact on the future of the electronics industry. Renk Fournier is studying atomic clusters238 and transition metal compiexes.239 He is using a combination of density functional methods, tightbinding models, and molecular simulations with empirical interaction potentials, as part of a research program designed to study materials by computations on simple model systems.

Simon Fraser University (1966) Simon Fraser University, founded in 1964, was one of many new universities established in Canada in the 1960s. It appears that Natalia Solony (a student of Fraser Birss) was the first theoretician at SFU, but she left after a year or two. In 1966 Margaret Benston, Gulzari Malli, and John Walkley joined the Department of Chemistry. Benston carried out research on the electronic structures of atoms and molecules for a few years, often in collaboration with Chong at UBC, before moving to the Department of Computing Science and then later to the Department of Women’s Studies. She died prematurely in 1981. John

270 The Development of Computational Chemistry in Canada

Walkley’s research interests over the years have focused on the thermodynamics and statistical mechanics of dilute solutions,24* intermolecular potential calculations, and Monte Carlo calculations. Gulzari Malli has maintained his interest in ab initio relativistic quantum chemistry since he joined SFU in 1966. In fact, the earliest work in Canada on relativistic quantum chemistry appears to have been carried out in 1965 by Malli and Serafin Fraga at the University of Alberta. Their earliest results were documented in a compact book.249 Malli’s interest in treating relativistic effects for many-electron systems containing heavy elements (2 = 75-112) led to the development of the first relativistic self-consistent field theory for closed-shell molecules.250 The formalism was tested on light diatomics. Later for heavy element molecular systems, Malli introduced the use of Gaussian basis sets in relativistic quantum chemistry. Next he and his coworkers developed a relativistic SCF theory for open-shell molecules and a multiconfiguration relativistic SCF theory. They showed that relativistic effects predict a bond shortening of about 0.45 bohr and a doubling of the binding energy for AuH. Malli and Pyper reported the first relativistic configuration interaction (RCI) calculations for AuH taking an all-electron, fully relativistic Dirac-Fock wavefunction as the reference.251 Ab initio all-electron, fully relativistic wavefunctions were also reported for a number of diatomics involving heavy elements; the effects of relativity on the dipole moments of various diatomics were reported for the first time.252 During the last decade, Malli and coworkers have investigated the effects of relativity and electron correlation using the fully relativistic Dirac-Fock SCF treatment and MP2, and most recently the relativistic coupled-cluster (REL-CCSD) and relativistic many-body perturbation theory (RMBPT) for atoms and molecules. Computational chemistry at Simon Fraser University received a major boost in 1990 when Saul Wolfe joined the Department of Chemistry. After obtaining his B.A. and M.A. degrees at the University of Toronto, Wolfe attended the University of Ottawa, where he was the first graduate student of Prof. Ray Lemieux and the first Ph.D. graduate in the Faculty of Science. Following postdoctoral work with Prof. Franz Sondheimer at the Weizmann Institute of Science, he joined Bristol Laboratories in 1959 as an organic chemist. In 1961 he moved to Queen’s University at the instigation of R. L. McIntosh who, as noted earlier, built up a strong theoretical group during his term as head of the Department of Chemistry. Wolfe remained at Queen’s until 1990, when he hung up his snow shovel and moved to Vancouver. In a review based on his 1992 Lemieux Award lecture,253 Wolfe presents a highly personal account of his research career and interests over a period of 33 years. His review provides an overview of his studies on the relative antibacterial activities of penicillins and cephalosporins and their differing abilities to complex to and react with penicillin-binding proteins. He describes how the insights from his analysis could be used to attempt the design of new structures targeted to the penicillin receptor. Besides providing glimpses of Wolfe’s signifi-

Departmental Histories 271 cant contributions to organic chemistry, his review describes how he became interested in computational chemistry through the work of Arvi Rauk,254 one of his first graduate students. Saul Wolfe has made important contributions on a broad range of topics in bioorganic chemistry and many aspects of theoretical organic chemistry. His special interests include the gauche effect,255 p-lactam compounds,256 and isotope effects.257

University of Manitoba (1966) The University of Manitoba appointed its first theoretical chemist in 1966. With postdoctoral experience in the groups of P.-0. Lowdin and A. D. Buckingham, Bob Wallace established a research program focused on the basic quantum theory of molecules, electromagnetic interactions, and the theory of molecular collisions and molecular processes. His interest in molecular collisions gave way with the passage of time to an emphasis on molecular processes.240 During the past decade he has had a special interest in large amplitude nuclear motion,241 including new methods for deriving internal coordinates.242 Bryan Henry joined the faculty of the University of Manitoba in 1969 and developed a research program based on the experimental and theoretical study of highly vibrationally excited molecules. At high enough energies, light interacts with molecules containing X-H oscillators to prepare states that are more localized than those expected on the basis of the traditionally accepted normal mode description of molecular vibrations. Henry and his group have developed the local mode description of such vibrational states, and this description has now gained general acceptance.243 States with overtone spectra, because of the localization, are extremely sensitive to the properties of X-H bonds. The spectra are measured with a variety of sophisticated spectroscopic techniques, including intracavity laser photoacoustic spectroscopy, and are used to study molecular structure and conformation. The time scale of the overtone experiment allows study of conformational processes that are much too fast to be observed by conventional spectroscopic techniques like NMR. The spectra also allow determination of intramolecular vibrational energy redistribution. Henry’s group is also involved in theoretical studies to determine sources of local mode overtone intensity. These investigators have developed a very successful approach that uses their harmonically coupled anharmonic oscillator local mode model to obtain the vibrational wavefunctions, and ab initio calculations to obtain the dipole moment functions. The researchers have applied these calculations to relatively large molecules with different types of X-H oscillator. Recently they have compared intensities from their simple model to intensities from sophisticated variational calculations for the small molecules H20 and H2C0. For example, for H 2 C 0 they generated a dipole moment function in terms of all six vibrational degrees of freedom.244 This comparison has allowed them to determine the quality of basis set needed to calculate dipole moment

272 The Development of Computational Chemistry in Canada functions that lead to reasonable values for fundamental and overtone intensities. Henry’s group also calculated intensities at different levels of ab initio theory and found a very surprising result. The calculations consistently show that although electron correlation in the calculation of dipole moment functions can be important for fundamental intensities, it is apparently unimportant for overtone intensities. Henry moved to the University of Guelph in 1987. Since joining the University of Manitoba in 1985, John Cullen has concentrated on quantum chemical methods for large molecules. He is interested in treating biological molecules by semiempirical methods.245 The research of Kathleen Gough, appointed in 1995, is concerned with theoretical and experimental studies of vibrational intensities and electronic charge flow. Gough and her coworkers use ab initio electron density distribut i o n and ~ ~calculate ~ ~ Raman scattering intensities to assist in the interpretation of observed spectra. One of the goals of their research is to quantify relationships between observed intensities in vibrational spectra, molecular structure and electronic charge flow in vibrating molecules.247

Carleton University (1970) Carleton University was one of many Canadian universities to appoint its first theoretical chemist in 1970. Since his arrival in Ottawa, Jim Wright has had a strong interest in unusual chemical structures258 that arose from his postdoctoral research with Lionel Salem on diradical mechanisms in the ring opening of cyclobutane. The C4H8 system was too big to explore using ab initio techniques at the time (even with a minimum basis set), and therefore Wright used the simpler isoelectronic model, cyclic 0,. Much to the researcher’s amazement the 0, ring was puckered, like cyclobutane. The lone pairs were acting as if they were localized, a representation that is hard to prove theoretically but which has great predictive value. Further work in this area led to predictions on cyclic ozone, nitrogen rings including the quasi-aromatic N6 ,and quasi-aromatic H, as a transition state for the H2 + D2 exchange reaction. Wright returned to this theme later when he examined the structure and stability of hydrogen rings.259 Recently Wright’s group has been examining polymeric nitrogen and oxygen analogs of saturated hydrocarbon chains: for example, how stable is HO(O),-OH or H,N-(NH),-NH,? The nitrogen chains turn out to be much more stable than expected (in the gas phase) and to adopt a helical conformation. It is interesting to note that Paul Gigdre at Lava1 identified H 2 0 3 and H204by matrix IR spectroscopy.260J61 Wright and his collaborators have studied extensively the doubly excited states of molecules.262 They have shown that promotion of electrons from antibonding MOs in diatomic molecules can lead to a drastic shortening of the bond length with an accompanying huge increase in the harmonic vibrational frequency. This new type of strongly bound, doubly excited state has now been observed experimentally in compounds such as B,, where the ground state

Departmental Histories 273 single bond is converted into what is essentially a triple bond in the doubly excited state. An offshoot of this work on heteronuclear diatomics led to the interesting discovery that some diatomics can be thermodynamically stable, even with a +2 charge.263 This can occur when charge-induced polarization overcomes the Coulomb repulsion. Wright is also interested in modifying potential surfaces for chemical reactions by the use of intense infrared laser fields. This interesting new area of research shows that not only is it possible to lower a barrier to chemical reaction, but the barrier can be caused to pulsate up and down with the frequency of the laser. Wright’s group looked at the H + H, exchange reaction as a first application and showed that the reactivity can be enhanced significantly with intense fields.264

Dalhousie University (1970) Despite being one of Canada’s oldest universities (founded in 1818), Dalhousie University did not formally begin a program of theoretical chemistry until Charles Warren was appointed in 1970. Initially Warren’s research interests were divided between spectroscopy and semiempirical MO calculations. By the end of the 1970s, he was primarily interested in laser Raman spectroscopy and related theoretical topics, including the Jahn-Teller effect. In the 1980s he turned his attention to computed-aided instruction, especially for undergraduates. He has written many interactive programs to illustrate the principles of quantum mechanics. Most of his programs feature excellent graphics, a consequence of his research on algorithms for computer graphics. Since arriving at Dalhousie University in 1975, Russell Boyd has established a broadly based research program. Boyd’s earliest work on electron correlation was concerned with developing physical pictures for the Coulomb and Fermi holes, initially in atoms and later in molecules; the research had interesting implications for many topics, including the interpretation of Hund’s multiplicity rule.265 Later the emphasis shifted from two-electron density distributions to one-electron density distributions. Boyd and his coworkers have made extensive comparisons between electron densities calculated from density functional methods and high level configuration interaction calculations.266 The long-term goal of this research is to use an accurate representation of the electron density as the basis for the development of new functionals for use in DFT. Boyd and his coworkers have studied several reaction mechanisms in detail including S,2 reactions.267 The group has published many papers on oxygen radicals (ROO and ROO’), which are of vital interest in biological systems because their attack on lipid biomembranes is related to many pathological events. Their interest in peroxyl radicals arose initially from the uncertainty surrounding the mechanism of the 1,3-migration in allylperoxyl radicals. Calculations268 showed that the experimentally observed stereoselectivity of

274 The Development of Computational Chemistty in Canada the rearrangement must result from a cage effect wherein diffusion of the allyloxygen pair is prevented because of caging by the solvent. Subsequent oxygen labeling and stereochemical studies of the rearrangement as a function of solvent stability and viscosity support the theoretical conclusions. The long-standing interest of Boyd and his coworkers in radicals and radical ions has led to many papers since 1993 on hyperfine structures. These papers have pushed the conventional multireference configuration interaction methods to the limits of the available computers, tested the predictive ability of various functionals commonly used in DFT calculations, and, among other topics, modeled the effect of a noble gas matrix on the hyperfine structures of radicals. Recent research focused primarily on radicals formed as a consequence of radiation damage to DNA. In the mid-1990s the Boyd group broadened their interests to include molecular dynamics simulations. One of the objectives of their research is to include quantum mechanical force fields directly or indirectly in MD simulations. One of the key issues underlying the success of a hybrid approach is an appropriate description of the coupling between the quantum mechanical and molecular mechanical force fields. A paper that reports the first ab initio MD simulation of liquid methanol269 showed somewhat surprisingly that the best overall agreement between the experimental and simulated results for vibrational frequency shifts is obtained with the weakest quantum mechanical/ molecular mechanical coupling. Peter Kusalik took up an appointment at Dalhousie University in 1989 and developed a research program focused on computer simulation studies of molecular liquids, solids, and solutions. As well as standard simulation approaches, he has explored nonequili brium molecular dynamics techniques and applied field simulations, the development of new models and methodologies being one aim of his research. A common focus throughout his work has been the examination of the interplay between microscopic structure and dynamics in condensed matter and their relationship to bulk properties. Kusalik’s group has published several key studies270 examining the local structure in liquid water. Using a unique spatial approach, their work has served to clarify greatly the understanding of the local molecular environment in aqueous systems. These investigators have also made exciting advances in modeling nucleation and crystal growth processes. Kusalik’s group reported the first successful computer simulation studies of the crystallization of molecular liquids, first with the electrofreezing of water271 and later with the crystallization of carbon dioxide272 under field-free conditions. The development of these techniques also allowed them to discover a possible new polymorph for ice.273

University of Guelph (1970) Theoretical chemistry began at the University of Guelph in 1970 with the appointment of Mike Zerner from the United States. It was a period of rapid

Departmental Histories 275 expansion of the relatively new university, which had been created in 1964 when the Ontario Veterinary College, the Ontario Agricultural College, and the MacDonald Institute joined with a new college of arts and science. Zerner quickly established a research program based on semiempirical treatments of large molecules, especially porphyrins and related systems.274 By the end of the decade Zerner and his coworkers had become well known for their work on the spectra of iron, cobalt, and copper complexes.275 The widely used ZINDO method was developed during the Guelph era of Zerner’s career and was applied to many systems.276 In 1980 Zerner made the first application to NSERC for a computer dedicated to computational chemistry. Zerner and his coapplicants (Jiri Cizek, Saul Goldman, Bob Le Roy, Fred McCourt and Joe Paldus) made a favorable impression on the site visit team but were told that computers were the domain of the university and that NSERC equipment grants were for experimental equipment only. The following year Zerner accepted a position at the University of Florida, and the NSERC application was resubmitted through the University of Waterloo. Bob Le Roy and his coapplicants were awarded $215,000 in 1982 and took delivery of a VAX 11/750 in the summer months. It is interesting to speculate about the extent to which Zerner’s resignation led to a reversal of the NSERC policy on funding computers. Although Zerner moved to the Sunshine State in 1982, his 12 years in Guelph had a lasting impact on the development of theoretical and computational chemistry in Canada. Indeed, his inspiring research and personality have had a large impact on computation chemists everywhere. After completing a Ph.D. degree and postdoctoral research in analytical chemistry, Saul Goldman joined the University of Guelph in 1972. As a reformed experimentalist, he developed a research program based on statistical mechanics,277 with special interests in supercritical fluid extraction, spectra of endohedral fullerenes,27*the transport of ions through biological channels, and the molecular basis for the properties of liquids and solutions.279 Upon joining the University of Guelph in 1983 John Goddard built up an applied quantum chemistry program spanning many aspects of main group inorganic chemistry and organic chemistry, with special interests in reaction intermediates and transition structures in excited state chemistry. Many of his papers in the 1980s resulted from a fruitful collaboration280 with two experimental colleagues, Richard Oakley and Nick Westwood, who joined the department at about the same time. This talented trio of chemists combined forces to synthesize, characterize, and study many interesting new compounds. Goddard and his coworkers have had a continuing interest in electronic, vibrational, and rotational spectra. This is evident from their papers on carbenes281 and other highly reactive species. Many of Goddard’s papers have studied the potential energy surfaces of thioformaldehyde282 and related species. Much of this research has resulted from his willingness to carry out high level ab initio calculations of interest to spectroscopists. The Goddard group

276 The Development of Computational Chemistry in Canada recently expanded the scope of their research to include effective core potentials and density functional methods for main group inorganic compounds.283 As described in connection with the University of Manitoba, Bryan Henry moved to Guelph in 1987 and has continued with his experimental studies of overtone spectra involving X-H stretching and the analysis of the results in terms of the local mode mode1.284 Indeed, most of his research on the intensities of overtone spectra was completed at Guelph.

University of Sherbrooke (1970) Andrk Bandrauk joined the University of Sherbrooke in 1970 following a NATO postdoctoral fellowship with C. A. Coulson and M. S. Child at Oxford, where he developed his interest in the theory of predissociation. His paper with Child reported the first application of scattering theory to spectroscopy.285 Bandrauk’s long-term research interests include the dressed-state representation of molecular spectroscopy. His contributions to the nonperturbative treatment of molecular spectroscopy from the weak field to strong field limits have been summarized in two chapters in a book he edited in 1993.286 Bandrauk and his coworkers published the first theoretical demonstration of the use of chirped pulses to effect laser bond breaking in less than a picosecond.287 His other firsts include the first prediction of molecular stabilization in intense laser fields288 and the first complete non-Born-Oppenheimer calculation of dissociative ionization of molecules in intense femtosecond laser pulses.289

COMPUTATIONAL CHEMISTRY IN CANADIAN INDUSTRY Canadian industry employs far fewer computational chemists than do companies in the United States, the United Kingdom, Japan, and many European countries. Nevertheless, there are a few successful cases, which are summarized briefly in this section.

Hypercube, Inc. Hypercube’s HyperChem is the best-known computational chemistry software package developed in Canada and marketed worldwide. Hypercube, Inc., the brainchild of Neil Ostlund, was incorporated in February 1985. The company has a headquarters in Waterloo, Ontario, adjacent to the University of Waterloo, but moved most of its operations to Gainesville, Florida, in 1997. The first phase of Hypercube’s existence involved contracting for Intel in association with Intel’s development of the first commercial highly parallel computer, the Intel iPSC hypercube computer. This is the origin of the com-

Computational Chemistry in Canadian Industry 2 77 pany’s name. Subsequently, Intel and Hypercube cooperated in the development and marketing of molecular modeling software for the commercial Intel iPSC. Later Hypercube focused on the internal development of a turn-key molecular modeling “instrument” called a Chemputer, consisting of a PC integrated with a parallel processing subsystem of transputers running HyperChem, Release 1. The market for such a product was small, and in 1990 Hypercube formed a joint venture by which Autodesk acquired an exclusive licence to manufacture, market, and sell HyperChem, with Hypercube doing all development and technical support for the products. The first Autodesk product, HyperChem Release 2, was shipped in March 1992. Release 3 was shipped in April 1993. The desktop computational chemistry software market turned out to be smaller than desired. In January 1994, Hypercube recovered its licence from Autodesk and brought all control for HyperChem and related products inhouse. Hypercube has since shipped Releases 4 and 4.5 of HyperChem and has delivered other new products such as ChemPlus and HyperNMR. The company is developing a range of new molecular modeling products for a range of platforms, although its dominant product remains HyperChem for Windows. The latest product is HyperChem Release 5 for Windows 95 and Windows NT, which includes many new computational features and new graphical abilities for the presentation of the results of calculations. HyperChem continues its wide popularity among PC users.

Ayerst Laboratories In 1982 Ayerst Laboratories in Montreal became the first company in Canada to install a commercial software tool (the SYBYL suite from Tripos Associates) to help in the development of pharmacophoric models from structure-activity relationships. The installation of the software was the second ever, worldwide, by a company and is a testimonial to the foresight of the director of medicinal chemistry, Dr. Leslie Humber, for having championed its installation. Dr. Adi M. Treasurywala, then an organic chemist with some experience in medicinal chemistry, became the first industrial computational chemist in Canada that year. The use of modeling approaches contributed in a minor but significant way to the discovery of the compound known as Tolrestat, which was an inhibitor of lens aldose reductase. This led to the acknowledgment of Treasurywala as a coinventor of the drug on several patents that were filed in this research area. Approximately in 1983, Ayerst closed down its discovery effort in Canada and moved to Princeton, New Jersey, where an expanded effort in the area of computational chemistry continues.

Merck Frosst Canada Inc. Merck & Company became involved in the early 1970s in applying computational chemistry to pharmaceutical research, an effort that continued and

278 The Development of Computational Chemistly in Canada expanded. Although Merck Frosst, the Canadian subsidiary of Merck & Company situated in Montreal, had these resources available to draw upon, there was no on-site expertise to facilitate collaborations. In 1992 Merck Frosst created a permanent senior-level position for a computational chemist. Christopher Bayly accepted the position, leaving a postdoctoral appointment in Peter Kollman’s group (University of California, San Francisco), where he had been doing force field development and free energy calculations. (Prior to that, Bayley had obtained his Ph.D. under the supervision of Fritz Grein at University of New Brunswick, having received an M.Sc. in synthetic organic chemistry with Pierre Deslongchamps at University of Sherbrooke in Quebec.) Recently, an additional junior-level position at Merck Frosst was filled by Daniel McKay (an M.Sc. graduate from Jim Wright’s group at Carleton University). Currently, the modeling effort involves the usual spectrum of methods: macromolecular simulation, small molecule force field and electronic structure calculations, with 2-D and 3-D database searching. Software resources consist primarily of proprietary Merck & Company modeling code, some of which has been published [e.g., the Merck Molecular Force Field (MMFF), SEAL, for fitting molecules together, and FLOG, a 3-D database docking program]. Other commercial codes are used, as well as academic code (e.g., AMBER, DeFT). The modeling group is integrated with the overall medicinal chemistry department from both a functional and organizational standpoint.

Xerox Research Centre of Canada During its relatively brief existence, the Materials Modeling and Simulation group at the Xerox Research Centre of Canada (XRCC)applied quantum mechanics and molecular dynamics simulations to the design of new materials for use in xerography. The group was headed by Tom Kavassalis, a theoretical chemist by training with a B.Sc. from the University of Toronto and a Ph.D. from MIT. The group disbanded a few years ago. In a 1995 overview of their research interests at the XRCC, Kavassalis290 summarized the role that computational chemistry plays in an industrial research laboratory. His overview describes the application of quantum mechanical methods to polymerization reactions and the role of crystal packing in determining the bond structure of organic photogenerators. He also shows how simulation methods have been incorporated into the treatment of problems that were until recently primarily in the experimental domain. Thus it is now possible to study polymer thermal and molecular properties and to estimate solvent diffusion coefficients by molecular dynamics simulations. The same methods are providing insight into polymer and surfactant solubility parameters, as well phase transitions. In the same article, Kavassalis summarizes the challenges and opportunities for the application of theoretical methods in materials science. It

Computational Chemistry in Canadian Industry 2 79 is clear that methods that were once the purview of academic researchers are making a significant impact on the development of new technologies.

ORTECH, Inc. ORTECH is a relatively new Canadian company in the computer-aided drug design field. From their Mississauga, Ontario, location, they have developed and marketed a program called MolScan, which does conformational analysis of ligands to identify pharmacophores. The resulting pharmacophores can be used to search databases of three-dimensional molecular structures for compounds that fit the criteria for potential biological activity. The group is headed by Robert Kirby, a 1990 Ph.D. graduate in organic chemistry from Carleton University in Ottawa. Kirby joined ORTECH after holding molecular modeling positions at BioChem Therapeutic, Inc., and Allelix Biopharmaceuticals, Inc.

BioChem Therapeutic In 1997 Alan Cameron, formerly with Eastman Kodak Company of Rochester, New York, and SynPhar Laboratories, Inc., of Edmonton, Alberta, joined BioChem Therapeutic, Inc., Lava], Quebec. He has been involved at BioChem in new compound identification and optimization of existing lead compounds directed toward various therapeutic agents including antithrombotics, opioid analgesics, and anticancer and antiviral compounds. He obtained his Ph.D. under the supervision of Mike Baird and Vedene Smith at Queen’s University and was a postdoctoral fellow with Mike Zerner at Florida. Dr. Miguel QuimpGre joined BioChem Therapeutic in 1990 as a synthetic organic chemist and has since moved on to data management and molecular modeling, with a particular view to diverse library design and combinatorial chemistry.

Advanced Chemistry Development, Inc. Advanced Chemistry Development, Inc., is also relatively new to the Canadian scene. The idea behind ACD originated with a small team of postdoctoral researchers working together at Moscow State University in Moscow, Russia. In 1990 these scientists brought together their talents in the areas of chemistry, spectroscopy, and computer science to develop chemical software tools focusing on accuracy and simplicity for the analysis of physicochemical properties. ACD continues to grow rapidly to keep pace with the demand for its products and innovations worldwide. From the original group of seven at Moscow State University, ACD grew to a staff of 70 chemists, algorithm specialists, programmers, and database compilation experts in Moscow. The Toronto headquarters has a staff of 20, providing sales, graphic design, market-

280 The Development of Computational Chemistry in Canada ing, product development, and technical support for the ACD product line. In addition to full-time staff, the Toronto office hires freelance programmers and scientific consultants. Broadly stated, ACD expertise lies in the area of data compilation (and critical analysis of these data), verification, correlation, prediction, and systematization of common scientific knowledge. The systematization falls into two categories: well-defined rules (e.g., nomenclature), which can be applied more uniformly on a computer than manually, and less rigorous ‘‘rules of thumb,” which must be devised, tested and parameterized, refined and then retested. ACD algorithms are proprietary and mainly based on a heuristic structure-fragment approach. The product line includes software for predicting NMR spectra, ionization coefficients, solubility, octanol-water partition coefficients, vapor pressure, Hammett-type constants, and related physicochemical phenomena, as well as software for chemical structure drawing and for assigning chemical nomenclature.

SynPhar Labs, Inc. SynPhar, a small-sized pharmaceutical R&D company located in Edmonton, is involved in the research and development of drugs against several therapeutic targets including infectious diseases and cancer. The company has its own internal research programs as well as collaborations (in various stages) with Japanese, European, and American multinational firms. The Molecular ModelingKomputational Chemistry group in SynPhar Laboratories, Inc., was set up in 1993 as part of a collaborative project involving molecular design with the Biotechnology Research Institute of the National Research Council in Montreal. At that time, the company had one computational chemist at SynPhar (Edmonton) and another at NRC (Montreal). In the molecular modeling area is one senior scientist, Dr. Sanjay Srivastava, who joined the company in 1993.

Bio-Mkga The computational chemistry group at Bio-M6ga is mainly involved in aiding the company’s medicinal chemistry efforts in the design of new antiviral agents. A variety of modeling techniques are used to help elucidate and understand the structure and conformation of inhibitors and how they might be acting in the target enzymes. The firm’s computational chemist, Dale Cameron (B.Sc., 1989, Ph.D., 1994, Queen’s University), was hired in 1995.

Astra Dr. Shi-Yi Yue joined the Astra Research Centre in Montreal in 1995 as a computational chemist. The Astra computational chemistry effort started with pharmacophore analysis and G-protein coupled receptor modeling and has

National Research Council of Canada 281

recently expanded to include library design, quantitative structure-activity relationships, and bioinformatics.

Other Examples It is impossible to make this account inclusive, but a few other examples of computational chemists working in Canadian industry should be mentioned. Liangyou Fan (Ph.D. Calgary 1992, under T. Ziegler) has been working for Nova Chemicals in Calgary since 1993 as a staff researcher conducting theoretical calculations on single-site olefin polymerization catalysis using density functional theory. Also Danya Yang (Ph.D. Calgary 1994, under A. Rauk) has been working since 1996 with Travis Chemicals in Calgary simulating corrosion with molecular mechanics and quantum mechanics methods.

HISTORY OF THEORETICAL CHEMISTRY AT THE NATIONAL RESEARCH COUNCIL OF CANADA Although the research of Gerhard Herzberg and his colleagues at the National Research Council of Canada needs no introduction to computational and theoretical chemists, the emergence of theoretical and computational chemistry at the NRC is a relatively recent development. In the 1950s, the Division of (Pure) Chemistry needed theoretical assistance in areas such as NMR and solid state chemistry. It acquired this assistance from long-term visitors, including John Pople (1998 winner of the Nobel Prize in Chemistry), David Buckingham, and Hugh Barron. Attempts to hire any of these visitors on a permanent basis failed. Therefore it was decided to start hiring recent PhDs to fill the gap. The first theorists hired were Willem Siebrand and Constantine Mavroyannis in 1963, soon to be followed by Raymond Somorjai and Michael Klein. Initially each theorist was assigned to an experimental group. After a few years, postdoctoral fellows joined NRC, and a new permanent member was added, David Peat. It was then decided that the theorists should form their own group, with Siebrand as group leader. The main areas of activity were molecular crystals and radiationless transitions (W.S.), elementary excitations in solids (C.M.), condensed phase structural simulations (M.K.), and quantum chemistry and biology (R.S.). A succession of postdoctoral fellows and (later) research associates (RA) joined the group, some of whom found permanent positions in other parts of NRC, while Marek Zgierski and Roger Impey were appointed to continuing positions in the group. An experimentalist turned theorist, John Tse, did not join the group but became a frequent collaborator. The changing political climate of the early 1980s, which ran against “pure” science and especially theory, coincided with the need of theorists to

282 The Development of Computational Chemistry in Canada obtain new and costly computational facilities. For these reasons, Michael Klein left for the University of Pennsylvania, and Roger Impey, Ray Somorjai, and David Peat went to other parts of NRC. The decimated theory group thus lost its group status and was “hidden” inside an experimental group. The restructuring of NRC in the early 1990s resulted in the formation of the Steacie Institute for Molecular Sciences and led to the resurrection of the theory group, as the Theory and Computational Program, consisting of Willem Siebrand and Marek Zgierski and two RAs. In recent years the group acquired four more staff members: Philip Bunker, who joined the group from the Herzberg Institute of Astrophysics; John Tse and Dennis Klug, who transferred from an experimental group in the institute; and Tamar Seideman, an RA appointed to the continuing staff. The group then consisted of six continuing staff members with three RAs and three postdoctoral fellows and was led by John Tse, Willem Siebrand having become director of research of the Steacie Institute. The research programs are focused on material sciences (J.T. and D.K.), spectroscopy (P.B.), quantum chemistry and biology (M.Z. and W.S.), and chemical dynamics (T.S.). Most recently, Dennis Salahub has taken over the reins of Steacie Institute. This augurs well for the status of computational chemistry in Canada.

HIGH-PERFORMANCE COMPUTING IN CANADA Canada’s computational chemistry community has not had the level of support for high performance computing that has been commonplace in the other leading industrialized nations for the past 15 years. Throughout the 1960s and 1970s, Canadian computational chemists worked on central mainframe computers, not unlike those used by other university-based researchers throughout the world. With the introduction of the VAX 11/780 in 1978 and similar minicomputers in the following few years, there was a strong migration to smaller dedicated computers. As discussed above, the Natural Sciences and Engineering Research Council of Canada started providing equipment grants to researchers for the purchase of computers in the early 1980s. It soon became apparent that university administrations no longer saw the provision of high performance computing to be a high priority. A recent study of high performance computing (HPC) in Canada291 included a detailed assessment of the Canadian situation and a comparison of the Canadian environment with that of other countries. The study showed that the United States, Europe, and Japan have multi-billion-dollar government programs in place to support the infrastructure requirements of high performance computing and communications, whereas Canada has not adopted a national strategy to support and develop such activity. This situation is dramatically illustrated by the fact that in August 1994 only one Canadian facility, the

Major Conferences 283

Atmospheric Environment Service computer center in Dorval, Quebec, ranked in the top 176 HPC sites in the world. Given Canada’s large land area (9.2 million km2) and low population (30 million), it is not surprising that weather forecasting is a national priority. Indeed, Canada is a leader in numerical weather prediction research, and many other nations have adopted methods developed in Canada. The dominant industrial application for high performance computing in Canada is the processing of seismic data for the oil and gas industry, which is centered in Calgary. As the new millenium commences, an infusion of funds from the Canada Foundation for Innovation has led to a dramatic increase in HPC facilities in Canada. Moreover, the formation of networks of researchers and facilities has virtually eliminated the problem of accessibility.

MAJOR CONFERENCES The distinguished history of the Canadian Symposia on Theoretical Chemistry (CSTC)was summarized earlier in this chapter. Several other significant conferences should be mentioned.

Fifth International Congress on Quantum Chemistry The awarding of the Fifth International Congress on Quantum Chemistry (ICQC) to Canada in 1985, a mere 20 years after the historic conference convened by Fraser Birss and Serafin Fraga in Edmonton, signaled that Canada had arrived in the field of computational and theoretical chemistry. The first four ICQCs were held in France (Menton, 1973), the United States (New Orleans, 1976), Japan (Kyoto, 1979), and Sweden (Uppsala, 1982). The Fifth ICQC was held at the University of Montreal in August 1985 with Camille Sandorfy as president and Willem Siebrand and Vedene Smith as vice presidents. The congress was the largest scientific meeting devoted exclusively to theoretical chemistry held in Canada up to that time. More than 450 participants attended the congress, which was a great success. The proceedings were published in the International Journal of Quantum Chemistry in four parts, spanning more than 1500 printed pages.292 The proceedings concluded with a general survey of the current (1985)state of the art of quantum chemistry by Per-Olov Lowdin.

Second World Congress of Theoretical Organic Chemists An even larger conference took place in Toronto in July 1990, when more than 550 participants attended the Second World Congress of Theoretical

284 The Development of Compufational Chemistry in Canada Organic Chemists at the University of Toronto. The First WATOC Congress had been held in Budapest in August 1987. Both congresses were chaired by Imre Csizmadia. The Toronto Congress, like the earlier one in Budapest, attracted many quantum chemists and computational chemists whose interests were not restricted to compounds containing carbon. At a general discussion relating to the focus of future congresses, it was decided that the name should be modified to better reflect the actual interests of the participants. The original name was, however, retained for the Third WATOC Congress, which was chaired by Keiji Morokuma in July 1993 in Toyohashi, Japan. The name was officially changed to the Fourth World Congress of Theoretically Oriented Chemists (WATOC 96) starting with the July 1996 meeting in Jerusalem.

Canadian Computational Chemistry Conference In recognition of the rapid expansion of computational chemistry in the 1980s, Andrt Bandrauk and Andrt Michel of the University of Sherbrooke organized the First Canadian Symposium on Computational Chemistry in May 1991 in Orford, Quebec. The conference included invited papers on dynamics, density functional methods, molecular modeling, Monte Carlo methods, and topics in quantum chemistry and statistical mechanics. About half of the invited speakers were from abroad (mostly from the United States). A subsequent conference was convened in May 1994 at Queen’s University in Kingston, Ontario. The name was changed to the Second Canadian Computational Chemistry Conference. The conference was organized by Ken Edgecombe and Vedene Smith with programming assistance by Axel Becke, Dave Wardlaw, and Don Weaver. The Third CCCC was held in July 1997 at the University of Alberta in Edmonton. The Organizing Committee was chaired by Mariusz Klobukowski. A preliminary presentation of the material in this chapter at the Edmonton CCCC meeting led to the recollection of many anecdotes from earlier conferences in Canada. At least one story is worth recalling. At the CSTC conference in Banff in 1989, a prominent quantum chemist was asked when one of the major computer manufacturers would introduce parallel computers. With hardly a pause, the speaker replied that he would rather have one ox to plough a field than 10,000 chickens!

SPREADING THEIR WINGS Numerous references have been made throughout this chapter to the large number of outstanding computational and theoretical chemists who have been attracted to Canada. This history would not be complete without at least some

Spreading Their Wings 285 coverage of the equally remarkable Canadian computational and theoretical chemists who have pursued their careers in other countries. We include only a few notable examples. Rudolph A. Marcus is perhaps the most famous theoretician to be raised in Canada. He has received many awards, most notably the 1992 Nobel Prize in chemistry. Marcus was born in Montreal. He received a BSc. degree in chemistry from McGill University in 1943, and a Ph.D. degree from the same institution in 1946. After doing postdoctoral research at the National Research Council of Canada and at the University of North Carolina, Chapel Hill, he became a professor at the Polytechnic Institute of Brooklyn from 1951 to 1964 and at the University of Illinois from 1964 to 1978, when he was named the Arthur Amos Noyes Professor of Chemistry at California Institute of Technology. His seminal contributions to the realms of electron transfer theory and intramolecular dynamics continue to earn him honors, including the 1997 ACS Award in Theoretical Chemistry. Marcus is a coauthor of the Rice-Ramsperger-Kassel-Marcus theory of molecular reactions. However, his theoretical work in the 1950s on electron transfer reactions started him on his pathway to eminence. Back then, chemists knew electron-transfer processes were occurring during chemical reactions, but they had no way to predict the speed of a reaction or to develop strategic chemical experiments. In addition, some reactions that were predicted to proceed rapidly instead poked along at a snail’s pace. In a series of papers between 1956 and 1965, Marcus solved much of the mystery by outlining a description of the probability of fluctuations in the geometry of reactants and their solvents. These fluctuations lead to changes in the energy barriers that the reactants must surmount before an electron can be transferred from one molecule to another. Marcus extended the theory to other systems, such as electrochemical rate constants at electrodes, and to chemiluminescent electron transfer reactions. The by-now famous “inverted effect” is a consequence of his theory: after a certain point, adding more energy to an electron transfer reaction actually slows the process. Scientists believe photosynthesis can occur because of the inverted effect. The RRKM theory is a ubiquitous tool for studying dissociation or isomerization rates of molecules as a function of their vibrational energy. Still highly active in the theoretical field, Marcus has tackled such issues as the semiclassical theory for inelastic and reactive collisions, devising reaction coordinates, new tunneling paths, and exploring solvent dynamics effects on unimolecular reactions in clusters. Vincent McKoy, a long-time colleague of Marcus at Caltech, was also educated in Canada. McKoy graduated from the Nova Scotia Technical College (now part of Dalhousie University) and went on to earn his Ph.D. degree at Yale University. Attila Szabo and Neil Ostlund, authors of a standard quantum chemistry text,293 graduated from Canadian universities. After receiving his undergradu-

286 The Development of Computational Chemisty in Canada

ate education at the University of Saskatchewan, Neil Ostlund earned his Ph.D. under the guidance of John Pople at Carnegie-Mellon University in 1968. Attila Szabo graduated from Montreal’s McGill University in 1968 and completed his Ph.D. at Harvard University in 1973. Several of the authors of the Gaussian software package were educated in Canada. Berny Schlegel graduated from the University of Waterloo and received his Ph.D. from Queen’s University (Saul Wolfe), while a few years earlier Mike Robb294 completed his Ph.D. at the University of Toronto (Imre Csizmadia). Schlegel is a long-time faculty member across the border at Wayne State University in Detroit whereas Robb is at King’s College London. There are many other examples of individuals who are not computational chemists but who are well known for their scientific contributions. Walter Kohn, a pioneer of density functional theory, graduated from the University of Toronto (B.A., 1945; M.A., 1946) and received his Ph.D. in physics from Harvard in 1948. He has received many awards (including the 1998 Nobel Prize in Chemistry) for his research in the theory of solids during his long association with the Department of Physics at the University of California, Santa Barbara. Charlotte Froese Fischer, well known for her computations on atoms,295 graduated from the University of British Columbia (B.A., 1952; M.A., 1954) and earned her Ph.D. in 1957 for research carried out under the supervision of D. R. Hartree at Cambridge. She returned to UBC in 1957 and rose through the ranks to become a professor of mathematics. She later became a professor of computer science, briefly at Waterloo, then at Penn State, and more recently at Vanderhilt University.

ACKNOWLEDGMENTS This chapter is the result of exchanging about 700 messages over the Internet and a similar number of conversations with many of the people named herein. Even Canada Post contributed by delivering an estimated 300 pieces of mail to the correct addresses. A full acknowledgment would be tedious to read but two individuals, more than anyone else, contributed to the completion of the project. Deanna Wentzell heroically transformed many annotated letters, articles, and messages into a readable text. My life-long partner and frequent scientific collaborator, Susan, had to forego many Sunday walks for a year while I tried to turn thousands of bits of information into something that might actually be read. Susan also acted as a critical but friendly referee. To everyone, thank you very much. I apologize, in advance, to anyone who may have been overlooked or whose achievements may have been misrepresented. I look forward to the continued development of computational chemistry in Canada and abroad.

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a

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Reviews in Computational Chemistry, Volume15 Edited by Kenny B. Lipkowitz, Donald B. Boyd Copyright 02000 by Wiley-VCH, Inc.

Author Index Abou-Rachid, H., 290 Acree, W. E., Jr., 210 Adamo, C., 207 Adamson, R. D., 209 Ageev, S. Z., 292 Ahlrichs, R., 85, 144 Al-Laham, M. A., 144,206,207 Albright, T. A,, 77 Alexandratos, S., 85 Allen, W. D., 84 Allendorf, M. D., 206, 208 Allinger, N. L., 205 Allnatt, A. R., 296 Allnatt, E. L., 296 Altenbach, H.-J., 80 Amarouche, M., 146 Amos, R. D., 84 Anderson, S. G., 295 Anderson, K., 144 Andres, J. L., 144, 206, 207 Andzelm, J. W., 76, 209, 289 Anklam, E., 82 Apeloig, Y., 83, 84 Arduengo, A. J., 111, 84 Arkos, G. G., 289 Arratia-Perez, R., 297 Ashfold, M. N. R., 146 Askenazi, G., 145 Asmus, K.-D., 82 Atabek, O., 290,299 Atchity, G. J., 143 Attfield, B., 299 Aubanel, E. E., 299 Avery, J. A,, 77 Ayala, P. Y., 144, 206, 207 Ayton, G., 289 Baboul, A. G., 207 Bach, M. A. G., 145

Bachrach, S. M., 84, 85, 295 Bader, R. W. F., 85, 294, 295 Baerends, E. J., 76, 77, 78, 79, 80, 82, 85, 203 Baggott, J., 141 Bagus, P. S., 78 Bailey, S. M., 210 Baird, N. C . , 82, 296 Baker, J., 144,206 Balasubramanian, K., 8 3 Baldridge, K. K., 144 Bally, T., 206 Bandrauk, A. D., 287,290,299 Bankmann, M., 81 Barclay, L. R. C., 298 Barone, V, 207 Barrett, J., 289 Bartlett, R. J., 143, 203, 204 Bartmess, J. E., 81 Bartolotti, L. J., 77, 203 Basile, A. G., 295 Bauernschmitt, R., 144 Bauschlicher, C. W., Jr., 204, 205, 206,209 Bearpark, M. J., 142, 143, 145, 146 Becke, A. D., 77, 81, 203,208,209,294 Becker, R. S., 146 Beddall, P. M., 295 Beenakker, J. J. M., 291 Belghazi, A., 83 Bell, A. J., 142, 146 Ben-Nun, M., 145, 146 Bender, C. F., 287 Benesch, R., 293 Benson, S. W., 210 Bentzian, J., 144 Berard, D. R., 289 Berkowitz, J., 83, 202 Bernard, M., 295 Bernardi, F., 78, 141, 142, 143, 145, 146, 299

301

302 Author Index Bertran, J., 82 Beveridge, D. L., 209 Bickelhaupt, F. M., 76, 78, 79, 80, 81, 82, 85, 203 Bingham, R. C., 209 Binkley, J. S., 84, 144, 203, 205, 206, 208 Binning, R. C., Jr., 84 Birss, F. W., 289 Birss, V., 295 Bishop, D. M., 289 Bissonette, C., 292 Blais, N. C., 145 Blaudeau, J.2,206, 207 Blochl, P. E., 296 Blokker, V. M., 296 Blomberg, M. R. A., 144, 207 Boatz, J. A., 144 Bock, H., 81 Boese, R., 80 Bohme, D. K., 296 Bonacic-Koutecky, V., 141, 146 Borden, W. T., 206 Botschwina, P., 81 Bottoni, A., 78, 142 Boyd, D. B., 77, 82, 85, 143, 144, 202,203, 206,289,295,299 Boyd, R. J., 287,297,298 Boyd, S . L., 298 Bramley, M. J., 288,297 Brewer, L., 202 Brewer, S. E., 293 Brotherton, T. K., 80 Brouwer, A. M., 144, 145 Brown, F. B., 207 Brown, R. J. C., 293 Brumer, P.,291 Bruna, P. I., 298 Buijse, M. A., 78 Buncel, E., 297 Bung, C. E, 294 Burant, J. C., 207 Burdett, J. K., 77 Burgstahler, A. W., 145 Burke, K., 209 Buzek, P., 83 Cade, P. E., 294 Cammi, R., 207 Car, R., 145 Carbb, R., 291 Carey, F. A., 82 Carpenter, G. S., 205 Carrington, T., Jr., 288, 297

Carter, J. T., 288 Caselli, M., 144 Casida, K. C., 144, 288 Casida, M. E., 144, 288 Cederbaum, L. S., 81,143 Celani, P., 142, 143, 145, 146 Chabalowski, C. F., 209 Chakrabarti, A,, 210 Challacombe, M., 144, 206, 207 Chandler, G. S., 205 Chandrasekhar, J., 84, 206 Chang, T. C., 292 Chase, M. W., Jr., 202 Chatgilialoglu, C. G., 83 Cheeseman, 1.R., 144,206, 207 Chelkowski, S., 299 Chen, C., 297 Chen, H., 289 Chen, W., 144,206,207 Chen, Z., 291 Chevary, J. A., 77 Chikos, J. S., 210 Child, M. S., 299 Choi, J. H., 289 Choi, Y. S., 142 Chong, D. P.,288,297 Christensen, R. L., 142, 146 Christiansen, J. J., 83 Chu, C., 297 Chuaqui, C. E., 292 Churney, K. L., 210 Ciccotti, G., 290 Ciganek, E., 80 Cioslowski, J., 85, 144, 206, 207 Cizek, J.. 203, 291 Clark, T., 82, 85, 205,206 Clarke, A. S., 289 Clement, S. G., 146 Clementi, E., 76 Cleve, R. H., 297 Clifford, S., 142, 207 Cohen, L. K., 290 Coleman, A. J., 293 Collins, J. B., 85 Collman, J. P., 82 Colpa, 1. P.,294 Coltrin, M. E., 208 Combariza, J. E., 85 Constas, S., 290 Coope, J, A. R., 288 Cooper, D. I.., 85 Corchado, J. C., 207,208 Cordes, A. W., 298

Author Index 303 Corkum, P. B., 299 Cornelisse, J., 145 Corongiu, G., 76 Correa de Mello, P., 298 Cosgrove, B. A., 297 Cossi, M., 207 Coulson, C. A., 287,293 Cox, J. D., 210 Craig, R. A., 296 Crawford, T. D., 82, 143,203 Crerner, D., 83 Crossan, R., 299 Crowell, J. E., 83 Csaszar, P., 290 Csizmadia, I. G., 290, 296 Cui, Q., 207 Cullen, J. M., 297 Curl, R. F., Jr., 287 Curtiss, L. A., 77, 84, 8 5 , 202, 203, 205, 206, 207,208 Cyr, D. R., 141 Dachsel, H., 144 Daniels, A. D., 207 Danovich, D., 82 Dantus, M., 145 Dapprich, S., 207 Das, K. K., 8 3 Dauben, W. G., 145 Daudel, R., 287 Davidson, E. R., 143, 144, 146,203,287 Davies, C. A., 202 Davies, P. B., 83 Davis, N. E., 206 Davis, W. M., 299 De Almeida, W. B., 81 de Kanter, F. J. J., 80 de Koning, L. J., 79, 80, 82 de Lange, C. A,, 80 de Oliveira, G., 204 de Visser, S. V., 79, 82 DeFrees, D. J., 144, 205, 206 DelMedico, A., 297 Delos, J. B., 289 Demain, A. L., 297 Demeio, L., 289 Deng, Y., 82 Dennington, R. D., 210 Deslongchamps, P., 293 Desouter-Lecomte, M., 141 Devlin, F. J., 209 Dewar, M. J. S., 209, 210 Dham, A. K., 292

Diefenbach, A., 79 Dilabio, G., 298 Ding, Y. H., 81 Dirac, P. A. M., 77 Ditchfield, R., 208 Dixon, D. A., 84, 204 Doerksen, R. J., 293 Doig, S. J., 141 Domalski, E. S., 210 Domcke, W., 143, 146 Donaldson, D. J., 299 Donovan, B., 141 Dorigo, A. E., 84 Dostaler, S., 294 Downey, J. R., Jr., 202,210 Dreizler, R. M., 76 Drewello, T., 82 Drowart, J., 81 Du, I?, 146 Duadey, J. P., 145 Duffy, P., 288 Duke, B. J., 206 Durnont, R. S., 295 Dunietz, B. D., 208 Dunitz, J. D., 76 Dunning, T. H., Jr., 143, 204 Dupuis, M., 144 Durand, P., 145 Durant, J. L., 206 Durocher, S., 290 Durup, J., 146 Dutler, R., 295 Dwyer, J. R., 297 Eckwert, J., 80 Eeken, P. J. K. M., 80 Ehlers, A. W., 79 El-Bakali Kassimi, N., 293 El-Gharkawy, E. R. H., 209 Elbert, S. T., 144 Ellis, D. E., 78 Elschenbroich, C., 82 Endredi, G., 290 Engels, B., 293 Epa, V. C., 289 Erdahl, R. M., 293 Eriksson, L. A., 298 Ernzerhof, M., 209 Erskine, R. W., 78 Eschrig, H., 82 Esskn, H., 295 Eu, B. C., 292 Evans, W. H., 210

304 Author Index Eveland, S. E., 210 Eyring, H., xuii Farhat, H., 292 Farkas, O., 207 Fast, P. L., 204, 207, 208 Fato, M., 145 Fawzy, W. M., 292 Feinberg, M. J., 78 Feller, D., 203, 204 Ferguson, H. D., 210 Fermi, E., 77 Ferrario, M., 290 Fielder, S. S., 297 Filatov, M., 209 Finke, R. G., 82 Fiolhais, C., 77 Fleming, D. G., 288 Fleming, I., 77 Fliszar, S., 287 Fliigge, J., 81 Flurchick, K., 77, 203 Fokkens, R. H., 80, 82 Fonseca Guerra, C., 78, 85 Foreman, J. B., 143, 144, 206, 207 Forst, W., 290 Forte, L., 296 Fournier, R., 297 Fox, D. J., 144, 203, 206, 207 Fraga, S., 289, 297 Francl, M. M., 205 Freedman, E., 210 Freedman, K. A., 146 Frenking, G., 83, 84 Frey, R. F., 143 Friedrich, B., 287 Friesner, R. A., 208, 288 Frisch, A., 143 Frisch, M. J., 84, 143, 144, 205,206, 207, 208,209 Froese, C., 299 Froese, R. D. J., 206 Frurip, D. J., 202, 203, 204,205,208,210, 21 1 Fujimoto, H., 78 Fiilscher, M. P., 144 Fuss, W., 142 Gadea, F. X., 146 Gal, 1.-F.,82 Gallant, R. T., 289

Gao, J., 144 Garavelli, M., 142, 143, 145, 146, 299 Gasteiger, J., 205 Gaunt, D. S., 291 Gauthier, J. M., 299 Gay-Lussac, L. J., 79 Gerry, M. C. L., 80 Gigusre, P. A., 298 Gilbert, A., 141 Gilheany, D. G., 84 Gill, P. M. W., 82, 144, 206, 207, 208,209 Gillespie, R. J., 84 Gimarc, B. M., 77 Gingerich, K. A,, 81 Givens, R. S., 145 Glukhovtsev, M. N., 202 Gobbi, A., 84 Goddard, J. D., 298, 299 Goede, S. J., 80 Goldberg, N., 79 Goldman, S., 298 Golub, I., 291 Gomperts, R., 144, 206, 207 Gonzalez, C., 144, 145, 206, 207 Gordon, H., 298 Gordon, M. S., 141, 144,205, 207 Gordy, W., 83 Gough, K. M., 297 Grabandt, O., 80 Grant, D. J. W., 211 Grant, M., 292 Gray, B. F., 296 Gray, C. G., 298 Green, W. H., 84 Grein, F., 292, 293 Greiner, G., 146 Grev, R. S., 202 Gritsenko, 0. V., 77, 78 Gross, E. K. U., 76 Grover, R., 82 Guan, J., 288 Guerra, M., 76, 83 Guest, M. F., 202 Gum, J. R., 288 Gupta, S . K., 210 Guttman, A. J., 290 Hackett, P. A., 288, 292 Hafner, K., 76 Hall, M. B., 84 Halow, I., 210 Halstead, T.K., 292 Hanack, M., 84

Author Index 305 Handy, N. C., 84,206 Harcourt, R. D., 82 Harder, J. M., 296 Hargittai, I., 84 Hariharan, P. C., 205 Harrison, B. K., 210 Harrison, R. J., 144, 202, 204 Hartree, D. R., 299 Hayden, C. C., 141 Head-Gordon, M., 143, 144, 203, 204,206, 207 Healy, E. F., 209 Hearing, E. D., 210 Hegedus, L. S., 82 Hehenberger, M., 298 Hehre, W. J., 202, 205, 208 Heinicke, J., 83 Henneker, W. H., 294 Henry, B. R., 297, 299 Herman, K., 298 Herman, Z . S., 298 Herring, F. G., 288 Herrington, T. M., 293 Herschbach, D. R., 77,287 Hertel, G. R., 210 Herzberg, G., 143, 286, 287 Hiberty, P. C., 82, 85 Hicks, M. G., 210 Higuchi, T., 211 Hinchcliffe, A., 81 Hirota, E., 83 Hirschfelder, J. O., 288 Hirshfeld, F. L., 85 Ho, P., 208 Hobbs, R. H., 83 Hofelich, T. C., 210 Hoffmann, R., 77, 78, 79 Hohenberg, P., 76 Holbrook, N. K., 296 Holder, A. J., 210 Holmes, J. L., 81 Hopkinson, A. C., 83, 290, 296 Hori, Y., 81 Houk, K. N., 76 Hout, R. F., 205 Howe, J. D., 146 Huang, X. R., 81 Huber, K. P., 286 Hudgens, J. W., 83 Humbel, S., 82, 206 Hunter, G., 296 Hutchinson, D. A., 293 Huzinaga, S., 289

Illies, A. J., 82 Irikura, K. K., 203, 204, 205, 208, 210, 211 Ito, M., 141 Ito, S., 76 Ivanov, Y. M., 294, 298 Jackel, G. S., 83 Jackson, K. A., 77 Jacobs, H. J. C., 145 Jacobs, P. W. M., 296 Jacobsen, H., 79 Jain, S., 295 James, M. A., 82 Jamorski, C., 144 Jankowski, K., 291 Janse van Rensburg, E. J., 291 Jauregui, R., 294 Jaworski, W., 294 Jayatilaka, D., 84 Jean, Y.,77 Jenneskens, L. W., 80 Jensen, F., 81, 203 Jensen, J. J., 144 Jensen, S. E., 297 Jie, C., 210 Jimmo, S., 294 Jin, A., 294 Joentgen, W., 80 Johnson, B. G., 144, 206, 207, 208, 209,298 Johnson, R. D., 111, 83 Jones, C., 203 Jordan, K. D., 81 Joslin, C. G., 298 Kafafi, A., 209 Kandori, H., 141 Kantorovich, L., 296 Kaplansky, M., 292 Kapral, R., 289, 290 Kari, R., 290 Karlstrom, G., 144 Karplus, M., 288 Kass, S. R., 209 Katagiri, S., 287 Katsuta, Y., 141 Kaufmann, E., 85 Kavassalis, T. A., 299 Kawaguchi, K. J., 83 Keith, T. A., 144, 206, 207 Kendall, R. A., 202 Kennedy, C., 293 Kerr, J. A., 210 Kestner, N. R., 85

306 Author Index Ketelaar, J. A. A., 294 Kim, C.-K., 297 Kimball, G. E., xvii Kinoshita, M., 289 Kirby, S. P., 210 Kirchner, R. F., 298 Kitaura, K., 78 Kittel, C., 85 Kjaergaard, H. G., 297, 299 Klein, J., 85 Klein, S., 146 Klesing, A., 80 Klessinger, M., 144 Kleyn, A. W., 78 Klobukowski, M., 289, 291 Kmetic, M. A., 296 Kock, R., 80 Kohler, B. E., 142 Kohler, W. E., 291 Kohn, W., 76,203,208 Kolbuszewski, M., 298 Kollman, P. A., 205 Komaromi, I., 207 Kompa, K. L., 142 Kondo, A. E., 296 Koppel, H., 82, 143, 146 Kordis, J., 81 Koseki, S., 144 Kost, D., 85 Kotomin, E. A., 296 Kouba, J. E., 296 Kowari, K., 288 Kozlowski, P. M., 144 Krishnan, R., 203, 205 Krogh-Jespersen, M.-B., 84 Kroto, H. W., 287 Kuchitsu, K., 81 Kudin, K. N., 207 Kusalik, P. G., 289, 298 Kuscer, I., 291 Kutzelnigg, W., 84 Laaksonen, A., 298 Labanowski, J. K., 76 LaBarge, M. S., 210 Ladik, J., 290 Laidlaw, W. G., 295 Laird, B. B., 76, 77 Lambert, C., 84, 85 Langhoff, S. R., 205 Langkilde, F. W., 144 Laria, D., 290 Larsson, S., 298

Lawetz, V., 293 Lawley, K. P., 141, 143, 145 Le Roy, R. J., 292 Lebrilla, C. B., 82 Lee, C., 208 Lee, T. J., 204 Lehn, J.-M., 76 Leigh, W. J., 143 Lekkerkerker, H. N. W., 295 Leroy, J. P., 297 Leroy, O., 80 Leung, F., 294 Leung, K., 288 Lever, A. B. P., 297 Levi, A. C., 291 Levine, R. D., 81, 146 Levy, M., 78 Lewis, G. N., 202 Li, K. B., 292 Li, Y.S., 288 Li, Z. S., 81 Lias, S . G., 81 Liashenko, A., 207 Lidiard, A. B., 296 Liebman, J. F., 81, 210 Lien, M. H., 290, 296 Light, J. C., 287 Lim, T. K., 292 Lin, M. C., 206 Lindh, R., 144 Linn, W. J., 80 Lipkowitz, K. B., 77, 82, 85, 143, 144,202, 203,206,289,295,299 Lipscomb, W. N., 287 Littlefield, R. J., 202 Liu, G., 207 Liu, W., 288 Lo, D. H., 209 Lochbrunner, S., 142 Loew, G. H., 298 Loftus, E., 296 Lohrenz, J. C. W., 296 Lombos, B. A., 287 Longuet-Higgins, H. C., 143 Lorquet, J. C., 141 Lowry, J. T., 288 Lu, G., 83 Luke, B. T., 84 Lynden-Bell, R. M., 293 Lynn, J. W., 80 Magnusson, E., 84 Mahy, J. W. G., 80

Author Index 307 Maier, G., 80 Maki, A. G., 81 Malevanets, A., 290 Malick, D. K., 207, 208 Malkin, V. G., 288 Malkina, 0. L., 288 Mallard, W. G., 81 Malli, G., 297 Malmqvist, P.-A., 144 Malrieu, J. P., 145 Manghi, F., 290 Mangini, A., 78 Manoli, S., 292 Manthe, U., 146 March, J., 82 Marchioro, T. L., 11, 295 Marcus, R. A., 294 Margl, P., 296 Maria, P.-C., 82 Markus, M. W., 80 Maroulis, G., 293 Marshall, C. H., 288 Martin, J. M. L., 202, 204 Martin, R. L., 144, 206,207 Martinez, A., 288 Martinez, T. I., 145, 146 Marynick, D. S., 84 Mathies, R. A., 141 Matsunaga, N., 144 Mattar, S. M., 293 Matusek, D. R., 298 Matuszewski, B., 145 Mavroyannis, C., 293 Mayer, P. M., 207 Mayhew, C. A., 202 Maynau, D., 145 McAllister, M. A., 290 McCourt, F. R., 288, 291, 292 McDonald, K. M., 289 McDonald, R. A,, 202 McDouall, J. J., 144 McEwen, K. L., 289 McGrath, M. P., 206, 207 McKee, M. L., 82, 143 McKellar, A. R. W., 292 McKelvey, J. M., 85 McLean, A. D., 205 McWeeny, R., 79 McWilliams, D., 288 Mead, C. A., 146 Meath, W. J., 292, 296 Mebel, A. M., 206 Meister, I., 86

Melius, C. F., 206, 208 Mennucci, B., 207 Merrill, G. N., 209 Messmer, R. P., 287 Mezey, P. G., 289, 290 Michels, H. H., 83 Michl, J., 141, 142, 146 Middlemiss, K. M., 291 Millam, J. M., 207 Minato, T., 78 Mitchell, D., 78 Mitchell, K. A. R., 288 Moc, J., 83 Moffat, J. B., 291 Moir, R. Y., 297 Moise, A., 296 Meller, K. C., 81 Momicchioli, F., 144 Monge, A., 288 Montgomery, 1. A., 144, 204, 205, 206, 207, 208 Moore, C., 207 Mooyman, R. A., 80 Moraal, H., 291 Morino, Y., 81 Morokuma, K., 78,206,207 Mueller-Westerhoff,U. T., 298 Muller, A. M., 142 Miiller, T., 83 Mulliken, R. S., 8.5, 287 Murphy, R. B., 208 Myers, C. E., 81 Nanayakkara, A., 144,206,207 Nau, W. M., 146 Naumkin, F. Y.,292 Naylor, R. D., 210 Negri, F., 144 Neogrkdy, P., 144 Newbold, B. T., 287 Nguyen, K. A., 144,207 Nguyen, M. T., 81 Nguyen-Dang, T. T., 290,295 Nibbering, N. M. M., 79, 80, 82 Nicolaides, A., 202, 208 Nieplocha, J., 144 Niki, H., 292 Nilar, S . H., 296 Noglik, H., 297 Noodleman, L., 82 Nooijen, M., 143 Norton, J. R., 82 Nusair, M., 81, 208

308 Author Index Nuttall, R. L., 210 Nyalhszi, L., 83 Oakley, R. T., 298 Oberhammer, H., 80 Ochrerski, J. W., 202, 204, 205, 207, 208 Olivucci, M., 141, 142, 143, 145, 146, 299 Olsen, J., 144 Olson, J., 295 Olsson, L., 83 Oosteroff, L. J., 141 Operti, L., 82, 83 Oreg, J., 297 Orlandi, G., 144 Orlandini, E., 291 Ortiz, J. V., 144, 206, 207 Osamura, Y., 78 Ostlund, N. S., 299 Ottani, S., 142 Ottosson, H., 83 Pacchioni, G., 78 Page, M., 143 Paldus, J., 291, 295 Palmer, 1. J., 142 Pandey, P. K. K., 295 Pang, T., 297 Papai, I., 287 Parker, V. B., 210 Parkinson, C. J., 207 Parmigiani, F., 78 Parr, R. G., 76, 203, 208 Parrinello, M., 145 Partridge, H., 204, 206 Pasztor, A. J., Jr., 210 Patey, G. N., 289 Paul, R., 295 Pauling, L., 79 Peard, P. J., 291 Peasley, K., 144 Peat, F. D., 293 Pederson, M. R., 77 Pedley, J. B., 210 Peerey, L. M., 210 Peng, C. Y., 144, 206,207 Perczel, A,, 290 Perdew, J. P., 77, 81, 82, 208, 209 Peric, M., 293 Perisanu, S., 210 Peschke, M., 82 Peteanu, L. A., 141 Petek, H., 142, 146 Peterson, K. A., 204

Petersson, G. A., 144, 202, 204, 205, 206, 207,208 Peyerimhoff, S. D., 293 Piche, L., 292 Pidun, U., 83 Piecuch, P., 291 Pietro, W. J., 205, 297 Piskorz, P., 207 Pitzer, K. S., 202 Poirier, R., 290 Poling, B. E., 210 Pomelli, C., 207 Ponterini, G., 144 Popkie, H. E., 291 Pople, J. A., 77, 84, 143, 144, 202, 203, 204, 205,206,207,208,209,287 Popov, v. s., 77 Porter, N. A., 298 Post, D., 78, 82 Postigo, J. A., 143 Poulin, N. M., 297 Prass, B., 294 Prausnitz, J. M., 210 Preston, R. K., 145 Prigogine, I., 143, 146, 207 Pritchard, H. O., 296 Proynov, E. I., 288 Pullen, S., 141 Purvis, G. D., 204 Pyper, N. C., 297 Qin, Z., 296 Rabezzana, R., 82 Rabinowitz, J., 145 Rabitz, H., 297 Rablen, P. R., 86 Rabuck, A. D., 207 Radius, U., 79 Radom, L., 82,202,206,207,208,209 Radzio-Andzelm, E., 289 Ragazos, I. N., 142, 143, 146 Raghavachari, K., 77, 83, 85, 144,202, 203, 204,205,206,207,208 Ramos, A. F., 297 Randall, M., 202 Rappoport, Z., 80 Rassolov, V., 205, 207 Ratajczak, H., 83 Rau, H., 146 Rauk, A., 77, 78, 79,202, 295, 297 Ravenek, W., 79 Raymond, K. N., 76

Author Index 309 Raymonda, J. W., 287 Rayner, D. M., 288 Reatto, L., 290 Redfern, P. C., 77, 202,203,205,206,207 Reed, A. E., 84, 85 Rees, C. W., 76 Reguero, M., 142 Reid, P. J., 141 Reid, R. C., 210 Reisenauer, H. P., 80 Replogle, E. S., 144, 206, 207 Reuter, W., 293 Ricca, A., 205 Rice, S. A., 143, 146, 207 Ridley, J. E., 144 Riopelle, R., 294 Ritchie, J. P., 85 Rivail, J.-L., 290 Robb, M. A., 141,142,143,144,145,146, 206,207,299 Robertson, S. H., 294 Robin, M. B., 287 Rodriguez-Santiago, L., 82 Rodriquez, C. F., 8 3 Rohde, C., 85 Rohlfing, C. M., 206 Rondan, N. G., 21 1 Ronis, D., 292 Roos, B. O., 141, 144 Rosa, A,, 79 Rose, T. S., 145 Rosker, M. J., 145 Ross, G . , 294 Ross, R. B., 76, 77 Roux, B., 288 ROY,KN., 288 Rozas, I., 294 Rozendaal, A., 79 Rudzinski, J. M., 83 Ruedenberg, K., 78, 143 Ruiz, E., 288 Ruiz-Morales, Y., 295 Ruscic, B., 83, 202 Rycerz, Z . A., 296 Sadlej, A. J., 144 Said, M., 145 St-Amant, A., 77, 287, 289 Sakai, Y.,289 Salahub, D. R., 83, 144, 287, 288 Salzer, A., 82 Sana, M., 80 Sanchez, M. L., 204, 207,208

Sanctuary, B. C., 292 Sandorfy, C., 287 Sandrone, G., 204 Santry, D. P., 287, 295 Sapse, A. M., 84 Sasabe, H., 141 Sauvageau, P., 287 Schaefer, H. F., 111, 81, 82, 84, 143, 202, 203, 205,292 Scheiner, P. R., 205 Scheller, M. K., 81 Schiffer, H., 85 Schipper, P. R. T.,78 Schlegel, H. B., 142, 143, 144, 145, 203, 205, 206,207 Schleyer, P. v. R., 82, 83, 84, 85,202, 205, 206 Schmider, H. L., 209, 294 Schmidt, M. W., 144 Schoenlein, R. W., 141 Schoffel, K., 146 Schreckenbach, G., 295 Schriver, G. W., 85 Schumm, R. H., 210 Schiitz, M., 144 Schwarz, H., 82 Schwarz, M., 83 Schwarz, W. H. E., 86 Scott, A. P., 202 Scuseria, G. E., 207, 209 Sebald, P., 8 1 Seeger, R., 205 Segal, G. A., 287 Seibert, J. W. G., 80 Seijo, L., 144 Selmani, A., 83 Seminario, J. M., 203 Sension, R. J., 141 Serrano-AndrCs, L., 144 Shaik, S., 82 Sham, L. J,, 76,208 Shamovsky, I., 294 Shank, C. V., 141 Shapiro, M., 291 Shavitt, l., 291 Shen, D., 296 Shepard, R., 144 Sherman, A., 289 Sherrill, C. D., 81 Shi, Z., 297, 298 Shizgal, B. D., 288, 289 Shnitman, A., 291 Shore, J. D., 292

310 Author Index Showalter, K., 290 Shushin, A., 294 Sichel, J. M., 292 Siegbahn, P. E. M., 144,207 Siehl, H.-U., 83 Sierakowski, C., 80 Sim, F., 287 Simard, B., 292 Simpson, W. T., 287 Singh, D. J., 77 Singh, S. R., 293 Sinke, G. C., 211 Slater, J. C., 79, 208 Slenczka, A., 287 Sloan, J. J., 299 Smith, B. J., 206,209 Smith, B. R., 142, 143, 145, 146 Smith, D. M., 83, 207 Smith, V. H., Jr., 293, 294 Smoes, S., 81 Snider, N. S., 293, 294 Snider, R. F., 288 Snijders, J. G., 76, 85 Sodupe, M., 82 Sofer, I., 291 Soteros, C. E., 291 Spitznagel, G. W., 206 Splendore, M., 83 Srivastava, H. K., 297 Stahl, M., 83 Stanton, J. F., 143, 203 Stashans, A,, 296 Stefanov, B. B., 144, 206, 207, 208 Stehlik, D., 294 Steiner, E., 79 Stephens, P. J., 209 Stewart, J. J. P., 144, 203, 206, 209 Stock, G., 146 Stoicheff, B. P., 81 Stone, A. J., 78 Storer, J. W., 211 Stowasser, R., 78 Strain, M. C., 207 Stratmann, R. E., 207 Streitwieser, A., Jr., 84, 85 Stroh, F., 80 Stull, D. R., 21 1 Stumpf, T., 80 Su, S., 144 Suba, S., 292 Suckow, R. A., 210 Sumners, D. W., 291

Sun, C. C., 81 Sundberg, R. J., 82 Sunil, K. K., 81 Sutter, D. H., 80 Svensson, M., 206, 207 Svishchev, I. M., 298 Syverud, A. N., 202 Szabo, A., 299 Szktsi, S. K., 83 Tal, Y., 295 Tanimoto, M., 81 Tarantelli, F., 81 Tatewaki, H., 289 Taylor, K. F., 296 Taylor, P. R., 202 te Velde, G., 76, 77 Teller, E., 142, 143 Tensfeldt, T. G., 205 Tesi, M. C., 291 Thachuk, M., 294 Thakkar, A. J., 293, 294,296 Thang, L., 296 Theil, W., 203, 209, 210 Thiem, J., 76 Thomas, L. H., 77 Thorson, W. R., 289 Thuraisingham, R. A,, 296 Tomasi, J., 207 Tonachini, G., 78 Torrie, G. M., 289, 290 Tounge, B. A,, 142 Trinquier, G., 83 Trucks, G. W., 144, 203, 206, 207 Truhlar, D. G., 141, 145, 146, 204, 207, 208 Trulson, M. O., 141 Tsai, B. P., 83 Tschinke, V., 79 Tseng, T. J., 292 Tsipis, C. A,, 77 Tully, J. C., 145 Turnbull, D. M., 299 Turner, R. E., 288 Ursenbach, C., 79, 295 Vaglio, G.-A., 82, 83 Vaida, V., 146 Valleau, J. P., 290 van den Hoek, P. J., 78 van der Does, T., 80

Author Index 31 1 Van der Lugt, W. T. A. M., 141 van Eikema Hommes, N. J. R., 78 van Leeuwen, R., 77, 79 Van Vleck, J. H., 289 van Voorhis, T., 209 van Wezenbeek, E. M., 79 Vatsya, S. R., 296 Veillard, A., 79 Vela, A., 288 Veszprkmi, T., 83 Vietmeyer, N. D., 145 Vilcu, R., 210 Vinette, F., 291 Visser, O., 76 Viviani, W., 290 Vlietstra, E. J., 80 Vogel, E., 80 Vogtle, F., 76 Voityuk, A. A., 209 Volatron, F., 77 Volpe, P., 82, 83 Vosko, S. H., 81,208 Vreven, T., 142, 143, 144, 146 Vu, D. T., 288 Wagrnan, D. D., 210 Walker, L. A., 11, 141 Walker, P. D., 290 Walkley, J., 297 Wall, F. T., 210 Wall, G. C., 293 Wall, J., 146 Wallace, R., 297 Walsh, R., 210 Walter, W., xvii Wan, J. K. S., 293 Wang, J., 294, 298 Wang, Q., 141 Wang, Y., 208 Wardlaw, D. M., 294 Warrington, C. J., 287 Weaver, D. F., 294 Webster, 0.W., 80 Wei, D., 289 Wei, H., 288 Weikert, H. G., 81 Weinberg, N., 297 Weinhold, F., 85 Weinstock, R. B., 85 Weis, I., 80 Wendschuh, I? H., 145 Weniger, E. J., 291

Western, C. M., 146 Westlake, D. W. S., 297 Westrum, E. F., Jr., 21 1 Westwood, N. P. C., 298 Wetmore, R. W., 289 Whangbo, M.-H., 77, 78 Whitehead, M. A., 292 Whiteside, R. A., 205 Whittington, S. G., 290, 291 Wiberg, K. B., 86,202 Wickham, S. D., 141 Widmark, P.-O., 144 Wilbrandt, R., 144 Wilk, L., 82, 208 Wilkie, J., 291 Willetts, A., 84 Williams, J. W., 85 Wilsey, S., 142 Wilson, S., 291 Wilson, W. G., 208 Wimmer, E., 209 Windus, T. L., 144 Winnewisser, B. P., 80 Winnewisser, G., 80 Winnewisser, M., 80 Wolfe, S., 78, 297 Woloschuk, K. J., 293 Wong, K. C., 288 Wong, M. W., 144,206,207 Wong, S.-K., 293 Woo, T. K., 296 Woon, D. E., 204 Wright, J. S., 298 Wrinn, M., 288 Wu, T. H., 292 Xantheas, 5. S., 143 Yacowar, M. M., 297 Yadav, J. S., 299 Yamada, C., 83 Yamada, K. M. T., 80 Yamamoto, N., 142, 144, 146 Yang, C., 208 Yang, D., 295 Yang, D.-S., 288 Yang, W., 76,203 Yang, Y. M., 288 Yarkony, D. R., 141,205 Yates, J. H., 81 Yates, K., 290, 296 Ye, X., 289

312 Author Index Yogev, A., 291 Yoshihara, K., 142, 146 Yu, J., 210 Yuan, J.-Y., 292 Zachariah, M. R., 208 Zakrzewski, V. C., 144, 206, 207 Zerbetto, F., 144

Zerner, M. C., 144,203,298 Zewail, A. H., 141, 145 Zgierski, M. Z., 288 Ziegler, T., 76, 77, 78, 79, 82, 295, 296 Ziesche, I?, 82 Zimmerman, H. E., 142 Zoebisch, E., 209 Zuo, T., 299

Reviews in Computational Chemistry, Volume15 Edited by Kenny B. Lipkowitz, Donald B. Boyd Copyright 02000 by Wiley-VCH, Inc.

Subject Index Computer programs are denoted in boldface; databases and journals are in italics. Ab initio molecular dynamics, 119 Ab initio molecular orbital theory, 4, 152, 155, 162 Academic appointments in Canada, 223, 225, 234 Accuracy of G2 and G3 methods, 168, 175 ACES, 153 Acetaldehyde, 173 Acetamide, 173 Acetic acid, 173 Acetone, 173 Acetyl chloride, 173 Acetyl fluoride, 173 Acids, 194 Acrylic acid, 196 Acrylonitrile, 173 Acylcyclopropenes, 123 ADF (Amsterdam density functional package), 153 Adiabatic reaction, 89, 121 Alberta Symposium on Quantum Chemistry, 218,241 AICI,, 171 Alcohols, 194 Aldehydes, 194 AlF, , 171 Alkanes, 194 Alkenes, 194 Alkynes, 194 Allene, 172 Allinger, N. L., x Ally1 radicals, 123 AM1, 152,183,184,201 Ambident radicals, 42 American Conference on Theoretical Chemistry, 220 Amines, 194

AMPAC, 153 Analytical gradients, 109 Antisymmetrization, 17 Aromatic compounds, 90,92 ASPEN, 189 Atomic enthalpies of formation, 149 Atomic natural orbitals (ANO), 157 Atomic orbitals (AO), 12 Atomic species, 167 Atomic structure, 214 Atomization energy, 151, 155 Atoms in molecules approach (AIM), 65 Aufbau principle, 109 Austin Model 1 (AMl), 183 Avoided crossing, 90, 91, 99, 121 Aziridine, 173 Azoalkanes, 121 B3LYP functional, 38, 152, 154, 156, 168, 179, 180, 181, 182 B3P86 functional, 182 B3PW91 functional, 180, 182 BAC-M04 method, 152 Bader, Richard, 220 Barrier controlled reaction, 94 Barrierless reaction path, 94, 129 Barriers, 92, 95 Basis set superposition error (BSSE), 70 Basis sets, 148 6-31+G(d), 186 6-31G(d), 158, 161, 164, 165, 168, 175, 178,186 6-31G(d,p), 178 6-31 l++G(3d2f,2df,2~), 188 6-311+G(3df,2p), 161, 180, 185, 186 6-311G, 164 6-311G(2d,d,p), 177

3 13

314 Subject Index Basis sets (contznued) 6-311G(2df,p), 185 6-311G(d,p), 159, 161, 163, 185, 186 6-31G(2df,p), 187 641(d), 164 DZ+d, 123 TZ2P, 52 BCI,, 171 Becke-3-Lee-Yang-Parr (B3LYP) functional, 38, 152, 1.54, 156, 168, 179, 180, 181, 182 Becke-88 functional, 52 Becke-88-Perdew-86 (BP86) functional, 38, 58,74,180 BeH, 170 Benson group, 194, 195, 196 Benson’s group additivity method, 152, 193, 197 Benson’s rules, 154 Benson’s second-order group contribution, 197 Benzene, 105,106,122,155,172,196 Benzvalene, 105 Beowolf clusters, 148 BF,, 171 Bicyclobutane, 172 Biological molecules, 245 BLYP functional, 152, 179, 182 Bond additivity approach, 179 Bond density, 18 Bond dissociation energy, 70, 236 Bond dissociation enthalphy (BDE), 36 Bond energy, 51, 155,190 Bond energy decomposition, 4, 44,51, 68 Bond energy tables, 191 Bond length, 21 Bond strength, 191 Born repulsion, 12 Born-Oppenheimer approximation, 96, 243, 244 BP86 functional, 182 BPW91 functional, 180, 182 Branching space, 100, 101 Brock University, 229, 232 Brueckner doubles, 163 Buckingham, A. D., 235,271 Buckminsterfullerene, 219 Burnett effects, 252 Butadiene, 122, 172 Butane, 172 t-Butyl radical, 174

C,C14, 172 C,F4, 172 CZH,, 170 C,H4, 170 C,H6, 170 c60, 219 CADPAC, 153 Canadian Association of Theoretical Chemists, 222 Canadian demographics, 223 Canadian Journal of Chemistry, 218, 242 Canadian Journal of Physics, 214 Canadian Society for Chemistry, 243 Canadian Symposium on Theoretical Chemistry, 220, 242 Canadian universities, 213 Canonical orbitals, 20 Carbon monoxide, 20 Carbon-lithium bond, 65, 71 Carbon-nitrogen dimers, 15, 17, 23, 30, 32, 35, 36, 40,44 Carbon-phosphorus dimers, 35, 39,40,44 Carboxylic acids, 194 Carleton University, 220, 229 Carotenoids, 88 Case-Western Reserve University, 217 CASSCF/MP2,109 Catalysis, 236 Catchment region, 136 CBS-4, 156, 162, 170, 175, 176 CBS-lq, 156 CBS-Q, 152, 156, 162, 169, 170, 176 CBS-q, 162, 169 CBS-QB3,156,169,170,176 CBS-RAD, 177 CCl,, 171 CCNN, 36,40 CCPP, 40 CF3CN, 172 CF,, 168, 171 CH,CI,, 148, 173 CH,F,, 173 CH,CI, 171 CH3Li, 65, 66, 70, 73 CH,, 170, 185, 186 Chaos, 249 Charge distribution, 73, 129, 132 Charge transfer, 24,28,42, 130 CHCI,, 173 Chem3D, 153 Chemical Abstracts Service (CAS), xi Chemical accuracy, 155, 156

Subject Index 315 Chemical bonding, 2, 11, 18, 240 Chemical Institute of Canada, 215 Chemical oscillations, 254 Chemisorption, 236,240 Chemistry in Canada, 239 CHETAH (Chemical Thermodynamic and Energy Release), 197, 198, 200 CH,SH, 171 CHF, , 173 Chirality, 245 Chlorobenzene, 396 Chlorocyclopropane, 195 cis-trans isomerization, 90 CI,, 148, 171 CIF,, 172 CINO, 172 CNCN, 35,44 CNDO/INDO, vii CNNC, 23,30,35,44 CO, 170 CO,, 155,171 COF,, 172 Coherent control, 249 Cold excited state, 118 Collecting funnels, 92 Collision theory, 243 COLUMBUS, 109 Combustion measurements, 148 Complete active space self-consistent field (CASSCF), 8,25, 105, 140 Complete basis set limit, 152, 157 Complete basis set methods (CBS); 149, 169 Complete neglect of differential overlap (CNDO), 183 Comptes Rendus de 1’Academte des Sciences, 217 Computational chemistry in Canada, 213, 223 Computational thermochemistry, 201 Computer-aided drug discovery, 245 Concordia University, 233 Condensed phase therrnochemistry values, 200 Configuration interaction (CI), 91, 100 Configuration interaction - singles (CIS), 108, 140 Conical intersection funnel, 94 Conical intersection optimization, 110 Conical intersections, 89, 90, 91, 95, 96, 98, 100, 101, 102, 105, 112, 113, 115, 122, 123, 133 Conjugated hydrocarbons, 121, 122 Constrained geometry optimization, 111

Contour plot, 17, 43, 64, 72 Core orbitals, 13, 19,20, 32 Core-valence effects, 157, 175 Correction for the gas phase, 199 Correlation consistent basis sets, 148, 157 Correlation effects, 3, 4, 6, 7, 8, 154, 159, 163,175,179 Correlation functional, 180 COS, 171 Cotton-Mouton effect, 245 Coulomb hole, 9, 26 Coulomb repulsion, 3 Coulson, Charles A., 235 Counts, Richard W., v, vii, x Coupled clustered theory, 156, 163, 250 Coupled electron pair approximation, 38 CPCP, 35,39,44 CPPC, 35,39,44 CPU times, 15, 175 Cr(CO),, 27 Critical phenomena, 248 CS,, 171 Csizmadia, Imre, 220 Current Contents, 242 Cyanogen, 35,173 Cyclobutane, 172 Cyclobutene, 91, 172 Cyclohexadiene, 107, 116, 122, 135 Cyclohexadiene/hexatrienephotochemical interconversion, 117 Cyclohexane, 192 Cyclohexene, 192 Cyclopentenes, 124 Cyclopropane, 172 Cyclopropene, 172 Dalhousie University, 214, 229, 232 Davidson, Ernest, 218 Decay channels, 92, 94 Decay paths, 112 Decay region, 134 deMon, 237 Density derivatives, 6 Density functional theory (DFT), 1 , 2 , 152, 154, 163, 168, 180, 201, 237, 254, 258, 264 Density functionals, 237 B3LYP functional, 38, 152, 154, 156, 168, 179, 180,181,182 B3P86 functional, 182 B3PW91 functional, 180, 182

316 Subject Index Density functionals (continued) Becke-3-Lee-Yang-Parr (B3LYP)functional, 38, 152, 154, 156, 168, 179, 180, 181, 182 Becke-88 functional, 52 Becke-88-Perdew-86 (BP86) functional, 38, 58,74, 180 BLYP functional, 152, 179, 182 BP86 functional, 182 BPW9I functional, 180, 182 LY functional, 180 Perdew-86 functional, 52 Perdew-Burke-Ernzerhof exchangecorrelation functional, 181 Perdew-Wang-91 exchange-correlation functional, 52 SVWN functional, 180, 182 X-alpha functional, 52 Desktop computers, 148 Determinantal wavefunction, 8 Deviations of calculated enthalpies of formation, 170 Dewar, Michael J. S., viii, 266 Diagonalization, 20 Dialkyl sulfide dimer radical cations, 51 Diatomic molecules, 96, 99, 214 Diazoalkenes, 123 Diazomethane, 123 Dicarbon, 36 Diffuse functions, 159, 165 Dimethyl sulfide, 173 Dimethyl sulfoxide, 173 Dimethylamine, 173 Dimethylether, 173 Dissociation energy, 151 Donor-acceptor interactions, 24, 28, 30, 36 Double cone, 100 Double-stranded polymers, 249 Dynamic electron correlation, 108 Effective external potential, 3 Effective Hamiltonians, 9, 238 Ehrenfest force, 120 Electron affinities, 151, 155, 162, 175 Electron correlation, 11, 148, 156, 250 Electron density, 2, 5 , 17, 43. 64, 72, 154 Electron density of proteins, 246 Electron pair bond, 30, 34, 46, 65 Electron-electron Coulomb repulsion, 5 , 8 Electron-nuclear attraction energy, 5, 7, 8 Electronic effects, 57 Electronic excitation energy, 28 Electronic spectroscopy, 216

Electronically excited state reaction paths, 88 Electrostatic potential, 3 , 4 Elliptic cone model, 116 Elliptic conical intersection, 114 Elliptic double cone, 98 Empirical methods, 201 Empirical parameters, 152 Empirical schemes, 152 Energy absorption, 89 Energy corrections, 160, 166, 176 Energy decomposition, 4, 33 Energy gap law, 91 Energy minimization, 4 Enones, 121, 123 Enthalpies of formation, 147, 149, 150, 155, 162, 175,185,190 Enthalpies of fusion, 200 Enthalpies of reaction, 190 Enthalpies of vaporization, 200 Enthalpy corrections, 151 Entropies of fusion, 200 Entropies of vaporization, 200 Entropy, 151 Equation-of-motion (EOM), 108 Equilibrium statistical mechanics, 256 Error bars, 155 Esters, 194 Ethanethiol, 173 Ethanol, 173 Ethers, 194 Ethyl chloride, 173 Ethylamine, 173 Exact kinetic energy, 7 Exchange energy, 3, 6 Exchange functional, 180 Exchange holes, 9 Exchange repulsion, 12 Exchange-correlation potential, 3, 6 Excitable media, 254 Excitation energies, 12 Excited Kohn-Sham determinants, 11 Excited state charge distribution, 132 Excited state density, 12 Excited state dynamics, 88 Excited state energy barriers, 94 Excited state potential energy surface, 88, 103, 111 Excited state reaction paths, 119, 122 Excited state structures, 88 Excited states, 96, 123 Expo 67,218 Extended Hiickel theory (EHT) method, 10, 217

Subject Index 31 7 External local potential, 5 Extrapolation methods, 156 F,, 171 F,O, 172 Faraday effect, 245 Fast radiationless decay, 90, 92 Femtosecond spectroscopy, 95 Fermi golden rule, 90 Fermi hole, 9 Fermion wavefunctions, 17 FH, 170 First-order saddle point, 40 Fluorescence, 90 Fluorescence lifetime, 92 Fock operator, 9 Formic acid, 173 FORTRAN, vi Fourth-order Msller-Plesset perturbation theory, 159 Fourth-order saddle point, 40 Fraga, Serafin, 220 Fragment molecular orbitals (FMOs), 12, 20, 50 Fragment orbital interactions, 53 Franck-Condon factors, 90 Franck-Condon point, 92, 108 Franck-Condon region, 104, 126 Franck-Condon structure, 95 Free energies of formation, 151 Frontier orbitals, 13, 33, 34, 40, 132 Frozen core approximation, 164, 187 Fullerenes, 218, 219 Fulvene, 105, 106 Funnel region, 88 Funnels, 92, 95, 124 Furan, 174 G2, 149, 156, 158, 160, 162, 168, 170, 179, 185 G2 energy, 159 G2 test set, 155, 170, 171 G2 theory, variations, 161, 164 G2(B3LYP/MP2/CC),163 G2(BD), 163 GZ(CCSD), 163 G2(MP2) theory, 156, 160, 161, 162, 170, 179,186 G2(MV2,SVP) theory, 156, 160, 161, 162 G2/97 test set, 155, 161, 168, 175 G2Q, 164 G3, 149, 152, 156, 158, 162, 165, 166, 168, 170, 179, 186,201

G3 energy, 167 G3 theory, variations, 168 G3(MP2), 156, 162, 166, 168, 170, 179 G3(MP2)//B3LYP,175 G3(MP3), 156 G3/B3LYP, 175 G3Large, 165 GaF,, 164 GAMESS, 109, 153 Gas phase thermodynamic data, 192, 200 Gaseous atoms, 150 Gauche effects, 193 Gaussian 76, vii Gaussian 94, 109, 111, 161, 177 Gaussian 98, 139, 153, 169, 177, 188, 201 Gaussian basis sets, 242, 248 Gaussian-1 theory (G l ), 152, 158 Gaussian-2 theory (G2), 149, 152, 156, 158, 160,162,168, 170,179,185 Gaussian-3 theory (G3), 149, 152, 156, 158, 162,165,166,168,170, 179,186,201 Generalized gradient approximations (GGA), 3 Generalized valence bond-localized MsllerPlesset method (GVB-LMP2), 179 Geodesic design, 219 Geometry optimization, 110 Glyoxal, 173, 251 Gradient difference vector, 100 Gradient optimization method, 94 Gram-Schmidt orthogonalization, 19 Gramicidin channel, 237 Graphite, 148, 149, 150 Ground state, 88, 89, 96 Ground state energy, 2 Ground state relaxation paths, 108, 122, 133 Group contribution methods, 193, 194 Group orbitals, 217 H,CO, 171 H,COH, 171 H,NNH,, 171 H,S, 52 H,, 105,114 Hagstrom, Stanley A., vi Hamiltonians, 4, 119 Harmonic frequencies, 158 Hartree, D. R., 286 Hartree-Fock determinantal energy, 6 Hartree-Fock exchange potential, 3 Hartree-Fock kinetic energy, 7 Hartree-Fock (HF) molecular orbital theory, 4, 75, 163

318 Subject Index Hartree-Fock orbitals, 4, 5 Hartree-Fock wavefunctions, 8, 11 Harvard University, 255 HCI, 192 HCN, 170 HCO, 170 Heat capacities, 189 Helium molecule, 18 Hepta-2,4,6-trieniminium cation, 137 Heptatrienyl radicals, 123 Herzberg, Gerhard, 214, 221 Hessian, 109, 110 Heterosymmetric biradicaloid, 132 Hexatriene, 107, 116, 122, 135 Higher level correction, 159, 165 Hirshfeld atomic charge, 73 Hirshfeld, Joseph O., 235, 265 HOCI, 171 Hoffmann, Roald, 221 Hohenberg-Kohn-Sham theory, 11 Hohenherg-Kohn theorem, 2, 5, 6 Hole density, 9 HONDO, 245 HOOH, 171 Hot system, 112 Housanes, 124 HS, 174 Hybrid quantum mechanicaVforce field (MM-VB), 119 Hydrides, 157 Hydrocarbon photochemistry, 123 Hydrocarbons, 90, 121, 122, 155, 162, 168, 169, 175,182,184,217 Hydrogen bonds, 237,243 Hydrogen gas, 150 Hydrogen molecule, 9, 18, 21, 34, 148, 149, 150,174 Hyperchem, 153 Hyperline, 100 Illinois Institute of Technology, 241 Imaginary vibrational frequencies, 58, 101, 114, 138 Independent particle model, 5 Indiana University, Bloomington, Chemistry Department, v Industrial chemistry, 192 Initial excited state geometry, 90 Initial molecular motion, 134 Initial relaxation direction (IRD), 104, 114, 115 Inorganic hydrides, 155, 162, 168, 175, 182, 184

Integrated projected population (IPP), 65 Intermediate neglect of differential overlap (INDO), 181 Internal conversion (IC), 89, 92, 104, 124 International Congress of Quantum Chemistry, 220 Internet, x , xiii Interpolation and extrapolation, 198 Intersection space, 100 Intersystem crossing (ISC), 124 Intraorbital interference factor, 176 Intrinsic reaction coordinate (IRC), 104, 112 Inversion barrier, 56, 57, 58 Ionization potentials (IP), 151, 155, 162, 175 Irreversible thermodynamics, 254 Isobutane, 172 Isobutene, 172 Isodesmic reaction schemes, 152, 158, 179 Isomeric effects, 193 Isopropanol, 174 52, 152, 179 Jaguar, 153 jahn-Teller, 56 Jahn-Teller degeneracy, 102, 105 Jahn-Teller distortion, 169 Journel of Physical Chemistry, 220 Ketene, 173 Ketones, 194 Kinetic energy operator, 3 Kinetic repulsions, 12, 18 Kohn, Walter, 286 Kohn-Sham (KS) density functional theory, 1, 4 Kohn-Sham determinant, 23 Kohn-Sham Hamiltonian, 9, 25 Kohn-Sham kinetic energy, 7 Kohn-Sham molecular orbital method, 75 Kohn-Sham orbitals, 5 , 23 Kohn-Sham potential, 3, 4 Koopmans’ theorem, 240 Kyushu University, 242 Landau-Zener model, 89 Laplacian, 6 Laplacian of electron density, 262 Laurentian University, 229, 232 Lava1 University, 225, 231, 246 Li,, 22, 170 LiF, 170 LiH, 170

Subject Index 319 Linear scaling methods, 245 Lithium molecule, 21, 32 LOADER, 197 Local chemical bond, 34 Local density approximation (LDA), 3, 6, 51, 52 Local density gradients, 180 Local DFT, 119 Local potential, 3 Local spin density functional, 180 Localized exchange-correlation hole, 10 Low energy electron diffraction, 240 Lowdin, P.-O., 271 LY functional, 180 Magnetic field effects, 252 Magnetic fields, 245 Many-body problem, 2 Many-electron correlation problems, 250 Marcus, Rudolph A., 285 Markov chain, 247 Mass-weighted Cartesian coordinates, 136 Mass-weighted internal coordinates, 116 Massively parallel computer systems, 148 Maxima, 96 McGill University, 213, 218, 224, 225, 227, 230,232,253 McKoy, Vincent, 285 McMaster University, 220, 227, 231, 232, 244 Mechanistic organic photochemistry, 121 Memorial University, 231 Methane, 149, 150 Methyl cyanide, 173 Methyl ethyl ether, 174 Methyl formate, 173 Methyl nitrite, 173 Methyl silane, 173 Methylamine, 173 Methylene cyclopropane, 172 Methylenecyclopentene diradical, 134 Methyllithium, 65, 66, 70, 74 Methyllithium oligomers, 73 MINDO, 152, 183,184 MINDO/l, 183 MINDOR, 183 MIND0/3,152,183,184 MINDOW2,183 Minima, 96 Minimum energy path (MEP), 88, 112, 130 Missing groups, 197 Mixed state dynamics, 120

MNDO, 109 MNDO/d, 183 Modified Neglect of Diatomic Overlap (MNDO), 38,183 MOLCAS, 109 Molecular clusters, 253 Molecular dynamics (MD), 237, 256 Molecular engineering, 245 Molecular mechanics (MM), 110, 152 Molecular mechanics valence bond (MM-VB), 110,119 Molecular orbital (MO) model, 2, 152, 155, 162 Molecular shape complementarity, 245 Molecule-dependent parameters, 156, 178 Molecule-independent parameters, 156, 201 Msller-Plesset calculations, 38 Msller-Plesset (MP) perturbation theory, 38, 74, 158, 165 MOLPRO, 153,161 Monte Carlo simulations, 247 MOPAC, viii, 153 Mount Saint Vincent University, 232 MP2 method, 109, 163 Mulliken population analysis, 27, 68, 129 Mulliken, Robert S., 235 Multicoefficient correlation method (MCCM), 178 Multiple funnel, 127 Multiple spawning method, 120 Multireference configuration interaction (MRCI), 109 N,O, 172 NaC1, 171 National Research Council of Canada (NRC), 214,216 National Resource for Computational Chemistry (NRCC), oiii National Sciences and Engineering Research Council of Canada (NSERC), 215,224 Natural population analysis (NPA), 65, 73, 74 NCCN, 15,17,23,30, 35,44 NCNC, 23 Neglect of diatomic differential overlap (NDDO), 181 NF,, 172 NH, , 170 NIST, 197 NIST Standard Reference Database, 197 Nitrogen molecule, 7, 36 Nitromethane, 173 NMR chemical shifts, 236

320 Subiect Index NMR relaxation, 238 NMR spin dynamics, 255 NO, 171 NO,, 174 Nobel Prize in chemistry, 214, 219, 221, 286 Nodal patterns, 10 Nonadiabatic coupling effects, 96 Nonadiabatic events, 104, 119 Nonadiabatic radiationless decay, 89 Nonadiabatic reaction path, 92 Non-interacting electrons, 6 Non-transition-metal elements, 164 Noncrossing rule, 91, 96, 97, 99 Nonequilibrium statistical mechanics, 248, 254 Nonlinear dynamics, 248, 254 Nonlinear optics, 244 Nonlocal corrections, 237 Nonlocal DFT, 3, 38, 58 Nuclear attraction, 9 Nuclear dynamics, 120 Nuclear field, 9 Nuclear magnetic resonance (NMR), 236, 238,254,255 Nuclear potentials, 5 Nuclear quadrupole resonance spectroscopy, 254 Nuclear spin relaxation, 256 Nucleus-nucleus Coulombic repulsion, 21 NWCHEM, 153

0,, 171 0 3 ,172 Octatetraene, 93, 103, 122 One-electron wavefunction, 4, 5 Optical data storage, 88 Orbital interaction, 23, 40 Orbital interaction diagram, 29, 32,45, 61, 67,70 Organic molecules, 123 Organic photochemistry, 92 Organolithium chemistry, 65 Organolithium oligomers, 65 Overlap integral, 19, 176 Overlap repulsion, 12 Oxford University, 247 Oxirane, 173 P,, 171 Pair bond wavefunction, 30 Parameterized correlation method (PCI-X), 178

Parametric method number 3 (PM3), 152, 183,184,201 Pattern formation, 248 Pauli antisymmetry principle, 12 Pauli exclusion principle, 14, 20, 21, 55 P a d repulsions, 4, 12, 14, 23, 29, 34, 47, 49, 54, 55 PCCP, 35,39,44 PCI-X, 152 PCMODEL, 153 Peaked conical intersections, 103 Pentadienyl radicals, 123 Percolation theory, 249 Perdew-86 functional, 52 Perdew-Burke-Ernzerhof exchangecorrelation functional, 181 Perdew-Wang-91 exchange-correlation functional, 52 Permanganate ion, 26 Permutation operator, 16 Personal computers (PCs), 148 Perturbation, 4, 250 PF3, 172 Phase transitions, 255 Phopholipid bilayers, 238 Photobiological systems, 88 Photochemical mechanisms, 108 Photochemical reaction coordinate, 139 Photochemical reaction path, 95 Photochemical reactions, 87, 89 Photochemistry, 87, 95 Photoelectron spectroscopy, 10, 35 Photoisomerization, 129, 130, 137 Photolysis, 128 Photophysical process, 89 Physics in Canada, 2 15 Pi bonding, 47 Pi-electron approximation, 217 Planck, Max, Institute, 258 PM3,152, 183,184,201 Polar bonds, 66 Polar molecules, 65 Polarization, 24, 28 Polarization functions, 158, 159, 165, 169 Polyaromatics, 194 Polyatom, 269 I’olyatomic molecules, 96, 214 Polyene radicals, 105, 122 Polyenes, 92, 121, 122 Polymer adsorption, 249 Polysaccharides, 249 Pople, John, vit

Subject Index 321 Potential energy surface (PES), 88, 92, 94, 96, 110,178 Potential optimized discrete variable representation, 237 PPDS (Physical Property Data Search), 197 Primitive basis functions, 27 Princeton University, 256 Product formation, 89 Projection operator, 112 Propane, 172, 188 Propyl chloride, 174 Propylene, 172 Propyne, 172 Protein complexes, 88 Protein folding, 246 Protein structure prediction, 238 Proton affinities, 151, 155, 162, 175 Protonated imines, 129 Protonated Schiff bases, 129, 137 Pyramidalization, 56, 62, 71 Pyrazine, 194 Pyrazoline, 124 Pyridazine, 194 Pyridine, 174, 194 Pyrimidine, 194 Pyrrole, 174 Q-Chem, 153 QCPE Bulletin, vii Quadratic configuration interaction, 157, 159, 163,165 Quadrature discretization method (QDM), 240 Qualitative MO theory (QMO), 4, 11 Quantum chemical methods, 155 Quantum chemical programs, 153 Quantum chemistry, 1,152,216,240 Quantum Chemistry Program Exchange (QCPE), v, vii, x i Quantum interference effects, 249 Quantum yields, 119, 121, 136, 140 Quasi-ergodic problems, 247 Queen’s University, 220, 224, 225, 227, 228, 231,232,233,255 Radiationless deactivation, 90 Radiationless decay, 90, 121, 134 Radiationless decay channel, 92 Radiationless transitions, 104, 121 Radicals, 155, 162, 168, 175, 177, 182, 184 Reaction coordinate, 90, 95 Reaction dynamics, 88 Reaction enthalpy, 148, 189

Reaction kinetics, 256 Reaction mechanisms, 87 Reaction path, 88 Reaction path branching, 95 Reaction-diffusion equations, 248 Real crossings, 90, 121 Real surface crossing, 91 Reduced density matrices, 255 Relativistic effects, 157 Relaxation channels, 114 Relaxation paths, 108, 116, 118 Renner-Teller degeneracy, 102, 111 Renormalization group methods, 255 Resonance formula, 132 Resonance integral, 31 Response theories, 108 Retinal chromophore, 90 Retinals, 88 Ring strain, 190, 193, 195, 196 Rotation, 151 Royal Military College of Canada, 230 Rydberg orbitals, 217 Saddle points, 92, 95, 96, 110, 112 Saint Mary’s University, 228 Salahub, Dennis, 216,221, 282 Salem, Lionel, 272 Sandorfy, Camille, 216, 218, 223 Scaling all correlation (SAC) energy, 177 Scaling methods, 177 Schiff bases, 121, 129, 137 Schlegel, H. Bernard, 286 Schrodinger equation, 3, 97, 152 SciFinder, xi Search engines, xiii Second-order group contribution method, 193 Second-order Jahn-Teller effect, 56 Second-order Mraller-Plesset perturbation theory, 38, 105, 158 Second-order saddle point, 39 Secular equation, 98 Self-consistent field (SCF), 140 Self-interaction correction, 9 Semi-ab initio method 1 (SAMl), 183 Semiclassical trajectories, 95, 108, 118 Semiempirical calculations, 4 Semiempirical molecular orbital methods, 11, 109,152,154, 181,201,254 Senftleben-Beenakker effect, 238, 252 Shape analysis, 245 Shull, Harrison, v, x Si,H,, 171 SiCl,, 172

322 Subject Index Sit.’,, 168, 172 SiH,, 170 Simon Fraser University, 220, 226 Single-determinental wavefunction, 4 Singlet-triplet intersections, 90 Singly occupied molecular orbitals (SOMO), 13, 33, 34, 40, 132 Slater determinant, 4, 19 Slater, John C., 235 Slater-Condon rules, 30 Sloped conical intersection, 94, 95, 103, 110, 120 Smith, Vedene, 220 SO,, 155, 171 Software, v, 108 Software piracy, xii Spartan, 153 SPECTROS, 245 Spectroscopy, 2 16, 243 Spin contamination, 164, 169, 178 Spin-orbit effects, 157 Spin-orbit energy corrections, 164, 167 Spin-projected Marller-Plesset theory, 163 Spin-restricted Hartree-Fock (RHF), 158 Spin-spin coupling constants, 237 Spin-unrestricted Hartree-Fock (UHF), 158 Spiropentane, 172 Split-valence plus polarization basis, 161 Standard enthalpies of formation, 148, 149 Standard states, 151 Statistical mechanics, 238 Steepest descent, 110, 114 Stereochemistry, 248 Steric crowding, 23 Steric effects, 195 Steric repulsion, 12, 14, 18, 28, 55, 64 Stewart, James J. P., viii STN databases, xii, xiu Stochastic theory of chemical rate processes, 248 Styrene, 110 Sulfides, 194 Supercomputer, 148 Superminicomputers, viii, 275 Surface chemistry, 250 Surface crossings, 89 Surface hop, 104, 119, 134 Surface-hopping method, 96 SVWN functional, 180, 182 Swiss Federal Technical Institute (ETH),255 Symmetrized fragment orbital (SFO),28 Symmetry, 245 Symmetry breaking, 251

Terminology, xiii Test sets, 154 Theoretica Chimica Acta, 253 Theoretical chemistry, 213, 215 Theoretical Chemisty Accounts: Theory, Computation, and Modelling, 253 Theoretical predictions, 155 Theoretical thermochemistry, 148 THERWEST, 197 Therrnochemistry, 147 Thermodynamic cycle, 150, 151 Thiirane, 173 Thiophene, 174 Three-electron bond, 4, 49 Three-electron conical intersections, 122 Time-dependent density functional theory (DFT), 108, 140 Time-dependent equation-of-motion (EOM), 140 Time-dependent Schrijdinger equation, 119, 120 Toxicological risk assessment, 245 Trajectory-surface-hopping-algorithm,119 Transition metal clusters, 236 Transition metal (TM) complexes, 10, 20 Transition state, 88, 101, 164 Transition structure, 11 1, 114 Transition vector, 114 Translation, 151 Transport properties, 252,254 Trent University, 233 Trial wavefunction, 12 Trimethylamine, 174 Two-center, one-electron bond, 34 Two-center, three-electron bond, 34 Two-center, two-electron bond, 34 Two-electron integrals, 154 Ultrafast laser techniques, 95 Ultrafast photochemical processes, 95 Ultrafast radiationless decay, 122 Umbrella sampling, 247 Unavoided crossing, 91 Unimolecular reactions, 246 Unimolecular rearrangements, 37 University of Alberta, 218, 224, 225, 228, 232,241 University of British Columbia, 220, 224, 225,226,227,229,230,233,238,247 University of Calgary, 226, 227, 228, 229, 230,241 University of Chicago, 214, 242 University of Guelph, 228, 229, 230, 231

Subject Zndex 323 University of Illinois, 256 University of Lethbridge, 230, 232, 233 University of Manitoba, 227, 228, 231, 232 University of Moncton, 229 University of Montreal, 216, 218, 223, 225, 226,230,231,232,233,236 University of Nancy, 246 University of New Brunswick, 225, 230, 231 University of Northern British Columbia, 233 University of Ottawa, 220, 225, 231, 233, 244 University of Saskatchewan, 214, 225, 230, 245 University of Sherbrooke, 228, 231, 246 University of Szeged, 216 University of Toronto, 213, 220, 224, 225, 226,228,229,231,233,246 University of Victoria, 227 University of Waterloo, 220, 225, 227, 228, 229,230,250 University of Western Ontario, 224,226,227: 228 University of Windsor, 227 University of Winnipeg, 231 Unrestricted Hartree-Fock (UHF) method, 178 Valence bond (VB) structures, 30,42 Valence bond (VB) theory, 34,49, 110, 238 Valence electrons, 159, 187, 217 Valence orbitals, 13, 19, 20 van der Waals dimers, 253 Vaporization enthalpies, 192

Variational principle, 2 VAX 111750 computer, 275 Vertical excitation energies, 109 Vinyl chloride, 173 Vinyl fluoride, 173 Voronoi deformation density (VDD), 66, 73

W1 method, 152, 157 W2 method, 152, 157 Wahl, Chris, 218 Walden’s rule, 200 Waldmann-Snider equation, 238 Water molecule, 20 Wave mechanics, 2 Wavefunction propagation, 119 Websites, 153 Whitehead, M. Anthony, 218 Wilfred Laurier University, 23 1 X-alpha functional, 52 X-alpha method, 3, 6,236 York University, 226,228, 232, 233 Zeolite channels, 255 Zerner, Michael C., x , 233, 274 Zero point correction, 159 Zero point energy, 70, 124, 151, 158, 184 Zero point vibrational energy, 57, 176 Zero-differential-overlap (ZDO), 181, 217 ZINDO, 109