Preface Dear Reader, Thank you for your decision to spend some time on reading this book, which has a little to do with quantum mechanics and much more with the subject of thermodynamics. The first is considered as extremely complicated by many non-physicists, but I promise that you need not be afraid of the small amount of quantum mechanics in this book. Thermodynamics may be less complicated than quantum mechanics, but it is often considered as dry and boring stuff. Indeed, during my own study of theoretical physics in GSttingen I considered thermodynamics as the most boring of the four fundamental theory courses on mechanics, electrodynamics, thermodynamics, and quantum mechanics. For me quantum mechanics was much more exciting, and hence I enjoyed working on solid state quantum mechanics in my diploma and Ph.D. theses, and I appreciated even more being able to study and practice molecular quantum mechanics and quantum chemistry during the 12 years when I worked in the computational chemistry department of Bayer AG in Leverkusen. However, although I learned parts of quantum chemistry on the job, I still lack a rigorous and systematic education in this field. Although, meanwhile I hope to have a reasonable knowledge of the subject, this lack may show in some places in the first chapters of this book. I ask the experts for their pardon if they notice any defects. While caring for the calculation of important physicochemical properties based on quantum chemical methods, I realized that I got into central regions of thermodynamics without really noticing it - I just memorized as little of thermodynamics as I unavoidably needed for the next step. Indeed, I often hardly recognized the relationship of the rather empirical thermodynamics notations used in physical chemistry to the bare theoretical framework I had been taught in the theoretical physics course. My confusion and frustration was even greater when I worked on chemical engineering thermodynamics and ix
x
Preface
noticed that they use a different language again. Meanwhile, I am confident that I have found a robust framework of thermodynamics that satisfies the needs of the topics I work on in the border region of physics, chemistry, and chemical engineering, and a language that can be reasonably understood for all of them. Nevertheless, I am sure that some thermodynamics experts from the different areas will disagree with the notations and conventions used in this book. Since I do not see a way to satisfy them all, I must ask them to be patient at this point. The situation is similar regarding my chemical knowledge. Because there is a large distance between the chemistry campus and the physics institutes in GSttingen I finished my study of physics with the minimum of chemistry required. Indeed, I also was not much interested in the kind of "cooking" taught to us on occasion. However, during my work at Bayer I learned about molecules and their properties, and I am able to understand parts of chemistry and to discuss it with chemists in their language. Even here, my lack of a systematic scientific education in this field may show to the experts. I apologize for potential insufficiencies in that regard. I am an autodidact in several areas that have to do with the work described in this book. This has been very helpful for finding some of the major methodological steps, because I was not biased by the knowledge of all the work that had been done on related topics by others before. By looking at the problems from the perspective given by my personal education and experience, I have been successful in finding novel and useful approaches twice or three times during the development of COSMO and COSMO-RS. I know that I have been quite lucky in that regard and I do not wish to preach that one should forget about the work of previous researchers and rely on your own ideas. Science would definitely not make continuous progress if everyone worked in this way. Also, the work presented in this book depends on achievements in quantum chemistry made by generations of theoretical chemists, and on the fundamental concepts of thermodynamics mainly introduced by Boltzmann more than 150 years ago. However, from time to time, it can definitely help to look at a problem with the fresh and unbiased mind of a newcomer. Originally I was educated in the belief that scientific articles and textbooks should be objective and hence that any personal
Preface
xi
aspects and sentiments in them are inappropriate. Meanwhile, I have seen the extent to which many scientific developments and approaches are influenced by subjective experience and knowledge. Science itself has to be objective in that the same results must be reproducible independent of the scientist, but I think that the development of science is driven by individual, subjective ideas. Since the description of the motivation leading to a new scientific development is a common and essential part of scientific writing, I am no longer convinced that it is useful to hide these personal ways of thinking behind anonymous pseudo-objective phrases. So many articles have been written in an objective style that e x p r e s s e s - intentionally or n o t - very subjective views. Therefore, I much prefer to describe the motivation for scientific developments in a personal and more vivid style, and thus I aim to use such a style throughout this book. I know that some scientists, among them some very good friends, dislike extremely such subjectivity, and I ask them for their patience. This is the first book I have written, and this may cause some insufficiencies and imperfections. I nevertheless hope that readers will enjoy reading this book, and that many will consider it useful. Finally, I wish to thank several people who have collaborated with me during my development of COSMO and COSMO-RS. My former and present colleagues Volker Jonas, John Lohrenz, Frank Eckert, Martin Hornig, Michael Diedenhofen, and Karin Wichmann must be mentioned specially here. I express my thanks to my wife Birgit and my children Angela, Johanna, and Lukas for their patience with my scientific work, which sometimes took much more of my time and of my mind than it should have done.
Chapter 1
Introduction Following the development of quantum theory by Heisenberg [1] and SchrSdinger [2] and a few further discoveries, the basic principles of the structure of atoms and molecules were described around 1930. Unfortunately, the complexity of the SchrSdinger equation increases dramatically with the number of electrons involved in a system, and thus for a long time the hydrogen and helium atoms and simple molecules as H2 were the only species whose properties could really be calculated from these first principles. In 1929, Dirac [3] wrote: The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. In the 1950s, theoretical chemists started to take advantage of evolving computer technology and were able to achieve approximate solutions for the wave functions and electron densities of other small molecules. The rapid development of computer capabilities as well as of efficient theoretical approximations and of computer codes, allowed the science of quantum chemistry to make enormous progress in the following decades. As representatives of a large number of brilliant researchers, John A. Pople and Walter Kohn were awarded the Nobel Prize for their achievements in quantum chemistry in 1998. However, despite rapid progress, quantum chemistry remained an academic science for about 40 years. Even by using large supercomputers it was only possible to calculate the well-known properties of two or three atomic molecules with acceptable accuracy, while real predictions for new molecules of industrially relevant size were impossible. The assessment that quantum chemistry is unable to
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treat real-world problems became so manifest during these years that many in the chemical community did not realize the change in this situation that took place around 1990. This change can be mainly attributed to the development of gradient corrected density functionals, which allowed the application of density functional theory (DFT) to molecular systems [4,5], after it had been applied successfully in solid-state theory about 10 years before. At about the same computational cost as for the meanfield theory of Hartree and Fock (HF) [6,7], DFT takes the electron correlation into account quite accurately and thus provides a large increase in accuracy over the HF level. In its most efficient implementations, DFT today allows for goodquality ground-state geometry optimizations and property calculations of molecules up to 40 atoms within a day or less on inexpensive PC hardware. By efficient combination of DFT geometry optimizations and frequency calculations with higherlevel single-point calculations, e.g., coupled cluster calculations, reaction enthalpies can be calculated for industrially relevant compounds, with chemical accuracy in a few days on standard parallel computer systems. Thus, quantum chemistry has achieved a state where it can provide useful information about new molecules at lower cost and in shorter time than experiments can do. It has the potential to be a predictive tool, and the speed of computer development will rapidly expand its predictive power in the foreseeable future. The reader interested in more details of the development of quantum chemistry should read one of the many textbooks and reviews~an excellent, recent one being that by Levine [8]. Despite the tremendous progress made in this field, there is still a severe drawback. The quantum chemistry developed by theoretical chemists tools are primarily suited for isolated molecules in vacuum or in a dilute gas, where intermolecular interactions are negligible. Another class of quantum codes that has been developed mainly by solid-state physicists is suitable for crystalline systems, taking advantage of the periodic boundary conditions. However, most industrially relevant chemical processes, and almost all of biochemistry do not happen in the gas phase or in crystals, but mainly in a liquid phase or sometimes in an amorphous solid phase, where the quantum chemical methods are not suitable. On the one hand, the weak intermolecular forces,
Introduction
3
i.e., dispersive, or van der Waals (vdW) forces, which play an important role in condensed systems, require a very high level of quantum theory for an appropriate representation, and DFT is not sufficiently accurate for this. On the other hand, the representation of condensed liquid systems by large ensembles of molecules converges only very slowly with an increase in ensemble size, even if artificial periodic boundary conditions are applied to avoid artifacts arising from the surface. Together with the fact that DFT calculations still scale at least quadratically with the system size, and that higher-order methods are even more demanding, there is no realistic prospect that quantum chemistry alone will be able to solve practical problems of liquidphase chemistry in the near future. Even the well-known Car-Parrinello method [9], which is a combination of approximate DFT and molecular dynamics simulation with periodic boundary conditions, still requires external constraints, and yields only short snapshots of relatively small condensed ensembles of molecules~and with very high computational costs. Thus, this approach may be helpful in generating some qualitative insight into mechanisms acting in liquid systems, but it is far from being a predictive tool for practical chemical questions. The situation of computational chemistry for the liquid phase can be illustrated by the pictorial analogy as shown in Fig. 1.1 that we will use from time-to-time throughout this book. In Fig. 1.1 we compare the vacuum with the south pole of a planet. Powerful quantum chemical methods have been developed for molecules in the clean environment of the south pole and its close surrounding, i.e., for molecules in vacuum or in dilute gases. However, almost all the continents and islands on this planet are far away from the south pole, mostly in the northern hemisphere. Hence, the practical value even of the most powerful quantum chemical methods is quite limited, unless viable methods are found to make use of quantum chemistry in the dirty environment of the northern hemisphere. The most obvious way to achieve a theoretical description of molecules in the liquid state is the path from single molecules, through small and large clusters of molecules, to very large ensembles of molecules. Even at times when realistic quantum chemical simulations were impossible for molecular sizes of
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Fig. 1.1. Schematic illustration of the different computational approaches for liquid-phase property prediction.
practical relevance, people started in this direction by developing classical approximations for the intra- and intermolecular interactions, often called force-field (FF) or molecular mechanics (MM) methods, instead of the much more appropriate, but impractical quantum theoretical description. The large number of parameters appearing in the sophisticated functional expressions of such force fields for bond lengths, bond angles, torsion angles, electrostatic forces, dispersive forces, and hydrogen bond interactions have been adjusted mainly to experimental data, and sometimes even to high-level quantum chemical results for small molecules in the last 15 years. Liquid states are, to a large degree, stabilized by the entropy of configurational disorder, and fluid-phase simulations must take this disorder into account, either by averaging the relevant properties over a sufficiently long time in a molecular dynamics (MD) simulation of a system that is representative for the liquid system [10] or by taking thermodynamic averages over a large number of randomly chosen configurations of such a system, the so-called Monte Carlo (MC) simulations [11,12]. For
Introduction
5
thermodynamic equilibrium simulations, where dynamic information is not requested, the latter technique has turned out to be more efficient. Powerful algorithms have been developed to sample the thermodynamically relevant configurations efficiently. Thus, using large supercomputers, very reliable MC simulations can now be performed for alkanes and other non-polar liquids and mixtures. However, for most classes of other compounds such simulations still suffer from the inaccuracies of the classical force fields required for the evaluation of the intermolecular interactions [13-15]. Since the basic idea of the MC and MD calculations is quite simple, and has been known for a long time, we will compare these techniques with rowing boats. People started to use them long ago, and some discoverers have reached remarkably distant coasts with such boats, but it is questionable whether they will ever become the method of choice for routine trips to the latitudes of solvation. Since the need for predictions of thermodynamic properties in the fluid phase is so strong, several more pragmatic physical and organic chemists, and also some chemical engineers, tried other pathways for property predictions in the liquid state. The most successful approach evolving from these attempts was the idea of group-contribution methods (GCMs) that was developed independently by chemists and by chemical engineers. The chemists, Hansch and Leo [16] introduced the idea that the free energy of transfer, and hence the logarithmic partition coefficient of a molecule between two phases A and B, can reasonably be approximated as a sum of contributions from the structural atom groups of which the molecule is composed. With a plausible definition of groups, and after fitting group-contribution values (or increments) to large sets of available experimental data, such methods can be used to predict partition coefficients of new compounds. On the basis of a wealth of experimental octanol-water partition coefficients for about 20,000 compounds, the calculated logarithmic (octanol-water) partition coefficient (CLOGP) method of Hansch and Leo [ 17] for the prediction of the octanol-water partition coefficient has been developed to highest perfection by adjusting some thousands of parameters. CLOGP and related methods are widely used in drug design and in environmental studies to assess the lipophilicity of new compounds. Less elaborated, but still useful, GCMs are in use for
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many other thermodynamic properties in these areas of research. However, none of these has the chance to become as elaborate as CLOGP, because for no other partition coefficient is there even approximately as much experimental data available. Since in these approaches only the solute is decomposed into groups, while the solvent phase and the temperature are kept fixed, we will refer to these methods as "one-dimensional group-contribution methods." In contrast to physical and organic chemists, who are often pleased with an estimate of a partition property for fixed solvent phases and temperature, chemical engineers typically require predictions of thermodynamic properties in fluid phases of variable composition and at variable temperature. Thus, the one-dimensional GCMs are quite useless for them. Instead, they have developed the concept of two-dimensional GCMs, where solute and solvent molecules are treated on the same footing and both are described as ensembles of groups. As a consequence, it is necessary to fit a squared matrix of group-interaction parameters instead of a single value per group. Typically, these groupinteraction parameters are interpreted as a kind of surface interaction energy of groups in a lattice model of fluids. After combining this idea with approximate concepts for the thermodynamics of lattice fluid models the large number of parameters had to be fitted to experimental data, before predictions for new compounds or mixtures could finally be made on the basis of such models. By making these parameters temperature-dependent, which typically required two additional parameters per pair of groups, reasonable predictions became available at variable temperatures. Much effort has been put into the development of such GCMs for about 30 years now, and as a result the most elaborate of these methods, which are different implementations of UNIFAC [18,19] and ASOG [20], represent the state of the art for structure interpolating thermodynamic property prediction in the liquid phase in chemical engineering, since about 1990. All GCMs suffer from two major problems of principle. First, they can only handle molecules and mixtures for which all required group parameters or group-interaction parameters have been fitted before. This limitation is more severe for the twodimensional methods because parameters for all combinations of groups have to be available. This is more unlikely than just
Introduction
7
having parameters for each group available separately. In these methods, more exotic groups can be less simply parameterized, because the parameterization procedure requires a large number of data for the combinations of groups, and these are often not available. The second, and perhaps even more severe limitation of GCMs, arises from their basic assumption that a functional group has approximately the same interaction properties in different chemical environments. Although being reasonably correct to a first order, this concept completely disregards the changes of the functional group properties that can arise from the intramolecular environment by electronic push-pull effects, or by intramolecular hydrogen bond formation, or by steric effects. As an example, a nitro group attached to an aromatic ring, which would typically be considered as a functional group in GCMs, will normally be in-plane with the aromatic ring and hence pull electrons from the ring by conjugation. But the polarity of the nitro group may change considerably if it is hindered from being in-plane by a bulky group in the ortho position. Alternatively, an electron-donating group such as the amino group, especially in the para position, may considerably increase the polarity of the nitro group, as shown in Fig. 1.2. Conversely, the oxygens of the nitro group in p-dinitrobenzene are much less polar than in nitrobenzene, but now the nitrogen atom takes a considerable opposite polarity, as shown in Fig. 1.2. Finally, a hydroxyl group in an ortho position may form a quite stable intramolecular hydrogen-bond 6-ring, as shown for the case of salicylic acid in Fig. 1.3. As a result, the polarity and
Fig. 1.2. COSMO surface polarity of nitrobenzene, 4-amino-nitrobenzene, and p-dinitrobenzene ( r e d - negative, g r e e n - neutral, blue = positive polarity). Due to intramolecular electron redistributions the oxygen atoms of the nitro group are much more polarized in 4-aminonitrobenzene, and much less polarized in p-dinitrobenzene.
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Fig. 1.3. COSMO surface polarity of salicylic acid, with and without an intramolecular hydrogen bond. accessibility of the carboxy group and of the hydroxyl group would be modified dramatically. Not all intramolecular effects of these kinds can regularly be taken into account in GCMs, leading to a general limitation of their accuracy and general applicability. To some degree this has been overcome in one-dimensional GCMs by more and more specialization in the definition of groups, and by the introduction of cross-corrections for special intramolecular interaction situations, leading to more than a thousand group types and more than 2000 adjusted parameters in a method such as CLOGP. However, even with this large number of parameters, only standard types of intramolecular interactions can be taken into account by mean corrections. In two-dimensional GCMs such a degree of sophistication is generally impossible, because it would lead to an exploding demand for interaction parameters. Thus, the concept of group contributions leads to a principal limitation in the resolution of intramolecular effects, and hence to a limitation in the resolution of isomer differences, arising from the differences in the positions of groups in similar molecules. In a sense, molecules just do not like to be cut into pieces. In our global figure we compare the GCMs with the use of rockets, which can fly tremendously fast from the south pole to any point on the globe. But the molecules do not like the extreme conditions in the rockets and degenerate into ensembles of groups, and there is no way to recover the full information about the molecules once they have degenerated under this brutal treatment. This book is mainly about what we might call a third method of transportation from the south pole of isolated molecules to the latitudes of solvation. Indeed, it will be a two-step method, using a
Introduction
9
kind of modern long-distance airplanes for the flight from the south pole to the north pole, and flexible small planes for the connection from the north pole to any point in the northern hemisphere. The long-distance airplanes in this picture are known as dielectric continuum solvation models (CSMs). Here, we will specially consider one kind of these models, the Conductor-like Screening Model (COSMO) that I and some coworkers invented in the early 1990s [C1]. The small commuter planes connecting the north pole with almost any place in the solvation region represent a special thermodynamic treatment of the molecular interactions, which is based on the information gained from the COSMO calculations. When finding this method I gave it the name COSMO for Realistic Solvation (COSMO-RS) [C9], although meanwhile I like the name COSMO thermodynamics (COSMOtherm) much better. But since it is hardly possible to change the name of an already published method, COSMO-RS remained the name of this thermodynamic extension of the COSMO approach, and COSMOtherm has become the name of our COSMO-RS program. As seen in Fig. 1.1, and as I will explain in detail in this book, the dielectric continuum solvation model COSMO and the subsequent COSMO-based thermodynamics COSMO-RS are two clearly separable and very different steps. However, I have found that many researchers in this field refer to both methods as "COSMO," which is both inaccurate and confusing. To avoid this confusion, I find it necessary to emphasize the importance of using the correct notations~COSMO and COSMO-RS-for these methods in all discussions and written literature on these subjects. In this book I will try to explain the different steps of the entire COSMO-RS method in detail, and give various examples of applications. For this purpose, we will start with the COSMO method in the next chapter.
Chapter 2
Dielectric continuum solvation models and COSMO
2.1
THE BASIC IDEA AND ITS DEVELOPMENT
In 1920, Max Born, a Nobel Prize winner, .published some work on the free energy of solvation of ions, AG~~ [21]. He conceived the idea of approximating the solvent surrounding the ion as a dielectric continuum. Defining a spherical boundary, between the ion and the continuum by an effective ion-radius, R ran, he got the simple result AV~On
t:S- 1 Q i~ eS 2Rio n
(2.1)
Here, we introduce the convention of denoting the solvent by a suffix, and the solute by a superscript index..Hence, es stands for the dielectric constant of the solvent S, and Qmn is the total charge of the ion. Surely, Born was aware that this is a crude approximation, but his formula led to a qualitative understanding of the experimentally observed values of solvation energies. Indeed, it was later used to define ionic radii R i~ and thus it became in some way a self-fulfilling prophecy. Born's idea of the dielectric continuum solvation approximation became very popular, and many researchers worked on its further development. Hence a brief overview of the most important development steps will be given, but it is impossible to mention all the different modifications and all workers who have been contributing to this field. Readers who seek a broader overview are referred to some reviews on continuum solvation methods, e.g., by Cramer and Truhlar [22] or by Tomasi and Persico [23]. The goal of the history given here is to enable 11
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the reader to understand the conductorlike screening model (COSMO) method and its relationship to earlier and other continuum solvation methods. Born's idea was taken up by Kirkwood and Onsager [24,25], who extended the dielectric continuum solvation approach by taking into account electrostatic multipole moments, M x, i.e., dipole, quadrupole, octupole, and higher moments. Kirkwood derived the general formula: ~c
1
AG~ -
X2
Mt
2 E f l(es)Rx~---------~
(2.2)
/=1
with
fl(e) -
e+xl
(2.3)
and xl =
1-1 1
(2.4)
In the above expression we can see the general structure of the energetic response in a dielectric continuum: all contributions are squared with respect to the solute multipole moment, because the multipole moment is, on the one hand, the source for a reaction field, and on the other it interacts with the reaction field. The factor -89 in front of all terms arises from the fact that in any linear response theory, one-half of the interaction energy of the source with the medium is required for the generation of the response, i.e., in this case for the polarization of the continuum. We can also see that the higher multipole contributions decrease with a high power of the inverse radius of the sphere chosen as interface with the continuum. Therefore, as a rule of thumb, it is mostly sufficient to take into account the highest non-vanishing multipole moment. Finally, all contributions scale with a simple function of the dielectric constant, es, that rapidly approaches unity for higher dielectric constants. Kirkwood also introduced the expression of the reaction field (RF) for the general dielectric polarization response. Since the first electrostatic moment, M x, is the total charge, the first term in Kirkwood's multipole expansion is identical with Born's result for ions. The second term, i.e., the
Dielectric continuum solvation models and COSMO
13
dipole term 1 e s - 1 p x~ AV~=2es+lRXa
(2.5)
has become well known as Onsager model for the solvation energy of dipolar molecules. It became widely used for the estimation of solvation energies of dipolar molecules, and was used later for models for solvatochromic effects [26,27]. Here, px is the dipole moment of the solute molecules. The main reason for stopping at the dipole contribution has most probably been the fact that dipole moments of molecules can be reasonably inferred experimentally from the gas-phase dielectric constant and its temperature dependence, while experimental information about higher electrostatic moments of molecules was not available at that time. Only later, when calculated multipole moments became available from quantum chemical methods, could Kirkwood's formula be exploited beyond Onsager's dipole level. So far, the multipole moments generating the electrostatic field of the solutes have been treated as fixed properties of each solute X. While for the monopole moments, i.e., for the ion charge Qion, this is not a problem, the higher moments are changed by the RF of the solvent, and thus it is not really obvious which values should be used in RF calculations. In the 1980s, therefore, researchers started to incorporate the RF method into the developing quantum chemical programs, and took into account the effect of the back-polarization of the solute by the solvent selfconsistently, which typically increases the solute-solvent interaction by ~20%. This can be done quite simply in an iterative way by using an initial guess of the molecular dipole moment, typically the quantum chemical gas-phase result, for a first quantum chemical RF calculation, and then iteratively taking the calculated dipole moment from the previous RF calculations as input for the subsequent calculation, until convergence is achieved. These models have come to be known as self-consistent reaction field (SCRF) models [22,23,28]. More efficient implementations have been worked out, which included the RF directly into the central quantum chemical equations (e.g., by addition to the Fock operator). Initially, there was some discussion about how to do this in a theoretically most accurate way. Using SCRF models
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on a semi-empirical quantum chemical level, Zerner and his co-workers considered different variants in their pioneering work on the calculation of solvatochromic effects, i.e., on the shift of the absorption spectra of dyes and other chromophores owing to solvation [29]. For them, an efficient direct integration of the RF in the quantum chemical kernel was important because they needed to take the RF into account in the configuration interaction (CI) calculations required for the calculation of excited states. Iterative solutions as usually used for groundstate calculations are neither feasible, nor theoretically consistent in the context of CI. The calculation of solvatochromic effects raised the problem of how to treat non-equilibrium solvation on the SCRF level. A dielectric solvent can only be expected to relax fully with its reorientational and electronic degrees of freedom in the long-living initial state of the solute, typically the ground state. However, only the fast electronic polarizability is able to respond to a lightinduced electronic excitation during the timescale of light absorption. Hence, the solvent does not reach full equilibrium with the solute in such short times. Therefore, Zerner [29] and others viewed the slow, re-orientational part of the RF as being frozen during light absorption, and considered only the residual electronic part for the relaxation of the solvent with respect to excitation. From basic electrodynamics, it is well known that the fast electronic polarizability corresponds to a dielectric constant of n~, where ns is the refractive index of the solvent. This is often called the optical dielectric constant, t~opt.Since the refractive index is well known for most solvents, it was simple to find the RF of the solvent in the absence of re-orientational polarizability by just replacing the macroscopic es by n~ in Eq. (2.5). Unfortunately, Zerner and many others, considered in the next step the difference between the full RF arising from es, and the optical RF arising from n s, 2 as the frozen part during excitation. In this assumption they made a severe error, because the electronic contribution is smaller, typically by a factor of i0, in the simultaneous presence of reorientational polarizability. Later, I criticized this physical inconsistency, and it turned out that a physically consistent treatment of the frozen part ends up with an even simpler formula for solvatochromic effects in dielectric continuum so]vation models than the wrong expressions used before [C5].
Dielectric continuum solvation models and COSMO
15
Although the solvatochromic effects are not the central focus of this book, it is worthwhile to introduce some typical values for total and optical dielectric constants and for the corresponding dielectric scaling functions. Water has a dielectric constant of 80.1 at 20~ hence its scaling factor f(e) is greater than 0.975 for all multipole orders, and it is 0.98 for dipoles (see Fig. 2.1). For a hypothetical, infinitely strong dielectric, or for a conductor, the scaling function is 1.0, so we can say that the solvent water is very close to the ultimate dielectric response limit. Methanol has a dielectric constant of 33 but still has a scaling factor of 0.96 for dipoles, and phenol with e = 12 has a scaling factor of 0.88. Solvents without a permanent dipole moment, such as cyclohexane, mostly for symmetry reasons only have electronic polarizability, which can be approximated quite well by a value of 2.0 for organic liquids. Inserting this value in Eq. (2.3), the dielectric scaling factor has a lower limit of about 0.5 for ions, 0.4 for dipoles, and 0.33 for the highest multipole moments. The typical range and behavior of the dielectric scaling function is shown in Fig. 2.1. It can be clearly seen that only ions really deserve a separate scaling factor, while all non-ionic solutes can be well approximated using the same scaling factor for dipoles and higher multipole contributions--especially if we take into account the fact that the higher multipoles typically give a small contribution
Fig. 2.1. Dielectric scaling functions for different multipole orders.
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to the solvation energy. We will make use of this finding later, in the development of COSMO. The dielectric scaling factor introduces a reasonable distance metrics for our pictorial analogy in Fig. 1.1. Having a vacuum with a dielectric constant of 1, and hence f(e) - 0, as our idea of the south pole and considering the infinite dielectric or the conductor with f(D - 1 as the north pole of our globe, alkanes are located slightly south of the equator and organic solvents and mixtures generally are in the northern hemisphere. The results of the SCRF models depend strongly on the radius R x used for the definition of the spherical interface between the solute and the solvent. Unfortunately, the dielectric theory does not provide an answer for the question of which value is appropriate for this radius. Owing to the implicit assumption of the dielectric continuum models that the electron density of the solute should be essentially inside the cavity, any value of R x below a typical van der W a ~ s (vdW) radius would not be meaningful. On the other hand, at least at the distance of the first solvent shell, i.e., typically at two vdW radii, we should be in the dielectric continuum region. However, there is no clear rationale for the right value between these two limits other than empirical comparison of the results with experimental data. Among others, the choice of spherical cavities which correspond to the liquid molar-solute volume has proved to be successful. Nevertheless, the use of spherical-cavity SCRF models finally led to a dead end, because no meaningful spherical cavities can be defined for non-spherical molecules. Even the attempt to overcome this problem by an extension of the Onsager model to ellipsoidal cavities did not really solve the problem, because this introduces additional fit parameters, while only a small portion of real molecules can still be considered as approximately ellipsoidal. Thus, the need for molecular-shaped SCRF models became more and more obvious. A problem of molecular-shaped SCRF models is the absence of an analytical solution for the reaction field. One line of development was the search for an approximate expression for the dielectric interaction energy of a solute in a molecular-shaped cavity, without the need for explicit calculation of the solvent polarization. These models were summarized as generalized Born (GB) approximations [22,30]. The most popular of these models
Dielectric continuum solvation models and COSMO
17
are the AMSOL and SMx models by Cramer and co-workers [22,31]. By fitting various empirical parameters of such models to exact numerical solutions for the corresponding polarization problem, GB models have become able to treat the dielectric response for typical molecular shapes reasonably, but they are always in danger of generating electrostatic artifacts as soon as less-common shapes of solutes are involved. Most GB models also suffer from the approximation that the solute charge distribution has to be represented by atomic partial charges in order to keep the number of adjusted parameters in a reasonable range. By the corresponding neglect of details of the electron density such as lone-pairs and bond densities, the electrostatic resolution of the GB models is quite limited. However, as a big advantage, GB models are very fast in comparison with other molecular-shaped SCRF models, and hence they can be used very efficiently in semiempirical quantum chemical models and in force-field calculations. In the latter case, since the molecular electrostatics is represented on an atomic partial charge level, the loss of electrostatic accuracy does not play a major role. Furthermore, analytic expressions for the derivative of the solvation energy with respect to the atomic coordinates are quite easily available, enabling efficient geometry optimization of the solute in the presence of the dielectric medium. There are two other main directions for the calculation of the electrostatic interaction between the solute and a surrounding dielectric continuum for molecular-shaped cavities. Both require intensive numerical calculations and are thus slower than GB methods. The first direction is the direct numerical solution of the Poisson equation for the volume polarization P(r) at a position r of the dielectric medium: ~(r)- 1 P(r) = ~ E ( r ) 4~
(2.6)
on a grid surrounding the solute, where e(r) is the local dielectric constant in the solvent and E(r) is the electric field. However, such a grid has to be rather fine, at least in the neighborhood of the solute, in order to avoid major errors, and on the other hand it has to reach far out in space to avoid artifacts from the unavoidable cutoff at some distance Rmax. The only real strength of this method is the ease of the grid construction, because the
18
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interior of the solute cavity can be defined quite simply by a minimum distance to each of the atoms. Another aspect sometimes stressed by the authors of such volume grid methods is the ability to include the effects of ionic strengths into the calculations by solving the linearized Poisson-Boltzmann equation instead of the Poisson equation. Therefore, these methods are sometimes referred to as Poisson-Boltzmann (PB) methods, but applications including ionic strengths are rare in the literature. Another volume grid method has been developed by Warshel and co-workers [32], who are considering explicit re-orientational dipoles on the grid positions instead of a homogeneously distributed dipole moment density. In principle, their approach is therefore not a dielectric continuum approach, and it is capable of taking into account saturation effects beyond the dielectric limit. However, when explicit dipoles of realistic size are taken, the results depend strongly on the relative position of the solute to the grid. Thus, the grid is usually made sufficiently fine, and the individual dipole moments are correspondingly made unrealistically small, so that this method becomes essentially equivalent to a volume grid PB method. The second direction for the numerical calculation of the interaction of a solute with a dielectric continuum is most important for the understanding of COSMO and COSMO-RS and hence it is treated in a separate chapter. A common feature of almost all molecular-shaped cavity models is the need to define the cavity by some distance criteria. This is mostly done by element- or atom-type-specific radii. The values reported in various papers vary considerably, ranging from 1.0 to 1.4 vdW radii. Considering the strong R -3 dependence of the solvation free energy of dipolar solutes on the average cavity radius, a radius variation of a factor 1.4 should roughly cause differences of the solvation energies of a factor of 2-3. Nevertheless, all workers have calibrated their radii with essentially the same experimental data and are solving essentially the same dielectric model. How can that be? The origins of the large variations are the different sources for the calculation of the multipole moments, or more generally for the electrostatic field. Some groups are using Hartree-Fock (HF) calculations as the quantum chemical method. This procedure is known to overestimate dipole moments by ~16% [C3], resulting in an
Dielectric continuum solvation models and COSMO
19
overestimation of the solvation energies by 35%. Therefore, larger cavity radii are required in this context to achieve agreement with experimental data. Density functional method (DFT) calculations yield good agreement with experimental dipole moments. Hence, the DFT electrostatics can be considered as reliable, and the cavity radii of c. 1.2 vdW radii can be considered physically most meaningful. Some groups have used HF gasphase electrostatics in order to generate atomic point charges, which are then used for dielectric continuum solvation calculations. Here, the overestimation of the HF method is partly compensated by the lack of back-polarization in fixed point charge models, and therefore this approach also ends up with radii close to 1.2 vdW radii. Finally, some workers are using good DFT electrostatics in connection with non-polarizable point charge models. Owing to the lack of back-polarization, they need to use smaller radii in order to achieve rough agreement with experiment, and thus they can use the vdWs radii without scaling.
2.2
APPARENT SURFACE CHARGE MODELS
According to basic electrostatics, the polarization of a homogeneous dielectric continuum can be described either by the volume polarization density P ( r ) or by the surface polarization charge density a(r) on its surface. By considering the dielectric continuum surrounding a solute as infinitely extended, the polarization charge density, a, on the outer surface can be neglected. As a result, only the polarization charge density, a, on the interface between the solute molecule and the dielectric medium is relevant for the solute-dielectric interaction. Thus, there is no cutoff required and the error of the calculation only depends on the accuracy of the construction of the grid of small surface segments, also called tesserae, which build the cavity. A few aspects of the technical details of the cavity construction will be given in Section 2.4. Given a cavity segmentation by m segments i, of sizes si and centered at positions ti, the dielectric surface polarization charge densities, ai, and the corresponding apparent surface charges, qi = siai, can be calculated from the exact dielectric boundary
20
A. Klamt
condition (EDBC) as 4uts~ - E x + En((~) - - E x + W$~
(2.7)
where we introduce an m-dimensional vector description for the properties of the segments, and a matrix description for the corresponding m • matrices. Hence, ~ is the set of unknown polarization charge densities on the segments. E x is the vector of the outward normal components of the electrostatic field produced by the solute X on the cavity segments, and E n ( ( ~ ) the corresponding set of electrostatic field components generated by the polarization charges. After introducing a diagonal matrix $ for the segment areas, and a matrix W as Wij
-
(ti - ti)ni
-
IIti -
ti
II 3
(2.8)
Eq. (2.7) takes the form of a standard set of linear equations. Finally, the interaction energy of the solute X with the polarization charges q i can be calculated from E~ - oXs~ : OXq
(2.9)
and the corresponding total free-energy gain, AG~, is half of the interaction energy, E x, owing to the energy required for the polarization of the continuum. The apparent surface charge model that is most widely developed and used is the polarizable continuum model (PCM), originally introduced by Miertus et al. [33], and later developed further by Tomasi and several co-workers. Originally they solved the coupled problem of the quantum chemical self-consistent field (SCF) equations and the dielectric boundary equations (Eq. (2.7)) by an alternative iterative procedure, starting with a gas-phase SCF yielding a start value for the electric field E x, followed by an iterative solution of Eq. (2.7), substituting the resulting polarization charges q as external charges into the next SCF step, and iterating until full self-consistency is achieved. Since, the quantum chemical SCF had to be repeated several times, the original PCM procedure unavoidably increased the time requirements to a large multiple of the costs of a gas-phase calculation. Because no analytic expression for the geometry dependence of the dielectric interaction energies was available from this iterative procedure, a geometry optimization in the dielectric
Dielectric continuum solvation models and COSMO
21
continuum was practically impossible. Later, after the corresponding developments had been introduced in the COSMO model [C1], the iterative solution of Eq. (2.7) was replaced by a direct matrix algebra solution that could be inserted directly into the SCF. Thus, essentially no additional SCF steps were required for a PCM calculation. Analytic expressions for the first and second derivatives have been developed on the basis of this matrix algebra, enabling efficient geometry optimization and calculation of properties. Meanwhile, a large number of quantum chemical methods and tools, which are available for gas-phase calculations, have been combined with PCM, making it the most complete package presently available for dielectric continuum solvation calculations [34]. For a detailed overview of the different developments and the innumerable publications see [35]. One of the greatest problems of the apparent surface charge models is the outlying charge, i.e., the fact that the tails of the wave function in a SCF calculation unavoidably reach out of the cavity if distances of ~1.2 vdW radii are used for the cavity construction, as required by energetic arguments. Since the electric field on a closed surface is mainly determined by the enclosed charges, the outlying charge almost does not contribute to the polarization of the continuum. However, because the positive nuclei are always inside the cavity, this outlying electronic charge leads to the situation that the dielectric continuum effectively sees a slightly positive solute, even if the solute is neutral. If only 1% of the electron density is outside the cavity, this leads to a lack of 0.5 atomic charge units (e) for a molecule as large as hexane, resulting in a strong artificial interaction with the dielectric continuum. In order to reduce such artifacts, Tomasi and co-workers have developed various correction methods for the polarization charges, qi, which enforce a global compliance with Gauss's law [35] m
es-1
~--~. qi -- ~ Q t o t i=1
x
(2.10)
~S
where QtXt is the total charge of the solute molecule. Nevertheless, each of these global corrections did not guarantee a correct placement of the correction charges in those areas where the
22
A. Klamt
outlying charge was really causing a problem, and a wrong placement could cause even larger energetic artifacts than the absence of a correction [C10]. The outlying charge problem has the additional nasty behavior that it undermines the achievement of a basis-set convergence, because better and larger basis sets cause an increase of the outlying charge and thereby an increase of outlying charge errors. The outlying charge problem in the apparent surface charge models remained essentially unsolved until it was considerably reduced by COSMO [C10] and similar boundary conditions derived later from this concept (see section 2.3).
2.3
THE CONDUCTORLIKE SCREENING MODEL (COSMO)
In May 1991, my colleague Gerrit Schfifirmann read some papers on the SCRF models by Zerner et al. [29,36]. Since he was developing quantitative structure-activity relationships (QSAR) for environmental properties and toxicology based on descriptors from semi-empirical quantum chemical calculations he wanted to get access to such methods in order to take solvation effects into account during the calculations of QSAR descriptors; but the SCRF methods were not implemented in the semi-empirical packages available at that time. I do not remember why he asked me whether I would be able to implement such SCRF methods in a semi-empirical quantum code. Since I am an educated metalphysicist, and that I had only recently learned a little bit about quantum chemistry, this was almost like asking a car mechanic to develop an enhancement of a computer. However, I found the subject at least interesting, and I sat down with Zerner's papers, took a piece of paper and tried to understand the physics of SCRF models, which I had never heard of before. Having forgotten the complicated dielectric boundary conditions, I only remembered the respective equations for a conductor, and wrote down the formula for the interaction of a dipole in the center of a conducting sphere. In order to get the solution, I used the method of mirror charges [37], which immediately led me to a general exact expression for the interaction energy of an arbitrary charge distribution within a surrounding conducting sphere of
Dielectric continuum solvation models and COSMO
23
radius R"
AE--
1 2~-~ ,/R ij V__ 4
--
qiqjR 2R2rirj ~- r~ir~j
(2.11)
Here, ri denotes the position vector of the charges qi with respect to the center of the sphere, and ri the distance from the center. By applying the dielectric scaling function for dipoles (Eq. (2.3)), which~as we have seen in Fig. 2.1~is also a good approximation for most other multipole orders, it was immediately clear that the idea of using a scaled conductor instead of the EDBC leads to a considerable simplification of the mathematics of dielectric continuum solvation models, with very small loss of accuracy. It may also help the finding of closed analytic solutions where at present only multipole expansions are available, as in the case of the spherical cavity. Thus the Conductor-like Screening Model (COSMO) was born. Eq. (2.11) makes it very evident that a spherical-cavity model must diverge if the solute-charge distribution approaches, or even reaches out of, the cavity. This is much harder to see in Kirkwood's formula (Eq. (2.2)), although both must be equivalent for ~ = c~. On the second view, it is mathematically also obvious that a spherical-cavity model cannot work for charge distributions reaching close to, or even outside, the cavity because the proof of convergence of a multipole expansion is only given if the extension of the charge distribution is smaller than the distance to the point where the potential is evaluated. However, the spherical-cavity SCRF model had been used for molecules with dimensions larger than the cavity radius, and that should not work in general. On the same evening, I wrote down the equations for the interactions of a charge distribution with an arbitrarily shaped closed conducting surface, having in mind some kind of molecular-shaped cavity. Given a segmentation by m segments, the boundary condition of vanishing potential on and within a grounded conductor immediately yields the equation: 0 - 9 x + O(q) = 9 x + Aq
(2.12)
where O x denotes the surface potential arising from the charge distribution of a solute X inside the cavity, and A is the Coulomb
24
A. Klamt
interaction matrix of the surface segments, i.e., Aij
-
I]ti - t j
I1-1 and
Aii
~
1.07si-~
(2.13)
where the latter is an approximation for the self-energy of a charged sphere segment, which turned out to improve considerably the convergence of the results with respect to the number of segments. By scaling the solute potential with a dielectric scaling factor f(e), we obtain the equation Aq - - f ( D O x
(2.14)
for the scaled conductor model, which can easily be solved by inversion of the symmetric, positive-definite matrix A, or more efficiently by a Cholesky factorization [38]. The solute-conductor interaction energy thus reads E~ -- (lDXq- -f(~)OXA-10 x
(2.15)
and the total free-energy gain, AGx, is -~E~. Next we introduce an operator B that generates the potential (I)~ from a charge-density representation QX. This operator can be very different in different QC codes, because there are several different representations of the electron density used in different codes. Some use a representation in atomic orbitals, others in molecular orbitals, some have plain waves, and others use a representation on a space grid. Nevertheless, such kind of operator must be readily available in any quantum chemical program, because it is required for the evaluation of the interaction energy of the electrons with the nuclei. Using this operator results in the expression, V~ - - 89
x = 89
(2.16)
where, t denotes matrix transposition. The matrix D defined by the last step in Eq. (2.16) can be considered as Greens' function of the dielectric continuum. By comparing this expression with the Coulomb interaction energy of the charge density QX, which is x Ecoulom b -- 89
(2.17)
where C is the Coulomb matrix, we immediately see that the presence of a dielectric continuum surrounding a charge distribution can be represented as an additional charge-charge interaction, which is just mediated by the dielectric continuum.
Dielectric continuum solvation models and COSMO
25
This formal analogy is very helpful, since in this way, a dielectric continuum can easily be implemented in a quantum chemical program as an addition to the Coulomb interactions. It is also immediately clear that the analytical derivatives of the total energy should be calculable in such a close analogy. If the solute charge density QX is variationally optimized with respect to a quantum chemical operator (the Fock operator in HF or the Kohn-Sham operator in DFT), the derivative of the energetic COSMO contribution with respect to any geometry coordinate x would--according to the Hellmann-Feynman theorem~just be given as the expectation value of the derivative of the COSMO Greens' function, i.e., d vX - 21Qx ( ~ - D ) Q X -
1 ~f(~)Qx(~-(BtA-IB)) QX
= - ~f(DQ 1 XBt(2A-1 ( d
B) _A_ 1(~---A)A -1 B)QX
1 = q ( d B) QX + 2fiei q ( d A ) q.
(2.18)
Since the derivative of the B operator is also readily available in quantum chemical programs, only the derivative of the A matrix needs to be coded for the calculation of analytic gradients. These immediate and simple findings motivated me to accept Gerrit Schiiiirmann's request and to implement COSMO as a new kind of SCRF model in the semi-empirical quantum chemistry package MOPAC [39]. Shortly afterwards, I met Jimmy Stewart, the author of the MOPAC package, in a European Computational Chemistry Workshop in Oxford, where he was available as a supervisor for a entire workshop. I gave a short presentation of my COSMO ideas and he was interested to get COSMO as the first solvation model in MOPAC. Therefore, he introduced me to some extend to the MOPAC program code, and we identified the places where COSMO would have to link in. The first step of the COSMO implementation was the development of a cavity-construction scheme. Since I did not know about the PCM method at that time, which might be considered now as an inexcusable ignorance for a scientist working in that area, I developed an independent code for the cavity construction. Nevertheless, I started from the same idea,
26
A. Klamt
i.e., to use a method of iterative refinement of sphere triangulation, which I had read about somewhere before, probably in the context of the GEPOL algorithm that was available as a tool for surface and volume calculation in QCPE [40]. However, since the technical details of the cavity construction are not of great importance for the entire understanding of COSMO and COSMORS, I will discuss the details in a separate section (section 2.5), which can be skipped by readers who are not interested in technical details. Alongside several other projects, the implementation of COSMO in MOPAC took about a year. I should gratefully mention the help of two younger colleagues at B a y e r ~ T h o r s t e n PStter and Felix Reichel~who were both quantum chemists by training and thus could help me when I did not understand what was going on in a q u a n t u m chemical code. It was Felix Reichel who pointed me to the analogy of COSMO to the PCM model that he had been using in his Ph.D. thesis. At that time, the concept of COSMO and its implementation was already complete, and it was apparent that our new model would have several inherent advantages over the PCM model, since PCM was still using an iterative algorithm, and as a result did not have analytic gradients available. In the summer of 1992, we had a first working implementation of COSMO in MOPAC6, and in 1993, COSMO was officially released as part of MOPAC7 and MOPAC93 [41]. About the same time---after being rejected by two journals as irrelevant--our first publication on COSMO appeared, presenting the basic theory, some details of the MOPAC implementation, and a few application examples as a proof of principle. Despite the lack of a detailed parameterization, this initial MOPAC-COSMO implementation worked reasonably well, and became widely used by the community of MOPAC users. Some years later, this implementation was implemented in two other widely used semi-empirical quantum chemical programs, AMPAC [42] and MNDO [43], on the basis of public-domain code in MOPAC7. A first quantum chemical ab initio implementation of COSMO within the quantum chemistry program Gaussian94 was published in 1995 by Troung and Stefanovich [44] under the name Generalized COSMO (GCOSMO), although no generalization of the theory was presented and although the idea and concept for
Dielectric continuum solvation models and COSMO
27
an ab initio extension of COSMO was already given in our paper. In parallel, I was collaborating with Kim Baldridge on an ab initio implementation in GAMESS-US [45,C2], and with Jan Andzelm on a first implementation in the DFT code DMol [46,47,C3]. In the GAMESS implementation, we decided to use an atomcentered multipole representation of the density for the calculation of the potential vector (I)x, as it was already used in MOPAC, owing to the usual electrostatic standards of that semi-empirical program. Such representation automatically treats all charges as if they were inside the cavity, and hence it avoids the problem of outlying charges. Taking into account all multipoles up to hexadecapoles, it turned out that good convergence is already achieved at the atom-centered octupole level but, on the other hand, octupoles are definitely required for an accurate representation of the important effects of lone pairs. In the DMol implementation, we used the more straightforward strategy of a direct calculation of the potential vector (I)x from the density, which is represented on a grid in this program. As a result, we had to consider the outlying charge effect, because small parts of the density were now definitely located outside the cavity. To our own surprise, in a careful analysis we found that the COSMO algorithm is much less sensitive to the outlying charge than the EDBC. The reason is that the electrostatic potential is much less sensitive to the outlying charge than the electrostatic field, and that COSMO only uses the potential, while EDBC uses the field. From a simple analysis for spherical molecules one can easily show that COSMO is approximately 10 times less sensitive to the outlying charge error (OCE). The details of this analysis are reported in [C5]. In that paper, we also introduced the first local outlying charge correction method, in order to avoid artifacts from global corrections and to get reliable values for the local polarization charge density, which we already needed for the COSMO-RS algorithm at that time. Later, we implemented COSMO in TURBOMOLE [48,C4], COLUMBUS [49], and MOLPRO [50] following the same concepts as in DMol. Meanwhile, many more quantum chemical programs have been extended by the COSMO solvation model. Owing to the very fast and reliable density-fitting approximation in TURBOMOLE, this is the fastest COSMO-DFT program presently available. Despite the speed of the TURBOMOLE program by itself, COSMO
28
A. Klamt
calculations only provide a computational overhead of about ~15% over gas-phase calculations. Thus, the TURBOMOLE/ COSMO combination is clearly the fastest tool for good-quality dielectric continuum solvation DFT calculations, enabling solvation calculations in the same timescale as the (already extremely fast) gas-phase DFT calculations. At this point, the reader who is mainly interested in the fluidphase thermodynamics aspects should have sufficient information about COSMO to understand the theory of COSMO-RS. Thus, one may skip the following two or three sections, which present more details on COSMO and other continuum solvation models.
2.4
MORE COSMO DETAILS
The fortunate finding of the strong reduction of the outlying charge effect by COSMO boundary conditions also motivated other groups to use these boundary conditions in their codes. Thus, COSMO became implemented within the PCM code (CPCM) [51], and nowadays many of the PCM applications use COSMO. Since PCM is implemented in the Gaussian program package [33], which is the most widely used package for ab initio calculations, the implementation made COSMO available to a large section of quantum chemists. Nevertheless, it should be noted that the C-PCM implementation differs slightly from my own implementations regarding cavity construction and outlying charge correction. In addition, the default for the dielectric scaling function is set to the monopole term fl(E) in Eq. (2.4), in order to achieve compliance with Gauss's laws, instead of the dipole-scaling factor f2(D. Considering the fact that the vast majority of dielectric continuum solvation applications are for neutral molecules, and that essential deviations between the scaling functions only occur anyway in low-dielectric media, where ions are extremely rare, I still consider the dipole-scaling factor as the best choice for general COSMO applications, and this has also been confirmed recently by some of the authors of the PCM program [52]. The integral equation formalism (IEF) introduced by Cances, Mennucci, and Tomasi [53] in PCM also has much to do with the COSMO boundary condition. Indeed, it can be shown to be
Dielectric continuum solvation models and COSMO
29
equivalent to a model in which the polarization charge-density is split into a contribution, a~, which is calculated with a non-scaled conductor boundary condition--and thus suffers little from O C E - - a n d a difference-contribution, O"d ----O'(~;)--O'oc, that gets calculated from EDBC at finite e and at e = ~c. This boundary condition is favorable in strong dielectrics, because it combines the small OCE of COSMO with the exact dielectric deviation from = co. However, in low dielectrics it becomes even worse than EDBC and COSMO, because here the OCE-affected difference contribution, 6d, becomes large. Therefore, it would be much better to define the first contribution as the scaled-COSMO screening charge density, and the second contribution as a difference between EDBC a(~) and the scaled infinite dielectric, a~ - a(~)-f(~)a~. This definition combines the small OCE of the scaled conductor with the small correction for the dielectric error of COSMO in the entire range of e. It can even be expressed in such a way that the electrostatic field of the solute, E x, no longer appears in this formalism, leading to a simplification of the algorithm compared with EDBC and IEF. I developed these ideas after the IEF had been published, but apart from several private communications I have never published them. Meanwhile, Chipman [54] published a new boundary condition for dielectric continuum solvation models named SS(V)PE. Although derived from different arguments, his method is equivalent to the method proposed above. It can be considered as the most accurate boundary condition presently available for dielectric continuum solvation models at lower values of e, but it is algorithmically much more demanding than COSMO and it is questionable whether the small gain of accuracy justifies the additional costs.
2.5
CAVITY CONSTRUCTION AND DERIVATIVES
The efficient construction of proper and sufficiently accurate segmentations of a molecular-shaped cavity is an important technical aspect of apparent surface charge models, because it has a strong influence on both the accuracy and speed of the calculations. Before going into details, some common features will be discussed.
30
A. Klamt
Given the coordinates and atomic numbers of all N atoms of a molecule, the usual concept for the definition of a continuum solvation model (CSM) cavity starts with the assignment of minimum distances, R~, for all atoms a. This assignment can either be according to the atomic numbers, using element-specific radii, or according to more details of the chemical environment, i.e., by the introduction of atom-type-specific radii. The latter procedure obviously introduces more adjustable parameters, with the potential of a more accurate reproduction of experimental data, but also with the danger of loss of the predictive power of the model. Some workers even introduce a dependence of the radii upon the partial charge of the atom or the total charge of the molecule, but this leads to obvious consistency problems, because the final partial charges depend on the cavity again. After assigning the radii, the union of the corresponding atomcentered spheres is considered as the interior of the molecule. This is shown as the blue region in Fig. 2.2a. Such cavities usually
Fig. 2.2. Schematic illustration of the cavity construction methods.
Dielectric continuum solvation models and COSMO
31
have some sharp crevices and even sharp cusps. For example, a benzene molecule would have two sharp inward-directed cusps in the middle of the ring. Such crevice and cusp areas are fatal for the dielectric continuum approach, because the electrostatic field in front of a dielectric cusp becomes infinite and the boundary conditions can no longer be solved. Thus, it would be mathematically and physically unreasonable to assume that the dielectric continuum solvent fills such sharp crevices and cusps. Therefore, any sphere-based CSM cavity construction requires a smoothing algorithm for such situations. In computational chemistry, a solvent radius R~olv is often used to define a smooth solventexcluding surface (SES) (in the sense of Connolly [55]) by the loci of the surface of a ball of radius R~olv rolling over the atomic spheres, as shown in Fig. 2.2a. However, the surface-segment construction on the Connolly smoothing region in three dimensions is geometrically very demanding, and therefore, other strategies are used in CSMs. These will be discussed later. As a next step, the resulting surface must be divided into segments. For the subsequent calculation of the electrostatic interactions, it is important to know the area, shape, and position of each segment. In addition, it is desirable to achieve reasonably homogeneous segment areas in order to reduce the total number, M, of segments on the molecule, because the time requirements of the algorithms for the solution of the CSM boundary equations typically scale with the third power of M and the memory-demand scales with M 2. The method most widely used for CSM cavity construction is the GEPOL algorithm, which is also used in PCM [39]. Starting from the atom spheres, GEPOL introduces auxiliary spheres between all pairs of atoms, which would have a sharp crevice otherwise on the basis of a minimum-allowed value for the cutting angle (see Fig. 2.2b). The procedure is repeated, including the new spheres, until all sharp crevices are removed. Next, the cavity is defined as the union of the spheres, and each sphere gets individually segmented. This is done by an almost regular triangulation of the full sphere into k - 4 i . 60 triangles ( i 0, 1, 2,...). The simplest triangulation corresponds to a pentakisdodecahedron, which can be visualized by considering a normal soccer ball or a C6o-Buckminsterfullerene ("Buckyball", see Fig. 2.4d), marking the center points of all 32 faces, and
32
A. Klamt
Fig. 2.3. Refinement of sphere triangulation starting from the icosahedron (12 corners, 20 triangles). Second is the pentakisdodecahahedron (32 corners, 60 triangles) used as default in PCM. Further, we see triangulations with 80, 180, and 240 faces. The last picture shows the triangulation used as a COSMO basis grid with 1082 corners and 2160 faces. The 12 corners of the original icosahedron are marked in red. Only the second and fifth triangulation are available in GEPOL, while the COSMO code can generate all of them.
connecting t h e m to 60 triangles. The centers of the triangles are just the 60 vertices of the Buckyball. Each triangulation can be refined by a factor of four by the introduction of the original edgemidpoints as new vertices. Some triangulations are shown in Fig. 2.3. Since k - 240 already leads to a very large n u m b e r of segments and hence very high time demands, the value k = 60 is usually used in PCM/GEPOL cavity constructions. As a second step, all triangles which are totally in the interior of other spheres are withdrawn. In PCM, partially exposed triangles are counted as segments and treated by spherical trigonometry to access their area and center of mass, and the fully exposed triangles are considered as normal segments. In this way, for typical molecules with N atoms the final n u m b e r of segments is given roughly by -~N. k. Then the center of each segment is used as a representation point, t/, and associated with the corresponding segment area, s/. Finally, the electrostatic interaction matrices W and A can be calculated using this information, depending on the needs of the special boundary conditions.
Dielectric continuum solvation models and COSMO
33
Analytic derivatives of the segment areas with respect to the atom coordinates can be calculated using spherical trigonometry for the partial triangles, but the expressions become very complicated due to the implicit dependence of the positions of the auxiliary spheres on several atom positions. The GEPOL cavity construction appears to work reasonably in its PCM implementation, but a few concerns should be mentioned. First, for a typical atom radius of 2 A, the default value of 60 triangles per sphere leads to a triangle side-length of 1.4 A, and to a distance of 0.8 A from segment corner to the segment center. These values are of the order of a bond length, and hence this cannot be regarded as a particularly fine segmentation. A second concern is the way of smoothing. In principle, the smoothing regions should have one direction of positive curvature and another direction of negative curvature, like the inside of a torus, while the auxiliary spheres have all positive curvature. This unphysical smoothing curvature appears to be questionable especially in the context of the EDBC, because here the direction of the segment normal vector plays an important role. Finally, the sudden introduction of auxiliary spheres depending on a critical threshold causes a discontinuous behavior of the segmentation and of the corresponding dielectric interaction energy. The cavity construction in the original COSMO implementation in MOPAC93---called OLDCAV in our following discussion, according to the corresponding MOPAC keyword--starts with a union of spheres of radius R~ + R~olv for all atoms ~. Then each sphere is triangulated, but in contrast to GEPOL, we start from a regular icosahedron with 20 triangles as an initial triangular grid. Then we use two different refinement steps. The first is the addition of the triangle edge-midpoints as new vertices. This increases the number of triangles by a factor of four, as in GEPOL. The second is the addition of the triangle centers as new vertices, leading to a new set of triangles and increasing the number by a factor of three. This second refinement step has been overseen by the authors of GEPOL, although it is just the operation that generates their start grid of 60 triangles from the icosahedron. Later, we realized that the edges of the triangles can obviously be subdivided by any integer n, leading to an increase factor n 2. Thus, we can generate triangulations with k - 20 93 i 9n 2, with i being 0 or 1, and n any positive integer. The lower values of k thus are o
o
34
A. Klamt
20,60,80,180,240,320, . . . . Apparently this offers considerably more choices than the GEPOL algorithm (see Fig. 2.3). Finally, we do not consider the triangles as segments, but the corresponding hexagons (and 12 pentagons), which we get when we consider each vertex of the triangular grid as a center, and connect all triangle midpoints of the six or five surrounding triangles of the vertex. Thus, we use the 12 faces of the dodecahedron as segments in the case of k = 20 triangulation, and the 32 faces of the buckyball for k = 60. Indeed, the explicit construction of the hexagons and pentagons is not required, because we only need the known positions of their centers and the corresponding surface area (see Fig. 2.4). At the same corner-center distance, the number of hexagons and pentagons is k' = k / 2 + 2, i.e., approximately half of the number of triangles, leading to sphere segment numbers of k' = 10 93 i 9n 2 + 2 = 12, 32, 42, 92, 122,.... In addition, during the evaluation of the electrostatic interaction matrix the center approximation is better justified for these near-circular segments than for triangular segments. In order to achieve a fine sampling of the surface, a basis grid of NPPA (number of points per full atom) equal to 1082 is initially projected on each of the atom spheres of radius R~ + Rsolv. Then all points in the interior of another sphere are discarded. Next, all remaining points are projected downward to the sphere of radius R~. By this construction, no points end up in the problematic smoothing regions. Next, the points on each sphere are connected to form larger segments. For this, a grid of NSPA (number of segments per full atom) segment centers is projected on the inner sphere where NSPA is one of the values k' - 10 93 ~ 9 n 2 + 2. N o w , the points from the basis grid are associated to their nearest grid
Fig. 2.4. Complementary segmentations: icosahedron with 20 triangles, and dodecahedron with 12 pentagons, as well as pentakisdodecahahedron (60 triangles) and "buckyball" (12 pentagons and 20 hexagons).
Dielectric continuum solvation models and COSMO
35
centers, and those grid centers getting zero points are discarded. Then all segment centers are redefined as the center of area of their associated points. Finally, this association procedure is repeated once in order to achieve nearest-neighbor association after the move of the centers. Finally, each segment is characterized by a center position, ti, a total area si, and a set of basis grid points with associated positions and areas. Then the Coulomb interaction matrix elements, Aij, are calculated as a sum of the corresponding partial contributions of the associated basis grid points of segments i and j, if their distance is below a threshold DISEX. The center approximation is used otherwise. The diagonal elements Aii are calculated in the same way using the basis grid points, and even for self-energies of the basis grid points a reasonable approximation of 1.07 9s -~ (see Eq. (2.13)) is used, where s denotes the area represented by a basis grid point. In this way, the A-matrix for the segments is evaluated with very high precision, while keeping the number of segments M, which is finally crucial for the numerical costs of the COSMO algorithm, at a relatively low level of ~ N 9 NSPA/3. In semi-empirical quantum methods a default value of NSPA = 42 is suggested for nonhydrogen atoms, which is still a finer segmentation than the PCM GEPOL defaults, which corresponds to 32 in our methods. For hydrogen atoms we consider a coarser segmentation as justified, and hence use a default of 12. Considering the rigor used for the sphere segments, a crude approximation was made for the smoothing region" we just did not put any segments into these regions, i.e., we left them electrostatically open. Indeed, this worked surprisingly well and generated good correlations of the calculated solvation energies with the experimental results. Nevertheless, we later realized the need for a surface closure, because some small segments could get extremely high values of if they were isolated by open regions from the rest of the cavity. To avoid such artifacts, a surface-closure algorithm has been developed that is the standard for our COSMO cavities (NEWCAV) today. No changes were made for the spherical part. For the surface closure, first all pairs of intersecting spheres are considered. For each pair, the intersecting solvent-increased spheres generate a ring. If this ring is partially within other spheres, the corresponding angle regions are marked as invalid.
36
A. Klamt
Then the ring is projected downward by Rsolv toward each of the two atom centers, leading to two opposing rings. The valid regions of these rings are now filled with a series of triangles, each having two corners on one of the rings and one on the other. Finally, the sole corner of each triangle moves a little toward the center of the ring that holds the two other corners of the triangle. By that procedure the triangles become slightly inclined, mimicking the shape of the Connolly surface. The degree of inclination is a function of the solvent radius, the two atom radii, and their distance. In this way, a continuous and differentiable behavior of the surface between two atoms is achieved, even in the difficult case close to their complete dissociation. Since the change of the surface according to a change of the distance between the atoms can be approximated reasonably well by a stretching and change of inclination of these triangles, the derivative of the surface area with respect to geometry can be calculated from the analytic behavior of these triangles. In the final step, the triangular regions belonging to triple points, i.e., intersections of three spheres, get paved with additional triangles. Each of the triangles generated in this way is considered as an individual segment. For NSPA = 92 and NSPAH--32, which is the present recommendation for DFT calculations, the surface-closure unfortunately enlarges the initial number of segments by approximately 70%. Nevertheless, we consider the corresponding increase in calculation time as justified by the resulting increase in physical consistency. Since all segments can be considered to move approximately with their nearest atoms, and area-changes have to be taken into account only for the closure triangles, the derivative of the A-matrix with respect to the geometry is quite simple. This derivative is needed for the calculation of analytic derivatives of the COSMO interaction energy according to Eq. (2.18). In addition to good analytic gradients, a stable cavity-construction algorithm is required for a good convergence of the geometry optimization algorithm. Artificial fluctuations of the total energy caused by small changes of the geometry must be kept as small as possible, although they can never be completely avoided in a discretized cavity model. The COSMO algorithm presented here works satisfactorily in this regard, achieving geometry
Dielectric continuum solvation models and COSMO
37
convergence in about the same number of steps as required for gas-phase geometry optimizations.
2.6
PRESENT STATE AND FUTURE DIRECTIONS
We have now achieved a situation in which dielectric continuum solvation models in general, and especially COSMO, are quite well established for SCF ground-state calculations of organic molecules. Many of the methods, tools, and properties available for gas-phase calculation can also be performed in a dielectric continuum solvation model. The PCM model including C-PCM provides the greatest breadth of implemented functionality. Initially, many CSMs, including COSMO, have been developed on the semi-empirical quantum chemical level. Nevertheless, the dipole moment is not a property which is normally considered during the parameterization of semi-empirical Hamiltonians, and as a consequence the electrostatic results of the most commonly available Hamiltonians such as AM1, PM3, MNDO, and even the recent PM5 [56-59] are not very reliable. While they are remarkably accurate for molecules composed of carbon, hydrogen, oxygen, and even halogens (see Table 2.1), yielding an rms error of only about 0.15D for gas-phase dipole moments of these classes, nitrogen atoms and hypervalent elements such as sulfur and phosphorous often introduce large dipole errors of 0.4 D and more. Since dielectric CSM results have a quadratic dependence on the solute electrostatics, the relative error of calculated dielectric interaction energies is twice as large as the error in the dipole moments. Thus, if we consider typical dipole moments of 2D for polar molecules and typical resulting dielectric interaction energies of 30kJ/mol, the error introduced by deviations of 0.4 D is of the order of 10 kJ/mol or more. This is not useful for predictive methods. The AMSOL model and the related SMx methods [22] are based on semi-empirical quantum chemical calculations. Normally these models use a GB approximation for the dielectric contribution of AG of solvation, but COSMO has also been used for one parameterization. In order to overcome the large electrostatic errors of bare semi-empirical methods, "charge models" have been developed, which improve the electrostatic
38
A. Klamt
TABLE 2.1 Accuracy of dipole moments (in Debye units) calculated by different QC methods for 64 compounds of elements C, H, and O (for more details see [Cl0]) Method
Program
DFT S-VWN DMol BPW91 DMol BP86 Gaussian94 BP86 Gaussian94 B3LYP Gaussian94 Ab initio HF Gaussian94 HF Gaussian94 MP2 Gaussian94 Semi-empirical AM1 Gaussian94
Basis set
RMS
Slope
Unscaled
Scaled
DNP DNP 6-31G(d) SVP24,25 6-31G(d)
0.1649 0.1380 0.1500 0.1374 0.1479
0.1585 0.1300 0.1483 0.1379 0.1447
1.0200 1.0228 0.9755 0.9927 1.0124
6-31G(d) 6-311G(d,p) 6-31G(d)
0.3293 0.3137 0.1953
0.1682 0.1401 0.1919
1.1630 1.1648 1.0124
0.1902
0.1909
0.9769
quality by using a large set of bond dipole corrections fitted to ab initio results [60,61]. The resulting dielectric results are combined with an extensive parameterization of atom-type-specific radii and different kinds of atom-type-specific, non-electrostatic energy contributions. In addition, they need some parameterization, of the solvent to take into account its effective dielectric constant and its surface tension for cavity formation. By this parameterization, many of the systematic errors of semi-empirical methods, and also the loss of electrostatic accuracy going along with the partial charge representation in the GBA are compensated, and good agreement between the calculated and experimental AG of solvation has been achieved for a wide range of organic compounds. Typical rms errors of 2.5kJ/mol can be expected for recent models. Nevertheless, because of the hierarchy of approximations, corrections, and fitted atom-typespecific parameters, such methods are always in danger of failing unpredictably in situations not represented in the parameterization dataset. Luque and Orozco [62] have developed a parameterization of the PCM model for semi-empirical methods, but that has not been
Dielectric continuum solvation models and COSMO
39
used widely. Meanwhile, the authors are apparently using more accurate quantum chemical levels as well. Being aware of the considerable problems, I have never started a quantitative COSMO parameterization on the semi-empirical level, and hence it is not recommended for quantitative prediction of solvation free energies. Nevertheless, the COSMO implementations in MOPAC and AMPAC can well be used to infer trends occurring in solution, and to calculate properties and QSAR descriptors that are closer to the reality in solution than the corresponding values taken from gas-phase calculations. And, the geometries resulting from semi-empirical gas-phase and COSMO calculations in general are quite close to results from higher-level quantum theory. Therefore, such geometries can often be used for a final single-point DFT/COSMO calculation, thus saving more than an order of magnitude of computation time otherwise spent in DFT geometry optimization. Another application for semi-empirical methods is for very large biomolecules such as enzymes. Here, linear-scaling methods have been developed that also include linear-scaling COSMO algorithms [63,64]. The linear scaling of COSMO has been achieved by consequent hierarchical treatment of long-range electrostatics combined with a conjugate-gradient algorithm for the calculation of the polarization charges by Eq. (2.15) [65-67]. Given the fact that gas-phase calculations for biomolecules are almost meaningless, the availability of linear-scaling semi-empirical COSMO codes may become important in the near future. However, some parameterization will be required to improve the electrostatic reliability of semi-empirical Hamiltonians for such typical biomolecules. These are discussed later in this book. HF-level ab initio calculations have often been used for CSM calculations. Combined with good basis sets providing sufficient polarization functions, which are essential for good electrostatic properties, good correlations between the calculated and experimental gas-phase dipole moments can be achieved, but with a general overestimation by 16% (see Table 2.1). As a consequence, good HF parameterizations of CSMs have been reported, which compensate for the electrostatic overestimation by increased cavity radii. Nevertheless, since better DFT methods are available with about the same computational costs, and with more
40
A. Klamt
consistent inherent electrostatics, the use of HF-level continuum solvation calculations cannot really be recommended at present. DFT calculations combined with polarized basis sets generally provide good quality gas-phase dipole moments (see Table 2.1). The rms error from experiment is about 0.1 D, without any need for scaling. The differences in electrostatic properties resulting from different gradient-corrected density functionals are generally small. Because DFT is not significantly more demanding than HF calculations, its reliable electrostatics makes it the method of choice for CSM calculations. We have developed efficient DFTCOSMO implementations in the programs DMol [46,47,C2] and TURBOMOLE [48,C4]. The implementation of COSMO in GAMESS [45,C3] meanwhile has also been extended to DFT as well. C-PCM is also parameterized on the DFT level. Another COSMO-DFT implementation has been reported in the ADF program. [68]. Very recently, a first implementation of COSMO in a plane-wave DFT code has been reported [69] and another one is in preparation [70]. Both implementations are intended for quantum molecular dynamics calculations with implicit solvation, by COSMO. From the different publications on DFT-COSMO applications it can be concluded that the free energy of hydration of polar neutral compounds can be calculated with an rms error of 3-5 kJ/mol, depending on the degree of parameterization. One of the most detailed parameterization exists for C-PCM. Fitting diverse non-electrostatic contributions, cavity formation energies, and even atom-type-specific cavity radii, Barone et al. [51,71] achieved an rms error of about 2.5 kJ/mol. This appears to be the ultimate lower limit presently achievable with DFT-COSMO calculations. Even this accuracy is a little surprising, taking into account the fact that a dipole-moment error of 0.15 D should result in an error of about 3 kJ/mol for typical polar molecules having a 2D dipole moment and 30kJ/mol solvation energy. Together with our results for DFT-COSMO-RS calculations (see chapter 7), which show an rms of only 1.4 kJ/mol for AGhydr, we may conclude that the intrinsic accuracy of DFT dipole-moment calculations is probably better than 0.15 D (rms), and that the apparent error is mainly due to experimental inaccuracy. Otherwise, the more accurate results with DFT-COSMO can hardly be explained. Nevertheless, it appears that the
Dielectric continuum solvation models and COSMO
41
electrostatic accuracy is of the same magnitude as the presently achievable accuracy of DFT-COSMO calculations. Thus, for further improvement in the results of COSMO calculations more accurate quantum chemical levels appear to be warranted. Combinations of COSMO with second-order Mr perturbation theory (MP2) [72] have been reported [73,74], but despite the few available results there are no indications that MP2 might be more accurate for CSM calculations than is DFT. Most likely the use of coupled cluster (CC) methods is required to achieve a significant improvement over the DFT quality of electrostatics. The formalism for some higher correlated calculation methods has already been developed for C-PCM in Gaussian [75,33], but the calculations are too demanding to generate a sufficiently large set of calculated data for typical organic molecules that would be required for a proof of the electrostatic capabilities. As a preparation for future work in this direction COSMO is presently being implemented in the MOLPRO package [50,76], which is known for its high efficiency in CC calculations. Summarizing, the reliable electrostatics of gradient-corrected DFT methods provides a good basis for CSMs. Nevertheless, advanced CSMs such as C-PCM or COSMO have achieved an accuracy that is mainly limited by the electrostatic accuracy of DFT, but the next better quantum chemical levels are presently much too expensive for most practical applications. Therefore, we will have to live with the acceptable DFT accuracy for the next years, and no big improvements of the CSMs beyond the present accuracy should be expected, until more accurate quantum levels as CC become practically useful.
Chapter 3
F u n d a m e n t a l criticism of the dielectric continuum approach In the previous chapter, we have seen how Born's simple and successful idea of a dielectric continuum approximation for the description of solvation effects has been developed to a considerable degree of perfection. Almost all workers in this area have been trying to obtain more efficient and more precise methods for the solution of dielectric boundary conditions combined with molecular electrostatics, but the question of the validity of Born's basic assumption has rarely been discussed. This will be done in the following sections, with a surprising result. 3.1
NON-POLAR SOLVENTS AS DIELECTRIC CONTINUUM
We start with a consideration of non-polar solvents having negligible dipole and quadrupole moments. Alkanes or perfluoroalkanes are probably the most typical representatives of such solvents. The only electrostatic interaction, and hence the total solvent polarization, originates from the electronic polarizability that can be approximated by a dielectric constant of e = n 2 ~ 2.1, where n is the refractive index. The polarizability of alkanes is quite homogeneously distributed within each molecule and hence within the entire solvent. Thus an alkane solvent looks quite homogeneous from the perspective of a solute molecule (see Fig. 3.1), making a continuum description at least plausible. Furthermore, the electronic polarizability of alkanes--and more generally, of most organic compounds--behaves rather linearly, even up to the extremely large electric field strengths, typically of 1011 V/m, occurring on the surface of polar solute molecules. Since the dielectric theory is a linear-response theory, such linear 43
44
A. Klamt
Fig. 3.1. Schematic illustration of the situation of a polar solute molecule (water) in hexane (left) and in water (right). behavior is crucial for the applicability of the macroscopic dielectric theory. In summary, for alkanes and other similarly nonpolar solvents without relevant dipole and quadrupole moments, the necessary r e q u i r e m e n t s of homogeneity and linearity are reasonably well met and the dielectric continuum approximation for the description of the electrostatic behavior of such solvents appears to be justified, even on the molecular scale.
3.2
T H E S I T U A T I O N FOR POLAR S O L U T E S IN POLAR SOLVENTS
The situation is very different for polar solvents, i.e., solvents t h a t have a relevant p e r m a n e n t dipole moment. In such solvents the greatest part of the dielectric response originates from the slight reorientation of the applied external field, and only a small part from electronic polarization. For water, with e = 78.4 (at 25 ~ the electronic polarizability contribution is only
Pel Zel ~el -- 1 n2 _ 1 1.77 - 1 0.77 = = ~ = = = --- 1% Ptot X t o t t~tot -- 1 ~ t o t - 1 7 8 . 4 - 1 77.4 -
(3.1)
If the polarization is mainly due to dipoles, we m u s t recognize t h a t the solvent does not look very homogeneous from the viewpoint of an embedded solute, because the distance between the individual dipoles is similar to the dimensions of the solute (see Fig. 3.1). Thus, a continuum description for the nearest-neighbor solvent atoms in a polar solvent appears to be inadequate.
Fundamental criticism of the dielectric continuum approach
45
If we consider a typical polar solute having a dipole moment of ~2 Debye (D) in a solvent such as water, also having a dipole moment of about 2 D, the electrostatic reorientation energy, i.e., the energy difference between the "right" and "wrong" orientation of the solvent dipole in the field of the solute is ~22 kJ/mole (assuming a typical nearest-neighbor distance of 3.5/~ between the two dipoles). For the re-orientational polarization response the linear part arises from the first term in the Taylor expansion of the Boltzmann factor. Hence, we observe linearity only if the reorientational energy is small compared to the thermal energy, k T, which is about 2.5 kJ/mol at 25 ~C. Thus the reorientational energy in the first-neighbor shell is almost nine times larger than k T ~ i n s t e a d of being small compared to k T--and we cannot assume a linear response of the re-orientational polarization in the first- and even the second- and third-neighbor shells. Instead, we must realize that the re-orientational polarization is almost in saturation in this limit. This is also confirmed by the general knowledge of chemists that the first-solvation shells in polar solvents are highly ordered. If there is already a high degree of order, a virtual amplification of the solute dipole moment by a factor of two can impossibly lead to twice the degree of order, i.e., twice the reorientational polarization from the near-neighbor shells. But this is exactly what would be required in a linear-response theory. Some people replied to these arguments that the reorientational energy difference is strongly reduced by the collective effects in a polar solvent, i.e., by the fact that the other solvent dipoles would respond to the mis-orientation of a solute and thus considerably reduce the energy of this state. This argument is partly correct, but even if we calculate the reorientational energy difference of a solvent molecule in the presence of a perfect conductor surrounding the solute-solvent pair it still has a value of ~8 kJ/mol. The conductor is definitely an overestimation of the neighbor responses, and nevertheless the reorientational energy stays far beyond the linear response limit. Thus, without any doubt, the situation of a polar solute in a polar solvent is very different from the situation of a solute in a dielectric continuum. The solvent is neither homogeneous, nor does it give a linear polarization response. Hence, the considerable success of dielectric continuum solvation models for the
46
A. Klamt
solvent water must be considered as a real surprise, and as some kind of fortunate coincidence.
3.3
ANALYSIS OF THE SITUATION
In chapter 2 we learned that state-of-the-art dielectric continuum solvation methods such as PCM or COSMO have come to provide computationally efficient and accurate ways to extend the quantum chemical methods originally developed for molecules in the vacuum or gas phase toward molecules in dielectric media. In our global picture we may compare them with airplanes that provide a convenient means of transportation from the south pole to the latitudes of solvation. Unfortunately, these models have only one essential parameter, the dielectric constant ~s, which has a value of 1 (unity) at the south pole and infinity at the north pole, i.e., in the limit of a conductor. This essential one-dimensionality binds the planes to one longitude of the globe. Because of the different boundary conditions the longitudes of COSMO and PCM differ by one or two degrees, resulting in an observable deviation of their routes at the equator but, considering the entire width of the world, this difference does not really appear to be important. As discussed in section 3.1, the route of the dielectric continuum solvation models definitely leads across the non-polar solvents like alkane, but apart from these we cannot really expect to find any polar solvent on this route. Dielectric continuum solvation models definitely cannot describe the differences between solvents that have the same dielectric constant, e.g., owing to the electrostatic interactions by quadrupoles appearing in benzene in contrast to alkanes having the same macroscopic value of e. Furthermore, we should not expect the dielectric continuum solvation model to be able to describe any polar solvent correctly, and especially not water. Apparently, dielectric continuum solvation models do a reasonably good job in describing water. This may be because the solvent water, for reasons other than its high dielectric constant, really is located close to the north pole of our globe. Because the distances are small in the vicinity of the pole, even if one is at the opposite longitude, it may be that the dielectric theory leads us close to a valid description of the solvent behavior of water, nevertheless.
Fundamental criticism of the dielectric continuum approach
47
Having recognized the theoretical inadequacy of the dielectric theory for polar solvents, I started to reconsider the entire problem of solvation models. Because the good performance of dielectric continuum solvation models for water cannot be a result of pure chance, in some way there must be an internal relationship between these models and the physical reality. Therefore I decided to reconsider the problem from the "north pole of the globe," i.e., from the state of molecules swimming in a virtual perfect conductor. I was probably the first to enjoy this really novel perspective, and this led me to a perfectly novel, efficient, and accurate solvation model based upon, but going far beyond, the dielectric continuum solvation models such as COSMO. This "COSMO for realistic solvation" (COSMO-RS) model will be described in the remainder of this book.
Chapter 4
M o l e c u l a r i n t e r a c t i o n s a t the north pole: A v i r t u a l e x p e r i m e n t What does it mean to "look at the problem from the north pole?" Let us try to understand this by the following virtual experiment, in which we consider a liquid mixture of molecules. In all the figures in this chapter we just take a mixture of water and CO2 as a simple example. Step 1: The first step of the experiment is to bring all molecules to the north pole. This means to perform a COSMO calculation with e = oc for each molecule, or more precisely for each type of molecules. The COSMO cavities are defined in the usual way by smoothed spheres with atomic radii, which are ~20% larger than the usual vdW radii. Such cavities have proved to give the most reliable results in dielectric continuum solvation models. For the following discussion it is important to note that the volume of such COSMO cavities corresponds approximately to the liquid molar volume of the components in standard conditions. As a result of these COSMO calculations we know the total energies of the molecules in the virtual conductor, and we know the distribution of conductor polarization charges, i.e., the polarization charge density, a, on each point of the COSMO surface. The quantity a is a very good local measure of the molecular polarity and can thus be used to quantify and color-code molecular polarity on the surface. Such color-coded a-surfaces of water and CO2 are shown in Fig. 4.1. Many more will be shown and discussed in more detail in the remaining sections of the book. Before moving on to further discussions, we shall say a few words on dispersive or vdWs interactions. These are composed of an attractive force that arises from a sophisticated quantumchemical short-range interaction of electrons, and an even shorter range, i.e., almost hard sphere repulsion of the cores of the atomic 49
50
A. Klamt
Fig. 4.1. COSMO surfaces of water and CO2 color coded by the polarization charge density a. Red areas denote strongly negative parts of the molecular surface and hence strongly positive values of ~. Deep blue marks denote strongly positive surface regions (strongly negative ~) and green denotes nonpolar surface. electron distributions, the so-called Pauli repulsion. The first term is due to the correlation in the motion of electrons that leads to a lowering of the energy of electrons. It decreases with r -6 with an increase in the distance, r, of two electrons. Electrons in the outer region of the electron cloud of a molecule in vacuum lack correlation partners in half of the space directions. Only when another molecule gets in close contact, do both molecules gain the additional electron-correlation energies in their contact-surface regions. The resulting strong attraction is balanced by the extremely strong Pauli repulsion, which avoids a strong inter-penetration of the electron clouds of non-bonded molecules. As a consequence, the molecules experience a very steep energy valley if they are about at vdW-distance from each other. Only 1 A further out, the vdW energy gain is almost lost, as shown schematically in Fig. 4.2. Therefore, molecular surfaces experience a kind of binary decision: "Either get into intimately close contact with the surface of another molecule, or stay away and enjoy your freedom." This is the reason why we usually observe such a clear first-order phase transition between gas phase and condensed phases, and not just a continuous increase of the average distance of the molecules with increasing temperature. Therefore, we may assume that in a liquid system almost each piece of molecular surface has a vdW-interaction partner in about the vdW-distance. The strength of the vdW interaction can be expected to be roughly
Molecular interactions at the north pole: A virtual experiment
51
Fig. 4.2. Orders of magnitude and distance behavior of molecular interactions (schematic illustration). proportional to the electronic polarizability of the p a r t n e r s . Since the electronic polarizability in a first-order approximation can be considered as a constant in organic molecules, corresponding to an optical dielectric c o n s t a n t of ~r = n 2 --- 2, the a m o u n t of vdWs interaction per contact area is about - 4 0 kJ/mol/nm 2 or about - 2 0 k J / m o l / n m 2 per molecular surface area, if we take into account the fact t h a t in each contact t h e r e are two molecular surfaces of the same size. To a second order, t h e r e will be variations, and these are k n o w n to depend mainly on the elements involved in a contact. Therefore, to the second order we can express the vdWs surface energy between two elements, e and e', in the form evdw(e, e')
0 w + 5evdw(e) + 6evdw(e ,) -- ~vdw(e) + rvdw(e ,) 2evd
(4.1)
U n f o r t u n a t e l y , the exact values of the element-specific vdW surface energies r(e), are not simple to calculate. Therefore, we have to a s s u m e t h a t these are k n o w n from some fit to experimental data. As we can see in this equation, the vdWs energy i s ~ t o the second order of a p p r o x i m a t i o n ~ j u s t the sum of two i n d e p e n d e n t contributions of the surfaces involved in the interaction. T h u s we can a s s u m e t h a t this vdWs energy is gained individually by each molecule w h e n it is b r o u g h t from v a c u u m to the virtual continuum, which is our reference state for molecules in the liquid state. Hence, we assume t h a t this c o n t i n u u m behaves like a conductor with respect to electrostatics, and like an average organic vdW p a r t n e r with respect to dispersion. In this way, we do not have to
52
A. Klamt
consider vdWs interactions explicitly beyond the continuum representation, although they are qualitatively and quantitatively a very important part of the interactions in the liquid phase. Corrections beyond this picture are discussed in chapter 7. With no further approximations we can now consider an ensemble of molecules, which represents the components of the mixture under consideration as swimming around individually in the infinitely extended virtual conductor. Because the electric field of the molecules is perfectly screened off by the conductor, there is no interaction of the molecules. We know the total energy of this ensemble by just summing the total energies of the individual molecules in this virtual, but distinguished state, which we consider as the north pole of our globe (Fig. 1.1). Step 2: Obviously, the interactions in a real fluid are much more complicated than in the ideal and clean condition at the north pole. In order to come closer to a realistic picture, we now start to squeeze out the conductor from between our molecules, until we finally end with a system having about the right liquid density. Let us assume that during this process there is always at least an infinitesimal thin film of conductor left between two molecules. Such a thin film is sufficient to ensure perfect screening of the electric fields of the molecules. Since there are no interactions between the molecules in the conductor, the squeezing of the conductor does not result in any change of energy as long as the molecules can keep their original shape and cavity. Finally, we need to slightly deform the individual molecules by pressing them together in order to get a close packing with about the right density. Let us assume that the cavities are slightly deformable if the entire volume is conserved under the deformation. Such deformation will cause the cavity to get a little closer to the atom centers at some part of the cavity, but a little further away in other parts, in order to conserve the volume. As a result, the dielectric interaction energy will increase slightly in some parts, but decrease in neighboring parts. In summary, we may assume that this kind of deformation costs only very little energy, and that on average the energies and polarization charges of the molecules stay the same as in the conductor. Because the volumes of the original COSMO cavities correspond well with the molar volumes in the liquid phase, there is almost no conductor left in the system when we finally reach the right density. We only have
Molecular interactions at the north pole: A virtual experiment 53 molecular cavities closely nestled against each other, and a thin film of conductor separating them. This situation is shown schematically in Fig. 4.3, where the grey lines may represent the thin film of the conductor. With the approximations mentioned above, we may assume that this system still has the same total energy as the systems of molecules swimming in the bulk conductor. Step 3: Although our system now looks much more like a liquid system, the thin film of conductor separating the molecules is an artifact. In n a t u r e there is no conductor between molecules, and hence we have to get rid of the conductor to get closer to the real situation. For doing this it is helpful to assume t h a t all polarization charges are now frozen, which does not make a real difference but may help our imagination during the following steps. Let us consider what happens if we take away the conductor on a small piece of molecular contact of size, aeff. Let the average polarization charge densities arising from the two neighboring molecules on this piece of contact surface be a and ~'. In this
Fig. 4.3. Schematic picture of an ensemble of molecules with COSMO cavities after squeezing out the bulk conductor. The grey lines indicate the residual thin film of conductor separating the cavities.
54
A. Klamt
situation, the net charge density on the contact under consideration is just a + a'. If a + a' is zero, as shown for one segment in Fig. 4.3, we have an electrostatically ideal contact. The conductor does not need to contribute to the electrostatic screening on this piece of contact, because the two molecules would already screen each other in the same way. Therefore, we can take away the conductor in this situation without any electrostatic energy difference. Although such pairing of opposite a-values is energetically optimal, the normal situation will obviously not involve such ideal pairing of exactly opposite polarities because, in reality, thermal fluctuations, steric hindrance, or even the non-availability of the right partner, will cause some electrostatic misfit, as shown for another contact in Fig. 4.3. Since the electrostatic potential on each piece of molecular surface is zero, as long as it has the conductor present, the removal of the conductor on a piece of surface corresponds to bringing the countercharge density of + ~' from infinity to this surface area. In a non-polarizable environment this would cost an electrostatic energy:
0 ~ (~ -4- 0.~)2 Emisfit((~, (~') -- aeff -~
(4.2)
The self-energy coefficient can be calculated simply from basic electrostatics. It depends on the size of the contact, and slightly on the shape, but assuming nearly circular contacts should be a good approximation. Since the electronic polarizability of the molecules will respond to the new situation, the electrostatic misfit energy is better approximated if we take this into account by reducing the energy by a portion, fdielectric(8~ ~-- 2) ~ 0.4. Thus, we obtain the expression
emisfit((~, (T') -- (1 --f(~cr
~ ((~ + a,)2 __ ~ ( a + a,)2
(4.3)
for the misfit energy density per unit of contact area. The functional dependence on the polarization charge densities is beyond any question, and for the coefficient ~' we have a good physical estimate as soon as we know the size of the contact area, ae~. Thus, we may assume for the moment that the electrostatic misfit surface energy density is known. Because we have already included the vdW energies in the continuum energy up to second order, only higher-order contributions would arise from the replacement of the continuum by a real
Molecular interactions at the north pole: A virtual experiment 55 molecular partner. Thus, we can neglect vdW interactions in this step of local conductor replacement. There is a third kind of important interaction which needs to be taken into account, i.e., the hydrogen-bond (hb) interactions. We shall consider a contact of a very polar hydrogen atom, an hbdonor, with an strongly charged electron lone-pair, an hb-acceptor, as shown in the upper right in Fig. 4.3. After removing the conductor, the hb-donor will start to penetrate into the hb-acceptor density. This goes along with a large enthalpy gain, and a specific re-orientation of the donor relative to the acceptor. Since the latter causes a considerable loss of entropy compared with less specific normal interactions we have to express the hb-interaction as a free-energy gain. Any attempt to quantify the hb-interaction energy by quantum-chemical means would require tremendous effort, because a very high methodological level is needed to achieve a good description of hydrogen bonding, even in vacuum. Therefore, we try to parameterize it empirically with the information we have available in our model. Hydrogen bonding only appears if two surface segments with strong and opposite polarity get in contact, and the hb energy should be stronger as both partners become more polar. Such behavior can be described reasonably by the formula ghb(0", 0") -- Chb(T) rain(0, rain(a, a') 4- tThb) max(0, max(a, a') - tThb )
(4.4) which expresses the hb free-energy per unit surface area gained in a contact of two molecular surfaces with polarization-charge densities a and a'. Other, simpler or more complicated formulae can be used as well, with only small differences in the results. The form given in Eq. (4.4) is the one presently used in our parameterizations. In this expression, min(a, a') denotes the more negative a, i.e., the more positive polarity of the two pieces of surfaces. This surface would obviously take the part of the hbdonor. Only if this donor is more polar than a certain threshold, i.e., if its a is more negative than ahb, can it act as a donor in a hydrogen bond. Conversely, its partner must overcome a certain threshold polarity to act as an acceptor, i.e., max(a, a')>(Thb is required. The parameters Cab and 6hb, and the detailed temperature dependence cannot be derived from theoretical arguments. They have to be determined later by fitting to experimental data.
56
A. Klamt
Let us for the moment assume that parameters of the hb-interaction energys are already known. We have thus now expressed the energy difference resulting from the local removal of the conductor between two molecules as a function of the two surface polarities, a and a', of the interacting molecular surfaces. All relevant kinds of intermolecular interactions, i.e., vdW interactions, electrostatic interactions, and hydrogen bonding are taken into account. Summing the local contributions up over all contact surfaces should finally lead us to the total energy difference of the real system without a conductor to the ensemble of perfectly screened molecules, which was our energetic reference point. Since we know the energy of the latter from the initial COSMO calculations, we then have the total energy of a realistic ensemble representing our liquid system. At this point it should be noted that an additional electrostatic contribution results from the interactions of the misfit charges, i.e., Ecorr 1 misfit "-- 2
~ ij
aiaj
(ai + a~)(aj + aj) _+ _ + II ti -- tj II
(4.5)
Here the summation goes over all contact surfaces i and j, and ai, oi a n d aj, a n d ti are the area, polarization charge densities, and center position of the contact segments, respectively. Fortunately this energy contribution essentially sums up to zero, if the misfit charges are not correlated, because in this case we have about the same positive and negative contributions. At least in liquids consisting of neutral compounds there is no reason for any preferential sign of the misfit charges in certain regions and hence we can safely neglect this contribution in these cases. Whether this is also true for electrolyte systems, where some local sign correlation may be important due to the presence of ions, must be kept as an open question for the moment. We have seen in this chapter how the total energy of a liquidlike ensemble of molecules can be calculated starting from COSMO calculations and taking into account deviations from the simple conductor-like electrostatic interactions as well as hydrogen bonding and vdW interactions as local pair-wise interactions of molecular surfaces. This is a very different way of quantifying the total energy than the ways usually used in all kinds of
Molecular interactions at the north pole: A virtual experiment 57 molecular mechanics (MM) or quantum mechanics (QM) calculations, where the total energy is evaluated by demanding summations of non-local pair-wise interactions of atoms, or of nuclei and electrons. Although chronologically incorrect, since I myself did not realize these relationships at the time of development of COSMORS, it is worth mentioning the similarity of the COSMO-RS view of liquid-phase interactions to that of the widely used empirical liquid-phase thermodynamics models, as there are the Flory-Huggins theory [77], Guggenheim's quasi-chemical theory [78], UNIQUAC [79], UNIFAC [18], and many others. All these theories work with a kind of local group interaction parameters, or more precisely local group surface interaction parameters, but all of them treat these parameters as empirical fit parameters to be adjusted to experiment. In some way, the COSMO-RS picture of first bringing the molecular surfaces close to each other within a conductor, and finally removing the conductor for the first time, provides the chance of connecting the models for surface interactions with real molecular interactions, and of calculating the surface interaction parameters from molecular structures. But the COSMO-RS picture allows for a much more detailed picture of the local surface interactions, since it does not require an averaging of the interactions over entire functional groups that is required in group contribution methods for a limitation of the degrees of freedom of the fit. Nevertheless, we are left with a big problem at this point: the energy of a single configuration of an ensemble is almost worthless for the evaluation of fluid-phase thermodynamics, because the total free energy must be evaluated as a statistical average of a large number of different configurations in order to take into account all the different ways in which molecules can contact and interact with each other. This part is usually solved by MD or MC techniques in the context of QM or MM calculations. Unfortunately, our COSMO-RS concept only provides us with the concept of how to evaluate the energy of a given liquid-like configuration, but not of generating such a configuration because, owing to the missing distance dependencies, we have no concept of the forces in our model. Having no forces, it is impossible to apply MD, and even finding reasonable configurations required for MC statistics is not feasible. Hence, if we wanted to follow the classical
58
A. Klamt
pathways of statistical molecular mechanics, we would have to combine COSMO-RS with a kind of force-field MC approach to generate configurations, which would then be energetically quantiffed by COSMO-RS. In this way we appeared to be in danger of losing most of the benefits we had achieved previously with our novel way of description of interactions in the liquid phase. Therefore, I decided to try another way out, which finally proved to be very successful. This is described in the next chapter.
Chapter 5
Statistical thermodynamics of interacting surfaces 5.1
THE STARTING POINT AND NOTATION
The basic idea for overcoming the necessity of a time-consuming statistical thermodynamics treatment by a molecular MC simulation method arises from the fact that we have only local pairwise interactions of surfaces in our COSMO-RS interaction model. Hence, for the evaluation of the interaction energies we only need to know which piece of surface is an interaction partner of which other, while in the usual 3D interaction models we need complete information about the coordinates and distances of all particles. Thus, one day I had the idea of testing an approximation in which we replace the complicated statistical thermodynamics of the 3D molecules (with all their complicated geometrical constraints) by the much simpler statistical thermodynamics of an ensemble of independently pairwise interacting surface segments. In such an ensemble we would consider only the pairing statistics of molecular surface segments of approximately equal size, aeff, which are cut out from the surfaces of the molecules in the liquid ensemble. We would completely disregard any kind of geometrical constraints and geometrical correlation that result from the fact that, in reality, the molecular surface segments have fixed neighbor segments in the molecules. I myself considered this approximation as very crude, but wanted to know where it leads to. When I discussed this idea with computational chemistry colleagues, I experienced strong resistance because molecular 3D geometry is usually considered as most important for all kinds of property calculations in computational chemistry. People could not believe that the approximation could make any sense, and, later, one reviewer of the first COSMO-RS paper called the idea 59
60
A. Klamt
weird and crazy. At that time, neither they nor I was aware of the fact that models of pairwise interacting surfaces had been quite popular and successful in chemical engineering thermodynamics for decades. The internal relationship between my proposal and the empirical approximations was not apparent, because these models, which do not deal with "real molecules," were almost unknown to the computational chemists. I am afraid that such situations are not uncommon, in which separate communities of scientist are working on closely related problems but ignore each other, since they have developed very different approaches and notation, which are almost not understandable for the members of the other community unless they put serious effort into it. Since the readers of this book will come from different communities, I have tried to develop a notation here which will be acceptable for most readers. After that we go on and consider the different empirical approaches and the novel COSMOSPACE approach for the statistical thermodynamics, which was specially developed for COSMO-RS. In the following discussion we will consider an ensemble of molecules that represents a liquid system. Assuming there is no correlation between molecular interactions and geometrical restraints, the partition sum Z of an ensemble can be factorized into three contributions" Z -- z ~
(5.1)
R
If N is the total number of molecules, and N i - Nxi is the number of molecules of species i in the system, then the first factor is the ideal entropic contribution arising from permutations of identical particles. In the thermodynamic limit of large particle numbers this is well approximated by lnZ ~ - N ~
xi l n x i
(5.2)
i
The second factor on the right-hand side of Eq. (5.1) is called the combinatorial factor. It is the partition sum of the equivalent ensemble of molecules that interact only through steric restraints. The combinatorial factor takes into account all size and shape effects of the molecules. There is no exact expression for Z c but, by fitting the simplified theoretical models to thermodynamic data of
Statistical thermodynamics of interacting surfaces
61
alkane-alkane mixtures, reasonable approximations have been derived. The Staverman-Guggenheim (SG) expression [78,80], In Z~sG - - N
~.
[xi In (Oi) z ~-/ +-~xiqi
(oi)]
In ~//
(5.3)
is used in many chemical engineering models, e.g., in UNIQUAC. The variables xi, r and Oi denote the mole, volume, and surface fractions of species i, respectively, with the volume and surface fractions defined in terms of the relative volume, ri, and relative area, qi, respectively. Both ri and qi are based on van der Waals cavities and normalized in a consistent way. r and q values have been tabulated for many molecules [81]. The UNIFAC group contribution method can also be used to estimate the molecular r and q parameters on the basis of a summation of group increments [82,83]. Obviously, if we have full 3D models of the molecules available from QM/COSMO calculations, we can also use volumes and surfaces derived directly from these. One should be prepared to find some differences in the empirical scales of UNIQUAC and UNIFAC, because the latter has some of these parameters adjusted in order to achieve better agreement with liquid thermodynamics data. The coordination number, z, is commonly assumed to be 10. For mixtures of compounds that differ by less than a factor 5 in size, the combinatorial factor is usually of moderate importance, and the SG approximation can be considered to be sufficiently accurate for the purposes of this chapter. Other functional forms of the combinatorial contribution have been suggested and are used in some models, but at present it is sufficient for us to assume that reasonable models for this part are available. The third factor, Z R, in Eq. (5.1) is called the "residual" contribution in the chemical engineering notation and it arises from all kinds of non-steric interactions between molecules, i.e., usually from vdW, electrostatic, and hydrogen bond interactions. Despite its name, it is the most important contribution in most liquids. The basic assumption of surface-pair interaction models is that residual--i.e., non-steric--interactions can be described as local pairwise interactions of surface segments. The residual contribution is just the partition sum of an ensemble of pairwise interacting surface segments.
62
A. Klamt
If we consider a system having Vmax different types of surfaces, the total number of surface segments ni on a molecule i is given by ,, n i --
ni-~-~ v
(5.4)
qi
aeff
where qi is the total surface area of molecule i, and n~' denotes the number of segments of type v on molecule i. Please note that we here and in the following are using the summation convention, that a summation index always runs over all of its possible values, unless noted otherwise. In the case of Eq. (5.4) this means that the summation is over all possible segment types v, i.e. for v= 1, 2, ..., Vmax. aeff is the size of the surface segments. There is no simple way to define a~ff from first principles and it must be considered to be an adjustable parameter. The total number, n, of segments in the system is given by n-
~
(5.5)
Nini i
In the same way, the number of segments of type v is given by n " -- R
(5.6)
Nine' i
The relative number of segments of type ~' is n r
(5.7) n From elementary statistical thermodynamics we know that the Gibbs free energy of the system is G--kTln Z (5.8) O~' ~
The chemical potential of species i in the mixture is given by ~G 0 In Z P i : ON---~= - k T c~N~ (5.9) In view of our earlier assumption that the residual partition sum Z R depends only on the segment composition, we have pR_ _kT01nZ R ONi - - k T E = - kT ~01n
ZR 0 In c~n" ,. c~n" c~Ni
ZR c~n,------:-n~'- E
n~'tl"
(5.10)
Statistical thermodynamics of interacting surfaces
63
where pR is the residual chemical potential of compound i, and pv the pseudo-chemical potential of a segment of type v in the ensemble of interacting surface segments. Following Ben-Naim [93], the pseudo-chemical potential is just defined as the true chemical potential minus the ideal entropic contribution, kT ln x", where x" is the fraction of species v in an ensemble. Thus, the pseudo-chemical potential is well defined at infinite dilution, while the true chemical potential diverges at this end, owing to the quantity k TIn x v. Although it is not really needed at this point, we introduce the concept of activity coefficients, which are very common for chemical engineers, but much less for theoretical and computational chemists. In the standard definition, the activity coefficients make reference to the pure liquid compound, i, at the same temperature and pressure. Hence, the activity coefficient 7i of species i is defined by kTln
(5.11)
7i - Pi - Pii
where Pii denotes the pseudo-chemical potential of compound i in a system with xi - 1, i.e., in pure compound i. Note that in some situations, when the pure compound is not liquid, other reference systems are chosen, e.g., for gases or electrolytes, but we will not discuss these complications at this point. In the same way as for the molecules we have kTln
(5.12)
7 ~' - pv
for the activity coefficient of a segment of type v, where we avoided the introduction of the reference segment chemical potential pr,. at O " - 1 , because it would cancel out in Eq. (5.13), and hence is of no relevance. If we define Piv and ~," [i to be, respectively, the chemical potential and activity coefficient of a segment, v, in an ensemble of pure compound, i, and apply these definitions to Eq. (5.10), we obtain /Aa
lnv R=
R
pv
~ k T- Pii -- ~
ni"
v
k T- Pi
V
,, ( p , _
-
Zni
= ~
pr,) _
kT
n~'(ln 7 " - I n 7~') V
(5.13)
64
A. Klamt
for the residual part of the activity coefficient of compound i. With the development of Eq. (5.13) we have reduced the thermodynamics of a system of chemical compounds to the thermodynamics of an ensemble of pair-wise interacting surface segments. We now consider this ensemble in more detail. Let p and v denote different kinds of surface segments, and let e~,v be the interaction energy of a pair, Zv. For a given configuration P of the ensemble, the total interaction energy Etot c a n be written as a sum of pairwise interactions of segments: Etot(P)- ~
P,r(P)et, r
(5.14)
pv
where p~,r(P) is the total number of pairs of kind pv. The partition sum Z R of this ensemble is zR = ~-~ eXp{p
_
~,~p,,, } (P)e,r k T
(5.15)
where P samples all possible total pairings of the segments. Unfortunately it is extremely complicated to evaluate all the different configurations with respect to the number of pairs of kind pv, and hence the computation of Z R is non-trivial. Several different approximations have been introduced, most of them being known as separate models. 5.2
PREVIOUS APPROACHES: FLORY-HUGGINS THEORY AND QUASICHEMICAL THEORY
The simplest model for the residual part of the partition sum uses the assumption of random mixing (RM). This corresponds to p~v(P)- nOlO r (5.16) i.e., to the approximation that the probability of finding a contact p, is just proportional to the product of the segment fractions of and v. This is the interaction model developed by Flory [77]. In the terminology of computational chemistry this would be called a "mean field approximation," because each segment just experiences the mean value of its interaction energy with all others, without any correlation due to specific interactions. Since the
Statistical thermodynamics of interacting surfaces right-hand side of Eq. (5.16) is independent configuration, P, Eq. (5.15) thus takes the form In ZRM =
n
2 ~
~I,,' O"O"~--~ -- In N p
65
of a specific
(5.17)
where N p is the total number of configurations that can be generated from n segments. With this simplification we easily find k T l n ~'R"iM
- - flRMr __ /2RM""
- - ~
O" ~I,,'k-Te''''
(5.18)
P
However, the random mixture approximation is reasonable only for weak interactions, i.e., as long as the differences in interaction energies between different segments are small compared with the thermal energy k T. Otherwise, we get preferential pairing of segments with relatively low interaction energies, i.e., the contact probability for the neighbors of a segment, v, deviates from the overall segment composition as given by O '. This effect is sometimes called "local composition" in chemical engineering literature, although it does not really have to do with a local composition, but with preferential pairing of segments that like each other. Guggenheim [78] was the first to develop a theoretical approach that tries to take the local composition into account. His approach has become famous as the "quasi-chemical approximation" (QUAC). For the derivation of his model, Guggenheim used the atLxiliary concept of a lattice, but did not specify a concrete lattice geometry. He considered a simple binary system with molecules A and B, each occupying a single lattice site and each having z faces. Guggenheim considered the contact probabilities, p~,,., for all combinations of surface types, p and v, i.e., in his case for AA, AB, BA, and BB pairs, as independent variables. Thereby he neglected "arrangement effects" as illustrated in Fig. 5.1. Although Guggenheim used a slightly more complicated argument, his approach is equivalent to the ad hoc assumption that the total free energy of the system can now be written as a sum of contributions from all quasi-chemical reactions (p, p) + (r, r) --+ 2(p, v): GR _
1~-~. pt,,.(kTln(p~,,.)+ u,,,.)-89 ~-~p,,,.kTln(Pl-~'"~ pr
pr
\'C p~,]
(5.19)
66
A. Klamt
Fig. 5.1. Arrangement effects in a quadratic 2D lattice. By placement of three blue objects one site is generated that has two blue neighbors. with ui,,' -
u~,,.
being half of the reaction (free) energy:
(2el,,'-
el, l , -
~,.,.)/2
(5.20)
and introducing the widely used interaction parameter notation ~ l,,' -
e x p { - k Tu l'''
}
(5.21)
Since the sum of all contact probabilities of a segment type v is equal to the surface fraction O", i.e., p ~ , . - O"
(5.22)
p
and owing to the symmetry condition, p t,,. - p , . t , , the total number of independent contact probabilities is rmax (rmax--1)/2. Finally, the independent, unknown contact probabilities, p~,,. with r < p, can be found by variational minimization of the total free energy G, i.e., from the set of equations: c3 GR _ 0
O~yr
(5.23)
Thus, in the case of Vmax- 2, as originally considered by Guggenheim, there is only one independent contact probability,
Statistical thermodynamics of interacting surfaces
67
i.e., PAB, and minimization of Eq. (5.19) leads to a quadratic equation with solution: 1/2 A
V/1 + 4oA(1
];QUAC --
-
oA)(T~]~
--
1) + 1 - 2(1
-
O A)
( V / 1 1 + + 4 o A ( 1 - oA)(r~ -- 1))O A (5.24)
for the residual activity coefficient of segments of type A. The GEQUAC model of Egner et al. [84] is the only model that uses the generalized QUAC for an arbitrary number of segment types, Vmax, without additional approximations by numerical solution of the minimization problem given by Eq. (5.21) for Vmax (rmax- 1)/2 independent variables. Only recently did I note that they are using an algorithm proposed by Laarsen and Rasmussen [85] for the solution of the resulting equations, which is equivalent to the COSMO surface pair activity coefficient equations (COSMOSPACE) algorithm derived independently in the context of COSMO-RS and to be described later. To our best knowledge, all other models use additional approximations. The UNIQUAC approximation of Abrams and Prausnitz has become very popular [79]. Starting from the QUAC and introducing additional heuristic approximations they ended up with a much simpler set of equations, which yields explicit expressions for the activity coefficients: In ];UNIQUAC ~ --
--
, r~,~.O~, - ~
In ~
O,.r'~.,. + 1 ~,r'~,,.O~,
(5.25)
ll
with the interaction parameter convention
l This definition of interaction parameters is asymmetric, i.e., the interaction parameter of a segment i with a segment j is different from the interaction parameter of j with i. This is physically counter-intuitive, and has indeed caused a lot of discussion in the literature because it leads to a mathematical inconsistency in the contact probabilities of i and j [86]. Its explicit solution for general number of surface types made UNIQUAC easy to use in the times when computers where rare and slow. Therefore, UNIQUAC
68
A. Klamt
Fig. 5.2. Comparison of the exact quasi-chemical results (CS) and UNIQUAC (UQ). For details see [C1]; r and q are UNIQUAC volume and surface parameter, respectively.
became widely used as an empirical thermodynamic model for the correlation of experimental mixture thermodynamics data. Nevertheless, it must be pointed out that, even by using the same model parameters, UNIQUAC deviates considerably from the original QUAC for strong interactions, i.e., for interaction parameters much smaller or much larger than unity, as illustrated in Fig. 5.2. The group contribution method UNIFAC [18] is based on the UNIQUAC thermodynamic model as well. It thus suffers from the same thermodynamic approximations as UNIQUAC, especially for strong interactions in the infinite dilution limit.
5.3
THE COSMOSPACE APPROACH
As mentioned earlier, I was not aware of the details of the chemical engineering models and Guggenheim's QUAC approximation, when I developed the COSMO-RS model. From a brief glance at that part of the literature I had got the impression that all these models are basically lattice theories, while I wanted to
Statistical thermodynamics of interacting surfaces
69
have a solution for the problem of pairwise interacting surfaces, without having any lattice in mind. However, I did not see a clear relationship at that time and hence followed my own thoughts. The model which I had in mind was just a model of independently pairwise interacting surfaces with an arbitrary number, Vmax, of surface types with surface fractions, O~., and a given symmetric interaction matrix, e~,.. The problem was, how to find the thermodynamically correct solution for the pairing statistics of such a simple ensemble. I was not experienced in statistical thermodynamics at all and took a few text books and the basic knowledge which I remembered from the thermodynamics course at university. I tried perturbation theory starting from a zerotemperature solution, and several other approaches, but the problem turned out to be harder than expected. I am grateful to my colleagues Jannis Batoulis and Peter Hoever, who spent some minutes every morning for fruitful discussions about the latest progress or failures. One morning I awoke at 4 a.m. and had a new idea, which I immediately recognized as the breakthrough. We shall call this idea the COSMO Surface Pair Activity Coefficient Model (COSMOSPACE) in the following discussion, although this name was only introduced many years later. The concept of COSMOSPACE is as follows. Let us assume that all surface segments in our ensemble are distinguishable, i.e., we have only one segment of each surface type, v, but a very large number of surface types, M = Vmax. Starting from our COSMO-RS view of molecular interactions this assumption appears to be reasonable, because in a given snapshot of the ensemble of molecules, each molecule will have a slightly different geometry due to vibrations, and hence there should be minor differences between all segments, even if they were taken equivalently from molecules of the same kind. These M segments can form M/2 pairs, and hence there must be M/2 contact sites in our system to place the pairs, each contact site providing two distinguishable segment sites. We do not need any assumption about the spatial distribution of the contact sites, i.e., no kind of lattice architecture. Let us introduce a numbering for the segment sites, in which each consecutive odd-even pair denotes a segment pair. A unique state of the ensemble can now be characterized by a placement P of the M segments on the M segment sites. Obviously, there are M! different placements. In
70
A. Klamt
this way, we can write the partition sum as Z-
~
P
exp
-1
M/2
}
-k-~Egv(2i_l;p)r(2i;P) i=l
(5.27)
where v(i;P) denotes the segment type residing on site i in placement P. On the other hand, we can construct the partition sum in the following way. First choose any segment, ;.. There are M possibilities for placing this segment on the segment sites. Then we can characterize a state by the partner v of )~. After having specified 2 and v, we are left with the placement of the remaining M - 2 segments on the remaining M - 2 sites. If we denote by Z_;._,, the partition sum for the segments 2 and v, we can Z-
y ~ M exp{-C';"}Z_;._,. kT
(5.28)
Since the chemical potential of a species p is defined by the derivative
;. _ 0N;. c~ G - - k T #tot
~ c3 In Z
(5.29)
and
in (Z_z_,.) _ in(Z_;.,_,.) _ ln(Z) ~
//tot -~-//tot
c3 ln Z _ ~ c3 In Z c~N;.
(5.3o)
kT
we can divide both sides of Eq. (5.28) by Z and rewrite it as 1-
~
M exp
{--g;.v } Z_;.,-v kT
Z
1,'
- R
,.
M exp{ -~;v + pt;~ W p[~ } kT
(5.31)
Note that the small approximation indicated in Eq. (5.30) becomes exact in the thermodynamic limit of large ensembles. Re-introducing the pseudo-chemical potential as //tot- k T l n O;,
Statistical thermodynamics of interacting surfaces
71
and using O; - O v - M -1 and Eq. (5.12) we obtain the equation -1
(5.32) and the corresponding Gibbs energy is Ot'[ln(7 ~') + ln(O')]
V R - kT ~
(5.33)
p
These equations are called the COSMOSPACE equations. They provide a set of Vmax non-linear equations for the Vmax unknown segment activity coefficients, 7;- Empirically we can say that the COSMOSPACE equations can be converged rapidly to a unique solution using a recursive procedure or other solvers of non-linear equations, although we do not have a mathematical proof for the uniqueness of the solution. A more detailed derivation of COSMOSPACE is given in [C17] and in Appendix A, taking into account the degeneracy of segments, but this does not change the final result. A proof of the Gibbs-Duhem consistency of COSMOSPACE results and the development of analytic derivatives are given in Appendices B and C, respectively. Since we have made no additional assumptions during the derivation of COSMOSPACE, we can be sure to have an exact thermodynamic solution for an ensemble of independently pairwise interacting surface segments, rigorously derived from the partition sum. This is an important difference from all other segment interaction models considered so far, because none of them is rigorously derived from a well-defined ensemble, but makes additional approximations. Thus it turns out that, being ignorant in the field and therefore not influenced by previous work, I found an exact solution for a long-standing problem. 5.4
EQUIVALENCE OF COSMOSPACE AND QUASICHEMICAL APPROXIMATION
As a response to an early draft of our COSMOSPACE publication someone told us that our analytical solution for the COSMOSPACE equations for the case of only two surface types, as presented in Appendix C of that paper, is equivalent to Guggenheim's result of the QUAC. Although they appear quite
72
A. Klamt
different, both can be transformed into each other by simple algebra. Since Guggenheim's QUAC is a lattice theory while COSMOSPACE does not use any lattice concept, we considered this to be as an accidental coincidence. Only when starting a collaboration with Klaus Lucas two years later, did he mention that COSMOSPACE should be equivalent to the generalized QUAC, as for the example used in GEQUAC [84]. Thus, we tried to substitute the COSMOSPACE solution given in Eq. (5.32) into the generalized QUAC expression for the total free energy given in Eq. (5.19). For that we need to express the contact probabilities, p/,,., in the notation of COSMOSPACE: p . , . - "/'0~'7"0"1:.,
(5.34)
Inserting this into Eq. (5.19) and using the COSMOSPACE equation (5.32) yields
G R - 89
p~,,.kT ln(P~"~ - 8 9 ~ pv
= 89
k, r/Iv J
~
= kT ~
7"O"7"O"r,,,.[ln(7") + ln(O") + ln(7")+ ln(O")] 7"|
ln(|
~
7"| v
y
= kT ~
kT',./'O/",'"O"r,,,, ln(7"0'7"0 ')
fly
7"Ol'[ln(7 ") + ln(O")](7") -1
P
= kT ~
O"[ln(7") + ln(O~)]
(5.35)
P
Thus, we see that the QUAC expression for the Gibbs free energy as given in Eq. (5.19) becomes identical with the respective COSMOSPACE expression (Eq. (5.33)), when the COSMOSPACE results for the contact probability of segments is used. Therefore the COSMOSPACE equations also provide a solution for the QUAC model, and both models are in general equivalent. Shortly later, before we could publish this finding, it was published by Panayiotou [87], who had started to work on COSMO-RS only a few months before. This result is somewhat surprising. It reveals clearly that the concept of a lattice is not required for the QUAC. COSMOSPACE shows more clearly the model approximations than QUAC. The only approximation is the assumption that all surface segments
Statistical thermodynamics of interacting surfaces
73
can form pairs absolutely independent, without any restraint by neighborhood effects. The statistical thermodynamics of such an ensemble of independently pairwise interacting surfaces is then solved exactly. Even if we have to realize that the COSMOSPACE thermodynamics model is equivalent to Guggenheim's theory, its independent derivation as an exact solution of a non-lattice model shed new light on it, and also stimulated discussion about a broader usage as a thermodynamic regression model in chemical engineering simulation. Indeed, now that we know the relationship between COSMOSPACE and QUAC, there is an even simpler way to derive the COSMOSPACE equation. If we assume that all segments are independent objects, then there must exist segment activity coefficients ~,,~' for all kinds of segments, so that the activity of segment type p is equal to 0 ~' 7 ~'. The probability of finding a "reaction product" pv, i.e., a segment pair pv, is given by the product of the activities of p and v multiplied by the Boltzmann factor for the reaction energy, r~,~.. This relationship is expressed in Eq. (5.34). By combining this with the boundary conditions for the sum of the contact probabilities of segments, as given by Eq. (5.22), we obtain o " -
p,,,. P
-
(5.36)
P
P
By dividing this by 0~' 7v we immediately have the COSMOSPACE equations. In this way the COSMOSPACE model appears to be almost trivial, nevertheless it was not so simple to find it.
5.5
COMPARISON WITH LATTICE MONTE CARLO SIMULATIONS
Lattice Monte Carlo (LMC) simulations are another approach for the study of the principles of liquids thermodynamics. In these simulations it is assumed that molecule-like objects can occupy one or more sites of a predefined regular lattice. Mostly, only nearest-neighbor interactions of the objects are allowed, resulting in a kind of surface interaction, eij, between objects, i and j, on neighboring lattice sites. Given certain mole fractions xi of the different objects, the configuration space of such a lattice
74
A. Klamt
ensemble can be sampled quite easily with a simple MC algorithm, in which basic moves are defined, typically as interchange of two randomly selected objects. Such a move is accepted, if the total energy of the system is lowered as a result. If the total energy increases by AE, then the move is accepted only with a probability corresponding to the Boltzmann factor, i.e., with probability exp{-A/kT). As proved by Metropolis et al. [12], the statistical averages of the total energy and entropy of the ensemble converge toward the thermodynamically correct values. Usually, periodic boundary conditions are used to avoid surface artifacts. Thus, the thermodynamics of lattice models can be solved without any further approximation, if LMC simulations are performed for sufficiently long times and sufficiently large lattice boxes. Such simulations have only become practical during the past 10-15 years, because they can be quite computerintensive, but nowadays they can be made quite simply on PCs in a few hours. In this way, they can be used to test the approximations made in different surface-interaction models in comparison with the "exact" results of the corresponding LMC model, but one has to keep in mind the fact that the LMC results are only exact thermodynamic results for the underlying lattice fluid model, which may itself be a very crude approximation for a real fluid. Nevertheless, it is interesting to compare the COSMOSPACE results with those of LMC simulations in order to learn about the error introduced by the assumption of independently pairwise interacting surface segments. Given any lattice fluid model with nearest-neighbor interactions, this can be easily translated into the notation of COSMOSPACE by counting the different kinds of interaction sites of the objects and translating them into surface-fractions. Using the same interaction energies, c.ij, as in LMC, we have all we require for COSMOSPACE and can simply solve the set of Eqs. (5.32), finally yielding the 7'' for all kinds of surface, v. Using Eq. (5.13) we then obtain the activity coefficients of the lattice objects. Now, all thermodynamic properties can be easily evaluated. Wu et al., [88] compared several local composition models with LMC simulations for lattice mixtures. The models tested included: UNIQUAC, the AD model for lattice fluids of Aranovich and Donohue [89], and the Born-Green-Yvon (BGY) model of Lipson [90].
Statistical thermodynamics of interacting surfaces
75
Following Wu et al., we calculate the internal energy and the internal energy change of mixing from U ~ H-
\~1-~]
(5.37)
y~v~
Expressions for the internal energy of the models tested by Wu et al. are given in their paper. For the homogeneous COSMOSPACE model we obtain a particularly simple expression (with "cii -- 1) Z u
-
(53s)
i
j
The mean field internal energy, Umf, is given by Z Umf - ~ ~ l
~. j
(5.39)
xixj~,ij
The (dimensionless) internal energy change of mixing ( A U / R T ) is obtained by subtracting the ideal mixture internal energy from the internal energy calculated as given above. For a cubic lattice mixture of monomers, we may set all the q's to unity, thus forcing the .segment fractions to equal the appropriate mole fraction ( O ' = xi). The coordination number, the number of nearest neighbors, is 6 in such a lattice mixture. By using specified values of the interaction energies, ell, ~ 1 2 e21, and ~22, for a number of such systems, we calculate the COSMOSPACE interaction parameter from, T12 -- 1=21 --
exp[--(~12 --2!(t;11
+
e22))/kT]
(5.40)
Figs. 5.3 and 5.4 show the internal energy change of mixing for lattice mixtures of monomers with properties as specified in the figure captions (these correspond to the systems in Figs. 2 and 4 of Wu et al. [88]). In addition to the results obtained with COSMOSPACE, we also show the results from our own MC simulations as well as those obtained using UNIQUAC (which here is identically equal to the Wilson model), the AD model, and the BGY model. From the results shown here (which appear to be in complete agreement with those of Wu et al. [88]) we see that the original UNIQUAC is the worst of the models (when compared to the MC simulations). The BGY method (which is the proper way to use UNIQUAC) is significantly better than UNIQUAC, but it is not a particularly good approximation to the
76
A. Klamt 0.30 0.25 [ 0.20 COSMOSPACE I-
0.15
9
o.lo
~..~AD BGY UNIQUAC
\\
o.o5
"
0.00 T ~ ' - 00 0.2 -0.05
0.4
MC-results
\
-
0.6
0.8
110
Mole fraction x l
Fig. 5.3. Internal energy for a binary monomer mixture. The energy parameters are e 1 1 / k T = - 0 . 6 5 , ~ 1 2 / k T = e 2 1 / k T = -0.2, ~ 2 2 / k T = -0.1. (From Fig. 2 of Wu et al. [88].)
0.01 .-]. 0.00 -'
,
-0.01
COSMOSPACE --
-
-
- UNIQUAC
I~" -0.01
BGY
-0.02
oa 9
-0.02
AD
9
M C-res uIts
/
I9
p#
"
"
-0.03 Mole
fraction
x 1
Fig. 5.4. Internal energy for a binary monomer mixture. The energy parameters are ~ 1 1 / k T - -0.4, e 1 2 / k T = , . 2 1 / k T - -0.36, e 2 2 / k T - -0.3. (From Fig. 4 of Wu et al. [88].)
Statistical thermodynamics of interacting surfaces
77
MC simulations. It is impossible to distinguish between COSMOSPACE and the AD model in all cases shown here, and both models are in extremely close agreement with the MC simulations. Our results for two other systems, those in Figs. 1 and 3 of the paper of Wu et al., show equally good agreement between COSMOSPACE, the AD model and the Monte Carlo simulations. Figs. 5.5 and 5.6 show the deviation in the activity coefficients predicted by COSMOSPACE and the BGY model from those obtained directly from the MC simulations using an addition to our MC code, which allows us to evaluate the activity coefficients of the components. We see from these results that COSMOSPACE is in much better agreement with the MC simulations than the BGY model. We have not calculated the activity coefficients for the AD model since it is not a model for the excess Gibbs energy. Figs. 5.7 and 5.8 show the internal energy change of mixing for polymer/solvent mixtures, again with properties as specified in the figure captions, which are for the systems in Figs. 7 and 9 of Wu et al. [88]. The coordination number for the pure solvent is 6, but the coordination number for the polymer is 4+2/C, where C is the polymer chain length. The coordination number for the mixture is the volume-fraction-weighted sum of these numbers. As seen earlier, COSMOSPACE and the AD method are in very close 0.01000 f P ~
,% 0.00000
-) - -
/ /0'2"
- -
I I
-0.02000
i C
-O.0aO00
"~,,~ , f , 0.6"~ ~ . . - - ~ . 8 "
/
-0.01000
"~
, 0.4
I I COSMOSPACE
-0.04000
--- - . -
BGY(=UNIQUAC/z=2)
-0.05000
Mole fraction X 1
Fig. 5.5. Deviation of activity coefficients from COSMOSPACE and the BGY model from MC simulations for the system shown in Fig. 5.3.
78
A. Klamt 0.00005
0.00000
f r
/ A
0.2
0.4
~.8"-~-
-- ~.8 ~
/
ca - 0 . 0 0 0 0 5
/
C ! C
]
I I
-0.00010
COSMOSPACE -0.00015 --
--
BGY(=UNIQUAC/z=2)
-0.00020
Mole fraction x 1
Fig. 5.6. Deviation of activity coefficients from COSMOSPACE and the BGY model from MC simulations for the system shown in Fig. 5.4.
0.20 0.18 0.16 0.14 Itr :~
COSMOSPACE
0.12
I 1
9" ~ "
0.10
~"
0.08
9 " -"
--BGY
9
0.06 -
AD
UNIQUAC MC-results
0.04 0.02
I
0.00 '~ 0.0
i
i
i
I
9
0.2
0.4
0.6
0.8
1.0
Volume
fraction
Fig. 5.7. Internal energy for a polymer/solvent mixtt~re. The energy parameters are e11/kT--0.5, e12/kT- ~21/kT--0.3, e22/kT--0.4. The length of the polymer chain is 5. (From Fig. 7 of Wu et al. [88].) agreement with each other, and show very good agreement with the MC simulations. The BGY model also gives reasonable predictions, and UNIQUAC is by far the worst of the models. Note here that COSMOSPACE can be used to calculate the internal energy at any volume fraction, whereas it appears that the MC method fails at some value of the volume fraction above 0.8 (see Figs. 7 and 9 in Wu
Statistical thermodynamics of interacting surfaces
79
0.2o I 0.18
-
0.16
-
0.14 I-
~~
0.12
%~~999
COSMOSPACE ----
-"
0.10
---
--BGY
0.08
= = "
UNIQUAC
9
MC-results
0.06
AD
0.04 0.02
I
0.00 i -
0.0
0.2
0.4
0.6
0.8
1.0
Volume fraction
Fig. 5.8. Internal energy for a polymer/solvent mixture. The energy parameters are ell/kT =-0.5, el2/kT = e21/kT =-0.3, e22/kT =-0.4. The length of the polymer chain is 10. (From Fig. 9 of Wu et al. [88].) et al. [88]), because the polymers can hardly be randomly mixed without overlap on a lattice at high-polymer concentrations. It is worth asking the question: are COSMOSPACE and the AD model always in close agreement? The answer is "yes," as long as the absolute values of the interaction energies (eij/kT) are less than unity. For interaction energies greater than unity the models can differ very considerably, although differences begin to appear at interaction energies lower than one. Fig. 5.9 shows the deviations from MC simulations for COSMOSPACE and the AD model as a function of el2/kT for a lattice mixture of monomers in which ~11 - - ~:22 = 0. The mole fraction is kept constant at 0.02 for these calculations. We see in Fig. 5.9 that up to ~12/kT = 0.2 both methods are within the statistical noise of the MC calculations. At higher interaction energies the deviation from MC simulations of the AD model increases exponentially (a factor of 2 for each 0.1 increase in interaction energy) while COSMOSPACE stays within the noise up to e l 2 / k T - 0.6, after which the deviations increase, probably owing to agglomeration effects that start to appear in the MC simulations. Phase separation occurs at el2/kT = 0.7. If we calculate a liquid-liquid equilibrium for a cubic lattice mixture of molecules we find that phase-separation occurs, starting at el2/kT = 0 . 4 2 and we find a liquid-liquid phase boundary at Xl = 0.02 with ~12/kT = 0.7. This explains why we
80
A. Klamt 5.00 ,-
4.00
cs
I m --AD
I I
3.00
I
O
!
/
2.00
I
I
t I
1.00
j l '/ J
0.00
I
0.2
0.4
,
0.6
0.8
-1.00 Interaction energy parameter
Fig. 5.9. Deviations from MC method for COSMOSPACE and AD model as a function of ~12/kT. observe deviations from the MC method starting from that interaction energy. These results show that COSMOSPACE is more accurate than the AD model. Furthermore, in the singlephase region, COSMOSPACE is in close agreement with the MC simulations for all systems. COSMOSPACE cannot account for differences between 3D-cubic lattices and 2D-hexagonal lattices both with z = 6, because the model does not include any lattice structure information other than the number of faces belonging to one unit (molecule), i.e., other than z. A pair of A-A hexagons creates two neighboring sites that have two A-neighbors, i.e., where a third A-hexagon can get two favorable interactions at once. The situation of correlated two-site interactions is not covered by COSMOSPACE and, hence, it fails earlier in such lattices than it does for a cubic lattice where only special triples of A-particles form a favorable double A-neighbor site (see Fig. 5.1). After finding the COSMOSPACE solution in 1994, my colleague Jannis Batoulis, a student R. Merkt, and I started very similar LMC calculations to test the surprisingly simple COSMOSPACE model, but that work was never published. We even
Statistical thermodynamics of interacting surfaces
81
considered cubic objects having different kinds of faces and thus could consider a larger variety of more realistic lattice fluid models. We found extremely good agreement between the LMC and the corresponding COSMOSPACE results in all cases, as soon as the system was beyond its freezing point, i.e., beyond a phase transition to long-range order. From all these examples we can conclude that the mixture thermodynamics described by the COSMOSPACE equations is in very good agreement with LMC simulations, unless one is very close to a critical point of phase transition or phase separation. This means the geometrical constraints that arise from neighborhood relations of the surface segments on the pseudo-molecules, i.e., on the cubes in the case of the simulations discussed before, and the lattice constraints are of negligible importance for the mixture thermodynamics over a wide range of concentrations and temperatures.
5.6
STATISTICAL THERMODYNAMICS CONCLUSIONS
From the different considerations in the previous subsections we can derive the conclusion that the thermodynamic model of independently pairwise interacting surface segments can be exactly and efficiently solved by the COSMOSPACE equations. The surprising equivalence of our off-lattice model to the older lattice model of Guggenheim in his quasi-chemical approximation gives additional support to the reliability of our approximation. Also, the extremely good agreement of COSMOSPACE results with lattice MC simulations proves that the neglect of any kind of neighborhood information, and of steric constraints induced by the packing of molecules in a lattice, appear to be of surprisingly small importance over a wide range of thermodynamic conditions. Since the geometric packing constraints should be even more random and hence less important in a real liquid system than in a hypothetical lattice fluid, the deviations observed close to the phase transitions are likely to average out to a considerable degree in real liquids. Thus, we can be optimistic that the COSMOSPACE model should be able to describe the physics of interacting molecules quite well, if the interactions of the molecules really can be expressed as local pair-wise interactions of surfaces as described in section 4.
Chapter 6
The basic C O S M O - R S We have now collected almost all the pieces required for a first version of COSMO-RS, which starts from the QM/COSMO calculations for the components and ends with thermodynamic properties in the fluid phase. Although some refinements and generalizations to the theory will be added later, it is worthwhile to consider such a basic version of COSMO-RS because it is simpler to describe and to understand than the more elaborate complete version covered in chapter 7. In this model we make an assumption that all relevant interactions of the perfectly screened "COSMO" molecules can be expressed as local contact energies, and quantified by the local COSMO polarization charge densities and a' of the contacting surfaces. These have electrostatic misfit and hydrogen bond contributions as described in Eqs. (4.31) and (4.32) by a function for the surface-interaction energy density
~'
e(a, a') - -~(a +
(T,)2+ Chbfhb(T) min(0, rain(a, a') + (Thb)
max(0, max(a, a') - ahb)
(6.1)
The parameters appearing in Eq. (6.1) are known. The exact values used in this section are a - 1385 kJ nm2/mol/e 2, ( ~ h b - 0.79 e/nm 2, and 19424kJ nm2/mol/e 2, and a temperature-dependence:
fhb(T) -
Tln(1 + exp{-20 kJ/mol/(kT)}/200) 298.15 Kln(1 + exp{-20 kJ/mol/(k298.15 K)}/200)
(6.2)
which is derived from plausible physical assumptions about the energy gain and the entropy loss during the formation of a hydrogen bond. The size of a thermodynamically independent contact, aeff, is 0.0767 nm 2. These values correspond to a basic COSMO-RS parameterization which is based on DFT/COSMO calculations with Becke-Perdew (BP) [150,151] functional and a triple-zeta valence plus polarization (TZVP) basis set. A simple expression for 83
84
A. Klamt
the combinatorial part of the chemical potential is used. For details see Appendix C.
6.1
6-AVERAGING
The polarization charge-density, a, on the molecular COSMO surface is the only property required for the evaluation of the surface interaction energies according to Eq. (6.1). In principle, the charge density qi/si from COSMO could be used for the value of a on a segment i. However, the COSMO segments have areas that range from 5 9 10 -3 nm 2 down to about 5 9 10 -5 nm 2. Thus, the COSMO charge densities are much more local than the average values on the effective contact surfaces, a e f f - 0.0767 nm 2. Hence, it appears to be reasonable to use values of a that are averaged over larger areas. For this, we introduce an averaging radius rav and define the COSMO-RS polarization charge density by a local average
ai =
qJ
exp{ - d 2 / d v }
sj
expl- / / vl
j sj-t-Sav
E
j Sj-'l-Sav
(6.3)
where Sav is the area of a circle of radius rav. Compared to the most trivial averaging, this slightly more complicated formula better takes into account the very different finite areas sj of the segments j contributing to the average. Since thermodynamically independent contact segments have an area of about 0.07 n m 2 corresponding to a radius of about 0.15nm, it would be most satisfying to use this value for the averaging radius, rav, i.e., if the averaging would be over an area corresponding to a thermodynamic contact area. Unfortunately, in repeated a t t e m p t s to average over so wide an area, a significantly lower accuracy in the COSMO-RS parameterization has resulted. It appears t h a t by use of such large radii, m a n y of the relevant details of the a-distribution get averaged out. We found that the optimum value of rav is about a factor of 3 smaller, i.e., at a b o u t rav ~0.05 nm, but we do not have any good explanation for this discrepancy between the radius of a thermodynamic contact and the optimum value for the averaging radius. In a re-implementation of COSMO-RS, Lin and Sandler [91] claimed to have
The basic COSMO-RS
85
removed this problem, but it turned out that they had missed the fact that in the COSMO files the segments coordinates were reported in Bohr units. Thus, the value they used was 1.5/~ for the thermodynamic radius, but a value of 1.5 Bohr ~ 0 . 8 , ~ - 0.08 nm for the averaging radius. Hence, they in fact used a radius much closer to our optimum than to the effective contact radius. Considering their results further, it appears that they may have lost some detail by using a larger averaging radius than 0.05 nm. This confirms our finding that an averaging radius of about 0.05 nm is quite optimal and thus we are using this value in all COSMO-RS parameterizations since 1998.
6.2
q-PROFILES
Since the interaction energies of the surfaces depend only on the local polarization charge-densities, a, only the net composition of the surface of a molecule X with respect to a is of importance for the statistical thermodynamics of local pair-wise surface interactions. Thus, we have to reduce the full 3D information about on the molecular surface to a histogram pZ(a), which tells us how much of a surface we find in a polarity interval [ ~ - d~/2, + da/2]. We will refer to such a histogram as the a-profile of the molecule X. As an example, let us consider water, the most important solvent of all. Its surface polarization charge density and the resulting a-profile are shown in Fig. 6.1. The entire a-profile of water spans the range of • 2 e/nm 2, and we will see that this is about the range of a-values generally found for stable organic and inorganic molecules, including most ions. It is dominated by two major peaks arising from the strongly negative polar regions of the electron lone-pairs of the oxygen atom and from the strongly positively polar hydrogen atoms, respectively. In the color coding of the surfaces these regions can be recognized clearly as deep red and deep blue. Note that owing to the sign inversion of the polarization charge density, a, compared to the molecular polarity, the peak from the negative lone-pairs is located on the right side of the a-profile at about 1.5 e/nm 2, while the peak arising from the positively polar hydrogens is located on the left side, at about - 1 . 5 e / n m 2. Both peaks are beyond the hb threshold of _O'hb- • 2, i.e., large parts of the surface of water
86
A. Klamt
Fig. 6.1. a-Profile of water.
molecules are able to form more or less strong hb's. Since hydrogen bonding is weak up to • 2, we will generally consider the a-regions beyond • 1 e/nm 2 as strongly polar and potentially hydrogen bonding, and the rest as weakly polar or non-polar. Very few surface areas on the water molecule are located in less polar ~-regions, i.e., between - 1 and 1 e/nm 2. One important feature of the a-profile is its remarkable symmetry with respect to a. There is about the same amount of strongly positive and equally strong negative surface area. This enables energetically very favorable pairings of positive and negative surfaces and formation of strong hb's without any lack of adequate partners. As we will see later, this is an almost unique feature of the liquid water, which causes its relatively high boiling point for such a small molecule. The polarization charge densities and a-profiles of additional characteristic solvents are shown in Figs. 6.2-6.4, and are always together with the a-profile of water, which acts as a type of reference. For the reader interested in an even more vivid 3D visualization of a on the surface of the molecules, all virtual reality mark-up language (VRML) files of all
The basic COSMO-RS
Fig. 6.3. a-Profiles of common compounds.
87
88
A. Klamt
Fig. 6.4. a-PROFILES of common compounds.
these molecules are given as supplementary material [92]. It is worthwhile, and hopefully also interesting to consider these in more detail, in order to become familiar with the typical features of molecular surface polarity. Let us start with hexane, which is a typical representative of alkanes. As can be seen in the inset, the top left of Fig. 6.2, the surface of hexane is mainly non-polar, i.e., green, with a tendency to blue-green on the hydrogens and to yellow-green on the carbon regions. The corresponding a-profile of hexane ranges from - 0 . 5 to +0.6 e/nm 2, with a maximum at - 0 . 1 e/nm 2 arising from the hydrogens, and a shoulder at about +0.2 arising from the exposed surfaces of carbon atoms. The a-profiles of other alkanes look very similar, and mainly differ in height according to the differences in the total surface area. The surface of benzene (inset below hexane in Fig. 6.2) shows more pronounced polarity. The negatively charged u-face appears yellow, corresponding to a moderately positive ~, and the
The basic COSMO-RS
89
hydrogens are light blue, corresponding to moderately negative a. In the a-profile, this results in two clearly separated peaks centered at _+0.6e/nm 2. The a-profile of benzene is almost exactly symmetric with respect to a. Next we consider methanol. On the surface (top right inset in Fig. 6.2) we can dearly identify the polar hydrogen as a deep blue area, the area of the oxygen lone-pairs as a deep red region, and the methyl group, which appears slightly more blue than the methyl groups of hexane. In the a-profile we see that methanol has about the same shape as water in the positive a-range, arising from the sp3-oxygen in both molecules. In the strongly negative a-region, methanol has about half the intensity of water because it has only one polar hydrogen compared with two in water. The methyl group appears alkane-like, with one higher peak from the hydrogens and a shoulder from the carbon, but the position is dearly shifted to the negative a-range, expressing the polarization of this methyl group by the neighboring oxygen. Acetone (propanone, see inset below methanol in Fig. 6.2) has an sp2-oxygen, which clearly shows deeply red polarity maxima in the direction of the two lone-pairs. The shape of the a-profile of this oxygen is very different from that of the sp3-oxygens in methanol and water. The six hydrogens are well polarized and give a peak at - 0 . 6 e / n m 2, almost identical with the hydrogen peak of benzene, while the three carbon atoms form a peak at 0.0 e/nm 2. The a-profile of acetone is very asymmetric. The a-surface of acetic acid is most colorful. The sp2-oxygen is slightly less polar than that of acetone, and the hydroxyl oxygen is less polar than in water, but the acidic hydrogen is much more polar than those of water and alcohols, causing a change into violet on the a-surface. The methyl group is quite similar to those of acetone. In Fig. 6.3, we see a-profiles of some nitrogen compounds and a sulfur-containing compound. Methylamine shows a very strong lone-pair on the nitrogen. In the a-profile we see that its polarization charge-density a ranges up to 2.7 e/nm 2, while water ends at about 2.1 e/nm 2. This causes extremely strong hb's of this lone-pair with polar hydrogens, and we will see that these extremely strong polarities cause some problems in the correct quantification of the hydrogen-bond energy of amines. We can also see that the two hydrogens of the amine group are moderately polar with a values
90
A. Klamt
at about - 1 . 0 e / n m 2. Compared with the hb donors of water or alcohol, they are only weakly hydrogen bonding. The methyl group of methylamine is less polarized than that of methanol. Nitromethane is shown below methylamine in Fig. 6.3. The two sp2-oxygens appear mainly yellow, each showing two more polar, red regions in the lone-pair directions. The shape of the oxygen peak is similar to that of the sp2-oxygen of acetone, but it is clearly shifted to lower a-values, i.e., lower polarity. On the other hand, the methyl group is strongly polarized and hence centered at almost - 0 . 8 e/nm 2 in the a-profile. The ~-orbitals of the nitrogen are electron-depleted by the two oxygens and appear slightly blue. Although the shapes on the positive and negative sides are a little different, the a-profile of nitromethane is overall quite symmetric. In acetamide the carbonyl oxygen has the typical shape of an sp2-oxygen in the a-profile, but is located at higher a-values than in acetone. It is about as polar as the oxygen of water. The two Nhydrogens appear as deeply blue as the hydrogens in water, and indeed their peak is located at about the same position in the a-profile. The methyl group is moderately polarized, the ~-orbital of the carbonyl carbon is quite neutral (green), and that of the nitrogen is slightly negatively polar, resulting in a somewhat positive a and a yellow color. Methanethiol, despite being iso-electronic with methanol, has a a-surface and a-profile, which look quite different. The sulfur lone-pairs are much weaker than those of the oxygen atom in methanol, but they are still weak hb acceptors. The hydrogen bound to sulfur barely overcomes the hb threshold. Some halogenated compounds are shown in Fig. 6.4. First we consider the non-polar compound tetrachloromethane. As can be seen in the upper left inset, the a-surface mainly appears green, but in the opposite bond directions the chlorine atoms show some blue regions, owing to the fact that the electrons of the px-orbitals of the chlorine atoms are involved in the covalent bond and hence are shifted toward the central carbon atom. As a result, the a-profile of tetrachloromethane is dominated by a strong peak at 0.2 e/nm 2, balanced by a peak at -0.2 e/nm 2 and another peak at -0.6 e/nm 2. Chloroform (trichloromethane, see inset below tetrachloromethane) has three chlorine atoms showing basically the same
The basic COSMO-RS
91
features as those in tetrachloromethane, but being slightly more polar because the electron density is withdrawn from the hydrogen. As a result, the hydrogen is strongly polarized and appears as deep blue on the a-surface. This is centered at about 1.4e/nm 2, and is thus almost as polar as the hydrogens of water. The a-profile of chloroform is therefore very asymmetric. Bromodifluoromethane is very similar to chloroform, but it can be seen that the smaller fluorine atoms do not show an electron lack (blue area) in the opposite bond direction, while the larger bromine clearly shows a yellow ring and a blue spot at the cap. Finally we consider formyl fluoride. Here, we see a relatively weakly polar carbonyl oxygen with two lone-pairs in orange. The fluorine is competing for electrons with the oxygen atom and becomes slightly more polar than in chloroform. The two electronwithdrawing neighbors cause the ~-orbital of the carbon to be electron-deficient and thus it appears as a clearly blue region on the a-surface. The hydrogen atom is strongly polarized and gives a peak similar to that in chloroform, but is slightly less polar. We could continue this discussion of a-surfaces and a-profiles with many other interesting and colorful examples, but this would exceed the limits of this book. From the representative examples discussed so far, the basic principles of the surface polarities of organic compounds expressed by the polarization charge densities, a, should have become clear. We leave it to the reader to study additional examples in the supplementary material.
6.3
WHY DO SOME MOLECULES LIKE EACH OTHER AND OTHERS NOT?
Before turning to more quantitative applications, it is worthwhile to consider a few examples of binary mixtures on a qualitative basis, just by viewing their a-profiles. From such considerations we can learn quite well why some molecules like each other so much, while others do not. We start with the most striking example, of a mixture of acetone and chloroform. Although they have been discussed separately before, their a-profiles are shown together in Fig. 6.5. Both have very asymmetric a-profiles, so they do not feel very comfortable in their pure liquids, because the oxygen in acetone does
92
A. Klamt
Fig. 6.5. a-Profiles and mixture properties for the mixture of propanone and chloroform. not find appropriate polar counterparts, and the polar hydrogen in chloroform does not find a partner with a reasonably positive a. Hence, in each pure liquid there is a considerable amount of electrostatic misfit. But if we mix both liquids, the a-profiles complement each other in an almost ideal way. The polar hydrogen of chloroform matches very well with the polar oxygen surface polarity of acetone, and even the other peaks in both a-profiles find almost perfectly corresponding peaks. In this way, the misfit energy is strongly reduced in the mixture, resulting in a strongly negative heat of mixing (see inset in Fig. 6.5), i.e., the mixture becomes warm, when both liquids are poured together. In some way this acetone-chloroform example appears to contradict the ancient chemical recipe of similis similibus solvuntur, i.e., "like dissolves like," since acetone and chloroform dissolve each other so exceptionally well as a result of their complementarity, and not their similarity. A typical example of similis similibus solvuntur is the mixture of acetone and dimethyl ether (see Fig. 6.6). The a-profiles of both solvents are very similar. Both are asymmetric in the same sense, and hence they do not feel very well in themselves, nor in the mixture. However, due to the entropy gain of the mixing process
The basic COSMO-RS
93
Fig. 6.6. a-Profiles and mixture properties for mixtures of propanone and dimethyl ether. both dissolve each other quite well. Since the energetic situation regarding the misfit does not change much during mixing, there is no significant enthalpy gain or loss in this mixture. Such mixtures, in which the situation for the molecules does not change, either in a positive or negative direction, are generally called "ideal" mixtures, although the situation for the molecules need not be ideal at all, as seen in this case. Another example of similis similibus solvuntur is the mixture of benzene and 1,3-pentadiene, as shown in Fig. 6.7. Here, both solvents have reasonably similar, symmetric a-profiles, and hence both feel well in themselves, as well as in the mixture. In a mixture of acetone and benzene, as shown in Fig. 6.8, the ~-profiles of both solvents are quite dissimilar. Acetone has a rather polar "unmatched" oxygen and does not feel well in itself, while benzene is only weakly polar, being quite comfortable in its pure liquid. Since in the negative region the a-profiles of both are quite similar, the pairing situation for the surface segments with positive a stays quite unchanged if both acetone and benzene are mixed. Therefore, these solvents form an ideal mixture for entropic reasons although they have quite different polarities. As a last example we consider the mixture of water and hexane, as shown in Fig. 6.9. As already discussed, water has a broad and
94
A. Klamt
Fig. 6.8. a-Profiles and mixture properties for the mixture of propanone and benzene. symmetric a-profile, and hexane has a narrow a-profile around a = 0. Both solvents are very satisfied in their pure liquids. If we try to put a hexane molecule into water at ambient temperature, the non-polar surface pieces of hexane cannot break the strong
The basic COSMO-RS
95
Fig. 6.9. a-Profiles and mixture properties for the mixture of water and hexane. pairs between water segments with strongly positive and negative a, which in addition to their good electrostatic match also form strong hydrogen bonds. There is no sufficient thermal energy available to break these pairs. Hence, the only choices for the hexane segments are the few less-polar surface segments of water. With these, hexane can pair without significant energy costs, and hence the excess heat of solution of hexane in water indeed is almost zero. However, the restriction to a small portion of the segments means a strong loss of entropy for hexane in water, and hence there is a strongly negative mixing entropy, resulting in the low solubility of hexane in water. On the other side, a water molecule in hexane does not find any good partners for its polar surface segments. All hexane segments are almost equally inappropriate for the polar water segments. This results in a large positive excess heat of solution and a low solubility of water in the alkane, but a positive excess entropy of solution, because the partner choice is much less selective than it was in pure water. As shown in more detail in [22], this COSMO-RS picture of the unusual mixing behavior of alkane and water is in good qualitative and quantitative agreement with experimental data. To summarize, we have seen that the a-profiles provide a detailed and realistic understanding of the mutual solubilities of
96
A. Klamt
solvents and overcomes old heuristic recipes, such as similis similibus solvuntur. 6.4
6-POTENTIALS
We shall now consider the solvent properties in a more quantitative way by applying the COSMOSPACE thermodynamics to an ensemble of solvent surface segments of composition n
xip~(6)A6 Os(a) - ps(a)Aa - i=1
n
(6.4)
~-~.xiqi i=1
with respect to the polarization charge density, a. Here, X i is the mole fraction of a component i of the solvent, pi(6) the corresponding a-profile, and qi the surface area. Apparently, for a pure solvent, the solvent a-profile, ps(a), is nothing but the a-profile of the solvent normalized to unity, i.e., a-profile divided by the total surface area qs. Applying Eq. (5.12) in combination with the COSMOSPACE Equation (5.32), we thus obtain
where the final identity follows in the limit of infinitely small a intervals, Aa. Indeed, this was the original form of the COSMORS thermodynamics. Here, ps(a) is the chemical potential of a unit piece of surface of kind a in the solvent S. We call this the a-potential of the solvent. It tells us how much a solvent S likes polarity of kind a. As noted for the COSMOSPACE equation, Eq. (6.5) also, in general, requires an iterative numerical solution, but it converges well starting with the initial assumption of ps(a') on the right-hand side. Only in the case of negligible hydrogen bonding and for a Gaussian-shaped solvent a-profile ps(a) an analytical solution is available. In that case, as shown in the appendix of [C9], the a-potential corresponds to a simple parabola. Obviously, the a-potential is a function of composition and temperature. In Fig. 6.10 we compare the a-potentials of some
The basic COSMO-RS
97
Fig. 6.10. a-Potentials of representative solvents at 25~ (and 100~C). representative solvents at room temperature. The plot of the a-potential of hexane is close to a parabola. Since hexane is not hydrogen bonding, and since the a-profile of hexane is a narrow, almost Gaussian peak, this parabola corresponds reasonably to the analytical solution of Eq. (6.5) derived for Gaussian a-profiles. A parabolic a-potential also corresponds to "dielectric behavior" in the sense that any solvent behaving like a dielectric on a molecular scale would show a parabolic a-potential, and the curvature would depend on the dielectric constant, e. Since we have already stated in section 4 that alkanes should behave like real dielectrics on a molecular scale, because of the absence of relevant permanent molecular electrostatic moments and the quite linear behavior of the electronic polarizability up to molecular electrostatic field-strengths, we may identify the curvature of the a-potential of hexane of 26.6 kJ nm2/e 2 with a dielectric constant of e ~ 2.1, i.e., the macroscopic dielectric constant of alkane. Apparently tetrachloromethane has almost the same apotential, because it is also rather non-polar. The slightly larger extension on the negative side of the a-profile cause a slightly smaller curvature on the positive side of the a-potential. The asymmetry of the a-potential is much more pronounced for chloroform, because this has a much more asymmetric a-profile, as discussed before.
98
A. Klamt
The a-potential of benzene also appears to be an almost perfect parabola. Apparently, the parabolic solution holds even for nonGaussian a-profiles, if these are reasonably symmetric and not hydrogen bonding. The curvature of the a-potential parabola of benzene is smaller than that of an alkane. Because of its broader a-profile, benzene is more tolerant to polar surfaces than hexane, resulting in a curvature of the a-potential of 20.6 kJ/mol nm2/e 2. By using the rough approximation curvature = const(1 - f(D) ~ const (1 - ~~+- 0_5) 1
(6.6)
and determining the proportionality constant from the values for hexane, we find an effective dielectric constant of ~ e f f - - 2.85 for benzene, while (because it has no dipole moment) its macroscopic dielectric constant is equal to its optical dielectric constant, and has the value 2.1. Thus, a dielectric continuum solvation model itself would have led to the wrong result that the solvents benzene and alkane behave almost identically, while COSMO-RS correctly reflects the fact that benzene is more tolerant with respect to electrostatic polarity than alkane. The a-potential of water shows an almost linear behavior in the non-hydrogen-bonding region. The very small curvature corresponds to a high effective dielectric constant, in agreement with its high macroscopic dielectric constant. For non-hydrogen-bonding compounds this almost linear behavior would extend further out, as indicated by the dashed line in Fig. 6.10. The small effective curvature caused by the almost symmetric a-profile, means that electrostatically water behaves roughly like a strong dielectric. This is the reason for the macroscopic dielectric continuum solvation models working roughly for the solvent water. As we will see later, the constant value of PH20 (~ --0) corresponds to a surface tension, which is usually fitted into the nonelectrostatic part of CSMs, and the slight slope of the curve does not affect the chemical potential of neutral solutes. Nevertheless, it is apparent from Fig. 6.10 that water has an exceptionally high ~-potential in the non-polar a-region. All other solvents have much lower values in this range and, as a result, a non-polar surface hates to be in water and likes to escape into any other solvent. This is exactly the behavior that is usually called "hydrophobicity." In most other models, hydrophobicity has to be
The basic COSMO-RS
99
introduced as an additional type of interaction, but in COSMO-RS it follows directly from the thermodynamics of the surface pairs as a result of the exceptionally strong and balanced interactions of the water segments, without any additional assumptions. Furthermore, we see that in both hydrogen-bonding regions the apotential of water turns into an almost linear descent, expressing water's ability to form some additional hydrogen bonds with the hb-donors and hb-acceptors of solute molecules. The a-potential of methanol clearly expresses strong differences from water. Owing to the presence of the weakly polar methyl group, it has good partners available for non-polar surfaces and hence has quite a low a-potential in that region. Owing to the lack of one polar hydrogen compared with water, it is more attractive for hb-donors than is water, which is expressed by the lower a-potential, in the range a < - 1 e/nm 2, but it does not like additional acceptors as much as water does. As can be seen in Fig. 6.10, only acceptors with a > 1.5e/nm 2 have a chance of making hydrogen bonds in methanol, because this solvent has an excess of strongly hydrogen-bonding acceptor surface area itself. It can also be seen in Fig. 6.10 that the solvent acetone has an almost parabolic a-potential over a wide ~-range, but this turns into a strong descent in the donor region, because acetone has a strong lack of hb-donors. As can be seen in the a-profiles, its hb-acceptors are less polar than those of methanol, and therefore acetone is less attractive for very strong hb-donors than is methanol. In the a-potential plot for methylamine it can be clearly seen that the moderately polar hydrogens of the amine group are not sufficiently polar to cause any significant attraction for hb-acceptors. However, the extreme ~-hotspot on the lone-pair of the amine nitrogen causes very attractive interactions with hbdonors. While, without any doubt, this is qualitatively correct it must be emphasized again that this very strong hydrogen bonding of amines brings the hb-interaction parameters of COSMO-RS to its quantitative limits and hence causes larger errors. As a result of its broad and symmetric, but still mainly nonhydrogen-bonding, a-profile, the solvent nitromethane has an almost zero a-potential in the non-hydrogen-bonding a-range. Here it behaves closest to the perfect conductor limit. Because it has no
100
A. Klamt
hb-donors, but very weak hb-acceptors, it shows a slight descent in the hb-donor ~-range, and a slight increase in the opposite region. In Fig. 6.10 we also see the a-potentials of water and hexane at 100~ (dashed lines). Hexane shows almost no temperature dependence, in good agreement with its almost temperature-independent macroscopic dielectric constant. For water, we see that the non-hydrogen-bonding part is also quite insensitive to temperature. In contrast, in the hydrogen-bonding regions the ~r-potential increases strongly with rising temperature, because hydrogen-bonding becomes less important at high temperatures, as expressed by the temperature-dependent scaling factor fhb(T) in Eq. (6.2). To summarize, in this chapter, we have seen that the a-potentials ps(a) are significant fingerprints of the solvent behavior of pure and mixed solvents. They express the affinity of a solvent to electrostatic polarities, hb-donors, hb-acceptors, and to nonpolar surfaces, and include the concept of hydrophobicity in a natural way.
6.5
CHEMICAL POTENTIAL OF SOLUTES AND PHASE EQUILIBRIA
Translating the thermodynamic concept of interacting surfaces to our basic COSMO-RS, the residual part of the chemical potential of a compound i in a solvent S is found by a summation of the chemical potentials of the surface segments of i. Starting from Eq. (5.10) we have pR(s; T) - ~
ni'tzv(S; T) -
jpi(a)t,s( )d
(6.7)
l'
where pi((r) is the a-profile of the solute i and ps((r) is the crpotential of the solvent S at temperature T. Since pi(~r) gives the amount of surface with polarity a, Eq. (6.7) expresses the residual part of the chemical potential of compound i in solvent S as a surface integral of the a-potential over the surface of the solute i. Combining this with a simple, and usually small, combinatorial contribution (see Appendix C) and with the trivial concentration
The basic COSMO-RS
101
dependence kT In X i we obtain the expressions
~i(S; T)
--
Ft~(S; T) -~-k T l n x i - pR(s; T) + pC(s; T) + k T l n x i
- fpi(cr)ps(~; T)da + pCi(s; T) + k T l n x i
(6.8)
for the chemical potential p i ( S ; T ) and the pseudo-chemical potential p~(S; T) of an arbitrary solute i in a pure or mixed solvent S. Since the pseudo-chemical potential p~(S; T), as introduced by Ben Naim [93], does not include the contribution proportional to lnxi, it can be calculated irrespective of whether the solute i is a minor or major part of the solvent S, or whether it is just a solute at infinite dilution, a fact that makes it more usable than the chemical potential itself in many situations. Having the chemical potential of arbitrary compounds in arbitrary solvents and mixtures, we are able to express any kind of liquid-liquid equilibria (LLE) between different liquid phases, S and S', using the necessary condition that the chemical potential of each compound i must be equal in both phases. Partition coefficients of a compound i between two phases S and S' can be calculated directly from Eq. (6.8). If we assume that the mole fractions of i in both phases is very low, i.e., close to infinite dilution, we may disregard it in the composition of the phases S and S'. Then the partition coefficient, i.e., the ratio of the mole fractions, is
, , fx (S) } , , k T l n K i ( S , S ; T) - ln[xi(S, ) - pi(S'; T) - pi(S'; T)
_ / pi(o.)(pS,(a;T) +
T) -
- ps(O-; T)) da
C(S; T)
(6.9)
It should be noted that Eq. (6.9) gives the partition coefficient in terms of mole fractions, as is common in the chemical engineering literature. We therefore denoted it as K*. In chemistry it is usual to report partition coefficients in mol/1 units, and to denote them by K. Although the partition coefficient is dimensionless in both conventions, it differs by the ratio of the molar volumes of the two solvents. As we will see in more detail in the next chapter, logarithmic partition coefficient data have been used heavily for the parameterization of COSMO-RS. The octanol-water partition
102
A. Klamt
coefficient, Kow, is the most widely considered and best-measured partition coefficient [94]. It should be noted that the octanol phase in equilibrium with water (wet octanol) contains about 25 mol% of water. The logarithmic octanol-water partition coefficient, logKow is often used as a measure of molecular lipophilicity. From Eq. (6.9) we see that logarithmic partition coefficients in general, and also log Kow are mainly given as a surface integral, because the differences of the combinatorial contributions are usually small. Hence, the difference in the a-potentials between the two phases can be considered as the local contribution to the partition coefficient, and hence the partition coefficient can be visualized as a surface property, as is done for the molecules o-cresol and methylimidazole in Fig. 6.11. The total logKow of o-cresol is calculated as 1.73 (exp. 1.95), and for methylimidazole, basic COSMO-RS gives 0.26 (exp. 0.24). From the difference of the two a-potentials shown in Fig. 6.11, we see that negative, hydrophilic contributions to the logarithmic octanol-water partition coefficient arise only from the positive polarization charge-densities a, i.e., mainly from strong hb-acceptors, while neutral and positively polar surface regions give a constant positive contribution favoring the octanol phase. Hence, only the acceptor regions of the oxygen in o-cresol and, in a more extreme form, the nitrogen
Fig. 6.11. a-Potential difference of water and wet (water saturated) octanol.
The basic COSMO-RS
103
Fig. 6.12. Surfaces of o-cresol and methylimidazole colored by log Kow, i.e., by the o-potential difference of octanol and water. lone-pair in methylimidazole are visible as green and blue areas, i.e., as being hydrophilic on the molecular surfaces in Fig. 6.12. For normal mixtures the activity coefficient of a compound i in a mixture S can be calculated as
~' ~S'~ , T ) - e x p {
p*(S; T) -
T)}
(6.10)
where p~(i, T) refers to the pseudo-chemical potential of the compound i in the pure liquid i. Here, it is assumed that this pure liquid exists in the same conditions as the mixtures. This, at least, is required for the experimental determination of the activity coefficient, while we would be able to treat the virtual pure liquid in any case with COSMO-RS. Owing to this experimental difficulty, activity coefficients are defined with other reference states for gases and for electrolyte solutions. Some caution is therefore warranted when the expression "activity coefficient" is used in chemical or chemical engineering literature. If we also know the chemical potential of the compounds in the gas phase or in a solid phase, then we can also calculate the equilibria between the vapor and the liquid phase (VLE), or between solids and the liquid phase (SLE). If the vapor pressure pvap(T) of the pure compound i is known, the partial pressure of i in the gas phase is given by _vap
Pi - P i
,
,
(T) ",i(S; T)*xi
(6.11)
104
A. Klamt
as long as the gas is sufficiently ideal, so that fugacity corrections are small. For SLE calculations, one can usually use an extrapolation of the free energy of fusion, AGfus, starting from the melting t e m p e r a t u r e Tfus, the enthalpy of fusion AHfus, and change in the heat capacity Acp,fu~ between the liquid and the solid, that gives / /tsolirr~ 99 ( i (.~ ~ - Pi (t, T) - AGfus ~ (AHfus + TfusAcp,fus) 1 + Tfu~Acp,fu~
In ~
(6.12)
For t e m p e r a t u r e s reasonably close to the melting point, Acp,fus often can be neglected. 6.6
SOME EXAMPLES OF BINARY MIXTURES
As a demonstration of quantitative LLE calculations, we now consider in more detail some of the binary mixtures that we have discussed qualitatively in section 6.3. In Fig. 6.13 we see the excess Gibbs free energy of mixing G ex, the heat of mixing H ~x, and the excess entropy of mixing S ex for mixtures of acetone and
Fig. 6.13. Binary mixture data for acetone and chloroform at 298 K.
The basic COSMO-RS
105
chloroform. The figure shows the large negative heat of mixing of about -1.35 kJ/mol and the exceptionally strong negative values of the logarithmic activity coefficients. The excess entropy of the mixture is negative (i.e., - T S E is positive), which results from the higher degree of order introduced by the very favorable, and hence more specific, interactions in the acetone-chloroform mixture. Fig. 6.14 shows the same excess mixture properties and logarithmic activity coefficients for a mixture of acetone and dimethyl ether. Since this mixture behaves ideally (see discussion in section 6.3) the excess properties and logarithms of activity coefficients are very close to zero (note the scale of the axes) in this case.
The situation is quite different for a mixture of 1-propanol and n-hexane (see Fig. 6.15). Here, we find a significant positive excess heat of mixing. The excess entropy of mixing shows a change of sign at about 10% n-hexane. This feature has also been found experimentally in many alcohol-alkane mixtures. As a last example, we consider the binary phase diagram of water and 1-butanol (Figs. 6.16 and 6.17). There is a negative heat of mixing, H E, but a positive excess Gibbs energy of mixing, GE. The infinite dilution activity coefficient of 1-butanol in water is very
Fig. 6.14. Binary mixture data for acetone and dimethyl ether at 298 K.
106
A. Klamt
Fig. 6.16. Binary mixture data for water and 1-butanol at 298 K.
high. Experimentally, the 1-butanol-water is the first in the alkane-water series to show a miscibility gap at room temperature. The existence of a miscibility gap is reproduced well by the COSMO-RS calculations, as shown in Fig. 6.17. Since a miscibility
The basic COSMO-RS
107
Fig. 6.17. Plot of mutual activities for the binary mixture of water and 1-butanol at 298 K. gap, i.e., a LLE, means that there are two compositions, Xl and x~' for which both compounds have the same activities, the existence of such an LLE can easily be checked by plotting the activities a2 vs. a~, where ai is x:,'i. If the resulting curve shows a closed loop, the binary mixture has a miscibility gap, and the two concentrations corresponding to the self-intersection point of the curve are the LLE compositions x~ and x~'. From Fig. 6.16 we can see that, in this case, there is an LLE between x ~ t ~ ~ 0.38 and Xwate r' ~_ 0.992
Chapter 7
Refinements, parameterization, and the complete COSMO-RS As discussed in the previous chapter, the COSMO-RS theory derived up to this point allows for a good qualitative understanding and a reasonable quantitative description of fluid-phase thermodynamics. Nevertheless, in order to improve the theory further, we need to break with the simple concept that the interactions of molecular surfaces can be described fully by the polarization charge densities, a, alone, i.e., by just one descriptor for each molecular surface segment. Therefore we will introduce a generalization of the COSMO-RS approach to multiple descriptors, and a list of additional descriptors that may be useful for a more refined expression of molecular interactions in this chapter. However, the reader should keep in mind the fact that the additional descriptors are only responsible for small corrections, and a will remain to be of dominant importance. Hence, all the qualitative considerations made above remain valid and the simple "one-descriptor" picture remains a valuable and helpful concept for the qualitative understanding of fluid-phase thermodynamics on a molecular scale. For the sake of higher accuracy, the "multidescriptor" COSMO-RS theory loses some of the simplicity of the preceding theory as additional descriptor dimensions are introduced. Thus, I recommend keeping the simpler "a-only" picture in mind for the qualitative understanding of mixture thermodynamics and let the multidescriptor evaluations be the task of computers. Fortunately, these have no significant problems with the multidescriptor generalization. Beyond the generalization, we will introduce a short section that introduces the subject of chemical potential in the gas phase. This is required for using vapor-liquid equilibrium data and vapor pressures as part of the data set for the parameterization procedure, which will be described in section 7.4. 109
110 7.1
A. Klamt ADDITIONAL SURFACE DESCRIPTORS
In section 6.1, we introduced the COSMO-RS polarization charge density, a, as a local average of the COSMO polarization charges over a region of ca. 0.5 A radius. In this section, we will introduce a list of other local surface descriptors. Some of them have already proved to be useful for improving the accuracy of COSMO-RS, while others are candidates for future improvements. Obviously the list given here only reflects the present state of our ideas, and it is open for good additional ideas.
7.1.1
Surrounding polarization charge density
The primary polarization charge density a only reflects the average COSMO charge density in the vicinity of r a v = 0.5 A around a segment center, but we will see below that the radius of a thermodynamically independent surface contact is about three times larger than that. Therefore, it appears reasonable to have a second descriptor available that describes the polarization charge density in a larger surrounding of a surface point that can be used to generate corrections of the electrostatic misfit interaction energy. We found that it is reasonable to use twice the o r i g i n a l averaging radius in Eq. (6.3) for this second averaged polarization charge density, a ~ Obviously the value of a ~ is highly correlated with the original value of a. In order to obtain a linear independent descriptor with small inter-correlation, we evaluated a and ~o over a large set of surface points of diverse compounds, and defined the perpendicular component of ao as a new descriptor a_L. o
ff~ ~ ffi -- ~ ( T i
~
0"~ -- 0 . 8 1 6 a i
(7.1)
This can now be used to improve the expression for the misfit energy expression of Eq. (4.2) as
if' + 6')((6 + 6' + f• (6 -1-+ ~• emisfit(6, 6• 6'6 -k,) -- -~(6
(7.2)
where f• is a parameter to be adjusted on a large data set, and is in the range of 2.5. The quantity a • is used regularly in COSMO-RS since 1998 [C10], and its introduction reduces the
Refinements, parameterization, and the complete COSMO-RS 111 root-mean-squared error (rmse) by about 10% compared with parameterizations without this descriptor.
7.1.2
The u n d e r l y i n g e l e m e n t
One rather obvious additional "surface" descriptor is the chemical element of the atom to which segment i belongs. We will denote this descriptor of a surface segment i as ei. With the availability of this descriptor, it is possible to perform element-specific parameterizations. For example, it is possible to use elementspecific hydrogen-bond thresholds crab in Eq. (4.3), although this options is not used fully in the present parameterizations. We only define the donor threshold to be infinity, unless the element of the donor segment is hydrogen. Thus, only a hydrogen surface can act as a donor in a hydrogen bond. Otherwise it could have happened that the COSMO-RS hydrogen-bond energy term applies to contacts between small cations as Li + and lone-pairs of heteroatoms, which obviously makes no physical sense. More importantly, the availability of the element descriptor, e, allows for the expression of the vdWs interactions as segment interactions: gvdw(e; e') = Cvdw(T)~(e; e') ~ Cvdw(T)(~(e) + r(e'))
(7.3)
where CvdW (T) reflects the overall temperature dependence of vdW interactions. If the vdW interaction energy between two elements e and e' is assumed to be the sum of two element-specific vdW tensions, as indicated by the right-hand side of Eq. (7.3), the total amount of vdW energy in an ensemble of segments becomes independent of the concrete pairing of the segments, and therefore this part of the energy can be factorized out of the partition sum of COSMO-RS. This corresponds exactly to the approximation made in the initial versions of COSMO-RS, in which the vdW energy is considered to be part of the continuum energy. However, we see clearly that the more general formulation given in Eq. (7.3) gives us the flexibility to use more sophisticated expressions for the vdW interaction energy. The most recent parameterizations of C O S M O t h e r m partly take advantage of this.
112 7.1.3
A. Klamt Local polarizability
Although only a i and e are used as additional descriptors so far, some others might be useful in the future. One of the most apparent is the local electronic polarizability of the molecule in the vicinity of a surface segment i, which we may represent by a local refractive index ni. Such local polarizability or local refractive index would allow for a refinement of the electrostatic misfit term, which presently only takes into account an average electronic polarizability of organic molecules. It is also likely that the local electronic polarizability is of importance to the strength of hydrogen bonds, which so far in COSMO-RS are only a function of polarity. Finally, from a physical perspective, the vdW interactions should be a function of the local polarizability as well. Unfortunately, the rigorous evaluation of a reasonable measure of local polarizability would take an extreme amount of additional computation time in the DFT/COSMO calculations. An approximate, but very fast q u a n t u m chemical calculation method for the local polarizability has recently been suggested by Politzer, Jin, and Murray [95]. The variations of the local polarizability on the surface of acetic acid are visualized in Fig. 7.1 according to this method. Its use for the improvement of COSMO-RS is just being tested.
Fig. 7.1. Approximate local polarizability visualized on the surface of acetic acid [95]. (Courtesy of P. Politzer and P. Jin.)
Refinements, parameterization, and the complete COSMO-RS 113 7.1.4
Local shape index
Without considering this list as final, we will discuss the local shape of the molecular surface in the larger surrounding of a segment i as our last example of potentially useful surface descriptors. While avoiding mathematical details, it should be possible to distinguish between normal, bulky, chain-like, and flat surface parts by an analysis of the intersection line of the COSMO surface of a sphere of a radius rshape ~ 3.5 A around a segment center (see Fig. 7.2). Based on such a shape index, it should be possible to introduce corrections to the vdW interaction energy, which result from the fact that differently shaped segments can attach to each other differently. For example, chain-like molecule parts can be closely packed with each other than with bulky, more spherical parts of molecules. With such corrections it should be possible to recover the lost 3D information into COSMO-RS, and to improve the ability of COSMO-RS to describe the differences of physicochemical properties of isomers with different shapes---as for 1-butanol and t-butanol. 7.2
COSMO-RS ALGORITHM FOR MULTIPLE DESCRIPTORS
Given a set of descriptors d,. for each surface segment, r, instead of just a single descriptor, a,., we can no longer calculate the
Fig. 7.2. Different shapes of a ring of radius 3.5/~ around segment centers: The chain end is more bulky (white) while the black ring corresponds to a typical chain segment.
114
A. Klamt
a-potential and the residual part of the chemical potential as integrals, as they are expressed on the right-hand sides of Eqs. (6.5) and (6.7). However, the first part of these equations, in which the segment potentials and the residual chemical potential of the solute are evaluated as sums over distinct segment types, stays essentially unchanged, apart from the fact that we have to replace a by d in Eq. (6.2):
p s ( d ) - k T In 7 s ( d ) - - k T ln(~-~,. Os(d,.)'/s(d,.)~(d,d,.) )
(7.4)
No other changes are required, and hence the extension of COSMO-RS to multiple descriptors is straightforward. 7.3
THE CHEMICAL POTENTIAL IN THE IDEAL GAS
So far we have only considered the chemical potentials of molecules in the liquid state. The state of the molecule in the ideal conductor has been taken as our virtual reference state. We calculated the chemical potential in the liquid state relative to this reference by taking into account the interaction energies arising from the surface interactions, i.e., electrostatic interactions, hydrogen bonding, dispersion energies (or vdWs energies), and the entropy arising from a number of different surface interactions that are possible at a given thermal energy, k T. However, as discussed at the beginning of this book, quantum chemistry is well able to calculate the total energy of a molecule in vacuum, i.e., at the "south pole of our globe". Indeed, this is the starting point of quantum chemistry. Hence, the total energy difference of the vacuum state and the state in a conductor can be calculated quite easily by doing one QC calculation for the compound X in vacuum, yielding the total energy EvacXand another for X in a conductor, yielding EcosMo. x We will denote this x difference as AECOSMO. It should be noted that a reasonable guess of EcosMo x can be achieved, on the basis of COSMO calculation alone. Since the main difference between the vacuum state and the COSMO state arises from the conductor screening, a simple expression x AEcosM O~ - 0.83E~ie 1
(7.5)
Refinements, parameterization, and the complete COSMO-RS 115 is reasonably accurate, where the empirical factor 0.83 arises from the fact that a part of the dielectric interaction energy is required to produce the increased polarization of the solute in the conductor as well as for conductor-induced geometry changes. Nevertheless, for more accurate calculations of free-energy change going along with the transfer to the gas phase, we recommend spending the extra computational costs of a QC calculation in vacuum. Since in the ideal gas (ig), there are no interactions of x the molecules, AEcosM o is the most important part of the chemical potential difference of the molecule between the COSMO reference state and the ig state. It is well known that the quantum-chemical and thermal free-energy corrections arising from zero-point vibrations and thermal, vibrational, and rotational excitations as well as from the translational and rotational energy of the entire molecule also contribute to the free-energy. These contributions can be calculated routinely in most QC programs, although with significant additional computation time, and they are of substantial importance for the calculation of reaction free energies. However, it appears that the largest part of these free-energy contributions is also present in the liquid state and hence cancels out in the difference of gas- and liquid-phase free energies of molecules. A simple, general ig free-energy correction ~ig(T) appears to be sufficient for most molecules. The change in vibrational free energies, which is likely to exist, can apparently be subsumed in the element-specific surface-proportional dispersion energies, and hence do not appear explicitly in our model. However, for cyclic molecules, a ring-correction, 09ring , o f approximately 0.8 kJ/mol is required for each ring atom. This is highly significant and appears to work independently of the ring-size and ring-type. It may be related to do some specific vibrational energy differences between chain parts of molecules and rings. To summarize, the simple expression X l/ig
X X -AEcosMo -~- O)ringnring
~- JTig(V)
(7.6)
appears to work well for describing the free-energy difference of molecules in the ideal gas with a reference state of 1 bar relative to the COSMO reference state.
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A. Klamt
Since the vast majority of organic molecules behave almost ideally in the gas phase for temperatures below 200~ Eq. (7.6) allows us to calculate the vapor pressure of such compounds from the chemical potential difference of the molecule in the ig phase and in the respective liquid state. As a special case, we can express the vapor pressure of the pure compound X as
X}
#X _ Pig
pvXp(T) - exp { - kT
(7.7)
It should be noted that this is only true for ideal gases, i.e., sufficiently below the critical temperature, Tc, of the compound X. As a rule of thumb, a value of Tig,max --- 0.7Tc, which is about the boiling temperature, is a safe limit for being in the ig range. For gases with low critical temperatures, or at high temperatures, reality corrections for the gas-phase free energy are required. Such corrections are known as fugacity corrections in chemical engineering thermodynamics, and several good models have been developed, including virial theory and equations of state. However, no single model is generally accepted as default for gas-phase reality corrections. Furthermore, all models require experimental data. Therefore, we do not intend to include such fugacity corrections as a routine part of the COSMO-RS model, and consider it as sufficient to be able to give a prediction of the free energy of transfer from the liquid state to the ig state. Empirical fugacity corrections have to be taken from other models, if they are required. It is very satisfying and useful that the COSMO-RS model~in contrast to empirical group contribution models~is able to access the gas phase in addition to the liquid state. This allows for the prediction of vapor pressures and solvation free energies. Also, the large amount of accurate, temperature-dependent vapor pressure data can be used for the parameterization of COSMO-RS. On the other hand, the fundamental difference between the liquid state and gas phase limits the accuracy of vapor pressure prediction, while accurate, pure compound vapor pressure data are available for most chemical compounds. Therefore, it is preferable to use experimental vapor pressures in combination with calculated activity coefficients for vapor-liquid equilibria predictions in most practical applications.
Refinements, parameterization, and the complete COSMO-RS 117 7.4
RESULTS OF THE PARAMETERIZATION
With the previous refinements of the basic COSMO-RS, a first quantitative parameterization of COSMO-RS has been published, which is based on DFT/COSMO calculations with the DMol program [46,47]. The details are given in [C10]. We shall now focus on a few important topics. The parameterization performed was based on 225 small- and medium-sized organic compounds composed of the elements H, C, O, N and C1. A set of 650 experimental room-temperature data for AVhydr(which is equivalent to the logarithmic Henry's constant), log(vapor pressure), and four log(solvent-water partition coefficients; solvent = n-octanol, hexane, benzene and diethyl ether) was used to optimize the parameters appearing in the COSMORS formalism. Two element-specific parameters were fitted, i.e., the COSMO radius used in the DFT/COSMO calculations and the vdWs coefficients ~(e) occurring in Eq. (7.3). The results for these parameters are given in Table 7.1, together with the results for six additional common elements frequently occurring in organic chemistry. It can be seen that the optimal radii are found in a narrow window of 108 to 120% of Bondi's vdWs radii [129]. An
TABLE 7.1 COSMO-RS optimized radii and vdWs coefficients for 11 elements Element
H C N O C1 F Br I Si P S
Cavity radius
Bondi radius
Ratio
1.30 2.00 1.83 1.72 2.05 1.72 2.16 2.32 2.48 2.13 2.16
1.20 1.70 1.55 1.52 2.05 1.47 1.85 1.98 2.10 1.80 1.80
1.08 1.18 1.18 1.13 1.17 1.17 1.17 1.17 1.18 1.18 1.20
(s
(s
vdWs coefficient (kJ/mol/nm 2) 3.82 3.48 2.21 3.66 5.14 2.65 5.50 5.80 4.00 4.50 5.10
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A. Klamt
approximate rule of 117(_+0.02)% of the Bondi radius appears to apply for most elements, with the exception of hydrogen. Hydrogen shows a significantly lower value, probably as a compromise between normal vdW's contacts and hydrogen bonding in which the distance between the hydrogen and the acceptor atom is much smaller than a normal vdWs distance. Thus, we conclude that the optimized COSMO radii stay quite close to the initial expectations. The vdWs coefficients scatter by a factor of 2, with a physically plausible increase from the first to the higher rows of the periodic system. It is only the reason for the extraordinarily low value for nitrogen that is not obvious to us. Within the general, non-element-specific parameters, the effective size a e f f of a thermodynamically independent contact is most important. Its value was calculated as 7.1A 2 in the original parameterization. Meanwhile, we obtain slightly smaller values, ~6.5A~ 2 . From the correlation of UNIQUAC q-parameters with the areas of the COSMO cavity surfaces, we can conclude that a UNIQUAC surface unit corresponds to ~40,~ 2 on the COSMO surfaces. Dividing this by the effective contact surface, we find that one UNIQUAC unit makes 5-6 independent contacts. This means that we find a coordination number of ~5-6. Given that the chemical engineering models work with values of z ranging from 2 to 10, this result appears to be plausible. The value of aeff~7 s p e r nearest-neighbor contact also agrees reasonably well with the results of MD and MC simulations, although the criterion for counting nearest-neighbor contacts is not well defined there. The optimized value of the electrostatic misfit coefficient, ~', in Eq. (6.1) is ~20% smaller than the simple electrostatic estimate including a mean polarizability correction. Considering the number of approximations included in the electrostatic misfit picture, this is a satisfactory agreement. For the other parameters, no simple plausibility checks can be made. Hence, we refer to [C10] for the values and for a discussion. Only the threshold value of 6 h b - - 0.82 e / n m 2 may be of interest here, since it tells us that only surfaces that are more polar than • 2 can make hydrogen bonds. By looking to the a-profiles given in Figs. 6.1-6.4, one can easily see that only the contributions of typical donor and acceptor atoms usually fall in these regions.
Refinements, parameterization, and the complete COSMO-RS 119 This initial parameterization was made exclusively for roomtemperature data. Although COSMO-RS has an intrinsic temperature dependence of the results arising from the statistical thermodynamics, we cannot expect to achieve a good description without temperature-dependent hydrogen-bond and vdW parameters, because both interactions are compromises of enthalpy gain by falling into a narrow, deep well and the entropy loss going along with this. Later, we introduced a temperature dependence for both interactions and adjusted the additional parameters to vapor pressure data because vapor pressure is most sensitive to temperature, and reliable data are easily available for a wide range of compounds. While initially using a simple (A+BT) type expression for the temperature dependencies, we later switched to more statistical thermodynamic expressions as given in Eq. (6.2), which show a better limiting behavior at very high and low temperatures. Other expressions have been tested as well with similar results, but presently there is no need for a change.
7.5
CONFORMATIONAL AND TAUTOMERIC EQUILIBRIA
So far, we have treated molecules as rather stiff entities, having a well-defined geometry and hence well-defined total energy, COSMO cavity and COSMO polarization charges. While this is a good approximation for many simple chemical compounds, e.g., methane, ethane, propane, benzene, toluene, methanol, dimethyl ether, etc., most of the more complex molecules have more than one relevant conformation, i.e., they have relevant metastable energy minima in addition to the total energy minimum. For example, the fourth carbon atom in n-butane can form a dihedral angle of 180 ~ with respect to the other three carbon atoms, called the trans-(or anti-) conformation, or it can take an angle of • 60 ~ called the gaucheconformation. In this case, and in most other cases, the transgeometry is energetically favorable by about 2.0kJ/mol, i.e., it is the minimum energy conformation. There are two equivalent gauche-conformations, i.e., the multiplicity of the gaucheconformation is 2. For molecules with a larger number of rotatable bonds, the number of conformations can easily increase exponentially.
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A. Klamt
Fortunately, the conformational ambiguity can be disregarded in many cases for the calculation of chemical potentials and phase equilibria with COSMO-RS. This is the case if the a-profiles of the different conformations are very similar, as for trans- and gaucheconformations of alkane chains. In such cases, the free-energy difference between the conformations is rather independent of the environment, and hence the total chemical-potential lowering of the compound arising from the population of multiple conformations cancels out any chemical-potential difference. This means that in such cases, the thermodynamic equilibria are unaffected by the conformational ambiguity, and the compound can be well described by its minimum energy conformation. There are many other compounds for which multiple conformations are known, but for the same there is just a single conformation much lower in energy than the next ones. In these cases, it is often sufficient to consider this minimum energy conformation, because the population of the other conformations is negligibly small at moderate temperatures. Typically, an energetic separation of 9 kJ/mol is sufficient to be safely within this limit. However, we must be aware that the energy difference between the conformations may change in different environments. If the polarity of the conformations is very different, e.g., due to the presence or lack of an intramolecular hydrogen bond, then the free-energy difference may change strongly between a polar solvent such as water and a non-polar solvent or, even more, the gas phase. For example, we may consider salicylic acid shown in Fig. 7.3. It has mainly four relevant conformations: two having an intramolecular hydrogen bond between the hydroxyl group and one of the carboxyl oxygen atoms, and the two others being generated from the former by a 180 ~ rotation of the hydroxyl group. In the gas phase, the conformation with the hydrogen bond to the sp2-oxygen has by far the
Fig. 7.3. COSMO surfaces of the four conformations of salicylic acid.
Refinements, parameterization, and the complete COSMO-RS 121 lowest energy, followed by the other hydrogen-bonded conformation at a difference of about 17kJ/mol, and the conformations without a hydrogen-bond at a separation of 50 kJ/mol. In the solvent water, the calculated energy difference from the first to the second conformation decreases by 8 kJ/mol, and those of the nonhydrogen-bonded conformations by 33 kJ/mol. So we can see the free-energy difference may depend dramatically on the solvent environment. In this case, the minimum-energy conformation stays the same in all solvents, well separated by more than 8 kJ/mol even in water. So, it is generally sufficient to consider the lowest energy conformation of salicylic acid. We choose glycerol as an example of a molecule requiring multiple conformations for an accurate description. This has three hydroxyl groups that may or may not form intramolecular hydrogen bonds with each other. From the DFT/COSMO calculations we find six conformations of glycerol (three of them shown in Fig. 7.4) within 4 kJ/mol of the lowest-energy COSMO conformer, and about six additional ones up to 10 kJ/mol above the lowest conformer. Conformer 1 has three intramolecular hydrogen bonds, two of them being 5-ring hydrogen bonds and the third
Fig. 7.4. a-profiles and COSMO stwfaces of low-lying glycerol conformations.
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A. Klamt
being forming a 6-membered ring. Thus it is the least polar glycerol conformation, followed by conformer 2, which is 1.5 kJ/mol higher in DFT/COSMO. Conformer 2 has one of the s t r a i n e d ~ a n d hence less favorable--5-ring hydrogen bonds opened, and thus shows a free hydrogen-bond donor. In the gas phase, the energy difference between these two lowest conformers increases to 8 kJ/mol, owing to the greater polarity of the second conformation. Interestingly, the energy difference of conformer 4, which is about 2.5 kJ/mol above conformer 1 in DFT/COSMO, stays almost the same in the gas phase, so that it is ranked second in the gas phase. Applying the program for COSMO-RS calculations (COSMOtherm) to the individual conformations, we find that in the solvent water the seven lowest conformations are within a freeenergy window of only 1.2 kJ/mol, and that conformer 2 is slightly (0.5 kJ/mol) lower than conformer 1. In contrast, we find that in the solvent acetone, conformers 4 and 5 are about 1.2 kJ/mol lower than conformer 1. This is due to the extreme affinity of acetone for hydrogen-bond donors, which we have discussed earlier in this book. In summary, glycerol is a kind of a chameleon that can change its outside polarity by switching between different conformations, induced by properties of the environment. It should be mentioned that all sugar compounds are of the same kind. In order to demonstrate that multiple conformations can also be important for non-hydrogen bonding molecules, we now consider 1,1,1,3,3-pentafluorobutane (see Fig. 7.5). Normally, butane
Fig. 7.5. Trans- and gauche-conformations of 1,1,1,3,3-pentafluorobutane.
Refinements, parameterization, and the complete COSMO-RS 123 would prefer a trans-conformation with respect to the carbon chain. Indeed, this conformation is found as the lowest in DFT/ COSMO. But even the moderate, opposite polarities of the CH3and CF3-end-groups cause a sufficient gain in Coulomb energy to bring the gauche-conformation very close, i.e., it has only 0.4kJ/mol higher energy than the minimum conformation in DFT/COSMO. In the gas phase, the gauche-conformation is by 5.5 kJ/mol lower than the trans-conformation. Thus, neglecting the gauche-conformation would lead to an underestimation of the vapor pressure by about a factor of 10. Indeed, it was this example that taught us the importance of conformations, because we mispredicted the vapor pressure of this compound by almost a factor of 10 in a validation study. The problem was immediately resolved by introducing the second conformation. These considerable solvent-dependent shifts demonstrate that for molecules with such pronounced conformational ambiguity, the equilibration of conformations has to be taken into account in the calculation of thermodynamic phase-equilibrium data. In order to enable a consistent treatment, we have implemented an automated conformation equilibration scheme in C O S M O t h e r m . A compound X can be represented by a set of COSMO-files for the conformers, and a multiplicity ~x(i) can be assigned to each conformer based on geometrical degeneration aspects. Then the population of a conformer, i, in a solvent S is calculated as
s(i) -
~x(/)exp{ Ej ~
ZX~
(i'}
{ EXoskoO)+ xo)} (J) e xp T
(7.8)
according to the Boltzmann distribution between states of different free energies. If the compound X is itself a relevant part of the solvent S, the chemical potentials, g~(i), themselves depend on the conformational population. Therefore, in general, Eq. (7.8) has to be iterated to self-consistency, starting from an initial population guess based on p~(i) = O. Apart from conformational ambiguity, some molecules can have different states that differ by the position of a hydrogen atom in the molecule, and a corresponding change in the bond orders. If the exchange between these states is sufficiently fast to keep both states in thermodynamic equilibrium, the different
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A. Klamt
forms are called tautomers of a compound. One of the most famous examples is the keto-enol tautomerism illustrated for 2,4-pentanedione in Fig. 7.6. Normally the keto form is more stable. The enol form can be of importance only if the hydroxyl group is stabilized strongly by intramolecular hydrogen bonds, as in the case of 2,4-pentanedione, or by strong solvent interactions. Another common example of tautomerism occurs in conjugated heterocycles, especially if they include an amide group as part of the ring. The solvent dependence of such tautomers has been studied with COSMO-RS in [C4]. The formation of zwitterions as in glycine (see Fig. 7.7) can also be considered as a type of tautomerism. The situation is even more complicated by the fact that two different neutral conformations and two different zwitterionic species can be found. It turns out that the neutral conformer 1, having an intramolecular 5-ring hydrogen bond to the strong, lone pair of the amine group, is---in all kinds of liquid phases and in the gas phase--more than 8.5 kJ/mol lower in energy than the neutral conformation 2, which has the normal carboxylic acid conformation and a Coulomb stabilization between the amine hydrogens and the sp2-oxygen. The zwitterionic conformations 1 and 2 differ by the nature of the interaction between hydrogens of
Fig. 7.7. Two neutral and two zwitterionic conformations of glycine.
Refinements, parameterization, and the complete COSMO-RS 125 the NH~ group and one of the oxygens. In conformation 1, which is lowest in DFT/COSMO and in non-polar solvents, one of the N H 3+ group hydrogens forms a strained 5-ring hydrogen bond, while in the other zwitterionic conformation, two of the N H 3+ hydrogens form a Coulomb bridge and leave all three polar hydrogens available for hydrogen bonds with the solvent. In water, zwitterionic form 2 is more stable (3 kJ/mol) than zwitterionic form 1, and it is approximately 50 kJ/mol more stable than the neutral conformation 1. In octane the neutral species is almost 42 kJ/mol more stable than the lowest zwitterionic form, which in this case is ZWl, the latter being more than 17 kJ/mol lower than ZW2. In the gas phase, the zwitterionic forms are not stable, but fall back directly to the neutral conformation 1. Such tautomeric equilibria, which may be crucial for the understanding and appropriate description of phase equilibria, can technically be treated in the same way as conformational equilibria in COSMO-RS. Unfortunately, we must be aware that the total energy differences between the tautomers may be in error by 8 kJ/mol or even more, if we calculate them on our default DFT/COSMO level. To gain an accurate description of such phenomena, it is therefore necessary to correct the COSMO energy differences either by comparison with higher level QM calculations, or by treating them as adjustable parameters.
Chapter 8
COSMO-RS for chemical engineering thermodynamics Although the primary focus of the development of COSMO-RS was the prediction of partition coefficients and vapor pressures, as they are typically needed in computational chemistry for drugdesign and more general chemical product design, and for environmental property predictions, chemical engineering simulations have become the most important area of applications for COSMO-RS. I am extremely grateful to Iven Clausen, who first calculated binary-phase diagrams with the COSMO-RS program, which was intended to be used for the prediction of fish oil-water partition coefficients in his Ph.D. thesis. In 1998, he sent me some binary phase diagrams of alcohol-water systems, which looked extremely promising (see Fig. 8.1). This was the starting point for chemical engineering applications of COSMO-RS and there is no doubt that the credit for detecting the potential of COSMO-RS in general chemical engineering thermodynamics is owed to Iven Clausen and his supervisor, Professor Wolfgang Arlt from the Technical University in Berlin [96,97]. 8.1
PREDICTION OF BINARY INTERACTION PARAMETERS
Although COSMO-RS generally provides good predictions of chemical potentials and activity coefficients of molecules in liquids, its accuracy in many cases is not sufficient for the simulation of chemical processes and plants, because even small deviations can have large effects on the behavior of a complex process. Therefore, the chemical engineer typically prefers to use empirical thermodynamic models, such as the UNIQUAC and NRTL, for the description of liquid-phase activity coefficients with 127
128
A. Klamt
Fig. 8.1. Original diagrams of the first COSMO-RS phase-diagram calculations by Iven Clausen [96] for four alcohol-water mixtures (methanol at 60 ~ ethanol at 55 ~ 1-propanol at 60 ~ and 1-butanol at 60 ~
interaction parameters fitted accurately to experimental data. The UNIQUAC model is based on some pure compound parameters such as surface areas, qi, and volumes, ri, and on binary interaction parameters, rij, between molecular species. The equation for the excess Gibbs free energy in UNIQUAC reads [79] V ex -- R T
{zi
(ri) z
xi In ~
9
(xiqi )} i
xiqi In
- ~
rav
+ ~ + ~ xiqi In q-~v ri /
~av r/j
(8.1)
$
where "av" denotes the mole-fraction average of the surface areas and volumes, respectively. The first two contributions in the brackets are the combinatorial contribution according to Staverman and Guggenheim (see Eq. (5.3) and Appendix C). The
COSMO-RS for chemical engineering thermodynamics
129
corresponding, even more empirical NRTL equation reads [98]
ZXiX
9 j
E k xk exp(--(XT.kj)
}
(8.2)
where ~ is an adjustable parameter. Only if experimental data are not easily available for certain compounds and binary mixtures, will the chemical engineer rely on predicted thermodynamic data for the missing binary interaction information. This is the situation where a predictive model such as COSMO-RS comes into effect, especially if simpler methods such as group contribution methods are not applicable. In order to feed consistent predictions into this overall framework, typically, the activity coefficients are predicted for the binary phase diagrams, and the missing binary interaction parameters of the empirical models are derived by a fit to the predicted activity coefficients. As a typical example from industrial practice we consider the simulation of a process with the reaction of methylphosphinic acid and butanol to methylphosphinic acid butyl ester and water, which was modeled by Gordana Hofmann-Jovic at InfraServ Knapsack [C28]. Because of the lack of experimental data for the binary systems with phosphorous compounds, COSMO-RS was used for the prediction of the binary activity coefficients. Then the results were fitted by an NRTL equation and the entire process was modeled by a commercial process simulator. The resulting phase diagrams were in close agreement with experimental measurements obtained later (Fig. 8.2). The general reliability and the accuracy limitations of COSMO-RS for a wide range of industrially relevant binary systems have been evaluated systematically in several studies [C18, C23-C25,99,106,107].
8.2
COSMO-RS AS THERMODYNAMIC MODEL IN SIMULATIONS
It would be at least interesting to use COSMO-RS directly as the thermodynamic model in process simulations. This would be especially useful if simpler empirical models are unlikely to be able to describe more complicated molecular interactions such as
130
A. Klamt
Fig. 8.2. Simulation of the isobaric phase diagram for the reacting system methylphosphinic acid + butanol --, methylphosphinic acid butyl ester plus water based on COSMO-RS predictions [C28].
hydrogen bonding or, even more, conformational and tautomeric equilibria. Technically, COSMO-RS meets all requirements for a thermodynamic model in a process simulation. It is able to evaluate the activity coefficients of the components at a given mixture composition vector, x, and temperature, T. As shown in Appendix C of [C17], even the analytic derivatives of the activity coefficients with respect to temperature and composition, which are required in many process simulation programs for most efficient process optimization, can be evaluated within the COSMO-RS framework. Within the COSMOtherm program these analytic derivatives are available at negligible additional expense. COSMOtherm can also be called as a subroutine, and hence a simulator program can request the activity coefficients and derivatives whenever it needs such input. One of the concerns regarding the use of COSMO-RS thermodynamics directly in simulations is the considerably larger computation time that is required for the evaluation of the activity coefficients compared to simpler empirical models with
COSMO-RS for chemical engineering thermodynamics
131
explicit expressions for the activity coefficients. This may be a real issue for large-scale dynamic process simulations, but because of the speed of modern computers this is not a serious issue in the majority of simulations done by chemical engineers, if the COSMO-RS method is efficiently implemented as in COSMOtherm. As a proof of the feasibility of such direct COSMO-RS process simulation, Taylor et al. [100] have linked the COSMOtherm program into their simulation program CHEMSEP [101] for distillation separation processes. For a number of typical separation problems they report very satisfying results, which are comparable with simulations based on empirical models. The simulation times were only a factor of 2 greater than those using empirical models. The quality of the simulations was considered as comparable to empirical models, although those were based on fitted experimental data. Nevertheless, for a broader usage of direct COSMO-RS thermodynamics in process simulations, it will be necessary to enable a fine-tuning of the COSMO-RS results to all the experimental data available for the specific system under consideration. This can be done either by introducing small charge corrections for the atoms, i.e., by a fine-tuning of the a-profile, or by the introduction of binary interaction corrections for the molecules, which would then have to be added into the interaction energy expression of COSMO-RS. In this way it would be possible to combine the advantages of models fitted to experimental data with the advantage of having a physically based and predictive model as in COSMO-RS. However, this extension of the program COSMOtherm is still under development and no results are reported yet.
8.3
SOLVENT SELECTION
Another important area of chemical engineering applications is in the selection of solvents, co-solvents, or entrainers~generally summarized as solvents~in the early phase of a process design, development, or optimization project. For such tasks, beyond the fixed essential starting materials and products of the process, a large number of compounds has to be considered as potential
132
A. Klamt
candidates for solvents. The essential compounds may be quite special and hence require individual DFT/COSMO calculations which typically will take half a day. In contrast, the solvent candidates are usually standard solvents. Therefore, the DFT/ COSMO calculations for the solvents can be taken from a database of precalculated common compounds and solvents, and need not be calculated individually for each project. Such a database (COSMObase) with about 2700 compounds is available [C30]. Using this database, the COSMOtherm program enables the calculation of the required thermodynamic properties, which may be the solubilities of compounds, separation factors, or any other property accessible by COSMOtherm, within a few minutes. By sorting the resulting list with respect to the desired property combinations, a number of promising solvents can be selected from a wide range of very different candidates. Finally, experimental measurements will be required for the set of most promising candidates in order to validate the COSMOtherm predictions. This procedure of COSMO-RS solvent screening, meanwhile, is applied routinely in a number of large chemical companies. One successful COSMO-RS-based solvent replacement has been reported. This is already implemented in an industrial process and saves more than a million Euro per year [102]. Owing to the secrecy restrictions of most industrial projects, it is likely that other successful applications exist, but are unpublished.
8.4
IONIC LIQUIDS
Room-temperature ionic liquids, i.e., salts with melting points below or around room temperature, have attracted considerable interest in recent years as potential, very promising green solvents, i.e., environmentally benign solvents for reactions, extraction and separation [103]. The special advantage of these solvents is that they provide a wide range of solvent polarity and have almost no vapor pressure. By combinations of various large organic cations with a set of suitable anions, a large number of different ionic liquids can be generated and thus one can think of tailoring the most suitable ionic liquids for each separation application.
COSMO-RS for chemical engineering thermodynamics
133
One of the obstacles in this aim is the lack of experimental thermodynamic data for activity coefficients in ionic liquids, which could be a basis for such solvent selection. In the past years several groups have started to measure such data; however, there is a lack of data because the number of suitable anions and cations, and even more the number of ionic liquids, are rapidly increasing compared to the rate (or speed) of measurements. Reliable inter- and extrapolation schemes and group contribution methods are still missing. Thus the search for an appropriate ionic liquid for a certain task can, at present, only be made randomly or by systematic measurements. A few years ago, we decided to try whether COSMO-RS would be able to predict thermodynamic equilibrium data in ionic liquids. We calculated a series of typical anions and cations used for ionic liquids. Their selection is visualized in Fig. 8.3, and a-profiles of ionic liquids are shown in Fig. 8.4. The typical cations used in ionic liquids have aromatic rings which strongly delocalize the positive charge. As a result, the surfaces show only moderate polarization charge densities without any polar hotspot. Polarization charge delocalization is also achieved in quaternary ammonium cations. Here, the charge is quite localized at the central nitrogen atom, but this is not accessible from the surface. Hence, the polarization charges are delocalized over the large amount of alkyl surface. On the other
Fig. 8.3. a-Surfaces of typical ionic liquid cations (upper row: 1-butyl-2,3methylimidazohum, 1-butyl-3-methylimidazolium, and 4-methyl-N-butylpyridinium) and anions (tetrafluoroborate, perchlorate, hexafluorophosphate, and bis-(trifluoromethylsulfonyl)imide).
134
A. Klamt
Fig. 8.4. a-Profiles of typical ionic-liquid anions and cations.
hand, the anions used in ionic liquids mostly have high symmetry and thus delocalize the polarization charge over the four or six ligands as in BF 4, C104, or PF 6, or they delocalize the negative charge by conjugation over several atoms, as in bis((trifluoromethyl)sulfonyl)imide. From the a-profiles of Fig. 8.4, it is clear that the anions and cations of ionic liquids are less polar than water, which is shown as a dotted line for comparison. Indeed, it appears to be one of the necessary conditions for ionic liquids that no polarity hotspots are present, since these would lead to aggregation of ions and subsequently to crystallization. The application of COSMO-RS to the calculation of infinitedilution activity coefficients in ionic liquids was surprisingly successful. As shown in Fig. 8.5, the activity coefficients of neutral compounds in ionic liquids are very well described. This was achieved without any special adjustment of COSMO-RS, which was developed and parameterized for neutral solvents, just by describing the ionic liquid as a 50:50 mixture of anions and cations. We only needed to take into account the convention of chemical engineers of counting a pair of an anion and a cation as
COSMO-RS for chemical engineering thermodynamics
135
Fig. 8.5. Calculated vs. experimental infinite-dilution activity coefficients of organic solvents in the ionic liquid, 4-methyl-N-butylpyridinium tetrafluoroborate. Experimental data from Heintz et al. [104]. one molecule while we consider them as two independent species. This originally caused a systematic mis-prediction of activity coefficients in ionic liquids by a factor of 2. Indeed, we believe that our convention is physically more meaningful, because the ions behave quite independently in ionic liquids. We consider it as a remarkable proof of the physical robustness of the COSMO-RS model that it is able to predict activity coefficients in ionic liquids, although it was developed and trained on neutral compounds. Meanwhile, COSMO-RS has been applied to many more ionic liquid systems [105,C26] and it is widely considered as a powerful tool to select the right ionic liquid for a given separation or reaction problem. Nevertheless, it must be noted that COSMO-RS is, at present, not able to treat ions correctly at finite low ionic strength, because it cannot reproduce the Debye-Hfickel limit resulting from the long-range ion-ion interactions. Fortunately, these effects are usually small. More research on these aspects has been initiated. These deficiencies may be the reason for considerable deviations
136
A. Klamt
Fig. 8.6. Liquid-liquid equilibria of alcohol-ionic liquid mixtures [105]. The left side shows the LLE curves of 1-butyl-3-methyl-imidazoliumPF6 mixtures with alcohols (ethanol, blue; 1-propanol, red; and 1-butanol, green symbols). The experimental curves (solid symbols) show a shape different from the calculated LLE curves, but the upper critical-solution temperatures (UCST) are surprisingly well met. On the right side, the trends of the UCST with a modification of the 1-alkylgroup of the anion (butyl = 4, octyl = 8) is shown. Again, the COSMORS predictions (open symbols, same color code as on the left) are in surprisingly good agreement with the experimental data. from the corresponding experimental data, in the shape of the LLE curves calculated for mixtures of ionic liquids and alcohols. Despite the shape deviations, the predicted upper critical solution t e m p e r a t u r e s were in surprisingly good agreement with the experiment [105] (see Fig. 8.6). Thus, such COSMO-RS applications to ionic liquid-solvent mixtures may be useful despite the known deficiencies of COSMO-RS in this field, but much more experimental and theoretical work is w a r r a n t e d in order to draw final conclusions here.
Chapter 9
The 9.1
-moment a p p r o a c h THE CONCEPT OF O'-MOMENT REGRESSIONS
As we have seen before, COSMO-RS is able to describe the chemical potentials of compounds in almost any pure or mixed liquid phase, as long as the chemical composition of the liquid is known and as long as it can be considered as a chemically homogeneous phase. While this has opened an enormously broad range of applications in chemical engineering, these limitations exclude COSMO-RS from a number of important application areas in environmental simulations, life-science modeling, and product development. In physiological studies, partition coefficients between different liquid or pseudo-liquid phases are often important. One example is the blood-brain partition coefficient, mostly abbreviated as log BB in its logarithmic form, which describes the distribution of drug compounds between blood and brain liquid. Obviously, neither blood nor brain liquid are homogeneous. Both include many macromolecular ingredients and a large variety of dissolved small neutral and charged species. Nobody really knows the exact chemical composition of blood and brain liquid: this is slightly different for each person, and it may even change as a function of the time of the day, nutrition, and other variables. Nevertheless, the log BB can be measured reproducibly by physiological chemists with an accuracy of about half a log-unit, and it is an important property for the description of drug distribution in humans [108]. It appears that kinds of effective blood and brain phases do exist, but the chemical composition of these effective phases is unknown. Thus, we cannot calculate the solvent a-profiles of these phases and hence cannot calculate the chemical potentials and the log BB directly with COSMO-RS. Another example is the soil sorption coefficient, or more precisely the partition coefficient between water and a wet organic 137
138
A. Klamt
soil phase, which is often used in environmental distribution models to describe the amount of a compound which gets adsorbed to the soil or aqueous sediment. Since it is mainly the organic part of the soil or sediment, which is responsible for the adsorption, this quantity is mostly denoted as logKoc, where OC stands for the organic carbon content of the soil. While, in this case, the water phase is reasonably well known, the wet soil phase is a kind of effective pseudo-liquid phase of unknown composition, and hence the log Koc cannot be calculated directly [C20]. A quite related situation is the adsorption on activated carbon, either from an aqueous phase or from the air. Here, the surface layer of activated carbon can be considered as a phase with a large amount of irregularly shaped surface area of unknown chemical composition. With some goodwill, one can consider this random air- or water-saturated phase as a type of pseudo-liquid. Then, the adsorption constants are a type of partition coefficients between this effective surface phase and the surrounding water or air, but a direct calculation of the adsorption constants is impossible because of missing knowledge of the chemical surface composition [C16]. In order to be able to treat such partition phenomena with COSMO-RS, we have developed a slightly more empirical extension to the COSMO-RS theory, which is called the "a-moment approach", [see C14,C16,C21 for more details]. The basic idea of this approach results from the finding that the a-potential ps(~) for almost all solvents can be well represented as a linear combination of about six a-functions. These are the simple polynomials with exponent i = 0, 1, 2, and 3, f i(6)
--
o "i
for i >/0
(9.1)
and two hydrogen bond functions 0 f-2/-l(6) -- facc/d~176
-~
:t:.a -- a'hb
if • a
a'hb
(9.2)
which are required because the almost linear parts of the a-potentials in the hydrogen-bond regions would cause a bad convergence in a purely polynomial Taylor series. Here, the most appropriate value of the hydrogen-bond cutoff in these functions is r -- i e / n m 2, and a parabolic smoothing of the edge of these
The a-moment approach
139
functions turns out to be useful. By indexing these hydrogen-bond functions as " - 2 " and " - 1 " , we can use the formal notation m
dsfi(~)
ps(O) ----
(9.3)
i=-2
for the linear expansion of the a-potentials, with m = 3 usually being sufficient. In Fig. 9.1, we see the a-function expansion of six important solvents. Apparently the aprotic solvents are almost perfectly described, while there are some small deviations remaining for the protic solvents. By using Eq. (9.3) for the calculations of partition coefficient of a compound X between two solvents, S and S', one easily obtains the expression log K xS , S '
----
--
0.4343 RT
~_ - 6s,s' +
cs.s.+ f
pX(a )
1 ~i Cs,s,fi(a) da
i=-2 m
= 5s,S, + ~
5is,s,M x
(9.4)
i=-2
Fig. 9.1. Original a-potentials of common solvents (solid lines) and their representations by a-functions (dashed lines).
140
A. Klamt
with solute specific a-moments Mi x being defined by
M x - J p X ( a ) f i ( a ) da
(9.5)
Some of the a-moments have a simple physical meaning. The zero-order moment M x is the total surface area of the solute X. The first-order a-moment is the total COSMO polarization charge on the surface of the solute, i.e., it is the negative of the total solute charge. Since charged species, i.e., ions, cannot be observed without the presence of counterions, bulk partition coefficients cannot be defined for ions. Hence the measurement and calculation of partition coefficients is restricted to neutral compounds, X, and therefore the first moment vanishes in Eq. (9.4). The secondorder a-moment, M x, can be shown to be highly correlated with the total COSMO polarization energy, and hence it is a measure of the overall ability of the solute to interact electrostatically with a polarizable continuum. The third moment lacks a simple physical analogy and represents a kind of skewness of the a-profile of the solute, while the two hydrogen-bond moments, MX2 = MXcc and MXl - MXon, quantify the ability of the solute to interact as a hydrogen-bond acceptor and donor, respectively. Skipping the trivially vanishing first moment, Eq. (9.4) implies that any logarithmic partition coefficient between phases S and S' can be described as a linear combination of just five solute-specific a-moments, with coefficients depending on the differences of the phases S and S'. If we know the chemical composition of S and S', we can calculate the coefficients from COSMO-RS. But if we have a reasonable set of experimental values of the respective partition coefficient available, we can also derive the coefficients from a simple multilinear regression analysis with respect to the a-moments. The latter approach does not require any knowledge about the chemical composition of the phases S and S' because all information about the phases is covered in the coefficients arising from the linear regression. Therefore, this a-moment regression approach is applicable to partition coefficients involving chemically unknown or less well-defined phases such as log BB or log Koc, if we have a small number of reasonably accurate experimental data for the partition coefficient available. The only assumption we have to make is that for each of the unknown phases there exists a type of a-potential ps(a) that expresses the
The a-moment approach
141
preference of the phase for a certain surface polarity, in the same way as for normal liquids. 9.2
SOME A P P L I C A T I O N S OF T H E O'-MOMENT APPROACH
The a - m o m e n t approach has proved to be a very robust and valuable tool for the analysis of the partitioning of compounds between most diverse phases. A typical application is log Koc, the logarithmic soil sorption coefficient, which is of crucial importance in environmental exposition and distribution models. Since the experimental m e a s u r e m e n t of Koc is difficult, expensive, and often unreliable, good prediction models are especially imp o r t a n t here. On the basis of a t r a i n i n g set of 387 compounds collected by Winget et al. [110], we derived a COSMO-KOC model expressed as log Koc - 0.0168M x - 0.017M x - 0.040M x § 0.19MXcc- 0.27MXon § 0.37
(9.6)
with a correlation coefficient of r 2 = 0.72 and an rms-deviation of 0.62 log-units [C20]. In this model, all five coefficients, as well as the regression constant, were statistically significant. On a chemically extremely d e m a n d i n g test set of 53 compounds, this model achieved an rms deviation of 0.72. As discussed in more detail in [C20], it is most likely t h a t about 0.5 log-units of the residual error result from experimental error (see Fig. 9.2). Applications of the a - m o m e n t approach to the adsorption constants of small compounds onto activated carbon from an aqueous and gas phases have been reported by Mehler et al. [C16]. They found the equations, X
ln(Hwater) -
- 5.458 + 0.102M x - 0.0957M X - 0.566MXon + 0.357M X ace
(9.7)
with n = 24 and rmse = 0.8 In-units = 0.35 log-units for the adsorption from water and ln(HXas) -
- 20.096 + 0.146M x § 0.0568M x + 1.578MXon - O.177M xa c e
(9.8)
142
A. Klamt
Fig. 9.2. Experimental vs. calculated soil sorption coefficients according to Eq. (9.6)[C20].
with n = 19 and rmse = 0.87 ln-units (= 0.38 loglo-units) for the adsorption from the gas phase. In both cases, the third a-moment was omitted as being statistically insignificant. As shown in Fig. 9.3, pyridine was initially the largest outlier for the aqueous adsorption to activated carbon, based on literature data, until an in-house measurement yielded a different experimental value in perfect agreement with Eq. (9.7). The a-moment approach has also proved to be especially useful in flavor and fragrance research and development. Since the performance of flavors and fragrances is finally determined by the small amounts of the compounds that are present in the gaseous form of the nose or the mouth, it is important to optimize their partition behavior between the different phases--from the production of the product to the final application by the end user. The phases considered in such projects are rarely homogeneous liquids, but are typically chemically less defined bulk or surface phases, such as candle wax, soap, shampoo, skin, hair, washing powder, or chewing gum. By using the a-moments, it has proved feasible to generate robust partition models for many such problems [111].
143
The a-moment approach 10
o~
4
E
2
x
X
=) -I-
Adsorption from liquid phase 24 experimental I-~ Adsorbens: Activated Carbon Solvent: Water T = 300K R2 = 0.963 S = 0.800 Pyridine [40] ~ e
ee
Pyridine ( m e a s u r e d ) e ~e
0
~-2 -4 -6
y. -8
!
|
!
i
!
!
!
!
-6
-4
-2
0
2
4
6
8
10
In(H x theor.) [m=/kg]
Fig. 9.3. Experimental and predicted values for the adsorption constant onto activated carbon from the aqueous phase according to [C16]. (Ref. [40] in the diagram is: [K. Fukuchi, Y. Arai, Coll. Surf. 37, 387, 1989].
Obviously, the a-moment approach can also be applied to the partitioning between chemically well-known phases. As a performance test, we used the best investigated partition coefficient, the octanol-water partition coefficient, log Kow. On the basis of about 210 compounds, we found a a-moment model log Kow - 0.029M x - 0.043M x - 0.036M x + 0.058MXcc + 0.005MXon - 0.67
(9.9)
The rms deviation of this model is 0.34, which is only slightly larger than the rms deviation achieved by the direct prediction of logKow with COSMO-RS. The advantage of the a-moment approach compared with the direct COSMO-RS calculation of partition coefficients is its extremely broad applicability. However, it has the disadvantage that no temperature dependence is available and no extrapolation to finite concentrations can be made, because of the loss of the sound statistical thermodynamics. In his Ph.D. thesis [112], Mehler
144
A. Klamt
shows a method to obtain approximate a-profiles of the phases from the regression coefficients obtained in the a-moment regression. On the basis of such a-profiles, one could then start to apply COSMO-RS thermodynamics to the prediction of the respective chemical potentials in the unknown phases, including the ability to predict temperature dependencies and finite concentration effects. This would be of special value for adsorption coefficients where finite concentrations and temperature effects are of considerable practical interest.
9.3
COMPARISON OF O'-MOMENTS AND ABRAHAM DESCRIPTORS
During the last 15 years, Abraham and his co-workers have established a set of five descriptors for the general description of logarithmic partition coefficients by linear regression. Their socalled "linear free energy relationship" (LFER) descriptors E, S, A, B, and V are effective parameters for the polarizability, polarity, hydrogen-bond acidity and basicity, and volume of the solute molecules, respectively [113-116]. They are mainly derived from experimental refraction and partition coefficients of the solutes. Since the set of the five Abraham descriptors E, S, A, B, and V as well as the set of the five a-moments M x, M x, M x, MXcc, MXon appear to be almost generally applicable and hence reasonably complete sets of linear descriptors for logarithmic partition coefficients, it is immediately obvious that they must be spanning essentially the same multidimensional property space. The fact that both independently developed approaches for logarithmic partition coefficients between arbitrary liquid or amorphous solvent phases require about five descriptors suggests that the solvent space is approximately five-dimensional. Indeed, when Mike Abraham and I first met at a conference we agreed to investigate the relationships between our descriptors in more detail. A rigorous analysis based on 470 compounds [C21] proved that the information content of our descriptor sets is almost the same, but it turned out that his polarizability descriptor, E, contains some information that is missing in our a-moments, while our skewness descriptor, i.e., the third a-moment M x is less well-reflected by the Abraham descriptors. Thus, it might be that a set of six
The a-moment approach
145
descriptors extracted from both data sets might be the optimal basis for LFER studies on logarithmic partitioning. From a practical point of view, the use of Abraham descriptors appears to be preferable in cases where the experimentally derived solute descriptors of the compounds are well available for all solutes of interest. If partition coefficients between phases similar to those used for the derivation of the Abraham descriptors are considered, the Abraham approach is usually more accurate, because it only needs to predict the small differences from the experimental information intrinsically included in the descriptors, and it avoids the total prediction of the large free energy differences that often occur between different phases. On the other hand, the COSMO a-moment approach does not require experimental input for each individual solute, and hence can be applied even in cases where no experimental data are available for the solute or even when the solutes are still virtual candidates for drugs or other agents. Furthermore, the a-moment approach is more rigorously derived from physical chemistry and provides more insight into the driving forces for partition equilibria.
9.4
(7-MOMENTS AS QSAR DESCRIPTORS
The multivariate linear regression analysis and even more advanced statistical and mathematical techniques, e.g., artificial neural networks, are regularly used in computational chemistry to find QSAR or quantitative structure-property relationships (QSPR). There are several packages available that generate a large number (from some tens to hundreds) of molecular descriptors from the chemical structure and statistical techniques are used to decide which of the properties are most significant for the property under consideration. Many of the industrially and pharmaceutically relevant activities and properties include partitioning between different phases as one important aspect. Although in most cases the partition aspect can be well reflected by other combinations of such molecular descriptors, it is preferable to use a complete set of partition descriptors as the a-moments as part of the descriptor set. This ensures that all aspects of partitioning are well covered by the descriptor set. In addition, the use of such a
146
A. Klamt
rationally defined descriptor set ensures a better predictivity and transferability of the generated QSAR models compared to "blind" QSAR models, which just select the combination of descriptors based on the performance on a limited training set of experimental activity values. While the latter approach often yields better statistical performance on the training set and on limited test sets, such models often fail completely if they are applied to compounds with chemical functionality not seen in the training set. An advantageous nice feature of the a-moment approach is its ability to plot the dependence of the fitted logarithmic partition property as a function of the polarity, a. In Fig. 9.4, we show the respective curves for the log Koc model in comparison with the log Kow model. It is apparent that log Koc exhibits a striking similarity to log Kow, but it is much more sensitive to strong polarities at both ends of the a-scale. Remarkably, both partition coefficients increase with the amount of hydrogen-bond-donor surface. Finally, unlike most other QSAR approaches, the a-moment approach allows for the graphical visualization of the logarithmic partition coefficients as local surface properties of the solutes,
Fig. 9.4. a-Dependence of the log Koc and log Kow a-moment models.
The a-moment approach
147
Fig. 9.5. Surface of theophylline colored by the logKow a-moment model (Eq. (9.9)). Red areas indicate enhancement of log Kow, while blue surface areas indicate reduction. While a large part of the molecule shows a slightly positive contribution (yellow regions), the hydrogen-bondaccepting regions decrease the octanol-water partition coefficient, and the strong hydrogen-bond donor in the front strongly increases logKow. because all m o m e n t s and hence, also their linear combinations are surface integrals. An example is given for the logKow of the drug theophylline in Fig. 9.5.
Chapter 10
The wider range of COSMO-RS applicability Since COSMO-RS allows for a rather fundamental calculation of the chemical potential of molecules in various chemical environments, there are many application areas that go beyond the calculation of activity and partition coefficients, which are directly accessible from the differences of chemical potentials in different phases. Often, such applications require the addition of some empiricism to the model because they involve free-energy contributions that are not directly accessible by COSMO-RS. Nevertheless, in many cases, the COSMO-RS is a robust starting point for such empirical extensions, and the resulting models are still less empirical and more fundamental than many other approaches. Without going into details we will describe some extended application areas in this chapter. 10.1
COSMO-RS FOR REACTION MODELING IN THE LIQUID PHASE
Although substantial progress has been achieved in the past decade, even advanced ab initio quantum chemical methods, like coupled cluster calculations, still have problems in achieving an accuracy of 1 kcal/mol in the calculation of the total free-energy differences of starting materials and products of a reaction. But such "chemical accuracy" is required for many practical decisions in reaction design and development. Therefore, the application of quantum chemistry in reaction engineering is still a very demanding, but definitely evolving technique. Even more demanding is the accurate location and energetic quantification of transition states, which is required for the prediction of the kinetics of chemical reactions. However, with some computational and chemical experience such transition-state searches are 149
150
A. Klamt
feasible to a certain extent, and have sufficient accuracy to distinguish between reasonable and impossible reaction mechanisms, and also to give a rough estimate of the expected activation energies. Nevertheless, most quantum chemical reaction modeling tools are restricted, so far, to gas-phase reactions. Rigorous tools for the transfer of such techniques to the technically and biologically important fluid phase are still rare. While, at first glance, the chances for a successful quantum chemical reaction modeling in the fluid phase appear to be small, since it is still extremely complicated in the gas phase, it turns out that by using COSMO-RS, the prediction of the changes of the reaction free energies, reaction enthalpies, and activation barriers of the same reaction between different solvents is much simpler than the prediction of absolute reaction quantities in the gas phase. Therefore, COSMO-RS can be used well for the selection of appropriate solvents meeting the required solubility properties for reagents and products, and lowering of the activation barrier of a reaction. The basic concepts of the solvent free energy changes in reaction kinetics are shown in Fig. 10.1. Indeed, prediction of the change of the equilibrium constants of a chemical reaction in a variable liquid environment requires nothing other than the prediction of the chemical potentials or activity coefficients of the starting materials and products in the liquids. Thus, this task can be performed simply by using the standard COSMO-RS capabilities. Successful applications of this
Fig. 10.1. Schematic visualization of the free energies relevant for reactions in solution.
The wider range of COSMO-RS applicability
151
kind have been reported by several industrial users, and quite promising results of the application of COSMO-RS to the TAME synthesis reaction in varying liquid compositions will be published in the near future [117]. Since most industrial reactions are not performed under equilibrium conditions, the prediction of kinetic constants and activation barriers in different solvents is much more important than the prediction of the equilibrium constants. Unfortunately this is also much more demanding and requires more approximations. The search for the transition state is restricted at present to the gas phase, and yet cannot be performed reliably in the COSMO CSM. Nevertheless, the first applications to the prediction of the solvent shifts of the kinetic constants of different reactions have yielded promising results, although single-point DFT/COSMO calculations based on the gas-phase transition states have been used for the generation of the transition-state COSMO files. Thus it appears that meaningful predictions of the change of kinetic constants under varying liquid conditions can be done with COSMO-RS, and hence appropriate solvents with low activation barriers can be selected for a reaction. As an example, the results for the Menshutkin reaction [C32] are shown in Fig. 10.2. Although
Fig. 10.2. Predictions of the logarithmic relative rate constants of the Menschutkin reaction in various solvents [C32]. The COSMO polarization charge densities a of the transition state are visualized in the inset.
152
A. Klamt
not yet put into practice, the prediction of the right solvent for lowering the free energy of a transition state leading to a desired product relative to other transition states leading to unwanted side products, may be of great importance in enhancing the yield of a reaction. This should also be possible with COSMO-RS. There is a growing interest in the prediction of liquid-phase reaction thermodynamics with COSMO-RS and the first results have been published in the literature [118]. 10.2
COSMO-RS PREDICTIONS OF pKa" A MYSTERIOUS SUCCESS
Proton transfer and the associated dissociation constants of acids and bases in water and in other solvents are of great importance in many areas of organic, inorganic, and biological chemistry. Therefore the first principle prediction of pKa values has been a subject of many research projects and scientific papers. In the gas phase, dissociation constants are experimentally difficult to measure and good quantum chemical calculations, i.e., postHartree-Fock ab initio calculations or state-of-the-art density functional methods, can nowadays be as good or even more accurate than experimental measurements. In the aqueous phase-which is of real importance--a large number of reliable data for dissociation constants is available, but here a theoretical prediction is extremely challenging because of the very strong solvation effects of the neutral and ionic species involved in the reaction. Thus the prediction of pKa in some way is the ultimate challenge for quantum chemically based solvation models. A large number of attempts in this direction have been reported applying all types of solvation approaches, but a consistent and broadly applicable solution is still lacking. According to the fundamental relationship, pKa - s
AGdiss + const RTln(10)
(10.1)
the calculation of pKa primarily means the calculation of the Gibbs free-energy of dissociation, which is typically defined by the reaction AH + H20 ~ A- + H30 +
(10.2)
The wider range of COSMO-RS applicability
153
where HA is an arbitrary acid. The slope s in Eq. (10.1) has to be unity if AGdiss is calculated without any systematic error, and the constant is expected to be - l o g [ H 2 0 ] - - 1 . 7 4 . Thus the computational task is just the accurate calculation of the total free energies of the four species occurring in Eq. (10.2). Here it should be noted that the total free energies of the neutral and protonated water species are the same for all acid dissociations, and hence an error in these values, or even their total omission, would only affect the value of the constant in Eq. (10.1), but not the slope or the quality of the prediction model. This is important, because the true state of the aqueous proton is not really known, and the species H30 + is just a plausible model for it. As an illustration of a typical aqueous dissociation reaction, the four species involved are shown in Fig. 10.3 for the case of formic acid. Before continuing the issue of pKa prediction, I should point out a peculiarity visible in Fig. 10.3. The hydronium ion H30 + appears to be an amphiphilic molecule with one extremely polar face resulting from the three hydrogens on the one side, and a hydrophobic face at the electron-depleted oxygen lone pair on the other side. I became aware of this unexpected feature during the Conference on Current Trends in Computational Chemistry in Jackson, MS, in November 2002, when Srinivasan Iyengar reported results on some new q u a n t u m molecular dynamics simulations on water clusters including one hydronium ion [130,131]. After showing some movies of such simulations, he pointed out that the hydronium had moved from the center of the cluster to the surface, with the oxygen finally sticking outside. He concluded that hydronium must be in some way hydrophobic. Immediately I turned on my laptop computer, visualized the a-surface of hydronium as shown in Fig. 10.3, and realized that it is green, i.e., hydrophobic, in the presence of oxygen. Showing this figure to the
Fig. 10.3. Species involved in the aqueous dissociation of formic acid.
154
A. Klamt
audience during the subsequent discussion generated much interest, since the same unexpected finding was now immediately supported by two absolutely independent approaches. While it is not clear to me whether this one-sided hydrophobicity, i.e., this amphiphilic behavior of hydronium, has any physicochemical importance, e.g., by some preferential association of hydronium to hydrophobic surfaces, it would be interesting to learn about experimental consequences. Because of its wide practical importance, I tried the prediction of pKa in the very early phase of my development of the solvation model, which was based on MOPAC/AM1/COSMO calculations in 1993, in collaboration with my colleague Jannis Batoulis, at Bayer [132]. While we found quite a good correlation of the calculated differences in total heat of formation with the experimental pKa values of a large number of alcohols and carboxylic acids, we were not satisfied with the slope of the regression curve, which was close to s = 0.5 in Eq. (10.1). Having no experience on the charge dependence of COSMO radii, we tried to achieve a physical slope closer to unity by changing the COSMO radii of the anions. To our surprise, we needed to change them to values much smaller than the Bondi radii in order to approach the expected slope, while our physical intuition would suggest that anions should be slightly larger than neutral species, owing to their additional electron. Furthermore, making such a change worsened the correlation with experiment. We concluded that these changes were unphysical, and that our approach was too crude to be meaningful. After this failure, I did not consider pKa calculations with COSMO or COSMO-RS for many years. In 2002, Michael Beck from Bayer CropScience reported very impressive COSMO-RS results on the calculation of pKa values of heterocyclic pesticide compounds [133] (see Fig. 10.4), for which all other methods failed to give good predictions. We started a joint study on pKa prediction [C27]. This time we chose a wide variety of 65 alcohols, phenols, carboxylic acids, inorganic acids, and heterocyclic acidic compounds, spanning an experimental pKa values in the range 0 to 18 as shown in Fig. 10.5. We calculated AGdiss as the total free-energy difference of the four species of Eq. (10.2) according to our COSMO-RS high-level standards. However, obviously, we made no change to the radii for the ions, because we then had sufficient experience to be reasonably sure that element-specific
The wider range of COSMO-RS applicability
155
Fig. 10.4. Correlation of pKa of some pesticide compounds with COSMORS dissociation energies [133]. Confidential parts of the molecules are hidden. The three yellow symbols denote three carboxylic acids that were treated for comparison. All other methods failed on this data set.
18 17 16 15 14 13 12 z "' 11 10 w 9 x" w< 8 ,,.,
Z~alcohols
13 o
7 6 5 4 3
carboxylic acids
9 inorganic acids 9 subst, phenols 0 N-acids (uracils, imines)
-2.5 0.0
2.5
5.0
7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0 AGdiu(kcal/moI )
Fig. 10.5. Experimental pKa values vs. calculated free energy of dissociation from COSMO-RS [C27].
156
A. Klamt
COSMO radii should also apply for charged species. A linear regression yielded the result pKa - 0.58(•
AGdiss + 1.66(=t=0.10) RTln(10)
(10.3)
with a very good correlation of r 2 = 0.983 and rms = 0.5. The constant is not far from the expected value o f - 1.74, at least if we consider the large free-energy differences, of the order of 300 kJ/ mol, involved in this reaction. However, the slope is again much closer to 0.5 than to unity. Next we confirmed this unexpected and somewhat unwanted, "wrong slope" by trying different quantum chemical methods (HF, various DFT functionals, and MP2) and different basis sets, but the slope was almost uninfluenced by these changes. A literature study yielded the result that many papers on pKa predictions based on a solvation model had shown similar results, with a slope near 0.6 in the pKa regression. Some workers had tried to enforce "better" slopes by modifying their models, while others just reported the regression equation. However, none of them had such a high statistical significance for the low slope as obtained by us. Our result for the slope is specially significant, because it was achieved without any pKaspecific model adjustments. As a final attempt to understand the origin of the unexpectedly low slope, we split AGd~ into four different parts, i.e., the quantum chemical gas-phase dissociation, the COSMO-interaction energies of the neutral and the ionic species and the chemical potential difference from COSMO-RS. On the basis of these reasonably independent descriptors, we performed a multilinear regression of pKa yielding an almost identical regression coefficient and rms error as before with slopes ranging from 0.55 to 0.62 for the four contributions. Thus all contributions--although very different in nature--show the same unphysical slope with respect to the pKa. At this point, we had to conclude that the "wrong" slope must have a mysterious reason outside our calculation methods, either in the experimental pKa scale or in the theoretical interpretation of pKa, although both alternatives seem equally strange and unlikely. Nevertheless, from a practical point of view, Eq. (10.3) is the most widely applicable and most accurate quantum chemically based PKa equation presently available, and it may be of great practical value in predicting the dissociation constants of
The wider range of COSMO-RS applicability
157
Fig. 10.6. Results of pKa prediction in DMSO. new acidic compounds with demanding chemical functionality, where conventional pKa prediction models fail. Meanwhile, the approach has been extended to base protonation reactions. The respective regression equation based on 43 bases reads as pg a =
0.75(•
AGdiss - 1.8(• RTln(10)
(10.4)
with r 2 = 0.98 and a RMS deviation of 0.56. The constant is in extremely good agreement with the expected value o f - 1 . 7 4 , but slightly different from the regression for the acids. The slope is a bit higher than the one found for acids, but still significantly lower than the theoretically required slope of unity. It must be noted that special parameterization is required for secondary and tertiary amines. The rather fundamental and almost unparameterized COSMO-RS approach also allows for the calculation of pKa in aqueous mixtures and other solvents. As an example we applied it to DMSO (see Fig. 10.6), achieving about the same accuracy as in water, but a slightly higher slope. It is not clear whether the difference in slope is of physical significance. Thus, COSMO-RS may be used for pKa prediction in different solvents, but unfortunately a parameterization is required for each solvent. 10.3
COSMO-RS FOR POLYMER SIMULATIONS
The state of polymers above the glass transition is quite similar to a highly viscous liquid state. Therefore it is a straightforward idea
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to apply COSMO-RS for the calculation of the solubility of molecules in amorphous polymers. Since large polymer molecules cannot be treated by DFT programs, the a-profile of a polymer has to be approximated by a sum of the partial a-profiles of the central repeat units of oligomer molecules. For most polymers it is possible to calculate at least trimer molecules, and the a-profiles of such central trimer units have been proved to be reasonably accurate for the representation of the polarity distribution of long polymer chains. By the definition of atom-weight factors, the COSMOtherm program provides the technical features to extract partial a-profiles of molecules and thus allow for the simple definition of pseudo-polymers based on oligomer COSMO files, as shown schematically in Fig. 10.7. It should be noted that co-polymers can also be "constructed" easily in the same way, if the net composition with respect to the monomer units A and B is known. Even AB transitions can be mimicked as oligomers and these
Fig. 10.7. Flowchart for polymer treatment in COSMO-RS.
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units can be considered in the right concentration. It should be noted that the combinatorial contribution has to be switched off in such polymer calculations, since it is not suited for the large molecular sizes of polymers. The calculation of the solubility of gases and other small molecules in various polymers showed t h a t for each polymer the relative solubility of the compounds was well predicted, but one polymer-specific correction had to be fitted. The results of this study are shown in Fig. 10.8. The correction constant may have to do with the missing combinatorial term, or with free-volume effects in polymers, or with structural differences of polymers, or with a combination of several such effects. The lack of a t e m p e r a t u r e dependence suggests that it is mainly of entropic nature. Unfortunately, no clear relationship between the value of
Fig. 10.8. Calculation of the solubility of gases in 24 different polymers, ranging from polyethylene to nitrocellulose [120]. One polymer-specific constant has to be adjusted in order to achieve good results. Experimental data are taken from Pauli [121] (HDPE = high density polyethylene, POP = poly(oxyethylene glycol), PDMS = poly(dimethylsiloxane), PTFE = poly(tetrafluoroethylene)).
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the polymer correction and structural parameters of the polymers has been detected so far. Since most other modeling techniques for polymers are extremely demanding, the limited capabilities of COSMO-RS for efficient prediction of solubilities in polymers can be of great help in practical applications when suitable polymers with certain solubility requirements are desired. One application may be the selection of appropriate membrane polymers for certain separation processes. Predictions of drug solubility in polymers are sometimes of interest for pharmaceutical applications. Furthermore, it is most likely that COSMO-RS can also be used to investigate the mutual compatibility of polymers for blends. This aspect, and many other aspects of the potential of COSMO-RS for polymer modeling, still awaits systematic investigation.
10.4
COSMO-RS FOR SURFACTANTS, MICELLES, AND BIOMEMBRANES
Surfactants, i.e., amphiphilic molecules with strongly polar, hydrophilic head groups and non-polar lipophilic tails, play a major role in many areas of industrial chemistry and biochemistry. The special feature of such molecules is their tendency to self-organize at liquid interfaces in form of spherical, cylindrical, or lamellar micelles. Biomembranes, which separate biological cells from the surrounding environment, are such lamellar micelle structures formed from phospholipid surfactants. Tensides, which are the most substantial parts of soaps, shampoos, and most other cleaning agents, are other examples of surfactants having practical importance in many areas of liquid-phase chemistry. There are three quite different classes of surfactants. While the tail mostly consists of one or several hydrocarbon chains with a few unsaturated bonds, the polar head group may either be a strongly polar neutral group, such as the hydroxyl group, or a zwitterionic fragment as in the case of the phospholipids, or even a positively or negatively charged group with an appropriate counterion. Even though surfactants can have long lipophilic chains, and often have molecular weights of 500-1000 a.u., the DFT/COSMO calculations for these molecules do not normally present major
The wider range of COSMO-RS applicability
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problems. Some examples of a-colored COSMO surfaces are shown in Fig. 10.9. Hence COSMO-RS calculations can be preferred for dilute surfactant molecules in homogeneous solvents with COSMO-RS. However, since molecules with pronounced surfactant properties tend to assemble at all kinds of surfaces and phase boundaries, there are almost no experimental data available for such strong surfactants dissolved homogeneously at low concentrations, and thus a direct proof of the applicability of COSMO-RS at this end is not possible. On the other hand, many compounds with weak or moderate surfactant tendency, like longchain alcohols, have been considered in the parameterization and validation studies with no apparent problems. Unfortunately, COSMO-RS by itself is not capable of treating inhomogeneous liquid systems with micelles and surfactants. This is a result of the basic approximation of neglecting the 3D geometry of the a-surface and reducing it to an ensemble of independently pairwise interacting surface segments. Without additional
Fig. 10.9. Collection of ionic surfactants (without counterions). It can clearly be seen that the polarization charge densities of the charges are localized on the head groups, while the tails already look like normal alkane or fluoroalkane chains after two carbon units from the head group.
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assumptions, there is no way to describe a behavior in which the probability of finding polar surface segments is higher in certain space regions of a liquid system, while less polar, lipophilic surface segments have a higher concentration in other regions. It would be very desirable to have an efficient tool for the modeling of surfactant micelles or biomembranes, e.g., for the investigation of the partitioning and solubilities of fragrances or other small additives in aqueous surfactant systems, or for the calculation of the affinity of pharmaceutical agents for cell membranes. While MD/MC force-field methods have been applied successfully to surfactants and biomembranes in order to build reasonable 3D models of the structures of such systems [125,126], the computational demands of such calculations are extremely high. Thermodynamically equilibrated MD/MC calculations including additional solutes would be almost prohibitively expensive, if not impossible. This essentially prohibits the calculation of membrane partition coefficients of drug candidates with MD/MC. Also, it is questionable whether standard non-polarizable force fields can handle the crossover from polar to non-polar regions of such systems with the accuracy that is required for a reliable description of the distribution of partially polar molecules in such systems (Fig. 10.10). Since it is well known that micelles and biomembranes have a large degree of internal dynamics and disorder, it appears plausible
Fig. 10.10. MD simulation results for a lamellar biomembrane and a typical spherical tenside micelle, respectively [122,123].
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to consider them as layered liquids with a composition changing as a function of the distance from the center of the micelle or membrane. Therefore, in collaboration with Bayer AG and the University of Cologne, we tried to model phospholipid biomembranes with COSMO-RS as a series of liquid layers with ~-profiles pM((~,z) changing as a function of the distance z from the center of the bilayer of phospholipids molecules [126]. After performing the required DFT/COSMO calculations for typical phospholipids, for which we selected the lecithin zwitterion dimyristyrolphosphatidylcholin (DMPC), we used experimental information about the probability of finding certain atoms of these molecules at different distances from the membrane center. In this way we constructed the z-dependent a-profilePM(a,z ) and converted it into a zdependent a-potential pM(a,z) by solving the COSMOSPACE equations. Special care had to be taken, because the individual layers were not neutral---owing to the charge separation within the DMPC molecule. The resulting electrostatic ~-potential was taken into account as an additional potential energy during the calculation of the a-potential. The a-potential pM(6,Z) reflects the chemical potential of a surface segment of kind a in a certain depth z of the membrane. For a given center position and orientation of a solute molecule X in this field of the membrane, we know the exact position and hence the z-coordinate of each segment and can easily integrate the respective values of pM(a,z) for the entire solute surface, to yield the residual part of the free energy of X in this special position and orientation. By a systematic sampling of the different center positions and orientations of X we yield the entire partition sum of X, and this easily gives us the probability of finding the solute X at a certain depth and orientation in the membrane/ water system. In this way, we were able to achieve promising results for a set of m e m b r a n e - w a t e r partition coefficients. A schematic illustration of this approach is given in Fig. 10.11. Recently, we extended the same concept to spherical geometries in order to simulate the distribution of fragrance molecules in tenside micelles. Our preliminary results suggest that the described concept of the simulation of micelles and membranes as layered COSMO-RS liquids is a promising new approach to an efficient simulation of such systems, but more development and validation work is
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Fig. 10.11. Schematic illustration of solute molecules in the field of a surfactant bilayer. required to establish it as a standard tool in this area. A clear disadvantage is that it depends on external information about the geometrical composition of the inhomogeneous system and cannot simulate the self-organization directly. When this technique is developed it might be possible to achieve the information about the micelle or membrane composition recursively by starting from a reasonable guess at the composition, to yield a a-potential PM(a,z), applying this to the surfactant molecules, and thus generating a partition function and a new composition of the membrane or micelle with respect to the surfactant molecules. Finally, it should be mentioned that a combination of COSMORS with tools such as MESODYN [127] or DPD [128] (dissipative particle dynamics) may lead to further progress in the area of the mesoscale modeling of inhomogeneous systems. Such tools are used in academia and industry in order to explore the complexity of the phase behavior of surfactant systems and amphiphilic block-co-polymers. In their coarse-grained 3D description of the long-chain molecules the tools require a thermodynamic kernel
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that evaluates the residual part of the chemical potential of each chain bead in a volume element with a given overall bead composition. While simple Flory-Huggins terms are currently used for this purpose, they require a lot of parameterization and even composition-dependent interaction parameters for a reasonable description; COSMO-RS could easily provide such free-energy contributions based on the partial a-profiles of the beads.
Chapter 11
Life science applications of COSMO-RS The behavior and fate of chemical substances of natural or artificial origin in living organisms is determined and regulated by a large number of active and passive mechanisms of partitioning, transport, binding reactions, and metabolism. Since these mechanisms are governed by thermodynamics~although not always by equilibrium thermodynamics~and hence by chemical potentials, and they take place in liquid or at least quasi-liquid physiological phases, all of them are related to the subject of this book, which is about the calculation of chemical potentials of molecules in liquid phases. By this statement, I do not wish to claim that COSMO-RS can fully explain or even partially apply to all of these complicated biochemical mechanisms, but it has proved to be a useful tool, with others, in the extremely ambitious efforts to better understand and predict the basic biochemical events in living systems and to influence t h e m ~ w h e r e necessary~by carefully designed drug agents. Computational chemists working in the areas of life science and medicinal chemistry have traditionally followed either the structural-descriptor-based QSAR approach in order to find relationships between important physiological target properties and the chemical structure, or they applied force field methods to simulate the often, important 3D behavior of molecules in the structurally demanding environment of cell membranes, proteins, and enzyme receptors. Quantum chemical methods have long been of negligible importance in this field, since they were either too slow for large number of property predictions for relatively large drug molecules with typical molecular weights of 400 a.u., or because they were inapplicable to the very large systems required for the simulation of membranes and enzymes. Quantum 167
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chemistry has only been used in order to assign atom charges for the force field description of new molecules. Only since reliable DFT methods, in combination with advanced solvation models, became available, a growing number of workers began the study on quantum chemistry in order to overcome the relatively poor description of molecular interactions given by traditional force fields. Some of them have recognized that the novel qualitative and quantitative approach of COSMO-RS provides some new opportunities, because its a-based interaction model, on the one hand, provides an integral picture of electrostatics, hydrogen bonding, and hydrophobic interactions, which at present are mostly considered by three separate empirical frameworks, and on the other, it is qualitatively and quantitatively well validated in less-sophisticated areas of thermodynamics. In this chapter, I will consider the actual and potential application areas of COSMO-RS in the wide range of pharmaceutical, medicinal, and biological chemistry. Although this is the reverse of the workflow in the pharmaceutical industry, I shall do this in the order of drug development, ADME, i.e. physiological absorption, distribution, metabolism, and excretion aspects, and finally drug design aspects, because this allows us to progress from simpler and already existing COSMO-RS applications toward more ambitious application ideas. Obviously, some methods are of importance for more than one of the three fields.
11.1
COSMO-RS FOR DRUG DEVELOPMENT
In the drug development process, the demands for physicochemical property predictions are somewhat similar to those in classical chemical engineering, and indeed this field is partly covered by chemical engineers in industrial practice. However, they collaborate with physical, analytical, and pharmaceutical chemists. For drug candidates entering the phase of development, small amounts of compound usually already exist and a large number of physicochemical measurements and characterizations has to be performed as a prerequisite for further laboratory and clinical testing. Furthermore, the new drug has to be prepared in a pharmaceutical form, which can be used for the first clinical studies, and so has to meet several release conditions in the body. The above processes
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should be completed as fast as possible because the "patent clock" for the drug is already running and every day spent in development can cost large revenues at the end of the patent period of a successful drug; also, the processes should be as accurate as possible because failures in the characterization or formulation can cause severe accidents during the clinical test phase. In this dilemma of contradicting interests, COSMO-RS can be a valuable tool for the computational characterization of the solubility behavior of the drug candidate as well as of its dissociation constants. Both are of crucial interest since the small and very expensive amount of compound has to be dissolved and embedded in different solvents and environments for the various steps of purification, crystallization, analysis, and formulation. At present, empirical solubility parameter approaches are often used in order to classify and predict the solubility behavior of the new drug, but despite their poor physical foundation, they have the additional disadvantage that the experimental measurement of the solubility parameters of the new drug consumes time and compound. In contrast, the required DFT/COSMO calculations can be started before the compounds come to the development laboratory, and a COSMO-RS solubility and dissociation screening can be completed---even at optimal computational level~ when the work in the development department starts. Furthermore, none of the valuable substance is wasted in this step. The most straightforward application is the prediction of the relative solubility of the compound in a variety of solvents. A blind test of the COSMO-RS performance for 4 drugs and 14 solvents has been done in collaboration with the development department of Merck & Co. [119]. The results are shown in Fig. 11.1. All predicted solubilities are plotted against the solubility of the drugs in ethanol. This blind test shows that COSMO-RS predicts the trends of solubility of the drugs between different solvents reasonably well. The main outliers can be seen for the solvent triethylamine. This was already expected because of the known problems of COSMO-RS with the hydrogen-bond (hb) strength of tertiary amines, which was mentioned earlier. Large deviations are also found for the solvent heptane. This is surprising, considering the fact that the solubility in other nonpolar solvents is quite well met or even shows deviations in the other direction. We attribute these deviations to experimental
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Fig. 11.1. Blind test for the COSMO-RS relative solubility prediction of commercial drugs from Merck & Co., Inc. [119]: All values are calibrated against the experimental solubility in ethanol. The 14 solvents are: water, 1-propanol, 2-propanol, dimethylformamide, ethyl acetate, methanol, heptane, toluene, chlorobenzene, acetone, ethanol, acetonitrile, triethylamine, and 1-butanol. problems that may result from very slow solvation kinetics of alkane solvents. In other studies for simpler drugs such as aspirin, cycloserine, and paracetamol, where experimental solubilities in several solvents are published, we found comparable results. The experimental results reported for these drugs often showed inconsistencies of up to 1 log-unit between one another, and the error of the COSMO-RS predictions was definitely less than that. Thus, we conclude that COSMO-RS can quite safely predict the
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relative solubilities of drugs in various solvents with an accuracy that is probably no worse than the typical experimental error of the solubility measurements. Thus a solubility screening with COSMO-RS, covering as many solvents and solvent mixtures that are considered useful should provide valuable information. If absolute predictions of solubility are required before the first experimental solubility is measured, the relative solubility prediction can be supplemented by an estimate of the absolute solubility in water according to the method described in the ADME section below. Although adding uncertainty, such estimates may help to fix the orders of magnitude where the solubility is expected in the experiment, and hence may help in designing the first solubility measurements most efficiently, with minimum waste of time and substance. Beyond the bare values of predicted solubilities and dissociation constants, COSMO-RS can provide valuable knowledge about conformations to be expected in various solvents, and (after some training) a closer analysis of the a-profile and the a-colored surface of the molecule can help in understanding the physicochemical behavior of the drug compound toward potential dimerization or agglomerization tendencies. Also, a more refined analysis of the solute-solvent contact statistics can even help to find a diverse set of solvents, which interact most differently with the drug candidate and may thus trigger different polymorphic forms during the crystallization process.
11.2
COSMO-RS FOR ADME PREDICTION
Physicochemical properties such as aqueous solubility and physiological partitioning behavior are of crucial importance for the design of potent drug candidates, because even the best drug molecules, which show perfect binding to a target enzyme receptor must exhibit a suitable physiological solubility and partition profile to arrive at the right place in the organism. Therefore the characterization and optimization of the absorption, distribution, metabolism, and excretion behavior, usually given the acronym ADME, has become an increasingly important part of drug research. It has more and more shifted from the development phase to the earlier drug-design phase, in order to prevent wastage of
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work and money on drug candidates that have no chance of exhibiting an acceptable ADME profile. Since in this early phase there is typically no drug candidate available, or only a negligible amount, the ADME characterization has increasingly become an in silico prediction, i.e., ADME prediction by computational chemistry means. Among many properties considered in the ADME characterization process, the aqueous solubility, the intestinal absorption, and the blood-brain partition coefficient (log BB) are of special importance, because all drugs have to dissolve an aqueous phase in order to be distributed within the body. Most drugs, especially those taken orally, need to pass the intestinal mocusa, which separates the gut from the bloodstream, and some drugs will pass the blood-brain barrier, while others do not. In chapter 9.2 we stated that through the a-moment approach COSMO-RS is well able to predict reasonably the partition equilibria between complicated and chemically poorly defined phases. Here, we will show the application of the a-moment approach to the log BB and the percentage of intestinal absorption (log Kia). For these partitioning problems, other computational models of comparable accuracy already exist, and the development of better models is mainly limited by the number and accuracy of the available experimental data [134]. Hence the only argument for using the novel and computationally more expensive COSMO-RS approach can be its better physicochemical basis and the smaller number of parameters it uses, compared to the existing QSAR approaches, which should allow for a better transferability to novel compound classes. However, as long as no reliable additional test sets are available, this advantage is hard to demonstrate. The accurate prediction of the aqueous solubility of drugs and drug-like compounds is much further away from a satisfactory solution because the existing QSAR- or group-contribution-based solubility prediction models exhibit quite limited predictive power for new drug classes. Why is it that the extremely important problem of the prediction of aqueous solubility much less solved than the prediction of less important partition coefficients? The answer is that the development of prediction models for logarithmic partition coefficients is much simpler, because the molecule X under consideration only acts as a solute at infinite dilution in the two phases. Hence the task is only to calculate the free energy of
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transfer of X between two different phases. As shown in previous chapters, such free energy of transfer can be expressed well as a surface integral of the molecule X, which can be split into groupspecific parts, as log K~s, - Css, + Css' +
kT
-1 [tt~ In 10
~groups(X)
kT
-
psx]
In 10 Aps's
(11.1)
Therefore, logarithmic partition coefficients exhibit a rather linear behavior with respect to surface-composition or group-contribution variables, and many approaches of this kind have proved to be quite successful. In contrast, the calculation of the solubility of a compound X in a solvent S is to calculate the free energy of transfer of X between the solvated state in S and its pure compound state, which may be liquid or solid:
X phase(X,] --1 10 [pX -- FiX l o g S x-cS+kTln
(11.2)
Even if X is a liquid, the calculation of the solubility of X involves the calculation of the chemical potential, px of X, in its pure liquid state. Here, X acts as solute and solvent, which introduces a complicated non-linearity with respect to the composition variables. Therefore simple linear-regression approaches, and nonlinear models without a sound physicochemical foundation, have little chance of being generally applicable. Here COSMO-RS has the big advantage that because of its well-proved statistical thermodynamics it is well able to predict the chemical potential of X in its pure liquid state by solving the non-linear COSMOSPACE equations for X in X. Hence, COSMO-RS can be used to calculate aqueous and other solubilities of liquid compounds without additional approximations. Nevertheless, since almost all drug compounds are solid at ambient temperature, the calculation of aqueous drug solubility requires knowledge of the chemical potential, of the drug, X, in its solid, i.e., mostly crystalline state. However, at present, there is no computational method available to reliably predict this chemical potential. Such prediction would involve knowledge about the crystal structure of the compound, which must
flX,cryst,X
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generally be considered as unknown for drugs in the drug-design and early drug-development phase. Even for a given crystal structure it is extremely demanding to calculate the total freeenergy difference from the liquid state to an accuracy of 1 kcal/ mol or less, which would be required for a reasonable prediction of the solubility of a drug. In this situation, we decided to make use of the thermodynamic capabilities of COSMO-RS as far as possible, i.e., up to the virtual state of the pure liquid drug, and to consider the missing piece for a solubility prediction of solid drugs, i.e., the free-energy difference of fusion, AGXu~between the crystalline state and the virtual supercooled liquid state, by an empirical QSAR approach. Since even the chemical potential, px, in the supercooled liquid state is not simply available for most other methods, nobody had tried this approach before. We considered it plausible that the crystallization free energy involves the size, rigidity or flexibility, and polarity of the molecule. Hence we used just three molecular descriptors for these quantities, i.e., the volume, the number of ring atoms, and the chemical potential of the compound in water as descriptors for such QSAR [C15]. We took the data set of solubilities for 150 organic and drug-like compounds used by Duffy and Jorgensen [135] for a MC-based prediction of water solubility. By fitting seven parameters in a model of MC-based surface descriptors they had achieved an rms deviation of 0.72 log-units on this data set. By fitting the three coefficients we obtained the approximate equation x AGXus __ 12.2Vx _ 0.76Nringatom ~- 0.54px
(11.3)
for the crystallization free energy, at room temperature which in combination with the other COSMO-RS contributions, yields a general method for the prediction of the solubility of drug-like compounds in solvent S:
1
log S x = kT In (10)
[px_pX+min(O,AGXus)l
(11.4)
The rms deviation of this COSMO-RS drug-solubility prediction is 0.66 log-units on the training set. Considering the fact that the available data sets for aqueous drug solubility typically have an intrinsic experimental error of about 0.5 log-units (rms), this
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must be considered as a good result. It should be noted that the liquid-state chemical-potential difference already explains 67% of the variance of logS x, and is thus the most significant and important contribution in Eq. (10.4). McFarland provided us with a data set of 24 aqueous solubilities of drugs [136], split into a highquality data set and lower-quality data set. On the first half we achieved an rms deviation of 0.64 using our COSMO-RS solubility method as parameterized before. On the second half we obtained 0.86, (see Fig. 11.2). In several tests on compound sets that are structurally very different from the training set, including a larger set of pesticide molecules, Eq. (10.3) proved to be very robust and transferable. Recently, the method was applied to polychlorinated dibenzop-dioxins (PCDDs) and polychlorinated dibenzofurans (PCDFs) [137]. As shown in Fig. 11.3. it was found that the COSMO-RS predictions appear to be even more accurate than the experimental data, especially at the low-solubility end. While the experimental results for many compounds show a scatter of more than a
Fig. 11.2. Aqueous molar solubility log(SXQ)calculated with COSMO-RS: diamonds = training set (filled symbols-- solids; open symbols = liquids); triangles = test data set.
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Fig. 11.3. Experimental solubilities of polychlorinated dibenzo-p-dioxins (PCDDs) and polychlorinated dibenzofurans (PCDFs) from various sources vs. values predicted from COSMO-RS [137].
log unit, the COSMO-RS predictions mostly appear well within the experimental error bars. As a result we may conclude that, despite its empirical character, the expression for the free energy of fusion that COSMO-RS provides the most fundamental and transferable prediction tool for drug solubility, with the smallest number of adjusted parameters of all competing approaches. Mainly for this reason, there is an increase in the use of COSMO-RS in the ADME modeling community. Extensions of the COSMO-RS solubility prediction method to the very important class of saline drug compounds are under development, and the first promising results have been achieved. It turns out that the chemical potential difference between the ions in aqueous solution and the virtual supercooled melt of the salt, i.e., the virtual ionic liquid, is again the most important contribution to the logarithmic solubility. Care must be taken, however, because two ions are involved in a solution step, so the average chemical potential of the ions has to be used and not their sum. As expected, the empirical formula for AGfus is not transferable to salts and a new QSAR will have to be developed as soon as a sufficiently large and diverse set of reliable experimental solubilities for saline drugs is available. Unfortunately, this may be the major drawback for further developments in this area.
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In collaboration with Paul Connolly and Ron Jones from GlaxoSmithKline we have developed a a-moment model for the percentage of intestinal absorption of drugs based on a data collection by Zhao et al. [138]. In order to make the a-moment approach applicable to this property we converted the percentage scale into a logarithmic partition scale using the relationship log gia
--
( %abs ~ log 106 - %absJ
(11.5)
Since %abs values close to, or equal to, 0 or 100 would lead to numerical problems in this conversion, we introduced a special non-linear optimization procedure for the fitting of the coefficients, which was able to handle them appropriately. In this way we derived a model for log Kia: log
gia
-
0.0040M0 - 0.0053M2 - 0.0024M3 - 0.113Macc
-0.11656Mdon + 1.37
(11.6)
Back-transformed to the corresponding %abs scale, this model is still slightly more accurate than the model derived by Zhao et al., which is based on Abraham's linear free solvation parameters. By its construction, our model avoids the prediction of negative %absvalues or those greater than 100, which can easily occur in most other methods (see Fig. 11.4). For the log BB we have fitted a model based on a 103-compound data set taken from Kier's group [109]. The a-moment regression yields the model log
SS
-
0.0046M x - 0.017M x - 0.003M x + 0.194
(11.7)
(N = 103, r 2 = 0.71, and r m s e - 0.4) without significant contributions from hydrogen-bond moments. The results are shown in Fig. 11.5. Altogether, the accuracy of the model is satisfying, taking into account the fact that the intrinsic experimental error in the log BB data sets is 0.3 log-units [134]. Other QSAR models achieve an rms error of about 0.4. The largest error of 1.4 logunits occurs for oxazepam. Interestingly, at least two other models also have more t h a n a log-unit deviation for this compound, and it appears questionable whether the reported experimental value of 0.61 is valid. Perhaps a minus sign has been lost over the years!
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Fig. 11.4. Surface of lamivudine colored by the logarithmic intestinal absorption coefficient logKia [C31]. Red areas indicate enhancement of intestinal absorption, while blue surface areas indicate strong reduction.
Fig. 11.5. Experimental logBB values [109] vs. results of the a-moment model (Eq. (11.7)). Results using two different conformational treatments are shown. In this study we also considered the conformational sensitivity of the results. While we usually try to find the conformation with the minimal COSMO energy by a conformational search (see chapter 11.3 for details), we tried a direct AM1/COSMO
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optimization starting from the 3D geometries of the 2D-3D converter CORINA [139] for comparison. It turned out that the results differ only slightly, and there is no observable preference in the rms error for one or the other treatment. Nevertheless, this may be different for other properties with stronger hydrogen-bond influences. Very recently, we also fitted a model for another physiological partition coefficient, the adsorption to h u m a n serum albumin. Since the albumin protein has the potential to bind a large portion of drugs in the blood serum, a drug should not bind strongly to albumin in order to be available for the real target. We used a data set of 92 h u m a n albumin binding constants published in a white paper by Kier et al. [140] and achieved the model log KHSA -- 0.0081M x - 0.016M x - 0-013MX + 0.145MX~c - 0.883
(11.8)
with an rms error of 0.33 log-units. In Fig. 11.6 we compare the a-dependence of logKia and logBB with those of log Kow and log Koc. We see that log Kia and log BB
Fig. 11.6. a-dependence of various partition coefficients (derived from a-moment models).
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have a negative dependence on both the negative and positive surface areas, while we had found that log Kow and log Koc increase with a negatively polar surface and decrease with a positively polar surface. It is also clear that loggia mainly depends on the hydrogen-bonding moments, while log BB shows a smooth electrostatic behavior. Before concluding on the area of ADME prediction, we should mention at least three models described earlier in this book, which can be considered as ADME property predictions, i.e., the pKa and pKb calculation, the log Kow prediction models, either by direct COSMO-RS thermodynamic or by using the a-moment approach, and the model for membrane partition coefficients as published in a diploma thesis [126], although this is not readily available as a software.
11.3
COMPUTATIONAL ASPECTS AND SOFTWARE FOR COSMO-RS IN DRUG DESIGN
Before considering several new and exciting applications of COSMO-RS in drug design, we will discuss some important computational aspects for the application of COSMO-RS in ADME prediction and drug design, and some special methods and software developed to overcome the computational bottleneck. In drug design and ADME prediction, a large number of new and relatively large and flexible molecules is typically considered, while in chemical engineering projects only a small number of new compounds usually plays a role, eventually in combination with a longer list of standard solvents that are already available in our database, COSMObase. Thus the computational effort for COSMO-RS, which mainly results from the quantum chemical COSMO calculations for the new compounds, and increases strongly with their size, is much greater in drug design than in chemical engineering. In order to keep the computational costs for COSMO-RS within reasonable limits, we have developed a slightly less accurate but much faster QM/COSMO calculation level for drug applications than the high calculation level that we usually apply for chemical engineering problems. This is further justified by the different accuracy expectations of the respective user groups. While chemical engineers require predictions to be
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as accurate as possible, drug design modelers are mostly satisfied if solubility and other physiological partition properties are met within half a log-unit. In the drug design COSMO calculations, we replace the full DFT/COSMO optimization by a much faster semiempirical AM1/COSMO [56,C1] geometry optimization followed by a single-point DFT/COSMO calculation with a reduced basis set (SVP instead of TZVP) in order to obtain reliable ~-profiles. While typically saving a factor of 20 in calculation time, this level increases the rms error of COSMO-RS by about 20% for high quality experimental data and only by 10% for typically noisy life science data. The CPU time required for a typical drug molecule on a modern 2.5 MHz computer is about 30 min. Indeed, all drugproperty applications presented here have been developed on this drug level, with the exception of pKa and pKb, which demand the better basis set for an accurate description of the anions and cations. In order to achieve the most comparable results, the AM1/COSMO calculations should best be done with the MOPAC/COSMO program, using the corresponding closed-cavity construction and the same element-specific radii as for the COSMO-RS standards in DFT/COSMO-RS. This can be done either in the commercially maintained MOPAC2002 program [59] or in the public-domain program MOPAC7 [40]. Unfortunately, the COSMO implementation in the latter program had never been bug-fixed until I recently took a public-domain copy of MOPAC7 and implemented all the necessary corrections and modifications. Thus, a stable and up-to-date MOPAC7/COSMO version including all the latest COSMO improvements is now available on request. Although most of the AM1/COSMO geometries deliver good COSMO-RS results if supplemented with a single-point BP-SVP DFT calculation, during our work on thousands of AM1/SVP/ COSMO calculations we detected some severe problems. The first concerns partially conjugated rings, which tend to be optimized into planar ring geometry by AM1, while in reality they are twisted. This wrong geometry caused surprisingly large effects for the polarization charge densities a in the COSMO calculations. Therefore I developed a "molecular mechanics" (MM) correction for rings in MOPAC, following the concepts of the already existing MM correction that cures the problem of pyramidality of amide
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groups. Later I also added a similar correction for sulfonamide groups, which also showed large deviations from higher-level calculations in AM1. For chemical engineering projects, where only small numbers of compounds usually occur, an individual analysis and preparation of the relevant conformations is feasible. In contrast, the large number (hundreds or thousands) of compounds considered in drug design prompt the question of a reasonably automated generation of the geometries, and eventually conformations of these compounds, starting from 2D-structure information. For this, the first task to do is a 2D-3D conversion, i.e., the automated generation of a reasonable 3D geometry starting just from the topological knowledge about atoms and bonds. While such 3D generation is quite trivial for many simple molecules, it becomes a really sophisticated problem for general organic, and especially drug-like molecules, because these often have multiple rings and conflicting demands. A number of such 2D-3D converters are available. We decided to use the CORINA program [139], which is usually considered as one of the best. We meanwhile have performed about 70,000 of such 2D-3D conversions followed by AM1/ COSMO optimizations. In general, the CORINA geometries did a very good job in indicating start structures for further optimization. Only in a few cases did we find problems arising from bad CORINA geometries. The most important one was the problem with hydroxyl groups being oriented orthogonally to aromatic rings. While they mostly, more or less randomly fall into one of the planar orientations during the subsequent geometry optimization, they sometimes stick in the definitely unphysical orthogonal conformation. Nowadays, this problem is tackled by some post-processing techniques. Other pathways of 3D generation may be similarly successful, but we strongly recommend checks for larger errors, at best by comparison of outliers in a big data set, e.g., on the COSMO log Kow results achieved with the procedure. We detected a frequent error in a 3D-generation procedure employed before we changed to CORINA. That procedure often generated false torsion angles at ester oxygen bonds, resulting in about 2 log-units' error in log Kow predictions. To our surprise, in combination with the previously discussed MM corrections the use of AM1/COSMO-optimized geometries
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without further conformational search (apart from aromatic hydroxyl correction) yields very reasonable results for almost all drug-property predictions; especially water solubility, log BB, and intestinal absorption results appear to be quite insensitive to sophisticated conformational details and to the presence or absence of all plausible intramolecular hb's. The values of log Kow and log Koc are much more sensitive in this regard, and vapor pressure calculations or alkane-water partition coefficients are extremely sensitive. The reason is that in cases where there is only one hydrogen bonding and another non-hydrogen-bonding phase, the missing intramolecular hb's are in some way replaced by solute-solvent hb's in one of the phases. But this is not the case in the other phase, leading easily to errors of several log-units in the partition coefficients. Nevertheless, for the three key ADME properties, water solubility, log Kia, and log BB, a systematic conformational analysis appears not to be required, as demonstrated for the case of log BB in Fig. 11.5. No significant statistical difference is detectable between the two models, one of which is based on simple CORINA-based MOPAC/COSMO optimization and the other is obtained with a more demanding conformational search for optimal COSMO conformations. Nevertheless, in order to be able to predict conformational, sensitive properties for large numbers of compounds, we developed a conformational search procedure as an interplay of a structural analysis program (which is a part of the COSMOfrag software described below), the locally modified MOPAC7 program, some smaller standalone programs, and finally the TURBOMOLE program. Starting from CORINA 3D geometries, or other 3D-input files, first the structure is analyzed with respect to potential hb's, i.e., 5- and 6-ring arrangements of hb donor and hb acceptor atoms. Then a multiple job MOPAC input is written, which for every combination of potential hb's sets additional MM constraints to enforce the hb distances and orientations. Each soconstrained optimization is then followed by an unconstrained optimization, which will or will not optimize into the tested hb conformation, depending on whether the enforced arrangement is energetically reasonable or not. All optimizations are done with additional constraints, which would push wrong aromatic hydroxyl groups or wrong ester arrangements out of the wrong conformation, but have no effects otherwise. When the MOPAC job is
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completed, the resulting set of optimized conformations is reduced by a cluster analysis, and representatives of the lowenergy conformations are re-optimized in the presence of a dielectric COSMO environment representing water. Finally, the low-energy gas-phase and COSMO conformations are forwarded to TURBOMOLE for a single-point BP-SVP calculation or for a final optimization on the BP-TZVP level. Since the energy differences of the conformations can vary considerably between AM1 and DFT, the final selection of conformations to be taken into account is done on the basis of the TURBOMOLE DFT results. By this procedure we usually find most of the relevant conformations, but unfortunately a complete coverage cannot be guaranteed. Improvements are being developed steadily.
11.4
HIGH-THROUGHPUT SCREENING WITH COSMOFRAG
For high-throughput screening (HTS) drug-design projects involving several thousands and up to millions of new drug candidates, even the "COSMO-RS drug level", i.e., the AM1/BP-SVP level, which typically requires 10-15 min per drug compound, is too slow. For these applications we have developed a very fast bypass for the demanding DFT/COSMO calculations, called COS-
MOfrag. Since for most COSMO-RS applications only the a-profile and some information about the area and volume of the molecule is needed, the basic idea of COSMOfrag is the composition of the a-profile of new molecules from existing a-profiles of the locally most similar molecules, whose a-profiles have already been precalculated and are stored in a database. For this purpose we have developed a database of 40,000 (as of February 2005) pre-calculated COSMO files of smaller basic- and larger drug-like compounds, trying to maximize the diversity of the compounds represented in the database. Three steps have been most important in the development of the COSMOfrag method. The first was the development of a molecular perception routine, which analyzes a molecule with respect to the hybridization states of atoms, bond orders, rings, and ring properties like aromaticity, and even their stereochemical
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classification. This perception (COSMOcept) is able to start from most different, common electronic file formats, such as SDF files, SMILES code, XYZ files, and others. Some of them include bond tables with bond orders, others include elements and geometry only, some have explicit hydrogens, and other implicit hydrogen atoms that have to be analyzed. It is most important that such a perception routine is able to end with a unique internal representation of the molecule, independent of all the different ways the chemical structure can be represented in the original input. Furthermore, care must be taken that equivalent atoms in a molecule also have an equivalent description. As an example of such problems we may consider the nitro group, which is sometimes represented by a four-valent nitrogen atom with formal charge + 1, one single-bonded oxygen with formal charge -1, and a double-bonded oxygen atom, while another usual way of description is a neutral five-valent nitrogen with two neutral double-bonded oxygens. Although the former description may be chemically more meaningful, for our purposes we have to apply the second convention, because only this describes the two oxygen atoms as chemically equivalent. In the same way, all cases of partially ionic bond descriptions had to be reduced to a neutral multiple-bond representation. A unique representation of aromatic bonds is also of crucial importance, because the usual Kekul~ description of alternating single and double bonds leads to nonunique representations. Having resolved the molecular perception problem and achieved a unique representation of all atoms, bonds, and rings in the molecule, the second major step is the definition of the most useful measure for local similarity of atoms and atomic environment. For the purpose of COSMOfrag, we need to achieve the state that atoms are considered as most similar, if their partial molecular surfaces and surface polarities, i.e., polarization charge densities, are most similar. But since the latter is not known, at least for the new molecule under consideration, we have to ensure that the local geometries and the electronic effects of the surrounding atoms are most similar. Obviously, two similar atoms should at least be identical with respect to their element and their hybridization. Turning this information into a unique real number, a similarity index of the lowest order (zeroth order) can be defined for each atom from the atom element numbers and
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the hybridization. In the next step, a similarity index of firstorder can be defined by propagation of the zeroth-order similarity indices of the neighboring atoms to the central atoms and the addition of this new information to the zeroth-order similarity index of the central atom. In order to achieve unique results, this must be done by a non-linear algorithm, which also takes into account the bond orders of the bonds used for the propagation. If done in this way, first-order identity of two atoms ensures zerothorder identity of all neighboring atoms. In the same way we can now generate the next higher similarity indices out of the similarity indices of all neighbors and continue this up to any level we require. Identity of the ith similarity index will ensure chemical identity up to the ith neighbor spheres of the atoms. We need to add some more information, e.g., about ring sizes at a lower level of similarity. For example, a carbon atom of cyclohexane would be identical with a central atom of a long-chain alkane up to a higher order, if we only used the above algorithm, but we know that the characteristics of a long-chain alkane CH2-group is somewhat different from that in a cyclohexane. Therefore we feed the information about the minimum ring size to which an atom belongs, into the similarity indices, starting at zeroth order for 3- and 4-membered rings, and continuing in this way for the higher similarity indices with the next higher ring sizes. Information about the cis or trans position with respect to double bonds is taken into account at the second order to achieve a difference between cis and trans isomers. In a similar way, the information, whether a typical hb donor or acceptor atom can form a favorable intramolecular hb forming a 5- or 6-ring is included in the second order similarity indices, because these may strongly change the properties of atoms in a molecule, and are thus important for finding similar atoms in other molecules in the sense of COSMO-RS. For an example of 6-membered intramolecular hb bond rings see the lowest energy conformations of salicylic acid in Fig. 7.4. COSMOfrag makes use of these similarity indices in two ways. First, it calculates the sum of the similarity indices of order 7 for all atoms of a molecule. The resulting molecular similarity index is essentially an identity index, because to our best knowledge identical indices imply identity of the molecular structures, apart from stereochemical differences. This can be used to detect the identity of molecular structures in the database. Much more
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i m p o r t a n t is the use of the atomic similarity indices for finding the similarity of atoms. For t h a t purpose, each of the eight similarity indices of an atom (orders 0-7) is converted into a five digit word of ASCII symbols, and then these are combined to a 40-digit ASCII string. Now the identity of the atomic ASCII similarity strings up to the 5th digit ensures zeroth-order similarity and the identity up to 10th ensures first-order similarity, etc. The atom-similarity strings of all atoms of the COSMOfrag database are stored as a sorted ASCII list. For the 40,000 molecules with an average of about 17 atoms (excluding implicit hydrogens) this results in a list of 700,000 entries. For an atom out of a new molecule the most similar atom in the database can now be found by a very efficient strategy of repeated interval splitting. First, the new string is compared with the string in the middle of the list. If the new string would be sorted before the middle string, we only have to search the first half of the list, or the second half otherwise. Now the same procedure is repeated for the remaining list part, until finally the new string is uniquely ordered between two strings of the list. One of these two has the highest length of identity, which can be reduced to a m a x i m u m identity order. All atoms with the same identity order can be considered as equally well suited to act as database pictures of the new atom (Fig. 11.7). For a new molecule, a small n u m b e r of picture molecules is finally selected from the molecules containing the candidate ~-J:-0 ~-J~~@ ~-S--@ "-j~-@ *-5~@ *-J~~8 "-S~-@ "-j~-@ "-j~-@
~{f~j &{f~j &{foj &{f6j &{f6j &{f~j &{f6j &{f~j &{f0j
(m-~/)~:'6 (m--~/)a+'0 (m~~/)a+~Lp (m-~/)a+~p (m~~/)a+~[p (m~~/)a+q[p (m~~/)a+~• (m-~/)a+0~ (m~~/)a+~h
~.'!t ,5i~0 .5#'6 /;.Jl~ ~.',| ,S'y4 .gg'i / N > ~ ~ A + V -v-i~<< .qR'' /~s~" "~WwW-Fs 'A .rvoI /ap~K *~HwW-Es 'A .=^S{ /~Di6 "6 b~-VZu@ .s--'V / ~ 0 ~ "II-~ ,~DDu .O7r- /~zY~) "~'6~-V--!J .q~@? /~v-/ "i~i1--s .pS~, /~%3'
)IOF6 -*S(i~ WfRGSFI40I 3/0 )IOF6 -~Sfi~ BXNBISRRI I-. )IOF6 -~p~D P ~ J ~ 201 )IOF6 -'p~D RCTK'TYTN0 HEF )IOF6 -'p~D AUSTELPNK :34 )IOF6 -*p&D XEKFYZONI 301 )IOF6 - ' S ~ RACMRMI~QI 845 )IOF6-'p~D EJAAAUMNI-+, )IOF6 -'p~D HJGNJDNNI 301
F I > f 0 ? K 4 @
Fig. 11.7. Random section of the COSMOfrag database file: five-digit atom word are separated by blanks. The first 8 words are the atom similarity codes of order 0 to 7, followed by 2 by-codes containing additional information. At the end of each line the molecule of each atom is marked, using a nine-letter unique name constructed from the molecule identity index, followed by information on the neighbor atoms and implicit hydrogens. In this section all atoms are identical up to the second order, while atoms 5 and 6 are the most identical (up to fifth order).
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picture atoms and the atoms to be used out of each of the picture molecules are listed in a COSMO m e t a file (Fig. 11.8). Owing to the individual fragmentation based on the described concept of m a x i m u m similar substructure the accuracy loss of COSMOfrag is only about 0.05-0.1 log-units compared to direct DFT/COSMO calculations (Fig. 11.9) for typical life science data sets. Water solubility appears to be especially insensitive to the approximations of COSMOfrag. As shown in Fig. 11.10, the rms error only increased from 0.66 to 0.71 log-units for the dataset used in the development of the drug solubility method. The octanol-water partition coefficient is much more sensitive, as shown in Fig. 11.11. On a large dataset of 3,000 compounds ~= CO~qKO/JIJDAPDXLNI.ccf CFDB w={001011000111101100000000Ollllll} f= f= f= f=
COSM0/K/K%IOCLGROI.ccf COSM0/R/RXTPEWPNI.ccf COSHO/U/UBFLMDHNI.ccf COSMO/G/~.ccf
CFDB w={01001100000100000000100000000100010000001000000} CFI)B w={0000000000001100000011}
CFDB w-(lO0000000000000100000000} CFDB w=(00000000100000001010000000101)
Fig. 11.8. COSMO meta file coding the parts of database molecules to be used as pictures for the construction of a new molecule.
Fig. 11.9. COSMOfrag decomposition of the drug Enalapril into three fragments. The parts of the COSMO surface not used for the decomposition are colored in black.
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Fig. 11.11. Evaluation of COSMOfrag performance for logKow on a large data set. The original method yielded 0.39 log-units MAE, while COSMOfrag gives 0.57. When corrected for the systematic shift, the MAE of COSMOfrag reduces to 0.42.
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the mean absolute error of log Kow increases from 0.37 for the full-compound COSMOtherm method to 0.57 based on COSMOfrag. A small systematic shift of the values predicted by COSMOfrag toward smaller log Kow can be observed, resulting from the fact that individually optimized compounds, in general, will show smaller polarity than such constructed from fragments. If we correct for this shift, the mean absolute error of the COSMOfrag method reduces to 0.42 and is thus only slightly larger than that of the original method. With this shortcut, the calculation times per compound are reduced from hours to seconds, and thus COSMOtherm can be applied even in high throughput screening in drug design. Nevertheless, it must be pointed out that COSMOfrag is only applicable to solubility and partitioning properties of neutral compounds. Properties involving ionic species, such as pK~, or properties involving total energy differences, such as vapor pressure, are presently out of the scope of COSMOfrag.
11.5
COSMO-RS FOR DRUG-ENZYME INTERACTIONS
After presenting the COSMO-RS method and its applications in ADME property prediction to different groups of industrial drugdesign researchers, several of them raised the question as to whether this novel and apparently successful and broadly applicable description of molecular interactions in liquid phases would also be applicable to the prediction of binding constants of drug molecules in enzymes, i.e., to the holy grail of computational drug design. Having in mind the old lock-and-key concept of fitting of drug and receptor introduced by Emil Fischer (see Fig. 11.12), I used to answer this question with a "NO," because all the steric constraints required for a good fit of the drug molecule into the receptor pocket cannot be expressed in the geometry-free COSMO-RS a-framework. A second reason for not thinking too much about this application area was the inability of current DFT methods to treat molecules of the size of an enzyme. Enzymes typically consist of 5,000-30,000 atoms, and only very few are known in the range below 1,000 atoms, which at present would be the upper limit for DFT calculations.
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Fig. 11.12. Classic lock-and-key picture of receptor-drug binding. While many colleagues accepted my negative answer, Robert Klein, at that time working at the agrochemical company Agrevo, rejected it on the basis that the geometrical fit may be much less important than generally assumed, and that many other methods used in that field most likely overemphasize the geometrical aspects. During lunch, we developed two ideas together, on how COSMO-RS could eventually become a useful tool for the prediction of drug-enzyme binding constants, or at least a source for some useful information in this respect. The first idea was rather revolutionary. Assuming that we knew the a-profile of a receptor, we could consider the receptor as a pseudo-liquid solvent and simply calculate the chemical potential of a ligand molecule in such a pseudo-liquid receptor (PLR). Since the calculation of the chemical potential of the ligand in water is just a standard task for COSMO-RS, we would immediately have a partition coefficient of the ligand between the PLR and an aqueous phase, i.e., a kind of binding constant. The idea of the PLR approach is shown schematically in Fig. 11.13. This binding constant would reflect the overall polarity suitability of drug and receptor and may be a useful number in the selection of promising drug candidates. Since the receptor a-profile only has
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Fig. 11.13. Schematic visualization of the pseudo-liquid receptor approach for ligand-receptor binding. to be calculated once, large numbers of drug candidates could easily be screened, and even the COSMOfrag approach can be applied without problems, enabling HTS application of this PLR drug-binding approach. The second idea was more straightforward. COSMO-RS could be used in combination with a docking program for the energetic scoring of docked ligand geometries in the receptor. It would be the task of the docking program to generate plausible arrangements of a ligand in a receptor pocket. But while being very efficient in generating such arrangements, docking programs do not usually give a good measure of the free energy of such arrangement. It is the task of scoring functions to evaluate the free energy of binding, i.e., the free energy difference between the docked arrangement and an aqueous reference state of the ligand. The scoring algorithm must be very fast because typically several hundreds of docked arrangements for a drug candidate have to be evaluated. Many different, mostly very heuristic approaches, are typically used for such scoring, but it appears that there is still a strong need for more reliable scoring algorithms. In this situation the independently derived and widely validated COSMO-RS surface-interaction model may be very useful. If we assume that we know the COSMO polarization surface-charge densities a and a'
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of the receptor pocket and the ligand, respectively, we could write the interaction energy of ligand and receptor as an integral of the surface-interaction energy, Eint(6, 6',d) over the ligand surface, where a is the polarization charge density of the receptor next to the position of a'. Obviously we assume that the interaction energy functional is basically given by the misfit and hydrogen bonding contributions of COSMO-RS. However, while we could assume that we have close contacts between solute and solvent surfaces in the liquid phase, we must be aware that for a ligand docked into a receptor, this condition may be much less well met, and therefore we should be aware that some dependence of the energies on the distance, d, may have to be taken into account. The development of reasonable distance dependencies for the misfit and hb surface interactions will probably be the major part of the work required for achieving a good scoring function based on COSMO-RS (Fig. 11.14).
Fig. 11.14. Schematic illustration of a COSMO-RS-based scoring function. Starting from the known surface polarities, a and a', of nearby receptor and substrate surfaces, the interaction energy could be evaluated using the COSMO-RS expressions for misfit and hb energies.
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Anyway, the first step toward any receptor-based COSMO-RS calculations is the calculation of qualitatively acceptable a-profiles of the receptor regions of enzymes. In a performance test of a highly parallel version of the TURBOMOLE program on the supercomputer at the Research Center Jfilich [141], we could show that TURBOMOLE presently can handle single point, i.e., fixed geometry, BP-SVP DFT-calculations of enzymes up to about 1,500 atoms. On the basis of preliminary data, an enzyme of 1,000 atoms requires about 6 CPU h on 32 CPUs of a supercomputer cluster with a minimum quadratic scaling of CPU-time with the number of atoms of the enzymes. Thus for medium-sized enzymes we would require a minimum of 600 h on such a supercomputer, which would be rather expensive, even if all the technical problems arising at these molecule sizes would be solved. Therefore, brute-force DFT calculations appear to be unfeasible at present, but they may be possible in the future. Combinations of QM with MM, i.e., QM/MM methods, may be one way of achieving the aim outlined above. In such calculations only the core region of i n t e r e s t ~ i n our case the vicinity of the receptor pocket--would be described by a rigorous QM treatment, while the rest of the system would be treated by MM in order to provide a reasonable steric and electrostatic environment for the QM calculation. However, typically many consistency problems occur when the cut between the QM and MM description has to be made along covalent bonds in molecules, as would definitely be required in enzymes. Furthermore, a COSMO implementation combining a fast DFT code such as TURBOMOLE with MM is not readily available, and would require considerable programing to avoid the final COSMO part of the calculations from becoming the time-determining step. For all these reasons we did not try to follow this QM/MM approach, but I must admit that we cannot exclude the possibility that this pathway might be viable and fruitful. Instead, we developed a procedure for the COSMO treatment of enzymes that is based on linear-scaling, semi-empirical calculations. As discussed earlier, linear-scaling, semi-empirical QC programs combined with linear-scaling COSMO have become available since 1997. In the MOZYME algorithm developed by Stewart [63], linear scaling of the QM part has been achieved by localization of the molecular orbitals, and a linear scaling of
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the COSMO algorithm [64] has been achieved by a replacement of the Cholesky factorization by a conjugate-gradient algorithm following the concept of York et al. [65-67]. In its MOPAC2002 implementation, the MOZYME/COSMO algorithm allows for semi-empirical single-point calculations of enzymes with up to 40,000 atoms almost within 1 day on a single CPU. Since in the above-mentioned ideas of COSMO-RS-based calculation of ligand-receptor interactions the enzyme a-profiles have to be calculated only once, such calculation times appear to be absolutely acceptable. Nevertheless, we are left with the problem that the semi-empirical QC m e t h o d s ~ w i t h only little variation between the commonly used Hamiltonians AM1, MNDO, PM3, or PM5 [56-59]~in general do not deliver sufficient accuracy of the electrostatics to allow for quantitative COSMO-RS applications. Therefore, we adopted the idea of Cramer et al., to use "charge models" [60,61], i.e., a large set of bond-dipole corrections, in order to achieve higher accuracy electrostatics based on fast semi-empirical calculations. Since the target quality of electrostatics is the AMl-geometry-based BP-SVP COSMO-RS "drug" level, we developed such an electrostatic AM1-DFT fitting procedure. For this we first performed AM1/COSMO calculations and subsequent BP-SVP DFT/COSMO calculations for 161 tripeptides, which reasonably represent the typical amino acid sequences in biological proteins. Since the resulting COSMO files typically have 1,400 segments, we thus had altogether about 220,000 grid points, for which we could compare the electrostatic potential resulting from the AM1/COSMO calculation and the BP-SVP DFT/COSMO calculation. By defining bond types based on bond order and atom types of the neighboring atoms, we found 39 different asymmetric bond types in this set of peptides. For each asymmetric bond type, i.e., for each bond with different atom types at both ends of the bond, a dipole correction, i.e., a correction charge +~c on one of the atoms and a countercharge - 5 c on the other atom of the bond, have been fitted. For further improvement of the fit, we allowed for bond quadrupole corrections, represented by charges 5q on the center of the bond and -Sq/2 on both bond atoms. Thus, we finally had 78 correction charges to be fitted to the 130,000 potential differences. In this way we were able to reduce the original rms deviation of the electrostatic potentials between AM1 and
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o
BP-SVP from 0.0112e/A to 0.0058e/A for the AM1/DFT corrected potentials. Technically, the AM1/DFT correction is now implemented as a combination of the structure analysis tool COSMOlyze developed for COSMOfrag and the MOPAC2002 program [59]. COSMOlyze analyses the bond types of a new protein and writes an input file for MOPAC, which contains all the correction charges for atoms and bond centers. By a small modification, MOPAC2002 is now able to read these AM1/DFT correction charges and to add the corresponding corrections to the electrostatic potential on the COSMO surface at the end of a MOPAC/COSMO calculation. In a final COSMO call, the improved potential is converted into updated COSMO charges and a COSMO file with quality closer to BP-SVP quality is written. As an example, the resulting a-profiles of the tripeptide cysteine-glutamine-threonine are shown in Fig. 11.15. The improvement of the AM1/DFT-Fit-a-profiles toward BP-SVP quality is apparent, especially in the region of the hb donors. In the AM1
Fig. 11.15. Semi-empirical a-profiles of amino acids before and after AM1/DFT correction in comparison to the BP-SVP drug standard of COSMO-RS.
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calculations, these are much less polar than the corresponding BP-SVP donors, but by the bond corrections the agreement with BP-SVP is very satisfying. Thus, we can expect that COSMO-RS calculations based on AM1/DFT-Fit ~-profiles using the BP-SVP COSMOtherm parameterization should lead to reasonable results. On the basis of the described MOZYME/COSMO procedure with AM1/DFT corrections we have performed first calculations of binding constants applying the PLR approximation. The preliminary results on a test set recently suggested for the evaluation of drug receptor binding methods [142] show a positive correlation of experimental and predicted binding constants and a slight but significant differentiation between the active-compound data set and a random set of compounds used for comparison. Nevertheless, the presently available results do not allow for a clear decision whether the method will be successful in the end or not. Further research is required in this direction (Fig. 11.16).
Fig. 11.16. a-Surface and a-profile of the active site of enzyme matrix metalloprotease. The COSMO calculations are done with AM1 using the linear scaling MOZYME in MOPAC2002, followed by a DFT/AM1 bond multipole correction.
198 11.6
A. Klamt COSMO-RS FOR DRUG SIMILARITY SEARCHES
Owing to the rapidly growing capabilities of protein isolation and crystallization the number of structurally well-explored enzymes and their receptors has increased exponentially during the past 15 years. Therefore the protein structure-based drug-design approach has attracted much attention in academic and industrial pharmaceutical research. Nevertheless, the older, "active analog" approach is still very important. Like the receptor-based drug design, this approach is based on the lock-and-key principle. However, when lacking structural information about the lock, the only way to construct better keys is to try small variations of known keys. In the language of drugs and receptors, this means to search for compounds that have a large degree of similarity to active compounds, with respect to shape and polarity, in order to find promising candidates for new and potentially better drugs. Obviously, if we do not know the structure of the lock, we will never be able to decide a priori whether a compound similar to an active drug will be more or less active than the precursor, or whether it may even fail completely. Nevertheless, the chance of finding better acting drugs, or to find patent-free drugs with comparable activity is definitely higher in a set of compounds with a higher degree of similarity than within a random set of chemicals. Therefore, the search for compounds similar to known active drugs is such an important task in drug design. This raises the crucial question about the best definition and measure of similarity for drug-design purposes. Many different answers can be given to this question. Some are based on structural similarity, i.e., on the presence of a large degree of structural identity between two compounds. "Molecular fingerprints," representing the presence of specific groups or other features in bit strings, have been developed for similarity measurement and have become quite popular. Also commonly used are similarity measures based on counts of interaction sites, i.e., of atoms acting as strong hb donors or acceptors, and of "hydrophobic" sites, which often are considered as special interaction sites in drug design. The term "2D similarity" is used, provided only the chemical structure, but no 3D information is used for the derivation of the similarity measure. If the distances between interaction sites are also used [143,144], we speak of a 3D-similarity measure.
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Compared to 2D similarity 3D-similarity calculation is usually much more demanding, because first a good 3D geometry has to be generated for each compound. Further, such measures often suffer from the conformational flexibility, since it is not obvious which of the conformations should be used for the similarity calculation. One of the most demanding similarity techniques is the "Comparative Field Analysis" (ComFA) [145,146]. This technique compares the mostly hypothetical interaction fields (steric field, Coulomb field, "hydrophobic interaction field", and "hydrogen bond field") on a grid surrounding of the molecules. The molecules have to be aligned such as to maximize the similarity of the interaction fields. A more efficient atom-based 3D-similarity approach (ComSIA) was suggested by Klebe [147]. A comprehensive review of current similarity-search methods for drug design has been given recently by Bender and Glen [148]. They demonstrate that no clear agreement has been achieved so far in the drug-design community, on which of the many similarity measures and techniques is superior, and point out that no universal advice for the best method can be given, since the selection of the most appropriate methods often depend on the special project. Nevertheless, they emphasize the importance of physical soundness and interpretability of the descriptors used for similarity studies. They also find that most similarity descriptors fail to describe the free-energy aspects arising from desolvation of the solute. Given these requirements, the COSMO-RS, 3D a-surfaces or the a-profiles appear to be very promising starting points for the definition of molecular similarity with respect to physiological behavior. As we have shown throughout this book, almost all kinds of relevant interactions in liquid or pseudo-liquid physiological phases, i.e., electrostatics, hydrogen bonding, and "hydrophobic interactions," can be represented as surface interactions on the basis of the interactions of COSMO polarization charge densities, a. As we have seen from the many examples throughout this book, a is a well-interpretable property and the physical soundness of the a-based calculation of physicochemical behavior has been widely proved. Since the chemical potential of molecules in water is simply accessible in COSMO-RS as a a-based surface integral, the treatment of desolvation, which is a problem in many other similarity measures, is easily included in a-based similarity methods.
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While we were aware of the latent potential of COSMO-RS for drug similarity searches, it was a customer, Michael Thormann from MORPHOCHEM, who did the first similarity searches based on COSMO-RS a-profiles. He tried the simplest approach of ~-profile comparison. Applying a generalization of Tanimoto's binary similarity coefficient for real numbers to a-profiles, the defined the similarity of two molecules X and X' as T(X, X ' ) -
min(AX,A x')
~
g~[pX(~ - i .
0.1 e/am2),
T"
pX'(a _ i 9 0.1 e/nm2)]/N(X, X ')
(11.9)
where A x and A x' are the surface areas of compounds X and X', respectively, N(X,X') is the number of intervals in one of the two ~-profiles having non-zero intensity, and the function g~ is defined as ~ ( a , b ) = 1 - la - bf a+b
(11.10)
On the basis of this simple definition he searched for bioisosteres of propionic acid. Using the COSMOfrag database of 28,000 compounds he found several interesting compounds within the top 20 most similar compounds of propionic acid (see Fig. 11.17). While six small carboxyhc acids are ranked as most similar to propionic acid, eight non-acidic compounds with quite different chemical skeletons follow, between ranks 7 and 19. Some of them are well known as suitable bioisosters of propionic acid, while the others may be tested as such. As a second example, Thormann considered four metabotropic glutamate receptor ligands with different scaffolds (see Fig. 11.18). In a similarity search for a newly identified active ligand [149] vs. an in-house database of 2 million drug-like compounds of MORPHOCHEM, he found one of the known active drugs ranked third: the other two known drugs also showed a substantial similarity, and were within the top 500 most similar compounds. This study was performed using the C O S M O f r a g method for producing approximate ~-profiles for the database, because obviously full compound DFT/COSMO calculations were not available for the 2 million compounds of the database. It should be noted that once the a-profiles have been calculated for a database, the
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Fig. 11.17. Search for bioisosteres of propionic acid based on a-profiles. The list of the top 20 most similar molecules is given on the left (SMILES code followed by a unique name, rank and Tanimoto coefficient), a-profiles and colored a-surfaces of propionic acid and six similar molecules with different scaffolds are shown on the right.
calculation of similarity coefficients is only a question of milliseconds per compound, so t h a t a-profile-based similarity searches over large databases can be as fast as with most other methods e.g., 2D-similarity methods. When we consider the fact t h a t the examples presented are the first applications of ~-profile-based similarity searches, the results m u s t be considered as very encouraging. Obviously, there are several possibilities to improve the method beyond its present state. One idea is to use a weighting function in the calculation of the Tanimoto coefficients, which gives a larger weight to the more polar ranges of a t h a n to the r a t h e r neutral a-range, because the interaction energies are much higher in the polar regions, and hence these are more i m p o r t a n t for similarity in the physicochemical behavior. Another idea is to use a definition of Tanimoto coefficients, which has some fuzziness regarding a, because two compounds should be considered as somewhat similar, even if their maxima in the a-profiles are slightly shifted. Thus a
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Fig. 11.18. Four metabotropic glutamate receptor ligands (A-D) and their structural dissimilar parts (a-d) [149]. For the latter, the COSMO a-surfaces are shown for comparison.
smearing function with a certain a-width should be applied in the similarity calculation. Alternatively, we might use a kind of pairing-algorithm, which assigns to each surface segment of molecule, X, a piece of surface of most similar polarization charge density a' on the other molecule, X', starting with the most polar segments, and stopping when one of the molecules has no surface left. If the resulting a-misfit is taken into account by an appropriate "a-misfit functional," e.g., by a "Boltzmann weight": exp{ - ~(a }-k a')2 T
(11.11)
the maximum sum of weights would correspond to the total surface area of the smaller of the two molecules, if optimal association is given for all surface segments, as would be the case when comparing two identical molecules. Hence, the sum of association weights divided by the larger of the two surface areas appears to
Life science applications of COSMO-RS
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be a reasonable measure of similarity. We will call this similarity measure ~-match similarity (SMS). Obviously, there is much room for further optimization of ~-based similarity searches. Since the simple a-profile-based similarity measures discussed so far do not directly involve the 3D geometry of the compounds, one might consider them as 2Dsimilarity measures. On the other hand, the a-profiles already include information about the 3D arrangements they are taken from, and thus the resulting similarity measures may be considered as a type of 2.5D similarity. In the future developments, one may also consider deriving real 3D-similarity measures based on the 3D a-surfaces. Nevertheless, such similarity measures will necessarily be computationally much more expensive. They also require either the full COSMO calculation for each molecule, or they require a composition algorithm that builds reasonable 3D COSMO surfaces from the fragments delivered by COSMOfrag. Finally, the use of a "a-polarity field" instead of different and more empirical electrostatic, hydrophobic, and hydrogen-bonding fields should lead to considerable advantages in molecular field analysis (MFA) techniques such as ComFA. On the one hand, includes all three aspects of interactions in a single field, and on the other hand, this interaction picture meanwhile is much better validated quantitatively than the empirical fields presently used in MFA. The only problem here is that a is a surface property and thus less a field in space. With some effort it should be possible to develop reasonable functions for the extension of a perpendicular to the molecular surface and thus to generate a kind of 3D a-filed as required for MFA.
Chapter 12
Summary, limitations, and perspectives At the end of this survey on the COSMO-RS method, it should have become clear that the combination of quantum chemical based continuum theory, with the statistical thermodynamics of pairwise interacting surface segments, opens a novel viewpoint and many new opportunities for the modeling of liquid, pseudoliquid, and biological systems. It appears that the COSMO state of molecules moving in a virtual conductor, i.e., the north pole of our globe, is a very good and reliable reference point for the exploration of fluid-phase thermodynamics. The combination of COSMO with statistical thermodynamics, i.e., the COSMO-RS method, is a fruitful compromise between the empirical and oversimplifying group contribution methods used for a long time in chemical engineering, and the basically much more rigorous molecular dynamics (MD) or Monte Carlo (MC) approaches, which try to simulate the entire liquid systems in molecular detail, but require severe approximations in order to describe at least some systems of practical relevance within reasonable time scales. The concept of surface polarization charge densities a, as local polarity measures, of a-profiles, and of a-potentials, provides a relatively simple, graphic, and intuitively understandable pathway from the molecular details to the macroscopic thermodynamics of liquid systems, which can be useful for teaching, understanding, and reasoning of fluid-phase thermodynamics. As demonstrated in the various application examples, the COSMORS theory provides good predictions of chemical potentials, partition and activity coefficients in its native area of neutral molecule chemical engineering applications, but it can be extended in many other directions by the addition of small pieces of empiricism. It proves to be very robust even in many extreme 205
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cases, where one would not really have expected good results. The list of application areas mentioned in this book is far from complete with respect to our own ideas, and I am sure other people will find many more useful application areas for COSMORS in the future than I have ever thought about. Despite the large application potential, one should be aware of several limitations of COSMO-RS, some being fundamental and others being eventually removable. The most fundamental limitation of COSMO-RS is its inability to handle dynamic properties such as viscosity and diffusion coefficients. At present, it is also unable to provide structural properties such as density of a liquid. Thus, COSMO-RS cannot give a complete picture of liquid-phase thermodynamics. In addition, at least at present, COSMO-RS has severe problems with a few, but important, solvents and solutes, e.g., secondary and tertiary amine systems, dimethyl sulfoxide (DMSO), or dimethylformamide (DMF) and dimethylacetamide (DMA). While for the amines the problem might be at least qualitatively understood, and some ideas for systematic correction have been developed, the bad performance of COSMO-RS for the solvents DMSO, DMF, and DMA is still not understood. As a further limitation, one should bear in mind that the calculation of solubilities of solids is strongly limited by the problem of accurate prediction of the chemical potential of molecules in their crystalline state. In this, no significant progress is to be expected in the near future. On the other hand, there are several clear perspectives for future improvements and extensions of COSMO-RS. One of the most obvious perspectives is the improvement of the underlying quantum chemical methods. While density functional theory appears to have reached its limit regarding the quality of the electrostatics, and hence of the COSMO polarization charge densities, there will be an increase in the availability of higher correlated ab initio methods like coupled cluster calculations at affordable computational cost. Quantum chemical calculation of local polarizability and eventually of suitable descriptors for dispersion forces should provide additional information about the strength of local surface interactions and can be used to improve the various surface interaction functionals. At the other end, the quantum chemical COSMO calculations for larger biomolecules and enzymes, which have just become available at reasonable
Summary, limitations, and perspectives
207
computational costs, are opening a wide area of application of the basic COSMO-RS concepts to important problems of biological and medicinal chemistry, e.g., the calculation of drug binding to receptors. As we have seen in the previous chapters, we are just at the initial stages, and much development can be expected in the coming years. Having reached the end of the book I thank the readers for their time and interest in my thoughts and developments. I also invite those who have become interested in the method, to use it on all occasions where they consider it applicable to their problems and in their projects. I especially invite those who feel the challenge to extend this novel theory toward new frontiers, or those who may already have developed ideas to improve the method while reading this book, to collaborate in the further development of COSMO and COSMO-RS.
List of Acronyms and Abbreviations ADME ASOG BB BP Car CC
(physiological) absorption, distribution, metabolism, and excretion analytical solution of groups, group contribution model blood-brain partition coefficient Becke-Perdew, a class of DFT functionals aromatic carbon atom coupled cluster method for high level ab initio
QC CI CLOGP COSMO COSMO-RS COSMOSPACE COSMO therm CSM DFT EDBC FF GBA GCM hb HF IEF IG KOC LFER LLE LOGKOW MC
configuration interaction calculated logarithmic (octanol-water) partition coefficient, a GCM conductorlike screening model COSMO for realistic solvation COSMO Surface Pair Activity Coefficient Equations program for COSMO-RS calculations continuum solvation model (electron) density functional method exact dielectric boundary condition force field generalized Born approximation group contribution method hydrogen bond Hartree--Fock integral equation formalism ideal gas partition coefficient between soil organic carbon and water linear free energy relationship liquid-liquid equilibrium logarithmic partition coefficient octanol-water, a GCM Monte Carlo simulation method 219
220 MD MM
Nav NRTL OCE PB PCM
Qc QM QM/MM QSAR QUAC RF SCF SCRF SES SLE SMS TS TZVP UNIFAC UNIQUAC vdW VLE VRML
A. Klamt molecular dynamics molecular mechanics Avogadro's n u m b e r - 6.023 x 1023 non-random two-liquid thermodynamic model outlying charge error Poisson-Boltzmann polarizable continuum model quantum chemistry quantum mechanics combination of quantum mechanic with molecular mechanics quantitative structure-activity relationship quasi-chemical model reaction field self-consistent field self-consistent reaction field solvent excluding surface solid-liquid equilibrium a-match similarity coefficient transition state triple-zeta valence plus polarization, a basis set universal functional activity coefficient, group contribution model, a GCM universal quasichemical model for mixture thermodynamics van der Waals vapor-liquid equilibrium virtual reality mark-up language, a standard for 3D visualization.
References COSMO AND COSMO-RS R E F E R E N C E S C1 A. Klamt and G. Schfifirmann, COSMO: A new approach to dielectric screening in solvents with explicit expressions for the screening energy and its gradient, J. Chem. Soc. Perkin Trans., 2 (1993) 799-805. C2 J. Andzelm, C. KSlmel and A. Klamt, Incorporation of solvation effects into density functional calculations of molecular energies and geometries, J. Chem. Phys., 103 (1995) 9312-9320. C3 K. Baldridge and A. Klamt, GAMESS/COSMO: First principle implementation of solvent effects without outlying charge error, J. Chem. Phys., 106 (1997) 6622-6633. C4 A. Sch~ifer, A. Klamt, D. Sattel, J.C.W. Lohrenz and F. Eckert, COSMO-implementation in Turbomole: Extension of an efficient quantum chemical code towards liquid systems, Phys. Chem. Chem. Phys., 2 (2000) 2187-2193. C5 A. Klamt, Calculation of UV/Vis-spectra in solution, J. Phys. Chem., 100 (1996) 3349. C6 A. Klamt and V. Jonas, Treatment of the outlying charge in continuum solvation models, J. Chem. Phys., 105 (1996) 9972-9981. C7 K. Bienert, A. Klamt, D. Krockenberger, F. Nader, B. Sewekow and R. Wittlinger, Zum Bioakkumulationspotential von Chlororganika, Z. Umweltchem. Okotoxologie 5 (1993) 228. C8 S. Maa~en, W. Arlt and A. Klamt, Vorhersage von GaslSslichkeiten und Verteilungskoeffizienten aufgrund einer vereinfachten molekulardynamischen Methode (COSMO), Chemie-IngenieurTechnik, 67 (1995)476-479. C9 A. Klamt, Conductor-like screening model for real solvents: a new approach to the quantitative calculation of solvation phenomena, J. Phys. Chem., 99 (1995) 2224-2235. C10 A. Klamt, V. Jonas, T. Bfirger and J. Lohrenz, Refinement and parameterization of COSMO-RS, J. Phys. Chem., 102 (1998) 5074-5085. C l l A. Klamt, COSMO and COSMO-RS. In: P.v.R. Schleyer (Ed.), Encyclopedia of Computational Chemistry, New York, Wiley, 1998. C12 A. Klamt and F. Eckert, COSMO-RS: A novel and efficient method for the a priori prediction of thermophysical data of liquids, Fluid Phase Equilib., 172 (2000) 43. 221
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A. Klamt A. Klamt and F. Eckert, COSMO-RS: A novel way from quantum chemistry to free energy, solubility, and general QSAR-descriptors for partitioning. In: Rational Approaches to Drug Design, Prous Science, Barcelona, 2001. A. Klamt, F. Eckert and M. Hornig, COSMO-RS: A novel view to physiological solvation and partition questions, J. Comp.-Aid. Mol. Design, 15 (2001) 355. A. Klamt, F. Eckert, M. Hornig, M. Beck and T. Bfirger, Prediction of aqueous solubility of drugs and pesticides with COSMO-RS, J. Comp. Chem., 23 (2002) 275-281. C. Mehler, A. Klamt and W. Peukert, Use of COSMO-RS for the prediction of adsorption equilibria, AICHE J., 48 (2002) 1093-1099. A. Klamt, G.J.P. Krooshof and R. Taylor, The surface pair activity coefficient equation: Alternative to conventional activity coefficient models, AICHE J. 48 (2002) 2332-2349. F. Eckert and A. Klamt, Validation of the COSMO-RS method: Six binary systems, Ind. Eng. Chem. Res., 40 (2001) 2371. F. Eckert and A. Klamt, Fast solvent screening via quantum chemistry: The COSMO-RS approach, AIChE J., 48 (2002) 369-385. A. Klamt and M. Diedenhofen, Prediction of soil sorption coefficients with COSMO-RS, Environ. Toxicol. Chem., 21 (2002) 2562-2566. A. Zissimos, M.A. Abraham, A. Klamt and F. Eckert, A comparison between the two general sets of linear free energy descriptors of Abraham and Klamt, J. Chem. Inf. Comput. Sci., 42 (2002) 1320-1331. A. Klamt, Prediction of the mutual solubilities of hydrocarbons and water with COSMO-RS, Fluid Phase Equilib., 206 (2003) 223-235. A. Klamt and F. Eckert, Prediction of vapor liquid equilibria using COSMOtherm, Fluid Phase Equilib., 217 (2004) 53-57. F. Eckert and A. Klamt, Prediction of halocarbon thermodynamics with COSMO-RS, Fluid Phase Equilib., 210 (2003) 117-141. R. Putnam, R. Taylor, A. Klamt, F. Eckert and M. Schiller, Prediction of infinite dilution activity coefficients using COSMORS, Ind. Eng. Chem. Res., 42 (2003) 3635-3641. M. Diedenhofen, F. Eckert and A. Klamt, Prediction of infinite dilution activity coefficients of organic compounds in ionic liquids using COSMO-RS, J. Chem. Eng. Data, 48 (2003) 475-479. A. Klamt, F. Eckert, M. Diedenhofen and M.E. Beck, First principles calculations of aqueous pKa values for organic and inorganic acids using COSMO-RS reveal an inconsistency in the slope of the pKa scale, J. Phys. Chem. A., 107 (2003) 9380-9386.
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A. Klamt and G. Hofmann-Jovic, Successful applications of COSMO-RS in process modeling, AIChE Conference, Indianapolis, November 2002. COSMOtherm program, COSMOlogic GmbH&CoKG, Leverkusen, Germany, 1999. COSMObase database, COSMOlogic GmbH&CoKG, Leverkusen, Germany, 1999. A. Klamt, M. Diedenhofen, R. Jones and P.C. Connolly, The use of surface charges from DFT calculations to predict intestinal absorption, J. Chem. Inf. Model., (submitted) A. Klamt and F. Eckert, Calculation of solvent effects on reactivity by COSMO-RS, Presented at AICHE annual meeting, Reno, November 2001, presentation 361 h.
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113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
A. Klamt S. Sonnenberg, A. Finke, A. Klamt, J.C.W. Lohrenz, T. Buerger and S. Matthiesen, US Patent Applications No. 20030008788, 20020152038, 20020146441. C. Mehler, Charakterisierung heterogener Feststoffoberfl~ichen durch Erweiterung eines dielektrischen Kontinuumsmodells, PhD Thesis: Technical University of Munich, 2003 (http://tumbl. biblio.tu-muenchen.de/publ/diss/mw/2003/mehler.pdf) M.H. Abraham, G.S. Whiting, R.M. Doherty and W.J. Shuely, J. Chromatogr., 587 (1991) 213-228. M.H. Abraham, J. Phys. Org. Chem., 6 (1993) 660-684. M.H. Abraham, Pure Appl. Chem., 65 (1993) 2503-2512. M.H. Abraham, Chem. Soc. Rev., 22 (1993) 73-83. S. Verevkin and A. Heintz, in preparation. R. Franke, C. Borgmann, D. Hess and K.D. Wiese, Zeitschrift fu'r anorganische und allgemeine Chemie 629 (2003) 2535-2538. We thank given to D. Bakken and Merck & Co., Inc. for providing us with the experimental data and for permission to show the results (2003). J.C.W. Lohrenz and A. Klamt (1998) unpublished results. S. Pauly, Permeability and Diffusion Data, In: J. Branderup and E.H. Immergut (Eds.), Polymer Handbook, 2nd ed. Wiley: New York, 1975, pp. 547ff. S.E. Feller, D. Yin, R.W. Pastor and A. MacKerrell, Jr., Biophysical J., 73 (1997) 2269-2279. A DPC micelle consisting of 54 lipids and a few thousand water molecules (shown schematically) taken from, http://indigol.biop.ox.ac.uk/tieleman/projects.html# micelles. U. Essmann and M.L. Berkowitz, Dynamical properties of phospholipid bilayers from computer simulation, Biophys. J., 76 (1999) 2081-2089. C.D. Bruce, M.L. Berkowitz and L. Perera and M.D.E. Forbes, J. Phys. Chem. B, 106 (2002) 3788-3793. T. Busalla, Berechnung von Membranverteilungskoeffizienten, Diploma Thesis: University of Cologne 1996 (in German). J.G.E.M. Fraaije, B.A.C. van Vlimmeren, N.M. Maurits, M. Postma, O.A. Evers, C. Hoffman, P. Altevogt and G. GoldbeckWood, J. Chem. Phys., 106 (1997) 4260. R.D. Groot and P. B. Warren, J. Chem. Phys., 107 (1997) 4423-4435. A. Bondi, Phys. Chem., 68, (1964) 441-451. G.A. Voth, Proton Transport in Aqueous and Biomolecular Systems, Conference on Current Trends in Computational Chemistry, Jackson, November 2002. M.K. Petersen, S.S. Iyengar, T.J.F. Day and G.A. Voth, J. Phys. Chem. B, 108 (2004) 14804. J. Batoulis and A. Klamt, Bayer research report 1993, unpublished.
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Subject Index 2D-3D converter 178, 182 3D visualization 86 Abraham descriptors 144-145 activated carbon 138, 141-143 activity coefficients 63, 67, 71, 73-74, 77-78, 103-105, 116, 127, 129-131, 133-135, 150, 205 ADME 168, 171-172, 176, 180, 183, 190 adsorption 138, 141-144, 179 agglomeration effects 79 agglomerization 171 Alkanes 5, 16, 43-44, 46, 88, 97 AMSOL 17, 37 Analytic derivatives 33, 36, 71, 130 analytical derivatives 25 apparent surface charges 19 approximations 1, 4, 16, 38, 52-53, 60-61, 64, 67-68, 71-72, 74, 112, 118, 151, 173, 187, 205 ASOG 6 association 35, 154, 202 atom-centered multipole representation 27 atomic orbitals 24 averaging radius 84-85, 110 back-polarization 13, 19 basis grid 32, 34-35 biomembranes 160, 162 Boltzmann factor 45, 73-74 bond dipole correction 38 Bondi radius 117 C-PCM 28, 37, 40-41 cavity 16, 18-21, 23, 25-33, 35-36, 38-40, 52, 118-119 center approximation 34-35
charge models 19, 21-22, 29 charge-density 24, 84, 89 Cholesky factorization 24, 194 CLOGP 5-6, 8 color coding 85 combinatorial factor 60-61 conformation 119-125, 130, 171, 177-178, 181-184, 186, 198 contact probability 65-66, 72 coordination number 61, 75, 77, 118 CORINA 178, 182-183 COSMO 7-9, 11-13, 15-19, 21-23, 25-29, 31-33, 35-37, 39-41, 46-47, 49-50, 52-53, 56-60, 67-69, 71-75, 77-81, 83-85, 87, 89, 91, 93, 95-103, 105, 107, 109-125, 127-138, 140-141, 143-145, 149-165, 167-177, 179-197, 199-203, 205-207 COSMO-RS 9, 18, 26-28, 47, 57-60, 67-69, 72, 83-85, 87, 89, 91, 93, 95-103, 105, 107, 109-119, 121-125, 127-138, 140, 143-144, 149-165, 167-177, 179-181, 183-187, 189-197, 199, 201, 203, 205-207 Coulomb interaction 24-25, 35 coupled cluster 2, 41, 149, 206 crystals 2 density functional theory (DFT) 2-3, 19, 25, 27-28, 36, 39-41, 83, 112, 117, 121-125, 132, 151, 156, 158, 161-162, 168-169, 180-181, 184, 187, 190, 193-195, 197, 201, 206 diagonal element 35 dielectric boundary condition 20, 22, 43 231
232
Subject Index
dielectric constant 11-17, 38, 43, 46, 51, 97-98, 100 dielectric continuum 9, 11-19, 21, 23-25, 27-29, 31, 33, 35, 37, 39, 41, 43-47, 49, 98 Dielectric Continuum Solvation Models 9, 11, 13-15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 45-47, 49, 98 dielectric interaction energies 20, 37 diffusion coefficients 206 dimerization 171 dipole 12-13, 15, 18-19, 22-23, 28, 37-40, 43-45, 98, 195 dipole moments 13, 18-19, 37, 39-40 dispersive 2, 4, 49 dissociation constants 152, 156, 169, 171 DMol 27, 40, 117 Drug Development 168 drug similarity searches 197, 199 drug-design 171, 173, 184, 190, 197-199 drug-like compounds 172, 174, 184, 189, 201 electrolyte systems 56 electronic polarizability 14-15, 43-44, 51, 54, 97, 112 element-specific radii 30, 181 entrainers 131 entropy 4, 55, 74, 83, 92, 95, 104-105, 114, 119 environmental 5, 22, 127, 132, 137-138, 141 enzyme receptors 167 force-field 4, 17, 58, 162 gas-phase 13, 19-21, 28, 37, 39-40, 116, 150-151, 156, 183 gases 3, 63, 103, 116, 159 generalized Born 16 geometrical constraints 59, 81 geometry optimization 2, 17, 20-21, 36-37, 39, 181-182
GEPOL 26, 31-35 GEQUAC 67, 72 Gibbs free energy 62, 72, 104, 128 Gibbs-Duhem consistency 71 Greens' function 24-25 grid construction 17 Group contribution methods 57, 129, 133, 205 Hartree and Fock (HF) 2, 18-19, 25, 39-40, 156 hb-acceptor 55, 99-100, 102 hb-donor 55, 99-100 high-throughput screening 184 hydrogen-bond (hb) interactions 7, 55, 89, 99-100, 111, 119-120, 122, 138-140, 144, 146-147, 169, 177, 179-180, 203 hydrophobic 98, 100, 153-154, 168, 203 hydrophobicity 98, 100, 154 ideal gas 114-116 infinite dilution 63, 68, 101, 105, 172 integral equation formalism 28 Interacting Surface 59-65, 67-69, 71, 73-75, 77, 79, 81, 100, 162, 205 interaction parameters 8, 57, 67-68, 127-129, 165 intermolecular forces 2 intermolecular interactions 2, 4-5, 56 intestinal absorption 172, 177-178, 182 intramolecular hydrogen-bond 7 intramolecular interactions 8 lattice 6, 65-66, 68-69, 72-75, 78-81 life-Science 137 limitations 129, 137, 205-207 linear-response theory 43, 45 linear-scaling 39, 194 linear-scaling methods 39 liquid ensemble 59 local composition 65, 74 Local polarizability 112, 206 Local shape index 112
Subject Index lock-and-key concept 190 lone-pairs 17, 85, 89-91, 111 Mr
perturbation theory (MP2) 1-9, 11-41, 43-47, 49-81, 83-86, 88-107, 109-125, 127-147, 149-165, 167-203, 205-207 methods 2-7, 9, 11-13, 17-18, 21-22, 30, 35, 37-39, 41, 43, 46, 57, 79, 129, 133, 149, 152, 154-156, 162, 167-168, 174, 177, 180, 190, 194-196, 199, 201, 205-206 molecular dynamics 3-4, 40, 153, 205 molecular mechanics 4, 57-58 molecular orbitals 24, 194 molecular-shaped cavity 16, 18, 23, 29 Monte Carlo 4, 73, 77, 205 MOPAC 25-27, 33, 39, 154, 181, 183, 194-197 multipole moments 12-13, 15, 18 non-electrostatic contributions 40 NRTL 127, 129 octanol-water partition coefficient 5, 101-102, 143, 147, 188 optical dielectric constant 14-15, 51, 98 outlying charge 21-22, 27-28 partition coefficient 5-6, 101-102, 117, 127, 137-140, 143-147, 149, 162-163, 172-173, 179-180, 183, 188, 191 partition sum 60, 62, 64, 69-71, 111, 163 pKa 152-157, 180-181, 190 plain waves 24 polarizable continuum model (PCM) 20-21, 25-26, 28, 31-33, 35, 37-38, 46 polarization charge densities 20, 53-54, 56, 83, 86, 91, 109, 133, 151, 161, 181, 185, 199, 206 polymers 78, 157-160 process simulation 129-131
233
proteins 167, 195 pseudo-chemical potential 63, 70, 101, 103 pseudo-liquid receptor 191-192 QSAR 22, 39, 145-146, 167, 172, 174, 176-177 quadrupole 12, 43-44, 46, 195 quantitative structure-activity relationships 22 quantitative structure-activity relationships (QSAR) 22, 39, 145-146, 167, 172, 174, 176-177 quantum chemistry 1-3, 22, 25-26, 114, 149, 168 quantum theory 1, 3, 39 Quasichemical Theory 64 reality corrections 116 refractive index 14, 43, 112 reorientational energy 45 residual contribution 62 ring-correction 115 rotational energy 115 scaling factor 15-16, 24, 100 scoring function 192-193 segmentation 19, 23, 29, 33-35 self-consistent reaction field (SCRF) 13-14, 16-17, 22-23, 25 semi-empirical 14, 17, 22, 25-27, 35, 37-39, 112, 180, 194-196 similarity indices 185-186 SMx methods 37 soil sorption coefficient 137, 141-142 solid phase 2, 103 solubility 95, 150, 158-160, 169-176, 180, 182-183, 187-190 solubility parameter 169 solvation energies 11, 13, 18-19, 35 solvatochromic effects 13-15 solvent selection 131, 133 solvent space 144
234
Subject Index
solvent-excluding surface 31 sphere triangulation 26, 32 Statistical Thermodynamics 59, 61-63, 65, 67, 69, 71-73, 75, 77, 79, 81, 85, 173, 205 steric hindrance 54 surface closure 35 surface polarization 19, 85, 205 surface potential 23 surfactants 160-162 Tanimoto coefficients 202 tautomerism 123-124 tautomers 123-124 temperature dependence 13, 55, 83, 100, 111, 118-119, 143, 160 tesserae 19 thermal fluctuations 54 transition states 149, 151-152
translational 115 TURBOMOLE 27-28, 40, 183-184, 193-194 UNIFAC 6, 57, 61, 68 UNIQUAC 57, 61, 67-68, 74-75, 77, 118, 127-128 vacuum 2-3, 16, 46, 50-51, 55, 114-115 van der Waals (vdW) 2, 16, 18-19, 21, 49-52, 54-56, 61, 111-114, 117-119 Viscosity 206 volume 16-19, 26, 49, 52, 61, 68, 77-78, 101, 128, 144, 165, 174, 184 volume polarization 17, 19 VRML 88 zwitterion 124-125, 160, 162 zwitterionic 124-125, 160
Appendix APPENDIX A:
DERIVATION OF THE COSMO-RS SELF-CONSISTENCY EQUATION
Consider an ensemble S of N objects, belonging to m classes (surface segment types in COSMOSPACE) of identical objects (surface segments in COSMOSPACE). Let Ni be the number of objects of class i. Then xi = N i / N is the relative concentration. Let there be N sites that can be occupied by the N objects. Each two of these sites form pairs. Hence, all objects occupying the N sites are paired. The interaction energy of the pairs will be described by a symmetric matrix, Eij, where i and j denote two different classes of objects. Let Z be the partition sum of the ensemble. From statistical thermodynamics, the chemical potential ~/i of objects of class i is given by
pi__kTdlnZ ln(z)-ln(z(_i)) dN-------~= - k T N - ( N - 1)
_kTln(Z) =
(A.1)
Z(_i)
where Z(_i) denotes the partition sum of the ensemble with decreased by one. In the same way we find
Z
Pi -[- Pj --
Z(_i)
-kTln Z-S_i)Z(_i_j)] - - k T l n
(z) Z(~_j) ~
Ni
(A.2)
where Z(_i,_j) denotes the partition sum of the ensemble with Ni and NN decreased by one. Let us now consider a second similar ensemble S' in which all objects may be distinguishable. Let us denote these objects by Greek indices ~ and ft. Let the objects still fall into m classes. Let i be the class of object ~ a n d j the class of ft. Let the multiplicities Ni of the classes be the same as in S. Let the pair interaction energies E'~Z depend only on the classes, i.e., E ' ~ - Eij. Due to the permutations of the distinguishable objects in S', each state of S corresponds to a number of states in S' that 209
210
A. Klamt
have the equivalent pairing of objects, and hence have the same energy. As a consequence, we have the relationship Z - Z'/Ndeg for the partition sums of the two ensembles, m
Ndeg --
YINi!
(A.3)
i=l
Applying Eq. (A.1) to this relationship we obtain
tti- _kTln( Z )-_kTln(Z'Ndeg--(-i)-) Z(-i)
\NdegZ'(-~)
= -kTIn(z,(Z' )-kTln(Ndeg----A(-~)) )
\ Ndeg
= p':~+ kTlnNi
(A.4)
where Ndeg(-i) is the degeneracy factor of the ensemble with one object of class i missing. According to Eq. (A.3) this is by a factor Ni smaller t h a n Ndeg. In the next step we derive a relationship for the chemical potentials in ensemble S'. For that, we first choose any object a. We can sort all states of the ensemble with respect to the p a r t n e r fl of ~. The partition sum of the system can be written as a sum over all partners fl, taking into account the Boltzmann weight exp(-E'~ /k T) of the interaction of ~ with fl multiplied by the partition sum Z(_~._~) of a system missing objects ~ and ft. A factor N/2 arises from the fact that a pair ~fl can be taken away from any of the N/2 pair positions of the ensemble, and a factor of 2 arises from the two possible orientations of that pair on each pair position. Hence we have Z
N -~ 2exp
!
~ Z' '-~.-/~,
N
(A.5)
/3-1
Appendix
211
where we have made use of the equivalent of Eq. (A.2) for ensemble S'. By dividing both sides by Z' and exp(p'~/kT), and on taking the logarithm, we have
p~ - - k TIn
exp
~-7~
-
kTlnN
(A.6)
B=I
The restriction fl # ~, which is caused by the fact that the object cannot be its own interaction partner, can be omitted in the thermodynamic limit of large N, because the resulting change is infinitesimally small. Doing this, replacing p'~ and p~ by Pi and pj according to Eq. (A.4), using the corresponding energies Eij instead of E'~, and replacing the summation with respect to fl by a summation with respect to classes j of multiplicity Nj, we obtain !
Pi - - kT In
{~~_~ (-EiN + PN- kTlnNN) } + k T l n N-i Nj exp kT N
= - kTln ~ e x p j:;
kT
+kTlnxi
(A.7)
The activity coefficient )'i of an object of type i in this ensemble is defined by k T In
~:i -
]ti -
Ftii -
kT In x i
(A.8)
where ~tii denotes the chemical potential of class i in an ensemble with xi - 1. In this case, Eq. (A.7) simply reduces to [dii -
(A.9)
~Eii
An interaction parameter
o.
rij can
be defined by
212
A. Klamt
Subtracting k T l n ( x i ) and ~.lii on both sides of Eq. (A.7) yields, in combination with Eqs. (A.8)-(A.10), k T ln "Yi xi
--- [d i - - [Aii - - k T I n
= -kTln
=-kTln
Eli
xj exp -Eij + (1/2)(Eii + Ejj) + pj - (1/2)Ejj - kTlnxj - (1~2)Eli kT e x P \ 2 k T j ~-'~xjzijexp\
kT
2
j=l
Thus, we have the general result m
(A.12)
1 _ E xjzij~) )'i j=l
APPENDIX B:
PROOF OF GIBBS-DUHEM CONSISTENCY
Here, we provide an independent demonstration that activity coefficients obeying Eq. (12) satisfy the Gibbs-Duhem equation. Multiplying Eq. (A.12) by xiTi followed by summation over the index i yields ].-- ~ X i i
-- EXi~Ti~Xj'2jT:ij i
(B.1)
j
Multiplying the above equation by N 2 and replacing xi by Ni yields N 2 -
~
(B.2)
Nj'/jzijNi'/i
ij
If we differentiate the above equation with respect to any object number N~ we obtain ~ N c~Nk
2 - 2N =
c~ ~ Nj;)T.ijNi.;i c~Nk zg ..
= 2E
j
0 ~ Nj'~:] c3Nk
gi),iz 0
(B.3)
Appendix
213
Division by 2N leads to 1-
ExjTj-~k~-~giTi-cij j i
-- ~-~xj~/j j
= 7k ~ XjTj~kj + ~ Ni c~Ti j
i
= 1 + ~ xi
aln 7i C~Xk
~
~ N k ~j
]'k'Ckj -+- ~ - ~ g i ' c i j ONk i
7k
Xj~/j~iiJ -- ~~'k +
~i Ni c~Ti 9 ~ c~Nk
(B.4)
where we have made use of Eq. (A.12) twice. Subtracting 1 from both sides yields 0 -- E
0 In ?i C~Xk
Xi ~
(B.5)
which is the Gibbs-Duhem relationship for activity coefficients. Hence, activity coefficients that satisfy Eq. (12) must be thermodynamically consistent. APPENDIX C-
SOME WORDS ON THE COMBINATORIAL CONTRIBUTION
The necessity of introducing a combinatorial contribution to the chemical potential is a result of the neglect of size effects in the thermodynamics of pairwise interacting surface models. It also appears in lattice models that do not allow for a realistic representation of molecular sizes and are often simplified to models of equally sized lattice objects. The task of the combinatorial contribution is to represent the chemical potential of virtually homogeneous interacting objects of different size in 1 mol of a liquid mixture of a given composition with respect to the size and shape of the molecules. Although I did not know about the concept of the combinatorial contribution, I recognized the need for such a correction even in the initial version of COSMO-RS [C9]. Since at that time I only had in mind the calculation of infinite-dilution partition coefficients and of vapor pressures, I only cared about a solvent-size correction in pure solvents. I thought of two different solvent-size effects influencing the chemical potential of a solute X in a solvent S. The first is quite obvious---in 1 mol of a homogeneous liquid S,
214
A. Klamt
a solute X has mainly a free choice of the first surface-contact partner. If the solute molecules have n s - - A s / a e f f independent surface segments, we have Nav* ns contact segments in 1 mol of S, and the free choice of the first contact gives a free-energy contribution of k Tln(Nav.ns). We thus get a contribution proportional to -kTln(As). On the other hand, larger molecules provide more geometric constraints, i.e., fewer places per mole of solvent that can really be occupied by a solute molecule of a certain size. This should lead to a contribution in the opposite direction. Although not having a concrete idea of the functional behavior of this contribution, we may assume that the decrease in the number of possible realizations is, to a first-order, proportional to the size of the solvent molecules. Using the surface area As as a descriptor of size, the corresponding free-energy contribution should also be proportional to k Tln(As). Therefore, I simply assumed that the solvent-size contributions to the free energy and to the chemical potential of a solute molecules, X, in 1 mol of solvent S is given by ).kTln(As), with ~. being a parameter to be adjusted to the experimental data. For simplicity, we defined the solvent surface area, As, as the average value of the surface areas of the components of the solvents, in order to be able to treat solvents that are not pure compounds, as is the case for the watersaturated octanol in the octanol-water partition coefficient data. Indeed, this solvent-size correction for the chemical potentials worked well in the initial COSMO-RS parameterizations, yielding quite small values of )., in the region of-0.14. Thus it appeared that the two different size contributions almost cancel out, i.e., only 14% of the obvious first contribution remains. Even large size differences of up to a factor of 10 in surface area only yielded corrections of 0.8 kJ/mol in the chemical potential differences between two phases; these corrections were only about half of our standard deviation at that time. Therefore, not much care was taken with respect to the combinatorial contribution during the next years of COSMO-RS development. All other contributions appeared to be of much greater importance. Indeed, Rolf Treckmann, then a colleague from the Chemical Engineering Department of Bayer AG, pointed out to me some expressions for combinatorial contributions of activity coefficients, as they are often used in chemical e n g i n e e r i n g ~ a n d especially to the Staverman-Guggenheim (SG) term given in
Appendix
215
Eq. (5.3). I tested COSMO-RS after replacing my simple solventsize expressions with these terms, but the overall COSMO-RS results became much worse, and I forgot again about combinatorial contributions. When Iven Clausen [96] first applied our COSMO-RS method to binary mixture thermodynamics, he obtained very realistic curves for the gas-phase compositions and activity coefficients of alcohol-water systems in good agreement with experimental data (see Fig. 8.1). Due to this good agreement, and since this was only a minor side-aspect of his Ph.D. thesis, neither he nor his supervisor Professor Arlt checked the Gibbs-Duhem consistency of the COSMO-RS results. I myself, having never heard about the G i b b s - D ~ e m consistency at that time, did not think about this aspect of thermodynamic consistency as well. Thus it was leI% to Gerard Krooshof in 1999 to detect the thermodynamic inconsistency in some binary mixture results of thermodynamical results of COSMO-RS that we had calculated for him on a contract basis. Since the mixtures of interest were almost ideal in his case, the thermodynamic inconsistency resulting from the size correction was quite apparent in plots of in 7. While the Gibbs-Duhem rule 0 -- E
i
Xi
c~In 7i ~J
= Xl
c3In 71 C7X1
c3In 72
(C.1)
-}- X2 ~
C~X1
apparently imphes that in a binary mixture a maximum of In 71 must coincide with a minimum of In 72 and vice versa, these plots showed just a maximum of one In 7 curve. I well remember my shock when I recognized that COSMO-RS was not thermodynamically consistent, just after having quit my safe position at Bayer AG and had started my own company. But within an hour I had located the origin of the inconsistency in the solvent-size correction, and within a day I found an expression for the size correction that ensures Gibbs-Duhem consistency, while leaving the limits of pure solvents unchanged compared with our original COSMO-RS expression:
c~Ni whereGcs-NkT2c[~isin2(~-~)lnA(-~oo) ]
x~ OGc,s PC,S = ,
-
+ ln(A0)
(C.2)
216
A. Klamt
The thermodynamic consistency of this expression follows directly from the fact that the chemical potential corrections are now calculated as composition derivatives of a thermodynamic potential. Since this expression left the limits of pure solvents unchanged, and since almost only these limits are of importance for our COSMO-RS parameterization data set, we could use the existing COSMO-RS parameterization in combination with the new Gibbs-Duhem-consistent solvent size correction. Unfortunately, I later realized that the heuristic expression given in Eq. (C.2) has some undesirable features. If one considers a mixture of two identical compounds of equal size, As, the solvent-size correction would not converge to the same result as given by just considering the liquid as a pure liquid. This is obviously nonsense since, for example, we would achieve a nonideal behavior of a water-water mixture. Therefore, during a subsequent re-parameterization of COSMO-RS, we tried to replace the solvent-size correction with an expression more similar to typical chemical engineering expression, especially to the most widely used SG expression of Eq. (5.3). On the other hand, we knew that the full SG expression did not lead to satisfying results. Scaling of the SG expression by a factor ~. would have been one straightforward way to proceed, but then we would have been left with a combinatorial expression that is zero in the limit of a pure compound; however, we prefered a combinatorial term that also contributes to the calculation of the chemical potential of a compound in its pure liquid state, and we see no reason why this should be zero. In most chemical engineering models only the excess free energy is of importance, and in that case the vanishing of the expression in the pure liquid is preferable; however COSMO-RS also calculates vapor pressures. In these, a pure compound contribution of the combinatorial part should be present. Furthermore, there is an open question about the right value of the coordination number, z, prior to the second contribution. In chemical engineering a value of z - 10 is often used. However,~ ~s discussed in Section 7.4, our effective contact size aeff --- 6.5 A clearly suggests a value of z ~ 5.5. The same value of z - 5.5 can be derived in different ways from the correlation of the molecular volumes and surface areas. Thus we are quite sure that the value of z - - 1 0 is unrealistically high. However, since the first and the second contributions of the SG
Appendix
217
expression have opposite signs, a change of the coordination number z prior to the second term strongly changes the behavior of the SG expression. Thus we conclude that the SG expression is most likely just one plausible functional form for a size and shape correction, but that some scaling of the different parts of the SG expression is definitely required to achieve good results in the context of the COSMO-RS model. We therefore introduced our own expression
Gcomb_ RT [2o E
xi ln r -- )'l ln ( E xiri -- ~t2ln ( ~ xiqi) (c.a)
The combinatorial contribution ~lZcombto the chemical potential of compound i is
ri lnrav) //comb-- 0Gc,s OXi =RT[).olnri+ ~ (1 rav +).2(1 qavqi lnqav)] i
qi
(C.4)
where ri is the molecular volume and the molecular area of compound i, both taken from COSMO cavities. Thus we have three adjustable parameters, 20, 21, and 22. If both 20 and ).1 would have the value of 1, then our expression would coincide with the first and most dominant term of the SG expression. However, in our parameterizations, we find these coefficients typically in the region of 0.16, i.e., again much smaller than the combinatorial contributions usually considered in other chemical engineering models. We have no clear answer, why COSMO-RS appears to require smaller combinatorial contributions than do empirical chemical engineering models.
