Computational Photochemistry

231 consistent field (CASSCF) method is used, including all electronic configurations consistent with a number of active orbitals and electrons. This is not required, but again avoids any unintended bias towards one of a pair of degenerate states. Unlike CASPT2, which adds dynamic electron correlation to CASSCF through perturbation theory, the CASSCF method is sufficiently inexpensive to be applicable for geometry optimizations and even dynamics calculations.16'29 However, such calculations generally omit solvent molecules or surrounding protein residues. This is a clear deficiency in the context of photobiology, but is being remedied in the most recent work of several groups, which we discuss below. It is critical to recognize that the choice of active space in CASSCF calculations is somewhat arbitrary. Although in principle it is true that a larger active space is better, this is not always borne out by calculations. The CASSCF method does include a varying degree of dynamic electron correlation as the active space is increased. This is clear because CASSCF becomes full configuration interaction (CI) (and hence exact within the chosen basis set) in the limit of an active space including all electrons and orbitals in the molecule. However, one is in practice always far from this limit, and the CASSCF method is well-known to be a very inefficient way of including dynamic electron correlation effects. Thus, we take a pragmatic view that the active space should be chosen on energetic criteria. One useful way to determine an initial guess for the number of electrons and orbitals in the active space is to compute the vertical excitation energies using EOM-CCSD. From analysis of the EOM-CCSD coefficients for the low-lying electronic states, one can deduce a good guess at the required number of electrons and orbitals in the active space. Then, several calculations are carried out varying the number of electrons and orbitals around the EOM-CCSD predicted values. From each of these, a CASPT2 calculation is carried out. If the active spaces are all reasonable, the vertical excitation energies computed by CASPT2 will be nearly independent of the underlying CASSCF. From this procedure, one can collect a set of different active spaces which have the electronic states ordered correctly and also are apparently equally valid as judged by the resulting CASPT2 vertical excitation energy. Ideally, one continues the verification procedure by tracing out reaction paths on the excited state using both CASSCF and CASPT2 with the active spaces which remain under consideration. The CASSCF active space which gives the best agreement with CASPT2 is the one which is then used. The number of electronic states included in the state-averaging procedure can also be varied in this process, using the same general criteria that a reasonable CASSCF calculation will reproduce the features predicted by CASPT2. This type of procedure is much more effective at determining a robust active space than one based on chemical intuition, which often ignores the influence of dynamic electron correlation. When used in the context of dynamics, it is necessary to repeat this process several times. Essentially, one first determines which active spaces are reasonable at the Franck-Condon point as discussed above and then carries out a few dynamics simulations. From the observed dynamics, one can extract coordinates which are important and construct paths along which the CASSCF and CASPT2 potential energy surfaces can be compared. Ultimately, one may supplant this procedure with dynamics and/or optimizations using CASPT2 (or perhaps an improved form of TDDFT)

232 directly. However, until this is computationally practical, the above-outlined approach provides a means for obtaining semi-quantitative agreement between CASSCF and CASPT2, which then allows for extensive dynamics calculations addressing the photochemical mechanisms. In Fig. 3. we demonstrate the kind of agreement which can be achieved using this technique for a biological chromophore - the same analog of the PYP chromophore presented in Fig. 2. The agreement is semi-quantitative, providing justification for the use of the chosen active space in geometry optimizations, reaction path calculations and dynamics studies.

00

ft ^ ^ S

SA-5-CASSCF(6/5) 10

10

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Mass-Weighted Distance / am u

12

Angstrom

Fig. 3. Comparison of CASSCF and CASPT2 for analog of the PYP chromophore. The geometries in the chosen path are chosen as in Fig. 2. The active space was chosen according to the procedure detailed in the text. Unfortunately, even CASSCF methods are quite expensive in the context of photodynamics of biological chromophores. Thus, there is incentive to seek alternative electronic structure methods with similar, or perhaps even better, accuracy but much less computational cost. Semiempirical methods are quite promising in this regard. However, most previous parameterizations have emphasized ground state properties. In the few cases where excited electronic states were considered,30'31 vertical excitation energies were emphasized and not global features of excited state PESs. Furthermore, most semiempirical methods are based on a single-reference electronic wavefunction. The MNDOC method32"34 has been employed using a multi-reference single and double excitation configuration interaction (MRSDCI) wavefunction, " but with molecular orbitals derived from a single determinant SCF calculation,

233 i.e. Hartree-Fock. An interesting multi-reference semiempirical model was proposed by Cullen, but only applied to ground states. ' We have recently explored the utility of a multi-reference semiempirical method which approximates the ab initio state-averaged CASSCF method. The floating occupation molecular orbital (FOMO) method40"42 is used to obtain molecular orbitals which avoid bias to a particular electronic state. These molecular orbitals are "best-compromise" orbitals for the low-lying electronic states and thus the procedure must be followed by CI. We usually perform a CASCI for this purpose, although other forms of CI may be used if necessary.40"42 Electron repulsion integrals are approximated using a "neglect of differential diatomic overlap" (NDDO) scheme.43 Our first investigations of the FOMO-CASCI semiempirical method focused on the predicted geometries and energetics of conical intersections. ' We found that MECI geometries were predicted quite well using this approach with standard semiempirical parameter sets such as MNDO,44'45 AMI, 46 and PM3. 47 In Fig. 4, we show an example of the agreement which is achieved for an So/Si MECI geometry in the case of the Green Fluorescent Protein (GFP) chromophore. The ab initio and semiempirical predicted MECI geometries are almost indistinguishable. However, the energetics predicted by the semiempirical FOMO-CI method is often qualitatively incorrect. For example, the MECI which is an absolute excited state minimum for ethylene using ab initio methods '4 ' was predicted to lie above the Franck-Condon point using the FOMO-CASCI method with the MNDO parameterization. This would lead to qualitatively incorrect dynamics, since the MECI is energetically inaccessible after photoexcitation. Nevertheless, the good agreement for MECI geometries suggests that the semiempirical FOMO-CI method may be useful if the semiempirical parameters are optimized for the specific molecule of interest.

Fig. 4. Comparison of ab initio (black) and semiempirical (grey) So/S i MECI geometries for the neutral form of the GFP chromophore. The ab initio results are obtained using a SA-2-CAS(2/2) electronic wavefunction with the 6-31G basis set and the semiempirical results are obtained with the FOMO-CI method using the AMI semimepirical Hamiltonian.

234 We apply high-level ab initio methods at the most important geometries of the molecule under study (ground and excited state minima and MECIs) for all the electronic states of interest. The semiempirical parameters are then optimized in order to reproduce the ab initio results. Such system-specific reparameterization was almost routine in early implementations of semiempirical methods, and was most recently revived under the acronym SRP ("specific reaction parameters") for ground state reaction rate calculations.50 A similar approach has been used in the context of photochemical reactions by Olivucci and coworkers under the rubric of the molecularmechanics/valence-bond (MMVB) method.51"54 However, the valence bond form of the wavefunction used in the MMVB parameterization is not expected to treat ionic states accurately, restricting its application to cases where only covalent states are involved. The best choice of what data to include in reparameterization of the semiempirical FOMO-CI method is a current topic of our research. In our work on the GFP chromophore,55 we have included stationary points (both local minima and transition states) on the ground and excited electronic states and also selected MECIs. At each of these molecular geometries, the ground and excited state energies were included in the data set to be reproduced. In Fig. 5, we show a representative example of the results obtained in this case. The reparameterized semiempirical FOMO-CI method is seen to be intermediate between the CASSCF and CASPT2 methods, at a computational cost which is significantly less than CASSCF. A simpler and more automatic approach to reparameterization may be possible. For benzene, we tried using only a few molecular geometries corresponding to local minima on the ground electronic state. For each of these geometries, both energies and gradients of the low-lying electronic states were included. This strategy was sufficient to produce reasonable agreement of both geometries and energetics of MECIs and also of potential energy surfaces along pathways connecting photochemically-important points such as the Franck-Condon point and MECIs.56

235

LLJ

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I-Twist Angle / Degrees Fig. 5. Comparison between reparameterized FOMO-CI (dotted lines) So and Si PESs along an Sicoordinate-driving path (optimizing all coordinates except for one of the torsion angles on the S, PES) and the corresponding PESs from SA-2-CAS(2,2) (solid lines) and SA-2-CAS(2/2)-PT2 (dashed lines). The molecule is the neutral form of the GFP chromophore and is depicted in the inset. The FOMO-CI and CASSCF paths are optimized separately. The CASPT2 results are obtained along the CASSCF-optimized paths. The zero of energy is in all cases chosen to be the So minimum at the respective level of theory. Although the reparameterized semiempirical FOMO-CI approach is much less computationally challenging than CASSCF methods, it is still too demanding for direct application in condensed phase and protein environments. However, the most significant effects of electronic excitation are often confined to a well-defined chromophore. Thus, a quantum mechanical/molecular mechanical (QM/MM) description may be appropriate. First implemented by Warshel and Levitt57, the basic idea in QM/MM methods is to describe a relatively small region of the system quantum mechanically (this could use either ab initio or semiempirical methods or even some combination), while the remainder is described with a classical force field. For example, in a photoactive protein, the QM region could be limited to the chromophore. The environment does exert an influence on the chromophore, through electrostatic and steric effects.

236 However, the electrons in the environment are not treated explicitly and are effectively treated adiabatically. The basic QM/MM method partitions the molecular Hamiltonian operator as follows: HTOT — tt(jM + HgM/MM

+ ti

MM

(1)

where HQM is the usual molecular electrostatic Hamiltonian dependent on the atoms and electrons in the quantum mechanical region, HMM contains the classical force-field terms for atoms in the molecular mechanics region, and HQMIMM couples the two subsystems. The QM/MM interaction usually contains both electrostatic and van der Waals terms:

^

(2)

where /and a are QM indices for electrons and nuclei respectively, m represents MM atoms, Z are nuclear (QM) or effective atomic (MM) charges and e and a are the van der Waals parameters. Only the first double summation in Eq. (2) contains the electronic coordinates, and this can be denoted as H^M/ml. For a given set of nuclear coordinates for both QM and MM atoms, one then carries out the usual QM optimization of the electronic wavefunction using the augmented Hamiltonian HQM +H^)MIMM in place of H0M . The procedure becomes considerably more complicated if the classical force field includes explicit electronic polarization effects. It can also be complicated when the QM and MM regions are connected by one or more covalent bonds, as is often found in protein environments. A number of solutions58"62 have been proposed for this "covalent embedding" problem, beginning with the "link atom" concept of Singh and Kollman.63 Recent work compared several of these approaches and found little advantage to the more complicated approaches. However, these comparisons were limited to semiempirical methods on the ground electronic state. It is not clear whether similar conclusions would be reached for excited states and/or ab initio methods. The QM/MM methods are intended to reproduce the results of a fully QM calculation on the same system. There will however be errors stemming from inadequacy of the classical force fields and neglected effects such as electronic polarization of the MM region. Thus, it is important to quantify the accuracy of the approximation. We show one example here for an So/Si MECI geometry obtained for the neutral form of the GFP chromophore surrounded by 27 water molecules (roughly one solvation shell.) This is sufficiently small that it is possible to carry out an MECI optimization with all atoms treated quantum mechanically using the semiempirical FOMO-CI method. We have also carried out the same calculation treating all but one of the water molecules using the SPC force field.65 The chromophore and the water molecule closest to the OH of phenol are treated with semiempirical FOMO-CI. Starting from the geometry determined quantum mechanically, we searched for an MECI using the QM/MM semiempirical

237

Fig. 6. Comparison of So/Si MECI geometries obtained for the neutral form of the GFP chromophore in a cluster of 27 water molecules using QM (black) and QM/MM (grey) methods. In both cases, the QM method is semiempirical FOMO-CASCI(12/8) using AMI parameters. The MM water molecules in the QM/MM method are modeled using the SPC force field. FOMO-CI method. The resulting QM and QM/MM geometries are compared in Fig. 6. The geometries are seen to be quite similar and a quantitative comparison can be given in the root mean square deviation of the structures, which is computed to be 0.28 Angstroms. Further comparisons including energetics and topography of PESs around MECIs can be found in our previous work. Several implementations of QM/MM using ab initio and/or reparameterized semiempirical methods for the QM region in the context of photobiology have been described recently. The ONIOM method of Morokuma and coworkers has been applied to photoisomerization of the retinal protonated Schiff base (RPSB) chromophore in the gas phase.66 Olivucci and coworkers have included dynamic electron correlation in the quantum mechanical region using the CASPT2 method in a QM/MM study of RPSB in the rhodopsin protein environment.67 This study was limited to geometries near the Franck-Condon point, but a more recent study of PYP using QM/MM with CASSCF for the QM region has also included excited state dynamics.68 The Schulten group69 has investigated the excited state dynamics of RPSB in bacteriorhodopsin (bR) using a QM/MM method with a CASSCF description of the QM region.

238 Warshel has used a limited reparameterization of the QCFF/PI semiempirical method in a followup7 to earlier QM/MM work on bR using QCFF/PI without reparameterization.71 We have used QM/MM methods with a semiempirical FOMO-CI description for the QM region in both condensed phase and protein environments.42'55 We have both optimized MECI locations and followed excited state dynamics including electronic quenching. 3. EXCITED STATE REACTION PATHS AND DYNAMICS The concept of a minimal energy path (MEP) connecting reactants and products is very useful for reactions occurring on the ground state, in spite of the paradox that it is only strictly valid in the infinite friction limit where activated reactions cannot occur. Likewise, the concept remains useful for excited state reactions even though it is only a model. In particular, the ultrafast nature of many photochemical reactions leads one to expect that vibrational energy relaxation rates could be bottlenecks to reaction. Olivucci and coworkers have used the MEP to connect points expected to be important in photochemistry,15'72'73 such as the Franck-Condon point, excited state minima, and MECIs (see Fig. 1). From these MEPs, reaction mechanisms can be inferred. However, ultimately dynamics becomes important in order to determine lifetimes for connection with experiment - how long does it take to reach a conical intersection and how efficiently is population quenched at the intersection? If the dynamics is to be modeled specifically, then one immediately faces the need for a quantum mechanical, or at least semiclassical, description of the nuclear degrees of freedom. Changes in electronic state signal the failure of the Born-Oppenheimer approximation and are most easily treated within a formalism which recognizes that the electrons are quantized. There are three basic approaches to model nonadiabatic effects which can be currently applied to large molecules in condensed phases: Ehrenfest dynamics, surface-hopping dynamics, ' and multiple spawning.9'77"80 We only sketch the methods here, as they have been adequately described and compared in the literature. In the Ehrenfest method, each classical trajectory moves on a potential energy surface which is a weighted average of all the electronic states. The weights in this average correspond to the probability of being on a particular state, which is determined by solving the Schrodinger equation along the prescribed trajectory. The drawback of this approach is that the potential energy surface can be quite unphysical when several electronic states which are very different from each other are populated. This has been discussed at length by Tully and others.76 Even when the electronic states in the problem are quite different, Ehrenfest dynamics can still be reasonable as long as population is transferred between electronic states quickly. The real problem comes when a trajectory propagates on an average of electronic states for an extended period of time and these states are very different from each other. Surface-hopping ameliorates the problems of Ehrenfest dynamics by demanding that the forces governing a trajectory are always derived from a single adiabatic electronic state. However, the electronic state with which a trajectory is associated changes discontinuously.

239 Typically this "hopping" occurs according to a probabilistic rule. In the "fewest switches" version, this rule is itself derived from an Ehrenfest-like dynamics, where a Schrodinger equation is solved for the electronic amplitudes along the trajectory. The multiple spawning method starts as a basis set description of wavepacket dynamics, and thus always has access to a nuclear wavefunction representing the system. Each nuclear basis function is like a classical trajectory in that its phase space evolution follows Hamilton's equations for a specified electronic state. New nuclear basis functions are created adaptively when the nonadiabatic matrix elements that signal failure of the Born-Oppenheimer approximation become large. Instead of trajectories hopping between electronic states, new trajectory basis functions are created and these are populated according to the solution of the nuclear Schrodinger equation in the expanded basis set. Since the new trajectory basis functions always begin with no population, i.e. they are initially "virtual" basis functions, the nuclear wavefunction evolves continuously with time. In principle, the advantage of this formalism is that it is guaranteed to converge to exact solution of the nuclear Schrodinger equation as the basis set is enlarged. Although exact numerical solution of the nuclear wavefunction may not be feasible for molecules with hundreds of degrees of freedom, it is nevertheless possible to test the convergence of specific observables as the basis set is expanded. This is not possible in either of Ehrenfest dynamics or surface-hopping, which are not easily cast as members of a hierarchy of methods culminating in exact solution of the nuclear Schrodinger equation. Furthermore, the availability of a nuclear wavefunction in multiple spawning means that correlation functions and hence spectra can be computed directly from the simulation results. ' When the multiple spawning method is used in conjunction with simultaneous solution of the electronic structure for the potential energy surfaces and their nonadiabatic couplings, we refer to it as "ab initio multiple spawning" (AIMS). 4. APPLICATION TO GREEN FLUORESCENT PROTEIN Now we turn to a representative application of the methods discussed so far. The Green Fluorescent Protein83 (GFP) is found in a variety of coelenterates, including the jellyfish A. victoria. The protein absorbs blue light emitted (approx. 470nm) in a chemiluminescent reaction by the protein aequorin and emits green light (508nm). It is still not entirely clear why the jellyfish should want or need to do this wavelength conversion, but the autofiuorescent nature of GFP has made it a useful tool in molecular biology. The GFP chromophore is formed autocatalytically from a Ser-Tyr-Gly tripeptide after expression and folding under aerobic conditions. Since there is no need for an external cofactor, chimeric proteins can be engineered such that GFP is attached to a desired protein and becomes a fluorescent probe reporting on its expression. Curiously, the chromophore of GFP (p-hydroxybenzylidene-imidazolinone, see Fig. 5) has been established to be virtually non-fluorescent when it is either synthesized or when the protein environment is disrupted. This raises the question of the mechanism for nonradiative

240 decay in the solvated chromophore and the means by which the protein prevents this decay. It is further known that the photocycle in the protein involves proton transfer,84 so that at least two protonation states of the chromophore are important. Early semiempirical studies were not very successful at determining which protonation states were involved,85'86 but the experimental evidence favors the neutral and anionic (phenolate) forms.87"90 Fluorescence is observed primarily from the anionic form and mutant GFPs have been engineered which stabilize the anionic form in order to increase fluorescence quantum yield.91 Previous theoretical work of Weber, et al. using semiempirical methods led to the suggestion that nonradiative decay occurred through conical intersections associated with torsion about one or both of the bridge bonds connecting the phenol (P-bond) and imidazolinone (I-bond) rings to the central carbon atom.92 These calculations also suggested the possibility of "hula twist" motion,93"95 i.e. concerted torsion about the I- and P-bonds, as an important coordinate leading to So/Si degeneracy. A number of subsequent theoretical studies concentrated on possible tuning of the electronic absorption of the GFP chromophore by the protein environment17 and also on the vibrational spectroscopy,96 partially in response to the early semiempirical work which shed doubt on the assignment of the chromophore protonation states involved in the photocycle. The most recent work using CASSCF and CASPT2 has focused on the anionic protonation state and suggested that while hula twist motion is disfavored on Si in the gas phase, it does lead to an So/Si MEC1 and was the only explanation which could be found for the nonradiative decay.97 As will be seen below, our calculations suggest that there are So/Si MECIs very close to two distinct but nearly-degenerate global minima on Si, which would explain the lack of fluorescence in the chromophore. What is more, solvent accelerates this decay significantly. As suggested in Fig. 5, photoexcitation of the neutral GFP chromophore is expected to lead to torsion about the "1-bond." Using the SA-2-CAS(2/2) method with a 6-31G basis set and searching for a Si/So MECI from the Si global minimum (SiMinG, twisted by 90° about the Ibond) leads to a conical intersection which is 7 kcal/mol above SiMino- The molecular geometry of the So/Si MECI is shown in Fig. 7, along with geometrical parameters for both Si Mine and the MECI. Torsion about the "P-bond," the bridge bond adjacent to the I-bond, is not favored in the neutral chromophore as might have been expected due to the propensity for bond alternation in the bright excited state of conjugated molecules. This situation changes in the anion form of the chromophore, where two twisted local minima exist one for each of the I- and P-bonds. These are essentially isoenergetic, and again the So/Si MECI geometries are energetically above the corresponding twisted Si minima. A schematic of the important points on the PES is shown in Fig. 8. Energetics obtained with both SA-2-CAS(2/2) and SA-2-CAS(2/2)-PT2 are shown, using the 6-31G* basis set. These results confirm the suggestion of Weber, et al. that torsion represents a viable pathway for nonradiative decay, although we should point out that the picture is quite different in details. Weber, et al. did not find a torsional deactivation pathway for the anion form of the chromophore which was energetically accessible from the Franck-Condon region, and even the inaccessible pathway which they found required simultaneous torsion of the I- and P-

241 bonds. However, these disagreements are not surprising in view of their use of an uncalibrated semiempirical method.

Fig. 7. The imidazolinone-twisted minimal energy conical intersection for the neutral form of the GFP chromophore. Included are heavy-atom bond lengths (A), bridge dihedrals and interior heavy-atom angles for the So/S| MECI and for the imidazolinone-twisted SiMinG (in parentheses). The relevant structures were optimized on S, using an SA2-CAS(2,2)/6-31G wavefunction.

242

Fig. 8. Fluorescence lifetime of neutral and anionic forms of GFP chromophore as predicted by CAS(2/2)/6-31G dynamics. There are four trajectory basis functions corresponding to the neutral form and three trajectory basis functions corresponding to the anion. The lifetime of the neutral is slightly shorter and the amplitude of the decay is larger. Larger sets of initial conditions are needed to make firm conclusions, but the results are in agreement with solution phase experiments. The immediate question which follows the preceding observations is whether there are any barriers to torsion in either of the neutral or anionic forms of the chromophore. Recent experiments have measured the fluorescence lifetime of both the neutral and anionic forms in different solution environments. The authors find a multiexponential decay of fluorescence with the shortest lifetime ranging from 70fs to 460fs. This suggests that any barrier which does exist must be quite small, at least in the solution environment. Furthermore, the lifetime of the neutral is found to be consistently shorter than the anion. We have carried out AIMS simulations to investigate this, using the SA-2-CAS(2/2) electronic wavefunction in the 6-31G basis set. Initial conditions are sampled from the Wigner distribution corresponding to the So minimum in the harmonic approximation. The resulting fluorescence is shown in Fig. 9, which is averaged over the trajectory basis functions used to represent the initial state. The neutral and anion are seen to be quite similar, with fluorescence from the neutral form decaying slightly faster. This is in broad agreement with the experimental results, but one should note that these simulations are in the gas phase and furthermore that the number of initial conditions used is not yet sufficient to

243 draw firm conclusions regarding the lifetime. An important point to note is that the observed fluorescence decay comes strictly from torsion of the chromophore and not from electronic quenching. As expected, the So/Si transition dipole moment governing the fluorescence intensity becomes very small when the molecule is twisted about either of the I- or P-bonds. Although Fig. 9 does include the proper weighting of the electronic state population, i.e. each trajectory basis function is weighted according to its population on Si, there is very little (< 10%) quenching to the ground electronic state in the time shown.

50

100

150

200

250

300

350

Fig. 9. Fluorescence lifetime of neutral and anionic forms of GFP chromophore as predicted by AIMS dynamics using CAS(2/2)/6-31G for the electronic structure. The square of the S]/So transition dipole moment at the center of each nuclear basis function, averaged over all trajectory basis functions, is shown. There are four trajectory basis functions corresponding to the neutral form and three trajectory basis functions corresponding to the anion. The lifetime of the neutral is slightly shorter and the amplitude of the decay is larger. Larger sets of initial conditions are needed to make firm conclusions, but the results are in agreement with solution phase experiments.

244 Thus, the decay of fluorescence should not match the rise time which would be measured by probing ground state recovery. However, we cannot say yet how much different these lifetimes should be, except to place a lower bound of lOOfs on the difference. Is there any significant effect from the solvent environment? In order to investigate this question, we turn to reparameterized semiempirical methods within which we are able to include an environment explicitly. Only the neutral form of the chromophore will be addressed here. We reparameterized the FOMO-CI semiempirical method for the GFP chromophore, using ab initio results for the calibration data.55 We refer the reader back to Fig. 5 for an example of the quality of the resulting PESs. Similar results are obtained for the neutral chromophore. The solvent environment is modeled using the SPC force field representation of water,65 with 51 water molecules surrounding the chromophore. Carrying out AIMS simulations with the reparameterized semiempirical FOMO-CASCI(12/8) method leads to the lifetimes shown in Fig. 10. These are now reflective of the population on Si, and cannot be directly compared to the fluorescence lifetimes shown in Fig. 9. What is initially striking about this data is the large decrease in excited state lifetime which is observed on solvation. Further investigation shows that this is a consequence of a significant change in the PES of the chromophore in solution as compared to the gas phase. In particular, the So/Si MECI in the gas phase is located approximately 7 kcal/mol above the I-bond twisted global minimum. Upon aqueous solvation, the So/Si MECI becomes the absolute minimum on Si, i.e. the Si Mine and So/Si MECI points become identical. Thus, access to the MECI becomes nonactivated, even for molecules which are able to dissipate energy rapidly. This is a consequence of the charge-transfer character involved in the states which are connected at the MECI, and is expected to be a general phenomenon. The lifetime we observe in the aqueous environment is in good agreement with the 70fs short time constant measured for the neutral chromophore in water, providing considerable support to this picture.

245 I

I

i

I

i

I

^Vacuum 0.8

-

^

-

^

^

^

O

0.6

-

Water

Q.

O CL

0.4

-

-

0.2

-

-

n

50

I

I

I

I

I

100

150

200

250

300

350

Time / fs Fig. 10. Population on Si as a function of time for neutral GFP chromophore in vacuum and water environments. The chromophore in both cases is modeled with the reparameterized semiempirical FOMO-CASCI(12/8) method. Water molecules are treated using a QM/MM method with the SPC force field. Finally, one can ask how the protein environment affects the excited state dynamics. Several X-ray structures are available99"101 and we use the wild-type A. victoria structure in this work. A truncated model was constructed which includes all residues within a 10 Angstrom radius of the phenol moiety of the GFP chromophore. This is depicted in Fig. 11, along with a cartoon representation of the GFP protein which shows the novel "y?-can" fold. We have used the

246

Fig. 11. Left: cartoon representation of the GFP protein, showing the chromophore buried inside the "(5can" structure. Right: Computational model used as a representation of the protein environment. The model has a total of almost 500 atoms, of which 39 are treated with the FOMO-CI method and the remainder with the AMBER force field. AMBER force field1 to represent the surrounding protein residues and the reparameterized semiempirical FOMO-CI method used for the chromophore. The chromophore is connected to the protein residues using "connection atoms" which have been reparameterized in order to reproduce the purely QM results for excitation energies in a smaller model including only the nearest protein residue explicitly. Our treatment of these connection atoms follows the strategy set forth by Antes and Thiel,103 and these connection atoms must be reparameterized (or at least the parameterization must be verified) for each new system. We first investigate the MEP for the chromophore in order to compare the vacuum, solvated, and protein environments. A simple way to generate an approximate MEP is by carrying out dynamics in a high-friction environment. We do this by using a Langevin dynamics:

dE(r) Pa =

~ d

YaPa

(3)

wherep a =mara is the conjugated momentum of the ra coordinate and ma is the associated mass. The usual Langevin equation also contains a term representing the random energy exchange between solute and solvent, which we ignore since our purpose is just to generate an

247 approximate MEP. In principle, the friction coefficients ya are related to the molecular diffusion coefficient D: D

-

^

-

(4)

2 However, for our purposes they do not have to represent any real solvent and we chose ya = 1014 sec"'. The numerical implementation has been coded in a development version of the MOPAC package, along with the other methods we describe in this paper.104 The resulting MEP is depicted in Fig. 12 for the neutral GFP chromophore in isolated, water, and protein environments. The distinction between the vacuum and aqueous environments we mentioned earlier is immediately evident. In the vacuum case, the energy gap between So and Si remains large when the minimum on Si is reached. In contrast, the So/Si gap vanishes even before the minimum on Si is reached for the aqueous environment. Thus, the chromophore is directed to a conical intersection seam and follows the seam toward the absolute minimum, which is itself a conical intersection. The rapid excited state decay in the aqueous case is thus easily explained. In the protein environment, the chromophore geometry does not change very much before a minimum is reached. This minimum is planar and is expected to fluoresce. Hence, the protein environment has induced a barrier to torsion which traps the chromophore in a fluorescent state. We have not yet determined the height of the barrier and dynamics simulations are underway to determine what fraction of the excited chromophores in the protein environment are trapped in this fluorescent local minimum. A further interesting question is the origin of the barrier, which could be dominated by steric or electrostatic effects.

248

Vacuum


& CD C

LLJ

10

15

20

25

30

35

40

Reaction Coordinate / amu1'2 Angstrom Fig. 12. Minimal energy path for the neutral GFP chromophore on Si in different environments. The water environment contains 51 water molecules modeled with the SPC force field. The protein environment corresponds to the truncated model depicted in Fig.ll, where the protein surroundings are modeled with the AMBER force field. In order to better compare the results, the reaction coordinate includes only the motion of the chromophore atoms, excluding the MM environment.

249 5. CONCLUSIONS We have provided an overview of methods to determine reaction mechanisms for photobiological systems. Early efforts using empirical potentials or restricted semiempirical methods have not been emphasized. Neither have we discussed in any detail the determination of reaction paths in isolated molecules of biological significance. Instead, we have focused on emerging methods which can include the environment in some detail while at the same time being able to provide a flexible and accurate description of the chromophore. Multireference electronic structure theory methods combined with classical force fields in QM/MM fashion seem to be the preferred way of simulating these systems. The methodology is still developing and the number of applications to date is rather few. However, there is every reason to believe that this will be a rapidly-growing field over the next few years and that we are on the verge of a new level of accuracy in photobiological simulation. Indeed, we have presented some applications where not only reaction paths but also dynamic evolution has been modeled, including quantum mechanical effects of both electrons and nuclei. Using GFP as a model application, we have shown how it is possible to characterize reaction paths and follow dynamics in a chromophore from the gas phase through to solution and protein environments. Electrostatic effects of the environment can change the photochemical outcome dramatically, providing an order of magnitude decrease in the excited state lifetime for the case of GFP chromophore in an aqueous environment. We have shown that our approach does predict a barrier to torsion in the protein environment, and therefore an expected increase in fluorescence in agreement with experiment. Future studies will characterize the nature of this barrier and quantify its effect on the excited state lifetime. The next step, which can be expected in the near future, is to begin designing complex environments for photochemical reactions. As the accuracy of the methods becomes quantified through widespread application and comparison to experiment and as computational power continues to increase, this will be a realistic and useful goal. ACKNOWLEDGEMENTS This work has been supported by the National Science Foundation and the Department of Energy. TJM is grateful to the Packard and Dreyfus Foundations for support through a Packard fellowship and Teacher-Scholar award, respectively.

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M. Olivucci (Editor) Computational Photochemistry Theoretical and Computational Chemistry, Vol. 16 © 2005 Elsevier B .V. All rights reserved

255

VIII. Development of Theory with Computation Howard E. Zimmerman Chemistry Department University of Wisconsin 1. INTRODUCTION - EARLY EFFORTS; THE BACKGROUND The present chapter describes the development of theory in our laboratory from the primitive beginnings at the outset to our present efforts. This includes both computational and theoretical aspects. The present author's interest in quantum mechanical and photochemical organic research began with his reading publications by Egbert Havinga in 1956 [1] and by Derek Barton in 1958 [2]. Thus, Egbert Havinga reported the curious behavior of p- and m-nitrophenyl phosphates in water. The para isotner hydrolyzed reasonably rapidly independent both in the dark and in light while the meta isomer was highly reactive in light but unreactive in the dark.

m NO2

hv H2O

(2) — No Acceleration Relative to the Dark Reaction

This solvolysis behavior on irradiation, of course, was in contrast to what we teach our undergraduates about ortho-para electron transmission. Egbert Havinga noted that there was no rationale in mechanisms of the time using qualitative resonance theory. It was clear to the present author that the limitation in understanding the phenomenon was not knowing the structure of the excited state of the phosphate esters.

256 Similarly, two years later Derek Barton had elucidated the structure of Lumisantonin, the product resulting from irradiation of Santonin. At the time, the best mechanism for this bizarre rearrangement (Note Eq. 3) consisted of "bond switching" which indicated which bonds were lost and which were gained. Again, it was clear that what was needed was a description of the excited state undergoing the reaction. But at the time there had been no real attempt to relate photochemical transformations to excited state structures.

These two examples - the m-nitrophosphate solvolysis and the Santonin to Lumisantonin rearrangement - which were typical of the time prompted the present author to investigate whether determination of the excited state structures involved would lead to an understanding of the experimental photochemical reactivity. 2. OUR EARLY EFFORTS ON THESE TWO PROBLEMS However, at the time, the best computational facility available was the IBM 1604 and the programming available was the Hiickel treatment employing a Jacobi diagonalization. Our first computation was for the benzylic species. With eight electrons this is isoconjugate with anisole, and the aim was to understand what transmission of unshared electrons occurs from an electron donor group, such as methoxy. Thus, Scheme 1 shows the local densities in the seven n MO's of the benzyl species. The ground state is populated as depicted with solid dots. The excitation process, in simplistic homo to lumo form, is represented by arrow B. It is seen that there is a large enhancement of the meta electron density by promotion of an electron from an MO which has no meta density to one which does.

257

+2.10

+ 1.26

T

,57

4^>.1 *14

o

-2.10

Scheme 1. Densities of Benzyl Species MO's. All solid arrows represent electrons in the case of an electron donor. Hollow arrows represent electrons in the case of a withdrawing group. In contrast, there is diminution of the para density. The simple Hiickel computation for anisole paralleled the computation for the isoconjugate benzylic anion as seen in Scheme 2. Thus we proposed the phenomenon of "meta electron transmission" in the first excited state which contrasts with the well-established ortho-para transmission so characteristic of the ground state chemistry.

Ground State

Scheme 2. Densities for So and Si Anisole

0.76 Excited State

258 With two meta methoxyl groups, the effect is enhanced and this led to study of the solvolysis of meta-methoxylated benzyl acetates. We discuss this chemistry below in the context of subsequent and more sophisticated computation of benzylic systems. With the Havinga report in mind, we again considered Scheme 1 but now with just six n electrons, since this corresponds to an aromatic ring with an electron withdrawing group. In this case, excitation corresponds to arrow A with the consequence of loss of meta electron density. The para site actually has an increase in density on excitation. Hence, again, we see meta electron transmission in the excited state. Having considered the basis of the Havinga phenomenon we turned to the esoteric rearrangement of Santonin described by Derek Barton (Note Eq. 3 above). A beginning point was the nature of the excited states of carbonyl compounds. While at the time the nature of the n-7t* and 71-71* excited states of formaldehyde, acetone and similar carbonyl compounds was in the physical chemical literature, organic chemists either were not aware of this or had not made the connection to organic photochemistry. The physical chemists, while certainly aware of these excited states, were not in a position to deal with the organic chemistry. Thus we noted that the three-dimensional n-Tt* excitation process (Scheme 3) provides two orthogonal systems with the ground state in electron distribution. First, the py (or n) orbital has lost an electron and is reminiscent of the orbital in oxy-free radicals. Second, there is a TC system that has an added electron and is isoconjugate with a metal ketyl (i.e. a radical-anion). It was suggested that, while thinking in three dimensions, the short-hand notation shown in Scheme 3 permits the organic chemist to draw excited state structures quickly. It was also suggested that excited state reactivity is guided by the reacting molecule selecting "energy valleys" just as observed for ground state reactions. Thus, the organic chemist's electron pushing may be applied to species on an excited state hypersurface. Thus, the Norrish II was described as in Scheme 3 in those early papers. It is seen that the n-Tt* excited state has an "electron hole" in the py orbital and this orbital is coplanar with the sigma bonds to the carbonyl carbon. With overlap between the py orbital and one of the sigma bonds, we see a system of three coplanar orbitals - the p y orbital and the two comprising the sigma bond. This linear system of three orbitals and three electrons is isoconjugate with the allyl radical species. The bond order between basis orbitals 1 and 2 in allyl is lower than unity; similarly, in the n-Tc* excited state here, delocalization of the electron-hole weakens the bond holding the alkyl group R with resulting scission and release of the alkyl free-radical (Scheme 4). This is one example of the role of the pv orbital in sigma system n-Tt* reactivity.

259

R R

hν O yy° o

R R

•• Oo y

SP HYBRID ELECTRONS

•

7i ELECTRONS

y

p y ELECTRONS

Scheme 3. n-jt* Excitation

Scheme 4. Acyl fission of an Alkyl group in the n-n* Excited State. Turning now to the Santonin to LumiSantonin rearrangement (Eq. 3), we found it easier to begin with 4,4-diphenylcyclohexadienone in Scheme 5, since this would provide new photochemistry and yet had the same 2,5-cyclohexadienone electronics. The proposed mechanism [3c] is outlined in Scheme 5 using the short-hand notation described above. The basic transformation involves four steps after excitation to S,.

260

p,(3-Bonding

hv, ISC

ISC

,© Pri

1,3-Shift

Ph

Scheme 5. The Type-A Rearrangement of 4,4-Dipheny lcyclohexadienone. The first is intersystem crossing to Ti. The second is bonding between the beta carbons of the dienone system. The third is radiationless decay with intersystem crossing to So in the form of a zwitterion. And the last is a cyclopropyl-carbinyl rearrangement involving a 1,3-shift. The beta-beta bonding found justification in simple Hiickel computations [4]. Note Scheme 6. While the beta-beta bond order in the ground state is negative, that for the singly promoted dienone is positive and bonding. This indicates that a perturbation with increased beta-beta bonding will lower the energy.

O -0.295 +0.173 +0.007 +0.053 P,p-bond order-0.070 i.e. antibonding

GROUND STATE

+0.419 -0.109 | -0.023 -0.131

EXCITED STATE

P,p-bond order+0.114 i.e. bonding

Scheme 6. Hiickel Computation of 4,4-Disubstituted 2,5-Cyclohexadienone; Electron Densities and Bond Orders

261 Application of the same mechanism to the Santonin to LumiSantonin rearrangement, not only leads to the correct product structure but also accounts for the stereochemistry. In this early work [3] we showed how the above type of reasoning accounted for the major fraction of photochemical reactions known at the time. This included the Norrish Types I and II, the Yang Reaction, a-expulsion of groups adjacent to carbonyl moieties, epoxyketone rearrangements, the Paterni-Biichi reaction, and the "Type-B Enone Rearrangement" which we had just discovered [5]. 3. THE MOBIUS-HUCKEL CORRELATION DIAGRAMS

CONCEPT

FOR

TRANSITION

STATES

AND

Hiickel computations proved useful again in 1966 when we proposed that pericyclic systems consisted both of the usual "Hiickel" variety and also of "Mobius" topology. See Scheme 7. At the time we derived [6] a circle mnemonic for Mobius systems, quite parallel to the FrostMusulin [7] for the Hiickel systems. We showed that for reacting molecules of the pericyclic variety, at half-reaction one had either a Hiickel (zero or an even number of basis set +/- overlaps around the orbital array) or a Mobius orbital array (with an odd number of +/- overlaps). The circle mnemonic afforded the MO eigenvalues and indicated the degeneracies. For each degeneracy one has a crossing of MO's. This then permits one to draw the correlation diagram easily. Hiickel computations confirmed the validity of the derivation.

: +1.00

+2.00 ' 0.0

o.o-2.00

-1.00

Hueckel Cyclopropenyl

Moebius Cyclopropenyl

MO 1 MO's1&2 Crossing ALLOWED WITH 2 ELECTRONS

ALLOWED WITH 4 ELECTRONS

Scheme 7. Example of Mobius-Huckel Determination of Degeneracies and MO Crossings Along the Reaction Coordinate.

262 This work noted the role of such degeneracies in giving conical intersections leading S, to So. In fact, it was in 1961 that the first organic reaction correlation diagram was put forth [8]. This was for the 1,2-shift of groups in the Grovenstein-Zimmerman Rearrangement of carbanions. There proved to be a real contrast between the 1,2-shift of an alkyl compared with a phenyl with one occupied bonding MO becoming badly anti-bonding at half-migration. With phenyl migrating, for several cases, that MO became only slightly antibonding. The MO's were drawn and the correlation was in words rather than being explicitly depicted. Also, in this study hybrid orbitals were required. These were obtained analytically and then used as a basis in the Huckel computation. This early philosophy used a truncated basis set consisting of p-orbitals and hybrid orbitals. The set selected was that which included basis orbitals involved in excitation and reaction for photochemistry and those sigma hybrids which were perturbed in the reaction studied. Thus far we had relied on Huckel (one-electron) computations. A comment is required regarding this. First, at the time, available computer facilities permitted nothing more elaborate. However, a further point is relevant. This is that Huckel theory is mathematically exact; it is identical with graph theory. The eigenvalues and eigenvectors for a set of points in space are fixed and determined by the topology of the species in space. This is independent of whether the figure subject to computation is a real molecule or a mathematician's graph. What is approximate are the assumptions that a one-electron operator is sufficient and ignoring electron-electron repulsion and exchange effects. Additionally, in our use of Huckel methodology the basis orbitals, such as p-orbitals and, possibly hybrids, are parametrized only roughly. Nevertheless, molecular topology seems most often to provide a semi-quantitative, or at least a qualitative guide, to molecular structure. Thus, while more sophisticated quantum mechanics can lead to precise numerical results, the simplicity of Huckel (graph) theory most often provides a guide to expectation of what greater sophistication will afford. Further, often a pad and pen will permit a quick assessment of molecular properties. 4. SUBSEQUENT DEVELOPMENT; EARLY SCF-CI COMPUTATIONS However, in about 1967 the author had purchased a PDP-8/I minicomputer. This was followed by a more powerful PDP-11/55 model. At the time we wrote our own programming which was based on a single Slater determinantal wavefunction for So. For Ti and S, the wavefunction was as in Eq. 4 (i.e. minus for S, and plus for Tj, all normalized). Here K refers to a bonding MO and L to an antibonding one. M O , ( 1 ) " M O 2 ( 2 ) ( i -• M 0 s ( N - l ) a M O

I l

( N ) ( ' | ± |MO, ( 1 ) " M O 2 ( 2 )

p

' MOK

(N-l)

P

MO,, (N) "| ( 4 )

263 Similarly, matrix elements between singly excited configurations were used. Thus configuration interaction was possible. We used an active space of thirteen or less and zero differential overlap. However, with limited memory, we used our hard-drive disk for virtual memory [9]. The computations were of K systems and included the py orbital with integrals taken from literature values. The main application was in obtaining wavefunctions, electron densities and energies of excited states. Interestingly, in the case of the 4,4-disubstituted cyclohexadienones, the beta-beta bonding of the n-n* triplet was positive and thus bonding, while that of the K-K* triplet was negative and antibonding. The earlier Huckel computations, of course, were spin-independent but similarly had predicted the positive beta-beta bond order. One rearrangement of particular interest was the Di-Tr-Methane Rearrangement of barrelene to semibullvalene [10]. The organic mechanism is given in Eq. 5. In 1967 the best treatment of the reaction we had made use [10] of extended Huckel computations [11]. The coordinates used were obtained by mechanical measurement. Remarkably, the basic surface obtained (note Scheme 8) is qualitatively similar to that obtained from our more sophisticated, subsequent ab initio computations subsequently (vide infra).

(5)

Acetone Barrelene, So

Dirad 1,T,

Dirad 2, T, Relaxed

Semibullvalene, So

Scheme 8. Extended Huckel Hypersurface. Circular dot for So and solid dots for the excited state.

264 5. SINGLE PHOTON COUNTING COMPUTATIONAL CHALLENGE

COMPUTATIONS

AS

ANOTHER

One other use of computation at that time was in our single photon counting measurements of very rapid excited state fluorescent decay [13]. In this we obtain the profile of the excitation lamp flash as shown in the I matrix in Eq. 6 with the subscripts referring to consecutive decay times. Also we get the profile of the fluorescence emission as in the E vector [Eo Ei E2]. The example given here has only three data points for simplicity; commonly there would be 512 or more. The D vector consists of the decay function values at the increasing time intervals. Io 0

0

I, Io 0

h h

Ir

Do D, D7

(6)

The decay function might consist of a simple negative exponential, ae"kl, or a sum of exponentials. Since the rate constant k is desired, it is the vector D which is the main unknown and wanted. However the lamp decay vector I matrix is singular due to the zero value of Io, and we cannot directly solve Eq. 6 for the D vector. However, we can make an initial assumption of the value of k, and a, and determine the error in the E vector obtained relative to experiment. By solving for the derivatives of the error with respect to k and a, we can determine what changes will minimize the error. With new values of k and a in the vector D, we can again determine the errors in the E vector. Our programming continues this iteratively until there is convergence. Thus, rather than a deconvolution, the method performs a convolution. With more data points than unknowns, the method is both accurate and useful. It also works with other decay functions. 6. A CURIOUS REACTION; A DIRADICAL REACTING IN TWO WAYS USES TWO STATES One unusual set of reactions is shown in Scheme 9 [14]. The reaction of the benzodiene A afforded photoproduct F while irradiation of F led to photoproducts C and A. Clearly a diradical of structure D must be involved in both irradiations. A superficial paradox is that a common species D led in two directions depending on the reactant. SCF-CI computations, summarized in Fig.s la and lb and in the notations in Scheme 9, clarify that there are two states of D which are involved. So leads selectively to A and C while S, affords product F. What is also revealed by the computations is that a conical intersection, or avoided crossing occurs in the D to F conversion. This is seen both at the MO as well as the state level; note Fig.s la and lb.

265 Another interesting aspect of the photochemistry in the F to C conversion is the dimethylcarbon of the three-membered ring moving along the surface of the % system. If one considers the hybridization of the three-membered ring as having two-sp5 orbitals aimed at the n-surface, this is then seen reminiscent of bicycling along the surface wherein the sp5 orbitals are the "bicycle wheels". In fact, this is a type reaction seen earlier in our research. Thus, a number of examples of the bicycle rearrangement are known [15]. One example is given in Eq.s 7a and 7b. The reaction is largely stereospecific with the Exo reactant giving the Anti-spiro product and the Endo reactant giving the Syn-spiro product. From the Endo-Bicyclic reactant there is a minor loss of stereochemistry. The topology of the reactions is depicted in Fig. 2. It is seen that the minor course (B) accounts for loss of complete stereospecificity.

Scheme 9. Rearrangements via Diradicals of Common Structure but Differing in States. Empty arrows for pathways originating from D as Si and dashed arrows for pathways originating from D as So.

266

Di-π A

-562

Spiro C Bicyclic F

-562

Fig.s la and lb. Correlation of MO's (in la) and States (in lb).

Ph

Ph

hu

H

hu (7a.b)

Ph'

Ph'

Exo-Bicyclic

Anti-Spiro (Major Product)

Endo-Bicyclic

Syn-Spiro

(Minor Product) from the Exo)

Fig. 2. Bicycle Topology; major (A) and minor (B).

267 Interestingly, minor amounts of benzenoid by-products derive uniquely from intermediate biradical intermediates along the topology shown. Thus the 2,5-diphenyltoluene, 3,4diphenyltoluene and 2,4-diphenyltoluene formed in minor amounts, give evidence for the reality of the three diradicals shown in Fig. 3.

From Main Path A

From Minor Path B

Fig. 3. Bicycle Derived Diradicals and Their Products. These 1,4-diradicals tend to undergo a Grob fragmentation, and we have shown the Grob fragmentation to occur only from So and not from S| diradicals [16]. Conical So-S, intersections were found for the diradicals by use of the SCF-CI computations discussed above. As a consequence we can conclude that the bicycle rearrangement occurs from Si diradicals D. Some of those subsequent diradicals which decay, rather than proceeding onward to the spiro products, are lost to the benzenoid by-products. Interestingly, the intermediate Diradicals, such as these in Fig. 3 and also D in Scheme 9, are the same species encountered in the Di-jt-Methane Rearrangement. In Scheme 9, the Si state of D actually comes from the Di-7t-Methane reactant A. The Di-71-Methane Rearrangement originally was discovered in the barrelene to semibullvalene rearrangement [10] as noted above. However, this case uses the triplet. Also, as noted, it utilizes diradical species, the cyclopropyldicarbinyl diradical, which are common to the bicycle rearrangement. Thus our SCF-CI computations have been applied to this rearrangement independently [14,15a, 15b, 16,17,]. The triptych mode of display of correlation diagrams has

268 been employed in a number of these studies. In these, conical crossings and/or avoided crossings have been uncovered of the type discussed for one example above. One example of the Di-71-Methane Rearrangement is shown in Eq. 8 [15c]. The regioselectivity favoring ring opening process "a" over "b" of the cyclopropyldicabinyl diradical is of intrinsic interest. The computations done on this system were carried out with just one carbomethoxyl group and, in place of the geminal diphenyls, we used single phenyl groups. The lower energy energy path corresponded to ring opening process "a" and also led to a conical intersection. Interestingly, the triplet process opening bond "b" rather than "a", and this was ascribed to the larger exchange integral for that process.

Ph

Ph Ph C O , Me CO,Me

7. QUANTITATIVE METHODOLOGY

Ph

SOLID

CO 2 Me

STATE

CO 2 Me CO 2 Me

STUDIES

AND

Ph

Ph

CO2Me CO 2 Me

COMPUTATIONAL

Solid state photochemistry had been known for a long time often to afford photoproducts which most often differ from those formed in solution [18]. Our aim was to put this photochemistry on a quantitative basis. With the coordinates of all atoms known from the X-ray data on each crystal, it was possible to define what we termed a "mini-crystal lattice", a portion of the crystal lattice large enough to be of use but small enough to permit computation. Our initial efforts utilized molecular mechanics (MM3) to assess overlap of alternative reaction intermediates with their neighbors and to determine the energy of the mini-lattice overall. A schematic mini-lattice is depicted in Fig. 4.[19,20] However, it was clear that molecular mechanics would include only steric but not electronic effects. One might argue that it is heresy to utilize MM3 with open-shell species such as triplet diradical intermediates. Thus an alternative was devised. A simple program, "Pairs", was written [21]. This obtained all distances between pairs of atoms, one atom in the reacting species in the mini-lattice and one atom in the surrounding molecules. These distances were then sorted with distance increasing. A cut-off was then selected and all the more distant atoms of the neighbors were computationally annihilated. This left neighboring atoms with "dangling" free valencies. The hydrogens were therefore computationally changed to helium atoms and the carbons, oxygens and nitrogens were changed to neons. Then the reacting intermediate imbedded in the inert gas shell was subjected to a Gaussian98 [22] ab initio computation with the shell kept fixed

269

/ • •

.-• /

/

/

/

,-".-'/

Fig. 4. A Mini-Crystal-Lattice. Each "R" represents a reactant molecule and "I" is an intermediate leading to one of the possible photoproducts. Other computational methods were least motion and molecular volume increase in formation of either product or the intermediate. These alternatives did not not correlate well with experiment. and the intermediate permitted to geometry optimize. These results, while including electronic effects, nevertheless led qualitatively to the same predictions of preferred reaction pathways. Nevertheless, this "inert gas shell" approach was of some interest [21]. One thing this does demonstrate is that steric influences are dominant over electronic factors and that in predicting the reaction course in the crystalline state, molecular mechanics approximation was not unreasonable. Nevertheless, one would like to explore this question further. At the time, Morkuma's Oniom QM/MM programming [23] had become available in Gaussian 98. We used this [24] with three concentric shells. The inner-most shell had the reacting molecule and was subjected to ab initio, Gaussian98 geometry optimization. The next layer was treated with molecular mechanics and also permitted to geometry optimize to account for relaxation effects. The outer-most layer was also treated with molecular mechanics but kept rigid. Additionally, the size of the middle layer was varied to ascertain how many molecules were needed in this shell for relaxation to lower the overall energy. It turned out that about one layer of molecules is really important. It was found that this ab initio/molecular mechanics treatment was in qualitative agreement with the original molecular mechanics and the subsequent inert gas shell methods. With these methods available we proceeded to study host-guest complexes [24] as well as further phenomena. One particularly exciting result was the finding that solid-state photochemistry proceeds in stages. [25] At a given point, commonly when each reactant molecule has one product molecule adjacent, there is a phase change and new products arise [26]. Often these are products not obtainable in other ways.

270 8. MORE SOPHISTICATED EFFORTS ON THE META-EFFECT. Our early work on the meta-effect had been limited by available computer methodology of the time. With Gaussian98 available, we carried out ab initio computations for S, of p-methylbenzyl acetate, m-methoxybenzyl acetate and 3,5-dimethoxybenzyl acetate systems at the CASSCF(8,8)/6-31G* level. Computations were carried on the ion-pairs and on the radical pairs. The separate cations and acetate energies were obtained and a method was devised for obtaining the ion-pairing energy. This total was compared with the sum of the benzylic and acetoxy radicals. The benzyl species were taken in the first excited state (i.e. S, and Di) while acetate anion and acetoxyl radical were taken in their ground states. The energetic preference was for heterolysis rather than homolysis. It was found that excited state heterolysis to form anion pairs is favored over homolysis to form a radical pairs in the case of the meta isomers. Also for heterolysis meta substitution is favored over para. Additionally, for homolysis, p-substitution leads to a slight energy lowering, in contrast with experiment. In the case of the 3,5dimethoxybenzyl cation a conical intersection of S, with So was encountered. [27] 9. THE DELTA DENSITY PREDICTION OF GROUND STATE AND EXCITED STATE REACTIVITY. The Delta-Density method was developed [28] to predict reactivity. Originally the method was applied to photochemistry but proved more general. The basic idea is to utilize the density matrix derived for a molecule, A, and then to consider that molecule perturbed in some fashion - by introduction of a photon, an electron to afford a radical anion, by loss of an electron to give a radical-cation, or by some small molecular deformation. The density matrix of the perturbed molecule, B, is also available from computation. Then the delta-density matrix is defined as in Eq. 9.

AB

or

DM D l 2 D r , D21 D 22 D 21

Dn D,2 D n D21 D, 2 D 2 3 D 3 , D32 D33

AD, | AD,2 AD,3 AD21 AD 2 2 AD 23 AD 3i AD 32 AD 33

AD r t = D b l t S b r t -D a r t S a r t

D3, D32 D33

B

(9a)

A

(9b)

271 For this we use the Weinhold natural hybrid orbitals [29] as our basis and to account for basis orbital orientation, the overlap between the hybrids r and t are included. The basic philosophy of the Delta-Density method is that the off-diagonal density matrix elements are effectively bond orders. On perturbation of molecule A, without geometry change, the resulting species B is formed with a number of bonds with less than ideal bond lengths or valence angles. These correspond to negative elements in the delta density matrix and to bonds which can relax by stretching. Positive elements indicate molecular relaxation by bond formation. Thus, one can predict which bonds are likely to be broken; these are the most negative ones. Similarly, new bonding is likely to occur with strongly positive delta density elements. In Scheme 10 there are listed fifteen reactants for the more common photochemical reactions. The most negative delta density matrix elements are labeled "a" and the next most negative elements are labeled "b". Near zero elements are labeled "c", while positive off-diagonal elements are labeled "f". One center positive diagonal elements are labeled "h". In each case, the bonds labeled "a" are, indeed those which are severed in the photochemistry. Thus, cyclopropyl ketones, as 1 and 6 undergo ring opening of with scission of the a-fS bond to the more substituted a-carbon. Ketones (e.g. 2, 7 and camphor 12) undergo the Norrish Type I cleavage. The cyclobutanones 3 and 8 undergo the Yates ring expansion to an oxacarbene.

t-Bu

/b

11

Scheme 10. Excited State Delta Density Predictions positive element.

272 The reaction has part of its mechanism involving a Norrish-like scission of the bond to the carbonyl carbon and is also predicted. Ketones bearing an a-substituent expel that substituent as an anion in polar media or as a radical in non-polar media. Again, this is the bond which has the most negative Delta-Density value (Note 5). The Di-71-Methane Rearrangement of barrelene 11 to semibullvalene has dissipation of two 71-bonds and vinyl-vinyl bridging as the initial step of its rearrangement, again predicted. Photochemical raeemization of optically active biphenyls results from planarization of the phenyl-phenyl bond in Si and here we see an increase in the aryl-aryl bond order in biphenyl species 4. It has been noted by a referee that in our research we had suggested that the course of reactions was determined by avoidance of excited state energy barriers and by the presence of conical intersections. The referee wondered then how this criterion based on the vertical Franck-Condon excited state could afford predictions. This is an interesting point. The answer is an important one, namely that the Delta-Density prediction merely gives the "primary photochemical step". This provides a necessary but not sufficient criterion for photochemical reactivity; without the primary step no photochemistry can occur. The Delta-Density method proved applicable to chemistry other than photochemical. Thus, introduction of an electron to afford a radical-anion or extraction of an electron to give a radicalcation leads to species with non-zero Delta-Density elements for the new species with the original geometry. These cases are quite parallel to the photochemistry examples, except that here the species in introduced or withdrawn is an electron rather than a photon. The application to the odd-electron species is illustrated in Eq. 10,

o + e"

(10)

Product

where on electron introduction it is the more substituted bond 2-3 which is severed. The carbonyl carbon of ketyl B is both electron-rich and odd-electron in character. But it is the odd-electron character which is dominant and leads to the more stable of two alternative opened species, this having a disubstituted carbon. Had the anionic character been controlling, ring opening to give a primary carbanion would have resulted.

273 10. SOME ADDITIONAL CHEMISTRY WITH INTERESTING INCLUDING SPIN-ORBIT COUPLING

COMPUTATIONS

The present author in early efforts had already suggested that the course of photochemical reactions was determined by excited state energy barriers and by conical intersections and/or points of intersystem crossing (vide supra). More recently he had added the requirement of a primary photochemical process of the Franck-Condon excited state (again note above). Thus, it was of interest to consider some of these factors further.

NB

Scheme 11 . Mechanisms of Benzobarrelene Rearrangements. The Di-ir-Methane Rearrangement of the barrelenes was selected for study [30]. The dimethylbenzobarrelene and the corresponding naphthobarrelene in Scheme 11 were selected for study. For reasons of space, structures are drawn just for the benzobarrelene (B).The basic mechanism is that given in our much earlier work. Two independent routes to the triplet "Diradical I" species (DR1). The first began with the barrelene B. The other began with the azo precursor A. We see that DR1 has two competing modes of three-ring opening, a and b, and this partition characterizes the species. Independent of precursor, B or A, in the naphthobarrelene case, the same ratio (8.3:1) of semibullvalene photophotoducts (SB1 and SB2) was formed. For the naphthobarrelene the triplet sensitizer, xanthone (ET = 74 kcal/mol) was used, while and for the azo precursor A benzil (ET = 53 kcal/mole) was employed. Both for the naphtho- and the benzo-barrelenes, the product ratio was independent of the reactant, B or A.

274 If DR1 were a transition state or a mere point on the triplet hypersurface then the partition between process "a" and "b" would differ depending on the direction of approach. Since the regioselectivity of the reaction of DR1 was independent of its source, we can conclude that it is a thermally equilibrated intermediate, Ti. Strikingly, with a higher energy xanthone (74 kcal/mole) sensitizer, the azo-naphtho reactant A afforded a quite different product ratio of 2.6:1. It is clear that the azo route with higher energy sensitizers proceeds via T2. Computations were carried out on the barrelene, benzobarrelene and and napthobarrelene triplets (Ti) using Gaussian98 and CASSCF((6,6)/6-31G*. A striking feature was one stretched double bond (1.51 A) with the remaining double bonds being more normal (1.34 A). The Jahn-Teller distortion results from a singly occupied degenerate pair in these cases. Computations for "Diradical 2" DR2 also led to an energy minimum. The triplets of DR1 and DR2 were found to cross the ground state surface and yet the quantum yields were of the order of 0.5, which signifies that Diradical I cannot be the site of major intersystem crossing to ground state. So of B is known to revert to reactant barrelenes. However, spin-orbit coupling computations utilizing GAMESS [33] revealed that spin-orbit coupling for DR1 is only l/4th of that for DR2. 11. SPIN-ORBIT COUPLING DISSECTION One question which arose was what structural features in a molecule or a reacting species gives rise to spin-orbit coupling. Our approach [31,32] to this problem involved doing some programming using the module in GAMESS [33] where the total SOC was computed. The approach was to take the loop as it obtained local contributions and collect these as basis orbital pairs. For this to be helpful, the Weinhold NHO (Natural Hybrid Orbital) basis [29] was used. What was found was that the main contributions come from geminal orbital pairs which tend to be geometrically orthogonal. In diradicals, the p-orbital with the odd electron and orthogonal NHO's of sigma bonds at the same center contribute the most. One interesting application of this approach was in the Type B cyclohexenone rearrangement. It was found that in this triplet process SOC become most heavy as the product geometry is reached with intersystem crossing then occurring as the last bond is being formed. Note Scheme 12.

275

Ph Closure with ISC

Phi

Ph

Scheme 12. Mechanism of the Type B Cyclohexenone Rearrangement.

12. CONCLUSION This presentation makes clear the role of computation in organic photochemistry. The computation ranges from simple Huckel to sophisticated CASSCF treatments. The upper level of computation has been a function of computer facilities available in a given period. However, it also is clear that Huckel treatments have not lost their importance in related sophisticated computations to molecular structure.

REFERENCES [1] Havinga, E.; De Jong, R. O.; Dorst, W., Rec. Trav. Chim., 1956,75, 378. [2] Barton, D. H. R.; DeMayo, P.; Shafiq, M., Proc. Chem. Soc, London, 1958, 205 which first clarified the reaction course. [3] (a) "A Mechanistic Approach to Organic Photochemistry", H. E. Zimmerman, Seventeenth National Organic Symposium of the Amer. Chem. Soc, Bloomington, Indiana, 1961, pgs. 31-41; (b). "Mechanistic Organic Photochemistry. IV. Photochemical Rearrangements of 4,4-Diphenylcyclohexadienone", H. E. Zimmerman and D. I. Schuster, J. Amer. Chem. Soc, 1962, 84 , 4527-4540; (c) "Mechanistic Organic Photochemistry. ", H. E. Zimmerman, Tetrahedron ,1963, Suppl. 2, 19, 393-401; (d) "A New Approach to Mechanistic Organic Photochemistry", Zimmerman, H. E., "Advances in Photochemistry", Editors: A. Noyes, Jr., G. S. Hammond and J. N. Pitts, Jr., Interscience, Vol. 1, 183-208, 1963; (e) "Interpretation of Some Organic Photochemistry", H. E. Zimmerman, Science, 1966, 153, 837-844.

276 [4] "Mechanistic Organic Photochemistry. VIII. Identification of the n-71* Triplet in Rearrangement of 4,4-Diphenylcyclohexadienone", Zimmerman, H. E.;Swenton, J. S. J. Amer. Chem. Soc. , 1964 , 86 , 1436-1437. [5] "A General Theory of Photochemical Reactions. VII. Mechanisms of Epoxy Ketone Reactions", Zimmerman, H. E.; Wilson, J. W., J. Amer. Chem. Soc., 1964, 86, 4036-4042. [6] "On Molecular Orbital Correlation Diagrams, the Occurrence of Moebius Systems in Cyclization Reactions, and Factors Controlling Ground and Excited State Reactions. I", H. E. Zimmerman, J. Amer. Chem. Soc., 1966 , 88 , 1564-1565. [7] Frost, A.; Musulin, B., J. Chem. Phys., 21, 572 (1953). [8] "Carbanion Rearrangements. II", Zimmerman, H. E.; Zweig, A., J. Amer. Chem. Soc, 1961 , 83 , 1196-1213. [9] Electronically Excited State Structures", Zimmerman, H.E.; Binkley, R. W.; McCullough, J. J; Zimmerman, G. A., J. Amer. Chem. Soc., 1967 , 89 , 6589-6595. [10] "Mechanistic Organic Photochemistry. XXIV. The Mechanism of the Conversion of Barrelene to Semibullvalene. A General Photochemical Process", H. E. Zimmerman, R. W. Binkley, R. S. Givens, and M. A. Sherwin, J. Amer. Chem. Soc., 1967, 89 , 3932-3933. [11] Zimmerman, H. E.; Zhu, Z. General theoretical treatments of solid-state photochemical rearrangements and a variety of contrasting crystal versus solution photochemistry. J. Am. Chem. Soc. 1995, 117,5245-5262. [12] Hoffmann, R.; Lipscome, W. N., J. Chem. Phys., 1962, 37, 2872. [13] "Mechanisms of Electron Demotion. Direct Measurement of Internal Conversion and Intersystem Crossing Rates. Mechanistic Organic Photochemistry, H. E. Zimmerman, K. S. Kamm and D. P. Werthemann, J. Amer. Chem. Soc, 1975, 97, 3718-3725. [14] "The Bicycle Rearrangement: Relationship to the Di-n-Methane Rearrangement and Control by Bifunnel Distortion. Mechanistic and Exploratory Organic Photochemistry," Zimmerman, H. E.;Factor, R. E. J. Amer. Chem. Soc, 1980, 102, 3538-3548. [15] (a) "Topology of the Photochemical Bicycle Reaction. Mechanistic and Exploratory Organic Photochemistry", Zimmerman, H. E.; Cutler, T. P. J.C.S. Chemical Communications, 1978, 232-234;(b) "Generality of the Photochemical Bicycle Rearrangement. Exploratory and Mechanistic Organic Photochemistry", Zimmerman, H. E.; Cutler, T. P. J. Org. Chem., 1978, 43, 3283-3303; (c) "Di-7iMethane Hypersurfaces and Reactivity; Multiplicity and Regioselectivity; Relationship Between the Di-7rMethane and Bicycle Rearrangements", Zimmerman, H. E.; Factor, R. E. Tetrahedron, 1981, 37, Supplement 1, 125-141; (d) "The Bicycle Rearrangement. A Review", Zimmerman, H. E. Chimia, 1982, 36, 423-428. [16] "Unusual Organic Photochemistry Effected by Cyano and Methoxy Substitution. Exploratory and Mechanistic Organic Photochemistry", Zimmerman, H. E.; Armesto, D.; Amezua, M. G.; Gannett, T. P.; Johnson, R. P., J. Amer. Chem. Soc., 1979, 101, 6367-6383. [17] "The Bicycle Rearrangement. A Review", H. E. Zimmerman, Chimia , 1982, 36 , 423-428. [18] Much elegant solid-state research preceded ours and is not surveyed here with out emphasis on our computational methodology. These references are found in our solid-state publications.

277 [19] (a) "Confinement Control in Solid State Photochemistry; Photochemistry in a Box", Zimmerman, H. E.; Zuraw, M. J., J. Am. Chem. Soc, 1989, 111 ,2358-2361. (b) "Photochemistry in a Box; Photochemical Reactions of Molecules Entrapped in Crystal Lattices; Mechanistic and Exploratory Organic Photochemistry", Zimmerman, H. E.; Zuraw, M. J., J. Am. Chem. Soc, 1989, 111 7974-7989. [20] (a)"A General Predictor for Photoreactivity in Crystal Lattices: Molecular Mechanics in Crystalline Media and Lock and Key Control; Reaction Examples" Zimmerman, H. E.; Zhu, Z., J. Am. Chem. Soc, 1994, 116 9757-9758; (b) "General Theoretical Treatments of Solid-State Photochemical Rearrangements; And, A Variety of Contrasting Crystal Versus Solution Photochemistry", Zimmerman, H. E.; Zhu, Z., J. Am. Chem. Soc. 1995, 117, 5245-5262. [21] (a) Zimmerman, H. E.; Sebek, P.; Zhu, Z. Ab initio computations of reacting species in crystal lattices; mechanistic and exploratory organic photochemistry. J. Am. Chem. Soc. 1998, 120, 8549-8550; (b) Zimmerman, H. E.; Sebek, P. Photochemistry in crystalline cage. Control of the type-B bicyclic reaction course: mechanistic and exploratory organic photochemistry. J. Am. Chem. Soc. 1997, 119, 3677-3690. [22] Gaussian 98, Revision A.6. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, Jr., J. A.; Stratmann, R. E.; Burant, J. C; Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C; Farkas, O.; Tomasi, J; Barone, V.; Cossi, M.; Cammi, R; Mennucci, B.; Pomelli, C; Adamo, C; Clifford, C; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J. V.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Gonzalez, C; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Andres, J. L.; Gonzalez, C; Head-Gordon, M.; Replogle, E. S.; Pople, J. A. Gaussian, Inc., Pittsburgh PA, 1998. [23] (a) Maseras, F.; Morokuma, K. IMOMM: a new integrated ab initio + molecular mechanics geometry optimization scheme of equilibrium structures and transition states. J. Comput. Chem. 1995, 16, 11701179; (b) Matsubara, T.; Sieber. S.; Morokuma, K. A test of the new "integrated MO + MM " (IMOMM) method for the conformational energy of ethane and n-butane. Int. J. Quantum Chem. 1996, 60, 11011109. [24] Zimmerman, H. E.; Alabugin, I. V.; Smolenskaya, V. N. Experimental and theoretical host-guest photochemistry; control of reactivity with host variation and theoretical treatment with a stress shaped reaction cavity; mechanistic and exploratory organic photochemistry. Tetrahedron 2000, 56, 6821-5831. [25] "Crystal Lattice Photochemistry Often Proceeds in Discrete Stages; Mechanistic and Exploratory Organic Photochemistry", Zimmerman, H. E.; Nesterov, E. E. Organic Letters, 2000, 2, 1169-1171. [26] Zimmerman, H. E.; Alabugin, I. V.; Chen, W.; Zhu, Z. Dramatic effect of crystal morphology on solid state reaction course: control by crystal disorder; mechanistic and exploratory organic photochemistry. J. Am. Chem. Soc. 1999, 121, 11930-11931. [27] "The Meta Effect in Organic Photochemistry; Mechanistic and Exploratory Organic Photochemistry", Zimmerman, H. E., J. Am. Chem. Soc. 1995, 117, 8988-8991. "The Ortho-Meta Effect in Organic Photochemistry; Mechanistic and Exploratory Organic Photochemistry", Zimmerman, H. E., J. Phys. Chem., 1998, 102 5616-5621.

278 [28] (a)"Excited State Energy Distribution and Redistribution and Chemical Reactivity; Mechanistic and Exploratory Organic Photochemistry", Zimmerman, H. E.; Alabugin, I. V. J. Am. Chem. Soc, 2000, 122, 952-953; (b) "Energy Distribution and Redistribution and Chemical Reactivity. The Generalized Delta Overlap-Density Method for Ground State and Electron Transfer Reactions; A new Quantitative Counterpart of Electron Pushing", Zimmerman, H. E.; Alabugin, I. V. J. Am. Chem. Soc. 2001, 121, 2265-2270 [29] (a) Foster, J. P.; Weinhold, F., J. Am. Chem. Soc, 1980, 102, 7211-7218; (b) Reed, A.; Curtiss, L. A.; Weinhold, F. Chem. Rev., 1988, 88, 899-926. [30] "Excited State Reactivity as a Function of Diradical Structure; Evidence for Two Triplet Cyclopropyldicarbinyl Diradical Intermediates With Differing Reactivity", Zimmerman, H. E.; Kutateladze, A. G.; Maekawa, Y.; Mangette J. E. J. Am. Chem. Soc, 1994, 116, 9795-9796. [31] "Novel Dissection Analysis of Spin-Orbit Coupling in the Type B Cyclohexenone Photorearrangement. What Controls Photoreactivity. Mechanistic and Exploratory Organic Photochemistry", Zimmerman, H. E.; Kutateladze, A. G. J. Org Chem. 1995, 60, 6008-6009. [32] "Novel Dissection Analysis of Spin-Orbit Coupling in the Type B Cyclohexenone Photorearrangement. What Controls Photoreactivity. Mechanistic and Exploratory Organic Photochemistry", Zimmerman, H. E.; Kutateladze, A. G. J. Org Chem. 1995, 60, 6008-6009. [33] Schmidt, M. W., et. al. J. Comput. Chem., 1993 14, 1347-1363. ACKNOWLEDGMENT It is a pleasure to acknowledge the support of the National Science Foundation without which these studies would not have been possible.

M. Olivucci (Editor) Computational Photochemistry Theoretical and Computational Chemistry, Vol. 16 © 2005 Elsevier B .V. All rights reserved

279

IX. Calculations of Electronic Spectra of Transition Metal Complexes K. Pierloot Department of Chemistry, University of Leuven, Celestijnenlaan 200F, B-3001 Leuven, Belgium 1. INTRODUCTION If this chapter would have been written ten-twenty years ago this introduction might traditionally have started with a sentence like: "Transition metals and their compounds still present a major challenge to ab initio quantum chemistry ...". Today, the magic surrounding these difficult elements has to a large extent been lifted. This of course has everything to do with the development of density functional theory (DFT), a method which turned out to work marvelously well also for molecules containing transition metals (TM), even in cases that were traditionally considered to be quite difficult to treat computationally (because of the presence of important nondynamic correlation effects), e.g. organometallic systems, with Cr(CO)6 and ferrocene as prototypes [1-3] . The possibility of treating TM systems on a routine basis with methods that could generally be considered trustworthy, at least until proven differently, has led to a drastic increase of computations in the field, especially focusing on organometallic and biological molecules. Molecular properties that are today treated on an almost routine basis include ground state structures, vibrational frequencies and reaction energies [4]. Furthermore, the availability of very fast computers has enabled such computations on systems of real chemical interest. Things are still quite different when considering electronically excited states. Theoretical calculations of electronic spectroscopy have in general today not reached a "black box" status. The specific electronic structure of TM complexes, originating from a partially filled d shell surrounded by a number of potential electron donating or accepting ligands, adds a number of additional intricacies to these computations, often making them accessible only to researchers with experience both in the field of coordination and computational chemistry. This is true even when applying DFT, be it under its time-independent formalism, usually denoted as ASCF [5], or its time-dependent formalism TDDFT [6-8]. Test results obtained with these methods in transition metal spectroscopy are certainly promising in some cases, but have also pointed to quite a number of remaining problems (e.g. the spectra of MnC% (see section 4) and the blue copper proteins (see section 5)).

280 An introductory overview of the various electronically excited states to be met in TM complexes most conveniently starts from a simple ligand field picture. According to this picture the TM d orbitals are split in energy by the presence of the coordinating groups, the so-called ligand field environment. Depending on the extent of the splitting the available electrons are spread over the different d-orbitals giving rise to a high spin ground state, or are placed pairwise in the lowest lying orbitals, thus producing a low-spin (singlet or doublet) ground state. By absorption of light the electrons are excited from the filled into the singly occupied or unoccupied d orbitals. Since the splittings of the d shell are critically dependent on the number, character and position of the ligands, these ligand field (LF) transitions provide sensitive probes of the electronic structure and geometric surrounding of TM ions. LF transitions are parity forbidden, and as such generally have a low intensity in an absorption spectrum (but can be intense in CD spectrum or low-temperature MCD spectra [9]). Overlapping the LF transitions and to higher energy in the visible/UV spectral regions are the charge transfer (CT) transitions, often electric dipole allowed and therefore more intense in absorption spectra than LF transitions. CT transitions either involve the excitation of an electron from one of the filled d orbitals into a ligand centered unoccupied orbital (MLCT), or the excitation of an electron from filled valence orbitals on the ligands into one of the singly occupied or empty d orbitals (LMCT). MLCT transitions are typically observed in organometallic compounds, containing organic ligands with low-lying (relative to the d shell) TT* orbitals. On the other hand, electronic spectra of TM containing biological molecules often contain one or several low-lying LMCT transitions. The position and intensities of CT transitions are sensitive probes of the character of specific metal-ligand interactions. Low energy CT transition are the reflection of close-lying ligand and metal valence orbitals. Moreover, the intensity of a CT transition increases with the overlap extent of the donor and acceptor orbitals. Low energy, intense CT transitions therefore usually reflect highly covalent metal-ligand bonds. The list of possible electronic excitations in TM spectra is not complete without mentioning also intra-ligand excitations. With some exceptions [10, 11] such transitions do not occur in the low-energy part of the electronic spectra. They are not further considered in this chapter. The intention of this chapter is not to provide a detailed overview of possible computational methods for the calculation of excited states in transition metal complexes. Such an overview has recently been given in a review by C. Daniel [12]. Instead we would like to present a few selected examples out of our recent work, which we believe clearly illustrate the distinct power of accurate computations for the interpretation of experimental spectroscopic features and the discussion of the specific electronic structure and metal-ligand bonding interactions giving rise to these features. The examples are taken from various fields of chemistry and include different types of excitations. In section 3 we will focus on the ligand field spectra of Co(II) and Cu(II) as probes of the coordination environment of these ions when bound to a zeolite surface. Section 4 contains a discussion of the long-standing and heavily debated spectrum of the permaganate ion, i.e. a pure charge transfer spectrum. In section 5 we wil discuss the electronic structure of two important classes of redox proteins, namely the blue copper proteins and a series of mononuclear oxomolybdenum enzymes, modeled by the (Tp)MoO(bdt) compound (with Tp =

281 hydrotris(l-pyrazolyl)borate and bdt = benzenedithiolate). In both cases, the considered electronic spectra are built from a mixtures of LF states and a number of low-lying LMCT states. All excited state calculations reported in this chapter were performed by means of the CASPT2//CASSCF method [13] as implemented in the Molcas software [14]. Ground state structures were either taken from experimental X-ray diffraction (XRD) data or calculated by means of the B3LYP-DFT method, using either the Mulliken [15] or Turbomole [16] codes. Whenever available, comparisons are made to the results obtained by other computational methods. In the following section, we will give a short introduction to the CASPT2/CASSCF method, focusing on those aspects that are of particular importance when applying the method for computations of electronic spectra of transition metal complexes. 2. CALCULATION OF ELECTRONIC SPECTRA OF TM COMPLEXES WITH MULTICONFIGURATIONAL PERTURBATION THEORY The CASPT2/CASSCF method is a mixed variational/perturbational procedure. In the first, C ASSCF step, a zero-order wavefunction is constructed by distributing a limited number of valence electrons, i.e. the active electrons, over a limited range of valence orbitals, i.e. the active space, in all possible ways consistent with the spin and symmetry of the considered state. By variationally optimizing both the orbitals and the CI coefficients of this (limited) full CI wavefunction the best possible treatment of strong correlation effects can be obtained. In the second, C ASPT2 step, remaining correlation effects can then be treated by a second-order perturbational approach. This second step is quite straightforward. However, since perturbational theory is not able to cope with strong, static correlation effects, it is of crucial importance that all such effects are treated at the CASSCF level by including the appropriate orbitals in the active space. In a recent paper [17] we have illustrated by a few examples (Fe(CO)5, Fe(II)-porphin) how the omission of just one important orbital from the active space may lead to errors of several thousands of wavenumbers on excitation energies calculated by the CASPT2 method. The validity of the CASPT2 treatment entirely depends on the quality of the CASSCF wavefunction. A thorough discussion of important static correlation effects in TM complexes has been presented in a recent book chapter [18]. These effects can be subdivided into two categories: • The so-called double-shell effect is an atomic, radial correlation effect, manifested in first-row TM atoms or ions containing a large number of electrons in a compact 3d shell (e.g. the Ni atom [19]). This effect should be treated at the CASSCF level by including in the active space a second d shell, usually denoted as 3d' or Ad. As was shown in ref. [18] a proper treatment of the double-shell effect is of greater importance when considering electronic transitions involving a change of the 3d occupation number (e.g. 3d —¥ As or charge-transfer transitions) than for transitions within the 3d shell (i.e. LF transitions). Furthermore, the effect is drastically reduced when going to the second and third TM series.

282 • Covalent metal-ligand bonds, i.e. bonds involving a significant admixture of ligand character into the metal d orbitals and vice versa, invariably give rise to strong correlation effects. These effects should be accounted for in the zero-order wavefunction by including in the active space both the bonding and antibonding combination of valence orbitals involved in the covalent bond(s). Qualitatively speaking, covalent metal-ligand bonds arise when the energy separation between the metal d and ligand valence orbitals is small and/or when both orbitals are strongly overlapping. For the same ligand L, M-L covalency and concomitant correlation effects therefore increase (a) with an increasing formal charge on the metal M, and (b) when moving from left to right in the TM series. Test calculations in ref. [18] also indicated a significant reduction of static correlation effects connected to M-L covalency when going to the second and third TM series. This fact, together with the strongly reduced double-shell effect in the higher TM series, explains the increasing succes of single-reference methods such as MP2 or MCPF for complexes containing these heavier metals [20-22]. As concerns the ligand L, M-L covalency generally increases as L becomes less electronegative and more polarizable or "soft". When considering electronic spectra, the selection of valence orbitals to be included in the active space should be based on two criteria, i.e. (a) all strong correlation effects should be incorporated, and (b) any orbital that gets a population number different from either zero or two in any of the considered states should be included. At first sight it might seem as if both criteria should in fact lead to the same selection of valence orbitals, i.e. the highest occupied and lowest unoccupied orbitals. This is however not always the case in TM chemistry, since the orbitals involved in static correlation effects either consist of a double d shell or of pairs of bonding and antibonding combinations of the metal d and ligand orbitals, whereas lowest-lying charge-transfer states may instead involve excitation of an electron from or into non-bonding ligand orbital(s). As an illustrative example, let us look at the octahedral Cr(CO)g complex. As is well known, the metal-ligand bonding in this organometallic compound is built from a-donation from CO into the empty Cr 3d orbitals, counteracted by vr-backdonation from the filled Cr 3d orbitals into CO TT* . Both bonding types are covalent and give rise to important correlation effects [23]. These effects may be treated at the CASSCF level by considering an active space consisting of the bonding an antibonding combination of CO a and Cr (d Z2 ,dx2_y2) within eg symmetry and of Cr (dxy,d.xz,dyz) and CO TT* within t2g symmetry. However, the lowest-lying excited states in the absorption spectrum of Cr(CO)e arise from MLCT transitions from the occupied t2g (predominantly 3d) shell into the lowest unoccupied i l u , t2u (CO TT*) shells [24]. The latter are non-bonding and therefore lower in energy than the antibonding t 2g (predominantly CO TT*) orbitals. A multiconfigurational treatment of the absorption spectrum of Cr(CO)e therefore requires an active space consisting of at least sixteen orbitals, which is at the limit of today's computing power. As a viable and cheaper alternative, the calculation of different excited states may be performed using different active spaces, by selectively including only those orbitals that are either populated or depopulated in each state considered [24]. Complexes containing transition metals with exceptionally high formal oxidation states (i.e. +VI or higher) present a special problem for CASPT2//CASSCF, as for other ab initio meth-

283

ods. Well-known examples are CrFG, CrO^ and MnO4 . This may again easily be understood by considering a qualitative orbital picture, in which now the metal 3d and ligand valence orbitals literally become near-degenerate [17, 18]. As a consequence, static correlation effects on the M-L bonds become extremely important in these complexes. Furthermore, these effects are no longer limited to the bonding-antibonding pairs of covalently interacting orbitals, but rather involve all ligand valence orbitals. In CrF6 for example important contributions to the ground state wavefunction are observed from excitations into Cr 3d out of all eighteen F 2p orbitals [17]. In section 4 we will discuss the results obtained for the electronic spectrum of MnO^", using CASPT2 based on a zero-order wavefunction comprising 2.2 million configuration state functions (9.5 million determinants), built from an active space of seventeen (five Mn 3d + twelve O 2p) orbitals. Such calculations are really at the limit of what can be handled with presently available computing hardware/software. Still it is important to note that even for exceptionally hard cases such as MnO J the combined variational/perturbational treatment offered by the CASPT2 method is quite successful in providing an accurate treatment of all low-lying excited states in the spectrum (see further in section 4). Once an appropriate zero-order wavefunction is obtained, the perturbational part of the calculation does not require much extra attention, but for the choice of appropriate basis sets to accurately describe dynamic correlation effects. In order to limit basis set superposition errors, core electrons are preferebly not included in the correlation treatment. In this respect, the metal semi-core orbitals (e.g. 3s,3p in first-row, 4s,4p in second-row TM, ...) may deserve some special care. It has been shown [25] that correlation of these electrons may give very important contributions to the relative energy of different states. As such, the semi-core orbitals should preferably be included in the CASPT2 correlation treatment. However, standard transition metal basis sets are not always well-suited for this purpose. The use of such basis sets without extra precautions may therefore give rise to huge basis superposition errors, when considering for example M-L bond dissocation energies [26]. The problem is less stringent in calculations of vertical excitation energies, but then of course semi-core correlation effects can not be described to their full extent unless the basis sets used contain the necessary functions to do so. This may be accomplished by uncontracting or adding some extra functions in the appropriate region of space. Even when the CASSCF active space was chosen large enough to include all strong correlation, it may happen that the perturbational treatment fails due to the occurrence of so-called intruder states. Such states may arise when the energy gap between the active and inactive or between the virtual and active orbital space becomes small or even negative (the orbital energies of the active orbitals being the eigenvalues of the zeroth-order hamiltonian Ho, which may in practice have different forms [27]). As long as the intruder states are weakly interacting with the zero-order wavefunction, they may be effectively shifted away by applying a special levelshift technique [28, 29]. (if they are strongly interacting they should instead be included in the reference space). The actual size of the level shift can be obtained by performing a series of test calculations [30]. For calculations of electronic TM spectra, the systematic application of a level shift 0.3-0.35 a.u. has been found appropriate. Practice learns that with a level shift of

284 this size all weakly interacting intruder states are systematicaly removed, whereas the relative energies obtained with CASPT2 are not significantly affected. Furthermore, applying a level shift also greatly speeds up the convergence of the perturbation treatment. 3. LIGAND FIELD SPECTRA OF CO(II) AND CU(II) COORDINATED TO OXYGEN SIX-RINGS IN ZEOLITES Zeolites are inorganic crystalline materials characterized by a regular structure of channels and cages of molecular dimensions. Their structure is built from tetrahedral units (TO4), consisting of a central atom (T) - mostly Si(IV) - coordinated to four oxygen atoms. Other atoms, most often Al(III), can replace the central Si(IV). The tetrahedral units are linked by sharing all the oxygen atoms, leading to a wide variety of materials, differing in framework structure and chemical composition. [31] As each Al(III) incorporated in the silicate framework leads to one excess negative charge, an equivalent amount of extra-framework cations must be introduced to neutralize the structure. These cations, mostly Na(I), K(I), Ca(II), are present inside the cages and channels of the zeolite together with intrazeolitic water. They are not covalently bound to the zeolite framework and can therefore easily be replaced by transition metal ions via conventional aqueous ion exchange. After dehydration, these transition metal ions become localized. They are dispersed over the large internal surface of the zeolite, coordinating to the framework oxygen atoms. However, in the absence of adsorbed molecules, the resulting TM coordination environment is often not saturated. This makes the TM sites interesting sites for adsorption and opens the possibility for catalysis. The microporous character of the zeolite material puts limits to the size and shape of the interacting molecules, thus enabling shape selective reactions. A crucial step in the investigation of the catalytic potential of these TM centers is the study of the different coordination possibilities of the metal in the zeolite and its accompanying electronic structure. A powerful tool in this respect is provided by electronic spectroscopy, in particular diffuse reflectance spectroscopy (DRS) and electron spin resonance (ESR) spectroscopy. With a partially occupied 3d shell in an ionic coordination environment, the lowest excitations in the electronic spectra can be expected to originate from transitions within the 3d shell [32]. Both the ligand field excitation energies and the ESR spectra, as manifested by the g tensors and hyperfine splittings, are sensitive to the surroundings of the transition metal ion. As such, these spectra provide "fingerprints" of the specific TM environment(s) in the considered zeolite. However, the interpretation of these spectroscopic data is not always straightforward, and in many cases still a subject of debate. Until some years ago, the spectroscopic fingerprints were mainly interpreted from X-ray diffraction (XRD) results or by means of semi-empirical calculations (ligand field theory) [33-41]. However, a major limitation of XRD in the study of TM exchanged zeolites is that it can only provide average information, making no distinction between Si and Al in the lattice nor between occupied and empty sites. As a consequence, the apparent symmetry of the coordination environment of TM ions obtained with XRD is often too high. All semi-empirical calculations inevitably started from these average XRD

285

structures and were therefore performed in fictitious high symmetry coordination environments. The possibility of symmetry reduction, for example by an asymmetric aluminum surrounding, was not considered. As will be shown in what follows, this factor does indeed play a crucial role in the metal-zeolite interaction. During the last ten years many computational studies, predominantly using DFT, have been reported [42-64] in which the siting and catalytic potential of TM ions (most often Cu(I)) bound to zeolite surfaces was studied by means of cluster models of different sizes. Especially worth mentioning in this respect is the work of J. Sauer and coworkers, employing a combined quantum mechanics/interatomic potential function technique [53, 65, 66], in which the cation exchange site is treated by quantum mechanical methods, while the periodic lattice is treated by an ion pair shell model potential. Using this method, the location, structure and coordination of isolated Cu(I) and Cu(II) ions in ZSM-5 were studied. [57, 59, 62]. In a series of recent studies [67-72] we have investigated the structure and spectroscopic properties of the Co(II) and Cu(II) ions in different zeolite environment (e.g. zeolites A, Y, ZK4, ZSM-5 and mordenite). In these studies, DFT structure optimizations on cluster models were combined with CASPT2//CASSCF calculations of absorption spectra and ESR (/-factors. An important plus point of such combined studies [62] is that the plausibility of the calculated ground state structures can be directly tested by a confrontation with experimental (spectroscopic) data. In the next sections we will present the basic methodology and discuss the results obtained for the ligand field spectra of Co(II) and Cu(II) coordinated at trigonal six-membered ring sites in zeolites A and Y. For a further discussion of Cu(II) coordination in pentasil zeolites (mordenite, ZSM-5) and the results obtained for ESR p-factors we refer to the literature [67-72]. 3.1. Calculated electronic spectra for trigonal cluster models The framework structures of zeolites A and Y are schematically drawn in Fig. 1. These zeolites are built from a regular stacking of so-called sodalite cages, each consisting of 24 T (Si or Al) atoms that make up a truncated octahedron and 36 bridging oxygen atoms. In Fig. 1 the vertices give the position of the T atoms, while the lines connecting the vertices represent the oxygen atoms. The sodalite cages can be connected to each other in two different ways, either through the oxygen four-membered rings or through the six-membered rings. This gives rise to two different zeolite topologies, respectively denoted as LTA (Fig. l.A) and FAU (Fig. l.B). Zeolite A belongs to the LTA topology, whereas zeolite Y is a member of the FAU = faujasite topology. Both zeolites also differ in their Si/Al ratio: strictly one in zeolite A, but about two or higher in zeolite Y. The latter difference was shown to be at the basis of the different ESR signals observed for Cu(II) bound to both zeolites [68, 72]. Structures obtained from single-crystal XRD are available for Co(II) and Cu(II) exchanged zeolite A [73, 74]. Both ions are found on the threefold axes in the hexagonal six-membered oxygen rings, denoted as site II in Fig. l.A. The coordination environment of both ions in this site is shown in Fig. 2. Both TM ion are coordinated to three close-by oxygen atoms (O^), with a second shell of three oxygen atoms (OB) at a considerably larger distance. The site symmetry is Cy,v, but with the OA-M-OA angle very close to 120° in both cases (119° for

286

A

B

Fig. 1. Schematic framework structure of zeolite A (A) and zeolite Y (B)

M = Cu and 117.4° for M = Co), indicating an almost trigonal planar (D 3 t ) coordination. For Cu(II) and Co(II) exchanged X-ray [75-77] and neutron diffraction [78] studies indicate that also in this case the sing-membered rings (indicated as I and II in Fig. 1) are the favored sites for both ions, although other possibilities (e.g. the center of the hexagonal prism, I, or, in case of Cu(II), the four-membered ring sites III and III) have been proposed. Both for M = Co and M = Cu the experimental DRS spectra of the M(II)-A and M(II)-Y combinations show ligand field transitions at similar energies. The spectra of Co(II)-A and Co(II)-Y consist of three main features [67, 79]: a broad band with a maximum at 7 000 cm" 1 (band I), a second broad band at 15 000-19 000 cm" 1 (band II) and a weaker feature at 24 000-25 000 cm" 1 (band III). All three features are split further by 1 000-2 000 cm" 1 . For Cu(II), both spectra consist of a main broad band with a maximum at 10 400 cm" 1 and a shoulder at around 12 500 cm" 1 , and a second, weak, band appearing at 14 700-15 400 cm" 1 [41]. A first series of CASPT2//CASSCF calculations was performed on a MOeSi3Al3Hi2~ cluster, representing the oxygen six-ring site in zeolite A, and using the average structure obtained from XRD (Fig. 2). With three Al and Si put at alternating positions (according to Loewenstein's rule [80] two Al should always be separated by at least one Si) the site symmetry is reduced from C3,, to C3. For M = Co, CASSCF calculations were performed with an active space containing thirteen electrons in eight orbitals, namely Co 3p, 3d. The Co-0 bonds are very ionic [67], such that contributions to the wavefunctions from O 2p -4 Co 3d excitations are not important in this case [18]. Instead, when including O 2p orbitals into the active space these orbitals spontaneaously rotate into Co 3p. The latter reflects the importance of these semi-core orbitals in the correlation treatment. Even if this is a dynamic rather than a static correlation effect, slightly superior results for ligand field transitions may be obtained with 3p in the active space [25]. On the other hand, in order to take care of the double-shell effect and the increased covalent character of the M-0 bonds for M = Cu, the active space was in this case constructed from two Cu d shells (3d,3-Cu 3d counterpart of the orbital

287

Al

Al

Fig. 2. M-OA and M-OB distances in the trigonal structures obtained from XRD on M(II)-zeolite A (M = Co, Cu)

which is singly occupied in the ground state (antibonding but predominantly Cu 3d [68, 72]). The effect of spin-orbit coupling on the ligand field transitions was taken into account by means of an effective one-electron operator [67, 68]. For Co(II), the effects of spin-orbit coupling are more limited. They are not included in the results reported here (but can be found in ref. [67]). For Cu(II) in a trigonal oxygen environment, the ligand field spectrum arises from a splitting of the sole d° term 2D into two 2E and one 2 A states, both 2E states being further split by spinorbit coupling. For Co(II), the states composing the ligand field spectrum correspond to two parent d7 states 4P and 4F, the latter being the ground state. The experimental 4 P- 4 F splitting in the free Co(II) ion is 14 561 cm" 1 [81]. In a C3 environment, these terms are split as follows: 4 F ->• 3 4A + 2 4E, 4P -)•4A + 4 E. In table 3.1 the results obtained from the CASPT2 calculations are compared to the experimental ligand field splittings in the zeolite environment. For the MOoSi3Al3Hi2~ cluster model at the crystal structure, the splitting of the free-ion states of both TM ions is obviously underestimated. For Cu(II) the predicted excitation energies do not reach higher than 8 500 cm" 1 , whereas the experimental spectra reveal the presence of ligand field states up to 15 000 cm" 1 . For Co(II), the calculated splitting of 3 500 cm" 1 for the 4 F state corresponds to only half of the experimental band maximum of band I. The calculated splitting of the 4P state, 7 000 cm" 1 , is closer to the experimental splitting of around 8 000 cm" 1 between bands II—III. The calculated excitation energy of the c 4E state, 15 719 cm" 1 , does correspond to the experimental band maximum of band II. However, the calculated result for the second state, 22 700 cm" 1 is too low when compared to band III. Such large errors are unlikely to be caused by deficiencies in the CASPT2 method. This strongly suggests that the origin must instead be found in structural deficiencies in the average XRD structures used in this first set of calculations. Obviously, the ligand field strength created by this fictitiously high symmetric coordination environment is much too low.

288 Table 1 CASPT2 spectral data for selected MO6Si(6-a.)Ala'Hi2~ clusters (M = Cu, Co) at crystal and DFT optimized structures exp. CuO6Si 3A13H12 cluster CuO6:3i3Al2Hi2 cluster B3LYP-DFT structure' spectrum'1 XRD structure" Energy (cm"1) State Energy (cm 1 )' ; State Energy (cm x )d 2 2 E A 0 0 2 1413 A 8411 2 2 E A 10 644 10 400 4 997 2 12 500 5 259 A 11303 2 2 A A 15 000 8 523 15 108 CoO6Si 3A13H12~ cluster CoO6:3i3Al2Hi2 cluster exp. B3LYP-DFT structure* spectrum^ XRD structure'' State Energy (cm : ) State Energy (cm 1) Energy (cm"1) 4 4 A 0 A 0 4 4 E A 562 63 4 4 E A 63 773 4 4 A 2 260 A 2 386 4 4 E 3 235 A 3 916 4 4 E A 3 235 4 576 4 4 A A 6 264 7 000 3 523 4 4 E A 16 000 15 519 15 886 4 4 E 15 519 A 16 410 4 4 25 000 A 22 793 A 25 163 "ref 74 ; see Figure 3.2; r;ref 41; ^including spin-orbit coupling; 'ref 73; 'ref 67

289 3.2. Structural deformations and concomitant ligand field strengthening In a second series of calculations, it was therefore decided to investigate possible local distortions of the zeolite surface produced by the coordination of the TM ions. This was done by DFT structure optimizations on clusters with the structural formula MO6Si6-a. Ala.(OH)i2(2~:i:)+ (x=0-3), containing the metal ion coordinated in the same sing six-ring site as above, but terminated by OH-groups instead of H. The starting geometry of the cluster models was based on the XRD data of M(II)-zeolite A [73, 74]. In order to allow full freedom of movement for the TM ions and the coordinating oxygens, these optimizations were performed without symmetry restrictions. However, in order to mimic the rigidity of the zeolite framework, restrictions were imposed on the orientation of the dangling OH bonds. Different aluminum contents and distributions were considered. Both the BP86 and B3LYP functionals were employed and produced similar results in case of Co(II) but not for of Cu(II), for which B3LYP proved to be superior [82]. All results included in this chapter were obtained using the B3LYP functional. In case of Cu(II), the accuracy of the obtained B3LYP-DFT structures was checked by performing additional restricted CASPT2 geometry optimizations for a few clusters. [68] Significant corrections were only obtained for the negatively charged CuO6Si3Al3(OH)12~ cluster model. Finally, the ligand field spectra corresponding to all optimized structures were calculated by means of the CASPT2 method. In order to reduce the computational effort, the terminal OH bonds were first replaced by hydrogens. Fig. 3 shows the results obtained for the structure of a representative cluster model containing two alumina. As compared to the crystal structures in Fig. 2, the largest changes are observed for the distances M-O^ between the central metal and the three distant oxygens. For M = Co two of these distances are shortened from 2.95 A to 2.67-2.42 A, while the third OB atom is driven away to 3.39 A. As such, the central Co(II) ion obtains a total coordination number of five instead of three, however with two bonds still signficantly weaker than the three original CO-OA bonds. On the other hand, for M = Cu two of the distant O B oxygens are pushed outwards, while the third O^ is moved towards the central Cu(II) to form a fourth strong bond with a Cu-O distance of 2.15 A (CASPT2 optimized; the corresponding B3LYP result is 2.21 A). This striving of Cu(II) to obtain a fourfold coordination environment when bound to a zeolite surface is a general given, observed in all our studies on Cu(II) bound to different zeolites, with and without external ligands [68-72]. It can be explained by considering the ground state d° electronic structure, with one singly occupied 3d orbital. Since this orbital has four, squarely oriented lobes, a maximum antibonding interaction with the ligand environment is obtained by a square-planar rather than by a trigonal oxygen environment. As the corresponding bonding orbital is doubly occupied, maximizing the antibonding interaction with the ligands leads to a stabilization. This explanation is in line with simple ligand field predictions. Co(II), with three singly occupied 3d orbitals, does not display this strong directional preference for one additional strong bond, but rather shows a more general tendency to increase its coordination number to four or five [67, 82]. However, it is important to note that in both cases (and all other cases included in our studies) the additional M-0 bonds are formed with O# oxygens that are bound to Al rather than Si. This indicates how the aluminum distribution in zeolites may

290 S Si iA

•• gf

29

an

AS>

^J9SI

Ji2-15

3.39

SSii

05 A 2 9M 9 2 67 - flT

A

0SlSi 12.42 02.4

3.39

Fig. 3. M-O^ and M-O# distances in the asymmetric structures obtained from B3LYP-DFT optimization of MO6Si4Al2(OH)12 (M = Co, Cu)

thoroughly affect the bonding properties and corresponding spectroscopic (electronic and ESR) properties of both ions in different zeolites. In particular, our calculations have indicated how different distributions of aluminums may actually lead to two distinctly different ESR signals in the zeolites Y, ZK4 and mordenite [68, 69, 72]. As for the ligand field spectra of Cu(II) and Co(II) coordinated to zeolites A and Y, similar excitation energies were obtained from CASPT2 calculations on optimized models with different numbers and distributions of aluminums. The results obtained for the model with two aluminums distributed as in Fig. 3 are included in Table 3.1. These data clearly reveal a much stronger ligand field surrounding of both ions as compared to the trigonal oxygen environment in the models with structures frozen at the XRD values. The splitting of the 2D ground state of Cu(II) is doubled, and the calculated transition energies now nicely correspond to the position of the two band maxima at 10 400 cm" 1 and 15 000 cm" 1 in the experimental spectra. For Co(II), the splitting of the 4F ground state is again almost doubled, but still remains slightly below the maximum of band I in the experimental spectrum. It is shown in ref [67] that the introduction of spin-orbit coupling leads to a further broadening by 800 cm" 1 of the transition energies belonging to 4F. Band II at 16 000 cm" 1 is now found to correspond to two distinct transitions, split by about 600 cm" 1 . This is in accord with the experimental splitting observed for this band, although the calculated splitting is still too small. Finally, the appearance of band III at 25 000 cm" 1 is well reproduced by the CASPT2 results. The experimental splitting of this band may at least be partly explained by spin-orbit coupling [67]. On the whole, the results presented here have indicated how the coordination of TM ions in zeolites may be succesfully explained and even predicted by means of cluster models and using a combined CASPT2//D FT approach. The CASPT2 excitation energies are especially useful since, in confrontation with experimental ligand field spectra, they provide a sensitive probe of the geometric distortions predicted by the DFT optimizations. Such distortions are also indicated by other spectroscopic experiments, e.g. IR, ESEEM and 27A1 NMR [83-85], but their actual extent has so far been impossible to obtain directly from experiment.

291

4. CHARGE-TRANSFER STATES: ELECTRONIC SPECTRUM OF THE PERMANGANATE ION The electronic spectrum of permanganate presents one of the theoretically most intensively studied spectra of transition metal systems. Based on a semi-empirical MO treatment, a basic interpretation of the electronic transitions was provided already in 1952 by Wolfsberg and Helmholtz [86]. During the seventies, numerous ab initio studies were reported, using either Hartree-Fock and limited configuration-interaction [87-91] or one of the earlier Hartree-FockSlater versions of DFT: HFS-SW [92], HFS-DVM [93]. More recently, MnOJ has been given the status of a prototype transition metal system, used as a benchmark for testing the performance of several newly developed methods for the calculation of excited states: DFT in its ASCF [94, 95] and time-dependent [96, 97] form and cluster expansion methods such as SACCI [98] and EOM- and extended EOM-CC [99]. This prototype status is however somewhat undeserved, given that when it comes to obtaining accurate electronic excitation energies for transition metal complexes, permanganate is an exceptionally hard rather than a representative case. The problem is related to the high (+7) formal charge on manganese in this molecule, giving rise to substantial static correlation effects involving excitations from all twelve doubly occupied orbitals originating from 0(2-) 2p, either bonding or nonbonding, into the five antibonding orbitals corresponding to Mn(7+) 3d, which are empty in the ground state [18] (see also section 2) . Moreover, these correlation effects change appreciably in the excited states [100]. Consequently, any single reference molecular orbital description of the permanganate ground state is quite dubious, and its shortcomings doomed to be reflected in the calculation of excitation energies using methods working with these ground state orbitals. The presence of important static correlation effects in permanganate also affects the DFT description of the excitation energies, with values that substantially differ between the ASCF and TD-DFT approaches, that fluctuate by up to 3 000 cm" 1 between different functionals, and that are on the whole considerably less satisfactory than the results obtained with the same methods for other transition metal systems [96, 97]. Another consequence if this general malaise is that the assignments of the permanganate spectrum have undergone a lot of changes throughout the years, and in fact a general consensus still has not been reached. Here, we will only discuss the most recent theoretical results, obtained with either TD-DFT or cluster expansion methods (an overview of the older results can be found in ref. [98]). Furthermore, we also present the results obtained from a CASPT2 treatment [101], based on CASSCF reference wavefunctions built from an active space comprising all seventeen valence orbitals originating from manganese 3d and oxygen 2p: 6a i, (l-2)e, lti, (5-7)^. CASSCF calculations with such a large active space have only recently become within the limits of computer power, but are still very time consuming. We report here the results obtained for the excitation energies of the lowest spin- and dipole-allowed 1Ai—>XT2 transitions in this tetrahedral molecule. A more detailed description of the spectrum will be presented in a separate paper [101]. In Table 4 calculated excitation energies of the four lowest XT2 states with different methods

Is) Is)

Table 2 Calculated electronic allowed transition energies (in cm" 1 ) of MnOj state Excitation energy (cm x)

Composition(CASSCF)(%) l*i

lr

a T2 bxT2

a

ASCF" LDA 21880 34 070 32 400 45 950

6

TDDFT BP/ALDA 22 825 31536 38 310 47 182

SAC-Cr E 20 728 30 003 28 874 46 940

ref 95; b ref 96-values differing by up

I CASPT2 18 066 29 035 29 600 46 779

18 389 27 906 31455 43 433

C

I 2e 48 3

6*2

l*i

6ai

6*2

2e 3 30 9

7*2

7*2

7*2

1 8 23

3 4

5 5

Assignment

l*i->7* 2 6*2^7*2

Experimental band position^ (cm"1) 18 300 28 000 32 200 43 956

293

are compared to the experimental band positions, shown in the rightmost column. Based on symmetry considerations it can easily be shown that the O 2p23—Mn 3d1 manifold can give rise to five lT2 states corresponding to one of the following excitations, or to a mixture of them: lti—>2e, 6£2—>2e, lii—>7t2, 6ai—>7t2, 6t2—>7£2- The calculated results in Table 4 have been ordered such that their principal character corresponds to the assignment in the second column from the right. One should however keep in mind that, with exception of the ASCF method, all methods describe the different excited states as a mixture of different orbital replacements. For example, with TDDFT the state b : T 2 which is assigned as 6£2—>2e in table 4 in fact consists of 63 % 6t2—>2e and 36 % l*i —>7t2 character, whereas the c *T2 state, assigned as lii—>7i2, is composed of 50 % lii—>7t2, 17 % 6t2—>2e and 20 % 6t2^-7t2 character [96]. Table 4 also shows the contribution of the five singly excited configurations to the CASSCF reference wavefunctions correponding to the calculated CASPT2 excitation energies. For one thing, these numbers reflect the extreme importance of nondynamic correlation and corresponding multiconfigurational character of the different state functions in permanganate, with a total contribution of singly excited character amounting to 51 % for the first a x T 2 state and decreasing further to only 9 % for the d XT2 state. The contribution of the HF configuration in the corresponding CASSCF ground state wavefunction is 58 %. All methods collected in Table 4 agree upon the assignment of the first band in the experimental spectrum to an excitation with predominantly lti—>2e character. Furthermore, as far as concerns single orbital replacements, the CASPT2/CASSCF results also agree with the other methods [96, 98, 99] that the fourth band in fact corresponds to an (almost equal) mix of the 6t2—>7t2 and (M\—>7t2 orbital transitions. However, the assignments of the second and third bands have so far remained controversial. The SAC-CI and ASCF methods assign the second band to \t\-^t7t2 and the third band to 6t2^2e. This also corresponds to the original assignment by Ballhausen and Gray [103], based on indirect evidence from various sources. On the other hand, according to the TDDFT, EOM-CC and also to the present CASPT2 results these assignments should be reversed. In view of the fact that (a) CA SPT2//CA SSC F is the only method that properly accounts for the strong multiconfigurational character of this molecule, and (b) the close correspondence (to within 1000 cm" 1 ) between the CASPT2 excitation energies and the experimental band maxima of all four bands, we hold it most likely that the assignment based on the CASPT2//CASSCF method is correct, and that the discussion on the assignment of the second and third bands in the permanganate spectrum can now finally be closed.

5. SPECTROSCOPY OF REDOX PROTEINS TM atoms or ions coordinated in proteins are usually found in ligand field environments consisting of very specific, sometimes remarkably complex ligands. The role of this environment is of course to provide the metal with the appropriate electronic structure to play its part in the catalytic cycle. In the case of redox active metalloenzymes an important task for the ligand environment is to modulate the metal reduction potential, the latter determining the driving force of the electron transfer reaction. The reduction potential is dependent on the ionization energy

294

of the reduced site as well as on the so-called reorganization energy, i.e the energy required to reorganize the ligand environment (inner-sphere reorganization energy) and the solvent and the rest of the protein (outer-sphere reorganization energy). Obviously the number, position and specific character of the ligands directly bound to the metal plays a crucial part in all this. Thus, strongly covalent interactions between the metal and one or more ligands will facilitate oxidation by lowering the effective charge on the metal. Moreover, the more strongly antibonding the redox active orbital the higher its energy and the easier ionization from this orbital. Since the same factors also determine the spectroscopic characteristics of the metal-ligand combination, electronic spectroscopy provides an excellent means of obtaining information concerning the catalytic potential of the metal active site in redox proteins. In the remaining of this section we will discuss the electronic structure of the active site in two important groups of redox active metalloenzymes, based on the computation and interpretation of electronic absorption spectra of model complexes. 5.1. Relation between the structure and spectroscopy of blue copper proteins Blue copper proteins are electron transfer proteins. The active site of all these proteins contains a single copper ion (with oxidation state varying between I and II), four- or five-coordinated with a typical short bond to the thiolate S of a cysteine residue. Two of the other ligands are N s atoms of histidine residues, and the fourth is normally a methionine thioether group. (Azurins have a fifth, weakly bound carbonyl oxygen ligand). A typical example is plastocyanin, a small protein involved in photosynthesis. In the oxidized form, the Cu(II) ligand surrounding in this protein is trigonal, the two histidine-N and cysteine-S forming a (distorted) trigonal plane, with the methionine S bound in an axial position at a large distance. Such a trigonal surrounding is quite peculiar for Cu(II): four-coordinated inorganic complexes of this ion are almost invariably (more or less distorted) square planar [32] (e.g. the square planar Cu(II) coordination in zeolites, discussed in section 3.2) . Instead, the trigonal Cu(II) surrounding in plastocyanin is closer to the (distorted) tetrahedral surrounding found for the reduced Cu(I) form of the same protein [104]. The close resemblance between the two geometries accounts for a small reorganization energy accompanying reduction, and hence to a high rate of electron transfer in this class of proteins [105]. This fact originally led to the suggestion that the unusual trigonal surrounding of the oxidized site is not the one preferred by Cu(II) itself, but is instead imposed upon it by the tertiary protein structure, in order to facilitate electron-transfer, the induced-rack [106, 107] andentatic state [108, 109] theories. However, DFT structure calculations on different realistic models of the Cu(II) surrounding but lacking the rest of the protein (e.g. Cu(imidazole)2(SCH3)(S(CH3)2)+) [9, 110, 111] have indicated instead that a trigonal structure is indeed what is preferred by the specific choice of Cu(II) ligands in plastocyanin. The ligand which is crucial in this respect is the Scv« thiolate group [112]. By forming a 7r-bond between the two lobes of its 3pw orbital and two of the lobes of the Cu 3d orbital in the trigonal plane, the S cy« ligand in fact plays the role of two ligands in an "apparent" square-planar coordination [113]. Electronic structure calculations further indicated that the Cu-Scy., yr-bond is highly covalent. The antibonding Cu 3d-S 3pw combination

295 involved is the orbital which is singly occupied in the Cu(II) ground state. Therefore, this is also the redox active orbital and its high covalency activates the blue copper site for its biological function as electron transfer site [9]. The highly covalent Cu-S^j,., 7r-bond in plastocyanin is also responsible for the appearance in the electronic spectrum of the "blue" band, i.e. an intense (e,,,.aa:=3000-6000 M~1cm~1) absorption band around 600 nm. This has been shown in several theoretical studies of the electronic spectrum of plastocyanin, where the blue band was found to originate from a charge-transfer excitation from the bonding to the antibonding Cu-thiolate IT orbital, gaining its high intensity from the large overlap between the two orbitals involved [114117]. 5.7.7. The spectrum of plastocyanin The electronic spectra of several blue copper proteins have been recorded by visual and nearinfrared absorption spectroscopy, circular dichroism and magnetic circular dichroism [9,115, 118-120]. For plastocyanin in total nine different absorption bands have been reported [115]. They all correspond to excitations of an electron into the Cu 3d-S 3pn antibonding combination. Four of these are LF transitions, i.e. where the electron originates from one of the other molecular orbitals with (predominantly) Cu 3d character. The other five are LMCT excitations, with the excited electron coming from a molecular orbital which is mainly centered on one of the ligands. The spectrum of plastocyanin was calculated by performing CASPT2/CASSCF calculations on different model systems, ranging from [Cu(NH3)2(SH)(SH2)]+ to [Cu(imidazole)2 (SC2H5)(S(CH3)2)]+ [117]. In order to include the important 3d radial correlation effects in the Cu2+ ion the active space of these calculation necessarily had to include a second d-shell (i.e. the double-shell effect; see also section 2). Since the model complexes which are closest to the real protein necessarily are unsymmetric, the size of the active space that could be used was limited to twelve orbitals, i.e two Cu d-shells, and the Scys 'ipa and 3pn lone pair orbitals. This active space allowed for the description of the lowest energy part of the spectrum, up to 20 000 cm"1, containing the ligand field states and the two most intense charge-transfer states, both originating from Scy!i- Two additional charge-transfer transitions at higher energy could be calculated for model systems that were forced into Cs symmetry, and were shown to originate from either imidazole or SMC*- For a more detailed discussion of these transitions and the corresponding experimental band positions we refer to ref. [117, 121]. The most accurate CASPT2 results for the spectrum of plastocyanin were obtained for the [Cu(imidazole)2(SH)(SH2)]+ model by making use of the plastocyanin crystal structure [122] but with CASPT2 optimized Cu-S^s a n c ' CU-SMC* distances, and after including the effect of the protein environment by means of a point-charge model [123]. These results are shown in Table 3. The ground state and the different excited states may most easily be characterized by analyzing the singly occupied natural orbital in each state. These orbitals are shown in Fig. 4. By choosing a coordinate system with the z axis along the Cu-S Met bond and the Cu-Sc,ys bond located in the xz plane, the Cu 3d orbitals involved can (approximately) be labeled as shown in the left column of Table 3.

Is)

Table 3 Comparison between the calculated and experimental spectra of plastocyanin and nitrite reductase " State Character Plastocyanin Nitrite reductase exp. exp. calc. calc. bA LF (Cu-S)o-*^(Cu-S)7r* 4 363(.0001) 4 408(.0000) 5 000(.0000) 5 600(.0000) 11 645(.0000) 10 800(.0031) 11 900(.0026) 12 329(.0003) cA LF Cu3dz2 —>(CU-S)TT* dA LF Cu3dyz -^(CU-S)TT* 12 981(.0025) 12 800(.0114) 13 500(.0086) 12 872(.0004) 12 671 (.0002) 13 95O(.OO43) 14 900(.0101) 13 873(.0028) eA LF Cu3dxz -KCU-S)TT* fA CT ( C U - S ) T T ^ ( C U - S K 15 654(.1162) 16 700(.0496) 17 550(.0198) 15 789(.0325) CT (CU-S)CT^(CU-S)TT* 21 974(.0012) 21 39O(.OO35) 21 900(.0299) 22 461(.1192) gA

Character LF (CU-S)TT*->(CU-S)CT *(+7T*)

LF LF LF CT(CU-S)TT^(CU-S)CT* (+7T*) CT(CU-S)CT->(CU-S)CT* (+7T*)

"Calculations were performed on a [Cu(imidazole)2(SCH3)(S(CH3)2)]+ model at the crystal geometry but with the Cu-Scya optimized at the CASPT2 level

and Cu-Sj^ e t distances

297

^

XA

bA

eA H Fig. 4. Plot of the singly occupied (natural) orbital in the ground state and the lowest excited states in [Cu(imidazole)2(SH)(SH2)]+ at the plastocyanin structure

As one can see, the ground state singly occupied orbital is indeed strongly delocalized between Cu 3dxv and Scy., 3p^, thus confirming the highly covalent character of this interaction. The four lowest excited states in the spectrum may formally be characterized as ligand field states. However, from Fig. 4 it is clear that all four in fact also contain a significant amount of charge-transfer character. Even if the departing orbitals in the second to fourth transition are almost pure Cu 3d, the accepting orbital is partly Scys 3pn such that these transitions involve a significant movement of charge from the copper into Scy.,. More importantly, the lowest excited state has a single electron in an orbital which is again strongly delocalized over the CuS<7y., bond, only now involving Cu 3dx2_y2 and Scya 3pCT. According to these CASPT2 results, the first band at 5 000 cm" 1 in the spectrum of plastocyanin therefore involves the excitation of an electron from the Cu-Sc,,., c* to IT* orbital. It should be noted that the CASPT2 assignment is different from the one reported by Solomon and coworkers [9,115], who instead assign the band at 5 000 cm" 1 as the excitation out of the Cu 3dz2 orbital and the second band at 10 800 cm" 1 as the excitation out of Cu 3d,l;2_y2. Fig. 4 also shows that the singly occupied orbitals in the highest two excited states included in the CASPT2 calculations are the bonding counterparts of the corresponding orbitals of the ground state and the first excited state respectively. As such, the first charge-transfer state at 16 600 cm" 1 in the spectrum of plastocyanin corresponds to a vr-to-Tr* Cu 3d-Scy« 3p^ transition. The large overlap between the two combinations involved explains the extremely high intensity of this transition. At higher energy a second charge-transfer band, corresponding

298

to a a-to-TT* transition, is predicted to appear with a considerably lower intensity, as is confirmed by the experimental spectrum. The CASPT2 results presented in Table 3 undoubtedly represent the most accurate set of computational results and analysis of the plastocyanin spectrum available today. Older theoretical studies, performed using the HFS-SW (Xa scattered-wave) DFT method [114, 115] and the semi-empirical CNDO/S method [116], could provide a qualitative but no quantitative explanation of the experimental spectrum. A DFT study of the spectrum was recently reported [9], containing results obtained with different approaches (ASCF, TD-DFT), functionals (GGA and hybrid) and basis sets (STO and GTO). Calculations were performed on the [Cu(imidazole)2(SCH3)(S(CH3)2)]+ model, fixed at the crystal structure of oxidized plastocyanin [122]. As it turns out, TDDFT completely fails in describing the electronically excited states of this model: all ligand field transitions are calculated much too high in energy, whereas the charge-transfer states are calculated much too low and in the wrong order. E.g. the lowestenergy band in the plastocyanin spectrum is predicted by TDDFT to be a SM<-J.—>Cu3d excitation ! According to the authors [9], the failure of DFT to describe the plastocyanin spectrum should be brought back to an overly covalent bonding description of the ground state, with a singly occupied HOMO containing too much S cys and too little Cu 3d character. A reasonable ASCF (BP86) description is obtained after introducing an adjusted effective nuclear charge on the central copper ion. 5.1.2. From blue to green Cu proteins The appearance of two S>cya—>Cu3d charge-transfer bands is characteristic of the electronic spectrum of all blue copper proteins, providing them with the label type 1 copper proteins (as opposed to type 2 copper proteins, the spectrum of which only contains a number of weak ligand field transitions in the same region). The band positions of the two transitions, around 600 nm (16 600 cm"1) and 450 nm (22 000 cm" 1 ), remain approximately the same for all proteins. However, the relative intensity of the two bands varies between different proteins [124, 125]. Thus axial type 1 proteins, like plastocyanin and azurin, show only little absorption in the 450 nm region, while the 450 nm band becomes much more prominent in rhombic type 1 proteins like pseudoazurin and cucumber basic protein (the classification of blue Cu proteins as axial or rhombic is based on their ESR characteristics). The increasing intensity of the 450 nm band in the latter proteins goes together with a decrease in intensity of the 600 nm band, so the sum of e^omn and eeoonm remains approximately constant [124]. A limiting case is nitrite reductase from Achromobacter cycloclastes. The intensity of the 600 nm absorption peak is in this enzyme reduced by a factor 3 compared to the classic proteins, and the 460 nm absorption has actually become the more intense, resulting in a green color of the enzyme. The gradual change in the spectral characteristics of different blue copper proteins goes together with a structure which is gradually more distorted with respect to the trigonal Cu(II) surrounding in plastocyanin. The distortion comes down to a flattening, and can most conveniently be quantified by considering the angle cj> between the planes formed by the N ffiil-CuNiH,, bonds and the S ^ - C U - S M C * bonds respectively [123]. In a strictly trigonal structure,

299 both planes are perpendicular ((/> = 90°). In practice, axial type 1 proteins have angles ranging between 77° and 89° (e.g. for the poplar plastocyanin described in the previous section =82°). On the other hand, in the rhombic type 1 proteins significantly smaller angles are found, ranging between 70° and 75°. Again, nitrite reductase is a limiting case: different crystal structures of this enzyme have cf> angles ranging between 56° and 65°. Together with the flattening of the structure subtle variations of two important bond distances are observed: the Cu-S Met bond is considerably shortened in nitrite reductase as compared to plastocyanin, from 2.88 A to 2.56 A, whereas the Cu-Scj,., bond distance is increased, from 2.11 A to 2.17 A. In order to (qualitatively) investigate the relation between the structural variations between different blue copper proteins on the one hand and the variations in the relative intensity of the 460 nm and 600 nm bands on the other hand, test calculations were first performed on a small model [Cu(NH3)2(SH)(SH2)]+. A series of (B3LYP-DFT + restricted CASPT2) geometry optimizations was performed on this model at fixed <j> angles, ranging between = 90° (i.e. a trigonal structure) and <j> = 0° (i.e. a square-planar structure). A CASPT2//CASSCF calculation of the electronic spectrum was performed at different points along the twisting path. From these small model calculations, several important points were noted, all of them highly relevant for the structure-spectroscopy relationship in the actual proteins. First of all, these calculations indicated that the energy curve along the <j> twisting angle is extremely flat. At the CASPT2 level two minima were found, one for =90 ° (i.e. close to the plastocyanin ) and one for — 50° (i.e. close to the nitrite reductase cj>). Yet, the barrier between these two minima is low: only 8 kJ/mol. This explains how different protein environments are in fact able to stabilize a continuum of Cu(II) surroundings [9], ranging from the trigonal structure in plastocyanin to a distorted tetragonal structure in nitrite reductase. A second point emerging from these calculations (see also Fig. 5) is that the optimized Cu-SH and CU-SH2 distances indeed show the trends also observed for the actual proteins: the Cu-SH2 distance is gradually decreased by more than 0.3 A as 0 is reduced from 90° to 50°, while the Cu-SH bond is simultaneously elongated by about 0.1 A. As for the electronic spectra obtained for these small models, the relative intensity of the two HS—>Cu transitions, also indicated in Fig. 5, indeed follows the trend observed for the proteins. This trend can now also be related to the electronic structure of the ground state at different 4> angles, as reflected by the nature of the singly occupied orbital. In the trigonal ( = 0°) structure this orbital displays a TT antibonding interaction between copper 3d and the SH ~ S 3p orbital, similar to plastocyanin (see also the previous section and Fig. 4). As such, an intense TT-to-Tr* transition is calculated for this structure, whereas at higher energy the second, a-to-vr* transition, is calculated with little intensity. However, as angles between 90° and 0° an intermediate electronic structure is obtained, with the ground-

300 25

I

210

60 Twisting angle ( )

80

Fig. 5. Cu-SH (squares) and CU-SH2 (diamonds) distances, and calculated quotient of the oscillator strengths of the transitions around 460 nm and 600 nm (bullets), as a function of the cj> angle for the [Cu(NH3)2(SH)(SH2)]+ model.

state singly occupied orbital gaining more and more Cu-SH a character (at the expense of TT) as the structure becomes more flattened. As a consequence of this changing ground state electronic structure, the second charge-transfer transition in the electronic spectrum, originating from the Cu-SH a bonding orbital, gains intensity at the expense of the first transition, originating from Cu-SH IT. The "crossover", i.e. the point where the second transition becomes more intense than the first, is for [Cu(NH3)2(SH)(SH2)]+ predicted at cf> around 74°. As such, even these small model calculations predict nitrite reductase to be green. Armed with the above qualitative considerations, the analysis and assignment of the lowenergy part of the experimental spectrum of nitrite reductase becomes straightforward. CASPT2/CASSCF calculations of this spectrum were performed in a similar way as for plastocyanin, i.e. using the [Cu(imidazole)2(SH)(SH2)]+ at the nitrite reductase crystal structure of Achromobacter cycloclastes [126] but with CASPT2 optimized Cu-S>cys and Cu-S^d distances, and a charge model for the protein environment [123]. The results are compared to the experimental spectrum [127, 9] and the corresponding data for plastocyanin in Table 3. The singly occupied natural orbitals characterizing the ground and each of the excited states in nitrite reductase are shown in Fig. 6. As one can see, the ground state singly occupied orbital in the nitrite reductase model has indeed become more of a* than of TT* type, and vice versa for the corresponding orbital of the first excited state. Interestingly, the corresponding bonding orbitals, depopulated in the two charge-transfer states, do retain their pure a or TT character. As such, the intense bands

301

Fig. 6. Plot of the singly occupied (natural) orbital in the ground state and the lowest excited states in [Cu(imidazole)2(SH)(SH2)]+ at the nitrite reductase structure

in the nitrite reductase spectrum should be assigned as Scys—>Cu 3d 7r-to-(<7* + TT*) (17 550 cm"1) and a-to-(a* + TT*) (21 900 cm"1), respectively. As for plastocyanin, four ligand field states appear at lower energies. The experimental ligand field transition energies are significantly increased in nitrite reductase as compared to plastocyanin, indicating a strengthening of the ligand field as the Cu(II) surrounding becomes more tetragonally distorted [9]. This trend is also confirmed by the theoretical calculations [123]. Finally, the observed interchange of a- and vr-character between the ground and the first excited state with a decreasing twisting angle indicates that the flat ground state energy profile along this angle is in fact the result of a (strongly) forbidden transition between these two states. This also provides further credit to the fact that the lowest-energy band in both spectra should indeed correspond to the exchange of an electron between a* and TT* within the Cu-S^., bond, as opposed to Solomon's assignment of this band as a ligand field transition originating from Cu 3dzz. In summary, the model calculations presented in this section have clearly demonstrated how subtle changes in the structures of these blue copper proteins have a profound effect on the electronic structure of the Cu(II) environment and as such also on their spectroscopic characteristics. We believe that this case presents a nice example of the strength of accurate theoretical methods for both ground and excited states in explaining and predicting the electronic structure properties of biologically important molecules.

302 O

H

2

N

,,

N

SH

N

O

4

3

Fig. 7. Structure of the pyranopterin derived from protein crystallographic studies (shown in protonated form).

5.2. Electronic structure of oxomolybdenum enzymes modelled by the (Tp)MoO(bdt) complex Apart from the nitrogenases, all molybdoenzymes are oxotransferase enzymes, i.e. they catalyze the net transfer of an oxygen atom to or from a substrate: X + H2O « X O + 2H+ + 2e" e.g. sulfite to sulfate, an aldehyde to the corresponding carboxylic acid, or nitrite to nitrate. During enzymatic turnover, the molybdenym center is proposed to shuttle through Mo(VI)/ Mo(V)/ Mo(IV) oxidation states. All these enzymes, further subdivided into three families based on structure and reactivity (xanthine oxidase, DMSO reductase, and sulfite oxidase), contain a mononuclear molybdenum center ligated by one or two molecules of a special pyranopterin, originally termed "molybdopterin" (MPT: see Fig. 7). MPT consists of a pterin core that is coordinated to the metal atom via the dithiolene group on the four-carbon side chain. The mere fact that MPT invariably is a part of the active site strongly suggest that this ligand plays an active role in the catalytic process, by acting as an electron transfer conduit from the metal to other prosthetic groups [128], and by modulating the Mo center reduction potential [128-131]. At least one terminal oxo ligand is associated with the Mo active center during catalysis. The application of electronic absorption spectroscopy to study the electronic structure of the molybdenum center in these enzymes is hampered by the presence of additional prosthetic groups (e.g. hemes, iron sulfur centers and flavins) whose intense electronic absorptions obscure any absorbance that might arise from the oxomolybdenum center of the pterin-containing molybdenum cofactor. On the other hand, systematic spectroscopic investigations and analysis of well-characterized model complexes containing the molybdenyl ([MoO]3+) fragment have since long [132, 133] been used to probe the electronic structue and molybdenum-ligand bonding in the considered enzymes. An important class of such model compounds are of the type (Tp*)MoO(S-S), where Tp* = hydrotris(3,5-dimethyl-l-pyrazolyl)borate and (S-S) is a dithiolate ligand which forms a five-membered ring with Mo(V). Experimental absorption and CD spectra of (Tp*)MoO2+ centers with various S-S ligands (e.g. bdt = 1,2-benzenedithiolate, tdt = 2,4-toluenedithiolate, edt = 1,2-ethanedithiolate, qdt = quinoxaline-2,3-dithiolate, bdtCl 2 = 3,6-dichloro-l,2-benzenedithiolate) have been reported and assigned [128, 134, 135]. In a forthcoming publication we will present the results of a comparative computational study of these spectra [136]. Here we report a representative case, namely (Tp)MoO(bdt) with bdt =

303

Fig. 8. B3LYP-DFT optimized structure of (Tp)MoO(bdt)

1,2-benzenedithiolate, whereas in Tp the 3,5 dimethyl groups of Tp* are replaced by hydrogen atoms to simplify the computations. The structure of this molecule was first optimized with B3LYP-DFT. These calculations used 6-31G(d) basis sets for all atoms except molybdenum, the core region (up to 3d) of which was described by a relativistic ECP [137], whereas for the valence region a triple-zeta basis set with one f-type polarization function was used. The optimized structure is shown in Fig. 8. For the calculation and description of the electronic spectrum the complex was oriented such that the axial M=O bond lies along the z-axis, whereas the mirror plane perpendicular to the ene-dithiolate plane corresponds to the xz plane. The ligand field splitting of the d-orbital manifold in (Tp*)MoO(S-S) is dominated by the presence of a short, strongly covalent axial M=O bond. In particular, a interaction between the Mo dz2 and O pz orbitals and IT interaction between Mo dXZ4)z and O px/!J orbitals give rise to bonding-antibonding pairs of molecular orbitals, thus resulting in a strong destabilization of the antibonding, primarily metal based combinations. In the equatorial plane the Mo dxy orbital is involved in and destabilized by a interactions with two N and two S-donor ligands. This then leaves the dx2-V2 orbital as the Mo d orbital which is least destabilized by the ligand field, and which therefore becomes singly occupied in the Mo(V) ground state. This is also the redox active orbital, changing its occupation number between zero and two during the catalytic Mo(IV)-Mo(VI) cycle. Next to the ligand field transitions, the electronic absorption spectrum of Tp*MoO(S-S) complexes is characterized by a number of low-lying LMCT transitions originating from the highest four filled dithiolate orbitals, primarily sulfur in character, and schematically presented in Fig. 9. Two of them (denoted as a'(S,;?,) and a"(S,j,)) are oriented within the dithiolate plane, while the other two (denoted as a (SOJ)) and a (Sop)) are orthogonal to this plane. The a "(S,;2,) orbital is involved in covalent a bonding with the Mo 4dxy orbital. In order to describe the entire low-energy part of the spectrum by means of a series of CASPT2/CASSCF calculations, the active space was chosen to include the five orbitals with

304

3 \&op) Fig. 9. Schematic plot of the highest occupied molecular orbitals of S-S = ene-l,2-dithiolate. The small amplitudes of the wavefunction on the carbon atoms are not shown

predominant Mo Ad character and the four dithiolate orbitals a (Sip), a (S,;,,), a (Sap), a (SO!,). Furthermore, important correlation effects connected to the strongly covalent Mo=O bond were treated at the CASSCF level by including also the relevant Mo 4d—O 2p bonding molecular orbitals. This sums up to a total of twelve active orbitals containing 15 electrons. The CASPT2//CASSCF calculations were performed using ANO type basis sets [136]. Oscillator strengths were obtained using the RASSI method [138]. The results obtained for the spectrum of (Tp)MoO(bdt) are shown in Table 4. A plot of all orbitals involved in low-lying excitations is provided in Fig. 10. Three different groups of orbital excitations may be distinguished in the table. First, there are the LF excitations of the single electron from Mo dx2_y2 into the orbitals with predominant Mo dxz, dyz and dxy character. It is noticeable that dz2 is missing from this list: this orbital is destabilized so strongly by a interaction with the axial oxygen that any excitation into it falls outside the range of the calculated spectrum, i.e. up to about 35 000 cm" 1 . A second group are the LMCT excitations out of the Sop and S,p orbitals into Mo dT2_y2. Such excitations give rise to four excited states, which can be clearly recognized in Table 4 as the states b 2 A', a 2A", e 2A" and f 2A . The third group consists of LMCT excitations out of the two SO2, orbitals into Mo dxz, dyz and dxy. Excitations out of Srp into the same orbitals occur at higher energies, outside the range of the calculated spectrum. Each of the excitations in the third group can occur with two different spin couplings (with the unpaired electrons on the metal either singlet or triplet coupled), such that the total number of configuration state functions arising from excitations within this group amounts to 12. Together with the first two groups this brings the total number of excited states included in Table 4 to 19. As one can see, most of the calculated excited states in fact correspond to a mixture of different orbital excitations. A detailed experimental study of the electronic absorption and MCD spectra of (Tp*)MoO(bdt) was presented in 1999 by Inscore et al. [128]. The low-energy region of the spectrum consists of three distinct features at 9 100 cm" 1 , 13 100 cm" 1 and 19 400 cm" 1 . This band pattern seems to be characteristic for all (Tp*)MoO(S-S) complexes reported to date, which possess five membered chelate ring formed between the ene 1,2 dithiolate and molybdenum atom [128, 130, 134, 135]. The bands at 9 100 cm" 1 and 13 100 cm" 1 were assigned as LMCT transitions from the Sop orbitals into the singly occupied Mo 4dx2_y2 orbital. This assignment is confirmed by our calculations, which now also further establish the energy ordering of the

305 Table 4 Comparison between the calculated spectrum of (Tp)MoO(bdt) and the experimental spectrum of (Tp*)MoO(bdt)"

b 2 A' a 2 A" b 2A" c 2 A' c 2A" d 2 A' d 2 A" e 2 A" f 2A" e 2 A' g 2A"

calculated results exc.energy* main character" (osc.str.) 10 912(.O168) a (S OJ) )^a ( d ^ . , / ) 85 % 14 326(.0147) a"(Sop)->a'(clr2_!/2) 88 % 16 457(.0018) a'(d^- y O^a"(d, y ,) 45% ; a'(S OJ) )^a"(d y J 38 % 18 670(.0106) a'(d3;2_!;2)->a'(d:l!j 48% ; a'(SOj,)^a'(da.z) 36 % 21 657(.0068) a'(SOI,)->a"(d,,J 58 % ; a(dx2_y2)^a"(dyz) 26 % 23 029(.0002) a"(S op )^a"(d,, z ) 57 % ; a"(SOJ,)^a"(d,,y) 20 % 23 131(.O257) a"(S O! ,)^a'(d ci ) 74 % ; a'(d3;2_y2)->a"(d.1/,) 5 % 23 276(.0023) a"(S,;j,)^a'(d:c2_,y2) 90 % 23 467(.0044) a'(S o} ,)^a"(d y J 71 % ; a'(dal2_!/2)^a"(da;!/) 6 % 23 985(.0431) a'(S OJ) )^a'(d,, 0 ) 64 % ; a'(d:i;2_,y2)^a'(d,,z) 21 % 25 655(.0038) a (Sop)^a" (dyz) 17 % ; a(Sop)^a"(dxy) 39 %

f 2 A' h 2 A"

26 274(.0006) 26 308(.0144)

State

a'(S,:j,)->a'(d3:2_v2) 89 % a"(S o ,,)^a"(d 3 , z ) 50 % ; a'(d c 2_,, 2 )^a"(d,, z ) 21 % ;

g 2 A' 27 030(.0111) a'(S OJ ,)^a'(d l;0 ) 71 % ; a'(d :l; 2_, y 2)^a'(d l;z ) 14 % h 2 A' 27 395(.0131) a"(S o ,,)^a"(d,,J 62 % ; a"(SOj,)-J-a"(d:,:iy) 18 % i 2 A" 33 270(.0146) a (So,,)^'(d^y) 72 % ; a'(da!2_v2)^a"(d;!1!y) 13% i 2 A' 34 121(.0043) a"(S o p )^a"(d : r ; / ) 66 % ; a " ( S q , ) - » a " ( g 13 % j 2 A" 36 715(.0011) a.'(Sop)^'(dvy) 45 % ; a'(da!2_!/2)^a"(d;I!y) 35% j 2 A' 37 284(.0016) a"(S o p )^a"(d : r ; / ) 63 % ; a"(S^^a (dyZ) 17 % "Calculations were performed on the B3LYP-DFT structure Excitation energies are given in cm ~ : r Only the main contributions (> 5%) are shown

experiment exc.energy h (osc.str.) 9 100(.0056) 13 100(.0033) 15 800 (-) 19 400(.0160)

22 100(.0170)

25 100(.0920)

306

a(d,.2_y2)

a {dxz)

a (SOJ))

a'(So,,)

a"(S, :p ) Fig. 10. Plot of the (natural) orbitals involved in the lowest electronic excitations in (Tp)MoO(bdt)

307

Fig. 11. Schematic plot of the three-center pseudo-a bonding and antibonding combinations formed by the Mo Ad.xi-yi and a'(S,;p) orbitals

excitations out of a (Sop) and a (Sop), the former being the lowest. The calculated transition energies, 10 912 cm" 1 and 14 326 cm" 1 are 1 200-1 800 cm" 1 too high, whereas the oscillator strength of both transitions is grossly overestimated by the calculations. Nevertheless, there can be little or no doubt concerning the character of the lowest two excited states in these spectra. The same is not true for the feature at 19 400 cm" 1 . Although this band was originally (for (Tp*)MoO(tdt)) assigned as the Mo 4dx2_y2->Adxy LF transition [134], all later studies [128, 130, 135] report the assignment of the 19 400 cm" 1 as the in-plane a'(Sj,,->Mo 4rfa.2_y2 LMCT transition. The latter is based on the relatively high intensity of this band, which supposedly originates from a large overlap between the donor and acceptor orbital involved. In order to explain such overlap it was suggested that both orbitals are involved in a highly covalent threecentered pseudo a bond, as shown schematically in Fig. 11. However, our calculations do not give any indication for the existence of such a three-centered bond. As can be seen from Fig. 10 the Mo 4rfiI2_y2 and Srp orbitals do not show any significant mixing. Correspondingly, the calculated oscillator strength of the a'(S,p)->Mo 4dx2_y2 excitation is very low, 6 x 10~4. This excitation is predicted to occur at 26 274 cm" 1 (state f 2A' in Table 4). The corresponding a"(S,;}))—>Mo 4dx2_y2 excitation is predicted at 23 276 cm" 1 with an oscillator strength of 2.3 x 10"3 (state e 2A" in Table 4). Both transitions should in the experimental spectrum be obscured by the occurrence in the same region of much more intense transitions originating from a (SO2,) and a"(Sop. The present results therefore throw a new light on the possible role of the pyranopterin dithiolate Sn, orbital in the catalytic activity of oxomolybdenum enzymes, either acting as an electron transfer conduit or modulating the molybdenum reduction potential. Facile electron transfer in general requires that there be a high degree of electronic communication between the donor and acceptor sites. In ref. [128] it was suggested that the highly covalent three-center a interaction between the a'(S,{)) orbital and the redox active Mo 4dx2_y2 orbital should be involved in and responsible for efficient pyranopterin mediated electron transfer. The in-plane nature of this interaction is convenient, since it may be expected to give a minimal contribution to the reorganization energy involving distortions along the apical M=O bond during electron transfer reactions. Furthermore, this effective in-plane covalency was also anticipated [128]

308 to play an important role in modulating the electrochemical potential of the active site, by destabilizing the energy of the redox active (predominantly) Mo Ad,r2_y2 orbital. The latter role was however put into perspective by a number of comparative studies of (Tp*)MoO(S-S) complexes with different (S-S) ligands, such as tdt, qdt or bdtCl2. The electronic absorption spectra of these complexes all contain the same band around 19 400 cm" 1 with a very similar transition energy and intensity. This would suggest (still assuming that this band indeed corresponds to a transition between the bonding and antibonding combinations of Mo d X2_y2 and a'(S,j,)) that the covalency of the three-center pseudo a bond and its destabilizing effect on the redox active orbital is indifferent to the character of the (S-S), in particular on the presence of electron withdrawing or donating groups. On the other hand, electrochemical and He(I) PES studies [130, 131, 135, 139] have indicated that both the ease of oxidation and reduction of the Mo(V) center is indeed strongly affected by the presence of such groups. In particular, these studies indicate that the redox active orbital is stabilized as the ene 1,2 dithiolate ligand becomes more electron withdrawing, thus making ionization more difficult and reduction easier. Based on these observations it was proposed that instead anisotropic covalency contributions involving only the Sop orbitals of the coordinated thiolate control the Mo reduction potential by modulating the effective nuclear charge of the metal. In refs 135, 140 it is suggested that a regulating factor of this interaction is the S-S fold angle of the ene dithiolate chelate ring, i.e. the angle between the S-C=C-S and S-Mo-S planes. Folding the S-C=C-S plane upwards in the direction of the M=O bond leads to an increased overlap between the the a (Sop) and the Mo idx2_y2 orbital, thus giving rise to a reduced effective charge on the metal. Such overlap is indeed manifested by the plot of the bonding a'(SOJ)) orbital in Fig. 10, although it is virtually absent in the corresponding antibonding orbital, i.e. the ground state singly occupied orbital a(d.j.2_y2). The role of the SOJ) orbitals and the effect of the S-S fold angle on the electronic structure of these (Tp*)MoO(S-S) complexes and pyranopterin Mo enzymes they stand model for certainly warrants further investigation. Returning to the spectroscopic data for (Tp)MoO(bdt) in Table 4 and to the band at 19 400 cm"1 in particular, our calculations indicate that this band should instead be assigned as the Mo dx2-y2—>dxz LF transition, which is however strongly mixed with a'(Sop)—>dxz LMCT character. This mixing may explain the intensity of the 19 400 cm" 1 band, although the calculated oscillator strength is significantly lower than the experimental value. A second ligand field transition, Mo dx2_y2—tdyz is predicted to occur at lower energy, 16 457 cm" 1 , again strongly mixed with CT character out of a'(Sop) into the same id orbital. The calculated oscillator strength of this transition is very low. This is in agreement with the appearance of a very weak feature (only discernable in low-temperature MCD) at 15 800 cm" 1 in the experimental spectrum, which is therefore assigned as such. The third LF exitation, Mo dx2_y2^dxy is situated at a considerably higher energy. The lowest excited state containing a significant (6 %) dx2_y2^dxy contribution is the f 2A" state at 23 467 cm" 1 . However, none of the states included in Table 4 can even grosso modo be identified as the transition to dxy, the largest contribution, 35 %, being found for the j 2 A" state at 36 715 cm" 1 . The high energy of the metal dxy orbital is of course related to the strong antibonding character (primarily with S,p but also with the equatorial N u-donor

309 ligands) of this orbital, as is obvious from the orbital plot in Fig. 10. Ranging from 21 000 to 27 400 cm" 1 a whole series of transitions is calculated, corresponding to excitations from the a'(SOJ,), a"(SOJ>) orbitals into Mo dxz, dyz. In the experimental spectrum, two intense bands are distinguished in this region: a shoulder at 22 100 cm" 1 and a very strong band at 25 100 cm" 1 . They are assigned to the two excited states for which the calculations predict the highest oscillator strength: d 2A", calculated at 23 131 cm" 1 and e 2A', calculated at 23 985 cm" 1 . Both states primarily correspond to an excitation into Mo dxz. Corresponding excitations to Mo dyz are calculated at slightly lower energies but with a much lower oscillator strength. Finally, excitations out of the a'(SOj,), a"(SO2,) orbitals into Mo dxy are predicted to occur between 33 000 and 37 200 cm" 1 , i.e. outside the range of the experimental spectrum. 6. CONCLUSION In this chapter we have described a few theoretical studies of electronic spectra of transition metal ions in different coordination environments. We believe that these studies have convincingly demonstrated the strength of accurate computational approaches: (i) in explaining and predicting the relative position and intensities of different types of excited states (LF or CT) in these spectra; (ii) in relating this information to the ground state electronic structure and bonding characteristics of different metal-ligand combinations; (iii) in relating to the ground state electronic structure and spectroscopic characteristics other important properties such as the geometrical structure of the coordination environment or the possibility for redox activity of the central metal ion. An illustrative example is presented by the calculations on the Cu(II) ion in two thoroughly different coordination environments: the ionic oxygen ligand surrounding of a zeolite surface on the one hand, leading to an almost strictly square planar Cu(II) surrounding and a typical ligand field spectrum, and the trigonal Cu(II) surrounding in blue copper proteins on the other hand, caused by the presence of a highly covalent Cu-S cVa "" bond, the latter also giving rise to a low-lying intense LMCT band, i.e. the "blue" band. The calculations on (Tp)MoO(bdt) have indicated that the Mo(V)-dithiolate bond in these model complexes is in fact highly ionic, as opposed to the earlier suggested presence of a covalent three-center S-Mo-S pseudo a bond. The question of course remains whether this complex is indeed a representative model for pterin-containing molybdenum enzymes and how to explain the redox activity of such enzymes. Finally, the calculations on MnO J have demonstrated how the CASPT2/CASSCF method, used throughout this work, is able to provide very accurate results also for the notoriously difficult-to-calculate spectrum of this molecule, provided that the active space is chosen large enough to include all (very important) nondynamic correlation effects.

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X. Perspectives in Calculations on Excited State in Molecular Systems Bjorn O. Roos Department of Theoretical Chemistry, Chemical Center P.O.B. 124, S-221 00 Lund, Sweden 1. INTRODUCTION The electronic Schrodinger equation has for most molecular systems an infinite number of bound solutions. Each of them corresponds to an electronic state with a specific electronic structure. Such a state gets populated when the molecule absorbs one or more quanta of light (photons). When a molecule is brought into one of the excited states, one of several things can happen. The energy surface for the excited state is different from that of the ground state, because the electronic structure has changed. The excitation process is fast, so the atoms will not have time to move. As a result, the molecule will find itself in a non-equilibrium position and will start vibrating. Ultimately, it may end up in the equilibrium geometry of the excited state (which in many cases means dissociation of one or several chemical bonds). That is, if there is time enough because the lifetime of the excited state is usually short and a competing process is emission of a light quantum and return to the ground state energy surface directly or via lower lying electronic states. But the dynamics may be even more complex. We show an example in Fig.l (for simplicity, the energy is shown for one geometrical coordinate only). A photon excites the system from the lowest vibrational level on the ground state energy surface, X, vertically to the first excited state surface, B. This state is bound but with another geometry than the ground state. Another state, A, crosses state B at a larger value for the geometry parameter. The crossing point lies below the energy that the molecule has adsorbed from the photon. So, the molecule has several possibilities. It can stay on surface B and loose the extra energy by thermal dissipation (possibly involving other molecules). Ultimately, it will then loose its energy by emitting a photon of lower energy (fluorescence). Or it can cross over the surface A and dissociate. It can also come back to surface B via the crossing point. Today it is possible to follow such dynamical processes experimentally using time-resolved laser spectroscopy. How can we deal with such problems theoretically and computationally? It is clearly not straightforward and simple. We need to be able to compute with good accuracy the energy surfaces for a number of excited states and we then need to be able to solve the dynamical problem for the nuclear motion on these surfaces. In this chapter we shall deal with the first of these problems. Chapters 5 and 7 will deal with different aspects of the nuclear dynamics.

318

hv

— X

Geometry

Fig.l. Energy surfaces in a hypothetical molecular system. Four electronic states are shown. The energies are given as functions of one arbitrary geometrical variable.

2. A BRIEF HISTORICAL BACKGROUND The history of electronic spectroscopy of atoms and molecules is older than quantum mechanics. Actually much of the experimental background that inspired the development of quantum mechanics in the beginning of the previous century was found in spectroscopy. It was a grand and immediate success of the Schrodinger equation [1] that it could explain with high accuracy the electronic spectrum of the hydrogen atom. The exact solution of this problem gave us the concept of the atomic orbital. It was almost immediately extended to many-electron systems by the introduction of the self-consistent field model, were the electron-electron interaction is modeled using a mean-field approximation. The model was introduced for atoms by R. D. Hartree in 1928 [2]. A similar model was actually proposed for molecules already in 1927 by F. Hund [3]. The approach was immediately used by R. S. Mulliken and others for the interpretation of electronic spectra of small (diatomic) molecules [4]. Today we know this approach as the Hartree-Fock (HF) method. The basic concept is the molecular orbital (MO) and its energy, leading to a shell model for molecules similar (but more complex in structure) to that of the atoms. This theoretical model explains the excited states of a molecule as follows: in the ground state the electrons reside in the MOs of lowest energy (the aufbau principle). If one electron absorbs a photon of the right energy it will move to an MO of higher energy, which is empty or only singly occupied in the ground state. This is a very powerful approach and forms still today the basis for our analysis of excited states in molecular systems. We now know that one has to go beyond the self-consistent field model to obtain quantitative results for excited states but we can always (almost) label them as excitations by one or more electrons from one MO to another. The development discussed above concerned mainly atoms and small molecules. A simple approach that was to become very influential for the development of the theory of electronic spectra of planar conjugated organic molecules was suggested by E. Huckel [5]. Huckel

319

Fig.2. The orbital energy levels for the 7t-electrons in the benzene molecule. simplified the problem of the many electrons by only considering the loosely bound electrons in the 7t-orbitals. The method that emerged was a simple semi-empirical scheme that could be used to compute the structure and energy of these 7i-orbitals. The use of the approach for the interpretation of electronic spectra was, however, limited because the important electronelectron repulsion terms were not accounted for. This problem was realized by M. GoeppertMayer and A. L. Sklar in their landmark study of the electronic spectrum of the benzene molecule in 1938 [6]. In order to understand the importance of the electron-electron interaction, let us take a closer look at the energy levels in benzene. Experimentally, four excited singlet states are known which originates from excitations within the valence 7i-orbitals with energies around 4.9, 6.2, 7.0, and 7.8 eV. The energy levels for the orbitals are shown in Fig. 2. Actually, we can form four excited configurations by moving one electron from an occupied to a virtual orbital and one might think that this corresponds to the four singly excited valence states seen in the electronic spectrum of benzene. This is, however, a too simple picture. Consider the singly excited configuration a ^ e ^ e ^ . We can form 16 Slater determinants for this configuration because of the double degeneracy of the e2U and eig orbitals in the D^ point group of the molecule. 12 of them correspond to triplet states and the four others to singlet states. A little bit of group theory shows that they are: 'B2U, 'BIU, and 'EH, . In a Hartree-Fock picture they have the same one-electron energy, but the electron-electron repulsion energies are different. Goeppert-Mayer and Sklar succeeded to calculate these energies, which was quite an achievement in 1938, even if they were forced to use crude approximations. We refer to their paper for details [6]. What about the fourth state at 7.8 eV? Today we know that this state has gerade symmetry. It can thus be formed from the single excitations aiu—>e2U or eig—»big . We know now that the wave function is a combination of these two configurations, but that also the double excitation efg —^e^ gives a sizable contribution. It is thus an oversimplification to assume that only single excitations are important for the excited states. We shall later see that double excitations are sometimes of crucial importance for the description of the electronic structure. It was to take another 15 years until further progress was made. A rather straightforward extension of the ideas of Goeppert-Mayer and Sklar would be the following: represent each of

320

the Ti-orbitals in a planar organic molecule with one basis function. Determine the MOs by solving a Hartree-Fock like problem. Then, expand the wave function for the excited states, V, as a linear combination of singly (and possibly multiply) excited configurations, O^ starting from the closed shell HF wave function:

(1)

^=1 ^

Use the variational principle to determine the expansion coefficients Q,. 50 years ago it was an impossible task to carry out such a calculation from first principles. Drastic approximations had to be made. One key problem was the calculation of all electron-electron repulsion integrals, which occur in the matrix element of the Hamiltonian:

(pg\rs)= \\^p{rl)^p — ^r{r2)Ur2)dVldV2 r

(2)

12

where <j>p etc. are the molecular 7t-orbitals. Even if each atomic basis function was a single Slater orbital, it was an impossible task to compute these integrals in 1950. Robert Parr then introduced a drastic approximation: the zero differential overlap, ZDO approximation [7]. All products of basis function, which where located on different centra were assumed to be zero everywhere. The only remaining integrals were then of the type (pp|rr) and the calculation of the matrix elements were drastically simplified. The integrals could be computed using a single Slater type orbital to represent the AOs. The one-electron matrix elements were taken from Hiickel theory. However, very accurate excitation energies were still not obtained. Rudolph Pariser [8] suggested that the one-center integrals for carbon should also be determined empirically from the valence state disproponation reaction: 2C->C~ +C+

(3)

The corresponding energy is (pp|pp) = IP - EA, where IP is the ionization energy and EA the electron affinity. Using experimental values for these quantities gave the value 11 eV for the one-center integral (pp|pp), a reduction of 6 eV from the theoretical value. These are the key ingredients in the, so called, Pariser-Parr-Pople (PPP) method to compute electronic spectra of planar conjugated hydrocarbons. It was later extended also to hetero-atoms and actually also to transition metal complexes with some success. For a nice historical review of the development we refer to articles by Pariser, Parr, and Pople in Ref. [9]. The PPP method was an immediate success and it became possible to assign a large number electronic states in conjugated organic molecules. Why was that? What was the underlying reason for the large reduction of the one-center two-electron integral? Actually, this was well understood immediately and Pariser wrote: the effect of the a-electrons can be approximately taken into account by changing the value of primarily one Coulomb repulsion integral [8]. Today we call this effect dynamic polarization. It is maybe easiest demonstrated on the Vstate of the ethene molecule C2H4. While the ground state of this molecule has two electron in the bonding 71-orbital, the V state has the electronic configurations (7ur*)s , where S stands for singlet coupling. It is easy to see that the state can be described (using localized orbitals) as a resonance between the two ionic states C+C~ and C~C+. The a-electrons will react dynamically to this polarization. One can actually show explicitly that if the o-system is

321 replaced by a polarizable medium, the effective Hamiltonian will have reduced values of the electron repulsion integrals. The one-center Coulomb integral is reduced from 16.2 to 11.5 eV, which is exactly what Pariser predicted [10]. It would take a long time before modern ab initio quantum chemistry was able to challenge the results obtained with the PPP method. However, chemistry is not only organic. What about the excited states of inorganic compounds. Actually, there was much less interest in this area with two exceptions, small molecules (mainly diatomics) and transition metal complexes. The chemical bond between transition metal ions and their ligands could not be explained using the classical valence picture. A different theory was suggested by Becquerel [11] and formulated more exactly by Bethe [12]. It was acronymed Crystal Field Theory. The applications in chemistry were initiated by the work of Van Vleck [13]. The metal ion was in this method assumed to be perturbed by an electric field from the surrounding ligands. The spectroscopy therefore was atomic in nature. Transition metal complexes often have high symmetry and much of the success of the theory was based on the possibility to use group theory for the analysis of the excited states. For octahedral complexes it was, for example, enough to introduce one single empirical parameter (lODq) to explain the perturbation on the d-type atomic orbitals by the ligands. The crystal field theory was, however, only moderately successful. It was clear that many experimental facts could only be explained if one assumed a delocalization of the open shell electrons onto the ligands. From the crystal field theory emerged Ligand Field Theory. The ligand orbitals directly interacting with the metal ion d-orbitals were introduced and a set of MOs were constructed as linear combinations of the d-orbitals and ligand orbitals. The open shell electrons are then delocalized onto the ligands and a number of spectroscopic properties could thereby be explained. Ligand field theory shares many of the features of the PPP method of organic chemistry both in its formulation and the semi-empirical parameterization. The two approaches have even been combined into one semi-empirical theory, which was used with some success to explain the spectra features of metal porphyrins and similar metalorganic systems [14]. For a detailed treatment of crystal and ligand field theory we refer to the classical book by Carl Ballhausen [15]. This was the general situation at the middle of the 60ies, which is the time when modern ab initio quantum chemistry started its development. We shall therefore leave the history here and continue with a description of the different methods to deal with excited states in molecules and complexes that have emerged during the last 30 years. 3. WAVE FUNCTIONS FOR EXCITED STATES The semi-empirical methods developed for excited state calculations were moderately successful in describing certain features of the electronic spectra. But they had severe deficiencies. The PPP method was, for example, only able to describe %—>%* excitations within the valence shell. It was not clear how to treat the Rydberg transition or n—>p* transitions that occur from a lone-pair orbitals («) in systems with hetero atoms. Today we know that such excitations form an important part of the electronic spectrum of conjugated molecules. Sometimes we also see strong interaction between valence and Rydberg excited states. A typical example is the V state in ethene, which was mentioned above. In transition metal chemistry we know that important excitations are of the charge-transfer type, which can hardly be treated using ligand-field theory. There was clearly a need for an unbiased nonempirical theory, which does not a priori make any assumption concerning the nature of the excited states. This was the aim, when the development of the ab initio methods started in the

322 middle of the 60ies. The first efforts were naturally concentrated to the ground states and the development for excited states came a little bit later. But let us start with the general formulation of modern quantum chemistry. 3.1. Orbitals and wave functions The quantum chemistry for the excited state is always formulated using molecular orbital theory. The basic entity is a set of one-electron functions, the molecular orbitals (MOs). They are multiplied with a spin function to yield a set of spin-orbitals (SOs). Usually, the MOs are expanded in a basis set consisting of functions centered on the different atoms in the molecule, the atomic orbitals (AOs). If we select« such functions, /p(p= 1, «), we can form n orthonormal MOs as linear combinations of the AOs (the LCAO method):

6- = y C- v

(4)

From the n MOs we can generate In SOs. This is the one-electron basis we use to build the wave function. Our results will depend strongly on the quality of the MOs and features not present here cannot be compensated anywhere else. A proper choice of the AO basis set is therefore crucial for the accuracy of the results. This is particularly important for excited states. Once the MO basis has been chosen we need to build a basis for the /V-electron wave function. We can generate In N Slater determinants, d^, by occupying the 2« SOs with N electrons in all possible ways. The total wave function can be expanded in these Slater determinants:

^ = EC /; O /(

(5)

The expansion coefficients can be determined using the variational principle, which leads to the well known secular problem:

J] Hfn,-ES/lr)Cr=0

(6)

When the AO basis set becomes complete, this equation becomes equivalent to the Schrodinger equation. In practice this is of course not possible, but we can (in principle) approach closer and closer to the exact solution by increasing the size of the AO basis set. This is the key feature of ab initio quantum chemistry in contrast to other approaches, like semi-empirical schemes, density functional theory, etc. With a finite basis set the, method is called Full Cl (FCI). It is the best (in the context of the variational principle) approximate solution to the Schrodinger equation that we can obtain with the chosen basis set. The problem is only that we cannot solve equation 6. The dimension is simply too large for anything but very limited basis sets and few electrons.

323

Therefore, a second approximation is needed. There are many of them. Actually, all of modern quantum chemistry (with the exception of density functional theory) attempts to approximate the FCI equation in one way or another. Many of the approximations concentrate on the ground state and treat excited states as a secondary problem. The simplest of these methods is obviously the Hartree-Fock method, which only keeps one determinant in the expansion 5. A more advanced approach is coupled cluster theory. The most important of the methods that treat ground and excited states in a balanced way, is the complete active space SCF method and the multi-reference CI methods. All these methods try to select the most important terms in the FCI expansion, using knowledge about the detailed structure of the electron-electron interaction in molecular systems. We shall take a look at some of the methods. There are essentially two different approaches used in excited state calculations. The first, attempts to determine wave functions and total energies for each state of interest. To this category belongs the configuration interaction based methods, which directly tries to approximate equation 6. The second approach starts with the ground state and considers the excitation process as a perturbation of the ground state wave function due to the interaction with the electromagnetic field. These are the linear response methods. Here we compute directly the excitation energies and the transition moments, but do not have access to a wave function for the excited states. To this category belongs time dependent density functional theory and coupled cluster linear response methods. Let us first take a look at the wave function based methods. 3.2. Hartree-Fock and configuration interaction The solution of the FCI equation 6 is independent of the form of the MOs because the FCI wave function is invariant to unitary transformations among them. It is no longer the case when we approximate the equation. We are then faced with two problems, to determine the most optimal form of both the MOs and the CI expansion coefficients. So, what MOs should we use? The most immediate answer to the question is to use the HF orbitals for the ground state (we shall assume that the reader is familiar with the HF method). The eigenfunctions of the Fock operator can be divided into one occupied and one empty (virtual) subspace of MOs. The occupied orbitals will, for a closed shell molecule, contain two electrons each (cf. Fig. 3.)

Ground State

Singly Excited States

Doubly Excited State

Fig.3. Occupied and empty orbitals in the HF model for excited states.

324

The HF method is quite successful in describing the electronic structure and energetics of many molecules. The reason is that the mean field approximation of the electron-electron interaction is able to recover almost all of the repulsion energy. The error in the total energy is in well behaved cases less than 0.5%. This is not always the case though. We shall later discuss cases, where it is not possible to start from a HF ground state, but here we shall assume that the HF wave function gives a good representation of the ground state electronic structure. The error in the HF energy compared to the energy of the exact electronic wave function comes from the mean field approximation for the electron-electron repulsion. Each electron will in HF theory experience the average electrostatic field from all the other electrons. The error is called correlation energy because it arises from the inability of the HF method to describe the correlated motion of the electrons, in particular when they come close together. The Pauli exclusion principle tells us that two electrons of the same spin will not occupy the same position in space. This Fermi correlation is included in HF theory through the antisymmetry requirement on the wave function. Thus, the major part of the correlation error comes from electrons of opposite spin. Only two electrons can simultaneously be close together. Thus, correlation can with a good approximation be described using pair-theories. With such a ground state, how do we describe excited states? It is rather obvious: by moving electrons from the occupied to the virtual orbitals. This will generate a set of electronic configurations, which we can use to describe the excited states (cf. Fig. 3). Will the configurations in themselves give a good representation of the electronic structure? Usually not. The reason is that the density of states is often large in the excited manifold and one can expect strong interaction between different configurations (the + and -combinations that occur in alternant hydrocarbons is a good example). So, we have to write the wave function as a linear configuration of the configurations (or more properly configuration state functions (CSFs) because we have to take linear combinations of Slater determinants in order to obtain functions with the correct spin symmetry, singlets or triplets). Our excited state wave function thus has the form:

^ = C 0 O 0 + X Ciaia + Z Cia,jbOiaJb

+....

(7)

where ®o is the HF wave functions, ia are singly excited CSFs and ®iajb are doubly excited CSFs, etc. Where do we stop the expansion? If we do not stop it at all, we shall recover the FCI wave function. We can expect that the singly excited CSFs will be most important, but we cannot exclude that in some cases also doubly excited CSFs will give important contributions. Actually, it is more common that one might expect that excited states dominated by double excited CSFs appear at low energies in the electronic spectrum. But if we believe that this is not the case, can we terminate the expansion after the singly excited CSFs? Such an approximation is today labeled CIS (CI singles). Let us consider an example, the V state of the ethene molecule. The dominating CSFs are presented in Table 1. The numbers are from an extensive multi-reference CI calculation [16]. All orbitals in the table are of cr-type, except those labeled n. The ground state configuration is (ITT)2. The excitations that involve a-orbitals describe dynamic polarization, which is the most important correlation effect for the V state in ethene (see the discussion in the previous section). We see in the table that the CSFs are both singly and triply excited with respect to the HF ground state, but they are, as expected, double excitations with respect to the main configuration of the V state. We conclude that CIS does not give a balanced treatment of the

325

dynamic correlation effects in the excited state [17]. Ideally, we ought to include up to triple excitations in the expansion, but such an expansion becomes quite long and difficult to handle computationally. There is another solution to the problem: instead of starting from a single HF configuration we add a second configuration (ITI*) 2 : ct>0 = a1(l^") + a2i}7r)

(8)

and determine the two expansion coefficients aj and 02 variationally. We then include in the CI expansion for the excited states singly and doubly excited states with respect to both the starting configurations. These are the multi-reference methods. Table 1 The most important CI coefficients for the V state in the ethene moleculea. CSF

Coefficient

lTT^lTI*

0.934

3a g —> 3biu

-0.101

S(D)

( l 7 i ) 2 3 a g ^ (171* ) 3b,u

0.071

T(D)

Ib 3 a ^2b 3 U

-0.064

S(D)

(17I) 2 ^(27C)(2TI*)

0.064

D(D)

Ib3g^3b3u

-0.063

S(D)

2b|U^4ag

-0.052

S(D)

(l7i) 2 3a g ^(27i*) 2 3b lu

-0.043

T(D)

2

Type

a

FromRef. [16].

b

S = single, D = double, T = triple. Within (): with respect to the V state.

But there is yet another way to view the table. Suppose we use the V state configuration (7i7i*)s , instead of the ground state, as the reference in the expansion 7. The most important CSFs will then be the double excitations. It is well known that the most important contributions to the correlation energy comes from the configurations, which are doubly excited with respect to a given reference state. So, it seems that the most natural way to approximate the FCI equation for a given excited state is to first find a good reference function, which describes well the qualitative features of the wave function and then keep in the CI expansion the reference function plus all doubly excited CSFs. For open shell systems, one also needs to include singly excited CSFs. Exactly the same result would be obtained if we tried to estimate the importance of the excited CSFs using perturbation theory. Only the singly and doubly excited CSFs will interact directly with the reference function and therefore appear in the low order corrections. The analysis points to a theory that first computes

326 reference wave functions for each excited state and then adds correlation using singly and doubly excited CSFs. This is the multiconfigurational approach. 3.3. Active orbitals and the CASSCF method The two configuration wave function given in Eq. (8) describes the ground state of ethene better than HF theory. Actually, the coefficients of the second configuration is as large as -0.17. The reason is the rather weak 71-bond, which leads to a large occupations of the antibonding orbital. It is clear that when such occupations become too large, the HF model will break down and we cannot describe the electronic structure with a single configuration. Such situations are obtained when bonds are weak or even broken (dissociation, transition states) and quite frequently in excited states. We need then to go beyond the single configurational model to describe the excited state. This type of electron correlation is called static. It is long range in contrast to the dynamic correlation, which describes what happens when two electrons come close together. The idea behind the Complete Active Space (CAS) model [18] is to find a wave function that is based only on the orbital concept (as is HF), but which can handle also situations where a single configuration is not sufficient to describe the electronic structure. This is achieved by partitioning the MOs into three subspaces: inactive, active, and virtual. The inactive orbitals are assumed to be HF like and thus doubly occupied. To them we add a number of active orbitals but we do not assume anything about their occupation. Instead we construct a full CI wave function in this orbital space. The number of active electrons is the total number minus the number occupying the inactive orbitals. The wave function is thus completely determined by the choice of active orbitals and the condition that it should be an eigenfunction of spin and have a given space symmetry (in the molecular point group): *V

=^C O

(9)

where the sum goes over all CSFs O^ that can be generated by occupying the active orbitals in all possible ways consistent with an overall spin and space symmetry. The coefficients of the MOs and the CI expansion coefficients are obtained using the variational principle. In the ethene case we would choose as active the n and TI* orbitals with two active electrons, which gives the wave function 8 for the ground state. The same active space can also be used to describe the V state and the corresponding triplet. If we start to rotate the molecule around the double bond we would find that also the, so-called, Z state, the singlet state orthogonal to 8 becomes interesting (see below). However, the choice of active orbital becomes less straightforward when we study excited states. We need to assume something about the nature of the excited states and the orbitals which are important for a qualitatively correct description of the wave functions. Take ethene as an example again. The problem to compute an accurate wave function and energy for the V state is related to a near degeneracy between the valence excited state and an excitation to a 3d Rydberg orbital. There is a qualitative difference between Rydberg and valence excited states. A Rydberg state has one electron in a very diffuse orbital which interacts only weakly with the other electrons. Thus, the correlation energy decreases and resembles more that of the positive ion. For the ionic V state, the correlation energy is, on the other hand, increased because of the strong dynamic polarization effects, which are more prominent here than they are in the ground state where the electronic structure has a more covalent character. In order

327

to balance the correlation effects, we therefore need very accurate wave functions where most of the correlation energy is included. In order to be able to describe both valence and Rydberg excited states we need to extend the active space with Rydberg orbitals. The first step is to extend the AO basis set with enough diffuse functions. How it is done will not be described in detail here. Instead we refer to a review that [19], where the choice of active orbitals for excited states of organic molecules are discussed in detail. Ideally, one would like to be able to solve Eq. (9) separately for each electronic state. It is in general not possible. Different electronic states can be close in energy and the corresponding energy surfaces may even cross. Separate optimization of two states, which are close in energy and of the same symmetry, could also yield wave functions, which are non-orthogonal. This is undesirable. The solution is state average optimization. Several roots are extracted from the secular equation. The average density matrices are constructed and a common set of orbitals is optimized. Such wave functions are per construction orthogonal to each other. The use of a common set of orbitals is normally not a problem. The wave functions will still be good reference functions, which describe the qualitative features of the electronic structure well. To summarize: the CASSCF method will determine a qualitatively correct wave function for each electronic state. This function can now be used as a reference function in the CI expansion 7. So, the next step would then be to solve this equation to obtain the final accurate description of the electronic states. 3.4. Dynamic correlation In the previous section we sketched a two step procedure for the calculation of wave functions for excited states. The first step consists of determining an optimal set of MOs together with a wave function, which describes the electronic structure of the excited state qualitatively correct. We saw that it could be achieved by extending the HF scheme to a full CI in a limited set of active orbitals, the CASSCF method. It should be noted that the method can, with a properly chosen active space, be used not only in the Frank-Condon region around the equilibrium geometry of the molecule, but also to map out full energy surfaces, describing chemical transformations, conical intersections and other curve crossings, etc. The HF method cannot be used in such situations. The second step consists of solving a large CI problem where the expansion comprises all CSFs that can be generated by single and double excitations in each of the reference configurations. We call this approach multi-reference CI (MRCI). It became feasible for large scale applications with the development of the direct CI methodology in the early 70ies [20,21]. It is probably the most accurate method available for general studies of excited states. Today it is possible to use more than 108 CSFs in the CI expansion. The most efficient implementation is probably the COLUMBUS code developed by H. Lischka and co-workers [22]. Even if modern MRCI programs can handle long CI expansions, it is easy to see that with a CASSCF reference function, the number of CSFs will easily reach 108, when the number of correlated electrons and the basis set increase. The calculations will then become quite timeconsuming and rather soon impossible. One way to reduce them is to use only a subset of the CAS configurations in the reference function, thereby introducing one more ambiguity. Another possibility, which has been used with some success, is to use the entire CASSCF wave function as the reference instead of the individual CSFs. This is the internally contracted form of the MRCI method [23,24].

328 The MRCI method has a deficiency: it is not a size-extensive method. To explain what this means, let us consider two non-interacting systems A and B with wave functions *PA and ^PB • The wave function for the total system is ^FAB = ^ A ^ B • Suppose now that the wave functions for the two sub-systems are of the MRCI type with singly and doubly excited CSFs from a given reference function (MR-SDCI). The total wave function VAB will then include up to quadruple excitations with respect to the product of the two reference functions. An MRSDCI calculation will not describe the total system with the same accuracy as the subsystems and the energy will not be the sum of the energies for A and B. It is not easy to correct for this deficiency of the MR-SDCI method. Several methods have been developed, which account approximately for the missing higher excitations, which are products of the double excitations, but none of them are completely satisfactory. The only completely efficient way out is to include the product terms in the formal expression for the wave function. This leads to the coupled cluster expansion, which however has so far only been developed into an effective computational method for a single determinant reference function. 3.5. Second order perturbation theory Another way to treat the dynamic correlation effects is to use perturbation theory. Such an approach has the virtue of being size-extensive and ought to be computationally more efficient than the MRCI approach. Moller-Plesset second order perturbation theory (MP2) has been used for a long time to treat electron correlation for ground states, where the reference function is a single determinant. It is known to give accurate results for structural and other properties of closed shell molecules. The approach was in the late 80ies extended to reference functions of the CAS type, the CASPT2 method [25,26]. The approach turned out to be very productive especially in molecular spectroscopy. A large number of applications has been made for organic molecules, transition metal complexes, and more recently also in heavy element chemistry [19,27,28]. A number of examples will be given in other chapters of this book. A CASPT2 calculation starts with a CASSCF calculation, which has been designed to include the electronic states of interest. Each of the CAS wave functions are then used as reference functions for a CASPT2 calculation. The first order wave function is generated as a sum over all single and double excitations with respect to the entire reference function. In this respect CASPT2 is similar to internally contracted CI, but the expansion coefficients, CM are only determined to first order and the energy to second order. They are determined by the equation: =-VOfl

(10)

Here, H{^ are matrix elements of a zeroth order Hamiltonian, which is chosen as a oneelectron operator in the spirit of MP2. Sf,v is an overlap matrix: the excited functions are not in general orthogonal to each other. Finally, VOu represents the interaction between the excited function and the CAS reference function. The difference between Eq. (10) and ordinary MP2 is the more complicated structure of the matrix elements of the zeroth order Hamiltonian. In ordinary MP2 it is a simple sum of orbital energies. Here it is a complex expression involving matrix elements of a generalized Fock operator combined with up to fourth order density matrices of the CAS wave function. We do not give further details here but refer to the original papers If one assumes the generalized Fock matrix to be diagonal, one can formally write the second order energy in the same form as for MP2:

329

(11) where K runs over a set of orthogonalized excited states and s are sums of generalized orbital energies. In ordinary MP2, the denominators are simply: £a+£b-s~£/, where a, b are virtual and i, j occupied orbitals. The CASPT2 denominators are more complex, but describes in principle the same energy difference. The CASPT2 approach has several advantages. It is computationally effective, as the many large scale applications have shown. It can use arbitrarily complex reference functions with maybe up to a million CSFs in the reference function. It is size extensive (a small deviation from size extensivity can occur with some choices of the active space, but the deviations are negligibly small). The generality of the reference functions makes it possible to treat, not only the Frank-Condon region but to follow excited states energy surfaces over energy barriers, through conical intersections, to dissociation of a chemical bond, etc. Other articles in the book will describe such situations in more detail. The accuracy of the excitation energies is high with errors generally smaller than about 0.2 eV. There are, however, some drawbacks with the method, which can sometimes cause problems. One is the limitation of the active space. It is today difficult to perform calculations with more than about one million CSFs in the reference function. This can in some cases make accurate calculations impossible because of requirements on the size of the active space that yields larger sizes of the CAS CI space. Another problem is caused by the, so called, intruder states. The denominator in ordinary MP2 will rarely become small because there is usually a large HOMO-LUMO gap in the orbital energies. It is not always the case in CASPT2 because the active orbitals have energies that are intermediate to the energies of the doubly occupied and virtual orbitals. A low order perturbation approach will become invalid when the denominators become small in comparison to the numerators in Eq. (11). The cause of such behavior is usually a too small active space, but it is not always possible to extend the space further. A level shifting technique has been developed that will remove the intruder state [29,30], but it should only be used for weak intruder states that will have only a minor effect on the computed correlation energy. The CASPT2 method also has a small systematic error, which is due to the definition of the zeroth order Hamiltonian. States with a closed shell character are favored in relation to open shell structures. This leads to too small excitation and dissociation energies in some cases. A modification of the zeroth order Hamiltonian that reduces the error considerably has recently been suggested [31]. The CASPT2 method relies on a well defined CAS reference function, which will not be strongly affected by the addition of dynamic correlation. Normally, this model is satisfactory, but it may happen in more complex cases that several states of the same symmetry appear close in energy. They may then interact strongly with large changes occurring in the reference functions. Typical situations are areas of avoided crossings, conical intersections, etc. A multi-state version of CASPT2 has been developed to handles such situations. This is multistate (MS) CASPT2 [32]. Here an effective Hamiltonian is constructed for a set of CAS reference functions that has the normal CASPT2 energies in the diagonal and includes interaction between the first order wave functions in the off-diagonal elements. A detailed

330

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

HCH Rotation angle Fig.4. Weights of the CSFs (n) (upper line) and (71*) for the N-state of ethene as a function of the rotation angle.

discussion of the possibilities and limitations of the MS-CASPT2 method is given by Merchan and Serrano in another chapter of this book [33]. 3.6. The ethene molecule As an example of the importance of a multiconfigurational treatment in photochemistry we consider a very simple but illustrative and important example: the rotation of one CH2 group around the double bond in the ethene molecule, C2H4. The calculations on which the results are based have been made with a small DZP basis set. Also, the only parameter that has been varied is the rotation angle. All other geometry parameters have been kept at the values of the equilibrium geometry, which is a serious approximation, in particular for the CC bond. We should therefore not pay much attention to the actual numbers, but merely to the qualitative features. We construct CASSCF wave functions for all states, which can be formed by occupying the two 71-orbitals, n and n* in all possible ways. This gives rise to four electronic states: The N-state, dominated by the CSF (nf The V-state, a singlet state with the configuration: (TC)(TT*) The T-state, a triplet state with the configuration: (TT)(7T*) The Z-state, dominated by the CSF (n*f

331

N-state Z-state V-state T-state

-1.00

S

1 -1.20

-1.40

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

Rotation angle

Fig. 5. The total energy of the N-, T-, V-, and Z-states as a function of the rotation angle

The two CSFs (TI)2 and (TT*)2 belong to the same symmetry and can thus mix, so both states will have some multiconfigurational character. The question is how it varies with the rotational angle. We show in Fig. 4 how the weights of the two CSFs vary with the rotation angle for the Nstate. The corresponding picture for the V-state is the mirror image in this simple model. For planar ethene there is not much mixing. The 7i*-orbital is high in energy, so the N-state will be quite HF like with one dominating CSF. But, when we start rotating one of the HCH groups around the double bond the energy drops and finally, at 90°, the two orbitals it and TI* become degenerate. The n bond is now completely dissociated. Thus, the two CSFs have the same energy and there will be complete mixing, which is typical of a dissociation process. The variation of the (CASPT2) energies with the angle is shown in Fig. 5. The large spread in energies at zero angle is decreased substantially at 90° and the four states become pairwise degenerate. The lower pair correspond to a singlet (S) and triplet (T) coupled biradical system. The upper pair of states are ionic with the two electrons resonating between the two centers. The description is oversimplified here. In reality, Rydberg like states in the same energy region interact strongly with the upper states and change the nature of the wave functions in the FC region. The two degenerate ionic states are split by a Jahn-Teller distortion. A pyramidalization of one HCH group will stabilize one of the states further. At the same time, the ground state is further destabilized and eventually a conical intersection is reached, which makes it possible to move from this surface back to the ground state, (a more detailed discussion of the subject can be found in Ref. [34]).

332

One might think that the above example is rather special and of little interest in photochemistry. On the contrary. This simple model is the key for the understanding of a large number of important photochemical processes in chemistry and biochemistry. Rotation around a double bond is a much used process in nature. The most striking example is probably the cis-trans isomerization of the retinal molecule in rhodopsin, the key molecular process in vision. A number of other examples could be given, for which we refer to other chapters in this book. We emphasize that these typical problems in photochemistry can only be dealt with using a multiconfigurational quantum chemical methods like CASSCF/CASPT2 or CASSCF/MRC1. With this example we finish the overview of the state specific methods. We shall next briefly discuss the linear response methods, where the focus is on the ground state wave function and the excitations are obtained through the response of the system to an electromagnetic field. 4. LINEAR RESPONSE THEORY An alternative way to treat excitation processes is to study the response of the ground state electronic structure perturbed by an electromagnetic field. The quality of such a method depends crucially on the quality of the ground state wave function. The first attempts along this line was based on Hartree-Fock theory and was named the Random Phase Approximation (RPA) [35]. The method is not much used today. Recent developments use more accurate wave functions and the two most popular approaches are based on Density Functional Theory (DFT) or the Coupled Cluster (CC) method. Because both these approaches assume the ground state to be single configurational HF like, the main area of application will be studies of electronic spectra in the FC region. It is not advisable to use LR methods far away from equilibrium where the ground state may be strongly multiconfigurational and doubly excited states are low in energy. The second limitation inherent in the methodology is that only excited states, which are dominated by singly excited configurations are included. LR theory cannot be use to study rotation around a double bond as in the ethene example discussed above. LR theory starts with the ground state wave function. A perturbation in the form of a time dependent electric field is applied: E(/) = E^cosiat)

(12)

The first order response function is then computed and the excitation energies are found as poles of this function while the transition moments are obtained from the residues. We shall not attempt to give a full account of the theory here but refer to the original papers. The method was first developed in CC theory by Monkhorst [36] and was extended into an effective computational tool by the Aarhus school (see Ref. [37] and further references therein). The CC-LR method is capable of giving a very accurate account of the excited states that are dominated by single excitations. It is available today in an integral direct implementation, which makes calculations with many basis functions possible. For HF like ground states and excitations dominated by singly excited CSFs it is probably the most accurate method available today. Due do its limitations it is, however, not useful for studies of photochemical processes. Linear response is in DFT a simple extension of the RPA approach. For a full account of LR in DFT we refer to the review by M. Casida where more references may be found [38]. The approach is available in most programs and is very simple to use. It gives a quick overview of

333

all singly excited states, provided that an adequate basis set is used. One needs, however, to remember the limitations. We shall illustrate the performance of both LR-DFT and LRCC in the next section. A detailed and critical evaluation of the approach is given in the chapter on Ab Initio methods by Merchan and Serrano-Andres [33]. 5. THE BENZENE MOLECULE Let us now for a while leave the theory and relax with an example. We choose the organic molecule, which has through the years been the prototype for testing methods for excited states, benzene. At the same time it will be a recapitulation of the history. In our historical review, we discussed the calculations of Goeppert-Mayer and Sklar in 1938 and the use of benzene for the testing of the PPP method in the 50ies. It is not surprising that these early successes would be a challenge for ab initio quantum chemistry when it became available for studies of excited states in the late 60ies. The first attempts were made using small CI expansions, involving only the Tt-orbitals. The first calculation was performed by R. Buenker et al. [39]. It was a (by today's standard) a small CI calculation involving only the six 7i-orbitals. The results are shown in the first column of Fig. 4 (BWP). The results were not very impressive for obvious reasons. Actually, the calculations resemble the PPP calculation by Pariser from 1952, but now without any empirical correction of the integrals. Thus there is no account for the dynamic polarization of the o-electrons, which is reflected in the errors, which are especially large for the more ionic states 'Biu and ' E i u . Some improvement was obtained in the calculation by Peyerimhoff and Buenker (PB in Fig. 7) by extending the basis set and increasing the number of 7i-orbitals from six to nine [40], but the errors are still large. Hay and Shavitt (HS) performed a calculation including 23 7t-orbitals in 1974 [41], which we might consider as limit of what can be obtained without explicit inclusion of the a-electrons. The errors are still larger than one eV for some of the states. One should remember that the largest error in the PPP calculation of Pariser was 0.5 eV for the singlet states and 0.9 eV for the triplets and later developments improved the results further.

Fig. 6. The benzene molecule.

334

eV 11 -,

10

8

•

BWP

PB

HS

R

MRM

Expt.

Fig. 7. Early CI results for the electronic spectrum of benzene. For the references, see the text. So, it was a hard task to improve on the semi-empirical methods. The first CI calculation that included the a-electrons was performed in 1976 by Rancurel et al. [42] in a minimal AO basis set (R in Fig. 7). The results did not improve, which is due to the too small basis set used, which did not allow for any orbital relaxation. A much larger CAS-CI calculation was performed by Matos et al. in 1987 [43]. It also included corrections for the size-consistency error. It was the largest Cl calculation performed on the benzene spectrum before 1990. As can be seen in Fig. 7, the errors in computed excitation energies are still sizable (more than 0,5 eV in some cases), but the quality is now at least comparable to that obtained with the PPP method 30 years earlier. The more recent developments of linear-response theory (both in DFT and CC theory) and the CASSCF/CASPT2 approaches have of course been tested on benzene. The first

335 CASSCF/CASPT2 calculation was performed in 1992 [44] and a more complete treatment of the vertical spectrum (also including the Rydberg states) was made in 1995 [45]. The results of these calculations are shown in column 2 of Fig. 5 (labeled CASPT2). There is now good agreement with experiments with errors of only 0.1-0.2 eV. One notices that most CASPT2 energies are slightly on the low side, reflecting the systematic error of the method. The CASPT2 results are in Fig. 8 surrounded by two results obtained using linear response theory. The first column shows the results of a TD-DFT calculation [46]. Again, we see improvement compared to the older CI results, but errors are as large as 0.5 eV for some states. Nevertheless, this very cheap method is able to recover most of the dynamic

eV

9

8

; 2g

1

B1u

6-

•3B.

•1B. 3

E<

'lu

DFT

CASPT2

CC3

Expt.

Fig. 8. More resent results for the benzene molecule. The references are given in the text.

336 correlation effects on the excitation energies. It is to an even larger extent true for the accurate coupled-cluster linear response studies (CC-LR) of Christiansen et al. [37] for the singlet states and Hald et al. [47] for the triplet states. The method gives energies that are slightly larger than the experimental results. Both TD-DFT and LR-CC rely on a Hartree-Fock like reference function and excited states dominated by single excitations. The first condition is reasonably well fulfilled for benzene. Among the excited states, one finds the largest contributions from doubly excited configurations for the 'E2g state. It is also the state that give the largest errors using the LR method, both in DFT and CC. So, today we have methods that allow an accurate description of the excited states of a molecule like benzene. It took ab initio quantum chemistry almost 30 years to achieve this goal and make quantitatively accurate calculations of excited states in larger molecules possible. The calculations presented above were all made at the ground state equilibrium geometry. The energies thus correspond to "vertical" excitations. This is not a measurable quantity. Instead, an electronic spectrum is resolved into a vibrational structure corresponding to excitations from a vibrational level in the ground state to a corresponding level in the excited state (very detailed spectra may also resolve the rotational fine-structure). The intensity distribution for allowed transitions is normally such that the most intense bands occur close to the vertical energy difference (the Frank-Condon region). But not exactly. And what happens for symmetry forbidden transitions? We can illustrate the problem for the lowest excited state in the benzene molecule, for which a calculation of the vibrationally resolved spectrum has been made using the CASSCF/CASPT2 method [48]. Such a calculation is more complex than a calculation at a single geometry. Now one has to compute not only energies, but also the equilibrium geometry of the ground and excited state, the force field (to obtain the vibrational frequencies) and the transition moment as a function of geometry (for the intensities). It was done for the two lowest bands in the benzene spectrum using the MULA program of the MOLCAS quantum chemistry software. This program solves the vibrational problem in the harmonic approximation and computes transition intensities for each vibrational mode [49]. The result for the !B2u band is shown in Fig. 9.

337 I.So-OS THEORETICAL

ee-oe

EXPERIMENTAL

I X

HiVrLCWOTH

Fig. 9. The theoretical and experimental spectrum for the 'B?u band in the benzene molecule.

The agreement between experiment and theory is spectacular. The intensity distribution among the vibronic bands are strongly dependent on the geometry difference between the ground and excited state. It is thus necessary to be able to predict the change in geometry with high precision. Here, the CASPT2 method was used, which for organic molecules normally predicts bond distances with an accuracy of a few thousands of an Angstrom. All intensity of the band comes from vibrations that break symmetry. The transition moment itself is zero in D6h symmetry. It was found that the vertical transition energy is about 0.1 eV larger than the energy of the most intense band, showing that relating vertical energies to band maxima is not a precise measure of accuracy. The example has shown that today we cannot only compute a set of excitation energies for a given geometry, but we can also compute the energy surfaces for the excited state. This

338 makes it possible to use theory to study photo chemical processes. A number of examples in other chapters of this book will show how this is done in practice. 6. HEAVY ELEMENT COMPOUNDS With heavy elements we shall here mean all atoms with an atomic number larger than 18 (Ar). Included among these atoms are the transition metals, the heavier main group elements, the lanthanides, and the actinides. Calculations of excited states for compounds containing such atoms are for several reasons more complex than for lighter atoms. The number of electronic states with low energy increases with increasing atomic number and the so do the orbitals involved. There is thus both a problem to choose an adequate active space and to know how many states one needs to include in the calculation. An additional problem is that one needs to account for relativistic effects. Relativistic quantum chemistry is an area where intense development is going on at the present time and many ideas have been suggested for treating relativistic effects ranging from four component theories based on the Dirac equation to simple addition of relativistic effects to a nonrelativistic method using perturbation theory. Most of these approaches has addressed mainly the ground electronic state. We shall not discuss all these methods in detail here but refer instead to a recent book where many approaches are discussed [50]. Instead we shall give some examples using an extension of the CASSCF/CASPT2 method to the relativistic regime. The approach is a compromise between the need to include relativistic effects and the requirement that one should be able to treat general electronic structures both in ground and excited states, the key feature of the multiconfigurational approach in quantum chemistry. In addition, the method should be applicable to large molecules. All these requirements rule out a four-component approach based on the Dirac equation. Instead a two step procedure has been introduced based on the Douglas-Kroll (DK) Hamiltonian [51] together with a meanfield approximation of the spin-orbit(SO) part of this Hamiltonian [52]. For a detailed discussion of this approach we refer to a recent paper where further references can be found [53]. The method can be characterized as a two step procedure: In a first step the scalar part of the DK Hamiltonian is introduced into a normal non-relativistic CASSCF/CASPT2 calculation. Specially designed AO basis sets are used, which account for the relativistic contraction of the inner shell orbitals. The method treats all electrons without the introduction of effective core-potentials, but deep core orbitals are described in a minimal basis set. Calculations are performed for a number of electronic states. The selection of the states depends on the specific problem and the importance of spin-orbit interaction. In a final step, the CASSCF wave functions for these electronic states are used as basis function in the construction of a matrix of the spin-orbit part of the DK Hamiltonian. The RASSCF state interaction method (RASSI) is used, which allows for the calculation of matrix elements of one-and two-electron operators in a basis of CASSCF wave functions, possibly using different (non-orthogonal) sets of orbitals [54]. The diagonal elements of the matrix are shifted to the CASPT2 energies in order to account for electron correlation. The eigenvalues of this matrix gives the final energies and wave functions including spin-orbit coupling. Can a method that treats relativistic effects in such an approximate way, really work? Actually it does. In connection with the development of a new set of relativistic ANO type basis sets, ANO-RCC, which also include correlation of the outermost core-electrons (which is necessary for heavy elements) we have studied the approach for all alkaline, alkaline earth, main group and noble gas atoms [55,56]. It turns out that errors in the spin-orbit splitting are small, with the possible exception of the heavier fifth row main group atoms (Bi-At). Ground

339 state potential curves for the dimer of Tl and Pb give spectroscopic constants in agreement with experiment [53]. The electronic spectrum of PbO could be successfully assigned with the present method [53]. Here we give an additional example from the third row transition metals. The energy levels of the platinum atom ANO-RCC basis sets for the transition metals are presently under development. In connection with the work we computed the electronic spectrum of the Pt atom. The ground state of Pt is (5df{6s), 3D, J=3. There is, however a large spread of the different J-level (J=l is located 10131 cm'1 above J=3), while the term (5df{6sf, 3F, has its lowest level, J=3, only 824 cm"' above the ground level. The term (5d)[0, 'S is found at 6140 cm"'. Is it at all possible to describe the spectrum theoretically? Let us see. We set up the following calculation: CASSCF/CASPT2 calculations are performed for all terms of the configurations (5
340

Table 2 CASSCF/CASPT2/RASSI results for the energy levels in the Pt atom (energies in cm"1). Level Theory Expt.a {5d)\6s\ 3D J=3 0 0 (5df(6s), 'D J=2

366

775

502

823

(5d)\6s), DJ=2

6105

6567

{5df, 'SJ=O

7278

6140

9932

10116

10204

10131

3

{5df(6sf, F J=4 3

3

(5df(6sf, F J=3 9

3

(5J) (fc), D J=l 3

13811

23

15494

15501

3

16853

16983

2 3

18522 22631 26647 48514

18567 21976 26638 -

(5df(6sf, P J=2 8

(5
fo) , P J=l (5df(6sf, 'G J=4 (5
The results are shown in Table 2. The agreement with experiment is impressive considering the many approximations involved. The largest error, 1138 cm"' (0.14 eV) is found for the (5d)10, 'S state. It is not surprising. It is well known that the different shape of the d-orbital and the strong correlation effects in the (5
341

3.5

3.0

2.5

1D 1G.

2.0 .3P

1.5

1.0 1D 0.5 3D

o.o L Theoretical

Experimental

Fig. 10. The electronic spectrum of the Pt atom computed without and with spin-orbit coupling.

6.1. Lanthanides and actinides The possibilities for lanthanides is not yet fully explored. Some promising results exist for small lanthanide molecules like SmO [60]. One problem is the size of the active space. At least the Af-, 6s-,and 6/?-orbitals needs to be included for the atom, which is already 11 orbitals. Adding ligand orbitals may easily extend the limits of the present implementations of the methodology. However, the chemistry of the lanthanides mainly involves metal in a high oxidation state. Then only the ^orbitals are needed. They are not involved in the bonding process and it would be interesting to develop a variant of the method, which allows inactive

342

open shell orbitals. It can easily be done at the CASSCF level, but is more complicated for CASPT2. But with the present programs one needs to include the whole 4f-shell in the active space. An additional problem is the number states that need to be treated. The spectrum is very dense and some experimentation is needed in order to find out how many states one needs to include for a given problem. A good example is the chemiionisation process studied for SmO as mentioned above. We shall certainly see some development of the multiconfigurational approach for lanthanide chemistry in the near future. More has been done in actinide chemistry. Experimental information is scarce and theory has a great opportunity to add to our knowledge about the chemistry of actinide compounds. Much of the early work in actinide chemistry dealt mostly with U(VI) compounds because of their closed shell structure. However, configuration interaction methods have also been used to study electronic spectra and other properties. We shall not discuss the earlier work here but refer instead to some recent reviews on the subject [50,61,62]. Instead, we shall illustrate the possibilities and problems that quantum chemistry faces in actinide chemistry by a recent example, the electronic spectrum of the UO2 molecule. For "normal" molecules one can usually determine the ground state electronic configuration using rather simple quantum chemical arguments based on standard theories of the chemical bond. This is not the case in actinide chemistry and UO2 is a striking example. The uranyl ion UO 2 + is maybe the most well studied species in uranium chemistry. It has a closed shell electronic structure with uranium, in the formal oxidation state VI, forming a triple bond to each of the oxygens. If we now add two electrons to form the neutral molecule, where do they go? Following the aufbau principle we would choose to place them in 5/orbitals on uranium with the resulting electronic state: (5f0)(5fo), JHg . However, is the aufbau principle valid here? The electronic configuration of the uranium atom is {5ff(6d)(7s)2, so we might suspect that there might be competition between different choices. As it turns out, the lowest configuration is (5f)(7s), 3u with the 3Hg state about 0.5 eV above [63]. But the story does not end here. Earlier studies of the vibrational spectrum in an Ne matrix had confirmed the electronic ground state to be 3u. But when these experiment were repeated in matrices of heavier rare gas atoms, a different spectrum was recorded [64] and guided by DFT and CCSD(T) calculations it was concluded that the ground state had changed to JHg. Measurements of the electronic spectrum in gas phase and in an Ar matrix gave, however, opposite evidence and suggested that the ground state is not changed but remains as 3 u also in the matrix [65,66]. The issue is yet unresolved, but it shows that in actinide chemistry we cannot take anything for granted. The density of states is so high that even weak interactions can change the electronic structure. It becomes quite complicated to study chemical transformations in such systems. We can illustrate the problem with a calculation of the electronic spectrum of the UO2 molecule that has recently been performed [67]. The calculations are of the same type as described for the Pt atom. In all 150 electronic levels were computed. The results are shown in Fig. 11. The two lowest levels with labels 2u and 3u come from the 3U state and the third level (at 0.20 eV) from the JHg state. As can be seen the spectrum is very dense. Actually, there are probably more lines in the spectrum at energies above about 2 eV. The calculations included in the active space (apart from orbitals from the UO bond) only the 5f, 7s and 7p orbitals because they give rise to allowed transitions from an ungerade ground state. However excitations to 6d orbitals on uranium will appear in the energy region above 2 eV. Electronic spectroscopy for actinide compounds is clearly not an easy task. If the electronic spectrum is so complex, what about dynamics and photochemistry? Here is a "simple" example to illustrate the problems one has to overcome if one wants to study the

343 TTieoretical{all)

eV

Theoretical(f>0.001)

Experimental Range

5.0

4.0

3.0

2.0

1.0

o.o-

a

Fig. 11. The electronic spectrum of the UO2 molecule. The first column shows all lines, the second those with oscillator strengths larger than 0.0001, and the third the measured spectrum. Grey lines are accessible from the 2u state and black lines from 3u.

dynamics of reactions involving heavy atoms. NUN is a stable molecule with a closed shell singlet ground state. It is produced by allowing uranium atoms to collide with nitrogen molecules in a molecular beam. The uranium atom has a Ls ground state (17-fold degenerate). A recent calculation shows that the transition state is one out of many close lying triplet states with a barrier that is completely determined by spin-orbit coupling. The reactions thus starts on a 17-fold degenerate surface, crosses over to a dense set of triplet states at the transition state region and finally settles on a singlet surface at equilibrium (for details, see Ref. [68]). Even with access to very accurate energy surfaces, how do we solve the dynamical problem?

344

7. THE FUTURE It is of course impossible to make any real predictions about the future developments of a research fields. Exploring the unknown cannot be predicted. The most we can do is to make some rather straightforward extrapolations from the present situation. As we have seen in this and other chapters of the book, the field is in quite a good shape. Many problems can be treated with reasonable accuracy and we can make sensible predictions in spectroscopy and in photochemistry. Quantum chemistry is today a valid and accurate tool for assigning electronic spectra of all sorts of molecules, ranging from organic systems and transition metal complexes to heavy element compounds involving main group elements, lanthanides and actinides. Photochemical reactions and excited state dynamics can be studied for organic systems even if we still have many things to learn here. The development is a little bit slower for transition metal complexes but progress is made also in this field. What are the major bottlenecks for extending the applications to larger and more complex systems? One is the standard problem of quantum chemistry: the integrals. Most methods used today for excited states still scale at least as n4 with the size of the basis set, n. Some progress is made in developing linear scaling models but the progress seems to be slow. Localization of orbitals offer another possibility for treating correlation only in an "active" part of a molecule. Other methods that will reduce the dependence on the size of the AO basis set are being developed. I have a feeling that a few years from now, this will no longer constitute a real problem. All the quantum chemical methods we are using today to treat excited states have inherent problems of one kind or another. The very cost-effective LR-DFT approach is not accurate enough and cannot treat all types of excited states in a balanced way, and it is restricted to a HF like ground state. It is possible that new functionals will become available in the future that will increase the applicability. However, another possibility seems to be more promising. One can formulate a density functional theory, which is not based on a single configurational model, but instead on a CAS wave function. Such a model would treat exchange exactly (A CAS model does not have the inherent problem of a delocalized exchange as HF has) and only use DFT to describe dynamic correlation. It would certainly open up new possibilities for excited state calculations. There will be no limitation on the electronic structure of the ground state, the electronic spin will be well defined for open shell systems, and all types of excited states can be treated. Many research groups have started research in this field and it is possible that we may have a working procedure soon (for a discussion of some aspects of the problem see for example Ref. [69]). In coupled cluster theory one would need a multiconfigurational analogue to the CC-LR methods available today. There are such developments but they are slow for molecules. The Fock space based methods can be very accurate for small systems like atoms, but are not general enough for photochemistry (See for example Ref. [70]). What about the CASSCF/CASPT2 approach. It is most likely going to continue to be useful tool for studies in photochemistry because of the general structure. In combination with methods that reduce the integral problem it should be able to treat very large systems. The method has, however, limitations that are not easy to overcome. The active space is limited to around 15 orbitals. This problem is solved if DFT is used instead of CASPT2 to treat dynamic correlation, because it is then possible to work with reduced CI expansions (RASSCF). Another problem with the CASPT2 method is its limited accuracy. In principle it may be increased by increasing the active space, but this is rarely an alternative in practical applications. There is hope that the error will be reduced in the near future by developments

345

of improved zeroth order Hamiltonians, which reduces the systematic error for open shell systems [31]. The development in heavy element chemistry is intense at present and we expect to see an increasing number of applications in this area in the future. The problem of the large active space needed for systems containing several heavy atoms (e.g. clusters of transition metals or actinides and other nanosystems) is still not solved and an accurate theoretical spectroscopy of such systems is not yet feasible. The same is true for many transition metal complexes where the metal is in a very high oxidation state. Generally, this area is full of challenging problems for theory, in particular, because it is needed to include relativistic effects, both scalar and spin-orbit coupling. But we are faced not only with the problem of computing accurate wave functions and energies. Most chemical processes (even in photochemistry) take place in an environment, a solvent, an active site in a protein, a crystal, etc. The methods we have today to treat environmental effects are rudimentary and need to be developed further. The problem of computing free energies instead of just enthalpies has not been solved in the general case. Will we in the future be able to do direct dynamics using accurate energies and forces, or will we instead see further development of global fitting procedures. We do not know today, but there are certainly many difficult problems to solve and exciting challenges to meet in order to develop the field into a true predictive tool in chemistry. Here is the ultimate challenge: develop theoretical tools that allow an accurate quantitative prediction of the entire process of transformation of solar energy into chemical energy in the photosystem of bacteria and plants.

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10602-10606, (2001). [64] J. Li, B. E. Bursten, and L. Andrews C. J. Marsden. J. Am. Chem. Soc, 126, 3424, (2004). [65] J. Han, V. Goncharov, L. A. Kaledin, A. V. Komissarov, and M. C. Heaven. J. Chem. Phys., 120,5155,(2004). [66] C. J. Lue, J. Jin, M. J. Ortiz, J. C. Rienstra-Kiracofe, and M. C. Heaven. J. Am. Chem. Soc, 126, 1812,(2004). [67] L. Gagliardi, M. C. Heaven, J. Wisborg Krogh, and B. O. Roos. J. Am. Chem. Soc, in press, (2004). [68] L. Gagliardi, G. La Manna, and B. O. Roos. Faraday Discuss., 124, 63, (2003). [69] S. Gusarov, P.-A. Malmqvist, R. Lindh, and B. O. Roos. Theor. Chim. Acta, 112, (2004). [70] A. Landau, E. Eliav, Y. Ishikawa, and U. Kaldor., J. Chem. Phys., 114, 2977, (2001)

349

Index 2,3-diazabicyclo[2.2.1]hept-2-ene(DBH) 16, 18,20 2,3-diazabicyclo[2.2.2]oct-2-ene (DBO) 20, 26 ASCF method 47, 279, 291, 293, 298 Absorption spectrum 76, 141-142, 144, 161,204,280,282,303 Actinides 338, 341, 344-345 Active electrons 37, 57, 281, 326 Active orbitals 14, 37, 45, 57, 193, 214, 215,231, 304,326-327,329 Adenine 116 Adiabatic approximation 97, 228 Adiabatic electronic - state 238 - wavefunction 140 Adiabatic transition(s) 66, 175, 178 Adiabatic excitation (energies) 76, 105, 111,204 Algebraic-Diagrammatic Construction (ADC) approach 49-50 Anisole 256-257 Antisymmetry (principle) 40, 324 Approximate propagator methods 49 Averaged Coupled-Pair Functional (ACPF) method 46 Avoided crossing(s) 4-6, 62-64, 82, 172, 195,206,264,268,329 Azoalkane 16, 18, 20 Azobenzene 112, 192, 203, 213 Azulene 2 Bacteriorhodopsin (bR) 7, 225, 237 Barrelene 263, 267, 272-274 Barrierless processes 204 Barrierless minimum energy path 210 Barrierless reaction path 208 Barrierless vs. barrier-controlled twisting path 217 Benzene 2, 20, 48, 80, 134, 141, 144, 148, 214-215,234,319,333-337 Benzenedithiolate (bdt) 281, 302-303 Benzobarrelene 273-274 Benzodiene 264 Bimolecular photochemical reactions 20 Biradicaloid electronic structure 5 Bi-(di-)radical 14, 17, 18, 26, 47, 208, 264, 267-268, 273-274, 331 Block iteration methods 103

Blue copper protein(s) 79, 279-280, 294295, 298-299, 309 Bottleneck 38, 104, 181, 195, 199, 238, 344 Butadiene 5, 68, 79 Cr(CO)6 279, 282 Carotenoids 116 Complete Active Space Self Consistent Field (CASSCF) method 37, 46, 57,82, 193, 198,206,213,231235, 326-327, 336, 338, 340 Complete Neglect of Differential Overlaps for Spectroscopy (CNDO/S) - Hamiltonian 150 - methods 298 Complete Active Space - Multireference Second Order Perturbation Theory (CASPT2) method 37, 46, 52, 5759,62,77,234,237,281,283, 287, 289, 328-329, 337, 339, 344 Charge transfer 7, 14, 20, 23-24, 116, 210, 228, 244, 280, 281, 295, 298, 300, 321 Charge transfer state(s) 79, 109, 208, 282, 291,295,297-298,300,306 Chlorophyll(s)109, 116 Cluster expansion method 291 Configuration Interaction (Cl) methods 44, 54, 323, 342 - full configuration interaction 36, 231,322 - multi-reference Cl (MRCI) 37-38, 42, 46, 54, 64, 74, 79, 83, 232, 327-328 - MRCI type with singly and doubly excited CSFs (MR-SDCI) 328 - singles and doubles configuration interaction (CISD) 37, 150 - singles configuration interaction (CIS) 37, 47-48, 51,81, 105, 109, 150 Configuration State Functions (CSFs) 44, 54, 56, 324-332 Conical intersection(s) 3-9, 12-16, 19-21, 24-25, 27, 63-65, 81-84, 111, 171177, 180-184, 187-188. 195-197, 199-201,204-211,213,216-218, 220, 226-227, 229, 230, 233, 238, 240-241, 247, 262, 264, 268, 270, 272-273,327,329,331. Coupled-Cluster (CC) methods 37-38, 4951, 53,79, 332, 336

350 - equation-of-Motion CC (EOM-CC) 50-53,78-79,228,291,293 - multireference coupled-cluster (MRCC) 52 Coupled-Electron Pair Approximation (CEPA) 37 Crossing point(s) 3, 5, 63, 198, 206, 209,

Ethene 68, 320-321, 324-326, 330-332 Ethylene(s) 6, 14, 171-172,233 Exact Exchange (EXX) functional 100, 109-110 Exchange correlation 41, 47, 95, 97-101, 103, 105, 109, 111,228, Exchange integral 46, 268

213,317 Crystal lattice 268 Cyanines 184, 186 Cyclobutane 14 Cyclobutanones 271 Cyclobutene 5, 14, 153-154 Cyclohexadienes 4 Cyclohexenone 274 Cyclooctatetraene (COT) 71-72, 74, 7678, 193,217-219 Cytosine 81-83, 116 Delta Density method 270-272 Density Functional Theory (DFT) 53, 67, 78, 93-128, 228, 279, 285, 289291,294,298,332,334,344 - Hartree-Fock-Slater versions of DFTHFS-SW291.298 - HFS-DVM 291 Derivative coupling 10, 29, 178 Di-7t-Methane Diarylethylene 179-181, 183 Dihydroazulene (DHA) 179, 182 Dipole allowed state(s) 161, 167 Dipole allowed transition(s) 74, 130, 144, 145, 147,280,291 Dipole forbidden state(s) 144, 161 Dipole forbidden transition(s) 72, 74, 130, 141-142, 144, 147, 150, 161 Direct-dynamics variational Multiconfiguration Gaussian (DDvMCG) method 179 Dynamic correlation 8, 45-46, 57, 62-64, 68-69, 76-78, 82-83, 283, 325-329, 344 Effective Hamiltonian 62-64, 83, 321, 329 Ehrenfest dynamics 177-178, 238-239 Electric field 49, 187, 321, 324, 332 Electron transfer 11, 23, 174, 239-295, 297, 307 Electrostatic effect 235, 247, 249 Electrostatic Hamiltonian 236 Electrostatic interaction 68 Electron Spin Resonance (ESR) spectra, g-factors 284-285, 290, 298

Ferrocene 52, 279 Floating Occupation Molecular Orbital (FOMO) method233 - FOMO-CASCI233, 244, 245-246 - FOMO-CI233-238 Fullerenes 94, 111-112, 130, 148, 150, 152, 159-160 Gaussian wavepacket 179 Gradient difference 10, 29 Graphitic materials 164, 167 Green's functions methods 49, 80 Green Fluorescent Protein (GFP) 7, 27, 233-234, 236, 239-240, 244-249 H2 42-44, 54, 100 Halorhodopsin 225 Hartree-Fock (HF) method 41-43, 318320, 323-324, 332 - coupled Hartree-Fock (CHF) 49, 53 - time-dependent Hartree-Fock (TDHF) (see also Random-Phase Approximation) 49, 50, 53, 97, 105, 106, 109,332 Hartree-Fock (HF) wavefunction 37 Hellmann-Feynman theorem 52, 96 Hexatriene(s) 4, 52 -1,3,5-trans-hexatriene 134-135,

155 Hybrid functionals 100, 105, 109-110, 114 Hybrid methods Hydrogen transfer 7, 20, 112, 297 Huckel methodology 262 Inactive orbitals 57, 193, 326 Indole 112 Initial Relaxation Directions (IRD) 199-200, 205, 208-209 Internal coordinates 7, 158 Intersection space 7, 12, 174, 180 Intrinsic Reaction Coordinate (IRC) method 198,200,205,208 Intruder states 60-61, 79, 110, 283-284,

329

351

Jahn-Teller coupling 80, 161 Jahn-Teller distorsion 274, 331 Lagrangian method 96 Lanthanides 79, 338, 341-342, 344 Level-shift CASPT2 (LS-CASPT2) method 61, Linear combination of atomic orbitals (LCAO)96, 101, 115, 322 Linear response method(s) 332, 336, 344 Luciferin 112 MnCV 279,283,291,309 Minimum energy path (MEP) 2, 12, 25, 82, 111, 188, 192,210 Molecular dynamics (MD) 12, 29, 115, 174-176, 186, 197 Molecular Mechanics/Valence Bond (MMVB) method 234 Mobius-Huckel theory 261 Mononuclear oxomolybdenum enzymes 280 Moller-Plesset (second-order) perturbation theory (MP2) 37, 49, 67, 153, 197, 203, 282, 328-329 Multiconfigurational self-consistent field (MCSCF) method 5-7, 37-38, 4546, 52-57 Multireference perturbation theory (MRPT) 37-38, 46, 62, 64, 79, 228 Multiple spawning method 238-239 - ab initio multiple spawning (AIMS) 239 Multi-state Complete Active Space Multireference Second Order Perturbation Theory (MS-CASPT2) 62-64, 6871,74,83-84,329,330 n-tetrasilane 67-70 N2O2 46, 52 Natural Orbitals (NOs) 46, 69, 98, 300 Non-dynamic correlation 45, 50 Normal coordinates 134, 139, 145-146, 149, 151, 159 Normal mode(s) 129, 136, 153 Ozone 50, 52 Pariser-Parr-Pople (PPP) method 2, 320-

321, 333-334 Penta-3,5-dieniminium cation 9, 184, 186 Pericyclic reaction 4, 261

Perturbation theory 35-37, 48, 50-51, 53, 58-59,79,94, 106, 111, 148,231, 325, 328, 338 Photoactive Yellow Protein (PYP) 7,117, 186-187,225,229,232,237 Photo- isomerization 9, 27, 82, 112, 150, 152, 157, 159, 208-218, 220, 237 -cis-trans 9, 10, 183-187, 192, 202, 208, 211, 216-217, 225, 332 Plastocyanin 294-300 Platinum atom 339-340 Polycyclic Aromatic Hydrocarbons (PAHs) 111, 164-165, 167 Polymers 94, 109-110, 115-116, 191 Polythiophene(s) 115-116 Porphyrin(s) 78-79, 94, 112-114, 321 Potential energy surface(s) 2-3, 5-9, 12, 26,81, 129, 135, 149, 153, 171, 176-178, 186, 188-189, 192-198, 206, 208, 211, 213, 216, 226, 229231,234,238-239 Protonated Schiff Base(s) (PSB) 4, 9, 14, 29, 82, 84, 153, 156-157, 192, 202, 237 Pyrazoline 20-26 Quantum Mechanics/Molecular Mechanics (QM/MM) approach 7, 27, 79, 107, 117, 235-238, 245-246, 249, 269 Radical anion 72, 77, 78, 80, 258, 270, 272 Radical pair 21, 23 Random-Phase Approximation (RPA) 49, 50,53,97, 105, 106, 109,332 Rrestricted Active Space State Interaction (RASSI) method 204, 304, 338-339 Reaction field (RF) approaches 79 Reaction path(way)(s) 1, 3, 5, 7, 9, 12, 19-

20,24,29-30,53,71,81-83, 111, 130, 150, 171-177, 179-184, 187188, 192-202, 204-205, 207-208, 213, 216-220, 226-227, 230-232, 238, 249, 269 Relativistic effect 80, 303, 338, 345 Renner-Teller coupling 80 Renner-Teller intersection(s) 226 Restricted active space self consistent field (RASSCF) 46, 80, 203, 214, 338, 344 Rhodopsin (Rh) 7, 9, 12, 27, 157, 159, 183, 202, 212-213, 225, 237, 332 Rydberg character 76, 107, 228

352 Rydberg excitation 101, 105 Rydberg orbitals 45, 68-69, 73, 326, 327 Rydberg (like) states 45, 61-63, 65, 68-69, 71-72, 76, 83, 109, 228, 321, 326327,331,335 Rydberg (-type) functions 61, 68-69, 71, 74-75 Rydberg transition(s) 69, 76, 321

Symmetry-allowed transition(s) 141 Symmetry-forbidden transition(s) 130, 141, 147-148, 161, 336 Surface hopping dynamics 177-178, 186,

188, 238-239e Tetraradical 14, 18, 217 Tamm-Dancoff approximation (TDA) 51,

105-106 Saddle point 14, 192 Salem correlation diagrams 2, 5-6 Second-Order Polarization Propagator Approach (SOPPA) method 49-50, 53 Semibullvalene 218, 263, 267, 272-273 Semi-classical dynamics methods 12, 29, 171, 177, 178, 179, 188, 196, 198, 238 Semiconductor clusters 94, 114-115 Semi-empirical methods 2, 74, 107, 232238, 244 - CAS/CI 7 - MNDOC 232 -QCFF/PI 150, 156, 167,238 Single-configuration methods 32, 47-48, 78 Slater-type orbital (s) (STO) 105 Solvation 112, 117,236,244 Spatial symmetry 37, 57, 103, 106, 195, 324 Spin-allowed transition(s) 141 Spin-forbidden transition(s) 130, 147, 150151, 161 Spin-orbit coupling 67, 80, 151, 273-274, 287, 290, 338, 340, 343, 345 Spin symmetry 37, 57, 103, 106, 195, 324 State-average CASSCF (method) 57, 68, 233, 327 Stationary point(s) 56, 101, 197-198, 200, 201,211,213,234 Stereochemistry 1, 6, 26, 220, 261, 265 Steric effect 235 Stilbene 112 Symmetry 4-5, 14, 35, 37-38, 40, 42-43, 50, 57,68-69,71-74, 102-103, 106, 110, 116, 130, 134-135, 139141, 143-144, 147-149, 151, 153, 159-161, 195, 203, 214, 226, 281282, 284-286, 293, 295, 319, 324, 326-327, 329, 331, 336-337, 339 Symmetry-Adapted Cluster Configuration Interaction (SAC-CI) 38, 50, 52-53, 78,291,293

Time-Dependent Density Functional Theory (TDDFT) 8, 13, 29, 38, 50, 53-53, 74-75, 78, 81, 93-128, 228-229, 231, 279, 291, 293, 298, 335-336, 342 Time-Dependent Kohn Sham framework (TDKS ) 94-95 - time-Dependent Kohn Sham eigenvalue problem (TDKS EVP) 97-98, 102, 105 Trans-1,3-butadiene 48 Transition metal compounds 94, 114, 340 Transition state 1, 3, 5, 20, 37, 65, 71, 77, 82, 180-182, 187, 193, 195-198, 204-205, 213, 216, 226, 234, 261, 274, 343 Transition state theory 197 Triplet state(s) 77, 148, 150-153, 155-156, 161,319,330,336,339,343 UO2 342 Urocanicacid 112 Variational principle 96, 179, 320, 322, 326 Vertical excitation energies 13, 47, 65, 74,

81, 101, 107, 109, 111, 115,203, 228,229,231-232,283 Vinylheptafulvene 179, 182 Woodward-Hoffmann rules 2, 5, 153-154, 172 Zeolites 284-285, 289-290, 294