Toward an accurate molecular orbital theory for excited states: the

John J. Nash and Robert R. Squires. Journal of the American ... Luis Serrano-Andrés, Markus P. Fülscher, and Björn O. Roos , Manuela Merchán. The ...
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9204

J. Phys. Chem. 1992, 96,9204-9212

circumstances, be adequately represented by a molecular cluster of finite size, suggesting that the electron density distribution in molecules and crystals is to a great extent dominated by local influences. This is not to say that long-range interactions in crystals are not important in terms of the energetics of crystal formation or stability, but it does imply that the fine details of the electron density redistributions which occur on bonding are dominated by near-neighbor interactions. We note in this regard that none of the present cluster calculations yield as satisfactory a description of the oxygen EFG tensor in corundum as do the CRYSTAL calculations of Salasco et a1.6 It would be interesting to ascertain whether this results from the exclusion of the longrange crystal field in the cluster calculations or, and this would appear more likely, from a difference between the two methods in describing the sharp quadrupolar deformations of the electron distribution close to the nucleus (perhaps due to basis set differences or choice of cluster) which tend to dominate the EFG tensor. The crystal field could, of course, be approximated in a molecular cluster calculation by the use of a carefully chosen set of point charges to mimic the long-range potential. However, the use of molecular clusters to obtain properties such as those pursued in this work will most likely become less attractive with improvements in crystal HartretFock d e s in the near future and the availability of full-potential LAPW codes for crystalline systems.*’ Acknowledgment. It is a pleasure to acknowledge discussions with Prof. G. V.Gibbs on the construction of molecular clusters and their usc in modeling mineral systems. A University of New England Research Scholarship (to A.S.B.) and financial support from the Australian Research Council (to M.A.S.) are gratefully acknowledged. Registry No. Corundum, 1302-74-5.

References pad Notes (1) Lewis, J.; Schwarzenbach, D.; Flack, H. D. Acra Crysrallogr., Secr. A 1982,38,733-739. (2) Kirfel. A.; Eichhom. K. Acra CmralloPr.. Secr. A 1990.46. 271-284. i3j Nanei. S:J. Phvs. 6: Solid Siate P h k 1 9 8 5 . 18. 3673-3685. (4) Ca&; M.; Dov&i, R.; Roetti, C.; Kotomin, E.; Saunders, V. R. Chem. Phys. Lerr. 1987, 140, 120-123.

( 5 ) Pisani, C.; Causa, M.; Dovesi, R.; Roetti, C. Prog. Surf.Sci. 1987,25, 119-137. (6) Salasco, L.; Dovesi, R.; Orlando, R.; Causa, M.; Saunden, V. R. Molec. Phys. 1991, 72, 267-277. (7) Dovesi, R.; Pisani, C.; Roetti, C.; Causa, M.; Saundera, V. R. CRYSTAL 88, Program 577, Quantum Chemistry Program Exchange, Indiana University. Bloominnton. IN. 1988. (8) The terms. “poor”, ‘good’”, and Very good” used to describe the computational parameters employed for a given calculation refer to the levels of approximation in the numerical treatment of the Coulomb and exchange series and the reciprocal space integrations in CRYSTAL 88. Errors related to the use of the poor computational conditions arc negligible for propertics such as the deformation electron density and the EFG tensor. (9) Brown, A. S.;Spackman. M. A.; Hill, R. J. Acra Crysrallogr., Secr. A, submitted. (10) Gibbs, G. V. Am. Miner. 1982,67,421-450. (11) Gibbs, G. V.; D’Arco, P.; Boisen, M. B. J . Phys. Chem. 1987, 91, 5347-5354. (12) Sauer, J. Chem. Rev. 1989,89, 199-255. (13) Velders, G. J. M.; Feil, D. Acra Crysrallogr., Secr. E 1989, 45, 359-364. ~. (14) Hehre, W. J.; Ditchfield, R.; Stewart, R. F.; Pople, J. A. J. Chem. Phys. 1970, 52,2769-2773. (15) Hehre, W. J.; Stewart, R. F.; Pople, J. A. J . Chem. Phys. 1%9,51, 2657-2664. (16) Ditchfield, R.; Hehre, W. J.; Pople, J. A. J . Chem. Phys. 1971, 51, 724-728. (17) Hehre, W. J.; Ditchfield, R.; Pople, J. A. J . Chem. Phys. 1972, 56, 2257-2261. (18) Francl, M. M.; Pietro, W. J.; Hehre, W.J.; Binkley, J. S.;Gordon, M. S.;DeFrees, D. J.; Pople, J. A. J. Chem. Phys. 1982, 77, 3654-3665. (19) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acra 1973,28,213-222. (20) Jensen, J. H.; Gordon, M. S.J . Compur. Chcm. 1991,12,421-426. (21) Schmidt, M. W.; Boatz, J. A.; Baldridge, K. K.; Koseki. S.;Gordon, M. S.;Elbert, S.T.; Lam, B. GAMESS. QCPE Bull. 1987, 7, 115. (22) Amos, R. D.; Rice, J. E. CADPAC: The Cambridge Analytic Derivatives Package, issue 4.0, 1987. (23) Clark, M. G. In MTP Inrernarional Review of Science, Physical Chemistry Series 2; Buckingham, A. D., Ed.; Buttenvorths: London, 1975; Vol. 2, pp 239-297. (24) Stewart, R. F. Acra Crysrallogr., Secr. A 1968, 24,497-505. (25) Epstein, J.; Swanton, D. J. J . Chem. Phys. 1982, 77, 1048-1060. (26) Hinchliffe, A. Compurarional Quantum Chemistry; Wiley: ChiChester, 1988. (27) See,for example, the description of the WIEN system and references therein: Blaha, P.; Schwarz, K.; Sorantin, P.; Trickey, S . B. Compur. Phys. Commun. 1990, 59, 399-415. (28) Brun, E.; Derighetti, B.; Hundt, E. E.; Niebuhr, H. H. Phys. Lerr. 1970, 31A, 416-417. (29) Ajzenberg-Selove, F. Nucl. Phys. A 1977, 281, 1-148.

Toward an Accurate Molecular Orbital Theory for Excited States: The Azabenrenes Markus P. Flilscher, Kerstin Andersson, and Bjiim 0. R m * Department of Theoretical Chemistry, Chemical Centre, P.O. Box 124, $221 00 Lund, Sweden (Received: March 30, 1992) A computational scheme recently proposed for ab initio calculations of electronic spectra of molecular systems is applied to the azabenzene molecules. The method has the aim of being accurate to better than 0.5 eV for excitation energies and is expected to provide structural and physical data for the excited states with good reliability. Applications are possible to molecules with up to about 20 atoms with good quality basis sets. The scheme is based on the complete active space SCF method (referred to as CASSCF), which gives a proper description of the major features in the electronic structure of the excited state, independent of its complexity, accounts for all near degeneracy effects, and includes full orbital relaxation. The remaining dynamic electron correlation effects are in a subsequent step added using second-order perturbation theory with the CASSCF wave function as the reference state. The approach is tested here in a calculation of the valence excited singlet states of the azabenzenes pyridine, pyrazine, pyrimidine, pyridazine, and s-triazine, using a (C,N,4~3pZd/H,3s2p) atomic natural orbital (ANO) basis. The ?M* excitation energies of the azabenzenes are computed with an average error of 0.14 eV. With the exception of one case, the n-r* excitation energies are computed with an accuracy of 0.32 eV or better in all cases where a comparison with reliable experimental data can be made.

1. Introduction

Optical spectroscopic methods are indispensable tools in numerous areas of research and technical sciences and are capable of providing, qualitatively and quantitatively, important information on the static and dynamic properties of the system under 0022-3654/92/2096-9204S03.00/0

study. As the application of spectroscopic methods, as well as the optical devices, becomes more refined, an increasing demand in theoretical support to analyze the spectra can be observed. For some time, we have attempted to gain insight into the correlation among spectral and structural features of aromatic systems

0 1992 American Chemical Society

The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 9205

Molecular Orbital Theory for Excited States

pyridine

pyrazine

pyrimidine

pyridazine

s-triazine

Figure 1.

through ab initio quantum chemical calculations on the electronic spectra of small- and medium-size (up to 20 first-row atoms) singly excited states moldea. Recently, we reported on the -* of benzene' as an illustration of a novel approach that has the aim of being accurate to better than 0.5 eV. In this paper, we report on the r-r* and n v * singly excited states of pyridine, pyrimidine, pyrazine, pyridazine, and s-triazine (Figure 1) using the same approach. The azabenzenes are important parent molecular systems for numerous compounds such as the biologically active nicotinic acid or the nucleotides cytosine, uracil, and thymine and have been subjected to extensive spactroscopic Since the very early days of quantum chemistry, these systems also have been used as test casea to describe excited states. The reason for this is that they are isoelectronic with benzene, are easily accessible, and therefore are preferred systems for systematic studies on the substitution effects in heterocycles. An excellent survey of experimental and theoretical studies on the electronic structure of the azabenzeneshas been given by Innes et al.2 (hereafterreferenced by IRM). Recently, a seriea of studies have been published by Walker et ala3+(here after referenced by WP), which includes vacuum ultraviolet absorption spectra, electron energy loss spectra,an extensive compilation of theoretical data, and the results from multireference configuration interaction (MRCI) calculations. In addition, a comprehensive collection of vacuum UV absorption spectra and theoretical data has also been presented by Bolovinos et ala7(hereafter referenced by BT). A first, quick glimpse of the spectra of the azabenzenes shows that they are remarkably similar to the spectrum of benzene; i.e., they show three bands located at approximately 5.0,6.5, and 7.5 eV, respectively. The substitution of a C-H group by a N atom, in general, introduces a blue shift of the dominating band corresponding to the El, band in benzene. Since the mono- and disubstituted azabenzenes belong to molecular point groups of lower order as compared to benzene, the degenerate orbitals of elg and e2" symmetry separate and the electronic states, which are degenerate in benzene, are split. In general, it seems to be difficult to give an experimental measure of this energy separation, and usually, only one excitation is extracted from the measured spectrum. In the low-energy region of the spectrum, additional n-r* states are seen. However, the analysis of the spectra is often very complex, and the assignments of the weak n-r* transitions remain controversial. Electronic structure calculations on excited states have, over the years, almost solely been performed using state approaches; that is, wave functions are evaluated for each of the individual states. In a different kind of approach, the propagator methods: transition energies, transition probabilities and other response properties are computed directly without knowing the individual states explicitly. Numerous theoretical studies of the excited states in the azabenzenes have in the past been b a d on semiempirical quantum chemical calculations, like CNDO/S9 and the PPP'O methods. Although the computed excitation energies are often very accurate, the approximations made in these methods make it difficult to draw firm conclusions based on the results of such calculations. To compute the excitation energiea from first principles, electron correlation has to be treated in a balanced way, and all states of the same symmetry, which are close in energy, have to be considered simultaneously. The most important correlation effects are describad by configuration state functions (CSFs) mixing the *-electrons among the *-orbitals. In addition, the lone pairs of heteroatoms may strongly interact with the v-electrons and need to be considered on equal footing. However, to obtain quantitatively correct results, the dynamic correlation effects also have

to be considered. These effects are dominated by the dynamic polarization of the u-electrons, which is described by c o n f i i t i o n s involving simultaneous u-u* and m r * excitations, and can differ substantially for states of different character. The situation has been nicely illustrated for the V state of ethylene and some aromatic systems where the inclusion of the u-polarization effects is indispensible to obtain reliable excitation Typically, large a-polarization effects are observed for states of strong ionic character and are substantially smaller for states dominated by covalent structures. Due to its inherent flexibility, the CASSCF appr~ximation'~ has proven to be particularly suited to cope with situations where the electronic structure varies strongly, e.g., in the close vicinity of transition states or in excitation processes. In this approach, the wave function is constructed by distributing the active electrons among the active orbitals in all possible ways, whereas the inactive orbitals are kept doubly occupied in all cofligurations. The strong configurational mixing, common to many excited states, is then automatically included in the wave function already at this level of approximation. Such an approach is a necessary prerequisite for a balanced treatment of the dynamic correlation effects. Both the inactive and active orbitals are optimized. Thus, the static response of the core orbitals is accounted for in addition to correlation effects involving the valence orbitals. The CASSCF method can, in principle, be brought close to the full CI limit, but this is, in most applications, not a practical approach. Instead, the dynamic correlation effects are treated separately either by means of multireference CI (MRCI) techniques or, as in the present study, by perturbation theory. Since the complex neardegeneracy effects have been included already at the CASSCF level, the treatment of the remaining correlation effects is normally without complication and a perturbation expansion is expected to converge fast. The most accurate way to include dynamic correlation effects is certainly to supplement a CASSCF calculation with an MRCI calculation. Very accurate results have been obtained for small systems with this technique. However, the applicability of the approach to larger molecular systems is severely hampered by the large number of CSFs needed to describe differential correlation effects. A convenient alternative, which has a much larger range of applicability, is a second-order perturbation treatment where the CASSCF wave function is taken as the reference functi~n.'~J~ The second-order perturbation approach, called the CASFT2 approximation,16has recently been applied successfully to compute a number of properties of the ozone molecule, including the vibrational ~pectrum.'~J* The same approach has also been used in studies of the excited states of the nickel atom19and the benzene molecule.1 The results obtained are surprisingly accuratein the case of benzene, the excitation energies were reproduced with an accuracy of 0.26 eV or better for all valence excited singlet and triplet states. In the present contribution, it is shown that a similar accuracy can also be obtained for the azabenzenes, including now the n-* states. With this development, we believe that ab initio quantum chemistry has finally reached a stage where theoretical data can be used in aiding the assignment of measured electronic spectra of canjugated organic molecules. So far, this has not been possible, and the computed excitation energies have often been in error by 1 eV or more, except maybe for systems small enough to be treated by accurate MRCI methods and large basis sets (see, however, the recent work of Graham and Fred7). The simplicity of the CASPT2 method makes it possible to use higher quality basis sets and thus to a larger extent avoid contamination of the results due to basis set deficiencies. The development of direct methods and gradient techniques will further extend the range of molecules, which can be treated with the praent approach, and will allow geometry optimization for excited states and transition states for photochemical reactions. In the present study, we report the results for the vertical excitation energies of pyridine, pyrazine, pyrimidine, pyridazine, and s-triazine using the CASSCF/CASFTZ approach. Details about the approach and the calculations are described in the next section. A discussion of the results for each moleucle separately

9206 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992

Fnlscher et al.

TABLE I: Det.uS of the Cdculations: Selected Active Sprees and Optimization Coaditio~ system transitions active spaces remarks 1 Rydberg orbital deleted for states of AI and B2 symmetry; average orbitals are pyridine T-T* used average orbitals are used n-# average orbitals are used pyridazine r-r* individually optimized state functions are used n i * , A2 symmetry individually optimized state functions are used n-r*, B, symmetry 1 Rydberg orbital deleted for states of A, and B2 symmetry; individually optimized pyrimidine T-T* state functions are used individually optimized state functions are used nq* 1 Rydberg orbital deleted for the second state of BI, symmetry; individually ~-r* pyrazine optimized state functions are used individually optimized state functions are used n-r* computations are done in CZvsymmetry; average orbitals are used s-triazine r-r* computations are done in CZvsymmetry; individually optimized state functions are n-s* used

then follows and is fmally summarized in the last section, which also include a discussion of the features common to all molecules.

2. Metbods and Computational Details A. Geometries and Basis Sets. The geometries used in the present calculations stem from various experimental sources and are copied from Table 5 in ref 2. Generally, contracted basis sets of the atomic natural orbital (ANO)typezo*21 are used. They have been obtained from (14s,9p,4d) and (8s,4p) primitive sets for the first-row atoms and hydrogen, respectively, and are contracted to the following final structure: (C,N, 4s3p2d) and (H, 3s2p). These basis sets are constructed to optimally treat the correlation and polarization effects and should be large enough to describe the electronic structure of the valence excited states within the desired accuracy. Diffuse functions are, however, missing in the hsis set, and we cannot expect to be able to treat Rydberg states with high precision. It is also not the purpose of the present work. For some of the molecules, we have encountered states which are of the Rydberg type. The corresponding energies are, however, not expected to be very accurate. When needed, the calculation of the valence excited states have been simplified by deleting the corresponding diffuse orbital from the molecular orbital (MO) space (seebelow). A possible real interaction between Rydberg and valence excited states has, as a consequence, been neglected, but such interactions are not expected to be of great importance for the excited states under consideration. For each molecule, a SCF calculation of the ground state was first performed. The corresponding orbitals were used as starting orbitals in the subsequent CASSCF calculations. The carbon and nitrogen 1s core orbitals were frozen at the SCF level and were not further optimized, and the 1s electrons were not included in the CASPT2 calculation of the correlation energy. B. CASSCF and CASSI Methods. Initially, multiconfigurational wave functions are determined at the CASSCF level of approximation.14 To compute the m*singlet excited states, at least the six valence *-orbitals have to be included in the active space. It was found in the study of the benzene molecule' that interference with a nearby Rydberg state in the CASPT2 calculations deteriorated the result for the 'Elustate when only six orbitals were used in the active space. The active space was extended by including the corresponding Rydberg orbital, with the consequence that the Rydberg state appeared as one root in the CASSCF calculation and was in this way separated from the valence state. We have in the present investigation followed the same recipe and used 12 active *-orbitals in the calculation of the r - ~ *excited states. Actually, preliminary calculations on pyrazine performed with 9 active orbitals gave almost identical results, so we are confident that the active space with 12 orbitals is saturated. We are in the present study only interested in the valence excited states. The basis set chosen is also, as pointed out above, not appropriate for treating Rydberg states. When such states occur as a solution to the CASSCF calculation, we have therefore carried through the calculation once more, now with the Rydberg orbital deleted. It was shown that this reduction in the MO space had a very small effect on the properties of the

valence states. The 12-active-orbital rule could not be applied to the pyridazine molecule, due to intruder states in the CASPT2 calculations of the u-u* Rydberg type. Instead an active-orbital space comprising four u-orbitals and eight *-orbitals was used. In addition, lone pair orbitals are also included in the active space to determine the n l r * singly excited state. Normally two orbitals (one strongly and one weakly occupied) per lone pair was used. The weakly occupied orbitals allow a description of the most important radial correlation effects in the lone pair already at the CASSCF level of approximation. Interference with Rydberg states was less common for the n l r * excited states, and the corresponding active space for the 7-orbitals could be kept smaller. Normally, six active orbitals were used. However, as can be seen in Table I, larger active spaces were used in two cases, pyridine and pyridazine. This was necessary to avoid intruder states of the u-u* and o7r* Rydberg type. The problem of the intruder states is an unfortunate feature of the approach, which occurs when extended basis sets containing some diffuse character are used. We have in later applications found that it is actually simpler to add Rydberg-type orbitals to the basis set and also include the Rydberg states in the calculation. The wave functions have been optimized for each state individually. However, in some cases, near-degeneracy between different states leads to convergence problems in the orbital o p timization. In such cases, it has proven useful to optimize a set of "average" orbitals; i.e., a single set of orbitals is determined which span a common MO basis for several excited states. In a few casgs, both methods have been used with very similar results for the excitation energies, ensuring that the state-average a p proach is accurate enough. Details on the selected active spaces and the way the wave functions are obtained are collected in Table I for each of the molecules treated. The wave functions obtained by optimizing individual states are not mutually orthogonal. The CASSCF state interaction (CASSI) methodz2 has been developed to compute transition properties from nonorthogonal state functions and is used here to compute the oscillator strength. In the formula for the oscillator strength, we used the energy differences corrected for by the second-order perturbation method. Such a strategy has been found to work well in a number of earlier applications. The transition moments are sensitive to details of the electron density of the two states, which is normally well-described at the CASSCF level of approximation but is not so sensitive to dynamic electron correlation. The energy differences, on the other hand, depend strongly on electron correlation and, therefore, need to be computed at a higher level. Oscillator strengths computed according to the present scheme were in a recent application to the pyrimidine molecule compared to the results obtained using the MRCI method.30 Good agreement was found between the two approaches. C. CASPT2 Metbod. The CASPT2 method16 computes the first-order wave function and the second-order energy in the full CI space, without any further approximation, with a CASSCF wave function constituting the reference function. The zerothorder Hamiltonian is defined as a Fock-type one-electron operator

The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 9207

Molecular Orbital Theory for Excited States

TABLE Ik CASSCF and CASFT2 Excitation Energies (in eV) and CASSCF Oscillator Strengths and Dipole Moments (in D)for Pyridine CASSCF ground state ('Al) P ? r * states IBdIBd lAI(LBlu) IA1!Bz('EIu) IBI,IAI(IEIJ n-r* states IBI 'A2

F'T2

5.03 7.48 8.91, 8.89

4.94 6.35 6.84, 7.49

8.14, 8.34

7.78, 7.84

5.42 6.12

4.91 5.21

exotla

4.99 6.38 7.22

4.59' 5.4Y

errorb

-0.05 -0.03 -0.05

+0.32 -0.22

ow strength comp exptl"

Pd

WC

0.89

2.36

0.88 0.86 0.83, 0.47

1.84 3 .OO 2.43, 1.19

0.04 0.006 0.73, 0.96

0.86, 0.85

2.31, 1.27

0.03, 0.26

0.88 0.88

0.5 1 0.90

0.02

other calculations

0.029 0.085 0.90

4.76: 5-44! 6.3,r 6.gh 6.99: 7.041 6.6,s 8Sh (8.2, 7.7): (8.2, 7.9)' (8.2, -),g (9.2, -)h

0.003 forbidden

4.5: 5.24; 6.2,g 7.2h 5.28,' 5.69/7.4,8 7.1h

aExperimental data from ref 7. bDifference between the CASF'T2 results and experiment. The (lAl,IB2) average values have been used for the IEluand IhB states. 'The weight of the CASSCF reference function in the first-order wave function. dDipole moment (experimental value for the ground state is 2.15 0.05 D (ref 27)). 'Results based on MRDCI calculations (ref 3). 'Results based on SAC-CI calculations (ref 25). 8Rtsults based on CIS calculations using a 6-3 1 +G* basis set (ref 26). As in g but with 'MP2" corrections added. Reference 24. Reference 3.

*

TABLE IIk CASSCF and CASP"2 Excitatioa Ewrgies (in eV) and CASSCF Oscillator Stmgtbs for Pyrazhe CASSCF" states IBzdIBzJ 5.10 (5.13) ~BId'BlU) 8.51 (7.78) I B I ~ , I B Z ~ ( I E ~9.65, ~ ) 9.55 (9.41, 9.28)

PT2

exptP

errof

ud

4.77 6.68 7.57, 7.75

4.81 6.51 7.67

-0.04 0.17 -0.01

0.77 0.76 0.69, 0.63

osc strength comp exptlb

other calculations

T-T*

'B3P,1AP(1EzP) 8.53, 8.40 (8.53, 8.39) n-r* states 93, 5.24 'A" 6.32 'B, 6.28 lBll 7.57

8.16, 8.27 3.58 4.37 5.17 6.13

0.07 0.08 0.76, 0.66

-0.25

5.19 6.10

-0.02 +0.03

0.77 0.76 0.77 0.79

4.77: 5-29! 5.16r 7.19: 9.91/8.81* (8.7, 7.9): (11.5, lO.0y (10.5, 9.8)g

forbidden

0.77, 0.78 3.83h

0.062 0.10 0.72

0.01

0.006 forbidden forbidden forbidden

4.93:, 5.57/ 4.728 5.36: 5-65! 5.3W 6.07,' 7.04,f 6.3W

"The numbers in parentheses give the results for the %only" MRCI calculation (for more details, see text). bExperimental data from ref 7. 'Difference between the CASPT2 results and experiment. The (1BIU,1B2U) and ('Bor,lA,) average values have been used for the IE and states. dThe weight of the CASSCF reference function in the first-order wave function. 'Results based on MRDCI calculations (ref 5). )Results based on single-reference CI calculations (ref 28). #Results based on CI calculations using a GVB reference function (ref 29). h Mtransition.

and is constructed such that a Mbller-Plesset-type perturbation theory is obtained in the closed-shell single-determinant case. In the present application, we have in most cases used the "diagonal only model"; that is, the off diagonal elements of the Fock operator are not included in the zeroth-order Hamiltonian. The model was thoroughly tested in the calculations on the benzene molecule,' and it was shown that the nondiagonal elements have a very small influence on the computed excitation energies (the largest computed difference was 0.07 eV). A further test was made here for the n-r* transitions in s-triazine. As can be seen from Table VI, the differences are also small in this case (the largest difference being 0.12 eV). This is an important result, since the diagonal variant of the CASPT2 method is an order of magnitude cheaper than the full model and can be extended much easier to larger systems. It should be emphasized, however, that it is only the nondiagonal approach that is invariant to rotations of the molecular orbitals. The full approach must therefore be used in cases where such invariance is important, for example, in calculations of potential surfaces. We have also noted in later applications that Rydberg states are more sensitive to the use of the full Fock operator than valence excited states. The CASPT2 equations are formulated exclusively in terms of one-, two-, and three-body densities and are therefore independent of the actual size of the reference function. The limiting factor is not the number of correlated electrons but the number of active orbitals, which determines the size of the density matrices. The current implementation of the CASPT2 method allows a maximum of 14 active orbitals. A larger number is rarely needed in any application and anyway could not be handled by the CASSCF program, except in cases with very few active electrons. The CASPTZ program also calculates the weight, w , of the CASSCF reference in the first-order wave function. This weight is a measure of how large a fraction of the wave function is treated variationally. The relative weight of w in different states then gives a measure of how balanced the calculation is. Normally, one requires w to be about the same for the ground and excited

states in order for the calculation to be balanced with respect to the treatment of electron correlation. In some cases, however, interference with nearby electronic states not included in the reference CI space deterioratesthis balance for some excited states. This happens for some of the molecules treated here and is in all cases due to interference with nearby Rydberg states. However, such a situation can be tolerated in cases where the interaction is very weak, and consequently, the effect on the second-order energy small. A large coefficient for the interfering Rydberg state may then appear in the first-order wave function with only a very small effect on the energy. A typical example is shown in Table I1 for the IB2 component of the "'El: state in the pyridine molecule and also for some of the states in s-triazine (Table VI). All calculations have been performed on IBM RS/6000 workstations (Models 530H and 550) using the MOLCAS-2 quantum chemistry software,23which includes as one module the CASPTZ program. Some timing data were presented in the recent paper on the benzene molecule.' 3. Results Calculated CASSCF and CASPTZ excitation energies for the five molecules are presented in Tables 11-VI, together with available experimental information and the mults of other ab initio calculations, which have been obtained using different variants of the configuration interaction technique. The CASSCF reference weights, w, are also included in the tables as well as the computed dipole moments for all excited states and the oscillator strengths obtained using the CASSI approach. The tables are organized such that the results for the T-T* states are given first followed by the n-* states. A. Pyridine. In the energy range of 3.5-5.8 eV, two optically allowed transitions have been otxerved with band maxima located at 4.5924(lBJ and 4.99' eV (I&), respectively. These values differ at most by 0.15 eV from the measurements published by WPS3 The first state is of the n-* type, while the second comsponds to the lB2"state of benzene. Our computed values, as shown in

9208 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992

FIllscher et al.

TABLE Iv: CASSCF and CASPT2 Exatation Energies (io eV) and CASSCF Oschtor S t m g t h ad Dipok M o w a b (io D)for PyllmMimC osc strength CASSCF PT2 exptl" erroP oc Pd comp exptl" other calculations ground state (IAJ 0.89 2.45 x-# states 0.84 2.14 0.001 5.12 -0.19 0.028 5.53 (0.024); 4.712 5.25 4.93 'M1B2J 0.88 4.01 0.05 6.7 +0.02 0.094 6.96 (0.07): 6 . 8 6 6.72 'Al('BIU) 7.68 -0.12 0.84, 0.83 2.61, 2.26 0.58, 0.79 0.72 (7.67 (0.76), 8.21 (0.99))' 9.26, 9.45 7.57, 7.32 7.57 'Al, IB2('EIu) (8.49, 8.33)' IB2, 'A,(IE2& 8.55, 8.54 8.31, 7.82 0.82, 0.79 1.17, 2.16 0.06, 0.14 (8.71 (0.01), 8.80 (0.16))l n-r* states 'BI 5.61 3.81 3.858 -0.04 0.78 0.60 0.02 0.005 3.82 (0.02): 3 . 7 6 I.42 6.22 4.12 (4.62) (-0.50) 0.77 1.05 forbidden 4.22: 4.19'

"Experimental data from ref 7. bDifference between the CASPT2 results and experiment. The (IAl, 'B2) average values have been used for the lEluand 'E, states. cThe weight of the CASSCF reference function in the first-order wave function. dDipole moment (experimental value for the ground state is 2.33 0.01 D (ref 31)). CResults based on CASSCF followed by MRCI calculations (ref 30)--oscillator strengths within parentheses. /Results based on MRDCI calculations (ref 4). 80-0 transition. TABLE V CASSCF and CASPT2 Excitation

ground state ('A,) PT* states IAi('B2J

'BA'BiJ IB2, lAl(lElu) n-** states lBI )A2 "42

IBI

eV) d CASSCF Oscillator Strengths a d Dipok Momeats (in D)for Pyridrziae osc strength other exptl errof ob PC comp expttd calculationd 0.81 4.37

(in

CASSCF

PT2

5.22 7.69 9.50, 9.31

4.86 6.61 7.39, 7.50

4.9," 5.F 6.2: 6.Y 7.1," 7.3e

~-0.1 =+0.1 =+0.1

0.77 0.74 0.42, 0.53

3.60 5.64 3.93, 5.09

0.009 0.003 0.75, 0.50

5.28 5.35 6.85 7.99

3.48 3.66 5.09 5.80

3.4d

+0.08

-0.21

1.59 1.97 1 .I4 1.74

0.01

5.3' 5.5-6.W

0.78 0.72 0.76 0.62

xhO.1

0.02 0.10

0.006 forbidden forbidden 0.008

5.02 6.10 7.25, 7.26 3.35 3.27 5.66

states. Comparison "Difference between the CASPT2 results and experiment. The (IAl, IB2) average values have been used for the 'El, and is made with the experimental data of WPS6 *The weight of the CASSCF reference function in the first-order wave function. 'Dipole moment (experimental value for the ground state is 4.22 D (ref 32)). dReference 2. 'Reference 6. fResults based on MRDCI calculations (ref 6). TABLE VI: CASSCF and CASPTZ Excitation Enemies (in eV) and CASSCF M i t o r Strellnths for s-Triazine osc strength CASSCF PT2" exptlb errof Wd comp exptlb other calculations' ground state (IAI') 0.81 ?rr* states ' A i ('B2J 5.54 5.33 5.70 -0.37 0.77 forbidden 5.80 'Ai'('BiJ 8.08 6.77 6.86 -0.09 0.77 forbidden 9.27 'E'('EiJ 9.79 8.16 7.76 +0.40 0.68, 0.82 0.61 0.73 9.39 'E' (IE2J 8.70 8.03 0.81, 0.86 0.21 n-r* states 1A If 6.19 3.81 (3.90) 0.70 forbidden 5.23 IA2If 6.05 4.00 (4.08) 4.59 -0.59 0.65 0.015 0.013 5.93 1Ett 5.79 4.24 (4.36) 3.97 +0.27 0.77, 0.77 forbidden 5.58 I EO 8.77 7.13 (7.15) (6.15) (+0.98) 0.57,0.71 forbidden 8.53 ~

"Values within parentheses have been obtained using the full Fock matrix in the CASPT2 calculations. bExperimental data from ref 7. cDifference between the CASPT2 results and experiment. dThe weight of the CASSCF reference function in the first-order wave function. eResults based on limited CI calculations (ref 33).

Table 11, are 4.91 (IBJ and 4.94 eV (IBj),respectively. The 'BI excitation energy is 0.32 eV larger than the value reported by BT? The calculated oscillator strength is larger than the (somewhat uncertain) experimental value.' The theoretical excitation energy and oscillator strength of the lBz state are in agreement with experimental data. Actually, the oscillator strength is somewhat larger than the experimental value, which is a little surprising (see below). WP3recorded electron energy loss spectra and detected three optically forbidden transitions in the same energy region (4.1,4.84, and 5.43 ev), but they do not give an amignment. These transitions have not been seen previously. Our results suggest that transitions to the 'Az state might give rise to the scattering features at 5.43 eV, whereas the peaks at 4.1 and 4.84 eV are possibly due to triplet states, since we have not found any singlet states in this region. From Table 11, it is seen that the CASSCF approximation overestimates the measured transition energies by O . M . 8 eV. It is also noted that the weights of the CASSCF reference functions indicate that all essential features are treated variationally. The main peak of the second band is located at 6.38 eV.7 The calculated excitation energy, 6.35 eV, is in excellent agreement

with the expenmental value with a deviation of only 0.03 eV. The computed oscillator strength is somewhat smaller than the experimental value. That is, however, of only little sigdbnce, since almost all the intensity in this and the *Bzu" band is induced vibrationally (the intensity of the correspondingforbidden transition in benzene is 0.09). The major peak in the strongest optical band is located at 7.22 eV7and is a composite of excitations with 'Al and IBZcharacter. In contrast to earlier observations, WP3 detected an additional weak band at approximately 7.6 eV, but they fail to assign its character. It is tempting to assume that this trace of a separate transition is nothing but the *B2state. The CASSCF estimates of theae excitation energia are 1-2 eV in error and become negligibly small after correcting for dynamic correlation effects (the arithmetic average of the El, states is used to compare). Similar to the excited state of El, symmetry in benzene, the corresponding state functions in pyridine are of strong ionic character. Because the CASSCF reference function accounts only in a very liited way for dynamic correlation effects, these states appear to be higher in energy than the states corresponding to

The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 9209

Molecular Orbital Theory for Excited States the Eg state in benzene. The difference in dynamic correlation is large for these two states, since one is mainly covalent (E2,), while the other is ionic, resulting in large relative effects from dynamic u-polarization. This correlation energy difference shifts the order of the states and places E,, above El, in all molecules studied here, except s-triazine. A broad absorption peak is seen in the range of 8-9 eV,' the fm structureof which indicates that the spectrum mainly is caused by transitions to Rydberg states. The calculated high-energy 'Al and IB, ( E d excitations, 7.78 and 7.84 eV, respectively, fall into this region. It is therefore likely that the corresponding bands are hidden under the Rydberg transitions. In ccmtrast to the states corresponding to the El, state in benzene, these states are characterized by a small dynamic correlation contribution. However, as shown in our reports on the excited states of benzene' and pyrimidine,'O the E, states exhibit strong configuration mixing, and a substantial part of the reference function, 30-4096, is characterized by double and higher excitations. The reference weight of the IBz state highest in energy is substantially smaller than all the others. This effect is due to interactions with a nearly degenerateRydberg state. There is in particular one c o d m t i o n state function which has a large coefficient in the perturbed wave function but whose contribution to the second-order energy is small. In Table 11, we have also included some earlier calculations covering a wide range of approximations. In general, the calculations by WP3 (MRDCI approximation) and Kitao and NakatsujiZs(SAC-CI approximation) are in agreement with the present results. One difference concerns the 'B, and 'Al states corresponding to the lElustate in benzene, which are predicted to be in opposite order compared to the present results. In contrast, the computed excitation energies reported by Foresman et a1.26 do not reproduce the experimental state ordering, and the errors in the computed excitation energies are spread over a wide range. All these approximations account for the u-polarization effects at various levels of theory. Therefore, as is to be expected, the calculations by WP3 and Kitao and NakatsujiZSgive excitations with lower energies than the CASSCF results, whereas the ad hoc procedure of Foresman et a1.26contradicts these trends. One way to access the accuracy of a wave function is to compare the dipole moments. The computed dipole moment for the ground state is 2.36 D and, thus, overestimates the experimental value (2.15 f 0.05 D)27by 0.21 D. The discrepancy is well within the limits to be expected using the present basii sets. Calculated values for the excited-state dipole moments given in Table I1 show that a blue shift can be expected for the n-r* states in solution, due to the decreased dipole moments. B. pvnzina The lowest ?M* excited states of pyrazine are located at 4.8,6.5, and 7.7 eV' and are of 'B,,,, IB1,, and lBlu 'Bh, symmetry. The computed values, listed in Table 111,are 4.77 (IB,,) and 6.68 eV (lBI,) for the first and the second band, respectively. For the same reason as in pyridine, the calculated oscillator strengths of the first and second bandsare smaller than the experimental values. The third band splits into two lines separated by 0.18 eV and are located at 7.57 (IB1,) and 7.75 eV ('B,,), respectively. The oscillator strengths of lBluand 'BzUin the third-band system are approximately equal in magnitude and differ from the experimental value for the composite band by only 4%. The IB3, and 'A states, corresponding to the states in benzene, are predicted to have energies 8.16 and 8.27 eV higher than the ground state. In the range of 8.0-9.0 eV, the spectrum recorded by BT' is rich in structure attributed to Rydberg excitations, and the E, states may well be hidden in that region. The dynamic correlation contribution exhibits the same pattern as was discussad above in the case of pyridine; i.e., it is small for the lowest state of B2, symmetry as well as for the states highest in energy and is largest for the El, states. In a series of preliminary calculations, we supplemented the CASSCF calculations with MRCI calculationsusing the '?r-only" approximation; that is, only single and double excitations from the active to the external *-orbitals are included in the CI expansion. The computed excitation energies are also given in Table

+

I11 (numbers in parentheses in the first column). It is seen that the states characterizedby a large dynamic correlation contribution at the CASPT2 level of approximation are also strongly affected by the MRCI calculations. However, it is also noted that the %-only" approximation is insufficient and, in particular, does not lead to the state ordering observed in experiment. Three of the n-u* transitions have been identified in experiments and are located at 3.8 (IB3,), 5.19 (IBt), and 6.10 eV ('B',).' The computed excitation energies are in excellent agreement with the experimental values (the errors are 0.25,0.02, and 0.03 eV, respectively). Only the lBju state is allowed with a measured oscillator strength of approximately half the size of the calculated value. In addition, we predict one state of 'A, symmetry between the IB3,, and lBZgstates, with a computed energy of 4.37 eV. Comparing our results with earlier calculation^,^^^^^^^ a good agreement at the qualitative level is found. One difference concerns the ordering of the absorption lines corresponding to the lElustate in benzene. In contrast to the earlier calculations, we predict the following order: IBlu C IB+. The experimental information at hand does not allow us to declde which of the orders are correct. However, the small overall errors obtained here strongly favor the present ordering. C. P y r b i d b The vacuum UV spectra presented by BT" show three r-r* states at 5.1 (IB2), 6.7 ('Al), and 7.5 eV ('Al 'BJ. The present study reproduces the vertical excitation energies with an error of less than 0.2 eV, and the third composite band splits into two lines separated by 0.25 eV. The final results are shown in Table N.The T-T* states highest in energy and corresponding to the E, state of benzene are predicted to be 7.82 and 8.31 eV higher in energy than the ground state. The lowest n-u* state, 'B1, is found at 3.81 eV. The experimental information is scarce. BT finds a 0 4 transition at 3.85 eV but does not give the vertical energy (the published spectrum has a peak intensity around 4.0 eV).' The calculated oscillator strength is again larger than the measured value. Transitions to the 'Az state lowest in energy are optically forbidden. The calculated excitation energy, 4.12 eV, is 0.50 eV smaller than the experimental information given by BT.7 The assignment is, however, very uncertain. The present results are qualitatively in agreement with earlier CI calculations4~30 but are overall much more accurate. The differences are observed with respect to the ordering and the splitting of the 'Al and 'Bz states corresponding to the 'El,, state in benzene. The present calculations estimate this difference to be 0.25 eV and the 'Bqstate to be lower in energy than the 'Al state. The line splitting is smaller than we computed previouslym by supplementing the CASSCF calculations with MRCI calculations, and the lines are in reversed order. The present results rather c o n f m the calculations by WP,4 who computed the same state ordering but a smaller line splitting. The experimental information at hand does not allow a choice between the two alternatives. A second difference concerns the location of the high-energy states corresponding to the optically forbidden 'E,, state in benzene. Compared to our previous MRCI calculations, the splitting of the two lines is enlarged to 0.5 eV, and they are shifted to the red by about 0.5 eV. Comparing the results obtained for the other P'K* states, we believe the present results for the states to be more accurate (a more detailed discussion of the situation is given in the next section). One of the major difFerences between the current and our earlier set of calculations30is that larger basis sets are used here (C, N, 3s3pld), (H, 2s) was used in ref 30). The change of the basis sets introduces only minor changes at the CASSCF level of approximation: The m r * excitation energies differ at most by 0.12 eV. The computed dipole moment for the ground state, 2.45 D, by 0.12 overestimates the experimental value (2.334 f 0.01 D)31 D and is identical to the dipole moment we computed previously at the MRCI level of approximation. Since the dipole moments are much more sensitive to small changes of the wave function than the energy, it is particularly gratifying that also the dipole moments in the excited states are in good agreement. They change

+

'5,

9210 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992

by at most 3%. except for the highest state of AI symmetry where the value is 15% smaller as compared to the MRCI result. The z2 component of the second Cartesian moment is also surprisingly stable. The relative changes are smaller than 2%, except for the highest state of B2 symmetry, which is computed to be 28.46 au as compared to 30.10 a d 0 obtained in the earlier CASSCF calculation. The larger basis set used in the present work has, of course, a more pronounced effect on the dynamical correlation energy. This is most probably one of the reasons why the MRCI energies in ref 30 are 0.34.6 eV larger than the present values. D. Pyridnzine. According to IRM? the r-r* excited states, correlating with the BIU,Bh, and El, states of benzene, are located at 4.9 (‘A’), 6.2 (’B2),and 7.1 eV (‘B2+ ‘Al), respectively. The measured excitation energies reported recently by W are 0.1-0.3 eV higher in energy. As shown in Table V, the position of the ‘Al line is nicely reproduced by our calculations, with a computed excitation energy of 4.86 eV. The second lowest computed T-T* transition energy is 6.61 eV and is 0.1 eV bigger than the experimental value reported by Wp6 but is 0.4 eV larger than the value compiled by IRM.* A line splitting of 0.1 1 eV is predicted for the third-band system, and the excitation energies are 0.1-0.2 eV higher than the measured values. Our results strongly indicate that the experimental data given by Wp6 are more accurate than those given by IRMqZ The computed ?M* excitation energies agree within 0.2 eV with the computed spectrum published by WP.6 The differential correlation effects are, as can be expected by comparison with the other diazabenzenes, large for the El, states and substantially smaller for the T-T* excitations lowest in energy. The reference weight of the states highest in energy is rather small. By inspection of the fmt-order c ~ r r ~ ~wave t e d function, it is found that two more u-orbitals and four more Ir-orbitals should be added to the active space, which would exceed the limits of the present code. Similar to the situation found for pyridine, the extra configurations not covered by the reference function are of the Rydberg type (PU* and .-.*). However, the contributions to the total energy are small. The deficiency of the active space is more severe for the E2gstates. The first-order wave functions are found to strongly deterioratetoward PU* excited states, and the results are therefore not included in Table V. No attempt was made to correct for this behavior by extending the active space, even if this certainly would have been possible. Only one n-r* state is firmly assigned and has an absorption maximum at approximately 3.4 eV.* Our calculations confirm the existence of this band, the calculated excitation energy being 3.48 eV, and predict a second, closeby line at 3.66 eV. n v * transitions to states higher in energy than the first ‘Bl state are not yet identified. Our best estimates for the next two lowest n-r* excited states are 5.09 (‘Az) and 5.8 eV (‘BJ. These values match nicely with some structure observed around 5.3 and 5.5-6.0 eV in the electron energy loss spectra published by WP.6 The reference weights indicate that the calculations are well balanced. The gas-phase dipole moment of pyridazine has been measured and is reported to be 4.22 D.32 The computed value is 4.37 D and overestimates the experimental value by only 0.15 D. As noted above, the discrepancy is well within acceptable limits. E. s-Triazine. The first of the three n-7r* absorption lines, which have been measured in the vacuum UV spectrum of striazine, is located at 5.7 eVa7 We computed this line to be positioned at 5.33 eV (cf. Table VI). The computed value is thus 0.37 eV too small. However, the experimental as well as the computed value is substantially larger than for benzene, the diazabenzenes, and s-tetrazine (experimental value 5.0 eV).2 In fact, the absorption band arising from transitions to the ‘ A i state is only seen as a small bump with faint structure on the highenergy flank of the first-band system. We conclude that the ‘ A i line has not been properly identified and is somewhere hidden in the first-band system. The second absorption maximum arising due to transitions from the ground to the lAl’ state is located at 6.86 eV. This transition is optically forbidden, and the calculated energy difference, 6.77 eV, is in excellent agreement with experiment.

Fiilscher et al. The present calculationshave been carried out in C , symmetry instead of the full symmetry of the molecule, D3,,. If no symmetry breaking occurs, the ‘Al and ‘B2states corresponding to the ‘E’ states in Djhshould build degenerate pairs. In Table VI, we give the average energies but list the reference weights separately. Although the orbitals were restricted to have full Djhsymmetry at the CASSCF level of approximation, symmetry breaking is observed in the CAS-CI calculation. This leads to differences of at most 0.1 eV in the CASSCF excitation energies for the two components of a degenerate pair, while somewhat larger splittings (at most 0.2 eV) are seen at the CASFT2 level. The difference in the reference weight is 0.14 for the two components of the El, state. The symmetry breaking leads to an artificial interaction between the upper and lower states having the same symmetry , . It is possible that this is the source of the somewhat larger in C errors obtained for s-triazine, as compared to the other azabenzenes: the first ‘E’state is 0.40 eV larger than experiment, while the ‘ A i state is 0.37 eV lower. In no other case have we found a a-r* excitation energy above the experimental value. We note, however, that the errors are still well within the proposed accuracy of the present approach, 0.5 eV. Two n v * transitions have been assigned and are located at In addition, we also find an 3.97 (‘E”) and 4.59 eV optically forbidden state of lA1” symmetry in the same energy region. Our calculations predict the ‘A1’’ state to be lowest in energy and located at 3.81 eV. The n v * excited state next in sequence, the ‘A? state, is computed to be 4.0-4.1 eV higher in energy than the ground state. Finally, an excitation energy of 4.24 eV is estimated for the ‘E”state. The experimental assignment of the allowed lA2/1 is probably correct, which gives an error in the computed value of 0.5-0.6 eV. This is the largest error found in the present work (see, however, below for the next n-** state of E” symmetry). It is therefore interesting to note that IRMz in their compilation give an energy of 4.09 eV for this state, in perfect agreement with the computed value. They further suggest that the first ‘Al’’ state lies in the same energy region, which is also in agreement with the theoretical results. However, the calculated energies are so close that we do not dare to draw any affirmative conclusions. In Table VI, we hav instead followed the assignments given by BT7 (see also the discussion in ref 2). The error in the computed energy for the ‘E” state is then -0.27 eV, a typical error for many of the calculated n-r* excitation energies. A second ‘E” state has been found with an energy of 7.15 eV. BT7 tentatively assign a band found at 6.15 eV to an excitation of either this or lAl” symmetry. The first alternative is not in agreement with the present results, and the second assignment would have to correspond to the second band of lAI” symmetry, since the first is located close to 4 eV. We have at present no definitive suggestion regarding the assignment of the structure around 6.15 eV. The only other ab initio study made on this is not accurate enough to make a comparison with the present results meaningful. 4. Discussion

The goal of the proposed computational method has been to compute excitation energies with an accuracy of 0.5 eV or better. This goal has been reached in all cases where an unambiguous comparison with experimental data can be made. The only exception is one n-r* excitation in s-triazine, where the experimental assignement is uncertain. The promising results obtained earlier for the benzene molecule thus hold also for the azabenzenes. If we add the benzene results’ and do not use the somewhat artificial (due to the symmetry breaking) results obtained for s-triazine, the average error (RMS) in computed excitation energies for the u-r* excited states is 0.1 1 eV. The maximum error is 0.20 eV (benzene and pyridazine). Adding also the s-triazine results increases the average error to 0.16 eV with a maximum error of 0.40 eV (the ‘Elustate in s-triazine). The average error in the calculated excitation energies for the n d * states is somewhat larger, 0.28 eV, but we have then included also those states for which the assignment is uncertain (but not

Molecular Orbital Theory for Excited States the second IE” state in s-triazine). This result must also be considered as very satisfactory. In two cases, however, the difference between the computed and the measured excitation energies is larger than the proposed accuracy: -0.50 eV for the ‘A2 state in pyrimidine and -0.59 eV for the ‘A? state in s-triazine (the error is reduced to -0.51 eV if the full Fock matrix formulation of CASPT2 is used). The experimental data are in both these cases very uncertain, and the assignments are not much more than guesses. The overall high accuracy of the present work can, of course, only be achieved with a large and flexible enough basis set. Problems associated with the basis set choice have not been addressed. Instead we have used extended ANO-type basis sets in order to avoid discussions about basis set deficiencies. One basic assumption of the current approach is that all essential electronic structure features should be covered by the CASSCF reference function. Such a condition is a prerequisite for a low-order perturbation theory to be able to give reliable results. Modifications of the electron density (the strongly occupied natural orbitals) due to dynamic correlation effects is a higher order effect involving, in addition, single and triple (and higher) replacements with respect to the main configurations of the reference state. This situation makes it, for example, difficult to use CASPT2 to compute electron affinities for negative ions that are not bound at the CASSCF level of approximation. In studies of excited states, similar problems arise due to artificial interactions between valence excited states and Rydberg states. Dynamic correlation is normally of less importance in Rydberg states. The ordering and spacing of the states at the CASSCF level are then incorrect, and an artificial mixing of Rydberg and valence excited states can occur due to an incorrect accidental near-degeneracy. Such situations can also lead to convergence problems in the CASSCF calculation. Normally the problem can be solved by a slight increase of the active space. This approach has been successfully used here. In more difficult cases, one possibility might be to shift the Rydberg states to higher energies by applying an external field. Rydberg states can also occur as intruders in the CASPT2 calculations in cases where the active space is not large enough to include all those which are nearly degenerate with the valence excited states. This is normally only a problem when the “energy denominator” in the perturbation expansion becomes very small. It has been shown that these singularities are very narrow,32and as long as the interaction is small, the effect on the second-order energy is in most cases negligible. This is almost always the case for interactions between valence excited states and Rydberg states. In cases where there is a large contribution, the corresponding Rydberg orbital has to be added to the active space or deleted from the MO basis. For pyrimidine, it has been possible to compare the present calculation with a previous MRCI which was designed to account for as much of the dynamic polarization effects as possible. The reference configuration selection was based on CASSCF calculations similar to those performed here. The CI wave function comprised all singly and doubly excited states with respect to the reference configurations, with the added condition that there was at most one hole in the inactive (all u-orbitals except the lone pairs) space. Thus, dynamic polarization of the a-orbitals was included, but no u-u pair correlation terms. It was shown in the CASPT2 treatment of benzene’ that the o-u pair comelation energy is indeed constant for all the excited states, and this result is confirmed in the present study to hold also for the azines (excluding again the lonapair electrons). A comparison between the CASPTZ and the MRCI results for pyrimidine shows that while CASPTZ gives n-r* excitation energies that are 0.0-0.2 eV lower than experiment, MRCI gives energies that are 0.2-0.4 eV too large. There are three main sources for this difference: (1) C A S E 2 probably overestimates the dynamic correlation energies to a small extent; (2) somewhat smaller basis sets were used in the MRCI study; (3) CASPT2 is based on a full CASSCF reference function, while MRCI uses a selection of the most important reference configurations. We

The Journal of Physical Chemistry, Vol. 96, NO. 23, 1992 9211 believe the last source to be the most important. It is difficult to achieve balance in the correlation treatment when a selection of a small number of reference configurations has to be made. Normally such a selection will favor the ground state and thus lead to excitation energies that are too large. A good example of these inherent difficulties of the MRCI method is a recent comparison between MRCI and CASPT2 in a study of the vibrational frequencies in The computed dipole moments of the ground states of pyridine, pyrimidine, and pyridazine are 0.1-0.2 D larger than the experimental estimates. They have been obtained at the CASSCF level of approximation, and it can be expected that the largest part of the error is due to correlation effects not included in the CASSCF wave function. The computed values for the excited states should have about the same accuracy. The latter can be used to estimate solvent effects on the excitation energies (a theoretical study of solvatochromatic shifts of the excitation energies in the azabenzenes has recently been publi~hed’~). A direct comparison of computed oscillator strengths with experiment cannot be made since the experimental values also include vibrationally induced intensities. Some conclusions can, however, be made. The lBzu bands have measured intensities varying from 0.013 (benzene) to 0.062 (pyrazine).’ Most of this intensity is vibrationally induced (the transition dipole is zero in benzene and s-triazine). The computed intensities should consequently be smaller as has also been found for all molecules, except pyridine. The same situation is obtained for the lBluband, where the experimental intensities are all close to 0.1, while the theoretical values are in the range 0.0034.09. The situation is different for the strongly allowed transition ‘Elu. Here the experimental values are in close agreement with theory, if the averages of the two component values are used in the comparison. The experimental values vary between 0.72 and 0.90, while the computed data are in the range 0.61-0.85. This agreement must be considered as highly satisfactory. We note also that the strongest component of the band has an appreciable intensity in pyridine, pyrimidine, and s-triazine (this band was not studied in pyridazine). It is therefore somewhat surprising that it has only been identified in benzene,36where it has an energy of 7.8 eV. We conclude from the present investigation that this band in the azabenzenes should be found in the energy range 7.8-8.2 eV. The calculated excitation energies should be accurate to about 0.2 eV. The situation is different for the n-* states. Most of the bands are symmetry forbidden, but for the allowed transitions, the computed oscillator strengths are larger than the measured values. In two cam, the difference is small (pyrazine and s-triazine), but for pyridine, it is an order of magnitude. We cannot claim that these small intensities are converged with respect to basis sets and correlation treatment. On the other hand, the measurements are also difficult to perform, and different experimental sources often give quite different values. 5. Summary and Conclusions

A theoretical method has been proposed for the calculation of properties of excited states in conjugated molecular systems. The model is a two-step procedure with static correlation effects included in the first step where the molecular orbitals are optimized. The remaining (dynamic) correlation effects are computed in the second step using a second-order perturbation approach. The model has been tested earlier in a study of the excited states of the benzene molecule, where it was shown to yield excitation energies with an accuracy of 0.25 eV or better for all valence excited singlet and triplet states. In the present contribution, we show that a similar accuracy is obtained also for the azabenzenes, now including excitations from the nitrogen lone pairs. No other ab initio calculation performedon these molecules has given results of similar accuracy. Graham and Freed have recently published the results for the tram-butadiene *-valence states obtained with an effective valence Hamiltonian method.37 The computed excitation energies have the same accuracy as those obtained here. Even though these calculations are carried to third order in the treatment of dynamic

9212 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992

correlation, they provide further evidence for the validity of a low-order multiconfigurational perturbation approach in calculations of electronic excitation energies. The present approach has recently been applied also to ethene, butadiene, and hexatriene3*and has for trans-butadiene given results almost identical to those obtained by Graham and Freed. The method proposed here is not a blackbox, which automatically gives results for a large number of excited states. Some care has to be taken in the selection of the active orbitals, which must be based on knowledge of the basic features of the electronic structure of the excited states. One more automatic way to help solve this problem is to start the study by, for example, a restricted active space (RAS) SCF calculation using a wave function comprising all singly and doubly excited states with respect to the HF ground-state wave function. By choosing a rather large active orbital space, it is possible to cover all excited states of interest in one state-average calculation per symmetry. The resulting natural orbital occupation numbers can then be used in aiding the selection of an active orbital space for the CASSCF/CASPT2 calculation. The results of the CASPT2 calculation must be controlled with respect to intruder states from excitations to orbitals not included in the active space (the control parameter is the reference function weight, a). When this happens, it might be necessary to move the corresponding orbital to the active space. Similar problems were encountered in the effective valence-shell Hamiltonian method of Graham and Freed.” However, these difficulties should not be overstated. A large number of applications have already been made with the present technique and in all cases the choice of the active orbital space was straightforward. The control parameter will always give an indication when the active space is too small. The CASPT2 natural orbitals can in such cases be used to extend the space. To make an estimate of the maximum size of a molecule that can be treated with the present approach is difficult, since it depends on many different factors, such as the basis set used, the number of hydrogen atoms in the molecule, the desired accuracy, etc. The active orbital space does not, however, constitute a bottleneck, since 14 orbitals would be sufficient in most applications, also for large molecules. The excitation process normally involves only a rather small number of molecular orbitals. As a small exercise, let us consider planar molecules of the type X a , , , where X is a first-row atom. Test calculations have shown that in order to obtain reasonable excitation energies (with errors less than about 0.5 eV), a basis set of the type 3s2pld for X atoms and 2s for H is a minimum requirement. Thus,30n basis functions are needed for a molecule of the type X2,H,,. The MOLCAS system has an upper limit of 255 for the basis set, which gives a maximum value for n of 8 (XI6Hs). Whether such a calculation can really be performed depends on the disk storage available. If there is no extra symmetry, the disk storage needed will be about 2 Gbytes, with the packing facilities of MOLCAS. Of course with a direct procedure, in which no A 0 integrals are stored, there will be no such size problem. Instead, the limit will be set by the CPU time available for the calculation. Development of a direct CASSCF/CASPT2 program is planned. Acknowledgment. The research reported in this paper has been supported by a grant from the Swedish Natural Science Research Council (NFR) and by IBM Sweden under a joint study contract.

FUlscher et al. Regiutry NO. Pyridine, 110-86-1; pyrazine, 290-37-9; pyrimidine, 289-95-2; py-ridazine, 289-80-5; s-triazine, 290-87-9.

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