Density functional calculations with configuration interaction for the excited states of molecules

30 August 1996

CHEMICAL PHYSICS LETTERS

ELSEVIER

Chemical Physics Letters 259 (1996) 128-137

Density functional calculations with configuration interaction for the excited states of molecules Stefan Grimme lnstitut fiir Physikalische und Theoretische Chemie der Universiti~'tBonn, Wegelerstrasse 12, D-53115 Bonn, Germany

Received 30 May 1996; in final form 21 June 1996

Abstract

Configuration interaction (CI) calculations restricted to single excitations with respect to a closed-shell ground state determinant have been performed using modified CI-Hamiltonian matrix elements. Shifted molecular-orbital (MO) eigenvalues from Kohn-Sham density functional theory (DVI') are used in the diagonal matrix elements. All Coulomb type two-electron integrals are scaled by an empirically determined factor. The approach, which is applicable to large molecules, is used with the ground state Kohn-Sham MOs expressed in extended Gaussian AO basis sets. The excited singlet and triplet states of a wide range of molecules including aromatic hydrocarbons as large as pentacene (C22Hjo) have been investigated. The errors of vertical excitation energies are in most cases below 0.2 eV, even for molecules for which traditional ab initio CI methods have substantial difficulties. The quality of the wavefunctions is examined by calculating the electronic circular dichroism spectra of systems with low symmetry (camphor, 4,5-dimethylphenanthrene) and good agreement with experiment is found.

1. I n t r o d u c t i o n Despite the progress in computational chemistry during the last decades the description of excited states of large molecules remains a challenging problem, Even if we restrict ourselves to the simplest cases where the ground state is well described by a single closed-shell Slater determinant and the excited states are dominated by single excitations with respect to the ground state reference, accurate excitation energies (i.e., errors < 0.2 eV) and transition moments for the description of the corresponding electronic spectra are difficult to achieve by theoretical methods. Since most organic (and many inorganic) systems fall in this class of molecules the subject is o f broad interest in theoretical and physical chemistry. From the theoretical point of view, there are two accurate ab initio methods for the topic under dis-

cussion: the multi-reference configuration interaction method ( M R [ S D ] - C I ) [1,2] and the complete active space CI with second order perturbative corrections (CASPT2) [ 3 ]. Unfortunately, these approaches have serious disadvantages with respect to the cornputational effort which exclude them from the general use in the case of larger systems (i.e., more than 10 non-hydrogen atoms): (a) both methods require large one-particle basis sets with high angular momenturn functions and (b) differential dynamical electron correlation effects which may be as large as 1-2 eV [ 3 ] require the inclusion of most of the valence electrons in the correlation treatment. Thus, the largest reported accurate CASPT2 calculation has been performed for naphthalene [4] (C10Hs) l but larger

a CASPT2 calculations reported for porphin (C20HInN4, D2h) are only possible to perform with small basis sets and restricted

000%2614/96/$12.00 Copyright (~) 1996 Elsevier Science B.V. All rights reserved. Pll S0009-26 14(96) 007 22-1

S. Grimme / Chemical Physics Letters 259 (1996) 128-137

homologues like tetracene (C18H12) or aromatic hydrocarbons with lower symmetry (e.g. [ n] helicenes) seem intractable with these methods. This is the reason for the fact that semiempirical molecular-orbital (MO) methods with minimal basis sets and small CI treatments are used in this field (see e.g. Ref. [ 5 ] ). However, these methods suffer from the inherent errors due to the integral approximations used. Furthermore, the choice of a minimal basis set is crucial for properties which depend on details of the wavefunctions (e.g. circular dichroism (CD) spectroscopy where electric and magnetic dipole transition moments are required simultaneously) so that their use is limited although success has been reported in some cases [6]. A comment on the ab initio Hartree-Fock (HF) CI method restricted to single excitations (termed CIS or SCI) [7] should be given here. Although the good performance of this computationally simple approach is often cited, the quality of CIS calculations is often questionable. For example, some excited states (nTr* states in carbonyl compounds) are described well but the nTr* states in pyridine are in error by 1-2 eV and for aromatic hydrocarbons the wrong energetic ordering of different rr~-* states with relative errors of more than 1 eV is found (see Table 1). On the other hand Kohn-Sham density functional theory (DFT) has been successful in the description of the electron correlation problem for molecules in their ground states [ 8 ] and much effort has been done to extend DFT to excited states (see e.g. Refs. [ 9-12 ] and references therein). In this Letter acomputationally simple approach for the calculation of molecular excited states is presented which uses an empirically corrected DFF ansatz with configuration interaction. The method consists of the following basic strategy: (a) modification of the CI single excitation method by shifted and empirically corrected CI Hamiltonian matrix elements to account for higher excitations, dynamical correlation and basis set deficiencies; (b) use of the Kohn-Sham eigenvalues as essential part of the excited state energies; (c) application of different standard AO Gaussian basis sets to gain enough flexibility for excited states of Varying character, The computational effort to do such a calculation is dramatically lowered compared to the MR-CI or CAS spaces [31.

129

CASPT2 methods since the number of configurations in the CI step scales only as Nocc × Nvirt ( N being the number of MOs in the respective spaces) and the N 5 AO to MO integral transformation step is reduced to a scaling of No2ce× N2irt × nAO. For the molecules and states under consideration (ground state with dominating closed-shell character, singlet and triplet excited states) the following results are obtained: (a) errors for vertical excitation energies do not exceed 0.2-0.3 eV in most cases; (b) Rydberg states or states with diffuse character (which can not be obtained from semiempirical methods) are treated on equal footing with similar accuracy compared to valence states; (c) singlet-triplet splittings are obtained with reasonable accuracy due to the usage of unscaled exchange terms in the CI step; (d) parts of excited state potential hypersurfaces, where the closed-shell ground state determinant is a good approximation (i.e., roughly where R < 1.5Re), can be investigated with the same reliability.

2. Theory The CI-Hamiltonian matrix elements for singly excited spin-adapted singlet configurations with substitutionof an occupied MO ¢~ by avirtualMO~br (g',~, ~0 is the closed-shell reference determinant with the HF expectation value E0 and ~ is the one- and twoelectron Hamiltonian) are (qt~l/:/- E01~) = (~brlf]~r) - ( ~ta~lla[~tr~lr)

--

(~alJ~l¢,o)

-~ 2 ( tha~l'r ]~atbr

) ,

( 1)

(q%l//lq' a) = x/2(~a[fl~Or) ,

(2)

(~alHl!gb) = (~lr[f[~lIs)~ab (~-Ialfl~tb)6rs - (~¢bl~r~Ps) + 2(~Oa~r [~Pb~/'s) ,

(3)

r

^

S

where f is the Fock operator and the other terms are the usual two-electron integrals in Mulliken's noration. The (~bIfl~/') elements correspond in the case of canonical HF-MOs to the orbital energies e. The matrix elements for the corresponding triplet states can be generated from the above equations by neglecting all exchange-type integrals K in Eqs. (1) and (3) (i.e., those in which the bra or ket contains MOs from the virtual and the occupied space; the Coulomb-type in-

S. Grimme/Chemical Physics Letters 259 (1996) 128-137

130

Table 1 Comparison of vertical singlet-singlet excitation energies AE ( D F r / s c I / B 3 - L Y E VTZP AO basis augmented with diffuse functions in the case of Rydberg states). The results are given for molecules/states included in the fitting procedure Compound

Excited state

CO H20 CH20 CH2S ethene trans-l,3-butadiene benzene naphthalene

pyridine

thiophene

a Ref. [24]. h Ref. [30].

b Ref. [25]. i Ref. [31].

c Ref. [26]. J Ref. [32].

b E (eV)

1 1II (n~'*) 1 IBl (n ~ 3s) 1 IA2 (nTr*) 1 IA2 (n~'*) 1 IB3u (7"r---* 3s) 11Bu (~rTr*) 2 ZAg ( ~ ' * ) 1 lB2u ('rrT"r*) 1 IBlu (Trgr*) 1 1B3u (~r~r*) 1 IB2u (~'~'*) 2 lAg (lrrr*) 2 IB3u (qT"qr*) 1 IBl (nTr*) 1 IB2 (~-Tr*) 1 IA2 (n~*) 2 IA 1 (qr'rr*) 2 1Al (~r~'*) 1 IB2 (~Tr*) d Ref. [27]. k Ref. [33].

e Ref. [28]. J Ref. [34].

tegrals J have MOs from the same space in the bra or ket). The magnitude of the excitation energies AE is often dominated by the energy gaps er - e~ of the dominating configurations. The energetic gap between the occupied and the virtual orbitals is large if the HF operator is applied (roughly 2-3 times AE) and is mainly reduced by subtraction of the integral Jar. In DFT theory, on the other hand, the gap of the KohnSham operator eigenvalues e Ks is much smaller, i.e., more closely resembling excitation energies. Thus, it seems a reasonable approximation to replace the term er-ea-2JarinEq.(1)by erKS - e ~KS -Cl J ~ where cl is an empirical parameter. In order to avoid an unphysical splitting of degenerate states the same correction factor cl must be applied to the Coulomb integrals of the off-diagonal elements also. It should be mentioned that O represents here an MO from the solution of the Kohn-Sham equations for the closed-shell ground state. These one-electron functions closely resemble HF-MOs in the occupied space but differ often substantially in the virtual part.

exp.

DFF/SC1

HF/SCI

8.4 a 7.4 b 4.2 c 2.5 d 7.1 e 5.9 f 6.2 g 4.9 h 6.2 h 4.0 i 4.6 i 5.5 J 5.9 i 4.6 k 5.0 ~ 5.4 m 6.4 1 5.3 n 5.8 n

8.24 7.29 3.80 2.37 7.26 5.98 6.41 4.90 6.09 4.07 4.31 5.80 6.15 4.88 5.04 5.40 6.32 5.54 5.80

9.08 8.64 4.58 2.70 7.17 6.41 8.40 6.06 6.20 5.29 5.15 7.37 7.13 6.11 6.15 7.42 6.46 6.81 6.17

f Ref. [291. m Ref. [35].

g CASFr2 results from Ref. [3]. n Ref. [36].

The virtual KS-MOs are more compact and similar to natural or CAS-SCF orbitals thus being better suited for CI calculations than their HF counterparts. Since we do not want to relax these KS-MOs to the HFMOs through a coupling of the g'a state functions with the ground state via the Fock-matrix elements (Eq. (2)) these terms are neglected. Consistently, the off-diagonal Fock-matrix elements between the singly excited configurations in Eq. (3) are also discarded so that the modified CI matrix elements in short notation read as follows: ( g r r l i 2 [ - E o I g l r ) = e r~s - e a KS _ Cl Jar -~ 2gar ,

(4)

(g'°[/:/I ~ r ) = 0 ,

(5)

(~Ya~[Ibl~llr~ts)

"~- 2(Oa0~l~/'b~bs) , (6) Test calculations with this ansatz and various gradient-corrected exchange-correlation functionals (B-LYP, B-P86 and B3-LYP) has shown encouraging results for zrrr* excited states (cl varies between 0.2 (~l/~/Ig'~)

= --C1

S. Grimme/Chemical Physics Letters 259 (1996) 128-137

and 0.4 depending on the functional used). In two cases, however, failures were observed: First, states in which excitations out of lone-pair or o- orbitals in MOs with little spatial overlap gain importance (i.e. nTr*, o-Tr* and Rydberg states), were calculated too low in energy (by 0.5-1.5 eV). These states are characterized by a small exchange integral K,r (<0.02 au) between the occupied and the virtual MOs of the leading configurations. Secondly, core-excited states as observed in XANES spectra, are calculated too high due to a too small energy decrease of the KohnSham eigenvalues as a function of nuclear attraction, Thus, an empirical shift (d) of the diagonal CI matrix elements as a function of Kar and e~ is applied. A two-parameter Gaussian-shaped function for the K a r depending part is used, where c2 is the maximum of the shift and c3 is the steepness of the decay of the shift (i.e., 7rrr* or o-o'* states with Kar > 0.03 au should remain essentially uncorrected). The ea values in the occupied space are corrected in a proportional manner by 2.5% of their value (this correction has only marginal effects for valence excitations, but is used for consistency in the description of core-excited states), KS KS ( g t , ~ l ~ - E o l q r S ) = e r - e~ - cl Jar -4-2Kar q-d(gar,

(7)

Ca) ,

A(K, e) = --0.025e + c 2 exp ( --C3 K 4 )

.

131

(DFT) with the nonlocal three-parameter gradientcorrected exchange-correlation functional of Becke and Lee, Yang and Parr including partially exact HF-exchange (B3-LYP) [ 13]. This functional has been found to be slightly superior compared to the B-LYP or B-P86 methods. Two Gaussian AO basis sets have been employed: The small basis A is of valence d o u b l e - ( ( V D Z ) quality (H: 4s--* [2s] ; C, N, O: 7s4p---~[3s2p]; S: 10s7p---~[4s3p]) [14]. The VDZ basis was augmented with polarization d-functions in the case of nitrogen, oxygen, phosphor and sulphur atoms to improve the description of lone-pair excitations (o~d ( N ) = 1.0, t~d ( O ) = 1.2, ad ( S ) --- 0.55, o/d (P) = 0.45). The large basis B is of valence triple-((VTZ) quality (H: 4s---~[3s]; C, N, O: 1 ls6p---~[5s3p]; S: 14s9p--~[6s4p] ) [14] with polarization functions on all atoms (ad(C) = 0.8, ap(H) = 0.8). For Rydberg states the basis sets were augmented with one set of diffuse sp-functions on each non-hydrogen atom whose exponents were chosen according to Dunning [ 15]. All DFT calculations have been carried out with a slightly modified version of the TURBOMOLE [ 16,17] program system. The geometries have been completely optimized at the B-LYP/TZP or B3-LYP/TZP level of theory. All valence electrons and virtual MOs have been included in the CI treatments.

(8)

In the case of degenerate electronic states K in Eq. (8) is averaged first over the configurations which are degenerate at the one-electron level to prevent symmetry breaking, In summary we arrive at three empirical parameters which remain to be determined by a least-squares fit of calculated and experimental vertical excitation energies of a selected (representative) set of molecules, This approach is completely in the spirit of DFT where a few global empirical parameters are used but it is different from the definition of empirical parameters in semiempirical MO theories where a series of atom and/or bond specific terms may be needed,

3. Outline of the calculations All calculations have been carried out in the framework of Kohn-Sham density-functional theory [8]

4. Results First, a reference set of molecules and states must be defined to determine the three parameters C l - C 3 . The selection of reference states (all of singlet multiplicity) is somewhat arbitrary but many test calculations have shown, that the parameters are very stable with respect to the actual choice. Furthermore, the results for the molecules not included in the reference set (see below) are rather insensitive to the actual parameters which demonstrates the inherent physical significance of the approach. The valence states employed comprise five nTr* type states of formaldehyde, pyridine, thioformaldehyd and CO and the two lowest ~'¢r* states of s-trans-1,3-butadiene, benzene, pyridine and thiophene. For the largest reference molecule naphthalene two higher valence ~rTr* states have been included in addition to the two lowest 7rTr* states. Although the approach is not specially designed to

132

s. Grimme/Chemical Physics Letters 259 (1996) 128-137

describe Rydberg states in small molecules (which can be treated accurately with traditional ab initio methods), two states of this type (the lowest Rydberg states of water and ethene) have also been included in the reference set used for the fitting procedure since the method should in principle be capable of a quantitative description of diffuse states also. The general procedure for the parameterization is to employ first the large basis set B to explore the real capability of the theoretical method (a VTZP basis gives DFT results for ground state properties near the basis set limit). In a second step the basis A is used to evaluate the transferability of the optimized parameters for the calculation of larger systems with this smaller basis set. The final DFT/SCI results with the parameters Cl = 0.317 c2 = 0.033, Eh and c3 = 1.27 x 107 Eh 4 are given in Table 1 in comparison with experimental and HF/SCI data. Generally, the experimental excitation energies (AE) refer to band maxima in the gas phase or non-polar solvent (the 0-0 transition is often nonvertical in the case of smaller molecules, i.e. the excited state geometry is quite different from the ground state structure while the difference AEmax - bE0-0 is below 0.1 eV for the larger aromatic molecules). In some cases comparison is made with accurate theoretical CASPT2 or MR-CI data from the literature due to the experimental difficulties in the precise location of the vertical AE. The standard deviation of mE between theory and experiment for all states considered in Table 1 is 0.16 eV for the DFT/SCI method ( 1.16 eV for HF/SCI) which is in the range of the experimental uncertaincy for the determination of AEvert. in larger molecules, The largest error is 0.3 eV but most deviations are below 0.2 eV. The errors for one-refererence (nTr* and Rydberg states) and multi-reference cases (rrTr* states) are of comparable magnitude demonstrating a balanced treatment of diagonal and off-diagonal CI matrix elements. The notorious difficult Rydbergvalence mixings in the ¢rTr* states of ethene and CHeO [18,19] (see Table 2) and the n~*/rrTr* splittings in pyridine are also described well. The results are really encouraging and of similar quality compared to CASPT2 or MR-CI data despite the relatively simple computational procedure. As will be shown below, the proposed DFT/SCI approach is capable of providing similar accuracy for larger systems with low

symmetry where traditional techniques can hardly be used. It should also be pointed out that this success is mainly due to the use of the e KS r - e aKS - Cl Jar term and that the remaining two parameters made much smaller changes. This emphasizes the physical significance of the KS one-particle energies which seems to be more pronounced than that of the HF eigenvalues although there is some controverse discussion about this point in the literature (see e.g. Ref. [ 12] ). Especially for the rrTr* states in the polyene and the polycyclic aromatic hydrocarbon (PAH) molecules the good performance of the method seems somewhat surprising. As is demonstrated by inspection of the poor HF/SCI results (see Table 1) double and higher excitations are very important for a proper description in a pure ab initio approach. This neglect has a strong effect for the splitting of the two lowest PAH states ( La and Lb in the nomenclature of the perimeter model) which is much too small (benzene) or even inverted (naphthalene) in the standard HF/SCI treatment. Furthermore, the 2 lAg state of butadiene (and other polyenes) shows a significant contribution ( ~ 40 %) from the H O M O - L U M O double excitation so that the HF-SCI result for this state is in error by more than 2 eV. The D F r / s c I method on the other hand reproduces the AE of these states not only in butadiene but also in higher polyenes and in benzene and naphthalene with an accuracy of 0.2-0.3 eV. Thus it is concluded, that the energetic effect of double and higher excitations is implicitely accounted for to some extent by the relative position of the diagonal CI matrix elements. Since most states under investigation are dominated by single excitations in MR [ SD] -CI wavefunctions the effect of the higher excitations can be interpreted as a shift of the leading configurations to their 'correct' energies. In other words, the DFT/SCI mimics a perturbative treatment of these effects with a very low computational effort. Finally it should be mentioned that differential dynamical o" - 7r correlation effects, which are quite large in the PAH states of covalent and ionic character [ 3 ], are also acounted for partially. The results for a wider range of molecules and excited states are presented in Tables 2 and 3 (singlettriplet excitation energies). In some cases higher-lying states of the reference molecules were considered also. Since the main object of this research are larger molecules, several PAH with ten or more carbon

S. Grimme/Chemical Physics Letters 259 (1996) 128-137

133

Table 2 Comparison of vertical singlet-singlet excitation energies AE ( D F F / S C I / B 3 - L Y P ) . The results are given for molecules/states not included in the fitting procedure. Compound

H2 NH3

AO basis

ethene

B B B

CH20

B

acetic acid trans-glyoxal

A A

trans- 1,3,5-hexatriene

A

trans-l,3,5,7-octatetraene

A

benzene

B

anthracene

A

tetracene

A

pentacene

A

phenanthrene

A

azulene

A

P406

A

Excited state

1~+ (~ro-*) 2 LAI (n ~ 3s) 1 IBig (~" -~ 3p) 1 IBlu (lrTr*) 1 IB2 (n ---+ 3s) 21B2 (n --, 3p) 41Al (Trrr*, n---, 3p) 1 I A" (nTr*) 1 1Au ( nTr* ) 1 IBg (nrr*) 1 I Bu (TrTr*) 2 l a g (Try'*) 1 IBu ( r r ~ * ) 2 l A g (Trqr*) 1 IElu (~'¢r*) 1 IElg ( r r - ~ 3s) 2 IE2g (~rcr*) I 1B2u (7r77",*) l tB3u (~'¢r,*) I IB2u (~'Tr*) I IB3u (~'~'*) 1 IB2u (~'~'*) 11B3u ('/7"77"*) 11Al (¢rTr*) 11B2 (Trot*) I IB2 (zmr ~) 2 IAi (TrTr*) I IT2 val. 21T2 val. 3 IT2 val.

AE (eV) exp.

DFT/SCI

12.9 a 6.3 I, 7.5 h 7.7 h 7.1 c 8.0 c 9.6 ,1 6.0 e 2.8 e 4.5 e 5.0 f 5.2 g 4.4 h 4.4 i 6.9 J 6.3 k 7.8 k 3.3 I 3.5 1 2.6 I 3.1 I 2.1 I 2.9 1 3.8 t 4.2 I 2.1 1 3.6 I 6.1 m 7.5 m 8.0 m

13.14 6.28 7.70 7.63 7.10 7.89 9.38 5.67 2.57 4.16 5.21 5.60 4.39 4.66 7.14 6.63 8.31 3.28 3.53 2.53 3.19 2.03 2.91 3.78 4.31 2.34 3.60 6.13 7.67 8.03

a Explicitly correlated calculations from Ref. [37]. h Ref. [28 I. c Ref. 126]. ~ MRD-CI results from Ref. [ 19], e Ref. [38]. f Ref. [39]. g CASPT2 results from Ref. [3]. h Ref. [40]. i CASPT2 results from Ref. [3]. J Ref. [301. k Ref. [41]. I Ref. [421. m Ref. [20].

atoms were investigated with the small VDZ basis set. Inspection of Table 2 shows that the errors for AE are also very small as found for the reference systems, i.e., 0.1-0.2 eV (the standard deviation for all states is 0.20 eV), with the largest deviation of 0.5 eV observed for the highest 0r~-* state of benzene (1E2g). In most other cases, where larger errors are found, the experimental results are not decisive since forbidden or strongly non-vertical transitions are involved (e.g. polyene 2 lAg states). A really satisfactory agreement is found for the two lowest La and Lb states of the PAH which show a strongly different size dependency,

The errors are below 0.1 eV up to the largest considered linear condensed acene (pentacene with five rings, 102 valence electrons). Also the non-alternant azulene molecule is described very well. In order to show that the approach is not only capable of a description of organic molecules one phosphoroxide cluster ( P 4 0 6 with Ta symmetry) has been investigated and a very good agreement of the theoretical data with the experimental UV spectrum [20] is observed (the forbidden transitions to A, E and T1 states, seen as shoulders in the spectrum, are not listed here but are also predicted quite reliable).

S. Grimme/Chemical Physics Letters 259 (1996) 128-137

134

Table 3 Comparison of vertical singlet-triplet excitation energies AE (in eV, DFr/SCI/B3-LYP) Compound

AO basis

Excited state

AE exp.

H2 CH20 ethene trans-l,3,5-hexatriene benzene

B B B A B

naphthalene anthracene

A A

a Explicitly correlated calculations from Ref. [37]. d Ref. [43]. e Ref. [441. f Ref. [421.

3E+ (tro-*) 13A2 (nTr*) 13BI u (zrTr*) 13Bu (zrTr*) 13BI u (,rrTr*) 13El u (zr~*) 13B2u ( ~ r * ) 13B2u (~'~*) b Ref. [38].

Furthermore, a very important result can be derived from the results in Table 2: the parameters obtained with the large basis set B can be used with minor loss of accuray in calculations employing smaller basis sets, i.e., no reparameterization procedure for different AO basis sets is nessecary. As a first application (whose results and implications cannot be discussed in detail here) the calculation of the circular dichroism (CD) spectrum of (M)4,5-dimethylphenanthrene with C2 symmetry is prysented. The molecule is the smallest member of the [n] helicene family in which a nearly regular cylindrical helix is formed through an all-ortho annelation of aromatic rings (see inset in Fig. 1 ). In this compound the helical structure is a consequence of the steric repulsion of the methyl substituents ( for details see Ref. [ 21 ] ). A recent semiempirical CNDO/S-SCI study of the CD spectrum of this compound [5] shows some agreement with the experimatal data; however, the CD sign of the band B a n d t h e intensity of the band F h a s not been predicted correctly. Prior experience with the calculation of CD spectra of chiral PAH with semiem-

10.5 3.1 4.2 2.6 3.9 4.8 2.6 1.8

DFT/SCI a b c d e e f f

10.28 3.08 4.14 2.27 3.89 4.29 2.76 1.81

c MRD-CI results from Ref. [18].

"-~ ~ ~ ~ <~

~000 I ~ ! 50.0!

c ,~ E

/, /,]

i

)'[,,'1"~'1\~_~ ~'

0.0 -500

,

. -J i D ~,, // ~ % ~

~ B

~ ,~

ii -~00.0 ,~ -~5o.O7oo ................ 200.0 230.0

260.0 290.0 320.0 350.0

~./nm Fig. 1. Comparison of the experimental CD spectrum of (M)-4,5-dimethylphenanthrene (reproduced from Ref. [451, full line) and the calculated rotatory strengths (dipole length form, DFr/scI/B3-LYE VTZ(C)/VDZ(H) AO basis) designated by sticks (in cgs, R has units of esu cm erg/G and this corresponds to 3.336 x 10-15 C 2 s - l m3). The filled points indicate excited states with near zero R values. The dashed line shows the simulated

CD spectrum obtained by summing rotatory strength weighted Gaussian curves with AFWHM = 0.3 eV for each calculated CD

pirical methods and a MRD-CI treatment in our laboratory has shown that such failures occur in some chromophores. Tentatively, this can be attributed to an inadequate weight of the configurations in the excited state wavefunctions (e.g. as found for naphthalene [22] ) which has only a small effect on AE but result in some cases in inverted CD signs. As can be seen from inspection of Fig. 1 the relative intensities of all resolved bands A-F are reproduced by the present calculation in which no integrals are neglected in the SCF

transition(scaled to the experimental intensity of band F). step as in semiempirical methods. The overall agreement of the simulated and experimental spectrum is remarkably good despite a slight blue-shift of the E and F bands in the theoretical spectrum. Another advantage of the DFT/SCI method with inclusion of all valence electrons should be discussed here by considering the CD spectrum of 1R-(+)camphor as an example (see Fig. 2). In the CASPT2

S. Grimme/Chemical Physics Letters 259 (1996) 128-137

4.0

-= ~

considered in the CI wavefunctions 2. The DFT/SCI calculation can offer a good explanation for the CD sign inversion in the 6.5-7.0 eV region when going from the gas phase to the polar solvent trifluorethanol (TFE). The second and third positive band in the gas phase spectrum originates from the transition to the

orc*/n3p(z) \ ~ \~/~

2o ~ 00 _2.0:~ I~,Cn'* . I / i ~ r l 3~p (

-4.0 -6.0 i II~,, t

135

°

-80 i ',' inTFE -10.0 i ~ ~ . . . . . . . 150.0 20o.0 2so.o 300.0 X /nm Fig. 2. Comparison of the experimental CD spectra of IR-( +)-Camphor (full bold line: gas phase, full line: solvent trifluorethanol) with the theoretical spectrum (dashed line, dipole length form, DFr/scI/B3-LYP; a VTZP+diff. AO basis on the atoms of the carbonyl group, a VDZP AO basis for the a-carbon atoms and a VDZ AO basis for all other atoms has been employed). The filled points indicate the position of the calculated excitation energies. The designation of the 3p Rydberg states corresponds to a local coordinate system of the carbonyl group (CO

along the z-axis, C-CO-C in the yz-plane).

method, which is currently the most reliable approach for the calculation of excitation energies in medium sized systems, only the energies are corrected in second order perturbation theory to account for dynamical correlation effects. The transition moments are calculated from the CASSCF wavefunctions which includes only the correlation of a few electrons. In larger chiral compounds, however, most of the valence electrons contribute to the rotatory strengths [23] (which are sensitive to details of the wavefunctions). These effects can only be accounted for if all valence electrons are treated explicitly in the CI as done in this work. A good example for such a system is the camphor molecule which is a bicyclic carbonyi compound (see inset in Fig. 2) with natural abundance and pharmacological importance. The n~* band at 4.3 eV, which has local C2~ symmetry with a small electric dipole transition moment (the magnetic transition moment is strong), is perturbed by the asymetric distribution of the electrons in the whole carbon skeleton. Hence, the correct sign of the CD of the lowest nrr* band is only obtained if at least 30-40 valence electrons are

n--+3s(AEcalc=6"47eV'Rcalc=W3"8x10-4°cgs)' n --* 3pv (AEcalc = 6.83 eV, Rcalc = + 0 . 6 x 10 -40 cgs) and n ~ 3px (AEcalc = 6.87 eV, Rcalc = - 2 . 4 x 10 -40

cgs) Rydberg states. At higher energies the calculation predicts three ~rTr* states (the first and second is strongly mixed with the n ~ 3pz Rydberg state; AEcalc = 7.00, Rcalc = +19.9 x 10 -40 cgs, AEcalc = 7.05, Rcalc = - 8 . 0 x 10 -40 cgs, AEcalc = 7.23, Rcalc = - 2 1 . 2 x 10 -40 cgs) with alternating CD signs. Summation of these CD intensities at the appropriate positions results in a very good agreement between the shape of the theoretical and experimental gas phase CD spectrum. Compared to the ground, nTr* and Rydberg states the highest o-Tr* state at 7.23 eV has a large dipole moment of 10.73 D (/x(1 1A) = 3.41 D, /z(2 1A) = 3.19 D) which explains the strong red-shift of this intense negative band down to 6.5 eV in TFE. To summarize not only the low-lying valence states but also the higher-lying transition energies and transition moments are obtained with good accuracy in this cornplex molecule.

5. Conclusions A new computational scheme for the calculation of excited states and electronic spectra of molecules with a closed-shell ground state is proposed. The method is based on the single-excitation configuration interaction (SCI) approach with emirically corrected CI Hamiltonian matrix elements and uses eigenvalues and molecular-orbitals from gradient-corrected density functional calculations. The resulting excitation energies are in most cases accurate to a few tenths of an electron volt even in cases where traditional ab initio CI methods have substantial difficulties (e.g. PAH molecules). Due to the use of flexible AO basis z In fact, the dependence of the CD sign of the nTr* band shows a oscillatory behaviourwith respect to the numberof correlated

electrons.

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S. Grimme/Chemical Physics Letters 259 (1996) 128-137

sets no restrictions to the spatial nature of the excited states are imposed and comparable accuracy is obtained for valence and Rydberg states. The overall results represent a major improvement compared to the standard HF/SCI procedure at no additional computational effort. This is attributed to the fact that dynamical correlation and the effects of double and higher excitations are implicitly accounted for to a large extent by the DFT components of the approach. T h e investigation of electronic properties such as cir-

cular dichroism spectra shows that the method is not only capable of providing good excitation energies but also yields a reliable description of the wavefunctions and transition moments. The computational bottleneck of the procedure is the AO to MO integral transformation task (and not the CI step) which limits the application currently to systems with ~ 300-400 AOs in C1 symmetry. However, the basis set requirements for the valence states of larger systems are not so severe so that the employment of valence double-( AO basis sets without polarization functions partially circumvents this problem. Approaches in which valence electrons or virtual orbitals are kept frozen in the correlation treatment cannot be recommended especially in the case of larger low-symmetry species since most of the valence electrons contribute to the excited state wavefunctions.

Acknowledgement The services and computer time made available by the Sonderforschungsbereich 334 ( 'Wechselwirkungen in Molekiilen') have been essential to this study which was financially supported by the Deutsche Forschungsgemeinschaft. I want to thank E Pulm and J. Hormes of the Physikalisches Institut der Universit'it Bonn for providing the experimental results of camphor prior to publication. References

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