Applications of level shift corrected perturbation theory in electronic spectroscopy

THEO CHEM ELSEVIER

Journal of Molecular Structure (Theochem) 388 (1996) 257-276

Applications of level shift corrected perturbation theory in electronic spectroscopy Bj6rn O. Roos a'*, Kerstin Andersson a, Markus P. Fiilscher a, Luis Serrano-Andr6s a, Kristine Pierloot b, Manuela Merch~n c, Vicent Molina c aDepartment of Chemistry, Lund University,P.O.B. 124, S-221 O0 Lund, Sweden bDepartment of Chemistry, University of Leuven, Celestijnenlaan 20OF, B-3001 Heverlee-Leuven,Belgium CDepartamento de Qulmica Ffsica, Universitat de Valencia, Doctor Moliner 50, Burjassot E-46100 Valencia, Spain

Received 13 March 1996; accepted 13 May 1996

Abstract

Multiconfigurational second-order perturbation theory (CASPT2) with a level shift technique used to reduce the effect of intruder states has been tested for applications in electronic spectroscopy. The following molecules have been studied: formamide, adenine, stilbene, Ni(CO)4, and a model compound for the active site in the blue copper protein plastocyanin, Cu(Im)2(SH)(SH2) ÷. The results show that the level shift technique can be used to remove the effects of the intruder states in all these molecules. In some cases a drift in the energies as a function of the level shift is observed, which however is small enough that the normal error bar for CASPT2 excitation energies ( -~ 0.3 eV) still holds. Keywords: Level shift technique; Perturbation theory; Electron spectroscopy

1. Introduction

It has recently been shown [1] that intruder states appearing in multiconfigurational second-order perturbation theory (CASPT2) [2-4] may be effectively removed by introducing a level shift combined with a back correction of the second-order energy to zero level shift, the so called level shift correction (LSCASPT2). The method was illustrated by calculations of spectroscopic constants of the ground and one excited state of the N2 and Cr2 molecules. Intruder states occur frequently in CASPT2 calculations of electronic spectra of organic molecules and transition metal complexes and it is not clear a priori whether * Corresponding author.

the LS-CASPT2 method will work in cases where several excited states are involved. We give in the present contribution some examples of the use of the approach in spectroscopy. Electronic spectra of the molecules formamide, adenine, stilbene, Ni(CO)4, and a model compound for the active site in the blue copper protein plastocyanin are studied. It will be shown that the LS correction can be used to remove the effects of the intruder states in all these molecules. Excitation energies vary only slowly with the level shift in regions where there are no intruder states. Thus the method seems to work well also in this type of application. This is an important observation, since one of the major areas of application of the CASPT2 method is in studies of excited states and electronic spectroscopy [5,6].

0166-1280/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved I'll S0166-1280(96)04712-4

B.O. Roos et al./Journalof MolecularStructure (Theochem)388 (1996) 257-276

258

Adenine and stilbene are examples of systems where the method works less well. They are aromatic molecules with delocalized ~r-electron systems. Any attempt to reduce the number of active orbitals below the number of valence r-orbitals leads to severe intruder state problems, which cannot be fully removed with the level shift technique. As a result, computed excitation energies will not be completely independent of the applied shift.

E~"S= i=lXV?Ci

2. The LS corrected CASPT2 approach

ciLSmci(l+ ei_;o+~)

Intruder states appear in the CASPT2 method when one or more of the state functions used to construct the first-order wave function (the FOI space) have eigenvalues of the zeroth-order Hamiltonian, lq0, close to that of the reference function. The level shift operator introduced in Ref. [1] modifies the zeroth-order Hamiltonian to lq0 + etch, where e is a small positive number and Pl is the projection operator for the firstorder interacting (FOI) space. The advantage of the level shift is that intruder states can be removed. The drawback is that the second-order energy will depend on e, a quantity that cannot be chosen on any physical grounds. The e dependence can, however, be removed to first-order by introducing the level shift (LS) corrected second-order energy M E2Ls - E 2 - e ( 1 - 1 ) = E 2 - e ~ 60 i-1

Ifi 12

(1)

where Ez, 60, and Ci are the second-order energy, the weight of the CASSCF reference function, and the expansion coefficients of the first-order wave function in the level shifted CASPT2 calculation, respectively. M is the dimension of the FOI space. In Ref. [1] we showed that E2Ls only changes a few miUihartree when increasing e from 0 to 0.5 a.u. for the ground state potential of N 2 where no intruder states are present. For the calculation of properties it is desirable also to have a level shift corrected first-order wave function. By introducing the right-hand side of the first-order equation, Vi ------(@ill2lllXIt0),where ~i are the eigenfunctions of izl0in the FOI space with eigenvalues ei and ~0 is the CASSCF reference function with eigenvalue E0, the second-order energy may be expressed as M

E2" ~, V*Ci i=1

(2)

The expansion coefficients can be written as Ci = - V i / ( e i - E o + e) and this together with Eq. (1) and Eq. (2) give

1 + ei -Eo + e

" i-1 • g*ciLS

(3)

where the expansion coefficients of the level shift corrected first-order wave function, CiLs, are defined as (4)

By using the definition in Eq. (4) one can define a level shift corrected weight of the CASSCF reference function, 60LS,as 1 M 60---~= 1 + i=lXIf/Lsl2

(5)

When increasing e from 0 to 0.5 a.u. for the ground state potential of N2, 60LSonly changes by 0.002. This should be compared with 0.012 which is the change in 60. Only 60 is computed in the application presented below.

3. Applications of level shifts in electronic spectroscopy Intruder states are common in CASPT2 calculations of excited states, especially in calculations using extended basis sets, which contain diffuse functions. It is then necessary to include a well defined set of Rydberg orbitals, which can be used to separate the Rydberg and valence excited states [5]. Corresponding Rydberg orbitals have to be included within the CASSCF active orbital space. The number of active orbitals is, however, limited (maximum 12-14) and it is not always possible to incorporate all Rydberg (or semi-Rydberg) type orbitals. Such diffuse states often have rather low zeroth-order energies and can act as intruder states in the CASPT2 calculation. Their direct interaction with the reference function is small and the effect on the second-order energy is only a result of an accidental near-degeneracy. The situation often occurs in calculations of electronic spectra of unsaturated organic molecules, The diffuse part of the FOI space contributes only little to the correlation energy and it should therefore be a safe procedure to remove

259

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

the intruder states by means of the level shift technique. The situation in transition metal complexes is more complicated. Also here do diffuse orbitals from the next shells cause intruder state problems. However, intruder state problems can also arise, for example, from charge-transfer excitations involving ligand orbitals not included into the active space. Such excitations may be important and cannot always be removed using level shifts. Sometimes it is also not possible to include them into the active space (which easily gets large for transition metal complexes). One then has to accept that the CASSCF/CASPT2 method cannot be used to solve the problem, since a too large active space is needed. Examples of such problems have been given in a recent review [6]. However, many times the intruder states are also here only weakly interacting and can be removed using the LS-CASPT2 method. We shall give below some examples from both organic and transition metal chemistry. 3.1. The electronic spectrum o f formamide

The UV/visible (VIS) spectrum of proteins exposes a strong band at about 7.4 eV which is, in general, attributed to the amide group. Detailed knowledge of the electronic properties of the monomers is of importance for the interpretation of the polymers, and therefore we decided to investigate the electronic spectrum of formamide. First, the ground state geometry was optimized at the MP2/6-31G* level of approximation, and the molecule was assumed to be planar (Cs symmetry). Second, calculations on excited states were performed with ANO basis sets contracted to C,N,O[4s3pld]/H[2s] [7] and augmented with an additional set of l s l p l d diffuse functions placed at the charge centroid. The 7r-system of formamide includes three centers: the C, N, and O-atom. Two 7r ---* a" valence excited states and a weak n ~ 7r* can be expected. In addition, a number of Rydberg states are likely to be recorded in the gas phase. CASSCF/CASPT2 calculations on excited valence states are best performed with an active space including two and four orbitals of a' and a" symmetry, respectively. Note, that two weakly occupied orbitals, one of each symmetry, have also been added to the active space, and are used to

correlate the lone pair electrons. Fortunately, formamide is small enough such that a set of l s l p l d Rydberg orbitals could also be included in the active space. The corresponding MOs were deleted in calculations on excited valence states. As explained elsewhere [8], such a procedure will minimize the erratic valence-Rydberg mixing that may occur in CASSCF calculations on excited valence states. Table 1 compiles the resulting CASPT2 excitation energies and weights of the reference CASSCF wave functions for the ground state, the two valence 7r ---* • "" states, and the valence n ---. 7r" states of the formamide molecule computed at different values of the level shift correction. As all the valence 7r space is included into the active space, the presence of intruder states can only be expected from the o skeleton or from high-lying diffuse orbitals with r character. The results, both for energies and weights, clearly show that no intruder states occur for the studied range of values of the level shift parameter in any of the states. The sensitivity of the computed energies to the correction is low, although there is a continuous small increase of the relative energies with the increase of the level shift parameter. This seems to be a general feature for level shift corrected excitation energies that is related to the fact that the percentage of correlation included in the calculation decreases with an increasing level shift. Table 2 compares the computed and experimental excitation energies and oscillator strengths for the formamide molecule. We have selected a value of 0.3 a.u. for the level shift correction for consistency with our study on larger amides [8]. However, as indicated in Table 1, other values of the level shift give virtually the same result. The gas-phase spectra Table 1 CASPT2 excitation energies (in eV) for the ~r --* ~r" (tA' symmetry) and n --, ~r" (1A"symmetry)excited states in the formamide molecule as a function of the applied level shift (the CASSCF reference weight is given in parentheses) LS a

0.0 0.1 0.2 0.3 0.4

llA '

-(0.90) -(0.91) -(0.91) -(0.92) -(0.92)

21A,

31A'

7.31(0.86) 7.33(0.87) 7.37(0.88) 7.41(0.89) 7.45(0.90)

10.42(0.86) 5.60(0.89) 10.44(0.88) 5.60(0.90) 10.46(0.89) 5.60(0.91) 10.50(0.89) 5.61(0.91) 10.54(0.90) 5.62(0.92)

Level shift parameterin a.u.

llA ,,

260

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

Table 2 Comparisonbetween CASSCF/CASFI2(LS = 0.3) excitation energies (in eV) and experimentfor the lr ---, r* and n ---* 7r"excited states in the formamide molecule State IIA" (n -.* lr') 21A' Qr ---, 7r*) 31A' (~r ---* ~r*) 13A"(n ---* 7r*)

CASSCF 8.30 8.83 11.93 6.72

CASPT2 5.61 7.41 10.50 5.34

Expt. 5.5 a 7.4 b

Oscillator strength Theory

Expt.d

0.001 0.371 0.131

0.002 0.30 0.1

5.30c

a Estimated value for acetamide in nonpolar solvents. Refs. [9,10]. b Vacuum transition, refs. [11,12]. c Traped electron spectrum of formamide in gas phase. Ref. [16]. Estimated values in solution. Refs. [9,10]. of the simple amides have been interpreted as fiveband systems including the transitions named W, R1, NV1, R2, and Q. In solution or solid films only two of the bands, apparently NV1 and Q, remain. The n ---* ~r 11A" state is computed at 5.6 eV with an oscillator strength of 0.001. This is the W band, estimated at 5.5 eV for acetamide in nonpolar solvents [9,10]. The presence of a number of intense Rydberg transitions in the vacuum formamide spectrum made the interpretation of its spectrum difficult. We have computed the most intense band of the spectrum at 7.4 eV with an oscillator strength of 0.37. The result suggests the assignment of the intense band observed at 7.4 eV in gas phase as the result of the ~" ~ 7r* transition [11,12], i.e. the NVI band. In larger amides the band will strongly shift to lower energies, due to both substitution and solvent effects. The second lr ---* 7r* band of forrnamide iscomputed at 10.5 eV for the isolated molecule, with an oscillator strength of 0.13. The presence of this transition, the NV2 band, at high energies rules out the suggested valence r --, 7r* character of the 7.7 eV band in different amides (see Refs. [12,13]), which we preferably attribute to sharp Rydberg transitions. The NV2 bands in amides can be related to the equivalent transitions in the carboxylic systems formaldehyde and acetone, in which the corresponding transition also occurs at high energies [14,15], but could not be observed in the experimental spectra. For a more detailed discussion of these aspects and the results for the Rydberg states in the formamide molecule we refer to ref. [8]. Finally, the 13A" n ~ ,r* state in formamide is computed at 5.3 eV, in agreement with the measured value in the gas phase [16].

3.2. The electronic spectrum o f the adenine molecule

The UV/VIS absorption spectrum of aqueous solutions of adenine includes two band systems. The low energy band (240-280 nm) is due to two electronic transitions with maxima at about 4.5 and 5.0 eV. The second band system starts out at about 6.0 eV and extends into the far UV. In the energy range 6 . 0 7.0 eV at least two electronic transitions can be identified. The structure of the spectrum is complex and remains unsolved. In particular, the number of transitions and their transition moment directions are controversial. The situation is further complicated by tautomerism: In aqueous solutions the N(9)H isomer dominates but about 20% of the molecules are believed to adopt the N(7)H form. To produce new information and hopefully resolve the spectroscopic problems, we have calculated the electronic spectra of both tautomeric forms. However, in the present context, we shall only discuss the results for the 7r ~ 7r* excited valence states of N(7)H adenine. The interested reader is referred to a forthcoming publication [171. The ground state geometry used for the present calculations was optimized at the MP2/6-31G* level of approximation, assuming Cs symmetry. The remaining calculations use an ANO-L type basis set [7] contracted to the following structure: C,N,O[4s3pld] and H[2s], plus an additional set of l s l p l d diffuse functions placed at the charge centroid. The ~r-system of adenine extends over 10 centers and includes 12 electrons. This orbital space was selected as the active orbitals for the subsequent

261

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

the following states. Despite the fact that their weights differ by more than 0.5 from the ground state weight, a comparison of the excitation energies with those obtained with the larger level shift parameters does not show any important variations. Even for the 71A ' state, where the intruder state presence leads to a weight as low as 0.07, the effect on the excitation energy is small. This behavior is not uncommon and shows that the coupling between the intruder state and the reference function is often small. If we go through the results for the different states obtained with level shift corrections, it is clear that the effect on the energy is more important. The reason is the back correction to zero level shift of the second-order energy (Eq. (1)). This formula is only valid when the reference weight is large and unaffected by intruder states. This is clear from the results in Table 3, which shows that the excitation energies do not stabilize until the level shift has reached a value of about 0.225 a.u. (cf. Fig. 1). The choice of a proper level shift correction to avoid intruder state problems in the computed states explicitly depends on the states studied and cannot be predicted in advance. The present value of 0.225 a.u. for the level shift seems to lead to balanced values for the reference weights (from 0.70 in the ground state to 0.68 in the 61A' state, cf. Table 3). Also the energies start to stabilize with this level shift. Increasing the level shift from 0.225 to 0.250 the variation of the excitation energy is less than 0.08 eV. The same

CASSCF calculations. As in formamide, preparatory calculations were first performed, where the Rydberg orbitals were initially included and then deleted from the active space to avoid the Rydberg-mixing problems, which sometimes occur in CASSCF wave functions for valence excited states. It was also found that the 1r-orbital lowest in energy could be kept inactive, because its occupation number was close to two for all studied excited states. Instead, an extra weakly occupied active orbital was added, yielding again an active space of 10 orbitals but now with only 10 active electrons. The electronic structure of adenine is complex and even if the Rydberg valence states mixing has been taken care of, intruder states were found. To remedy the problem we included stepwise two more a--orbitals into the active space. Table 3 lists the computed CASPT2 excitation energies and CASSCF reference weights for several valence r ---, r ' tA' states of the N(7)H-adenine molecule using different choices for the level shift corrections and active spaces. In addition, Fig. 1 shows the dependence of the CASSCF reference weights and the excitation energies on the level shift parameter for the different states. Focusing on the results without any level shift correction, the 21A ' and 31A ' excited states seem not to be affected by any severe intruder state problem. The weights of the reference functions are considerably smaller for

Table 3 CASPT2 excitation energies (in eV) for the r ---, 7r" excited states in the adenine molecule (7H isomer) as a function of the applied level shift (the CASSCF reference weight is given in parentheses) a LS

IIA '

Active space: 10 a" orbitals 0.000 -(0.69) 0.050 -(0.70) 0.100 -(0.70) 0.175 -(0.71) 0.225 -(0.72) 0.250 -(0.73) 0.300 -(0.73) Active space: 11 a" orbitals 0.000 -(0.69) 0.250 -(0.73) Active space: 12 a" orbitals 0.000 -(0.69) 0.250 -(0.73)

21A '

31A '

41A '

51A '

61A '

71A '

4.56(0.66) 4.51(0.66) 2.78(0.47) 4.58(0.70) 4.60(0.71) 4.61(0.72) 4.63(0.72)

4.83(0.60) 4.70(0.61) 4.84(0.67) 4.92(0.69) 4.95(0.70) 4.97(0.70) 5.01(0.71)

6.02(0.48) 2.52(0.25) 3.75(0.44) 5.65(0.65) 5.98(0.69) 6.02(0.70) 6.07(0.71)

6.30(0.48) 4.84(0.42) 5.17(0.55) 5.42(0.63) 6.08(0.69) 6.15(0.70) 6.23(0.71)

6.28(0.45) 2.82(0.24) -325.4(0.01) 5.89(0.64) 6.20(0.68) 6.32(0.69) 6.42(0.71)

6.93(0.07) -1687(0.00) 5.65(0.57) 5.70(0.62) 6.44(0.69) 6.49(0.69) 6.56(0.70)

4.54(0.66) 4.59(0.72)

5.00(0.63) 5.11(0.71)

6.12(0.11) 6.03(0.70)

6.20(0.13) 6.13(0.70)

6.33(0.49) 6.33(0.70)

6.63(0.53) 6.66(0.70)

4.58(0.62) 4.59(0.72)

5.04(0.61) 5.16(0.71)

5.92(0.17) 6.08(0.71)

6.44(0.41) 6.12(0.70)

6.77(0.25) 6.31(0.70)

6.80(0.08) 6.84(0.70)

a Active space: 10 electrons in 10-12 a" orbitals. Level shift in a.u.

B.O. Roos et al.IJournal of Molecular Structure (Theochem) 388 (1996) 257-276

262

7.00

,1~,.

I

E

I

I

I

'%

6.00

'y

\

5.00

.,.-'"

"\., I' i'

/, ,' ',

: --i-t----

/:. '/\

,

, .:......._.:...............-.---.-.-.-.---.--._.

..". xLL ............................

.........

', . /

". ., / '../

',2,...'

/!~

."

.

4.00

o

k

\,~ . !, , .,j',~,.,

A

,~ •

...... :--- ;: .......

,.

" " "

I

#,..,,,.',

"1'',

,

/

........

: ,

.."

;

3.00 ,'

! i

:~:

i

2.00

',

: ;

!

:

',

I ,

......... 4 A

5ZA '

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

7ZA '

I

i i i ! i ,

1.00

0.00 0.8

I

I

I

I

-~- . . . .

-,

-

•

-,

I

i~

"

/"'""'~'

'

I

I

- .....

_---L2L_--/.-.,.~.:

~.'f

'I

t,'~

~ / / ,, ,.

,'~ .

f

/

,,

:

y

,

T.

O. 0 5 0

I I A

.......

2ZA

-

.........

I"

/'

................. 4ZA

. i

,:z ."

i

i 'V'

........

~,'-

-. . . . . . . .

7ZA'

s' ,. '%

'

"~"%

/

.

y

0.0 O. 0 0 0

"

;~ j

,.-.

,

•

?" /'/

;

~, ;

"

"',' ,

" ',,

.

,

........... ;;." ,.,

.~.

'

I,

"~.

-'"

"~

,,

"....

'

,.,

'%

..-','--

"f- .

/'" . . . .

;

.

~

~'.,.,.', :.' \ /

'

!

.~

~'.

0.2

0.1

.

, \ . :'-'.....I "..l

",, 0.3

, "'"

., "#.?'

~ I "~

~2. . ~..

, x"

/

"..

0.4

I

/

,. ,,:i .,~

I

I

. . . . . . .,.-.-=-.- .-.-.--.-.-.-.. . . . -- ~'_~.'._~.:--.----. . . . . . . . . . . . .

%

0.5

: ;I

...........

0.7 0.6

', ', ,

O. I 0 0 Level

I 0. 1 5 0 Shift

I 0.200 (a.u.)

I 0.250

0. 3 0 0

Fig. 1. Plot of the level shift corrected CASPT2 excitation energy (upper diagram) for several ~r ---, x " excited states in the adenine molecule as a function of the applied level shift. The lower diagram gives the corresponding weight of the CASSCF reference function in each of the states as a function of the applied level shift.

stability is observed with further increases to 0.30 a.u. As occurs also in some of the other studied systems, a small linear increase in the excitation energies with the level shift correction is, however, observed. Therefore, we recommend the selection of the lowest level shift, which corrects the intruder state problems

using both the weight and energy stability criteria. In the adenine system a value of around 0.25 a.u. seems to be proper for the considered states. A better procedure than using the level shift to avoid intruder states, is to increase the active space. Table 3 also shows the results of extending the active

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

space by one or two additional Ir-orbitals. Focusing on the results without any level shift correction, the intruder states appear to shift to lower energies. However, two additional weakly occupied 7r-orbitals is not enough to remedy the problems as is illustrated by the numerous singularities shown in Fig. 1. By comparison, we find that adding a level shift correction of 0.25 a.u. leads to results consistent with those obtained for the smallest active space. The calculations presented here are part of a more extended study of the adenine spectrum. A detailed comparison between the computed data and experiment will be made in a forthcoming publication [17]. We note, however, that theory predicts one excitation at 4.6 and one at 5.0 eV in agreement with experiment; The calculations also predict at least four ~r ---, r transitions in the energy range 6.0-7.0 eV, two of which have considerable intensity. This is again in agreement with experimental findings.

3.3. The electronic spectrum of the stilbene molecule The electronic spectrum of stilbene is a challenge for the CASSCF/CASPT2 approach. The ideal active space comprises 14 valence ~'-orbitals with 14 active electrons. Such an active space gives rise to a large CAS CI space (about 1.4 x 10 6 configuration state functions in the C2h symmetry of the trans conformer). A CASSCF calculation with 14 active orbitals is at the limit of what the present technology can handle and is very time consuming. All the 14 r-electrons are, however, not directly involved in the lower excited states. It might therefore be tempting to try to reduce the number of active orbitals, either by moving some 7relectrons to the inactive space, or by moving some of the weakly occupied orbitals to the virtual space. In this section we shall show the results of such a calculation where the two most weakly occupied orbitals have been deleted from the active space. As a result, intruder states will appear, which interact quite strongly with the reference function. The level shift technique will remove the near degeneracy, but the excitation energies will not be stable with respect to the size of the level shift. This example is thus different from the other illustrations. It shows a case where the LS technique is used to remedy a too small active space, which is not a recommended procedure. The calculations were performed with the ANO-L

263

basis set contracted to C[3s2pld]/H[2s] [7] with two s-type and two p-type diffuse functions added, placed at the charge centroid. The geometry was optimized at the CASSCF level with 12 active orbitals, (0660) and 10 active electrons (two inactive 7r-orbitals) [18]. The molecule was assumed to be planar with C2h symmetry. Using this geometry, Restricted Active Space (RAS) SCF calculations were performed with all the 14 r-electrons active. The active space was divided up with the RAS1 space (0230), the RAS2 space (0220), and the RAS3 space (0320). Up to two holes were allowed in RAS1 and at most two electrons were allowed to occupy the RAS3 space. This active space was used in a state-averaged calculation including six states of 1Bu symmetry. The number of states was determined such that all excitations below 6.0 eV were included in the study. The resulting natural orbital occupation numbers for the 14 valence Irorbitals are presented in Table 4 for the six electronic states. It is clear from these results that the preferred active space in a CASSCF calculation for stilbene should include all the 14 valence 7r-orbitals. A similar calculation performed for t h e lAg states gave the same results. Experience shows that natural orbitals with occupation numbers in the range 0.02-1.98 should be included in the active orbital space. As shown in Table 4 there is no orbital for which the occupation Table 4 Natural orbital occupation numbers for the 14 valence ~'-orbitals in the 1Bu states of stilbene, as obtained from state-averaged RASSCF calculations 11Bu

21Bu

Symmetry 1.97 1.95 1.65 1.41 0.12 0.16 0.03 Symmetry 1.96 1.87 1.85 0.58 0.37 0.05 0.03

au 1.98 1.96 1.93 1.03 0.07 0.06 0.02 bg 1.97 1.94 1.93 0.98 0.07 0.03 0.02

31Bu

41Bu

51Bu

61Bu

1.97 1.93 1.62 1.25 0.18 0.06 0.03

1.97 1.93 1.55 1.34 0.14 0.05 0.02

1.97 1.90 1.49 1.67 0.21 0.31 0.03

1.97 1.94 1.36 1.74 0.11 0.26 0.02

1.97 1.83 1.94 0.77 0.38 0.06 0.02

1.97 1.83 1.95 0.78 0.38 0.06 0.02

1.96 1.77 1.69 0.55 0.33 0.08 0.03

1.97 1.88 1.75 0.25 0.67 0.07 0.02

264

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

Table 5 CASPT2 excitation energies (in eV) for the r ---, 7r* excited states in the stilbene molecule as a function of the applied level shift (the CASSCF reference weight is given in parentheses) States of lAg symmetry LS ~ 1 lAg 0.0 -(0.62) 0.1 -(0.64) 0.2 -(0.66) 0.3 -(0.67) 0.4 -(0.69) 0.5 -(0.70) States of lBu symmetry llBu 0.0 3.59(0.56) 0.1 3.63(0.61) 0.2 3.70(0.63) 0.3 3.77(0.65) 0.4 3.84(0.67) 0.5 3.91(0.69)

2 lAg 4.17(0.37) 3.80(0.59) _b 4.13(0.65) 4.22(0.67) 4.30(0.69)

3 lAg 4.84(0.58) 4.87(0.62) 4.91(0.64) 4.95(0.66) 5.01(0.68) 5.06(0.69)

4 lAg -b 5.15(0.60) 5.25(0.62) 5.30(0.65) 5.45(0.67) 5.54(0.68)

21Bu 3.94(0.59) 3.96(0.61) 4.01(0.64) 4.07(0.65) 4.12(0.67) 4.18(0.69)

31Bu 5.11(0.52) 5.22(0.60) 5.32(0.62) 5.42(0.65) 5.52(0.67) 5.62(0.68)

41Bu 5.30(0.32) 5.14(0.59) 3.41(0.51) 5.42(0.65) 5.52(0.67) 5.61(0.68)

51Bu 5.48(0.52) 4.78(0.55) 5.33(0.63) 5.46(0.65) 5.55(0.67) 5.63(0.69)

61Bu b

5.76(0.60) 5.84(0.63) 5.95(0.65) 6.03(0.67) 6.12(0.68)

a Level shift parameter in a.u. b Not computed due to severe intruder state problems.

number falls outside this range for all the states considered. Such a CASSCF calculation is, however, not feasible at present. A compromise is then to exclude the two orbitals with the lowest occupation numbers, thereby increasing the lower limit from 0.02 to 0.03. The alternative to move two orbitals to the inactive space leads to larger problems with intruder states. However, none of these alternatives are entirely satisfactory, as the results below will show. Stilbene is an illustration of a bottleneck of the present approach in spectroscopy that occurs in strongly correlated organic molecules with many r-electrons. It is in such cases difficult to limit the active space so that the CASPT2 calculation can be done and there is a need to extend the capabilities of the approach, so that larger active spaces can be used. A number of state-averaged CASSCF calculations were performed using the (0660) active space. Only r ---* 7r excited states were studied. One set of calculations included four states of lAg symmetry, and another included six states of ~Bu symmetry. The level shift, e, was varied between zero and 0.5 a.u. The resulting excitation energies are listed in Table 5 together with the weights of the corresponding CASSCF reference function. The weights for the 1 l A g ground state are also given for comparison.

It is clear that intruder states appear for almost all of the excited states in the calculations without level shift. Only when e is increased to 0.3 a.u. will the reference weights all be similar to that of the ground state. However, as in the case of adenine, the excitation energies depend on the level shift also in regions

Table 6 Comparison between CASSCF/CASPT2 (LS ffi 0.3 au) and experimental excitation energies (in eV) for the r ---, 7:' excited states in the stilbene molecule State

CASSCF CASPT2

Expt.

Oscillator strength Theory

llBu 21Bu 2lAg 3 lag 4lAg 31Bu 41Bu 51Bu 61Bu 13Bu

5.42 6.05 5.41 5.92 7.41 7.96 7.76 7.18 8.36 3.02

3.77 4.07 4.13 4.95 5.30 5.42 5.42 5.46 5.95 2.56

0.038 4.22 a 0.723 ~- 4.1 c 5.00 ~ 5.43 a 0.371 0.117 0.019 6.17 a 0.524 2.14 a

' Ref. [261. b Ref. [27]. c Ref. [28]. Experimental data for the 0 - 0 band, Ref. [29].

Expt. 0.74 b 0.29 b

0.41 b

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

where there are no apparent intruder states. All reference weights increase in magnitude with the level shift parameter, the lAg states somewhat less than the 1B u s t a t e s . This result shows that the intruder states interact strongly with the CASSCF reference functions. A detailed analysis shows that the largest interaction energies are of the order of 0.2-0.4 eV and involve semi-internal excitations where the external orbital is one of the missing valence lr-orbitals. The result is too low excitation energies when e is large enough to make all energy denominators positive. The application of the level shift technique as a substitute for a larger active space thus overestimates the correlation effects arising from excitations to the missing active orbitals. A comparison with experiment shows, however, that the errors are still within the limits expected for the CASPT2 method. These results are presented in Table 6, where the excitation energies obtained with e = 0.3 a.u. have been used (the lowest value, which gives equal reference weights for all states). We notice, however, that e = 0.2 a.u. would give a larger deviation from experiment, even if the weights are acceptable for all states except 2lAg and 41Bu. Included in Table 6 are also the results for the lowest triplet state, 13Bu. This state is not affected at all by the intruder states and the excitation energy is independent of the applied level shift (the variation is 0.07 eV with e varying between 0.0 and 0.5 a.u.). The calculated energy is 0.42 eV larger than the experimental value given in the table, which, however, corresponds to the 0 - 0 transition. A more detailed discussion of the electronic spectrum of stilbene will be given in a separate publication [19]. 3.4. The electronic spectrum of Ni(CO) 4 The electronic spectrum of tetrahedral Ni(CO)4 is built from charge-transfer excitations from the Ni 3d orbitals into the CO 7r*-orbitals. Within the point group Td the 3d orbitals transform as e + t2, while the eight CO ~r*-orbitals belong to the e + tl + t2 representations. A calculation of the full spectrum therefore requires an active space of 13 orbitals - 9t2, 2e with predominant Ni 3d character, and 10t2, 3e, 2tl with predominant CO 7r* character including 10 electrons. This active space encloses six possible singly excited configurations, giving rise to the following

265

singlet states: 9t2 9t2 9t2 2e 2e 2e

--* ---' ---* --, -'* ---.

10t2 2tl 3e 10t 2 2tl 3e

: : : : : :

1A1 IA2 1T1 1T 1 IT1 1A 1

+ + + + + +

1E + 1Tl + 1T2 1E + IT1 + 1T2 1T2 1T 2 1T2 1A2 + 1E

All singlet charge-transfer states were included in the calculations, except for the third 1A1 state, which could not be located and is expected to lie above several doubly excited states. Calculations were performed at the experimental Ni(CO)4 geometry, using the ANO-S type basis sets: (17sl2p9d4f)/[6s4p3dlf] for Ni and (10s6p3d)/[3s2pld] for C,O [20]. The symmetry used in the calculations is D2, although in all cases additional symmetry restrictions were imposed in the CASSCF step to prevent mixing between molecular orbitals belonging to different Td representations. This does not, however, fully prevent symmetry breaking for the degenerate E and T states, since no orbital equivalency restrictions are imposed between different tetrahedral subrepresentations. Table 7 shows how the different Td representations are reduced when lowering the symmetry to D2~ and further to D2. The T1 and T2 states belong to the same D2 representations. However, no matter which of the three components of these states is computed, the actual symmetry preserved during the CASSCF calculation is D2d, and the symmetry of the calculated state is A2 for the T1 and B2 for the T2 states, both corresponding to B 3 in D2. Changing to a different component simply changes the orientation of the Td "* D2d symmetry reduction, by maintaining a different $4 principal axis. The other Td representations A1, A2 and both E components reduce to the same A representation in D2. For the E states symmetry Table 7 Decomposition of the representations of T d relative to its subgroups D2d and D2

Td

D2d

D2

A1 A2 E T1 T2

A1 B1 AI+BI A2 + E B2 + E

A A A+A B3 + B1 + B2 B3 + B1 + B2

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

266

breaking can therefore be prevented by performing a state-averaged CASSCF calculation for its two components. The resulting CASSCF energies are strictly degenerate in all cases. In two cases, the averaging was actually performed over three states, including also the A 2 state belonging to the same configuration: both the A2 and E state being nearly degenerate, an individual orbital optimization for both states refused to converge. Except for these few exceptions, the CASSCF wavefunctions were, in a first set of calculations, obtained using individually optimized orbitals for the different states. Such an approach results in a set of active orbitals which thoroughly change character from one state to another in some cases, due to the presence of a very important 3d-4d correlation effect in Ni(CO)4 (the double shell effect). Thus for the 1A1 ground state the 3e and 10t2 orbitals contain, apart

from COlt* character, a considerable amount of Ni 4d character. The 4d character is also kept to a maximum extent in the calculations on the excited states, and it is only the orbital which actually gets occupied that loses its Ni 4d character and turns into a pure CO,r* orbital. This is the case for all excited states containing either a singly occupied 3e or 10t2 orbital, the CASSCF description of which is therefore slightly worse than for the ground state, since part of the Ni 4d character gets lost. States with a singly occupied 2tl shell are more in balance with the ground state, since in this case the Ni 4d character in the 3e and 10t2 shells is fully preserved. An alternative approach, used in a second set of calculations, consists of performing only one stateaveraged CASSCF calculation per symmetry representation. This is the procedure, which has been used in a very extensive series of CASSCF/CASPT2

Table 8 Effect of the level shift in the zeroth-order Hamiitonian on the non-corrected (AP,) and level shifted corrected (AE Ls) excitation energies (in eV), and CASSCF reference weights (w) of the charge-transfer states in Ni(CO)4. CASPT2 results obtained with CASSCF orbitals, optimized for each state individually

aELS(zXE) IA1, 1A2 states XIA1 E = 0.0 a.u. 0.00 e = 0.1 a.u. 0.00 = 0.2 a.u. 0.00 = 0.3 a.u. 0.00 1T1 states alT1 = 0.0 a.u. 4.42 e = 0.1 a.u. 4.43(4.55) = 0.2 a.u. 4.46(4.65) = 0.3 a.u. 4.49(4.74) IT 2 states alT2 = 0.0 a.u. 4.72 e = 0.1 a.u. 4.73(4.85) e - 0.2 a.u. 4.76(4.96) e = 0.3 a.u. 4.80(5.05) 1E states alE e = 0.0 a.u. 4.48 4.68 e = 0.1 a.u. 3.53(4.29) 4.04(4.26) = 0.2 a.u. 4.14(4.44) 4.14(4.44) e = 0.3 a.u. 4.20(4.57) 4.20(4.57)

~

t~ELS(t~E)

¢o

,XELS(zXE)

~

AELS(,aE)

¢o

aELS(,xE)

o~

.682 .699 .716 .730

blAl 4.41 2.92(4.97) 4.37(4.87) 4.57(5.07)

.380 .459 .672 .699

alA2 5.49 5.53(5.68) 5.58(5.81) 5.63(5.91)

.615 .673 .695 .712

blA2 6.51 5.95(6.14) 6.06(6.28) 6.12(6.38)

.152 .667 .695 .714

.655 .679 .698 .714

biT1 5.10 5.11(5.19) 5.13(5.26) 5.15(5.33)

.666 .686 .703 .719

clT1 5.30 5.31(5.39) 5.33(5.46) 5.35(5.52)

.665 .686 .703 .719

dlTl 5.96 6.00(6.13) 6.05(6.24) 6.09(6.33)

.638 .678 .698 .715

elT1 7.00 7.01(7.10) 7.04(7.18) 7.07(7.25)

.662 .684 .702 .719

.653 .679 .698 .714

biT2 5.45 5.46(5.55) 5.48(5.63) 5.50(5.70)

.663 .684 .702 .718

c1T2 5.75 5.76(5.85) 5.78(5.93) 5.81(6.00)

.661 .684 .702 .718

dlT2 6.20 6.24(6.40) 6.31(6.53) 6.37(6.62)

.623 .671 .696 .714

elT2 6.90 6.93(7.06) 6.97(7.17) 7.02(7.29)

.647 .677 .698 .715

.066 .021 .585 .662 .688 .688 .707 .707

blE 5.52 5.53 5.58(5.74) 5.58(5.74) 5.63(5.87) 5.63(5.87) 5.69(5.98) 5.69(5.98)

.526 .573 .672 .672 .694 .694 .712 .712

clE 5.99 6.03 6.11(6.25) 6.10(6.25) 6.17(6.37) 6.17(6.37) 6.22(6.46) 6.22(6.46)

.572 .615 .674 .674 .697 .697 .715 .715

B.O. Roos et al.Hournal of Molecular Structure (Theochem) 388 (1996) 257-276

267

also includes the weights of the CASSCF reference in the final first-order wavefunctions. The presence of intruder states in the non-shifted CASPT2 results can be most easily recognized by comparing the weights of the different states. Both the ground state and the excited T1, T2 states have weights which are about the same: 0.62-0.68, indicating a balanced set of calculations with no intruder state problem. However, most of the excited states ofA 1, A2 and E symmetry possess significantly lower weights. The origin of the intruder states is different for the different states concerned: for the alE, btA2 states for example they arise from excitations from CO~" orbitals into the active space, while for the blAt state the intruder appearing at the non-shifted level involves excitations out of the active space into a diffuse e orbital with predominantly C 3p character. The problem is most severe for the atE state, both components of which have a weight which has virtually dropped to zero. Furthermore, the weights are different for the different E components, as are the excitation energies, indicating that intruder states affect the two components to a different extent. When applying a level shift, the LS corrected second-order ground state energy gradually increases, although the effect is limited: 0.0018 a.u. for e ~- 0.1 a.u., 0.0065 a.u. for e ~ 0.2 a.u. and 0.013 a.u. for e ~0.3 a.u. The corresponding changes of the uncorrected energy are (of course) larger: 0.045 a.u., 0.086 a.u., and 0.124 a.u., respectively. Similar changes are

calculations on electronic spectra of organic molecules, where it leads to results which are not significantly different from the results obtained with individually optimized CASSCF orbitals [5,6]. In the case of Ni(CO)4 only two CASSCF calculations now have to be performed: one calculation including 10 roots of A1 symmetry (in D2), for the description of the 1mb 1A2 and tE states, and one calculation including 10 roots of B3 symmetry for describing the tTt, 1T2 states. Again, Td symmetry is preserved in the CASSCF calculations of the 1A1, 1A2, ~E states, the two components of the last state being strictly degenerate, while the actual symmetry for the 1T 1, 1T2 states is D2d. A drawback of this second approach is that the Ni 3d-4d correlation effect is no longer included in the CASSCF description since all active orbitals are now used to describe the different electronic transitions to the COTr* orbitals. In the CASPT2 calculations all valence electrons, originating from Ni 3d and C,O 2s,2p were correlated. It was shown [6,21] that inclusion of the Ni semi-core 3s,3p electrons in some cases has an important effect on the excitation energies. However, since correlating the 3s,3p electrons is not expected to affect the intruder state problem to a large extent, they were not included in the calculations presented in this work. The results obtained for the CASPT2 excitation energies with different level shifts, starting from the first series of CASSCF calculations with individually optimized orbitals, are presented in Table 8. The table

ai T2

~" 6.0

......

I 4.0

!

..~ 2.0 0.0

m

0.6

o~ 0.2 0.0 0.0

O.l

0.2

0.3

0.0

0.1

0,2

0.3

0,0

0.1

0,2

0..3

0.0

0.1

0.2

0.3

0.0

0.1

0.2

OJ

0.4

Level Shift (a.u.)

Fig. 2. Plot of the CASPT2 excitation energy of the a 1E, b ~A 1, a 1T 2 and b IA 2 states in Ni(CO)4, as a function of the applied level shift: the level shift corrected excitation energy AE Ls is shown as a full line, the non-corrected energy AE as a broken line. The lower diagram gives the corresponding weight of the CASSCF reference function (full line), and the ground state CASSCF reference weight (broken line).

268

B.O. Roos et aL/Journal of Molecular Structure (Theochem) 388 (1996) 257-276

observed for the 1T b 1T2 excited states, and, as the results in Table 8 indicate, the effect on the LS corrected excitation energies is very small for those cases where no intruder states appear at e = 0. The uncorrected excitation energies (mentioned within parentheses in Table 8) are significantly affected, with a difference of up to 0.4 eV between e = 0.0 and 0.3 a.u., but the changes are to a large extent removed by adding the LS correction to the second-order energy. For most of the ITs, 1T2 states the difference in excitation energy between e = 0.3 a.u. and e = 0.0 a.u. is less than 0.1 eV. Slightly larger changes are found for the states dlT1 (0.13 eV), dlTz (0.17 eV), and elT2 (0.12 eV). We notice however that for those states the weights at e = 0 are significantly lower than for the other T~, T2 states, which may point to the nearpresence of an intruder. The progress of the excitation energies and CASSCF reference weights as a function of the applied level shift was examined in more detail for the states a~E, blA2 and blA1, with severe intruder state problem at e = 0, and for the a~T2 as a reference case with no intruder states at e = 0. The results are shown in Fig. 2. For comparison, the CASSCF weight of the ~A1 ground state has been included in all plots. The ground state weight very smoothly increases with the applied level shift, confirming the absence of any intruder for this state. However, we now notice the presence of an intruder for the alT2 state, manifesting itself by a very sharp decrease of both o: and AE Ls at e -- 0.036 a.u. The orbital causing the intruder state is a diffuse t2 type orbital. Without level shift, the zerothorder energy of the CSF (Configuration State Functions) involving excitations into this orbital is actually lower than the energy of the lT2 state under consideration (negative energy denominator). However, their energy difference is large enough to prevent the intruder state causing a problem at this point. When introducing a level shift, the energy difference first moves to zero at e = 0.036 a.u. and then to positive values. However, apart from the sharp dip in o: and AE Ls at e = 0.036 a.u., both curves are very straight. The o: curve for the ground state and the alT2 state are almost parallel, while the AE Ls curve is almost completely flat, as was already indicated by the results for AE LSincluded in Table 8. Also confirming the results of this table, the curve of the uncorrected excitation energies has a considerably larger slope.

The same observations also hold for the alE and b 1A2 states. Apart from the fact that these states show

intruder states already with e = 0 they essentially behave in a similar way as the alT2 state when a level shift is applied. For both 1E components all intruder states have been removed at e = 0.15 a.u., and from this point both components become strictly degenerate. As for the alT2 state, the AE Ls curves of alE and blA2 become very flat once all intruder states have been removed. The blA1 state represents a more pathological case. The origin of the problem is already apparent from the non-shifted CASPT2 result, which contains a larger number of negative denominators than is observed for any of the other states. Plenty of orbitals are involved: essentially all COa, Tr type inactive orbitals as well as quite a number of diffuse orbitals with C 3p or Ni 4s character. Excitations from and into these orbitals give rise to a combination of intruder states appearing in the range e = 0.0-0.2 a.u., resulting in much broader dips in both the ~ and zIE Ls curves. However, even after all intruder states have been removed at around e = 0.2 a.u., the o~curve is still considerably below the corresponding ground state curve and only approaches it slowly. At the same time, the slope of the corresponding zIE Ls curve is more than four times larger than for the other states. Thus, between e -- 0.2 and 0.3 a.u., the level shift corrected excitation energy of the blA1 state is increased by 0.2 eV, while the corresponding increase for the alT2 state is only 0.03 eV (see Table 8). Apart from the blAl state, it is clear that a balanced set of CASPT2 results is obtained when applying a level shift of 0.2 a.u. All intruder states have disappeared, while at the same time only the (LS corrected) excitation energies have been thoroughly affected for those states where intruders did appear at the nonshifted level. The experimental spectrum of Ni(CO)4 is not well resolved, and a comparison with the calculated results is only possible for the five dipole allowed IA 1 ~ 1T2 transitions. Based on the present results, the three bands appearing at 4.6 eV, 5.4 eV and 6.0 eV in the most recently recorded gas-phase spectrum [22] can be assigned as the transitions to alT2 (calculated at 4.76 eV), b,clT2 (calculated at 5.48 and 5.78 eV), and dlT2 (calculated at 6.31 eV). An even better correspondence with experiment is obtained after including also the Ni semi-core 3s,3p electrons in the CASPT2 correlation treatment [6,21].

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

The results obtained from the second set of CASPT2 calculations, using state-averaged CASSCF orbitals, are presented in Table 9. When looking at the weights obtained without level shift, it is clear at once that the intruder state problem becomes more serious in this case. Except for the ground state, all states are now subject to intruders at the unshifted level. Furthermore, the results also indicate that a larger level shift is now necessary in order to remove all intruder states. For example, looking at the results for the alE state we notice that at e = 0.2 a.u., the two components still have very different (and low) weights, and are still far from being degenerate. At e ,= 0.3 a.u., all intruder states do seem to be shifted away: all states now have comparable weights and the degeneracy of all three IE states has been recovered. Fig. 3 shows the results obtained from a more

269

detailed analysis of the intruder state problem, performed for the 1At ground state and the five 1T2 states. As indicated by its co curve, the absence of Ni 3d-4d correlation in the CASSCF reference does not introduce an intruder state problem for the ground state. The effect of the level shift on the IN corrected second-order ground state energy is only slightly larger than for the calculations with individually optimized orbitals: at e = 0.3 a.u., the ground state energy has now increased by 0.015 a.u. (0.013 a.u. when calculated with optimized orbitals). This does not of course mean that the accuracy to be expected from the CASPT2 calculations does not suffer from deleting the 4d orbitals from the active space. The Ni 3d-4d double shell effect is an important non-dynamic correlation effect, which stems from a strong interaction between the calculated d" state and 3d --~ 4d excited states, even if the 4d orbitals are (now) found with a

Table 9 Effect of the level shift in the zeroth-order Hamiltonian on the level shift corrected excitation energies (AE Ls, in eV) and CASSCF reference weights (w) of the charge-transfer states in Ni(CO)4. CASPT2 results obtained with state-averaged CASSCF orbitals

z:kE~ ~A 1, ~A2 states X1A1 = 0.0 a.u. 0.00 e = 0.1 a.u. 0.00 = 0.2 a.u. 0.00 = 0.3 a.u. 0.00 ~T ~ states a~T1 e = 0.0 a.u. 5.05 e = 0.1 a.u. 4.22 e ~ 0.2 a.u. 4.66 e = 0.3 a.u. 4.85 1T 2 states alT2 = 0.0 a.u. 4.33 e = 0.1 a.u. e = 0.2 a.u. 5.05 = 0.3 a.u. 5.34 ~E states alE = 0.0 a.u. = 0.1 a.u. = 0.2 a.u. e = 0.3 a.u.

4.70 4.71 0.81 1.75 2.96 0.84 5.03 5.03

o~

z:~tEl's

¢o

zkE ts

6o

z:kEts

.665 .684 .702 .717

blA1 6.03 4.73 5.12 5.53

.448 .548 .648 .691

alA2 5.81 3.86 5.90

.559 .095 .551 .700

.439 .597 .663 .689

biT1 5.48 4.81 5.20

.530 .024 .646 .687

clTl 5.45 5.50 5.55 5.61

.616 .665 .688 .707

5.93 3.75 5.52 5.77

.047 .000 .655 .690

biT2 6.26 5.31 5.61

.076 .136 .652 .688

clT2 10.89 5.05 5.62 5.75

.003 .588 .675 .699

dlT2 6.40 3.78 5.68 6.19

.328 .288 .337 .380 .539 .446 .689 .689

blE 5.49 5.55 0.60 5.55 5.84 5.99 5.99

.073 .044 .292 .053 .650 .672 .697 .697

clE 7.00 7.00 4.84 5.01 5.65 5.68 6.14 6.14

.249 .248 .494 .511 .639 .641 .687 .687

¢o

z21,Ets

¢o

.592 .447 .663 .696

e~Tt 6.23 6.92 6.99 7.03

.065 .662 .687 .705

.562 .416 .642 .692

e~T2 6.59 2.21 5.86 6.40

.291 .307 .634 .684

blA2 6,86 5.80 5.89 6.30

.444 .600 .653 .697

dtT{

270

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

~" 6.0

ctT2

el T 2 ~ - - - - ~ ' - ~

4,0 2.0 0.@

0.6 0.4 0.2

o.o

0.0

0.1

0.2

0.3

0.0

0.1

0.2

0.3

0.0

0.1

0.2

0.3

0.0

0.1

0.2

0.3

0.0

0.1

0.2

0.3

0.4

Level Shift (a.u.)

Fig. 3. Plot of the level shift corrected CASPT2 excitation energy ,aE ls of the five IT2 excited charge-transfer states in Ni(CO)4 as a function of the applied level shift. The lower diagram gives the corresponding weight of the CASSCF reference function (full line), and the ground state CASSCF reference weight (broken line).

very high orbital energy in the external space. The 4d orbitals are therefore also not the orbitals involved in the intruder states appearing for the excited Ni(CO)4 states. Instead they are caused by excitations to the lowest-lying external orbitals, with predominantly C 3p or Ni 4s character, as well as by excitations from the highest-lying doubly occupied, CO a and 7r, orbitals into the active space.

Fig. 3 clearly illustrates the severity of the intruder state problem for the excited 1T2 states. Within the e --0.0-0.2 a.u. region, a whole series of intruder states appears for all five states, such that the weights never even come close to the ground state weight, and very low or negative values AE Ls are obtained throughout the region. For e values larger than 0.2 a.u. no more intruder states appear, yet the results are not at all

Table 10 Composition (in %) of the excited state CASSCF wavefunctions in Ni(CO)4, in terms of singly excited configurations: (A) with individually optimized CASSCF orbitals, (B) with state-averaged CASSCF orbitals State

alE blA1 alTi alT2 biT1 clT1 a~A2 blE biT2 clT2 dlT1 blAz c~E dlT2 elT2 elTj

A 9t2

9t2

9t2

2e

2e

2e

B 9t:

9t2

9tz

2e

2e

2e

10t2

2t~

3e

10t2

2tl

3e

10t2

2tl

3e

10tz

2t~

3e

37 30 78 43 11 1

16

91 30 92 92

51

92 94 5 3

86 86 92 94

93 93 93 93 6 16

14 21 14 1 30 2

88 78

1

3 16 18 1 59 66 31 5 55 24 1 19 11

29 51 1 9 53

14 2 76

6 24 3

25 3 22

3 60 4

1 1 59 51

23 22 5

6

38 42 76

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

stable even in the 0.3-0.4 a.u. region. Going from e 0.3 to 0.4 a.u., the LS corrected excitation energy increases by 0.13 eV for alT2, 0.16 eV for biT2, 0.08 eV for c l T 2 , 0.14 eV for dlT2, and as much as 0.21 eV for elT2. Obviously, introducing a level shift is not the final solution for the intruder state problem occurring for the excited states of Ni(CO)4, when performing CASPT2 calculations using state-averaged CASSCF orbitals. One important difference, which explains why the intruder state problem becomes so much more severe when performing CASPT2 calculations using stateaveraged CASSCF orbitals, may be found in the composition of the CASSCF wavefunctions resulting from both approaches, as shown in Table 10. Using fully optimized orbitals, almost all excited states can be characterized by one singly excited configuration. The single configurational character must be at least partly traced back to the fact that in this case the active space serves to describe both the excited states of interest and the Ni 3d-4d double shell effect. Due to the importance of the second effect, the number of CSFs involving a single excitation into either the 3e or 10t2 shell is kept to a minimum in the CASSCF wavefunction, thus allowing a maximum number of weakly occupied orbitals with a large Ni 4d contribution. On the other hand, when state averaging is

271

used the CASSCF wavefunctions of the excited states (in terms of the averaged orbitals) are more complex, showing extensive mixing between different singly excited configurations. As a consequence, the 10 electrons in the active space are now more spread out over the 13 active orbitals, several of which get occupation numbers significantly different from either zero or two. Such a situation results in a diminished energy gap between the inactive-active and active-external orbitals, thus creating an increased possibility for intruder states. We also notice that the blA1 state is the only excited state for which extensive mixing occurs between different excited configurations, even when calculated with fully optimized orbitals. Obviously, the interaction between 9t2 ---, 10t2 and 2e ---* 3e is for this state more important than the 3d-4d correlation effect. Consequently, both 3e and 10t2 shells are now again predominantly COa-* orbitals, with very little Ni 4d character. This explains why more intruders appear for this state than for any of the other excited states, when calculated with individually optimized orbitals. Maybe it should come as no surprise that the LS-CASPT2 excitation energies obtained with state-averaged CASSCF orbitals (for e ~ 0.3 a.u.) do not compare at all well with the first set of results, obtained with fully optimized CASSCF

Table 11 Comparison of the CASPT2 results obtained for the absorption spectrum of Ni(CO)4, using either fully optimized CASSCF orbitals (with e ~ 0.2 a.u.) or state-averaged orbitals (with e * 0.3 a.u.) Final state

Optimized orbitals Transition energy (eV)

alE blA1 alT1 a 1T2 biT1 clTl alA2 blE b 1T2 CiTE d 1T l b 1A2 clE d 1T: e 1Tz e 1T l

4.14 4.37 4.46 4.76 5.13 5.33 5.58 5.63 5.48 5.78 6.05 6.06 6.17 6.31 6.97 7.04

Averaged orbitals Oscillator strength

0.32

0.40 0.30

0.48 0.83

Transition energy (eV) 5.03 5.53 4.85 5.34 5.20 5.61 5.90 5.99 5.61 5.75 5.77 6.30 6.14 6.19 6.40 7.03

Oscillator strength

0.03

0.16 0.15

0.20 2.01

272

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

orbitals (using e = 0.2 a.u.). Both sets of results are compared in Table 11. Although for a few states a very similar excitation energy is obtained, the difference is very large (up to 1.0 eV and more) for most states. The general trend is that the lowest states are calculated at higher energies when using state-averaged CASSCF orbitals, while the highest states are calculated lower. Both the lack of 3d-4d correlation in the CASSCF reference wavefunctions and the instability of the CASPT2 results when applying a level shift should be held responsible for the unreliability of the second set of results, obtained with averaged orbitals. We notice, for example, that the band appearing at 4.6 eV in the gas-phase spectrum [22] is no longer reproduced by these results. Finally, we have also included in Table 11 the results obtained with both approaches for the oscillator strengths of the five allowed 1A1 ----, IT 2 transitions. The transition moments in the expression of the oscillator strength were obtained at the CASSCF level using the CAS State-Interaction method, while for the excitation energies the corresponding CASPT2 AE Ls results were used. As one can see, the two approaches lead to very different results for the oscillator strengths too. The difference can again be traced back to the difference in composition of the CASSCF wavefunctions obtained with the two methods (Table 10). As was shown in a recent study of the electronic spectrum of C r ( C O ) 6 [21], the transition moments obtained for the different charge-transfer states are indeed very sensitive to the interconfigurational mixing in their CASSCF wavefunction. In this respect, the oscillator strengths obtained with fully optimized orbitals should be considered less accurate than the results obtained for the corresponding excitation energies. Indeed, due to the presence of Ni 4d in the active space, the interconfigurational mixing between different charge-transfer configurations may be seriously underestimated at the CASSCF level. 3.5. Cu(Im)2(SH)(SH2) ÷ as a m o d e l for plastocyanin

Cu(Im)z(SH)(SH2) ÷ (Im = imidazole) serves as a simple model compound for the active Cu(II) site in plastocyanin, a typical blue copper protein. The electronic structure of the Cu(II) site in blue copper proteins is unique compared with "normal" square planar small-molecule Cu(II) compounds, due to

presence of a strongly covalent Cu-thiolate bond, in combination with a pseudotetrahedral environment of N and S ligands. In plastocyanin the Cu(II) active site is surrounded by a distorted trigonal plane composed of a cysteine and two histidine amino acids. In addition, an axial S ligand belonging to methionine is present at quite a large distance (about 3 A,). In Cu(Im)2(SH)(SH2) + imidazole serves as a model for histidine, while SH- and SH2 model the side chains of cysteine and methionine respectively. The structure of the model compound was optimized with density functional theory, using the B3LYP (Becke, Lee, Yang and Parr) hybrid functional. The resulting structure very closely resembles the actual geometry of the oxidized Cu(II) site in plastocyanin, indicating that the pseudotetrahedral Cu(II) environment in blue copper proteins is indeed connected with its unique electronic structure, rather than being imposed by the tertiary protein structure [23]. In order to minimize the computational effort necessary for the CASPT2 calculations of the electronic spectrum of Cu(Im)2(SH)(SH2) ÷, symmetry was imposed during the structure optimization. It was shown [24] that this restriction only has a minor effect on the calculated transition energies. Since Cu(Im)2(SH)(SH2) ÷ formally is a d 9 system, part of its electronic spectrum consists of 3d ---* 3d transitions, moving the hole in the 3d shell. However, the intense blue color of plastocyanin is caused by low-lying charge-transfer excitations from the surrounding ligands, in particular the cysteine sulfur atom, into the Cu 3d shell. In order to be able to include as many charge-transfer states as possible it was decided to perform the calculations of the spectrum of Cu(Im)2(SH)(SH2) ÷ using different active spaces for states of different symmetries, A' or A". In both cases 13 electrons were correlated in 12 active orbitals, consisting of five Cu 3d and five correlating Cu 4d orbitals, and, in addition, two ligand orbitals belonging to the symmetry representation of the calculated state. This approach allows for the calculation of the X2A" ground state as well as eight singly excited doublet states: five of A' and three of A" symmetry. Excited states included are the four "ligand field" states (b2A", a-c2A'), corresponding to an alternative d 9 configuration on Cu, and four ligandto-Cu 3d charge-transfer states, with a single electron in the S H r (c2A"), SHtr (d2A'), and in the symmetric

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

and antisymmetric combination of two imidazole 7rorbitals (d2A ", e2A'), respectively. All calculations were performed using state-averaged CASSCF orbitals for the four 2A" roots and five 2A' roots, respectively. The main reason for not considering individually optimized CASSCF orbitals in this case is that such an approach resulted in a very large deviation from orthonormality between the different states. During the CASPT2 calculations all valence electrons, originating from Cu 3d, S 3s,3p, C 2s,2p, N 2s,2p and H ls orbitals, were correlated. The Cu 3s,3p electrons were not included in the series of test calculations performed here. They were included in the final calculations of the spectrum of Cu(Im)E(SH)(SH2) ÷ and several other model compounds for plastocyanin [24]. ANO-S basis sets were used, contracted to [6s4p3dlf] for Cu, [4s3pld] for S, [3s2p] for N and C, [2s] for the H in SH and SH2, and [ls] for the imidazole H [20]. The CASPT2 excitation energies obtained without level shift, and the corresponding CASSCF reference weights, are presented in Table 12. From the low weights obtained for the states aaA ' and cEA" it is obvious that both states suffer from a severe intruder state problem at this level. The description of the other excited states is more in balance with the ground state, although the fluctuations between the weights of the different states are unsatisfactorily high. The effect of applying a level shift was studied in detail for all states considered. The results are presented in Fig. 4, Table 12 Comparison of the level shift corrected CASFF2 excitation energies (AELS), obtained with a level shift of 0.3 a.u., and the unshifted CASPT2 excitation energies of Co(Im)2(SH)(SH2) ÷ no LS State a XEA"(GS) a2A'(LF) baA'(LF) b2A"(LF) cZA'(LF) c2A"(CT) d2A'(CT) d2A"(CT) e2A'(CT)

AE (eV) 0.00 0.32 1.76 2.01 2.02 2.20 2.47 3.84 3.83

E ffi 0.3 a.u. w .608 .046 .548 .624 .636 .356 .590 .604 .594

AE Ls (eV) 0.00 0.64 1.74 1.94 1.91 1.83 2.52 3.90 3.89

~0 .695 .700 .700 .698 .701 .689 .689 .682 .676

a GS = Ground state; LF = ligand field excitation; CT = chargetransfer excitation.

273

showing both the level shift corrected excitation energies AE I-s (upper part) and the CASSCF reference weights (lower part, where the ground state weight was also included) as a function of e. All states, including the ground state, are now found to be the victim of at least one intruder state at some point along the e coordinate. All intruder states can be characterized as ligand-to-metal charge-transfer excitations originating from the highest-lying inactive ligand orbitals of the same symmetry as the calculated state. The number of intruders (within the range of values of the level shift parameter studied) increases with an increasing total energy of the calculated state, with a maximum of two for d2A" and four for e2A '. Denoting the different intruders by the ligand orbital from which the electron is excited, we find from left to right in the d2A" curve an imidazole a- and a--orbital, and in the e 2 A ' c u r v e imidazole a, S H 2 a, imidazole ~', and SH27r. All states concerned give rise to a negative energy denominator in the non-shifted CASPT2 solutions of the highest excited states. For the lower-lying states, the corresponding denominators gradually become positive, such that for the XaA" ground state and the a2A ' state only the right-most excitation from the corresponding list still behaves as an intruder. Apart from the limited number of well-separated intruders appearing as narrow dips in the AE I's and weight curves, both curves behave perfectly well with an increasing level shift: the energy curves are almost completely flat along the whole curve, while the corresponding weight curves almost perfectly coincide with the ground state weight curve at all points where no intruders are present. At e = 0.3 a.u., all intruders have disappeared and a balanced CASPT2 treatment of the spectrum is obtained. The calculated (level shift corrected) CASPT2 excitation energies at = 0.3 a.u. are compared with the non-shifted results in Table 12. Except for the two states, a2A ' and c2A ", which were subject to an intruder at the unshifled level, the effect of the level shift on the calculated excitation energies is very limited: a maximum energy difference of 0.11 eV is found for the c2A ' state. The experimental spectrum of plastocyanin [25] contains three intense bands at 1.60, 2.07 and 2.32 eV, which are responsible for its blue color. The bands can be assigned as the transitions to b2A ", c2A" and d2A ', calculated at 1.94 eV, 1.83 eV and 2.52 eV, respectively. The correspondence is not

274

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

perfect but, as is shown in Ref. [24], can be greatly improved by adding the Cu 3s,3p electrons to the correlation treatment and including relativistic corrections, as well as by substituting the hydrogens in SH2 and SH- by CH3 groups.

systems to which the CASSCF/CASPT2 method is currently applied. The results show that the level shift technique is not a global solution to the intruder state problem in all cases. Instead, it gives the expected but essential result that the level shift technique can and should only be used to remove intruder states, which interact weakly with the reference state. The result is obvious. If the interaction is strong, the intruder state will give an important contribution to the correlation energy, and since it is nearly degenerate with the reference state, perturbation theory cannot be used. The active space has instead to be increased such that the intruder state is moved into the CAS CI space where the interaction is treated variationally. Stilbene and Ni(CO)4 are the two most obvious examples. In stilbene, the two lowest 7r-orbitals give important contributions to the correlation energy and the best solution is to include them in the active space.

4. Conclusions The electronic spectra for some molecules have been studied with the CASSCF/CASPT2 method in order to see to what extent intruder states can be effectively removed by means of the recently introduced level shift technique [1]. Five different systems were chosen for the study: three organic compounds (formamide, adenine, and stilbene) and two transition metal complexes (Ni(CO)4 and a model compound for the blue copper protein plastocyanin). These examples are good representatives for most of the a2A ,

~ 3.0

b2A ,

c2A ,

b~"

2.0

I Ltl

.,~ l.O r~

0.0

0.6

i

0.4 0.2

t

i

0.0

~ 4.0

i

i

i

i

t

i

j~

1

. . . .

d2A ,

c~"

3.0

JL

i

1

i

i

. . . . . .

,"~ ~'~-

d2A ,,

2.0 i 1.0 0.0

p

t

i

n

i

i

i

f 0.6 i

"" ' i............. ~ "7"5"'"~--

~

i!

..•

0.4

.j

0.2 o.o

o.o

o.1

°.5

0.3

[i o.o

0.1

0.5

0.3

0,0

0.1

0.2

0.3

0.0

0.1

0.2

0.3

0.4

Level Shi~ (a.u.) Fig. 4. Plot of the level shift corrected CASPT2 excitation energy z ~ t's of all calculated excited states of Cu(Im)2(SH)(SH2) ÷ as a function of the applied level shift. The lower diagrams gives the corresponding weight of the CASSCF reference function (full line), and the ground state CASSCF reference weight (broken line).

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

The result illustrates a serious bottleneck of the CASSCF/CASPT2 method: the size of the active space. Calculations with 14 electrons in 14 active orbitals cannot be performed today, except for systems with D2h symmetry. The LS-CASPT2 approach makes the study of stilbene feasible, although the results might be afflicted with somewhat larger error bars than is common for the method. A similar drift found for the excitation energies in adenine is due to the same problem. One r-orbital was left inactive in this molecule. Ni(CO)4 illustrates the same dilemma but here in a slightly different context. It is well known from a number of examples [6] that when calculating charge-transfer excitations (or other processes that change the number of 3d electrons), a second d-shell has to be included in the active space for first row transition metals with many 3d electrons. This correlation effect is strong and cannot be treated by perturbation theory. In Ni(CO)4 the solution is to do separate calculations for each excited state. If instead averaged orbitals are used for many CT states, the active space will contain too many ligand orbitals and there will be no room for a second d-shell. As a result the first-order wavefunction will contain, next to a large number of intruder states, a number of states corresponding to 3d ~ 4d excitations. The latter are strongly interacting with the reference states, and cannot be effectively treated by the level shift technique. The Ni(CO)4 example illustrates another bottleneck for the CASSCF/CASPT2 approach: the difficulty of performing separate CASSCF calculations for each excited state. Normally, such calculations will not converge for excited states, especially if the symmetry is low and the density of states is high. Instead stateaveraged calculations have to be performed, which include many states of the same symmetry. Such an approach usually works very well, as a number of examples have shown, but the demands on the size of the active space are again increased. State-averaged calculations could have been used for Ni(CO)4 if the active space could have been extended to include the second d-shell in addition to all the ligand orbitals. This was, however, not possible. The calculations for Cu(Im)2(SH)(SH2) ÷ illustrate a case where stateaveraged calculations were necessary. However, in this case all charge-transfer states included in the calculations are ligand-to-metal, such that the second

275

d-shell remains available for correlation in all states, even when averaged orbitals are used. State-averaged calculations are routinely used in r-electron spectroscopy with good results. The reason is here that the 7rorbitals used to describe different valence excited states are not very different in character. It should also be noted that the CASPT2 method can to a certain extent correct for the use of non-optimal orbitals, since single excitations are included in the CI space used to construct the first-order wave function. The conclusion of this study is thus, that the level shift technique can in many cases be used to effectively remove intruder states in the first-order wave function. It is, however, necessary to ensure the condition that the intruders are only weakly interacting with the reference function. The best way to do this, is to study the stability of computed energies for values of the level shift, where no intruder states appear.

Acknowledgements This paper is dedicated to Jan Alml6f. He was one of the first to set the path for accurate calculations on large molecules. His efforts and his successes have been a great inspiration to all of us, who slowly try to follow along the trail he has marked. The memory of Jan is the memory of a great man, a great scientist, and a dear friend. The research reported in this communication has been supported by a grant from the Swedish Natural Science Research Council (NFR), in Spain by the DGICYT project PB94-0986, and by the Belgian Governement (DPWB). L.S. acknowledges a postdoctoral grant from the DGICYT of the Ministerio de Educaci6n y Ciencia of Spain. V.M. gratefully acknowledges a fellowship from the Generalitat Valenciana. K.P. thanks the Belgian National Science Foundation (NFWO) for a research grant.

References [1] B.O. Roos, K. Andersson, Chem. Phys. Lett., 245 (1995) 215. [2] K. Andersson, P.-.~. Malmqvist, B.O. Roos, A.J. Sadlej and K. Wolinski, J. Phys. Chem., 94 (1990) 5483. [3] K. Andersson, P.-.~. Malmqvist and B.O. Roos, J. Chem. Phys., 96 (1992) 1218.

276

B.O. Roos et al./Journal of Molecular Structure (Theochem) 388 (1996) 257-276

[4] K. Andersson, B.O. Roos, P.-,~. Malmqvist and P.-O. Widmark, Chem. Phys. Lett., 230 (1994) 391. [5] B.O. Roos, M.P. FiJlscher, P.-,~. Malmqvist, M. Merch~in and L. Serrano-AndrEs, Theoretical studies of electronic spectra of organic molecules, in S.R. Langhoff (Ed.), Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy, Kluwer Academic, Dordrecht, The Netherlands, 1995, p. 357. [6] B.O. Roos, K. Andersson, M.P. Ftilscher, P.-,~. Malmquist, L. Serrano-Andr6s, K. Pierloot and M. MercMn, Adv. Chem. Phys., 93 (1996) 219. [7] P.-O. Widmark, P.-,~. Malmqvist and B.O. Roos, Theor. Chim. Acta, 77 (1990) 291. [8] L. Serrano-Andr6s and M.P. F/ilscher, J. Am. Chem. Soc., in press. [9] L.B. Clark, J. Am. Chem. Soc., 117 (1995) 7974. [10] E.B. Nielsen and J.A. Schellman, J. Phys. Chem., 71 (1967) 2297. [11] K. Kaya and S. Nakagura, Theor. Chim. Acta, 7 (1967) 117. [12] H.D. Hunt and W.T. Simpson, J. Am. Chem. Soc., 75 (1953) 4540. [13] D.L. Peterson and W.T. Simpson, J. Am. Chem. Soc., 79 (1957) 2375. [14] M. MercMn and B.O. Roos, Theor. Chim. Acta, 92 (1995) 227. [15] M. MercMn, B.O. Roos, R. McDiarmid and X. Xing, J. Chem. Phys., 104 (1996) 1791.

[16] R.H. Staley, L.B. Harding, W.A. Goddard II1 and J.L. Beauchamp, Chem. Phys. Lett., 36 (1975) 589. [17] M.P. F~ilscher and L. Serrano-Andr6s, B.O. Roos, J. Am. Chem. Soc., submitted. [18] The active spaces will be labeled (abcd), where a, b, c, d are the number of active orbitals in symmetry species ag, au, bg, and bu, respectively, of the C2h point group. [19] V. Molina, M. MercMn and B.O. Roos, J. Chem. Phys., submitted. [20] K. Pierloot, B. Dumez, P.-O. Widmark and B.O. Roos, Theor. Chim. Acta, 90 (1995) 87. [21] K. Pierloot, E. Tsokos and L.G. Vanquickenborne, in preparation. [22] M. Kotzian, N. R6sch, H. Schr6der and M.C. Zerner, J. Am. Chem. Soc., 111 (1989) 7687. [23] U. Ryde, M.H.M. Olsson, K. Pierloot and B.O. Roos, J. Mol. Biol., 261 (1996) 586. [24] K. Pierloot, J.O.A. De Kerpel, U. Ryde and B.O. Roos, j. Am. chem. Soc., in press. [25] A.A. Gewirth and E.I. Solomon, J. Am. Chem. Soc., 110 (1988) 3811. [26] H. Suzuki, Bull. Chem. Soc. Jpn., 33 (1960) 379. [27] M.S. Gudipati, M. Maus, J. Daverkausen and G. Hohlneicher, Chem. Phys., 37 (1995) 47. [28] G. Hohlneicher and B. Dick, J. Photochem., 215 (1984) 231. [29] T. Ikeyama and T. Azumi, J. Phys. Chem., 89 (1985) 5332.