The origin of differences between coupled cluster theory and quadratic configuration interaction for excited states

Volume 2 18, number 1,2

CHEMICAL PHYSICS LETTERS

4 February 1994

The origin of differences between coupled cluster theory and quadratic configuration interaction for excited states Rudolph

J. Rico a, Timothy

J. Lee b and Martin Head-Gordon

a

a Department of Chemistry, University of California, Berkeley, CA 94720, USA b NASA Ames Research Center, Moffett Field, CA 94035, USA Received 28 September 1993; in final fonu 22 November 1993

By employing time-dependent linear response theory, the ground state quadratic contiguration interaction method with single and double substitutions (QCISD) can be generalized to excited states. However, the excited state QCISD method is numerically inferior to the analogous coupled cluster (CCSD) excited state method, although its formal properties of size consistency and exactness for two electrons are the same. Therefore a modified QCISD method which includes an additional quadratic operator term in the doubles equations, as in the CPMET-C approximation, is investigated. It is shown by calculations on a variety of small molecules to correct substantially the deficiencies of QCISD for excited states, yielding results that are close to full CCSD theory. This CPMET-C based excited state method is size-consistent, but unlike CCSD and QCISD, is not exact for separated pairs of electrons.

1. Introduction Multireference configuration interaction (MR-CI ) methods are the most widely used procedures for accurate calculations of electronic excited states (for recent reviews of the MCSCF approach to excited states, see for example ref. [ 1 ] ). This is principally because of the tremendous flexibility that reference configuration selection permits. However, the same flexibility can make MR methods challenging to apply: some chemical insight, not always evident a priori, is helpful in choosing appropriate reference configurations. We focus instead on the alternative approach of single reference (SR) excited state theories. Now the strengths are reversed. As no configuration selection is performed, SR methods are unambiguous to apply, but they may not have the flexibility to describe a wide range of excited states with uniform accuracy. For example, a method based on single orbital replacements #l cannot describe ex-

*’ Single excitation CI is the basis of pi electron PPP theory, ref. [ 21. Subsequently it has been quite widely applied in ab initio calculations, for example ref. [ 3 1.

cited states which are principally double excitation in character. We are interested in single reference excited state methods which include all single and double substitutions from the SR Hartree-Fock (HF) determinant. For ground states, the coupled cluster method with single and double substitutions (CCSD) [ 4,5 ] and the quadratic configuration interaction method (QCISD) [ 6,7 ] are successful theories of this type. Both are size-consistent and exact for two electrons, although QCISD does not have a well-defined wavefunction [ 6,7]. For excited states, generalizations of CCSD theory have been available for some time [ 8201. Equivalent equations can be obtained by timedependent linear response [ 8,19,20 1, or equations of motion (EOM) methods [ 9- 15 1, or by the symmetry-adapted cluster approach ( SAC-C1) [ 16- 18 1. Very promising applications have been reported [ 14,15,18-201, where the level of agreement with either experiment or exact calculation (full CI) is often as good as 0.1 eV for excited states that are primarily 1 electron excitations from the ground state. The ground state QCISD method, while having the same desirable properties as the CCSD method, is algebraically much simpler (although the computa-

0009-2614/94/S 07.00 0 1994 Elsevier Science B.V. All rights reserved. SSDI OOOOS-2614(93)El398-Z

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tional complexity of the two methods is virtually identical [ 2 I]: the simplicity of QCISD relative to CCSD involves the omission of terms which are not rate-limiting). It is of interest to investigate generalizations of QCISD to excited states. This has recently been done via the equations of motion approach [ 15 1, and by ourselves using linear response theory [ 201. These two approaches yield the same excited state equations, which are exact for two electrons and size-consistent, like the excited state CCSD method. Additionally there is a close and beautiful relationship to QCISD gradient theory, again akin to the CCSD method [ 8,15 1. However, the numerical performance of QCISD was markedly poorer than CCSD for some simple excited state problems. This was exemplified by the excited states of H20, which exhibited systematic errors of roughly 0.5 eV with QCISD, relative to only about 0.1 eV for CCSD [ 201. The first purpose of this Letter is to investigate the origin of the sometimes poor performance of the QCISD excited state theory. In particular, we propose that its main deficiencies can be remedied with the addition of a single quadratic operator term, T1T2, to the QCISD doubles equations. If so, then it is possible to correct the drawbacks of QCISD without proceeding to the full CCSD treatment. A second related objective is to take a step towards understanding the relative importance of the different terms which enter CCSD excited state theory. Such an understanding is a necessary prerequisite for developing effective perturbative approximations to the iterative response equations at the CCSD level (and possibly higher). If the substantial errors incurred in excited state QCISD are remedied by a single missing CCSD operator product then such a term is likely to be significant in any perturbation analysis. In section 2, the QCISD and CCSD excited state equations are reviewed, and the modified QCISD equations are motivated and presented. One motivation can be given immediately, which is that SACCI approximations to CCSD excited state theory include the leading part of the TITz term we shall add, and do not exhibit large errors for excited states of HZ0 [ 22 1. However SAC-C1 [ 16- 18 ] does not represent true linear response of the ground state except when carried to full CCSD. For the ground state, the modified equations we propose reduce to the existing CPMET-C approximation [ 23 1, and this method 140

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1994

is hence a CPMET-C based excited state theory. In section 3, this modified excited state theory is tested against QCISD and CCSD in a series of calculations on molecules ranging up to butadiene in size. We summarize our conclusions in the final section.

2. Theory In time-dependent linear response theory [ 241, we examine the first-order response of the ground state wavefunction to a time-dependent perturbation consisting of a single Fourier component of frequency o. As w is varied, one will occasionally find frequencies at which the first-order response diverges, which we identify as the Bohr frequencies of the molecule for transitions from the ground to excited states. We shall take the original CI interpretation of QCISD (although we could just as well begin from full CCSD theory and discard terms), and begin with a CISD-like wavefunction in intermediate normalization, Y(h, t)=exp[

-iE(h,

t)]

x[l+T,(h,t)+T,(h,t)l~~.

(1)

Ir is a formal parameter that we shall use to group reponses of different order (ho will simply yield the time-independent unperturbed problem). a0 is the HF single determinant, composed of orbitals which we treat as time-independent. Tl (h, t) and T,(h, t) are the complete single and double substitution operators, T, (h, t) = 1 a?(& ia

t)e ,

(2a)

(2b) which contain the independent variables: the individual single and double substitution amplitudes a:(& t) and azb(h, t) (as usual, the indices i, j, kand a, b, c denote spin orbitals which are respectively occupied and unoccupied in ao). We begin from the time-dependent Schrijdinger equation (in atomic units) : HY(h, t)=i\k(h,

t) .

(3)

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Suffkient conditions to determine all a4(h, t) and a$(h, t) are obtained by projecting against eq. (3) with the (time-independent) singly and doubly substituted determinants 0; and @$‘, while information about the energy (phase) follows from projection with aO. As we are interested in the wavefunction responses we eliminate the phase to leave:

(4a)

=ici~b+(cPoIHIT,Q>o)a~b.

(4b)

As usual, R=H-Em, where the Hartree-Fock energy is ( a0 I H I mo). Note that the left-hand side of eqs. (4) is not simply the projection of HY(h, t) in the space of single and double substitutions: that would be CISD. Instead, following the QCISD prescription [ 61, we have added a quadratic operator product, T, T2 to the singles, and 4 T2T2 to the doubles equations. The so-called disconnected part of the matrix elements of these operators in eqs. (4) cancels the second term on the right, so that we have

4 February 1994

T,(h, t) =T, +hU,e-iw’+...

,

0)

T,(h, t) =Tz +hUze-iw’+...

.

(7b)

Substituting eqs. (6) and (7) into eqs. (5) and separating the terms zeroth order in h yields the ground state QCISD equations. This involves replacing T,, T,, and the associated u by their time-independent zeroth-order values, which also zeros the right-hand side. Collecting terms first-order in 1 yields the following conditions for QCISD transition energies:

(~,~I~I(U,+U~+T,UZ+U,T~)~~)~ =wb,“, (~~bIRI(U1+U2+TzUz)~o)c=w~~.

(gal (8b)

The first-order expansion of the left-hand side for the time-dependent problem applies equally at w = 0, which will be the response to a static perturbation as is required in QCISD gradient theory [ 271. Coupled cluster theory for excited states [ 8-201 can be obtained analogously, beginning from the time-dependent exponential wavefunction

(5a)

xev[TI(L O+T2(L ON%, . ‘ab =lUij

.

(5b)

The subscript C indicates that only the connected part of the matrix elements is retained [25,26]. We expand the time-dependent quantities, a:(%, t) and uzb(h, t), in powers of h, anticipating free-oscillation solutions for the first-order amplitude by including e-‘O’ explicitly. Our labels for timeindependent first-order response amplitudes (b) are incremented by a letter relative to zeroth order (a), and we reserve italics for the general time-dependent variables roman for time-independent (i.e. quantities ) , a?@, t) =a; +hbpe-‘“I+... cQb(h, t) =a$b+hb$be-i”‘+...

,

(6a) .

(6b)

Substituting eqs. (6a) and (6b) for the perturbed amplitudes into eq. ( 3 ) , we define expansions of the complete substitution operators in analogous notation as

(9)

which permits retention of only connected terms in the operator matrix element expressions because of the linked cluster theorem. The general projection equations can again be written in fully connected form (note that in our previous paper [ 201, we chose to write these equations in a disconnected form) (~fIRI(T,+T,+tT:+T,T,+aT:)~o)c =i@

(loa)

(@$bIRI(1+TI+Tz+tT;+T,T2+;T: +tT:+tT:Tz+~T~)~,,)c=iri~b.

(lob)

The ground state equations are obtained as the zeroth-order terms of eqs. ( 10 ) , as described above for QCISD: the time-independent T and a replace the time-dependent T and a, which zeros the right-hand side. The first-order terms are collected to obtain the eigenvalue equations that define CCSD excitation energy conditions [ 8-201,

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(@~]Rl(U,+Uz+T,U,+T,Uz+UITz

(@cb In] (U, +Uz +TiUz +UiTz +T2U+&,)c =obf

+TJJ2+fT:U2+T1T2UI+~T:U,)~o)c = wbzb .

(lib)

The spin-orbital and spatial orbital form of this contraction is already available in the literature (for example, see ref. [ 28 ] ), since the left-hand side also will determine the response of the CCSD amplitudes to a time-independent perturbation, such as a nuclear response of the CCSD amplitudes to a time-independent perturbation, such as a nuclear displacement. We propose that a key difference between eqs. (8) for QCISD excitation energies, and eqs. ( 11) for CCSD excitation energies may be the absence of the T, T2 term in eq. (Sb). Inclusion of TIT, in the doubles equations gives rise to a consistent set of connected projection equations, although unlike before we must simply discard the disconnected part of T, T2 if we are to retain size-consistency,

. ‘ab =lUij

.

(12b)

For the ground state, where doubles (T,) are generally more important than singles (T, ), T,TI should have only a small effect on the amplitudes and energy. For the ground state, these equations comprise the CPMET-C approximation [ 23 1. For consistency with the form of the projection equations, ( 12), we choose to calculate the energy via the QCISD expression, rather than the CCSD expression originally used [ 23 1. This choice affects neither the ground state amplitudes nor the transition amplitudes and excitation energies. The crucial difference relative to the QCISD occurs at first order, where the excitation energy conditions are now

(~~IRI(U,+U~+T,U~+U~T~)~~)C =obi”, 142

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(13a)

.

(13b)

The UITz term in eq. ( 13b) will be a significant coupling term for one-electron excited states where singles (U, ) are dominant. Thus its inclusion may significantly affect the doubles amplitudes and excitation energies in these cases, which is precisely where QCISD appears to perform poorly [ 201. The possible importance of the additional terms in the excited state doubles equation ( 13b) also follows from the fact that they are included in the various quite successful1 SAC-C1 approximations [ 1618,221. But for these theories, the important relationship between time-independent ground state responses and the transition energies is lost, whereas it is preserved in eqs. ( 12) and ( 13). All three excited state methods, QCISD, CPMETC and CCSD, are size-consistent by virtue of the connectedness of the defining equations. We previously employed this property to explicitly prove sizeconsistency for QCISD [ 201, and the arguments are equally valid for other connected sets of equations. Additionally, the QCISD and CCSD equations are exact for two-electron systems, as follows immediately from the presentation in ref. [ 20 1. However CPMET-C is not, since we have added only the connected part of TIT2 [26] to eq. (12b), and this is non-zero (and opposite in sign to the “disconnected” part) for a two-electron system. This destroys the correctness of eqs. ( 12) for such systems. It does not appear possible to recover exactness for two electrons without putting back virtually all of the operator terms that constitute the CCSD method.

3. Applications

The CPMET-C ground and excited state equations have been implemented as extensions to the Titan programs #‘, as we have recently done for CCSD and QCISD [ 201. This implementation is currently restricted to closed shell ground states, and singlet excited states. It exploits the fact that the rate x2 Titan is a set of electronic structure Lee, A.P. Rendell and J.E. Rice.

programswritten by T.J.

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determining step of a Davidson-like diagonalization for the excited states [29,301is the same as iterative solution of the z vector equations [ 3 1] in ground state gradient theory (e.g. see. ref. [28] for CCSD). All calculations reported here employ this program. Consistency of the resulting programs for the ground state amplitudes and transition amplitudes was ensured by verifying that the CPMET-C analytical energy gradient was constructed correctly (since as noted above the same contractions are performed). We wish to assess whether the addition of the T,Tz operator to the QCISD doubles equations corrects the problems for excitation energies that were first noted in ref. [ 201. Accordingly we have performed a series of vertical excitation energy calculations using the QCISD method, QCISD with the TIT,correction (i.e. CPMET-C excited states), and CCSD. In table 1, we add CPMET-C results to a series of calculations we have reported previously [20], for which full configuration interaction results (FCI) are available [ 22,32-351. There are slight numerical differences for Be relative to ref. [ 201, where all or-

bitals were included, with 6d functions for CCSD and 5d for QCISD. Here we employ exactly the FCI orbital basis (5d functions with the highest s virtual deleted). This yields agreement with EOM-QCISD calculations on Be using the FCI basis [ 15 1, consistent with that being exactly the same as our linear response derived QCISD. With the exception of HzO, reasonable agreement was obtained between QCISD and CCSD for the molecules in table 1. By comparison with FCI, it is evident from table 1 that CPMET-C excitation energies are of comparable quality to CCSD for all species considered, including H20. While CCSD and also CPMET-C are of excellent quality for all one-electron excitations, all three single-double substitution methods show substantial errors of 0.5 to 1.4 eV for excited states which are primarily two-electron excitations from the ground state. Despite the fact that CPMET-C is not exact for pairs of separated electrons, it is evident that the extent of degradation of the results relative to CCSD is very slight indeed (usually less than 0.1 eV). For

Table 1 A comparison of excitation energies for singlet excited states, determined by the approximate QCISD, CPMET-C and CCSD single reference excited state methods, and by full configuration interaction. All excitation energies are in eV Molecule

Basis

Ref. *)

State b,

QCISD

Be Be Be Be

[ 9s9pSd] [ 9s9p5d] [ 9s9p5d] [ 9s9p5d]

1321 [321 1321 1321

‘P (le) ‘S (le) ‘D (2e) ‘P (le)

5.326 6.780 7.140 7.473

CH+ CH+ CH+ CH+

[4s3~12sl 14~2~12~1 14~2~I2~1 [4s2~12si

1331 1331 1331 1331

‘II (le) ‘A (2e) ‘E+ (2e) *Z+ (le)

CHz (rc) CH, (170”) CH2(1.5r,)

[4s2pldl2slp] [4s2pldl2slp] [4s2pldl2slp]

[341 1341 1341

SiHz

[6s4pld]2slp]

CPMET-C

CCSD

FCI ‘)

5.351 6.850 7.161 7.504

5.323 6.773 7.139 7.468

5.318 6.765 7.089 7.462

3.239 7.878 9.096 13.535

3.253 7.890 9.111 13.568

3.261 7.889 9.109 13.581

3.230 6.964 8.549 13.525

‘Ai (2e) ‘Ai (2e) ‘Ai (2e)

6.006 1.684 7.055

6.004 1.682 7.120

6.001 1.682 7.123

4.596 1.069 4.450

[341

‘Ai (2e)

6.504

6.507

6.499

5.332

1221 1221 [221 [221

‘Bi (le) ‘A2 (le) ‘Ai (lc) ‘Bz (le)

8.129 10.282 10.323 12.826

8.468 10.609 10.618 13.100

8.581 10.716 10.737 13.208

8.696 10.798 10.840 13.272

‘) For details of the molecular geometry, and full specification of the basis sets, see the references indicated. These references are also the source of the FCI results. All electrons are correlated for Be, CH+ and HsO, while for CHs and SiHr the core is frozen. 5d orbitals are employed for Be, and the highest s orbital is also frozen as in FCI [ 32 1, while previously for Be we did not freeze the virtual [ 2 I]. ‘) Spectroscopic designation of the excited state, and whether the dominant promotion from the ground state is one ( le) or two (2e) electron in character. 143

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HzO, CPMET-C corrects the principal deficiencies of QCISD, as we anticipated it might. About 80% of the difference between QCISD and CCSD is recovered by CPMET-C, which indicates that for this system at least, we have isolated the principal origin of the difference. Therefore although QCISD has the formal advantage of being both exact for pairs of electrons and size-consistent, it is not a well-balanced description of electron correlation in the ground versus excited states. In table 2, we present a second series of calculations on several low-lying singlet states of H20, H&O, C2H4, O3 and trans-C,Hb. Ozone is a highly

4 February 1994

correlated molecule [ 6 1, while the organic systems are also considered challenging for single reference theories [ 18,361. It is beyond the scope of this Letter to review the excited state literature on these molecules except to note that CCSD (or the SAC-C1 variant) calculations have been performed previously for HZ0 [ 18,191, H&Z0 [ 171, CzH4 [ 181, O3 [ 151 and C4H6 [ 18 1. Our calculations employ the polarized POLI basis [ 351, which is a [ 5s3p2d] basis on first row atoms, and [ 3s2p] for hydrogen. We have used Cartesian 6d functions. The POLl basis does not saturate the one-particle space [ 351, and therefore errors due to basis set incompleteness will gen-

Table 2 Excitation energies (in eV) of selected lower singlet excited states of water, formaldehyde, ethylene, ozone and tram-butadiene using the Sadlej POLl basis [ 35 ] Molecule a)

State

CIS

QCISD

CPMET-C

CCSD

Exp. ‘)

IIU,(CCSD)II =)

Hz0 Hz0 Hz0 Hz0

1 ‘B, 1 ‘AZ 2 ‘A, 1 ‘B2

8.675 10.344 10.987 12.649

6.733 8.468 9.159 10.878

7.237 8.992 9.659 11.400

7.375 9.134 9.808 11.555

7.4 9.1 9.7

0.871 0.868 0.872 0.869

CHrO CHxO CHIO CH>O CHrO

1 ‘A2 1 ‘B2 2 ‘B2 2 ‘A, 3 ‘A,

4.539 8.604 9.382 9.515 9.715

3.460 6.093 7.054 7.095 8.874

3.869 6.845 7.736 7.837 9.336

3.979 7.007 7.876 7.996 9.482

4.07 7.11 7.97 8.14

0.861 0.816 0.827 0.816 0.849

CxH4 CzH,

193,

7.149 7.705 7.762 7.915

7.095 7.812 7.776 7.795

7.359 8.023 8.035 8.071

7.303 8.006 7.981 8.017

7.15 7.66 7.83 8.0

0.890 0.892 0.910 0.883

I ‘A2 1 ‘B, 2 ‘A, 1 ‘B1

2.687 2.191 5.108

1.573 1.714 10.301 4.820

2.457 2.537 9.822 5.246

2.298 2.363 9.852 5.389

1.92 2.1 3.5 4.86

0.764 0.787 0.047 0.761

GH,

1 ‘B, 1 ‘B, 1 ‘A. 2 ‘A,

GH,

2’4

6.198 6.226 6.515 6.710 7.332

6.049 6.059 6.332 6.521 6.660

6.362 6.389 6.652 6.840 7.122

6.289 6.375 6.580 6.765 7.034

5.92 6.27 6.66 6.80 7.08

0.858 0.868 0.856 0.854 0.760

1 ‘B,u

C2H.

1 ‘B,,

C2H4

1 ‘B,

03 03 03 03

GHs G& GH,

a) Molecular geometries are taken from previous excited state calculations, as given below, together with the SCF energies (in hartree). Hz0 [20]: OH=0.9584 A, HOH=l04.45”, EHF=-76.054459. H2C0 [3]: CO=l.2122 A, CH=l.l044 A, HCO=121.94”, O3 [15]: OO= 1.272 A, Em=-113.901256. C2H, [3]: CC=l.338 A, CH=l.085 A, HCC=l21.41”, Em=-78.048070. ooO= 116.80”, Em= -224.328927. C,H, [36] (H2C&HC&IC.H2): C.C,= 1.343 A, C,C,= 1.467 A, CH= 1.094A, HC,C,,= 119.5”, C&C,= 122.8”, Errr= - 154.951176. Correlated calculations are performed with the core orbitals frozen for butadiene, and all orbit& correlated in the other cases. w References from which experimental excitation energies were obtained: Hz0 [ 19 1,H&O [ 3 1,C2H4 [ 31, O1 [ 15 ] and C,Hs [ 361. c, l]U, (CCSD) 11 is the norm of the singles part of the CCSD response function amplitudes. The singles and doubles transition amplitudes together are the eigenvectors of eq. ( 11) and are chosen to have unit norm. Thus ]]Ur(CCSD) IIapproximately measures the extent of one-electron character of an electronic transition, on a scale from 0 to 1.

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erally be comparable or larger than errors due to differences between the CCSD excited state method and full CI. However this will not be the case for less accurate excited state methods, and therefore comparison of the different theories against experiment at this level is useful. In addition to the QCISD CPMET-C and CCSD methods, we also report excitation energies obtained with the simple one-electron configuration interaction with single substitutions (CIS) method (see footnote 1). The deficiencies of the QCISD method for excitation energies are clearly evident again for all molecules considered in table 2, with the largest difference relative to CCSD being about 0.9 eV for several states of formaldehyde. As we observed previously, the QCISD-CCSD differences are somewhat systematic for a given molecule, but vary considerably from molecule to molecule. By contrast, in all cases the agreement between CPMET-C and CCSD is very satisfactory. In particular the RMS deviation between QCISD and CCSD excitation energies is 0.55 eV while the rms deviation between CPMET-C and CCSD is only 0.12 eV. On this small but diverse range of molecules and states, the T, T2term in the doubles is therefore primarily responsible for the differences between QCISD and CCSD in all cases. One significant failure of all methods including CCSD is evident in table 2: the lowest ‘A, excited state of ozone. This state can be characterized as a two-electron excitation from the ground state [ 37 1, and thus cannot occur in CIS. Within coupled cluster theory, it is not possible to properly obtain correlation contributions to the energy of a state of this type without including connected triplet substitutions. For example, crucial determinants for excited state correlation are double promotions from the primary doubly excited configuration, which will lead to double, triple and quadruple substitutions. The quadruples represent correlation of electron pairs not involved in the excitation, which can probably be represented by disconnected operator products of the U2T2ao form, but triple substitutions necessarily require connected IJ@,, terms as they involve electrons which participate directly in the electronic transition. This, rather than near-degeneracy effects, is the main reason that the lowest *Ai CCSD excitation energy deviates from experiment (and MR-CI calculations [ 371) so dramatically. It will be a chal-

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lenging test case for future single reference excited state theories with connected triple substitutions. To summarize, if we exclude the two-electron excitation of ozone discussed above, the RMS deviation between CCSD and experiment is only 0.2 1 eV. CPMET-C is nearly comparable with an RMS deviation of 0.27 eV. These results contrast with the much larger 0.80 eV deviation for the essentially uncorrelated one-electron CIS method, and 0.5 3 eV for QCISD. Consistent with other calculations [ 14-22 1, this illustrates that CCSD treats correlation effects in both the ground state and the electronic excited states in a well-balanced way, provided that the excited states are primarily one-electron excitations.

4. Conclusions The results discussed above show that the deticiencies of QCISD-based linear response theory for excited states are largely remedied with the addition of the connected part of the TIT2operator to the doubles equations. This is a somewhat surprising result, because the addition of this extra term has destroyed one of the desirable features of QCISD, namely correctness for separated pairs of electrons. For this reason, we do not necessarily advocate the use of the CPMET-C based theory for routine excited state calculations. It is probably preferable to employ full CCSD theory, which is correct for two electrons, and numerically very satisfactory. The main value of the results is that they demonstrate that the principal reason for the poor performance of QCISD for absolute excitation energies can be viewed as the absence of this term. We speculate that as perturbation expansions of the excitation energies akin to ground state Moller-Plesset theory become available (for a step in this direction, see ref. [ 38]), this term will prove to be of relatively low order. The results presented here have possible implications for ground state calculations, where the differences between the QCISD and CCSD methods are usually well below the level of chemical significance, and well below the residual errors in both theories compared to full configuration interaction. First, the fact that the poles of the QCISD frequency-dependent polarizabilities (i.e. the QCISD excitation ener145

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CHEMICAL PHYSICS LETTERS

gies) are sometimes significantly shifted relative to CCSD and full CI may mean that QCISD frequency dependent polarizabilities calculated off-resonance may be poorer than those obtained from CCSD. Second, in cases where Tr is unusually large (as measured for example by the ‘TV diagnostic [ 39]), some differences between QCISD and CCSD may ultimately emerge, which could be correct by the procedure discussed here. However, we emphasize that for the calculation of ground state energy and properties, QCISD remains an extremely effective method with accuracies exactly comparable to the CCSD method, as has been extensively documented. Acknowledgement RJR is the recipient of a Graduate Fellowship from the Office of Naval Research. This work was partly supported by the National Science Foundation (CHE-9357 129)) and was also supported by a grant from the UC Berkeley Committee on Research.

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