Calculation of frequency-dependent polarizabilities using general coupled-cluster models

Journal of Molecular Structure: THEOCHEM 768 (2006) 71–77 www.elsevier.com/locate/theochem

Calculation of frequency-dependent polarizabilities using general coupled-cluster models Miha´ly Ka´llay a, Ju¨rgen Gauss b,* a

Department of Physical Chemistry, Budapest University of Technology and Economics, P.O. Box 91, H-1521 Budapest, Hungary b Institut fu¨r Physikalische Chemie, Universita¨t Mainz, Jakob-Welder Weg 11, D-55099 Mainz, Germany Received 2 May 2006; accepted 3 May 2006 Available online 22 May 2006 Dedicated to Professor Debashis Mukherjee on the occasion of his 60th birthday.

Abstract An analytic scheme for the calculation of frequency-dependent polarizabilities within a response-theory approach has been implemented for the use within general coupled-cluster (CC) models with arbitrary excitations in the cluster operator. Calculations for CHC and CN demonstrate the fast convergence of the coupled-cluster approach when successively higher excitations are considered. Quadruple excitation effects on the frequency-dependent polarizabilities are found to be rather small except close to the poles. q 2006 Elsevier B.V. All rights reserved. Keywords: Coupled-cluster theory; Response theory; Analytic derivative theory; Frequency-dependent polarizabilities

1. Introduction Within coupled-cluster (CC) theory [1], frequency-dependent properties such as dynamical polarizabilities and hyperpolarizabilities can be computed using a responsetheory ansatz [2]. Monkhorst and Dalgaard [3,4] were the first to derive corresponding expressions in 1974, although a detailed theoretical derivation of CC response theory was carried out only much later by Koch and Jørgensen [5] and by Christiansen et al. [6]. A first implementation for the calculation of frequency-dependent polarizabilities was reported by Kobayashi et al. [7] in 1994 within the coupled-cluster singles and doubles (CCSD) approximation. Implementations for the computation of first and second hyperpolarizabilities, again within the CCSD approximation, were reported in 1997 and 1998 by Ha¨ttig et al. [8,9]. As CCSD calculations often do not provide results of sufficient accuracy, there has been a lot of interest to include higher excitations in the CC response-theory treatment. As the full inclusion of triple excitations via the CC singles, doubles, triples (CCSDT) model [10] is rather demanding, * Corresponding author. Tel.: C49 6131 3923736. E-mail addresses: [email protected] (M. Ka´llay), [email protected] (J. Gauss).

0166-1280/$ - see front matter q 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2006.05.021

corresponding implementations have been reported for CC schemes with an approximate treatment of triple excitations. As the CCSD(T) approach [11] with a perturbative treatment of connected triple excitations on top of a CCSD calculation turned out to be not suited for a CC response-theory treatment [6], iterative approximations to CCSDT have to be employed for this purpose. However, as the available CCSDT-n schemes [12] are based on perturbation-theory arguments that are not compatible with CC response theory, Christiansen et al. decided to design a new hierarchy of approximate CC models consisting of CC singles (CCS), CC2, CCSD, CC3, CCSDT, etc. [13,14] for the calculation of dynamical properties as well as excitation energies. The CC3 model [14] is within this hierarchy the first approach that includes triple excitations, though, unlike CCSDT, in an approximate manner and with a reduced computational scaling. Expressions for the CC3 response functions were consequently derived [15] and implementations were reported for dynamical polarizabilities [16,17] and hyperpolarizabilities [18,19]. The existing CC response-theory implementations have been all obtained in a traditional way, i.e. using hand-written computer codes based on formulas derived in a conventional manner. However, in recent years it has been realized that the use of automatized tools significantly facilitates the implementation of general CC methods with arbitrary excitations. First attempts in this direction were based on simple modifications of existing full configuration-interaction (FCI) programs

72

M. Ka´llay, J. Gauss / Journal of Molecular Structure: THEOCHEM 768 (2006) 71–77

[20–22], but later schemes which ensure the proper computational scaling of the implemented procedures were developed as well [23–26]. In particular, the combination of many-body and string-based techniques [23,24] has been proven efficient and not only allowed the implementation of general CC methods for energy calculations but also schemes for the analytic evaluation of the corresponding first and second derivatives of the energy [27,28]. More recently, such techniques have also been employed for the determination of excitation energies (within a linear response treatment) using general CC models [29] and to implement approximate schemes for the treatment of higher excitations [30]. As an alternative one might consider schemes, where computer programs are used to produce the actual computer code. This strategy has been mainly pursued within the tensor contraction engine (TCE) project by Hirata and others [25,31,32] and provided so far CC codes up to CCSDTQ. The capability to include higher excitations in CC calculations has turned out essential in improving the accuracy of quantum-chemical calculations beyond what is usually provided by CC methods restricted to triple excitations [33–38]. In the present work, we will extend the developed analytic techniques for the calculation of first and second-order static properties [27,28] to frequency-dependent properties. We will demonstrate that in this way a general CC implementation for these properties is possible as well. To be more specific, we will report in this paper for the first time the calculation of frequencydependent polarizabilities using general CC models. We thus extend existing implementations in two ways, namely, we report for the first time corresponding calculations at the CCSDT, CCSDTQ, etc. levels and we present for the first time calculations for high-spin open-shell systems, as all previous implementations have been restricted to closed-shell wave functions. Our implementation is also capable to provide frequency-dependent polarizabilities for general CI models up to FCI. We note that implementations for the latter have already been presented in the literature by Olsen et al. [39] in 1989 and by Koch and Harrison [40] in 1991. 2. Theory

T Z T1 C T2 C T3 C . C Tn ; Tm Z

1 X X abc. C C tijk. a ib j. m!2 i;j;k;. a;b;c;.

2.1. Coupled-cluster theory In CC theory, the wave function is parameterized via the following exponential ansatz (1)

(2) (3)

with n as the number of electrons. The individual excitations in Eq. (3) are given in a second-quantized form with the abc. amplitudes tijk. as the wave-function parameters to be determined. As usual, indices i,j,k. refer to occupied spin orbitals, while indices a,b,c. denote virtual spin orbitals. Equations for energy as well as the unknown amplitudes are obtained from the Schro¨dinger equation by multiplication with exp(KT) from the left and projection. Projection onto the reference determinant leads to an expression for the energy E Z h0jexpðKTÞH expðTÞj0i;

(4)

while projection onto excited determinants jFpi yields the so-called amplitude equations 0 Z hFp jexpðKTÞH expðTÞj0i

(5)

that need to be solved in order to determine the CC wave function. 2.2. Analytic derivative theory Due to the non-variational character of CC theory, a suitable starting point for the discussion of analytic energy derivatives in CC theory is the following energy functional [5,41] E~ Z h0jð1 C LÞexpðTÞH expðTÞj0i

(6)

which is obtained by augmenting the energy expression in Eq. (4) with the CC amplitude equations (Eq. (5)) multiplied by Lagrange multipliers lijk. abc. . The latter are combined to the L operator L Z L1 C L2 C L3 C . C Ln

A detailed discussion of CC response theory can be found in Ref. [6]. We will not repeat the derivation of the corresponding expressions for the response functions, i.e. the frequencydependent properties, rather we will emphasize in the following the close relationship between analytic derivative theory and response theory. In particular, we will demonstrate how an existing CC analytic second derivative code [28,42] can be used, with a few minor modifications, to compute frequency-dependent polarizabilities.

jJi Z expðTÞj0i

with j0i as a suitable reference determinant, often chosen as the Hartree–Fock Self-Consistent-Field (HF-SCF) wave function. The cluster operator T is defined as an excitation operator consisting of single, double, etc. up to n-tuple excitations

(7)

with Lm Z

1 X X ijk. C C l ai bj . m!2 i;j;k;. a;b;c;. abc.

(8)

to enable a compact notation. The energy functional in Eq. (6) abc. is required to be stationary with respect to the amplitudes tijk. ijk. and labc. , thus providing the already given amplitude equations, Eq. (5), as well as an additional new set of equations 0 Z h0jð1 C LÞ½expðKTÞH expðTÞ; tp
j0i

(9)

which often is referred to as the L equations. The operator tp in Eq. (9) is introduced for convenience and corresponds to a simple excitation operator, i.e. when applied to the reference determinant j0i the excited determinant jFpi is obtained. An expression for the first derivative of the energy is obtained by differentiation of Eq. (6) with respect to a

M. Ka´llay, J. Gauss / Journal of Molecular Structure: THEOCHEM 768 (2006) 71–77

perturbation x:

VðtÞ Z

dE vH Z h0jð1 C LÞexpðKTÞ expðTÞj0i: (10) dx vx The stationarity of E~ with respect to T and L is here used to skip all contributions that involve derivatives of T or L with respect to x. For the second derivative of the energy, further differentiation with respect to a second perturbation y yields [43]: d2 E vL vH Z h0j expðKTÞ expðTÞj0i dxdy vy vx v2 H C h0jð1 C LÞexpðKTÞ expðTÞj0i vxvy
 vH vT expðTÞ; C h0jð1 C LÞ expðKTÞ j0i: vx vy

(11)

(12)

and the first-order L equations  vL  expðKTÞH expðTÞ; tp j0i vy 
   vH  expðTÞ; tp j0i C h0jð1 C LÞ expðKTÞ vy  

vT ; tp j0i C h0jð1 C LÞ expðKTÞHexpðTÞ; vy

(14)

with X as the perturbation operator, 3X as the perturbation strength, uX as the associated frequency, and the sum running over all Fourier components of V(t). The corresponding properties are then given in form of response functions hhX;Y;Z;.iiuY ;uZ ;.

(15)

aXY ðuÞ ZK hhmX ;mY iiu ;

0 Z h0j

(13)

respectively, that are obtained by straightforward differentiation of the original zeroth-order equations with respect to the perturbation y. Note that for actual calculations, the given expressions for first and second derivatives need to be further rewritten. This is usually done in terms of density matrices [42,44]. In addition, the contribution due to the dependence of the molecular orbitals on the perturbation need to be separated; the corresponding contribution is treated either via coupled-perturbed HF theory [45] or the related zeroth- and first-order Z-vector equations [46,47]. 2.3. Response theory While analytic derivative theory enables the efficient calculation of time-independent, i.e. so-called static properties, response theory deals with time-dependent perturbations and is the appropriate tool for the treatment of dynamical properties. For this purpose, the time-dependent perturbation is assumed to have the following form

(16)

i.e. the linear response function with the dipole operators mX and mY as the corresponding perturbations. The required frequency-dependent response functions can be calculated in CC response theory as the corresponding derivatives of the time-averaged quasi-energy Lagrangian [6]. This, for example, means that the frequency-dependent polarizability is determined as 1 d2 fLCC gT  aXY ðuÞ ZK CGu ; u Z uY ZKuX 2 d3X d3Y 3XZ3Y Z0

vH expðTÞj0i vy


 vT C hFp j expðKTÞH expðTÞ; j0i vy

3X X expðKiuX tÞ

X

which describe the effect of the perturbations Y, Z,. with frequencies uY, uZ,.on the property X. For example, the frequency-dependent polarizability aXY(u) is then given by

While the gradient expression (Eq. (10)) is independent of derivatives of T and L, derivatives of T and L appear in the corresponding expression for the second derivatives. These derivatives have to be determined from the first-order amplitude equations 0 Z hFp jexpðKTÞ

X

73

(17)

with the quasi-energy Lagrangian LCC(t) given as the quasienergy   v QðtÞ Z h0jexpðKTÞ H Ki expðTÞj0i (18) vt augmented by the time-dependent CC equations   v 0 Z hFp jexpðKTÞ H Ki expðTÞj0i vt multiplied by Lagrange multipliers lp, i.e.   v LCC ðtÞ Z h0jð1 C LÞexpðKTÞ H Ki expðTÞj0i; vt

(19)

(20)

where the L operator defined in Eq. (7) is used to obtain a compact notation. The time averaging denoted in Eq. (17) by { }T is carried out via: 1 fLCC gT Z lim T/N 2T

ðT

LCC ðtÞdt

(21)

KT

Finally, to ensure real values for the response function, symmetrization with respect to complex conjugation together with a sign reversal of all frequencies is enforced via the operator CGu. The dependence of T and L on the perturbations X,Y,. can be expressed in first order via [6] X T ð1Þ Z T X ðuX Þ3X expðKiuX tÞ (22) X

M. Ka´llay, J. Gauss / Journal of Molecular Structure: THEOCHEM 768 (2006) 71–77

74

and L

ð1Þ

Z

X

X

L ðuX Þ3X expðKiuX tÞ

(23)

X

with TX(uX) and LX(uX) defined in the same manner as T and L as weighted sum of excitation and de-excitation operators, respectively. With these definitions for T(1) and L(1), the expression for the frequency-dependent polarizability given in Eq. (17) can be rewritten as [16] 1 aXY ðuY Þ Z CGu fh0jLY ðuY ÞexpðKTÞmX expðTÞj0i 2 C h0jð1 C LÞ !½expðKTÞmX expðTÞ; T Y ðuY Þ
j0igT :

(24)

with the first-order responses TY(uY) and LY(uY) as solutions of the following equations 0 Z hFp jexpðKTÞmY expðTÞj0i C hFp j½expðKTÞH expðTÞ; T Y ðuY Þ
KuY T Y ðuY Þj0i

(25)

and 0 Zh0jLY ðuY Þ½expðKTÞH expðTÞ; tp
C uY LY ðuY Þj0i C h0jð1 C LÞ½expðKTÞmY expðTÞ; tp
j0i

(26)

Y

C h0jð1 C LÞ½½expðKTÞH expðTÞ; T ðuY Þ
; tp
j0i: 2.4. Relationship between analytic-derivative and response theory Comparison of the expressions for the analytic second derivatives of the energy (Eq. (11)) and for the frequencydependent polarizabilities (Eq. (24)) reveals some similarities: the derivatives vT/vy and vL/vy in the second-derivative expression obviously correspond to the first-order responses TY(uY) and LY(uY), respectively, in the response-function. The equations for vT/vy and vL/vy differ from those for TY(uY) and LY(uY) only by the additional frequency-dependent terms in the case of the latter quantities. Furthermore, the term which includes the second derivative of the Hamiltonian does not appear in the response function, as for electric-field perturbation the Hamiltonian contains only first-order perturbations. From the discussion, it is clear that in the zero-frequency limit, the response-theory expression coincides with the corresponding second-derivative expression. However, one should note that in CC response theory only the cluster operator is allowed to respond to the perturbation [6], while reference determinant and molecular orbitals are kept fixed. This is necessary, as the response of the reference would introduce additional, unphysical poles in the response function [6]. On the other hand, both the response of the reference determinant and the cluster operator are usually considered within derivative theory. The former might be ignored in the case of electric perturbations, thus leading to what is usually referred to as unrelaxed derivatives [49]. There is no indication that results

obtained with the unrelaxed approach are inferior in the case of electric properties [48]. Indeed, the unrelaxed approach appears to be a viable option, as CC theory provides via the single excitations in the cluster operator an efficient treatment of orbital relaxation effects [49]. 3. Implementation Considering the similarities between analytic-derivative and response theory, an implementation that enables the calculation of linear response functions is rather straightforward when starting from a working analytic second-derivative code. Required are only the following modifications: † For the calculation of unrelaxed derivatives all terms that involve orbital-relaxation contributions need to be skipped. As in the presence of the perturbation the reference determinant is no longer a Hartree–Fock (HF) wave function within the unrelaxed approach, the corresponding so-called non-HF terms need to be included. Those are terms that contain the derivatives of the virtual-occupied block of the Fock matrix which no longer vanish. Concerning the density matrices, a significant simplification is that the unrelaxed approach does not necessitate the computation of the unperturbed and perturbed two-particle density matrices. On the other hand, unrelaxed derivatives require the calculation of the virtual-occupied and occupied-virtual block of the reduced one-particle density matrices. Those terms can be skipped in the case of orbital-relaxed derivatives, as their contribution is then treated via the Z-vector equations. † For going from unrelaxed derivatives to frequency-dependent properties, the corresponding frequency-dependent terms, i.e. KuYTY(uY) and uYLY(uY), have to be added to the first-order response equations. † If necessary, symmetrization with respect to complex conjugation together with a simultaneous sign reversal in all frequencies need to be carried out. Our starting point for implementing frequency-dependent polarizabilities is the analytic second derivative code for general CC models described in Ref. [28]. This code exploits the string-based technique [23] which highly facilitates the implementation of high-order CC methods and the corresponding derivative models. The basic idea is to store and process all quantities—such as cluster amplitudes, intermediates, etc.—based on strings of spin–orbital indices, thereby avoiding many-index tensors. Consequently, contractions of these quantities can be evaluated by a couple of simple, prewritten contraction routines [23,27] which enables the general implementation of the aforementioned approaches. As described in Ref. [28], the generalized CC/CI code MRCC [50] provides solutions of the unperturbed CC and L equations as well as of the corresponding first-order response equations and constructs the perturbed and unperturbed oneparticle density matrices. The MRCC program has been interfaced to the ACES2 suite of programs [51] that calculates

M. Ka´llay, J. Gauss / Journal of Molecular Structure: THEOCHEM 768 (2006) 71–77

the HF-SCF wave function, transformed molecular-orbital integrals as well as their derivatives. It is possible with the present implementation to compute linear response functions for general CC models, i.e. CCSD, CCSDT, CCSDTQ, etc. up to FCI. Calculations are possible for closed-shell cases (with a restricted HF determinant as reference function) as well as for high-spin open-shell systems (with an unrestricted HF reference determinant).

4. Applications As a first application, we have computed the frequencydependent polarizability for CHC, which has become a standard example for benchmarking high-accuracy schemes for the computation of polarizabilities [16,39,48]. Calculations have been performed at the CCSD, CCSDT, CCSDTQ, and FCI level in order to monitor the convergence with respect to the inclusion of higher excitations in the cluster operator. Our results for three frequencies (uZ0.0, 0.05, and 0.1 a.u., respectively) are summarized in Table 1 (note that some of the results, though with less digits, have been already given in the literature [7,16,39,48]). We observe that quadrupleexcitation effects (about 0.001–0.006 a.u.) are an order of magnitude smaller than the corresponding triple-excitation contributions (about 0.05–0.07 a.u.) and that the CCSDTQ results are very close to the FCI benchmarks. The only exception is the axx component for uZ0.1 a.u., for which the quadruple-excitation contribution is significantly larger with about 0.04 a.u. This is due to the fact that axx(u) exhibits a pole whose origin is the lowest 1SC/1P excitation at about 0.12 a.u. As the CCSDT excitation energy (0.1186 a.u.) is slightly too low compared to the exact value of 0.1187 a.u., CCSDT overestimates the corresponding component of the frequency-dependent polarizabilities, an effect which is largely corrected at the CCSDTQ level. In this context, it is interesting to note that the approximate CC3 model, unlike CCSDT, does not underestimate the 1SC/1P excitation energy (the CC3 value is 0.1191 a.u.) and that consequently the CC3 value for axx(0.1 a.u.) is lower than the corresponding FCI value. Furthermore, unlike the xx component, the zz component of Table 1 Static and frequency-dependent polarizability (in a.u.) for CHC as calculated at various levels of CC theory together with the excitation energy (in a.u.) for the lowest 1SC/1P transition

Exc. energy uZ0.0 a.u. xxZyy zz uZ0 0.05 a.u. xxZyy zz uZ0.1 a.u. xxZyy zz

CCSD

CCSDT

CCSDTQ

FCI

0.119829

0.118594

0.118684

0.118684

5.7544 8.2688

5.6959 8.3263

5.6901 8.3265

5.6900 8.3265

6.1134 8.3350

6.0513 8.3941

6.0436 8.3944

6.0435 8.3944

9.4756 8.5415

9.5381 8.6057

9.4986 8.6063

9.4983 8.6063

Calculations have been performed as described in Ref. [39].

75

a(u) is not affected by the 1SC/1P excitation and shows the usual behavior with respect to inclusion of higher excitations. As a second application, we have investigated the frequency-dependent polarizability for the CN radical (2SC) for various frequencies at the CCSD, CCSDT, and CCSDTQ level. The high computational cost of these calculations necessitates the use of a rather modest basis set. The d-augcc-pVDZ set [52,53] is here a reasonable compromise (64 functions for CN), as the diffuse functions provide enough flexibility for a decent description of electrical properties [2]. The calculations are performed within the frozen-core approxi˚ [54]. mation using the experimental CN distance of 1.1718 A Table 2 summarizes our results for the xx and zz component of the frequency-dependent polarizability together with the corresponding excitation energy for the lowest 2SC/2P obtained at the same computational levels. The frequency dependence of axx and azz is also shown in Fig. 1. For frequencies sufficiently far way from the 2SC/2P transition, triple-excitation effects amount to about 0.3–0.4 a.u. for the xx component and 0.6–0.7 a.u. for the zz component. Quadrupleexcitation effects are roughly by a factor of three smaller. It furthermore is noted that the contributions due to higher excitations increase with the frequency. This effect is particularly pronounced in the vicinity of the poles. There the contributions amounts to about 0.8 (uZ0.3 a.u.), 1.4 (uZ 0.4 a.u), and 45 a.u. (uZ0.5 a.u.) for the triples and to about 0.3, 1.0,and 221 a.u. for the quadruples. The different performance of the various CC models is here attributed to the different values for the 2SC/2P excitation energy which determine the location of the poles. As CCSD significantly and CCSDT slightly overestimate the transition energy, the corresponding polarizability components are for those two schemes Table 2 Static and frequency-dependent polarizability (in a.u.) for CN (2SC) as calculated at various levels of CC theory together with the excitation energy (in a.u.) for the lowest 2SC/2P transition

Exc. energy uZ0.00 a.u. xxZyy zz uZ0.01 a.u. xxZyy zz uZ0.02 a.u. xxZyy zz uZ0.03 a.u. xxZyy zz uZ0.04 a.u. xxZyy zz uZ0.05 a.u. xxZyy zz

CCSD

CCSDT

CCSDTQ

0.056942

0.051480

0.050310

15.878 25.587

16.211 26.269

16.327 26.432

15.982 25.637

16.346 26.331

16.473 26.495

16.335 25.790

16.822 26.520

16.991 26.691

17.118 26.056

17.966 26.853

18.267 27.036

18.994 26.455

21.364 27.359

22.315 27.560

26.525 27.018

71.871 28.088

293.087 28.318

Calculations have been performed for the experimental CN distance (rCNZ1. ˚ [54]) using the d-aug-cc-pVDZ basis and the frozen-core approxi1718 A mation.

76

M. Ka´llay, J. Gauss / Journal of Molecular Structure: THEOCHEM 768 (2006) 71–77

excitation energies (i.e. the actual pole position) with the different methods. The present work can be considered a continuation of our effort to provide efficient tools for the calculation of molecular properties for arbitrary CC models. It is planned to extend the present work to higher response functions (e.g. first and second hyperpolarizabilities) as well as to implement the corresponding schemes also for approximate treatments of higher excitations as, for example, provided via the corresponding CCn (nZ4,5,.) models. Acknowledgements This work has been supported by the Deutsche Forschungsgemeinschaft, the Fonds der Chemischen Industrie, and the Hungarian Scientific Research Fund (OTKA) under Contract no. D048583. References

Fig. 1. Frequency dependence of the computed xx (part a) and zz component (part b) of the polarizability for the CN radical.

in the vicinity of the pole smaller than the corresponding CCSDTQ value. Due to symmetry reasons, the 2SC/2P transition does not affect the zz component, though also here larger contributions due to higher excitations are observed with increasing frequency. 5. Summary and outlook In this paper, we have demonstrated how frequency-dependent polarizabilities, or more generally linear response functions, can be calculated using an analytic second-derivative code. The required modifications are rather minor and in the present work allow the computation of dynamical polarizabilities for general CC models, i.e. CCSD, CCSDT, CCSDTQ,. up to FCI. The implementation is based on our recently presented analytic second derivative code for general CC wave functions and in this way exploits modern techniques for implementing CC approximations with consideration of arbitrary higher excitations. Initial results for CHC and CN demonstrate the fast convergence in the calculations when successively higher excitations are introduced. Larger changes are only observed in the vicinity of a pole and there can be attributed to the different values obtained for the

[1] For recent reviews, see. J. Bartlett, J.F. Stanton, in: K.B. Lipkowitz, D.B. Boyd (Eds.), Reviews in Computational Chemistry, vol. 5, VCH, New York, 1994, p. 65ff; T.J. Lee, G.E. Scuseria, in: S.R. Langhoff (Ed.), Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy, Kluwer, Dordrecht, 1995, p. 47 ff; R.J. Bartlett, in: D.R. Yarkony (Ed.), Modern Electronic Structure Theory, World Scientific, Singapore, 1995, p. 1047 ff; J. Gauss, in: P.v.R. Schleyer, N.L. Allinger, T. Clark, J. Gasteiger, P.A. Kollmann, H.F. Schaefer, P.R. Schreiner (Eds.), Encyclopedia of Computational Chemistry, Wiley, New York, 1998, p. 615ff. [2] For a recent review, see O. Christiansen, S. Coriani, J. Gauss, C. Ha¨ttig, P. Jørgensen, F. Pawłowski, A. Rizzo, in: M. Papadopoulos, J. Leszczynski, A.J. Sadlej (Eds.), Nonlinear Optical Properties of Matter: from Molecules to Condensed Phases, Kluwer, in press. [3] H.J. Monkhorst, Int. J. Quantum Chem. Symp. 11 (1977) 421. [4] E. Dalgaard, H.J. Monkhorst, Phys. Rev. A 28 (1983) 1217. [5] H. Koch, P. Jørgensen, J. Chem. Phys. 93 (1990) 3333. [6] O. Christiansen, C. Ha¨ttig, P. Jørgensen, Int. J. Quantum Chem. 68 (1998) 1. [7] R. Kobayashi, H. Koch, P. Jørgensen, Chem. Phys. Lett. 249 (1994) 30. [8] C. Ha¨ttig, O. Christiansen, H. Koch, P. Jørgensen, Chem. Phys. Lett. 269 (1997) 428. [9] C. Ha¨ttig, O. Christiansen, P. Jørgensen, Chem. Phys. Lett. 282 (1998) 139. [10] J. Noga, R.J. Bartlett, J. Chem. Phys. 86 (1987) 1041; G.E. Scuseria, H.F. Schaefer, Chem. Phys. Lett. 132 (1988) 382; J.D. Watts, R.J. Bartlett, J. Chem. Phys. 93 (1990) 6104. [11] K. Raghavachari, G.W. Trucks, J.A. Pople, M. Head-Gordon, Chem. Phys. Lett. 157 (1989) 479. [12] M. Urban, J. Noga, S.J. Cole, R.J. Bartlett, J. Chem. Phys. 83 (1985) 4041; J. Noga, R.J. Bartlett, M. Urban, Chem. Phys. Lett. 134 (1987) 126. [13] O. Christiansen, H. Koch, P. Jørgensen, Chem. Phys. Lett. 243 (1995) 409. [14] H. Koch, O. Christiansen, P. Jørgensen, A.S. de Mera´s, T. Helgaker, J. Chem. Phys. 106 (1997) 1808. [15] O. Christiansen, H. Koch, P. Jørgensen, J. Chem. Phys. 103 (1995) 7429. [16] O. Christiansen, J. Gauss, J.F. Stanton, Chem. Phys. Lett. 292 (1998) 437. [17] K. Hald, P. Pawłowski, C. Ha¨ttig, J. Chem. Phys. 118 (2003) 1292. [18] J. Gauss, O. Christiansen, J.F. Stanton, Chem. Phys. Lett. 296 (1998) 117. [19] F. Pawłowski, P. Jørgensen, C. Ha¨ttig, Chem. Phys. Lett. 391 (2004) 27. [20] M. Ka´llay, R. Surja´n, J. Chem. Phys. 113 (2000) 1359. [21] S. Hirata, R.J. Bartlett, Chem. Phys. Lett. 321 (2000) 216. [22] J. Olsen, J. Chem. Phys. 113 (2000) 7140. [23] M. Ka´llay, R. Surja´n, J. Chem. Phys. 115 (2001) 2945.

M. Ka´llay, J. Gauss / Journal of Molecular Structure: THEOCHEM 768 (2006) 71–77 [24] [25] [26] [27] [28] [29] [30] [31] [32]

[33] [34] [35] [36] [37] [38] [39]

M. Ka´llay, P.G. Szalay, R. Surja´n, J. Chem. Phys. 117 (2002) 980. S. Hirata, J. Phys. Chem. A 107 (2003) 9887. J. Olsen, to be submitted for publication. M. Ka´llay, J. Gauss, G. Szalay, J. Chem. Phys. 119 (2003) 2991. M. Ka´llay, J. Gauss, J. Chem. Phys. 120 (2004) 6841. M. Ka´llay, J. Gauss, J. Chem. Phys. 121 (2004) 9257. M. Ka´llay, J. Gauss, J. Chem. Phys. 123 (2005) 214105. S. Hirata, P.D. Fan, A.A. Auer, M. Nooijen, P. Piecuch, J. Chem. Phys. 121 (2004) 12197. A.A. Auer, G. Baumgarten, D.E. Bernholdt, A. Bibireata, V. Chopella, D. Cociorva, X. Gao, R. Harrison, S. Krishnamoorthy, S. Krishnan, C.C. Lam, Q. Lu, M. Nooijen, R. Pitzer, J. Ramanujam, P. Sadayappan, A. Sibiryakov, Mol. Phys. 104 (2006) 228. T.A. Ruden, T. Helgaker, P. Jørgensen, J. Olsen, Chem. Phys. Lett. 371 (2003) 62. A.D. Boese, M. Oren, O. Atasoylu, J.M.L. Martin, M. Ka´llay, J. Gauss, J. Chem. Phys. 120 (2004) 4129. A. Tajti, P.G. Szalay, A.G. Csa´sza´r, M. Ka´llay, J. Gauss, E.F. Valeev, B.A. Flowers, J. Vazquez, J.F. Stanton, J. Chem. Phys. 121 (2004) 11599. Y.J. Bomble, J. Vazquez, M. Ka´llay, C. Michauk, P.G. Szalay, A.G. Csa´sza´r, J. Gauss, J.F. Stanton, J. Chem. Phys. in press. T.A. Ruden, T. Helgaker, P. Jørgensen, J. Olsen, J. Chem. Phys. 121 (2004) 5874. M. Heckert, M. Ka´llay, J. Gauss, Mol. Phys. 103 (2005) 2109. J. Olsen, A.M.S. de Mera´s, H.J.Aa. Jensen, P. Jørgensen, Chem. Phys. Lett. 154 (1989) 380.

77

[40] H. Koch, J. Harrison, J. Chem. Phys. 95 (1991) 7479. [41] H. Koch, H.J.Aa. Jensen, T. Helgaker, P. Jørgensen, G.E. Scuseria, H.F. Schaefer, J. Chem. Phys. 92 (1990) 4924; P.G. Szalay, Int. J. Quantum Chem. 55 (1995) 151 (see also J. Arponen, Ann. Phys. 151, 311 (1983)). [42] J. Gauss, J.F. Stanton, Chem. Phys. Lett. 276 (1997) 70. [43] J.F. Stanton, J. Gauss, in: R.J. Bartlett (Ed.), Recent Advances in Coupled-Cluster Methods, World Scientific, Singapore, 1997, p. 49. [44] J. Gauss, J.F. Stanton, J. Bartlett, J. Chem. Phys. 95 (1991) 2623. [45] J. Gerratt, I.M. Mills, J. Chem. Phys. 49 (1968) 1719; J.A. Pople, R. Krishnan, H.B. Schlegel, J.S. Binkley, Int. J. Quantum Chem. Symp. 13 (1979) 225. [46] N.C. Handy, F. Schaefer, J. Chem. Phys. 81 (1984) 5031. [47] J. Gauss, Chem. Phys. Lett. 191 (1992) 614. [48] O. Christiansen, C. Ha¨ttig, J. Gauss, J. Chem. Phys. 109 (1998) 4745. [49] E.A. Salter, H. Sekino, J. Bartlett, J. Chem. Phys. 87 (1987) 502. [50] MRCC, a generalized CC/CI program by M. Ka´llay, see http://www. mrcc.hu [51] ACES2 (Mainz-Austin-Budapest version), a quantum-chemical program package for high-level calculations of energies and properties by J.F. Stanton et al., see http://www.aces2.de [52] T.H. Dunning, J. Chem. Phys. 90 (1989) 1007. [53] D.E. Woon, H. Dunning, J. Chem. Phys. 100 (1994) 2975. [54] K.P. Huber, G. Herzberg, Molecular Spectra and Molecular Structure V. Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979.