Coupled-cluster singles, doubles and triples CCSDT calculations of atomization energies

(1)

Ž . Chemical Physics Letters 317 2000 116–122

www.elsevier.nlrlocatercplett

ž /

Coupled-cluster singles, doubles and triples CCSDT calculations of atomization energies

Keld L. Bak

^a,)

, Poul Jørgensen

^b

, Jeppe Olsen

^b

, Trygve Helgaker

^c,1

, Jurgen Gauss ¨

^d

aUNI-C, Olof Palmes Alle 38, DK-8200 Aarhus N, Denmark´

bDepartment of Chemistry, Aarhus UniÕersity, DK-8000 Aarhus C, Denmark

cDepartment of Chemistry, UniÕersity of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK

dInstitut fur Physikalische Chemie, UniÕersitat Mainz, D-55099 Mainz, Germany¨ ¨ Received 15 October 1999; in final form 15 November 1999

Abstract

Ž ¹ .

Atomization energies have been calculated for CO, H O, F , HF, N and CH₂ ₂ ₂ ₂ the A state using the coupled-cluster₁

Ž .

singles, doubles and triples CCSDT model as well as the coupled-cluster singles and doubles model with a perturbative

w Ž .x Ž .

correction for triples CCSD T . The CCSD T model provides an excellent approximation to the CCSDT model; at the

Ž .

cc-pV5Z basis set level, the CCSDT valence triples contribution is underestimated by 9.1% 0.8 kJrmol for CH₂ and

Ž . Ž .

overestimated for the remaining molecules by as little as 4.3% 1.3 kJrmol for F and as much as 8.4% 3.0 kJ₂ rmol for N . At the CCSDT level, the agreement with experiment is not improved, suggesting that some cancellation of error occurs₂ between the missing triples contributions at the CCSD T level and the contributions from the connected quadruples.Ž . q2000 Elsevier Science B.V. All rights reserved.

1. Introduction

For systems dominated by a single electronic configuration, dynamical correlation is described ef-

Ž .

ficiently using the coupled-cluster CC method with

Ž . Ž .

the Hartree–Fock HF self-consistent field SCF wave function as the reference state. The coupled-

Ž .

cluster singles and doubles CCSD model is the simplest and most common CC model 1 . Its compu-w x tational cost scales as N⁶ where N is the number of basis functions. At the next level in the CC hierar-

) Corresponding author. Fax: q45-8619-6199; e-mail:

[email protected]

1Department of Chemistry, University of Oslo, P.O. Box 1033, Blindern, N-0315 Oslo, Norway.

chy, triple excitations are introduced, resulting in the

w x ⁸

CCSDT model 2–4 , which scales as N . Because of the high cost of the CCSDT model, a number of approximate triples models have been proposed, all

7 w x Ž .

of which scale as N 5 . Among these, the CCSD T

Ž . w x

model CCSD with perturbative triples 6 is the most popular; it has been used to calculate a variety of molecular properties such as atomization energies w x7 , nuclear magnetic shielding tensors 8 , and equi-w x librium geometries 9 , with an accuracy comparablew x to or surpassing that of experiment.

In this Letter, we examine the performance of the CCSD T model for the calculation of equilibriumŽ . atomization energies. Comparing with CCSDT calculations, we identify those errors in the CCSD TŽ . model that arise from an incomplete treatment of the

Ž .

PII: S 0 0 0 9 - 2 6 1 4 9 9 0 1 3 1 5 - 9

(2)

connected triples. The errors arising from an incomplete one-electron basis set are minimized by carrying out sequences of calculations, using the correla-

w x

tion-consistent basis sets 10–13 in conjunction with

w x

two-point extrapolations 14,15 . The remaining errors of the CCSD TŽ . calculations of equilibrium atomization energies are mainly due to the neglect of higher connected excitations, in particular the connected quadruples. Recently, estimates of the connected quadruples contribution have been calculated

w x explicitly for smaller systems 16 .

We here report calculations of atomization energies for the six molecules CO, H O, F HF, N and₂ ₂ ₂

Ž ¹ .

CH₂ the A₁ state . These molecules are chosen because accurate experimental equilibrium atomization energies are available and because the molecules are small enough that CCSDT calculations can be carried out for correlation-consistent basis sets with large cardinal numbers. Both the valence and core contributions to the atomization energies are considered.

2. Computational details

For the SCF, CCSD, CCSD T , and CCSDTŽ . wavefunction models, valence-electron calculations are reported using the valence correlation-consistent

w x

basis sets cc-pVxZ, with 2(x(5 10–13 . For SCF, CCSD, and CCSD T , some cc-pV6Z calcula-Ž . tions are also reported. Core effects are examined by carrying out valence-electron and all-electron calculations, using the core-valence correlation-consistent

w x

cc-pCVxZ basis sets 10–13 . The molecular calcula- tions are carried out using spin-restricted wavefunctions, whereas the open-shell atomic calculations use spin-unrestricted wavefunctions. The molecular calculations have been performed at geometries opti- mized at the CCSD TŽ .rcc-pCVQZ level. These geometries are very accurate, differing by 0.1 pm or less from the experimental equilibrium geometries w17 . To obtain atomization energies near the basis-setx limit, extrapolations have been carried out using the

w x

two-point scheme of Refs. 14,15 . The geometry optimizations, the atomic calculations, and the CCSDT calculations have been carried out using the

w x

ACESII program 18,19 ; the remaining calculations have been carried out with a local version of the

w x

DALTON program 20 , containing the coupled-clus-

w x

ter code described in Refs. 21–23 .

3. Dependence of atomization energies on basis set and correlation treatment

3.1. Hartree–Fock andÕalence correlation contribu- tion

In Table 1, the valence equilibrium atomization energies are listed for the six molecules using the SCF, CCSD, CCSD T and CCSDT wavefunctionŽ . models. Except in a few cases at the SCF level, the atomization energy for a given wavefunction model always increases with the cardinal number. This behavior reflects the general rule that an improve- ment in the basis always favors the system of lowest total energy – that is, the molecule rather than its constituent atoms. For a given cardinal number, the atomization energy also increases in the sequence SCF-CCSDrCCSDTrCCSD T . In general, there-Ž . fore, the atomization energy increases both with improvements in the 1-electron basis and in the N-electron treatment.

Comparing the CCSDT and CCSD TŽ . valence atomization energies, we find that the perturbative treatment of the triples contribution at the CCSD TŽ . level provides an excellent approximation to the full triples treatment at the CCSDT level. For basis sets beyond the double-zeta level, the CCSD T modelŽ . overestimates the effect of the triples for all molecules except CH . Although the difference between the₂ CCSDT and CCSD TŽ . atomization energies increases steadily with the cardinal number, the increase from one cardinal number to the next be- comes smaller. The double-zeta basis is too small to provide an accurate description of the full triples correction. At the cc-pV5Z level, the CCSDT triples correction is underestimated by 9.1% for CH by the₂ CCSD TŽ . model but otherwise overestimated between 4.3% for F₂ and 8.4% for N . In absolute₂ terms, the atomization energy is underestimated by 0.8 kJrmol for CH , whereas the largest overestima-₂ tion is 3.0 kJrmol for N . Even though the magni-₂ tude of the triples correction is similar for N₂ Ž35.9

. Ž .

kJrmol and F₂ 30.0 kJrmol , the corresponding

(3)

Table 1

Ž .

Valence electron contributions to atomization energies kJrmol

D T Q 5 6

CH2 SCF 514.56 528.19 530.20 530.90 531.06

CCSD–SCF 171.03 202.26 211.48 214.09 215.17

CCSDT–CCSD 6.23 8.20 8.68 8.84

CCSD T –CCSDŽ . 5.16 7.28 7.84 8.03 8.11

CCSDT–CCSD TŽ . 1.07 0.92 0.84 0.80

CO SCF 710.25 727.13 730.32 730.09 730.13

CCSD–SCF 274.60 293.60 307.55 312.65 314.91

CCSDT–CCSD 23.70 29.83 30.72 31.05

CCSD T –CCSDŽ . 23.61 31.45 32.77 33.29 33.49

CCSDT–CCSD TŽ . 0.09 y1.62 y2.05 y2.24

F₂ SCF y170.01 y154.71 y155.40 y155.62 y155.33

CCSD–SCF 262.08 271.78 278.25 281.18 282.47

CCSDT–CCSD 19.33 27.37 29.29 29.99

CCSD T –CCSDŽ . 19.02 28.34 30.45 31.27 31.61

CCSDT–CCSD TŽ . 0.31 y0.96 y1.16 y1.28

H O2 SCF 619.83 645.30 650.17 651.93 652.30

CCSD–SCF 247.25 283.82 297.73 302.20 303.82

CCSDT–CCSD 6.13 11.87 13.12 13.62

CCSD T –CCSDŽ . 6.00 12.43 13.92 14.53 14.73

CCSDT–CCSD TŽ . 0.13 y0.56 y0.80 y0.91

HF SCF 380.53 401.00 404.40 405.50 405.72

CCSD–SCF 145.23 164.57 172.64 175.23 176.12

CCSDT–CCSD 2.75 6.88 7.83 8.21

CCSD T –CCSDŽ . 2.71 7.27 8.39 8.83 8.97

CCSDT–CCSD TŽ . 0.04 y0.39 y0.55 y0.62

N₂ SCF 451.17 477.88 481.94 482.55 482.92

CCSD–SCF 356.64 389.82 411.17 419.22 422.74

CCSDT–CCSD 27.40 34.20 35.40 35.94

CCSD T –CCSDŽ . 27.88 36.62 38.24 38.96 39.25

CCSDT–CCSD TŽ . y0.48 y2.42 y2.84 y3.02

difference between CCSDT and CCSD T are 3.0Ž . and 1.3 kJrmol, respectively. In short, the difference between the CCSD T and CCSDT triples contribu-Ž . tions does not appear to be correlated with the overall magnitude of the triples contribution.

The differences between the CCSDT and CCSD energies are much larger than the corresponding differences for CCSDT and CCSD T , as well as lessŽ . stable with respect to an increase in the cardinal number. Thus, whereas the CCSDT-CCSD difference for small cardinal numbers cannot be used to estimate the full triples contribution in a larger set, a useful estimate of the CCSDT-CCSD T differenceŽ . can be obtained beyond the double-zeta level.

The HF contribution to the atomization energy converges rapidly with increasing cardinal number.

For cardinal numbers 5 and 6, the difference between

the HF contributions is always smaller than 0.4 kJrmol and hence small compared with the overall accuracy of the calculated atomization energies. By contrast, the correlation contribution to the atomization energy converges slowly with the cardinal number, see Table 2 which shows that the largest difference in the CCSD T contributions between cardinalŽ . numbers 5 and 6 is 3.8 kJrmol for N . The slow₂ convergence arises from the well-known difficulties faced by orbital products in describing short-range

w x

electronic interactions 14,24 . Still, the systematic improvements obtained by the correlation-consistent basis sets enable us to extrapolate to the limit of a complete basis set. The two-point extrapolation of

w x

Refs. 14,15 has proved particularly successful. From Table 2, we see that the extrapolated numbers converge rapidly to the basis-set limit in accordance

(4)

Table 2

Ž .

Valence and extrapolated valence correlation contributions to atomization energies kJrmol

D T Q 5 6 DT TQ Q5 56

CH2 CCSD 171.03 202.26 211.48 214.09 215.17 215.40 218.22 216.82 216.65

CCSD TŽ . 176.19 209.54 219.32 222.12 223.28 223.58 226.46 225.06 224.86

CCSDT 177.26 210.45 220.16 222.93 224.43 227.25 225.82

CO CCSD 274.60 293.60 307.55 312.65 314.91 301.59 317.73 318.00 318.01

CCSD TŽ . 298.21 325.05 340.32 345.94 348.40 336.35 351.46 351.83 351.79

CCSDT 298.30 323.43 338.27 343.70 334.01 349.10 349.40

F₂ CCSD 262.08 271.78 278.25 281.18 282.47 275.86 282.97 284.25 284.25

CCSD TŽ . 281.09 300.11 308.70 312.44 314.09 308.12 314.96 316.38 316.35

CCSDT 281.40 299.15 307.53 311.16 306.62 313.65 314.97

H O₂ CCSD 247.25 283.82 297.73 301.20 303.82 299.22 307.88 306.89 306.04

CCSD TŽ . 253.26 296.25 311.66 316.73 318.54 314.35 322.90 322.05 321.04

CCSDT 253.39 295.69 310.86 315.82 313.50 321.92 321.03

HF CCSD 145.23 164.57 172.64 175.23 176.12 172.71 178.53 177.94 177.34

CCSD TŽ . 147.94 171.84 181.02 184.06 185.09 181.91 187.73 187.24 186.52

CCSDT 147.98 171.45 180.47 183.43 181.34 187.06 186.54

N2 CCSD 356.64 389.82 411.17 419.22 422.74 403.80 426.74 427.66 427.59

CCSD TŽ . 384.52 426.44 449.40 458.17 461.99 444.09 466.16 467.38 467.24

CCSDT 384.04 424.02 446.57 455.15 440.86 463.02 464.16

w x

with previous observations 25 – the difference between the extrapolated Q5 and 56 CCSD T ener-Ž . gies is only 0.14 kJrmol for N , while the largest₂ difference of 1.0 kJrmol is obtained for H O.₂ 3.2. Core correlation contribution

The core contributions to the CCSD, CCSD TŽ . and CCSDT atomization energies are listed in Table 3, along with the contributions obtained by two-point extrapolation. The largest core contribution of 4.87 kJrmol occurs for CO for the CCSDT model with TQ extrapolation.

CCSD gives the largest fraction of the core contribution but the effect of triples is substantial; for example, for N the Q5 CCSD core contribution is₂ 3.06 kJrmol and the TQ CCSDT triples contribution is 1.72 kJrmol. The double-zeta basis is too small to provide an accurate description; in particular, the triples contribution is strongly underestimated at this basis level. Extrapolation improves the core correlation contribution; for example, for CO, CCSD TŽ . gives for the cardinal number T 3.89 kJrmol, and for the DT extrapolation 4.51 kJrmol, which should be compared with a Q5 extrapolated value of 4.80 kJrmol.

For a given cardinal number, the CCSDT and CCSD T core-correlation contributions are similar –Ž . the largest difference occurs for N where the TQ₂ numbers differ by 0.25 kJrmol. Since the difference between the CCSD T and CCSDT core contribu-Ž . tions is small compared with the accuracy of the calculated atomization energy, it may be neglected and CCSD T used to obtain the triples contribution.Ž .

3.3. Comparison with experiment

In Table 4, we have listed calculated equilibrium atomization energies together with the experimental numbers. The experimental equilibrium energies,

w x

which were taken from Ref. 26 , have been obtained by correcting observed atomization energies for molecular zero-point vibrations and for first-order atomic spin–orbit splittings. The energies labeled ‘ x’

are obtained by adding to the valence equilibrium atomization energy of cardinal number x the core contribution obtained for cardinal number xy1; for xsD, the double-zeta core contribution is used.

Ž .

Likewise, the extrapolated energies xy1, x are obtained by adding to the valence extrapolated ener-

Ž . Ž .

gies xy1, x the xy2, xy1 extrapolated core

Ž .

contribution. Again, for xy1, x sDT, the

(5)

Table 3

Ž .

Core and extrapolated core correlation contributions to the atomization energies kJrmol

D T Q 5 DT TQ Q5

CH2 CCSD 1.28 1.37 1.50 1.55 1.41 1.59 1.60

CCSDT–CCSD 0.20 0.39 0.48 0.46 0.54

CCSD T –CCSDŽ . 0.17 0.27 0.33 0.34 0.32 0.37 0.35

CCSDT–CCSD TŽ . 0.04 0.11 0.15 0.15 0.18

CO CCSD 2.16 3.22 3.57 3.74 3.67 3.82 3.92

CCSDT–CCSD 0.26 0.75 0.93 0.95 1.06

CCSD T –CCSDŽ . 0.24 0.67 0.80 0.84 0.84 0.90 0.88

CCSDT–CCSD TŽ . 0.01 0.08 0.13 0.11 0.16

F₂ CCSD 0.08 y0.77 y0.97 y1.02 y1.13 y1.11 y1.07

CCSDT–CCSD 0.43 1.08 1.22 1.36 1.31

CCSD T –CCSDŽ . 0.41 0.94 1.04 1.07 1.16 1.12 1.09

CCSDT–CCSD TŽ . 0.02 0.14 0.17 0.20 0.19

H O2 CCSD 1.31 1.64 1.65 1.66 1.78 1.65 1.68

CCSDT–CCSD 0.11 0.27 0.31 0.33 0.34

CCSD T –CCSDŽ . 0.11 0.25 0.28 0.29 0.30 0.31 0.31

CCSDT–CCSD TŽ . 0.00 0.02 0.03 0.03 0.03

HF CCSD 0.57 0.79 0.78 0.78 0.88 0.78 0.78

CCSDT–CCSD 0.06 0.14 0.16 0.18 0.17

CCSD T –CCSDŽ . 0.06 0.14 0.17 0.17 0.18 0.18 0.18

CCSDT–CCSD TŽ . 0.00 0.00 y0.01 y0.01 y0.01

N2 CCSD 2.15 2.82 2.82 2.94 3.10 2.82 3.06

CCSDT–CCSD 0.49 1.30 1.54 1.64 1.72

CCSD T –CCSDŽ . 0.47 1.13 1.32 1.39 1.41 1.47 1.45

CCSDT–CCSD TŽ . 0.01 0.17 0.21 0.23 0.25

double-zeta core contribution is used. The all electron results are obtained in this way because the core-contributions at a given level are well repre- sented by the contributions at a level one lower seeŽ Table 3 , and the all electron results are therefore.

obtained with a computational cost that is dominated by the valence-electron calculations.

The difference between the CCSD T and CCSDTŽ . numbers are similar for different basis sets – with or without extrapolation applied. Since the extrapolated

Table 4

Ž .

Experimental and calculated atomization energies kJrmol

Exp.a D T Q 5 6 DT TQ Q5 56

Ž . Ž .

CH2 756.7 22 CCSD T 692.19 739.17 751.17 754.85 756.22 753.21 758.39 757.92 757.87

CCSDT 693.30 740.12 752.12 755.80 754.10 759.32 758.86

Ž . Ž .

CO 1086.0 5 CCSD T 1010.86 1054.58 1074.53 1080.39 1083.10 1065.88 1086.30 1086.63 1086.70 CCSDT 1010.96 1052.98 1072.56 1078.28 1063.56 1084.04 1084.36

Ž . Ž .

F2 163.2 6 CCSD T 111.57 145.89 153.47 156.90 158.81 153.90 159.60 160.77 161.04

CCSDT 111.91 144.95 152.45 155.79 152.42 158.48 159.56

Ž . Ž .

H O₂ 974.2 1 CCSD T 874.51 942.97 963.71 970.58 972.80 961.07 975.15 975.94 975.32

CCSDT 874.64 942.41 962.93 969.70 960.22 974.20 974.94

Ž . Ž .

HF 592.3 9 CCSD T 529.10 573.47 586.36 590.51 591.77 583.53 593.19 593.71 593.20

CCSDT 529.14 573.08 585.80 589.88 582.96 592.51 593.00

N₂ 955.72 CCSD TŽ . 838.31 906.94 935.30 944.87 949.34 924.59 952.61 954.21 954.68

CCSDT 837.84 904.54 932.63 924.06 921.37 949.70 951.25

aRef. 25 .w x

(6)

energies converge more rapidly, they are – for both CCSD T and CCSDT – in closer agreement withŽ . experiment than are the unextrapolated energies. It is noteworthy that a complete treatment of the triples at the CCSDT level does not in general improve the agreement with the experimental atomization energies. In fact, for four of the molecules, the Q5 extrapolated CCSDT numbers differ more from experiment than does the CCSD T numbers, indicatingŽ . an element of error cancellation at the CCSD TŽ . level. To improve upon the CCSD TŽ . agreement with experiment, it is not enough to include the fully relaxed CCSDT triples; other contributions – most notably contributions from the connected quadruples – must be included as well. This situation is reminis- cent of what we observe when we try to improve upon the MP2 description of molecular structures.

When the doubles are fully relaxed at the CCSD level, the agreement with experiment usually deterio- rates. Only when the triples are simultaneously ac- counted for at the CCSD T level does the calcula-Ž . tions improve relative to the MP2 level. We note that w x scalar relativistic effects were calculated in Ref. 25 molecules except CH and was found to be between₂ y0.1 andy1.1 kJrmol.

4. Conclusions

Equilibrium atomization energies have been cal-

Ž ¹

culated for CO, H O, F , HF, N and CH₂ ₂ ₂ ₂ the A₁

. Ž .

state at the SCF, CCSD, CCSD T and CCSDT levels using large correlation-consistent basis sets.

Two-point extrapolations have been applied to gen- erate atomization energies near the basis-set limit.

The basis-set convergence of both the core and valence contributions to the atomization energies have been examined, with and without extrapolation applied.

The CCSD T model gives an excellent approxi-Ž . mation to the full triples contribution. The error in the CCSD T triples correction never exceeds aboutŽ . 10%; it is underestimated for CH₂ but otherwise overestimated. Beyond the double-zeta level, the difference between CCSD T and CCSDT energies isŽ . relatively stable. The full triples contribution may therefore be estimated by adding to the CCSD TŽ . contribution the CCSDT–CCSD TŽ . difference ob-

tained with a smaller basis set. The quality of the CCSD T approximation is independent of the mag-Ž . nitude of the triples contribution. The triples effect to the core contribution is significant compared to the total core contribution and should be considered in calculations aimed at chemical accuracy.

The overall agreement with experiment is not improved by full relaxation of the triples, indicating some cancellation of error in the CCSD T modelŽ . between the missing relaxation of triples and the missing contributions from connected higher excitations. The missing relaxation of triples is more im- portant than scalar relativistic effects.

Acknowledgements

This work has been supported by the Danish

Ž .

Research Council Grant No. 9600856 .

References

w x1 G.D. Purvis, R.J. Bartlett, J. Chem. Phys. 76 1982 1910.Ž . w x2 J. Noga, R.J. Bartlett, J. Chem. Phys. 86 1987 7041.Ž . w x3 G.E. Scuseria, H.F. Schaefer, Chem. Phys. Lett. 152 1988Ž .

382.

w x4 J. Watts, R.J. Bartlett, J. Chem. Phys. 93 1991 6104.Ž . w x5 R.J. Bartlett, in: D.R. Yarkony Ed. , Modern ElectronicŽ .

Structure Theory, Advanced Series in Physical Chemistry, vol. 2, World Scientific, 1995, p. 1047.

w x6 K. Raghavachari, G.W. Trucks, J.A. Pople, M. Head-Gordon,

Ž .

Chem. Phys. Lett. 157 1989 479.

w x7 J.M.L. Martin, Chem. Phys. Lett. 259 1996 669.Ž . w x8 J. Gauss, J.F. Stanton, J. Chem. Phys. 104 1996 2574.Ž . w x9 T. Helgaker, J. Gauss, P. Jørgensen, J. Olsen, J. Chem. Phys.

Ž .

106 1997 6430.

w10 T.H. Dunning Jr., J. Chem. Phys. 90 1989 1007.x Ž . w11 D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 100 1994x Ž .

2975.

w12 K.A. Peterson, D.E. Woon, T.H. Dunning Jr., J. Chem. Phys.x

Ž .

100 1994 7410.

w13 A. Wilson, T. van Mourik, T.H. Dunning Jr., J. Mol. Struct.x

Ž .

388 1997 339.

w14 A. Halkier, T. Helgaker, P. Jørgensen, W. Klopper, J. Olsen,x

Ž .

Chem. Phys. Lett. 286 1998 243.

w15 T. Helgaker, W. Klopper, H. Koch, J. Noga, J. Chem. Phys.x

Ž .

106 1997 9639.

w16 P. Piecuch, S.A. Kucharski, R.J. Bartlett, J. Chem. Phys. 110x Ž1999 6103..

w17 K.L. Bak, P. Jørgensen, J. Olsen, T. Helgaker, J. Gauss, inx preparation.

w18 ACESII, an ab initio program system, authored by J.F.x Stanton, J. Gauss, J.D. Watts, W.J. Lauderdale, R.J. Bartlett.

(7)

w19 J.F. Stanton, J. Gauss, J.D. Watts, W.J. Lauderdale, R.J.x

Ž .

Bartlett, Int. J. Quantum Chem. S26 1992 879.

w20 T. Helgaker, H.J.A. Jensen, P. Jørgensen, J. Olsen, K. Ruud,x H. Agren, T. Andersen, K.L. Bak, V. Bakken, O. Chris-˚ tiansen, P. Dahle, E.K. Dalskov, T. Enevoldsen, B. Fernan- dez, H. Heiberg, H. Hettema, D. Jonsson, S. Kirpekar, R.

Kobayashi, H. Koch, K.V. Mikkelsen, P. Norman, M.J.

Packer, T. Saue, P.R. Taylor, O. Vahtras, DALTON an ab

Ž .

initio electronic structure program, Release 1.0 1997 . See http:www.kjemi.uio.norsoftwarerdaltonrdalton.html.

w21 H. Koch, O. Christiansen, R. Kobayashi, P. Jørgensen, T.x

Ž .

Helgaker, Chem. Phys. Lett. 228 1994 233.

w22 H. Koch, A.S. de Meras, T. Helgaker, O. Christiansen, J.x

Ž .

Chem. Phys. 104 1996 4157.

w23 H. Koch, P. Jørgensen, T. Helgaker, J. Chem. Phys. 104x Ž1996 9528..

w24 W. Kutzelnigg, J.D. Morgan III, J. Chem. Phys. 96 1992x Ž . 4484.

w25 J.M.L. Martin, G. de Oliveira, J. Chem. Phys. 111 1999x Ž . 1843.

w26 K.L. Bak, P. Jørgensen, J. Olsen, T. Helgaker, W. Klopper,x in preparation.