Coupled cluster studies. III. Comparison of the numerical behaviour of coupled cluster doubles with configuration interaction and perturbation theory. Basis set and geometry optimizations

355

Chemical Physics 110 (1986) 355-363 North-Holland. Amsterdam

COUPLED CLUSTER STUDIES. III. COMPARISON OF THE NUMERICAL BEHAVIOUR OF COUPLED CLUSTER DOUBLES WITH CONFIGURATION INTERACTION AND PERTURBATION THEORY. BASIS SET AND GEOMETRY OPTIMIZATIONS Wolfgang FGRNER’,

Laura PYLYPOW and Jil? CiiEK

Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI and Department of Chemistry, Faculty of Science, Universiq of Waterloo, Waterloo, Ontario, Canada N2L 3GI and Guelph - Waterloo Centre for Graduate Work in Chemistry, University of Guelph and University of Waterloo, Waterloo, Ontario, Canada N2L 3GI Received 12 June 1986

In the beryllium atom most of the valence electron correlation energy can be obtained with only three equivalent excitations. It is shown that this is possible also with only one primitive set of p-type gaussians, provided their exponent is carefully optimized. Further it is demonstrated that this is due to the near degeneracy of the ls22s2 and ls22p2 states of Be, because it does not occur in He and Hz. Moller-Plesset (MP) perturbation theory gives less correlation energy than coupled cluster doubles (CCD) in these cases but the same optimum exponent for the set of p functions. For the molecules CO and CO2 where MP2 is known to overestimate the correlation energy in comparison to CCD the convergence properties of MP are studied up to fourth order. It is shown that the results of MP4(DQ) and CCD are very similar also in these cases. For comparison results for ethylene are also discussed. Due to size cousistency CCD is superior over configuration interaction with double excitations (CID). Equilibrium properties computed with CCD and MP4(DQ) are shown to be as reliable as those given by CID, although the variational theorem does not hold for CCD and MP4(DQ). The effect of single excitations turned out to be negligible in these cases. In basis set optimizations all correlation methods studied worked out similar.

1. Introduction This paper is meant as continuation of a series of publications dealing with the numerical properties of the coupled cluster doubles (CCD) theory in its ab initio form. This method was originally many-electron theory called coupled pair (CPMET). The first paper [l] was concerned with the influence of various basis sets on the correlation energy obtained with CCD, the second one [2] with the results of a new approach to the calculation of the correlation energy using CCD and localized orbitals (LO). To avoid misunderstandings it should be pointed out, that LO in this paper denotes only locahzed orbitals obtained from the canonical Hartree-Fock molecular ’ Permanent address: Lehrstuhl fbr lheoretische

Chemie, Fried&h-Alexander Universitgt Erlangen-Nlkmberg, Egerlandstrasse 3, D-852 Erlangen, FRG.

orbitals (MO) by separate unitary transformations in the occupied and virtual MO space. This paper gives a systematic comparison of the numerical behaviour of CCD with configuration interaction truncated to single and double excitations (CISD) and to doubles only (CID), and with perturbation theory in Moller-Plesset partitioning in second (MP2), third (MP3), and fourth order including double and quadruple (MP4(DQ)) and also single (MP4(SDQ)) excitations. For this purpose a simultaneous optimization of the CO and CC distances and the exponents of the d polarization functions of a split valence plus polarization basis set has been carried out for carbon monoxide, carbon dioxide, and ethylene. Bartlett and Purvis [3] reported in a comparative study on many-body perturbation theory and coupled cluster the application of these methods to several small molecules. They found a close similarity between MP4(DQ) and CCD results

0301-0104/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

356

W. F6rner et al. / Coupled cluster studies. III

which would be very useful if beeing general. However, in the pathological case of N, (at large separations) they found a restricted Hartree-Fock reference function to be insufficient. Moreover the close resemblance between MP4(DQ) and CCD is lost in this case. A more detailed study on N, is given in subsequent papers by the same authors [4,5]. In ref. [3] it is stated that it is unknown, whether or not the similarity between CCD and MP4(DQ) persists as function of geometry in non-pathological systems. Meanwhile this persistence has been shown, e.g., in a computation of the quartic force field of water (61, for the ground and for excited states of formaldehyde [7], and for the C, molecule [8]. An example where coupled cluster fails to predict one of the minima of the potential curve while MP4 (DQ) is qualitatively correct is Be, [9] where full CI predicts two minima

WI* For N, and H, (at large separations) and also in other cases of near degeneracies a very slow convergence of the perturbation series has been reported [4]. For sufficiently large separations, e.g. in H,, the MP(D) series will even diverge. Since in infinite polyacetylene in the Hubbard model it can be shown that even the full perturbation series diverges [ll] it seems to be interesting to study some other systems with unusually high secondorder perturbation contributions to the correlation energy. Two of the molecules (CO and CO,) studied in ref. [3] show a larger MP2 correlation energy than the CCD one in their equilibrium geometry. Unlike Nz their potential curves have not been studied (at least to the authors’ knowledge) comparatively with different methods. Since MP4(DQ) is computationally far easier than CCD (being the infinite-order extension of MP4(DQ)) more information upon convergence of the perturbation series and the behaviour of non-variational methods compared to variational ones seems to be desirable for these systems. To have a molecule with no pathologies, treated on equal footing for comparison ethylene is also considered. Another question of importance is the optimization of basis sets on the correlated level. There are a lot of basis sets published obtained with correlation methods, e.g. with CI [12,13] or unrestricted MP2 [14,15] (these are meant as examples

not as complete list). Also comparisons of basis sets of different size or type are available (see ref. [12] for just one example). However, at least to our knowledge there is no study published upon how different variational and non-variational methods behave in exponent optirnizations within one basis set. For this reason we have also optimized the d-function exponents in the basis set applied to the molecules mentioned above using different correlation methods. Concerning basis sets it is interesting to note that in the case of Be with a near degeneracy between the ls22s2 and the ls22p2 states it has been reported that the valence correlation energy can be obtained almost completely from one excitation. This was observed for example by Watson [16] who used a complicated procedure to obtain the virtual orbitals and also with basis sets containing more than one set of p functions (see ref. [4])_ Thus it would be interesting to know whether or not this special feature can also be obtained with a carefully optimized primitive set of p-type gaussians, and to what extent behaviour like this appears also in other systems. A study related to this special problem is also reported very briefly.

2. Method The mathematical formulations of the methods used here are already described extensively in the literature, so it is not necessary to present the formulae here. Interested readers are referred to the literature (see for example refs. [17,18] for coupled cluster, ref. [19] for CI, refs. [20,21] for perturbation theory, and the review by Bartlett [22] on numerical applications of many-body methods). But it should be pointed out that CCD has two main advantages over the other methods. First of all it is size consistent in contrast to truncated CI expansions. This has been implicitly shown already by &ek [17] and since then pointed out by many workers in this field like Pople, Bartlett, and others. References can be found (e.g. refs. [17-211) in the’review articles by Bartlett et al. [22,23] and numerical determinations, e.g. in ref. [24]. Secondly, CCD is invariant to separate unitary transformations in the occupied and vir-

W. Edrner et al. / Coupled cluster studies. III

tual Hartree-Fock MO space (localization) what is not the case for the finite-order MP, if the two-electron integrals and diagonal Fock matrix elements in LO basis are used in the formulae for MP, thus neglecting the off-diagonal Fock matrix elements. This can be corrected by treating these matrix elements as an additional perturbation [21]. Very recently an orbital-invariant formulation of MP2 and MP3 has been developed and applied [25,26]. Also on the basis of the- CCD equation system an orbital-invariant Ml?? scheme can be obtained [27]. In this connection it should be pointed out that both CCD and CCSD are invariant to the kind of localization described above. Thus if the effect of single excitations can be neglected in the MO space it is also negligible in the LO space since it does not change. This is not contradictory to the results of Laidig et al. [28,29] who found in their treatment of localized orbitals that single excitations within the CCSD scheme are essential. The reason is that Laidig et al. used a non-SCF reference function and due to Thouless’ relation (301 the large amplitude of single excitations in this study is probably due to a correction of the reference function [29] towards the SCF reference function. The implications of these properties of CCD especially when dealing with biochemically important molecules are described in refs. [1,2,31] and references therein. On the other hand for CCD and perturbation theory the variational theorem does not hold. The numerical calculations for the present study have been performed with the program system GAUSSIAN 82 [32] on the VAX computer of the Department of Chemistry of the University of Waterloo. In the calculations on He, Be, and H, double-zeta basis sets augmented with p-type polarization functions have been used. For He and H, the 31G basis set from GAUSSIAN 82 and for Be the double-zeta basis published by Huzinaga et al. [33] have been used. The exponents of the p functions have been varied. For the CH bond lengths in ethylene 1.09 A was assumed and all bond angles have been 120”. In the calculations on CO, CO,, and C,H, the 6-31G basis set from GAUSSIAN 82 augmented by a set of six Cartesian d functions has been applied. The exponent of these d functions (equal

351

for C and 0; for the 6-31G* basis set its value is 0.8) has been varied as well as the CC and CO distances R in the molecules. The 6-31G basis set is of split-valence type (single-zeta for the core and double-zeta for the valence electrons). In the correlation calculations the core orbitals have been frozen. The optimization of the d exponents and of R has been carried out on the Hartree-Fock (HF) level and with the correlation methods mentioned in the introduction (CID, CISD, CCD, MP2, MP3, MP4(DQ), and MP4(SDQ)). The optimum d exponents a have been obtained by adjusting a polynomial of second degree to the lowest energy values E(a,R) with R closest to the experimental value, and by determining its minimum. To obtain the equilibrium properties of the molecules like nuclear separation R, correlation energies EC,total energies E,,and the force constants f, the energies for nuclear separations R, and R, f0.02 A (R,= 1.13 A for CO, 1.16 A for CO,, and 1.34 A for C,H,; d exponents a = 0.95 for CO, 1.00 for CO*, and 0.65 for C,H,) have been calculated and a parabola was adjusted to them. From these parabolae the equilibrium properties have been computed.

3. Results anddiscussion Before turning to larger molecules the results obtained for He, Be, and H, are briefly summarized. The HF limit for He is - 2.86167 hartree [34], the non-relativistic limit -2.90372 hartree [35]. The CCD calculations result in an optimum p exponent of 1.25 and a total energy of - 2.88760 hartree (HF energy: -2.85516 hartree). Thus the correlation energy amounts to - 32.44 mbartree (77% of the exact value of -42.05 mhartree). As Shull and Lowdin [36] already pointed out for the radial correlation, also here the optimum p function is local&d in the same space as the occupied 1s orbital. The correlation energy for Be is plotted in fig. 1 as a function of the p exponent a. The solid lines have been obtained with CCD, the dashed ones with MP2. The upper curves correspond to the first p function added to the basis. The exponent of this function is then fixed in the core minimum (a = 6.25) and a second set of p

W. Fiimer er al. / Coupled cluster studies. III

358

EC

0

0.05

0.1 0.5

1.0

1.5

2

4

6

6

10

30

50

-a Fig. 1. The correlation energy EC of Be obtained with the MP2 (- - -) and the CCD ( -) method as a function of the exponent (1!of a first p function (upper curves) and a second one (lower curves; exponent of first function fixed at 6.25) added to a double-zeta basis set containing only s functions.

functions with varying exponent (lower curves) is added. The minima in the core region are broad and shallow, while those in the valence region are very deep and sharp. Although in this case of near degeneracy [4,9,37,38] MP2 faces problems [4,37,39,40], it still gives the same optimal p exponents as CCD. The HF limit for Be is - 14.57302 hartree [16], the non-relativistic limit is - 14.66736 hartree [41], and thus the total correlation energy amounts to -94.31 mhartree. With the p exponents 0.20 and 6.25 the HF energy obtained is -14.57258 hartree and the CCD correlation energy is - 59.80 mhartree (roughly 60% of the exact one). In this case - 24.13 mhartree are due to core correlation and - 35.67 mhartree due to valence electron correlation. It should be pointed out that the valence correlation energy is up to 90% contained in the three equivalent excitations from 2s2 to 2$. Thus this result obtained by Watson with a virtual orbital constructed in a complicated manner [16] or by Purvis and Bartlett [4] with a 7s3p basis set can be obtained also with a primitive set of p functions (our total energy of -14.63239 hartree is only roughly 2.5 mhartree above the CCSD result of ref. [4]). Again the virtual orbitals are localized in the same space as the occupied ones (as illustrated in fig. 2) as

0.7

f(R) 0.6

I 0.5

0.4

0.3

0.2

0.1

0.0

i

i

i

-R(B) Fig.2.Thefunctionf(R)=R2(~(R)(*forthelsandthe2s atomic orbitals of Be and for one (p,, R along the x-axis) of the p functions added to the double-zeta basis set for the valence (a = 0.20) and the core (a = 6.25) minimum of energy.

W. Fiimer et al. / Coupled cluster studies. III

should be expected. In H, finally the HF limit is - 1.133668 hartree [42], the exact correlation energy is -40.81 mhartree [43]. The HF minimum in our calculations occurs for a p exponent of 0.9 with - 1.13133 hartree and for CCD the optimum exponent is 0.7 with a correlationOenergy of - 35.46 mhartree (HH distance 1.402 A [44]). To summarize, the valence electrons of Be are a unique case and the rather good results obtained for them with a simple basis set are due to the near degeneracy of ls22s2 and ls22p2 states and cannot be obtained in other systems by basis set optimization. For CO, CO,, and C,H, simultaneous variations of the d exponents in the basis sets and the CC and CO distances R (the CH, groups in C,H, have been kept rigid) have been carried out. In table 1 the results of the variations with R for one of the different exponent values are shown. The left column for each R value shows the correlation energies obtained (for higher-order perturbation theory the increments are given), the right column the total energies. The results of CID and CISD in table 1 compared to those of CCD and MP show in agreement with fig. 1 in Bartlett’s review [22] the size-inconsistent behaviour of the variational methods. In the case of CO, all MP2 correlation energies are larger than the CCD and CID values, while for CO this holds for most of the values. In C,H, the MP2 values are even slightly smaller than the CI ones. However, in all cases where MP2 gives a larger energy than CCD, the MP3 correction turns out to be positive as already reported for the equilibrium geometries [3]. Thus the CCD and the MP4(DQ) potential curves are almost the same in the cases studied. Thus it seems to be an almost general rule that MP4(DQ) can be used safely instead of its infinite-order counterpart CCD besides some pathological cases like N, for large separations [3-51. For cytosine in a split-valence basis the MP2 energy is reported to be larger than the CCD one [45,2] and the MP3 correction is negative [45]. This seems to contradict the conclusion stated above. However, the CCD result of -723.00 mhartree [2] is based on an approximation which is expected to yield 90-95% of the correct CCD energy. Thus the correct CCD energy should be

359

between - 800 and - 760 mhartree and may well exceed the MP2 result of -777.087 mhartree [45]. The difference should not be too large which would fit to the rather small MP3 correction of - 3.545 mhartree reported [45]. One may think of cytosine as an aromatic system and thus expect the higher-order corrections to be much larger. However, due to the heteroatoms of cytosine the aromaticity is far less pronounced than, e.g., in benzene. This is also shown by the fact that all molecular orbitals of cytosine can be localized to one- and two-center LOS using the procedure of Boys [46]. The shape of the E(R) potential curves for the three molecules under study is quite similar for CID, CCD, and MP4(DQ), while MP2 differs a bit for CO and CO,. The contributions from the single excitations are almost negligible since the nuclear seperations considered here are far off the bond breaking regions where the singles usually become important [23]. Table 2 gives for each molecule and method the correlation energy EC in the energy minimum, both in mhartree and in eV per valence electron, the total energy E,, the optimal value of the d exponent a, the optimum bond length R, and the force constant f of the harmonic symmetric stretching vibration. First of all it is obvious that CCD gives a larger fraction of the correlation energy than CI and almost the same as MP4(DQ). The difference between CID and CCD is roughly 20 mH for CO and C,H, and roughly 45 mH for CO, due to size inconsistency of CID. The MP2 correlation energy is smaller than the CID and CCD ones in the case of C,H,, nearly equal to CCD for CO, and roughly 15 mhartree larger than CCD for CO,. But through positive MP3 corrections for CO and CO, the correlation energies in fourth order (DQ) are almost the same as for CCD in all cases. The single excitations contribute between - 2.19 mhartree (CISD, C,H,) and - 5.37 mhartree (MPqSDQ), CO) to the correlation energies and are thus almost negligible in the cases considered. The correlation energies per valence electron are significantly higher in CO and CO, than in C,H,. This is probably due to the lack of polarization functions for the hydrogens in ethylene. The optimum values of the d exponents are

E,

R=l.OOA

* , _^ -

-

MP4 (DQ) MP4 WQ)

^

_z

ethylene (a = 0.7) HF CID CISD CCD ME? MP3

MP4 (DQ) MP4 (SDQ)

-

-

carbon dioxide (a = 1.00) HF CID CISD CCD MP2 MP3

carbon monoxide ( a = 1 .OO) HF CID - 258.39 CISD - 260.64 CCD - 273.17 h4P2 - 245.67 MP3 - 4% - 2.33 h@4 PQ) - 5.47 MP4 (SDQ)

Method

,

-

-

R,

I

,”

112.69510 112.95348 112.95573 112.96826 112.96086 112.%581 112.96815 112.97128

_

.1

268.21 271.51 285.30 282.01 0.62 -3.94 - 9.21

_

- 421.08 - 424.81 -460.16 - 465.51 8.21 - 2.66 - 10.26

-

R,

R=l.lOA

-

__.._

-

-

R,

187.62611 188.04719 188.05092 188.08628 188.09162 188.08341 188.086% 188.09367

112.73785 113.006@4 113.00935 113.02313 113.01974 113.01922 113.02316 113.02843

,.,

_.

277.53 282.28 297.09 299.46 8.58 - 6.68 - 15.46

-

,-

256.00 257.75 275.48 250.26 -21.87 - 2.39 -4.36

- 436.03 - 441.22 -481.02 - 495.86 20.36 - 5.37 - 17.86

-

E,

R=1.20A

___

-

-

-

__

78.00603 78.26203 78.26378 78.28151 78.25629 78.27817 78.28055 78.28252

187.61832 188.05435 188.05954 188.09934 188.11418 188.09382 188.09919 188.11167

112.71935 112.99688 113.00162 113.01644 113.01881 113.01023 113.01690 113.02571

Re

.__

-

-

-

-

259.19 261.25 280.00 254.76 - 21.59 -2.54 - 4.95

449.00 456.16 499.79 525.65 35.22 -9.84 - 29.96

286.29 292.93 308.44 318.09 19.37 - 11.16 - 25.53

R,

R-1.30i$

-

-

_

-

Ri

-_

78.02947 78.28867 78.29073 78.30947 78.28423 78.30582 78.30836 78.31077

187.54355 187.99254 187.99971 188.04351 188.06919 188.03397 188.04381 188.06393

112.67057 112.95686 112.96350 112.97900 112.98866 112.96930 112.98045 112.99483

I

-

--

262.83 265.26 285.21 260.21 - 20.96 - 2.75 - 5.74

R,

R-l&A

_,,..

-

R,

--

-

78.02284 78.28566 78.28810 78.30805 78.28304 78.30400 78.30676 78.30974

Table 1 The correlation energies EC (mhartrw) (for higher-order perturbation theory the increments are given) and the total energies E, (hartree) obtained with different methods for carbon monoxide, carbon dioxide, and ethylene as function of the bond length R (CO or CC bond; a is the d function exponent in the basis)

W. Fiirner et al. /

361

Coupledclusterstudies.III

Table 2 Correlation energies (in mhartree and in eV per valence electron), minimum total energies E,, optimum d exponents a, equilibrium C-O and C-C distances R, and harmonic symmetric stretching force constants f for carbon monoxide, carbon dioxide, and ethylene for various calculation methods, obtained as described in the text Method

E, wart=)

(eV/e)

-

-

E, (hartree)

a

R (A)

f (N/cm)

- 112.73843 -113.00819 - 113.18958 - 113.02612 - 113.02461 - 113.02158 - 113.02625 - 113.03266

0.922 0.974 0.975 0.910 0.984 0.910 0.910 0.970

1.109 1.129 1.132 1.134 1.145 1.130 1.135 1.142

21.14 21.47 21.26 21.14 20.81 21.69 21.14 20.38

1.128 ”

19.02 ”

1.138 1.152 1.157 1.162 1.173 1.159 1.162 1.168

38.03 38.55 38.35 38.41 31.49 39.12 38.25 37.16

1.162 b,

36.33 ‘)

1.321 1.337 1.338 1.343 1.344 1.342 1.343 1.345

10.35 10.25 10.22 10.13 9.92 10.03 10.24 10.03

1.337 d,

10.76 =)

carbon monoxide HF CID CISD CCD ME2 MP3 MP4 h4P4

(DQ) W-N)

210.12 214.69 289.19 289.33 284.19 289.42 296.19

0.131 0.141 0.181 0.187 0.773 0.788 0.808

experimental carbon dioxide HF CID CISD CCD MP2 MP3 MP4 (DQ) MP4 (SDQ)

-

429.41 434.23 413.18 481.64 468.26 412.99 484.91

- 0.130 - 0.739 - 0.805 - 0.829 -0.1% -0.804 - 0.825

-

181.63458 188.06271 188.06126 188.10527 188.11680 188.10091 188.10507 188.11551

0.848 0.999 0.999 0.995 1.009 0.991 0.994 0.996

experimental ethylene HF CID CISD CCD MP2 MP3 MP4 (DQ) MP4 (SDQ)

-

261.00 263.19 282.88 257.61 219.10 281.10 284.40

- 0.592 - 0.597 - 0.641 -0.584 - 0.633 - 0.639 - 0.645

-

78.02951 18.29021 18.29235 18.31179 18.28646 78.30808 18.31065 18.31321

experimental

0.792 0.690 0.693 0.618 0.689 0.613 0.616 0.681

‘) ref. [47]. b, Ref. [48]. ‘) f is calculated from f = 2mM(2nic)’ where c = 299192458 m/s (velocity of light), M = 1.6605655E - 27 kg (atomic mass unit), m = 16 (mass of oxygen), the factor 2 arises from the transformation between the normal coordinate and R, and V= 1388.11 cm-’ (from ref. [48]). d, Ref. [49]. ‘) Ref. [50].

nearly the same for all methods. Only the HF values differ somewhat from the correlated ones.

The equilibriumbond lengths are alI in reasonable agreement with the experimental ones. In all cases the correlation tends to increase the HF results. For CO and C,H, CID gives the best agreement with experiment, while for CO2 the CCD and MP4(DQ) values (almost identical) are the best ones. It should be mentioned that in the case of

CO and CO, MP2 gives too large values for R. The force constants are in moderate agreement with experiment. Generally the single excitations seem to increase the bond lengths somewhat and to decrease the force constants. This reduction of the force constraints by the single excitations is more pronounced for MP4 than for CI, and it also occurs if the MP4 result is already smaller than the experimental one.

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W. Fiirner et ul. / Coupled cluster studies. III

4. Conclusion

Acknowledgement

Calculations on Be have shown that most of the valence correlation energy can be obtained in a double-zeta basis with one set of primitive p functions. Corresponding calculations on He and H, show that this is true only for Be, and solely due to the near degeneracy occurring in Be. Computations on CO, CO,, and C,H, have again emphasized the size consistency problem for CID and CISD already in these rather small molecules. The results suggest that in most cases (besides pathological ones, e.g. related to large nuclear seperations), even when MP2 gives a larger correlation energy than CCD, the CCD results are of similar quality as those of MP4(DQ). Thus perturbation theory seems to converge also in these cases rather fast. The contributions of single excitations are almost negligible in these systems. Thus the results of this investigation suggest that CCD, being both size consistent and invariant to separate unitary transformations in the occupied and virtual HF MO spaces (localization), is the most suitable method for the calculation of the correlation energy in large systems using localized orbitals (in the abovementioned sense). However, if bond breaking regions are studied or non-SCF localized orbitals are used the inclusion of singles (CCSD) is essential [28,29]. The equilibrium properties of the investigated molecules computed with CCD are as reliable as those obtained with CID (which is variational but not size consistent) and very similar to those computed with MP4(DQ) (which is size consistent but not invariant to localization in the abovementioned sense if the off-diagonal Fock matrix elements are neglected). On the other hand, properties computed with the MP2 method differ somewhat from the CID, CCD, and MP4(DQ) values, although the MP2 energies are larger for CO and CO,. However, the positive MP3 corrections in these cases indicate that the large MP2 energies are artifacts of the very limited order in the perturbation expansion, and not a real improvement in the description of the electron correlation compared to the other methods applied.

One of us (WF) wants to thank all members of the Department of Applied Mathematics, especially his co-authors and Professor J. Paldus for their kind hospitality and continuing support during his stay at the University of Waterloo. Special thanks go to Mr. J. Shelley from the department of Chemistry for his help with all problems concerning the VAX computer on which the calculations for this work have been performed. This work has been supported in part by Natural Sciences and Engineering Research Council of Canada Grants in Aid of Research, which is hereby gratefully acknowledged.

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W. Fiirner et al. / Coupled cluster studies. III

J. PaIdus, in: New horizons P.-O. Lowdin and B. Pullman

[19]

[20]

[21] [22] [23]

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