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A study of the reliability of different many-body methods: Potential energy curve for the ground state of Be2
Chemical Physics 125 (1988) 255-260 North-Holland, Amsterdam
A STUDY OF THE RELIABILITY OF DIFFERENT MANY-BODY METHODS: POTENTIAL ENERGY CURVE FOR THE GROUND STATE OF Be2
Department of PhysicalChemistry, Faculty of Science, Comenius University, Mlynska Dolina, es-842 15 Bratislava, Czechoslovakia
Jozef NOGA Institute of Inorganic Chemistry, Slovak Academy of Sciences, DBbravsU cesta, CS-842 16 Bratislava, Czechoslovakia
Geerd H.F. DIERCKSEN Max-Planck-Institut ftir Physik unddstrophysik, Institut ftir Astrophysik, Karl-Schwarzschild-Strasse I, O-8046 Garching near Munich, FRG
and Andrzej J. SADLEJ Theoretical Chemistry, Chemical Centre, University of Lund, Box 124, S-221 00 Lund, Sweden
Received 9 November 1987; in final form 1 June 1988
The performance of different coupled-cluster methods is evaluated for calculations of the potential energy curve of the ground state of Be2. It is shown that for highly correlated calculations the corresponding conclusions strongly depend on the quality of the basis set employed. The incorrect predictions of the Be, potential by the CCSD mode1 are confirmed. However, in contrast to earlier calculations with a relatively small basis set, the present CCSDT-la results predict a correct shape of the Be, potential energy curve. The occurrence of the long-range minimum is found to be an artifact of the unsufficiently flexible basis set.
1. Introduction
There are two major factors affecting the accuracy of atomic and molecular data calculated theoretically: the level of treating electron correlation effects and the basis set truncation. Both of them reflect the approximate nature of computational methods for solving the Schriidinger equation for many-electron systems. If the problem under consideration involves large energy differences then either one of these factors or both may be of little importance. However, for problems whose solution follows from small differences in the calculated total energies, a careful consideration of both these factors is of utmost importance [ 11.
It is obvious that full-C1 calculations [ 2 ] have only little chance of being extended beyond the studies of relatively small systems and/or small basis sets. The next most advanced and promising class of computational methods is based on the exponential ansatz [ 3 ] for the generation of the exact eigenfimction form the given reference functions. The truncation of the cluster operator T= T,+ T2+ ...+ T,[3 ] at some size of electron clusters results in a variety of coupledcluster (CC) schemes which have been developed during the past decade [ 4-71. In general, the CC models which include T,,T2and, at least partly, T, [6,7 1, appear to be extremely successful in simulating the corresponding full-C1 energies [ 6,7]. On interpreting the CC methods in terms of the
0301-0104/88/% 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
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I. cernw*ak et al. /Reliability of different many-body methods
many-body perturbation theory (MBPT ) series, one finds that different truncations of the cluster operator correspond to infinite summations of certain classes of diagrammatic contributions to the electron correlation energy [4,8]. The interpretation of the approximate CC schemes shows both their advantages (recovering a very large portion of the correlation energy [ 6,7 ] ) and disadvantages (unbalanced treatment of different diagrammatic contributions in low-order approximations for T [91).The progress in developing higher-order cluster methods [ 5-7 ] will certainly diminish their disadvantages. However, most conclusions concerning the performance of different CC schemes are based on the comparison of CC and full-C1 calculations in relatively small basic sets [ 6,7,9] and may not be as valid as expected. In the present paper different CC approximations are examined from the point of view of their performance in predicting the ground state potential energy curve of the beryllium dimer. This problem has been recently studied by Lee and Bartlett [ 91 who compared their MBPT and CC results with full-C1 calculations. According to these authors the deticienties of their best CC (CCSDT-1 [ 6,7] ) interaction potential for Be2 can be, at least to some extent, attributed to the importance of the T4 operator which is neglected in the CCSDT model. As indicated by Lee and Bartlett [ 9 1, the approximate treatment of T3 in the CCSDT-1 scheme [6,7] is likely to be another important factor affecting the Be2 potential at short internuclear distances. The comparisons carried out by these authors seem to stress the effects of the truncation of the T operator in approximate CC models and are certainly valid for the basis sets employed in their study. However, according to our previous analysis of the basis set truncation effects on the shape of the Be2 interaction potential [ 1 ] one can expect that the performance of different CC models also strongly depend on the choice of basis set functions. For this reason we have carried out a series of different CC calculations of the Be, interaction potential using a large GTO/CGTO basis set whose quality has been examined previously in our MBPT calculations [ 11.
2. Results and discussion The attempts to predict the shape of the Be2 interaction potential have involved most of the currently available advanced methods of computational molecular physics and are surveyed in refs. [ 1,9 1. The controversy concerning the existence of either one or two minima on the potential energy curve and their position (inner and/or long-range minimum) has been solved finally by the experimental determination of the interaction potential [ 10 1. On analysing the reasons for the failure of different high-level methods in predicting the correct shape of the interaction potential we have found [ 1 ] that the basis set choice can affect considerably the final conclusions. In particular, it has been found that the complete fourth_orderMBPTmethodwitha [ 12.5.2.2/7.3.2.2] GTO/CGTO basis set for the beryllium atom can give almost quantitative agreement with the experimental data. This basis set (basis Dz of ref. [ 1 ] ) has been obtained by extending a series of carefully investigated smaller sets and is employed in the present CC study. The coupled-cluster methods considered in this paper have been described in numerous publications by Bartlett et al. [ 4-9 ] and correspond to the CCSD [ 4,5 ] and CCSDT- 1a [ 5-9 ] approximations for the amplitudes of the T operator. These two methods have been employed also by Lee and Bartlett in their studyofBez [9] with [7.3.1] and [7.3.2] CGTObasis sets for Be. In addition to the CCSD and CCSDTla models we have considered two approximations which are expected to provide a reasonably good estimate of the CCSDT- 1 results. These are the CCSD models approximately corrected for the contribution due to triple excitations. The effect of triple excitations can be estimated from the fourth-order MBPT results and then added to the CCSD energy (CCSD + T (MBPT4) ) [ 111. Though this approximation can give a reasonable estimate of the CC correlation energy [ 5-81, its validity depends on the rate of convergence of infinite summations involved in the CCSDT model and may not be applicable in the case of the Bez interaction potential. A much better approximation seems to follow from the evaluation of the contribution due to triple excitations with the amplitudes of T2 obtained at the level of the CCSD model ( CCSD + T (CCSD ) ) [ 6 1. Moreover, a com-
I. CernUrali et al. /Reliability of different many-body methods
parison of these results with a more extensive treatment of T3 in the CCSDT-la model can be used as an indication of the importance of the complete evaluation of T, [ 6,7 1. The numerical results obtained in this paper for the Be, interaction potential are listed in table 1. All these data correspond to the valence-shell approximation for Be. The different interaction potentials are compared in fig. 1 with CCSDT- 1a potential energy curves calculated by Lee and Bartlett [ 9 ] and with our previous complete fourth-order MBPT results [ 11. The latter curve is considered as a reference for the other calculations and according to our earlier analysis it follows closely the full-C1 results of Harrison and Handy [ 12 ] computed with a similar though slightly smaller basis set. Moreover, both the MBPT(4) [ 1 ] and full-C1 [ 12 ] curves are very close to the nearly exact interaction potential obtained by Lengstield et al. [ 131 with a large basis set of Slater-type orbitals. As discussed previously, this shows that the basis set employed in the present study is close to the saturation limit with respect to the calculation of the Be2 interaction potential. Both the [ 7.3.11 and [ 7.3.21 basis set results of Lee and Bartlett [ 91 are shown in fig. 1. Though their larger basis set gives a short-range minimum, it simultaneously increases the depth of another minimum in the range of larger internuclear distances. The possibility of the existence of this long-range mini-
mum has been clearly rejected by both the experimental [ 10 ] and accurate theoretical ‘[12,13 ] data. Before discussing the different results presented in table 1 and fig. 1 let us consider another possible source of errors common to all calculations of interaction energies with incomplete basis sets, i.e. the basis set superposition effects [ 141. The basis set employed in the present study has already been shown [ 1 ] to give perhaps fortuitously negligibly small superposition effects in the vicinity of the inner minimum at both the SCF and MBPT (4) levels. However, we have found that the superposition error dependent on MBPT (4) results [ 1 ] are due to the near-cancellation of relatively large basis set superposition effects in different orders of the MBPT expansion. Hence, the size of the basis set superposition error depends on the method used to study the electron correlation contribution to the interaction energy. The counterpoise estimates [ 14 ] of the basis set superposition effect have been calculated in the present paper for the CCSD interaction potential for several interatomic distances and found to be completely negligible. They range from 0.044 mhartree at 4.5 au to 0.023 mhartree at 7.0 au and decrease monotonically with increasing interatomic distance. Obviously, the same applies to the CCSDT-la results since in the valence-shell model of Be the basis set superposition effect on the contribution due to the triple excitations is equal to zero. The very small val-
Table 1 Distance dependence ofthe Be2 interaction energy, AE(R) =E(R; Be*) -2E(Be)“) are in mhartree and the interatomic distances in au Ra’ 4.5 4.625 4.75 4.878 5.0 5.25 5.50 6.00 6.50 7.00 8.00 9.00
10.00 100.00
257
in different approximations. The interaction energies
CCSD
CCSD+T(MBPT)
CCSDT-la
CCSD+T(CCSD)
2.890 2.261 1.845 1.569 1.397 1.188 1.042 0.717 0.356 0.063 -0.209 -0.219 -0.157 0
0.752 0.265 -0.013 -0.153 -0.202 -0.198 -0.120 -0.177 -0.238 -0.202 -0.418 -0.323 -0.209 0
- 1.022
- 1.517 - 1.921 -2.024 -1.995 - 1.892 - 1.588 - 1.292 -0.923 -0.790 -0.735 -0.591 -0.403 - 0.246 0
- 1.558 - 1.530 - 1.071 -0.781 -0.695 -0.671 -0.563 -0.243 0
‘) The CCSD reference energy of two isolated beryllium atoms is -29.237605 au.
258’
I. c”ernurhket al. /Reliability of differentmany-bodymethods
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Fig. 1. Distance dependence of the Be2 interaction potential A/T(R) =E(R; Be*) - 2E(Be) in different approximations. (0) PresentCCresults; (m) ref. [9] (CCSDT-la, [7.3.1]); (0) ref. [9] (CCSDT-la, [7.3.2]); (x)referenceMBPT(4)results [l].
ues of the counterpoise estimate of the basis set superpositon error obtained in the present calculations can be considered as a measure of the reliability of our evaluation of the performance of the different CC methods [ 14 1. As shown by the plots presented in fig. 1 the CCSD potential energy curve calculated in the present paper is perhaps even worse than those of Lee and Bartlett [ 91. Since in the present case the basis set truncation effects are presumably less significant than for the GTO/CGTO basis set used by the other authors the failure of the CCSD method to predict even qualitatively the correct shape of U(R) should be attributed solely to the deficiency of this CC scheme. This confirms the earlier conclusions of Lee and Bartlett [ 91 concerning the unbalancedtreatment of different diagrammatic contributions of the AE in the CCSD approach.
Hardly any improvement is achieved by adding to the CCSD results the fourth-order MBPT contribution due to triple excitations T(MBPT4). The corresponding curve has a minimum in the short-range region. However, in addition it has also two other minima at larger distances. Hence, the CCSD +T( MBPT4) model must be considered as a very poor approximation to the complete CCSDT approach. In contrast to the results of previous calculations [ 9 ] the present CCSDT- 1a interaction potential has qualitatively the correct distance dependence. The CCSDT-la potential energy curve obtained in our study has a well marked single minimum in the expected range of internuclear distances. Hence, a rather poor performance of the CCSDT- 1a method in calculations of Lee and Bartlett [ 91 follows mainly from the basis set truncation effects. The importance of using the f-type functions in calculations for Be, has been stressed in our MBPT studies of this system [ 11. As expected [ 61, the CCSD+T( CCSD) potential does predict as well the distance dependence of AE correctly and is even better than that obtained in the CCSDT-la model in comparison with the reference curve. In both cases there is only a single inner minimum in the range of distance covered by the most accurate CI [ 12,13 ] and experimental [ 10 ] data. On the other hand both CC curves seem to be a little deficient in the long-range region and show some remains of the long-range minimum incorrectly predicted by the CCSD model. According to the most accurate CI data of Lengstield et al. [ 13 ] the interaction potential of Be2 has only one short-range minimum. As regards the binding energy at the equilibrium internuclear distance neither the CCSDT-la nor the CCSD+T(CCSD) models are quantitatively sufficient. The first one gives only 1.0 kcal/mole ( % 1.6 mhartree) which is about 50 per cent of the full-C1 result of 1.86 kcal/mole computed with a very similar basis set [ 121. The corresponding MBPT (4) value obtained previously with the same basis set as that used in the present calculations amounts to 1.83 kcal/mole [ 11. The CCSD + T( CCSD) approach gives a little better result of 1.4 kcal/mole ( x 2.1 mhartree) in comparison with the reference data. However, in view of the very small value of De, both the CCSDT-la and CCSD + T( CCSD) results ap-
I. c*ernufaket al. /Reliability ofdifferent many-bodymethods
pear to be acceptable within the limits of what is usually considered as the “chemical accuracy” of quantum mechanical calculations for molecules. Certainly, one would like to achieve better agreement with the experimental value of D,=2.26 k&/mole [ 10 1. However, in view of a number of approximations involved a better result might be considered as more or less fortuitous. In this respect the very good performance of the MBPT( 4) method [ 1 ] is quite puzzling. The closeness of the MBPT (4 ) [ 1 ] and the full-C1 results of Bez can be interpreted in a number of ways. One may argue that the nature of the perturbation expansion for the correlation energy ensures a well balanced treatment of different levels of excitation and their contributions to AE. Some support to this interpretation is given by a comparison of the AE values obtained in different orders of the MBPT treatment. Obviously, one can always argue that for Be2 the MBPT (4) result for De follows from mutual cancellations of (non-negligible) higher-order contributions. Only little support to this interpretation is given by the recent higher-order MBPT calculations on Be, [ 15,16 ] which have been carried out with poor basis sets and may reflect again mostly the basis set truncation effects. The failure of the CCSD model to correctly predict the AZ?(R) function is undoubtedly the result of an unbalanced treatment of higher-order disconnected terms. The better performance of the CCSDT-la model which partly accounts for the effect of T3 shows that the T3 amplitudes restore to some extent the balance between different terms in the correlation contribution to AE. Some part of the difference between the full-C1 and CCSDT-la results of the present paper should be due certainly to approximations involved in the evaluation of the T3 amplitudes. The elucidation of this problem can be gained from recent extensions of the CCSD-1 model discussed by Nogaetal. [7]. Since in the valence-shell approximation for Be2 the complete CC wavefunction should involve the cluster operators through T,, one can argue about the importance of the effect due to T, amplitudes. Also the completeness of the treatment of the T3 amplitudes might be of some importance. Though Lee and Bartlett [ 9 ] do not exclude the possibility of a significant basis set truncation effect on their comparison of the
259
CC and full-C1 results, they appear to stress the influence of the incompleteness of the CCSDT- la model. Obviously, this model may be too approximate for a weakly bound system involving four valence electrons. According to the present study the difference between the CCSDT- 1a and full-C1 results for Be* depends significantly on the basis set. Hence, the performance of different approximate CC schemes cannot be unambiguously asserted by comparisons with full-C1 calculations performed with insufficiently flexible basis sets. The relative importance of the higher-order cluster contributions is basis set dependent. On the positive side, the present study shows that even in the case of the Be2 interaction potential, the CCSDT-la model is a priori capable of predicting its correct shape in the full range of internuclear distances. According to the present results a qualitatively correct potential can be obtained within CC models which do not involve T4. All calculations reported in this paper have been performed by using a set of programs designed at the MPI laboratory [ 17 ] and interfaced with the MBPT/ CC package of the Bratislava group [ 6,7,18].
3. Conclusions A series of different CC calculations of the potential energy curve of Be, reported in this paper shows that the evaluation of the performance of different advanced computational methods must be carried out in appropriately large and carefully selected basis sets. This conclusion is of particular importance for the numerical evaluation of the quality of highly correlated approaches. According to the present results the CCSD model does not appear to be sufficiently balanced for a reliable evaluation of small energy differences over a wide range of molecular geometries. Including approximately the T, amplitudes within the CCSDT-la scheme [ 5-91 results in a considerable improvement of the CC approach and restores to a large extent the balance between different small contributions to the correlation energy. However, it follows from our current computational experience that models based on the CCSDT approximation are extremely time consuming. Hence, there is only a little chance for their
260
I. CernuZak et al. /Reliability of diflerent many-body methods
routine application in molecular calculations without further approximation. At the CCSDT level reliable results can be obtained only with suffkiently large and flexible basis sets and in most cases this condition makes the corresponding calculations prohibitively time-consuming and expensive.
References [ 11 G.H.F. Diercksen, V. Kellii and A.J. Sadlej, Chem. Phys. 96 (1985) 59. [2] P. Saxe, H.F. Schaefer III and NC. Handy, Chem. Phys. Letters 79 ( 198 1) 202; R.J. Harrison and N.C. Handy, Chem. Phys. Letters 95 (1983) 386. [3] F. Coester, Nucl. Phys. 1 (1958) 421; F. Coester and H. Kiimmel, Nucl. Phys. 17 ( 1960) 477; J. Ciiek, Advan. Chem. Phys. 14 ( 1969) 35, and references therein. [4] R.J. Bartlett and G.D. Purvis, Intern. J. Quantum Chem. 14 (1978) 561; R.J. Bartlett, Ann. Rev. Phys. Chem. 32 ( 1981) 359, and references therein. [ 51 Y.S. Lee, S.A. Kucharski and R.J. Bartlett, J. Chem. Phys. 81 (1984) 4371. [ 6 ] M. Urban, J. Noga, S.J. Cole and R.J. Bartlett, J. Chem Phys. 83 (1985) 4041; Erratum 85 (1986) 5383.
[7] J. Noga, R.J. Bartlett and M. Urban, Chem. Phys. Letters 134 (1987) 126; J. Noga and R.J. Bartlett, J. Chem. Phsy. 86 (1987) 7041. IS] S.A. Kucharski and R.J. Bartlett, Advan. Quantum Chem. 8 (1986) 281. [9] Y.S. Lee and R.J. Bartlett, J. Chem. Phys. 80 (1984) 4371. [ 101V.E. Bondybey, Chem. Phys. Letters, 109 (1984) 436. [ 111 R.J. Bartlett, H. Sekino and G.D. Purvis, Chem. Phys. Letters 98 (1983) 66. [ 121 R.J. Harrison and NC. Handy, Chem. Phys. Letters 98 (1983) 97. [ 131 B.H. Lengsfield III, A.D. McLean, M. Yoshimine and B. Liu, J. Chem. Phys. 78 (1983) 1891. [ 141S.F. Boys and F. Bemardi, Mol. Phys. 19 (1970) 553; D.W. Schwenke and D.C. Truhlar, J. Chem. Phys. 82 (1985) 2418; G.H.F. Diercksen and A.J. Sadlej, Mol. Phys. 59 ( 1986) 889. [ 151P.J. Knowles, K. Somasundram, N.C. Handy and K. Hirao, Chem. Phys. Letters 113 ( 1985) 8, [ 161 W.D. Laidig, G. Fitzgerald and R.J. Bartlett, Chem. Phys. Letters 113 (1985) 151. [ 171G.H.F. Diercksen and W.P. Kraemer, Munich Molecular Program System, Max-Planck-Institut fur Physik und Astrophysik, Munich, FRG. [ 181 M. Urban, I. Hubac, V. Kellii and J. Noga, J. Chem. Phys. 72 (1980) 3378; J. Noga, Computer Phys. Commun. 29 (1983) 117.
A STUDY OF THE RELIABILITY OF DIFFERENT MANY-BODY METHODS: POTENTIAL ENERGY CURVE FOR THE GROUND STATE OF Be2
Department of PhysicalChemistry, Faculty of Science, Comenius University, Mlynska Dolina, es-842 15 Bratislava, Czechoslovakia
Jozef NOGA Institute of Inorganic Chemistry, Slovak Academy of Sciences, DBbravsU cesta, CS-842 16 Bratislava, Czechoslovakia
Geerd H.F. DIERCKSEN Max-Planck-Institut ftir Physik unddstrophysik, Institut ftir Astrophysik, Karl-Schwarzschild-Strasse I, O-8046 Garching near Munich, FRG
and Andrzej J. SADLEJ Theoretical Chemistry, Chemical Centre, University of Lund, Box 124, S-221 00 Lund, Sweden
Received 9 November 1987; in final form 1 June 1988
The performance of different coupled-cluster methods is evaluated for calculations of the potential energy curve of the ground state of Be2. It is shown that for highly correlated calculations the corresponding conclusions strongly depend on the quality of the basis set employed. The incorrect predictions of the Be, potential by the CCSD mode1 are confirmed. However, in contrast to earlier calculations with a relatively small basis set, the present CCSDT-la results predict a correct shape of the Be, potential energy curve. The occurrence of the long-range minimum is found to be an artifact of the unsufficiently flexible basis set.
1. Introduction
There are two major factors affecting the accuracy of atomic and molecular data calculated theoretically: the level of treating electron correlation effects and the basis set truncation. Both of them reflect the approximate nature of computational methods for solving the Schriidinger equation for many-electron systems. If the problem under consideration involves large energy differences then either one of these factors or both may be of little importance. However, for problems whose solution follows from small differences in the calculated total energies, a careful consideration of both these factors is of utmost importance [ 11.
It is obvious that full-C1 calculations [ 2 ] have only little chance of being extended beyond the studies of relatively small systems and/or small basis sets. The next most advanced and promising class of computational methods is based on the exponential ansatz [ 3 ] for the generation of the exact eigenfimction form the given reference functions. The truncation of the cluster operator T= T,+ T2+ ...+ T,[3 ] at some size of electron clusters results in a variety of coupledcluster (CC) schemes which have been developed during the past decade [ 4-71. In general, the CC models which include T,,T2and, at least partly, T, [6,7 1, appear to be extremely successful in simulating the corresponding full-C1 energies [ 6,7]. On interpreting the CC methods in terms of the
0301-0104/88/% 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
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I. cernw*ak et al. /Reliability of different many-body methods
many-body perturbation theory (MBPT ) series, one finds that different truncations of the cluster operator correspond to infinite summations of certain classes of diagrammatic contributions to the electron correlation energy [4,8]. The interpretation of the approximate CC schemes shows both their advantages (recovering a very large portion of the correlation energy [ 6,7 ] ) and disadvantages (unbalanced treatment of different diagrammatic contributions in low-order approximations for T [91).The progress in developing higher-order cluster methods [ 5-7 ] will certainly diminish their disadvantages. However, most conclusions concerning the performance of different CC schemes are based on the comparison of CC and full-C1 calculations in relatively small basic sets [ 6,7,9] and may not be as valid as expected. In the present paper different CC approximations are examined from the point of view of their performance in predicting the ground state potential energy curve of the beryllium dimer. This problem has been recently studied by Lee and Bartlett [ 91 who compared their MBPT and CC results with full-C1 calculations. According to these authors the deticienties of their best CC (CCSDT-1 [ 6,7] ) interaction potential for Be2 can be, at least to some extent, attributed to the importance of the T4 operator which is neglected in the CCSDT model. As indicated by Lee and Bartlett [ 9 1, the approximate treatment of T3 in the CCSDT-1 scheme [6,7] is likely to be another important factor affecting the Be2 potential at short internuclear distances. The comparisons carried out by these authors seem to stress the effects of the truncation of the T operator in approximate CC models and are certainly valid for the basis sets employed in their study. However, according to our previous analysis of the basis set truncation effects on the shape of the Be2 interaction potential [ 1 ] one can expect that the performance of different CC models also strongly depend on the choice of basis set functions. For this reason we have carried out a series of different CC calculations of the Be, interaction potential using a large GTO/CGTO basis set whose quality has been examined previously in our MBPT calculations [ 11.
2. Results and discussion The attempts to predict the shape of the Be2 interaction potential have involved most of the currently available advanced methods of computational molecular physics and are surveyed in refs. [ 1,9 1. The controversy concerning the existence of either one or two minima on the potential energy curve and their position (inner and/or long-range minimum) has been solved finally by the experimental determination of the interaction potential [ 10 1. On analysing the reasons for the failure of different high-level methods in predicting the correct shape of the interaction potential we have found [ 1 ] that the basis set choice can affect considerably the final conclusions. In particular, it has been found that the complete fourth_orderMBPTmethodwitha [ 12.5.2.2/7.3.2.2] GTO/CGTO basis set for the beryllium atom can give almost quantitative agreement with the experimental data. This basis set (basis Dz of ref. [ 1 ] ) has been obtained by extending a series of carefully investigated smaller sets and is employed in the present CC study. The coupled-cluster methods considered in this paper have been described in numerous publications by Bartlett et al. [ 4-9 ] and correspond to the CCSD [ 4,5 ] and CCSDT- 1a [ 5-9 ] approximations for the amplitudes of the T operator. These two methods have been employed also by Lee and Bartlett in their studyofBez [9] with [7.3.1] and [7.3.2] CGTObasis sets for Be. In addition to the CCSD and CCSDTla models we have considered two approximations which are expected to provide a reasonably good estimate of the CCSDT- 1 results. These are the CCSD models approximately corrected for the contribution due to triple excitations. The effect of triple excitations can be estimated from the fourth-order MBPT results and then added to the CCSD energy (CCSD + T (MBPT4) ) [ 111. Though this approximation can give a reasonable estimate of the CC correlation energy [ 5-81, its validity depends on the rate of convergence of infinite summations involved in the CCSDT model and may not be applicable in the case of the Bez interaction potential. A much better approximation seems to follow from the evaluation of the contribution due to triple excitations with the amplitudes of T2 obtained at the level of the CCSD model ( CCSD + T (CCSD ) ) [ 6 1. Moreover, a com-
I. CernUrali et al. /Reliability of different many-body methods
parison of these results with a more extensive treatment of T3 in the CCSDT-la model can be used as an indication of the importance of the complete evaluation of T, [ 6,7 1. The numerical results obtained in this paper for the Be, interaction potential are listed in table 1. All these data correspond to the valence-shell approximation for Be. The different interaction potentials are compared in fig. 1 with CCSDT- 1a potential energy curves calculated by Lee and Bartlett [ 9 ] and with our previous complete fourth-order MBPT results [ 11. The latter curve is considered as a reference for the other calculations and according to our earlier analysis it follows closely the full-C1 results of Harrison and Handy [ 12 ] computed with a similar though slightly smaller basis set. Moreover, both the MBPT(4) [ 1 ] and full-C1 [ 12 ] curves are very close to the nearly exact interaction potential obtained by Lengstield et al. [ 131 with a large basis set of Slater-type orbitals. As discussed previously, this shows that the basis set employed in the present study is close to the saturation limit with respect to the calculation of the Be2 interaction potential. Both the [ 7.3.11 and [ 7.3.21 basis set results of Lee and Bartlett [ 91 are shown in fig. 1. Though their larger basis set gives a short-range minimum, it simultaneously increases the depth of another minimum in the range of larger internuclear distances. The possibility of the existence of this long-range mini-
mum has been clearly rejected by both the experimental [ 10 ] and accurate theoretical ‘[12,13 ] data. Before discussing the different results presented in table 1 and fig. 1 let us consider another possible source of errors common to all calculations of interaction energies with incomplete basis sets, i.e. the basis set superposition effects [ 141. The basis set employed in the present study has already been shown [ 1 ] to give perhaps fortuitously negligibly small superposition effects in the vicinity of the inner minimum at both the SCF and MBPT (4) levels. However, we have found that the superposition error dependent on MBPT (4) results [ 1 ] are due to the near-cancellation of relatively large basis set superposition effects in different orders of the MBPT expansion. Hence, the size of the basis set superposition error depends on the method used to study the electron correlation contribution to the interaction energy. The counterpoise estimates [ 14 ] of the basis set superposition effect have been calculated in the present paper for the CCSD interaction potential for several interatomic distances and found to be completely negligible. They range from 0.044 mhartree at 4.5 au to 0.023 mhartree at 7.0 au and decrease monotonically with increasing interatomic distance. Obviously, the same applies to the CCSDT-la results since in the valence-shell model of Be the basis set superposition effect on the contribution due to the triple excitations is equal to zero. The very small val-
Table 1 Distance dependence ofthe Be2 interaction energy, AE(R) =E(R; Be*) -2E(Be)“) are in mhartree and the interatomic distances in au Ra’ 4.5 4.625 4.75 4.878 5.0 5.25 5.50 6.00 6.50 7.00 8.00 9.00
10.00 100.00
257
in different approximations. The interaction energies
CCSD
CCSD+T(MBPT)
CCSDT-la
CCSD+T(CCSD)
2.890 2.261 1.845 1.569 1.397 1.188 1.042 0.717 0.356 0.063 -0.209 -0.219 -0.157 0
0.752 0.265 -0.013 -0.153 -0.202 -0.198 -0.120 -0.177 -0.238 -0.202 -0.418 -0.323 -0.209 0
- 1.022
- 1.517 - 1.921 -2.024 -1.995 - 1.892 - 1.588 - 1.292 -0.923 -0.790 -0.735 -0.591 -0.403 - 0.246 0
- 1.558 - 1.530 - 1.071 -0.781 -0.695 -0.671 -0.563 -0.243 0
‘) The CCSD reference energy of two isolated beryllium atoms is -29.237605 au.
258’
I. c”ernurhket al. /Reliability of differentmany-bodymethods
AEilm 2!
1. ccslJ+
‘?
:
I
0.
-1.
-2. ‘f 1, ‘i’
\ :
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Fig. 1. Distance dependence of the Be2 interaction potential A/T(R) =E(R; Be*) - 2E(Be) in different approximations. (0) PresentCCresults; (m) ref. [9] (CCSDT-la, [7.3.1]); (0) ref. [9] (CCSDT-la, [7.3.2]); (x)referenceMBPT(4)results [l].
ues of the counterpoise estimate of the basis set superpositon error obtained in the present calculations can be considered as a measure of the reliability of our evaluation of the performance of the different CC methods [ 14 1. As shown by the plots presented in fig. 1 the CCSD potential energy curve calculated in the present paper is perhaps even worse than those of Lee and Bartlett [ 91. Since in the present case the basis set truncation effects are presumably less significant than for the GTO/CGTO basis set used by the other authors the failure of the CCSD method to predict even qualitatively the correct shape of U(R) should be attributed solely to the deficiency of this CC scheme. This confirms the earlier conclusions of Lee and Bartlett [ 91 concerning the unbalancedtreatment of different diagrammatic contributions of the AE in the CCSD approach.
Hardly any improvement is achieved by adding to the CCSD results the fourth-order MBPT contribution due to triple excitations T(MBPT4). The corresponding curve has a minimum in the short-range region. However, in addition it has also two other minima at larger distances. Hence, the CCSD +T( MBPT4) model must be considered as a very poor approximation to the complete CCSDT approach. In contrast to the results of previous calculations [ 9 ] the present CCSDT- 1a interaction potential has qualitatively the correct distance dependence. The CCSDT-la potential energy curve obtained in our study has a well marked single minimum in the expected range of internuclear distances. Hence, a rather poor performance of the CCSDT- 1a method in calculations of Lee and Bartlett [ 91 follows mainly from the basis set truncation effects. The importance of using the f-type functions in calculations for Be, has been stressed in our MBPT studies of this system [ 11. As expected [ 61, the CCSD+T( CCSD) potential does predict as well the distance dependence of AE correctly and is even better than that obtained in the CCSDT-la model in comparison with the reference curve. In both cases there is only a single inner minimum in the range of distance covered by the most accurate CI [ 12,13 ] and experimental [ 10 ] data. On the other hand both CC curves seem to be a little deficient in the long-range region and show some remains of the long-range minimum incorrectly predicted by the CCSD model. According to the most accurate CI data of Lengstield et al. [ 13 ] the interaction potential of Be2 has only one short-range minimum. As regards the binding energy at the equilibrium internuclear distance neither the CCSDT-la nor the CCSD+T(CCSD) models are quantitatively sufficient. The first one gives only 1.0 kcal/mole ( % 1.6 mhartree) which is about 50 per cent of the full-C1 result of 1.86 kcal/mole computed with a very similar basis set [ 121. The corresponding MBPT (4) value obtained previously with the same basis set as that used in the present calculations amounts to 1.83 kcal/mole [ 11. The CCSD + T( CCSD) approach gives a little better result of 1.4 kcal/mole ( x 2.1 mhartree) in comparison with the reference data. However, in view of the very small value of De, both the CCSDT-la and CCSD + T( CCSD) results ap-
I. c*ernufaket al. /Reliability ofdifferent many-bodymethods
pear to be acceptable within the limits of what is usually considered as the “chemical accuracy” of quantum mechanical calculations for molecules. Certainly, one would like to achieve better agreement with the experimental value of D,=2.26 k&/mole [ 10 1. However, in view of a number of approximations involved a better result might be considered as more or less fortuitous. In this respect the very good performance of the MBPT( 4) method [ 1 ] is quite puzzling. The closeness of the MBPT (4 ) [ 1 ] and the full-C1 results of Bez can be interpreted in a number of ways. One may argue that the nature of the perturbation expansion for the correlation energy ensures a well balanced treatment of different levels of excitation and their contributions to AE. Some support to this interpretation is given by a comparison of the AE values obtained in different orders of the MBPT treatment. Obviously, one can always argue that for Be2 the MBPT (4) result for De follows from mutual cancellations of (non-negligible) higher-order contributions. Only little support to this interpretation is given by the recent higher-order MBPT calculations on Be, [ 15,16 ] which have been carried out with poor basis sets and may reflect again mostly the basis set truncation effects. The failure of the CCSD model to correctly predict the AZ?(R) function is undoubtedly the result of an unbalanced treatment of higher-order disconnected terms. The better performance of the CCSDT-la model which partly accounts for the effect of T3 shows that the T3 amplitudes restore to some extent the balance between different terms in the correlation contribution to AE. Some part of the difference between the full-C1 and CCSDT-la results of the present paper should be due certainly to approximations involved in the evaluation of the T3 amplitudes. The elucidation of this problem can be gained from recent extensions of the CCSD-1 model discussed by Nogaetal. [7]. Since in the valence-shell approximation for Be2 the complete CC wavefunction should involve the cluster operators through T,, one can argue about the importance of the effect due to T, amplitudes. Also the completeness of the treatment of the T3 amplitudes might be of some importance. Though Lee and Bartlett [ 9 ] do not exclude the possibility of a significant basis set truncation effect on their comparison of the
259
CC and full-C1 results, they appear to stress the influence of the incompleteness of the CCSDT- la model. Obviously, this model may be too approximate for a weakly bound system involving four valence electrons. According to the present study the difference between the CCSDT- 1a and full-C1 results for Be* depends significantly on the basis set. Hence, the performance of different approximate CC schemes cannot be unambiguously asserted by comparisons with full-C1 calculations performed with insufficiently flexible basis sets. The relative importance of the higher-order cluster contributions is basis set dependent. On the positive side, the present study shows that even in the case of the Be2 interaction potential, the CCSDT-la model is a priori capable of predicting its correct shape in the full range of internuclear distances. According to the present results a qualitatively correct potential can be obtained within CC models which do not involve T4. All calculations reported in this paper have been performed by using a set of programs designed at the MPI laboratory [ 17 ] and interfaced with the MBPT/ CC package of the Bratislava group [ 6,7,18].
3. Conclusions A series of different CC calculations of the potential energy curve of Be, reported in this paper shows that the evaluation of the performance of different advanced computational methods must be carried out in appropriately large and carefully selected basis sets. This conclusion is of particular importance for the numerical evaluation of the quality of highly correlated approaches. According to the present results the CCSD model does not appear to be sufficiently balanced for a reliable evaluation of small energy differences over a wide range of molecular geometries. Including approximately the T, amplitudes within the CCSDT-la scheme [ 5-91 results in a considerable improvement of the CC approach and restores to a large extent the balance between different small contributions to the correlation energy. However, it follows from our current computational experience that models based on the CCSDT approximation are extremely time consuming. Hence, there is only a little chance for their
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I. CernuZak et al. /Reliability of diflerent many-body methods
routine application in molecular calculations without further approximation. At the CCSDT level reliable results can be obtained only with suffkiently large and flexible basis sets and in most cases this condition makes the corresponding calculations prohibitively time-consuming and expensive.
References [ 11 G.H.F. Diercksen, V. Kellii and A.J. Sadlej, Chem. Phys. 96 (1985) 59. [2] P. Saxe, H.F. Schaefer III and NC. Handy, Chem. Phys. Letters 79 ( 198 1) 202; R.J. Harrison and N.C. Handy, Chem. Phys. Letters 95 (1983) 386. [3] F. Coester, Nucl. Phys. 1 (1958) 421; F. Coester and H. Kiimmel, Nucl. Phys. 17 ( 1960) 477; J. Ciiek, Advan. Chem. Phys. 14 ( 1969) 35, and references therein. [4] R.J. Bartlett and G.D. Purvis, Intern. J. Quantum Chem. 14 (1978) 561; R.J. Bartlett, Ann. Rev. Phys. Chem. 32 ( 1981) 359, and references therein. [ 51 Y.S. Lee, S.A. Kucharski and R.J. Bartlett, J. Chem. Phys. 81 (1984) 4371. [ 6 ] M. Urban, J. Noga, S.J. Cole and R.J. Bartlett, J. Chem Phys. 83 (1985) 4041; Erratum 85 (1986) 5383.
[7] J. Noga, R.J. Bartlett and M. Urban, Chem. Phys. Letters 134 (1987) 126; J. Noga and R.J. Bartlett, J. Chem. Phsy. 86 (1987) 7041. IS] S.A. Kucharski and R.J. Bartlett, Advan. Quantum Chem. 8 (1986) 281. [9] Y.S. Lee and R.J. Bartlett, J. Chem. Phys. 80 (1984) 4371. [ 101V.E. Bondybey, Chem. Phys. Letters, 109 (1984) 436. [ 111 R.J. Bartlett, H. Sekino and G.D. Purvis, Chem. Phys. Letters 98 (1983) 66. [ 121 R.J. Harrison and NC. Handy, Chem. Phys. Letters 98 (1983) 97. [ 131 B.H. Lengsfield III, A.D. McLean, M. Yoshimine and B. Liu, J. Chem. Phys. 78 (1983) 1891. [ 141S.F. Boys and F. Bemardi, Mol. Phys. 19 (1970) 553; D.W. Schwenke and D.C. Truhlar, J. Chem. Phys. 82 (1985) 2418; G.H.F. Diercksen and A.J. Sadlej, Mol. Phys. 59 ( 1986) 889. [ 151P.J. Knowles, K. Somasundram, N.C. Handy and K. Hirao, Chem. Phys. Letters 113 ( 1985) 8, [ 161 W.D. Laidig, G. Fitzgerald and R.J. Bartlett, Chem. Phys. Letters 113 (1985) 151. [ 171G.H.F. Diercksen and W.P. Kraemer, Munich Molecular Program System, Max-Planck-Institut fur Physik und Astrophysik, Munich, FRG. [ 181 M. Urban, I. Hubac, V. Kellii and J. Noga, J. Chem. Phys. 72 (1980) 3378; J. Noga, Computer Phys. Commun. 29 (1983) 117.