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Comparison of full-configuration interaction and coupled-cluster harmonic and fundamental frequencies for BH and HF
6 July 2001
Chemical Physics Letters 342 (2001) 200±206
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Comparison of full-con®guration interaction and coupled-cluster harmonic and fundamental frequencies for BH and HF Helena Larsen a, Jeppe Olsen a, Poul Jùrgensen a,*, J urgen Gauss b a
Department of Chemistry, Aarhus University, Langelandsgade 140, DK-8000 Arhus C, Denmark b Institut fur Physikalische Chemie, Universitat Mainz, D-55099 Mainz, Germany Received 23 February 2001
Abstract The harmonic and fundamental frequencies are calculated for the potential-energy curves of BH and HF using the full-con®guration interaction model and two hierarchies of coupled-cluster wavefunction models. The anharmonic contributions are also obtained using second-order vibrational perturbation theory. A consistent and systematic improvement is seen for both the harmonic and anharmonic contributions when increasing the level of the correlation treatment. The changes are largest for the harmonic contributions. This is also the case when including valence or diuse functions in the basis set. Second-order perturbation theory gives a good approximation to the anharmonic contribution and introduces errors that are small compared to the ones that occur in the harmonic frequencies. Without signi®cant loss of accuracy the anharmonic contribution can be computed at a lower level of correlation and using smaller basis sets than for the harmonic contribution. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction During the last decade, the accuracy with which fundamental vibrational frequencies can be calculated have increased tremendously [1,2]. Thus, highly correlated wavefunction models such as the CCSD(T) model can now routinely be used to obtain force ®elds and anharmonicity eects. In a recent issue of Spectrochimica Acta dedicated to the calculation of vibrational frequencies, Lee thus states in the preface that ab initio vibrational frequencies can be determined on average to within
*
Corresponding author. Fax: +45-86-196199. E-mail address: [email protected] (P. Jùrgensen).
8 cm 1 in CCSD(T) calculations using large basis sets [3]. A more thorough understanding of the errors associated with the use of the CCSD(T) model can be obtained from a comparison with FCI results. The errors in the harmonic and anharmonic contribution to the fundamental frequencies can then separately be estimated and by carrying out calculations at higher correlated levels such as CCSDT and CCSDTQ, it can be deduced which level of approximation is required to obtain a given accuracy. The anharmonic contribution for larger molecules is usually determined using second-order vibrational perturbation theory starting from the harmonic oscillator rigid rotor approximation [4] as implemented in the SP E C T R O package [5] and more recently in AC E S II [6] using
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 5 6 4 - 4
H. Larsen et al. / Chemical Physics Letters 342 (2001) 200±206
analytical derivative techniques [7±10]. For diatomics, where it is simple to solve the nuclear Schr odinger equation and, therefore, to get the full numeric anharmonic contributions, the errors involved in getting the anharmonic contributions to second order may be estimated. In this Letter, we report calculations for the BH and HF molecules using the standard wavefunction models and the full-con®guration interaction (FCI) model. In particular, we examine the accuracy of the harmonic and anharmonic contributions at dierent levels of correlation, and compare the full numeric and the second-order estimates of the anharmonic contributions. We also consider the remaining error when using the CCSD(T) model by comparing with the higher-order correlation methods CCSDT and CCSDTQ. Our analysis is an extension of the one by Martin [11,12] who has compared CCSD(T) and FCI results for BH and HF and carried out CCSD(T) calculations with large basis sets to establish basis-set limit results. Two wavefunction model hierarchies have been considered: self-consistent ®eld (SCF) ± secondorder Mùller±Plesset perturbation theory (MP2) ± coupled-cluster singles and doubles (CCSD) ± CCSD with an approximate triples correction [CCSD(T)] ± CCSD with triples (CCSDT) ± CCSDT with quadruples (CCSDTQ) and SCF± CC2±CCSD±CC3±CCSDT±CCSDTQ, where the CC2 and CC3 models contain similar approximations in the doubles and triples equations, respectively [13,14]. In both hierarchies, the energy is determined through increasing order in perturbation theory. The MP2 [15] and the celebrated CCSD(T) models [16] are non-iterative models which only can be used for static ground-state properties as opposed to the iterative CC2 and CC3 models which may also be used to compute frequency-dependent ground-state properties as well as excited-state properties [13,14]. Thus, it is of interest to see how the CC2 and CC3 models perform compared to the MP2 and CCSD(T) models. We also examine the triples contributions recovered by the CC3 and CCSD(T) models in comparison with the CCSDT results, and we investigate how much of the remaining correlation is recovered by the inclusion of quadruples via the CCSDTQ model. The core-correlation eect is
201
investigated by comparing all-electron and valence-electron calculations. Finally, the eects of including diuse functions in the basis sets as well as the eects of increasing the cardinal number of the basis sets are investigated. 2. Computational details The harmonic and anharmonic contributions to the fundamental frequencies have been computed for the ground states of BH and HF using the wavefunction models SCF, MP2, CC2, CCSD, CCSD(T), CC3, CCSDT, CCSDTQ, and FCI. Harmonic frequencies were obtained from second derivatives which were computed for CCSDT, CCSDTQ and FCI by means of numerical dierentiation, while analytic techniques [7±10] have been used for the other wavefunction models. Fundamental frequencies were obtained in two dierent ways: 1. By solving the nuclear Schr odinger equation numerically employing the corresponding potential-energy curves. 2. By estimating the anharmonic correction via second-order (SO) vibrational perturbation theory [4] using analytic derivative techniques [17,18]. The correlation-consistent valence [19], corevalence [20,21] and augmented-valence [22] basis sets of Dunning et al. were used. All electrons were correlated, except in the HF calculations with the cc-pVDZ basis set where the 1s canonical HF orbital was frozen. Up to the CC3 level the energies for the potential-energy curves were obtained using a local version of the DA L T O N program [23]. The CCSDT and CCSDTQ energies were computed using the general coupled-cluster code [24] in the LU C I A program that was also used to obtain the FCI results [25]. The harmonic frequencies (up to the CC3 level) and all the second-order anharmonic contributions were obtained using a local version of the AC E S II program [6]. 3. Results and discussion In Tables 1±5 the results for the harmonic, anharmonic (exact), anharmonic (SO) and
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H. Larsen et al. / Chemical Physics Letters 342 (2001) 200±206
fundamental frequencies are given for the various wavefunction models and basis sets. For comparison, the experimental values of the harmonic and anharmonic contributions are 2366.9 and 98:78 cm 1 for BH and 4138.32 and
179:88 cm 1 for HF, respectively [26]. Although we have not used large basis sets, we still see that our results give reasonable values of the harmonic and anharmonic contributions. The deviations from the experimental values are 11.7 and
Table 1 Vibrational frequencies (cm 1 ) for BH obtained for various wavefunction models using the cc-pCVDZ basis set SCF MP2 CC2 CCSD CCSD(T) CC3 CCSDT CCSDTQ FCI
Harmonic
Anharmonic
Anharmonic (SO)
Fundamental
2489.1 2419.9 2413.8 2350.9 2342.1 2341.9 2339.6 2340.1 2340.1
)87.1 )87.3 )88.5 )94.8 )96.1 )96.2 )96.7a )96.6a )96.6a
)89.1 )89.0 )90.1 )96.1 )97.3 )97.3 ± ± ±
2402.0 2332.6 2325.4 2256.0 2246.0 2245.8 2242.9 2243.5 2243.5
a
Including one more digit the values of the anharmonic contributions are )96.69, )96.64 and )96.65 for CCSDT, CCSDTQ and FCI, respectively.
Table 2 Vibrational frequencies (cm 1 ) for BH obtained for various wavefunction models using the aug-cc-pCVDZ basis set SCF MP2 CC2 CCSD CCSD(T) CC3 CCSDT CCSDTQ FCI
Harmonic
Anharmonic
Anharmonic (SO)
Fundamental
2476.7 2405.3 2400.0 2333.5 2323.7 2323.6 2320.7 2320.8 2320.8
)84.4 )87.1 )88.0 )94.9 )96.2 )96.2 )96.7a )96.7a )96.7a
)85.8 )88.3 )89.1 )95.8 )96.9 )96.9 ± ± ±
2392.3 2318.1 2312.0 2238.6 2227.5 2227.5 2224.0 2224.1 2224.1
a
Including one more digit the values of the anharmonic contributions are )96.69, )96.65 and )96.66 for CCSDT, CCSDTQ and FCI, respectively.
Table 3 Vibrational frequencies (cm 1 ) for BH obtained for various wavefunction models using the cc-pCVTZ basis set SCF MP2 CC2 CCSD CCSD(T) CC3 CCSDT CCSDTQ FCI a
Harmonic
Anharmonic
Anharmonic (SO)
Fundamental
2481.6 2435.3 2428.6 2367.8 2357.3 2357.1 2355.5 2355.2 2355.2
)83.9 )86.9 )88.8 )94.7 )96.2 )96.1 )96.6a )96.6a )96.6a
)85.4 )89.5 )90.5 )96.3 )97.7 )97.7 ± ± ±
2397.7 2348.4 2339.8 2273.2 2261.1 2261.0 2258.9 2258.6 2258.6
Including one more digit the values of the anharmonic contributions are )96.63, )96.61 and )96.61 for CCSDT, CCSDTQ and FCI, respectively.
H. Larsen et al. / Chemical Physics Letters 342 (2001) 200±206
203
Table 4 Vibrational frequencies (cm 1 ) for HF obtained for various wavefunction models using the cc-pVDZ basis set SCF MP2 CC2 CCSD CCSD(T) CC3 CCSDT CCSDTQ FCI
Harmonic
Anharmonic
Fundamental
4440.8 4170.0 4145.2 4169.0 4150.2 4149.1 4148.5 4143.7 4143.5
)169.2 )172.0 )175.3 )180.1 )183.0 )182.8 )183.1a )183.3a )183.3a
4271.6 3998.0 3969.9 3988.9 3967.2 3966.3 3965.4 3960.4 3960.2
a
Including one more digit the values of the anharmonic contributions are )183.10, )183.32 and )183.34 for CCSDT, CCSDTQ and FCI, respectively.
Table 5 Vibrational frequencies (cm 1 ) for HF obtained for various wavefunction models using the cc-pCVDZ basis set SCF MP2 CC2 CCSD CCSD(T) CC3 CCSDT CCSDTQ FCI
Harmonic
Anharmonic
Anharmonic (SO)
Fundamental
4440.6 4170.8 4145.6 4171.7 4151.9 4150.8 4150.3 4145.5 4145.3
)169.4 )172.0 )175.4 )180.1 )182.9 )182.7 )183.2a )183.4a )183.4a
)174.1 )175.8 )178.9 )183.0 )185.6 )185.5 ± ± ±
4271.2 3998.7 3970.3 3991.6 3969.0 3968.1 3967.1 3962.1 3961.9
a
Including one more digit the values of the anharmonic contributions are )183.15, )183.40 and )183.39 for CCSDT, CCSDTQ and FCI, respectively.
2:2 cm 1 (BH/FCI/cc-pCVTZ) and 7.0 and 3:5 cm 1 (HF/FCI/cc-pCVDZ) for the harmonic and anharmonic contributions, respectively. In the following sections, we ®rst discuss variations of the harmonic and anharmonic contributions with the wavefunction models and basis sets as well as the validity of using the second-order approximation for the anharmonic contribution.
both the harmonic and the anharmonic contributions a smooth and consistent improvement is seen (except for CC2 in case of HF). Thus, for BH
3.1. Wavefunction models We ®rst consider the general convergence towards the FCI results within the SCF-MP2/CC2± CCSD±CCSD(T)/CC3±CCSDT±CCSDTQ hierarchies. In Figs. 1 and 2 the deviations from FCI have been plotted as a function of the wavefunction model for the harmonic and anharmonic contributions, respectively, for each of the molecules (BH/cc-pCVTZ and HF/cc-pCVDZ). For
Fig. 1. The deviation from FCI (model-FCI) (cm 1 ) for the harmonic and anharmonic contributions to the fundamental frequency of BH (cc-pCVTZ) as a function of wavefunction model.
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H. Larsen et al. / Chemical Physics Letters 342 (2001) 200±206
Fig. 2. The deviation from FCI (model-FCI) for each wavefunction model (cm 1 ) for the harmonic and anharmonic contributions to the fundamental frequency of HF (cc-pCVDZ) as a function of wavefunction model.
(CC3/cc-pCVTZ) as well as HF (CC3/cc-pCVDZ) the deviations are less than 0.1% and 0.4% for the harmonic and anharmonic contributions, respectively. For the anharmonic contribution, the improvements from CCSD (and in particular CC3/ CCSD(T)) and onwards are small compared to the improvements in the harmonic contribution. This is clearly seen in Figs. 1 and 2. When comparing the performance of MP2 to that of CC2, it is seen that the harmonic CC2 frequency for HF is (accidentally) very close to the FCI result while MP2 is about 25 cm 1 above. However, for BH the CC2 result is only slightly better than the MP2 result and both results are much further away from the FCI value than the results for the higher correlated models. The CC3 and CCSD(T) results have similar accuracy for both the harmonic and the anharmonic contributions. Thus, it seems that CC2 and CC3 appear to be well-suited, though computationally more expensive, substitutes for MP2 and CCSD(T), as it has already been discussed in other references [27]. We next consider the triples contributions recovered by CCSD(T) and CC3 in comparison with the CCSDT results. For HF/cc-pCVDZ, the CCSD(T) and CC3 models recover 93% and 98%, respectively, of the triples harmonic contribution, whereas for the anharmonic contribution the recovery is 92% and 86%, respectively. For BH/ccpCVTZ the recovery is 86% [CCSD(T)], 87% (CC3), 79% [CCSD(T)] and 74% (CC3), for the
harmonic and anharmonic contributions, respectively. Using other basis sets the results are more or less the same. Thus, the major part of the triples contributions to the harmonic as well as the anharmonic part of the fundamental frequency is recovered by both models. In some cases, CC3 seems to be better than CCSD(T), in other cases it appears to be the opposite. Finally, we consider the quadruples contribution as obtained with the CCSDTQ model. For BH/cc-pCVTZ and HF/cc-pCVDZ quadruples recover 99.9% of the higher-order correlation contributions for the harmonic as well as the anharmonic contributions, whereas for HF/ccpCVDZ the quadruples recover 96% and 99.9% for the harmonic and anharmonic contributions, respectively. The errors in the CCSDTQ results are less than 0:2 cm 1 for the harmonic and less than 0:02 cm 1 for the anharmonic contributions. 3.2. The anharmonic corrections The anharmonic corrections are as expected negative. The anharmonic contributions obtained from second-order vibrational perturbation theory are more negative than the contributions obtained from the numerical solution of the nuclear Schr odinger equations. The dierence between the SO and numerical contributions decreases, as correlation is included and is, as expected, larger for HF than BH. For CC3, the dierence is 2:8 cm 1 for HF and up to 1:6 cm 1 for BH. The dierences between the SO and the numerical contributions are much smaller than errors in the harmonic frequencies. Thus, the second-order approximation is very accurate at least for diatomics. The errors due to the use of the SO approximation even tend to cancel to some extent with the higher-order correlation errors. As basis-set and correlation errors are small for the anharmonic contribution, this suggests that the computationally more expensive anharmonic contribution might be computed at a lower correlation level and in a smaller basis set than the harmonic contribution. Finally, it should be mentioned that the errors that are introduced by not including any anharmonic correction at all are signi®cant and non-negligible (4±5%).
H. Larsen et al. / Chemical Physics Letters 342 (2001) 200±206
3.3. Basis-set investigation Core-correlation eects have been investigated for HF for the fundamental frequency by comparing all electron cc-pCVDZ and valence electron cc-pVDZ calculations. The core-correlation eect is, as expected, small (less than 0.1% compared to the total frequency). However, it increases with correlation level. Thus, it is signi®cantly larger for CCSD and the correlated models upwards compared to MP2 and CC2. Including the corecorrelation eect raises the harmonic vibrational frequency by about 2 cm 1 while the anharmonic contribution is hardly aected. The eect of including diuse functions has been investigated for BH by comparing aug-ccpCVDZ and cc-pCVDZ calculations. The harmonic frequency becomes smaller when diuse functions are included and the change is signi®cantly larger than for the core-correlation eect ( 19 cm 1 for FCI). The change in the anharmonic contribution due to diuse functions, however, is very small (0:01 cm 1 for FCI). Improving the valence description by increasing the cardinal number in the BH calculations from D to T increases the harmonic frequency by a similar amount (15 cm 1 for FCI), as inclusion of diuse functions (i.e., adding one augmentation level) decreases the corresponding values. The changes in the anharmonic corrections are again small (0:04 cm 1 for FCI).
205
seen beyond the CCSD(T)/CC3 levels. For CCSD(T), the deviation from the FCI results is less than 8 cm 1 and, therefore, in accordance with the error stated for the CCSD(T) model in [3]. The harmonic frequency is changed considerably when changing the basis set, whereas only a minor change is seen for the anharmonic contribution. Comparing the exact anharmonic contribution with the approximate one obtained from secondorder vibrational perturbation theory, it is seen that only small errors are introduced as compared to the ones seen in the harmonic frequency part. Extending the basis set to include more diuse functions or to obtain a better valence description by increasing the cardinal number leads to significant changes in the harmonic frequencies, whereas the corresponding changes in the anharmonic contribution are minor. In conclusion, the anharmonic contribution can be computed without a signi®cant loss of accuracy at the lower level of correlation using smaller basis sets than the harmonic part. Acknowledgements This work has been supported by the Danish Research Council (Grant No. 9901973). In addition, J.G. thanks the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie for ®nancial support. We would like to thank an anonymous referee for drawing our attention to essential references.
4. Concluding remarks In the hierarchies SCF±MP2/CC2±CCSD± CCSD(T)/CC3±CCSDT±CCSDTQ±FCI, a consistent and systematic improvement is seen when going to higher levels of the correlation treatment. The CC2 and CC3 models appear to be suitable, though computationally more expensive substitutes, for the MP2 and CCSD(T) models. A considerable improvement in the harmonic part is obtained when going from CCSD(T)/CC3 to CCSDT where the full triples contribution is included, and for HF also at the CCSDTQ level where quadruples are introduced. For the anharmonic contribution, little or no improvement is
References [1] S.R. Langho (Ed.), Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy, Kluwer Academic Publishers, Dordrecht, 1995, p. 47. [2] For further references, see the special issues Spectrochimica Acta, 53 A, (1997), Spectrochimica Acta 55 A, (1999). [3] Preface in Spectrochimica Acta, 53 A, vii (1997). [4] I.M. Mills, in: K.N. Rao, C.W. Matthews (Eds.), Modern Spectroscopy: Modern Research, Academic Press, New York, 1972, p. 115. [5] SP E C T R O , written by J.F. Gaw et al. [6] J.F. Stanton, J. Gauss, J.D. Watts, W.J. Lauderdale, R.J. Bartlett, Int. J. Quantum Chem. Symp. 26 (1992) 879. [7] J.A. Pople, R. Krishnan, H.B. Schlegel, J.S. Binkley, Int. J. Quantum Chem. Symp. 13 (1979) 225.
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[8] N.C. Handy, R.D. Amos, J.F. Gaw, J.E. Rice, E.D. Simandiras, Chem. Phys. Lett. 120 (1985) 151. [9] J. Gauss, J.F. Stanton, Chem. Phys. Lett. 276 (1997) 70. [10] J. Gauss, J.F. Stanton, Phys. Chem. Chem. Phys. 2 (2000) 2047. [11] J.M.L. Martin, Chem. Phys. Lett. 283 (1998) 283. [12] J.M.L. Martin, Chem. Phys. Lett. 292 (1998) 411. [13] O. Christiansen, H. Koch, P. Jùrgensen, Chem. Phys. Lett. 243 (1995) 409. [14] H. Koch, O. Christiansen, P. Jùrgensen, A. Sanchez, de Meras, T. Helgaker, J. Chem. Phys. 106 (1997) 1808. [15] C. Mùller, M.S. Plesset, Phys. Rev. 46 (1934) 618. [16] K. Raghavachari, G.W. Trucks, J.A. Pople, M. HeadGordon, Chem. Phys. Lett. 157 (1989) 479. [17] W. Schneider, W. Thiel, Chem. Phys. Lett. 157 (1989) 367. [18] J.F. Stanton, C.L. Lopreore, J. Gauss, J. Chem. Phys. 108 (1998) 7190.
[19] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [20] R.A. Kendall, T.H. Dunning Jr., R.J. Harrison, J. Chem. Phys. 96 (1992) 6796. [21] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 100 (1994) 2975. [22] D.E. Woon, T.H. Dunning, J. Chem. Phys. 103 (1995) 4572. [23] An electronic structure program, Release 1.0 (1997) written by T. Helgaker et al. [24] J. Olsen, J. Chem. Phys. 113 (2000) 7140. [25] LU C I A , a full CI, restricted active space program by J. Olsen, Aarhus, with contributions from H. Larsen. [26] K.P. Huber, G.H. Herzberg, Molecular Spectra and Molecular Structure. Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979. [27] H. Larsen, J. Olsen, P. Jùrgensen, O. Christiansen, J. Chem. Phys. 113 (2000) 6677.
Chemical Physics Letters 342 (2001) 200±206
www.elsevier.com/locate/cplett
Comparison of full-con®guration interaction and coupled-cluster harmonic and fundamental frequencies for BH and HF Helena Larsen a, Jeppe Olsen a, Poul Jùrgensen a,*, J urgen Gauss b a
Department of Chemistry, Aarhus University, Langelandsgade 140, DK-8000 Arhus C, Denmark b Institut fur Physikalische Chemie, Universitat Mainz, D-55099 Mainz, Germany Received 23 February 2001
Abstract The harmonic and fundamental frequencies are calculated for the potential-energy curves of BH and HF using the full-con®guration interaction model and two hierarchies of coupled-cluster wavefunction models. The anharmonic contributions are also obtained using second-order vibrational perturbation theory. A consistent and systematic improvement is seen for both the harmonic and anharmonic contributions when increasing the level of the correlation treatment. The changes are largest for the harmonic contributions. This is also the case when including valence or diuse functions in the basis set. Second-order perturbation theory gives a good approximation to the anharmonic contribution and introduces errors that are small compared to the ones that occur in the harmonic frequencies. Without signi®cant loss of accuracy the anharmonic contribution can be computed at a lower level of correlation and using smaller basis sets than for the harmonic contribution. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction During the last decade, the accuracy with which fundamental vibrational frequencies can be calculated have increased tremendously [1,2]. Thus, highly correlated wavefunction models such as the CCSD(T) model can now routinely be used to obtain force ®elds and anharmonicity eects. In a recent issue of Spectrochimica Acta dedicated to the calculation of vibrational frequencies, Lee thus states in the preface that ab initio vibrational frequencies can be determined on average to within
*
Corresponding author. Fax: +45-86-196199. E-mail address: [email protected] (P. Jùrgensen).
8 cm 1 in CCSD(T) calculations using large basis sets [3]. A more thorough understanding of the errors associated with the use of the CCSD(T) model can be obtained from a comparison with FCI results. The errors in the harmonic and anharmonic contribution to the fundamental frequencies can then separately be estimated and by carrying out calculations at higher correlated levels such as CCSDT and CCSDTQ, it can be deduced which level of approximation is required to obtain a given accuracy. The anharmonic contribution for larger molecules is usually determined using second-order vibrational perturbation theory starting from the harmonic oscillator rigid rotor approximation [4] as implemented in the SP E C T R O package [5] and more recently in AC E S II [6] using
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 5 6 4 - 4
H. Larsen et al. / Chemical Physics Letters 342 (2001) 200±206
analytical derivative techniques [7±10]. For diatomics, where it is simple to solve the nuclear Schr odinger equation and, therefore, to get the full numeric anharmonic contributions, the errors involved in getting the anharmonic contributions to second order may be estimated. In this Letter, we report calculations for the BH and HF molecules using the standard wavefunction models and the full-con®guration interaction (FCI) model. In particular, we examine the accuracy of the harmonic and anharmonic contributions at dierent levels of correlation, and compare the full numeric and the second-order estimates of the anharmonic contributions. We also consider the remaining error when using the CCSD(T) model by comparing with the higher-order correlation methods CCSDT and CCSDTQ. Our analysis is an extension of the one by Martin [11,12] who has compared CCSD(T) and FCI results for BH and HF and carried out CCSD(T) calculations with large basis sets to establish basis-set limit results. Two wavefunction model hierarchies have been considered: self-consistent ®eld (SCF) ± secondorder Mùller±Plesset perturbation theory (MP2) ± coupled-cluster singles and doubles (CCSD) ± CCSD with an approximate triples correction [CCSD(T)] ± CCSD with triples (CCSDT) ± CCSDT with quadruples (CCSDTQ) and SCF± CC2±CCSD±CC3±CCSDT±CCSDTQ, where the CC2 and CC3 models contain similar approximations in the doubles and triples equations, respectively [13,14]. In both hierarchies, the energy is determined through increasing order in perturbation theory. The MP2 [15] and the celebrated CCSD(T) models [16] are non-iterative models which only can be used for static ground-state properties as opposed to the iterative CC2 and CC3 models which may also be used to compute frequency-dependent ground-state properties as well as excited-state properties [13,14]. Thus, it is of interest to see how the CC2 and CC3 models perform compared to the MP2 and CCSD(T) models. We also examine the triples contributions recovered by the CC3 and CCSD(T) models in comparison with the CCSDT results, and we investigate how much of the remaining correlation is recovered by the inclusion of quadruples via the CCSDTQ model. The core-correlation eect is
201
investigated by comparing all-electron and valence-electron calculations. Finally, the eects of including diuse functions in the basis sets as well as the eects of increasing the cardinal number of the basis sets are investigated. 2. Computational details The harmonic and anharmonic contributions to the fundamental frequencies have been computed for the ground states of BH and HF using the wavefunction models SCF, MP2, CC2, CCSD, CCSD(T), CC3, CCSDT, CCSDTQ, and FCI. Harmonic frequencies were obtained from second derivatives which were computed for CCSDT, CCSDTQ and FCI by means of numerical dierentiation, while analytic techniques [7±10] have been used for the other wavefunction models. Fundamental frequencies were obtained in two dierent ways: 1. By solving the nuclear Schr odinger equation numerically employing the corresponding potential-energy curves. 2. By estimating the anharmonic correction via second-order (SO) vibrational perturbation theory [4] using analytic derivative techniques [17,18]. The correlation-consistent valence [19], corevalence [20,21] and augmented-valence [22] basis sets of Dunning et al. were used. All electrons were correlated, except in the HF calculations with the cc-pVDZ basis set where the 1s canonical HF orbital was frozen. Up to the CC3 level the energies for the potential-energy curves were obtained using a local version of the DA L T O N program [23]. The CCSDT and CCSDTQ energies were computed using the general coupled-cluster code [24] in the LU C I A program that was also used to obtain the FCI results [25]. The harmonic frequencies (up to the CC3 level) and all the second-order anharmonic contributions were obtained using a local version of the AC E S II program [6]. 3. Results and discussion In Tables 1±5 the results for the harmonic, anharmonic (exact), anharmonic (SO) and
202
H. Larsen et al. / Chemical Physics Letters 342 (2001) 200±206
fundamental frequencies are given for the various wavefunction models and basis sets. For comparison, the experimental values of the harmonic and anharmonic contributions are 2366.9 and 98:78 cm 1 for BH and 4138.32 and
179:88 cm 1 for HF, respectively [26]. Although we have not used large basis sets, we still see that our results give reasonable values of the harmonic and anharmonic contributions. The deviations from the experimental values are 11.7 and
Table 1 Vibrational frequencies (cm 1 ) for BH obtained for various wavefunction models using the cc-pCVDZ basis set SCF MP2 CC2 CCSD CCSD(T) CC3 CCSDT CCSDTQ FCI
Harmonic
Anharmonic
Anharmonic (SO)
Fundamental
2489.1 2419.9 2413.8 2350.9 2342.1 2341.9 2339.6 2340.1 2340.1
)87.1 )87.3 )88.5 )94.8 )96.1 )96.2 )96.7a )96.6a )96.6a
)89.1 )89.0 )90.1 )96.1 )97.3 )97.3 ± ± ±
2402.0 2332.6 2325.4 2256.0 2246.0 2245.8 2242.9 2243.5 2243.5
a
Including one more digit the values of the anharmonic contributions are )96.69, )96.64 and )96.65 for CCSDT, CCSDTQ and FCI, respectively.
Table 2 Vibrational frequencies (cm 1 ) for BH obtained for various wavefunction models using the aug-cc-pCVDZ basis set SCF MP2 CC2 CCSD CCSD(T) CC3 CCSDT CCSDTQ FCI
Harmonic
Anharmonic
Anharmonic (SO)
Fundamental
2476.7 2405.3 2400.0 2333.5 2323.7 2323.6 2320.7 2320.8 2320.8
)84.4 )87.1 )88.0 )94.9 )96.2 )96.2 )96.7a )96.7a )96.7a
)85.8 )88.3 )89.1 )95.8 )96.9 )96.9 ± ± ±
2392.3 2318.1 2312.0 2238.6 2227.5 2227.5 2224.0 2224.1 2224.1
a
Including one more digit the values of the anharmonic contributions are )96.69, )96.65 and )96.66 for CCSDT, CCSDTQ and FCI, respectively.
Table 3 Vibrational frequencies (cm 1 ) for BH obtained for various wavefunction models using the cc-pCVTZ basis set SCF MP2 CC2 CCSD CCSD(T) CC3 CCSDT CCSDTQ FCI a
Harmonic
Anharmonic
Anharmonic (SO)
Fundamental
2481.6 2435.3 2428.6 2367.8 2357.3 2357.1 2355.5 2355.2 2355.2
)83.9 )86.9 )88.8 )94.7 )96.2 )96.1 )96.6a )96.6a )96.6a
)85.4 )89.5 )90.5 )96.3 )97.7 )97.7 ± ± ±
2397.7 2348.4 2339.8 2273.2 2261.1 2261.0 2258.9 2258.6 2258.6
Including one more digit the values of the anharmonic contributions are )96.63, )96.61 and )96.61 for CCSDT, CCSDTQ and FCI, respectively.
H. Larsen et al. / Chemical Physics Letters 342 (2001) 200±206
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Table 4 Vibrational frequencies (cm 1 ) for HF obtained for various wavefunction models using the cc-pVDZ basis set SCF MP2 CC2 CCSD CCSD(T) CC3 CCSDT CCSDTQ FCI
Harmonic
Anharmonic
Fundamental
4440.8 4170.0 4145.2 4169.0 4150.2 4149.1 4148.5 4143.7 4143.5
)169.2 )172.0 )175.3 )180.1 )183.0 )182.8 )183.1a )183.3a )183.3a
4271.6 3998.0 3969.9 3988.9 3967.2 3966.3 3965.4 3960.4 3960.2
a
Including one more digit the values of the anharmonic contributions are )183.10, )183.32 and )183.34 for CCSDT, CCSDTQ and FCI, respectively.
Table 5 Vibrational frequencies (cm 1 ) for HF obtained for various wavefunction models using the cc-pCVDZ basis set SCF MP2 CC2 CCSD CCSD(T) CC3 CCSDT CCSDTQ FCI
Harmonic
Anharmonic
Anharmonic (SO)
Fundamental
4440.6 4170.8 4145.6 4171.7 4151.9 4150.8 4150.3 4145.5 4145.3
)169.4 )172.0 )175.4 )180.1 )182.9 )182.7 )183.2a )183.4a )183.4a
)174.1 )175.8 )178.9 )183.0 )185.6 )185.5 ± ± ±
4271.2 3998.7 3970.3 3991.6 3969.0 3968.1 3967.1 3962.1 3961.9
a
Including one more digit the values of the anharmonic contributions are )183.15, )183.40 and )183.39 for CCSDT, CCSDTQ and FCI, respectively.
2:2 cm 1 (BH/FCI/cc-pCVTZ) and 7.0 and 3:5 cm 1 (HF/FCI/cc-pCVDZ) for the harmonic and anharmonic contributions, respectively. In the following sections, we ®rst discuss variations of the harmonic and anharmonic contributions with the wavefunction models and basis sets as well as the validity of using the second-order approximation for the anharmonic contribution.
both the harmonic and the anharmonic contributions a smooth and consistent improvement is seen (except for CC2 in case of HF). Thus, for BH
3.1. Wavefunction models We ®rst consider the general convergence towards the FCI results within the SCF-MP2/CC2± CCSD±CCSD(T)/CC3±CCSDT±CCSDTQ hierarchies. In Figs. 1 and 2 the deviations from FCI have been plotted as a function of the wavefunction model for the harmonic and anharmonic contributions, respectively, for each of the molecules (BH/cc-pCVTZ and HF/cc-pCVDZ). For
Fig. 1. The deviation from FCI (model-FCI) (cm 1 ) for the harmonic and anharmonic contributions to the fundamental frequency of BH (cc-pCVTZ) as a function of wavefunction model.
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H. Larsen et al. / Chemical Physics Letters 342 (2001) 200±206
Fig. 2. The deviation from FCI (model-FCI) for each wavefunction model (cm 1 ) for the harmonic and anharmonic contributions to the fundamental frequency of HF (cc-pCVDZ) as a function of wavefunction model.
(CC3/cc-pCVTZ) as well as HF (CC3/cc-pCVDZ) the deviations are less than 0.1% and 0.4% for the harmonic and anharmonic contributions, respectively. For the anharmonic contribution, the improvements from CCSD (and in particular CC3/ CCSD(T)) and onwards are small compared to the improvements in the harmonic contribution. This is clearly seen in Figs. 1 and 2. When comparing the performance of MP2 to that of CC2, it is seen that the harmonic CC2 frequency for HF is (accidentally) very close to the FCI result while MP2 is about 25 cm 1 above. However, for BH the CC2 result is only slightly better than the MP2 result and both results are much further away from the FCI value than the results for the higher correlated models. The CC3 and CCSD(T) results have similar accuracy for both the harmonic and the anharmonic contributions. Thus, it seems that CC2 and CC3 appear to be well-suited, though computationally more expensive, substitutes for MP2 and CCSD(T), as it has already been discussed in other references [27]. We next consider the triples contributions recovered by CCSD(T) and CC3 in comparison with the CCSDT results. For HF/cc-pCVDZ, the CCSD(T) and CC3 models recover 93% and 98%, respectively, of the triples harmonic contribution, whereas for the anharmonic contribution the recovery is 92% and 86%, respectively. For BH/ccpCVTZ the recovery is 86% [CCSD(T)], 87% (CC3), 79% [CCSD(T)] and 74% (CC3), for the
harmonic and anharmonic contributions, respectively. Using other basis sets the results are more or less the same. Thus, the major part of the triples contributions to the harmonic as well as the anharmonic part of the fundamental frequency is recovered by both models. In some cases, CC3 seems to be better than CCSD(T), in other cases it appears to be the opposite. Finally, we consider the quadruples contribution as obtained with the CCSDTQ model. For BH/cc-pCVTZ and HF/cc-pCVDZ quadruples recover 99.9% of the higher-order correlation contributions for the harmonic as well as the anharmonic contributions, whereas for HF/ccpCVDZ the quadruples recover 96% and 99.9% for the harmonic and anharmonic contributions, respectively. The errors in the CCSDTQ results are less than 0:2 cm 1 for the harmonic and less than 0:02 cm 1 for the anharmonic contributions. 3.2. The anharmonic corrections The anharmonic corrections are as expected negative. The anharmonic contributions obtained from second-order vibrational perturbation theory are more negative than the contributions obtained from the numerical solution of the nuclear Schr odinger equations. The dierence between the SO and numerical contributions decreases, as correlation is included and is, as expected, larger for HF than BH. For CC3, the dierence is 2:8 cm 1 for HF and up to 1:6 cm 1 for BH. The dierences between the SO and the numerical contributions are much smaller than errors in the harmonic frequencies. Thus, the second-order approximation is very accurate at least for diatomics. The errors due to the use of the SO approximation even tend to cancel to some extent with the higher-order correlation errors. As basis-set and correlation errors are small for the anharmonic contribution, this suggests that the computationally more expensive anharmonic contribution might be computed at a lower correlation level and in a smaller basis set than the harmonic contribution. Finally, it should be mentioned that the errors that are introduced by not including any anharmonic correction at all are signi®cant and non-negligible (4±5%).
H. Larsen et al. / Chemical Physics Letters 342 (2001) 200±206
3.3. Basis-set investigation Core-correlation eects have been investigated for HF for the fundamental frequency by comparing all electron cc-pCVDZ and valence electron cc-pVDZ calculations. The core-correlation eect is, as expected, small (less than 0.1% compared to the total frequency). However, it increases with correlation level. Thus, it is signi®cantly larger for CCSD and the correlated models upwards compared to MP2 and CC2. Including the corecorrelation eect raises the harmonic vibrational frequency by about 2 cm 1 while the anharmonic contribution is hardly aected. The eect of including diuse functions has been investigated for BH by comparing aug-ccpCVDZ and cc-pCVDZ calculations. The harmonic frequency becomes smaller when diuse functions are included and the change is signi®cantly larger than for the core-correlation eect ( 19 cm 1 for FCI). The change in the anharmonic contribution due to diuse functions, however, is very small (0:01 cm 1 for FCI). Improving the valence description by increasing the cardinal number in the BH calculations from D to T increases the harmonic frequency by a similar amount (15 cm 1 for FCI), as inclusion of diuse functions (i.e., adding one augmentation level) decreases the corresponding values. The changes in the anharmonic corrections are again small (0:04 cm 1 for FCI).
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seen beyond the CCSD(T)/CC3 levels. For CCSD(T), the deviation from the FCI results is less than 8 cm 1 and, therefore, in accordance with the error stated for the CCSD(T) model in [3]. The harmonic frequency is changed considerably when changing the basis set, whereas only a minor change is seen for the anharmonic contribution. Comparing the exact anharmonic contribution with the approximate one obtained from secondorder vibrational perturbation theory, it is seen that only small errors are introduced as compared to the ones seen in the harmonic frequency part. Extending the basis set to include more diuse functions or to obtain a better valence description by increasing the cardinal number leads to significant changes in the harmonic frequencies, whereas the corresponding changes in the anharmonic contribution are minor. In conclusion, the anharmonic contribution can be computed without a signi®cant loss of accuracy at the lower level of correlation using smaller basis sets than the harmonic part. Acknowledgements This work has been supported by the Danish Research Council (Grant No. 9901973). In addition, J.G. thanks the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie for ®nancial support. We would like to thank an anonymous referee for drawing our attention to essential references.
4. Concluding remarks In the hierarchies SCF±MP2/CC2±CCSD± CCSD(T)/CC3±CCSDT±CCSDTQ±FCI, a consistent and systematic improvement is seen when going to higher levels of the correlation treatment. The CC2 and CC3 models appear to be suitable, though computationally more expensive substitutes, for the MP2 and CCSD(T) models. A considerable improvement in the harmonic part is obtained when going from CCSD(T)/CC3 to CCSDT where the full triples contribution is included, and for HF also at the CCSDTQ level where quadruples are introduced. For the anharmonic contribution, little or no improvement is
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