Explicit Electron Correlation by a Combined Use of Gaussian-Type Orbitals and Gaussian-Type Geminals Pal Dahle*'^, Trygve Helgaker^, Dan Jonsson** and Peter R. Taylor^ *Norwegian Computing Center, Gaustadalleen 23, N-0314 Oslo, Norway "^Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, P. O. Box 1033 Blindern, N-0315 Norway ** Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden ^Department of Chemistry, University of Warwick, Coventry CV4 7AL, UK Abstract. In MP2-GG« theory, the second-order Meller-Plesset (MP2) energy is recovered by a combined Gaussianorbital—Gaussian-geminal (GG) expansion of the first-order wave function. While the restriction of geminals to doubly occupied orbital pairs (MP2-GG0) provides a modest improvement on standard MP2 theory, their inclusion also in all singlyexcited (MP2-GG1) and doubly-excited (MP2-GG2) pairs recovers essentially all of the correlation energy in small (doubleand triple-zeta) basis sets. For several small systems, our MP2-GG« energies represent the best MP2 energies reported in the literature. Keywords: explicit electron correlation, Gaussian-type geminals, Meller-Plesset theory PACS: 31.15.Md, 31.25.-V
INTRODUCTION In standard orbital-based wave-function theory, the convergence of the correlation energy with respect to the size of the orbital-basis is very slow [1]. Specifically, if correlation-consistent basis sets such as aug-cc-pCVXZ are used for the expansion [2], then the error in the correlation energy is proportional to X^^. Although it is possible to improve on this convergence by extrapolation, a more fundamental solution is to improve upon the description of the electronic system by including the interelectronic distances exphcitly in the wave function [3]. In 1982, Szalewicz and coworkers proposed the Gaussian-type-geminal (GTG) method, where use is made of Gaussian correlation factors exp(-7vrj2) [4, 5]. With this model, a number of accurate calculations have been carried out on small systems but apphcation to larger systems is difficult because of a nonlinear optimization of the variational parameters and the exphcit evaluation of three-electron integrals. An alternative technique is the R12 method, introduced a few years later by Kutzelnigg and Klopper [6, 7]. Using a linear r\2 correlation factor and avoiding many-electron integrals through the resolution of the identity, the R12 method is applicable to large systems and has recently been generahzed to correlation factors different from ri2 [3]. We here discuss an alternative model, which uses a mixed Gaussian-type-orbital (GTO) and GTG expansion, avoiding the nonlinear optimization of the GTG method but otherwise closely following the approach of Szalewicz and coworkers, with an exphcit evaluation of three-electron integrals [8, 9].
THEORETICAL BACKGROUND In Moller-Plesset theory, the zero-order system is represented the Hartree-Fock wave function, whose spin orbitals 0, are eigenfunctions of the Fock operator F[0,(1) = e,0,(I). To first order, the electrons are correlated pairwise, perturbing the pair functions in the manner 0y(l,2) = 0,(1)0,(2)-0,(1)0,(2)
^
0y(l,2) + ei2My(I,2)
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CP963, Vol. 2 Part A, Computation in Modern Science and Engineering, Proceedings of the International Conference on Computational Methods in Science and Engineering 2007, edited by T. E. Simos and G. Maroulis © 2007 American Institute of Physics 978-0-7354-0478-6/07/$23.00
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(1)
where the first-order pair function Ujj may or may not depend on the interelectronic distance rn and where the strongorthogonality (SO) operator Qu ensures orthogonality to all occupied pairs QU = [1 -PoccW]
[1 -Pocci2)]
,
Pocc = I , m
{(p,\
(2)
Upon determining the first-order correction by minimizing the Hylleraas functional J[uij] = 2{uij iQui-u I 0y) + («y |6i2(Fi +F2 -£i-£j)Qn\uij)
(3)
the MP2 correlation energy is obtained as the sum of pair energies £'corr = l y % ,
Ey = ( " y 1612^2^ | 0 y )
(4)
where M,y is the minimizer of the SO functional Eq. (3). In standard MP2 theory, the pair function is expanded in virtual spin-orbital products M,y = Y.abC"f 0
(7)
which provides an upper bound to the SO functional W[u,j] >J[u,j] >e,j
(8)
Equality holds for the exact pair function and orthogonality is controlled by a penalty function. The GTG model is highly flexible but relies on a difficult nonlinear optimization of exponents and centers. To avoid this optimization, we have proposed a mixed GTO-GTG (GG) model [9]:
= labCfhab +
Ipgl^cff'^expi-yyu) "pq
(9)
using standard atom-fixed GTOs and fixed GTG exponents, with Cfj' and c^^'" determined from the WO functional. There are three distinct levels of theory: lnMP2-GG0 theory, only "ground-state" geminals Qxp^-yvr^jj^ki are used; in MP2-GG1 theory, we include also "singly-excited" geminals Qxp^-yyrljj^af, in MP2-GG2 theory, all "doublyexcited" geminals exp(-7vrj2)0afe are included. The integer n in MP2-GG« thus indicates the excitation level of explicit correlation. These levels of theory have been implemented in the GREMLIN [10] program.
FIGURE 1. The MP2 correlation energy (in m^h) recovered by standard MP2 theory (full), MP2-GG0 (dotted), MP2-GG1 (dashed), and MP2-GG2 (dash-dotted) in the aug-cc-CVXZ basis sets
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TABLE 1. The proportion of the all-electron MP2 correlation energy (%) recovered for Ne, HF, and H2O, using standard MP2 theory (with and without extrapolation) and MP2GG« theory standard MP2
extr MP2
MP2-GG0
MP2-GG1
MP2-GG2
68.2 88.3 94.8
96.8 99.6
87.8 96.3 98.5
97.2 99.8
99.2 100.0
aug-cc-pCVDZ aug-cc-pCVTZ aug-cc-pCVQZ
NUMERICAL ILLUSTRATIONS We have applied the MP2-GG« method to a number of small systems [9]—see Fig. 1, where we present the Ne results. The MP2-GG0 model reduces the standard MP2 correlation error by a factor of three or more—^perhaps less than hoped for However, when exphcit correlation is introduced into excited pairs at the MP2-GG1 level, a dramatic improvement is observed relative to standard MP2 theory. The MP2-GG2 model recovers more than 99% of the correlation energy in the smallest basis. In Table 1, we have hsted the average proportion of the correlation energy recovered for Ne, HF, and H2O in the aug-cc-pCVXZ basis sets. As seen from this table, the MP2-GG0 model performs no better than standard MP2 theory with X^^ extrapolation. By contrast, the MP2-GG2 model recovers 99.2% and 100.0% of the correlation energy, respectively, in the double- and triple-zeta basis sets. In Table 2, we compare our results with the available literature data. For Ne, H2 and HF, our MP2 energies are more accurate than previous literature data. A comparison of MP2-GG« and R12 theories has been presented in Ref [9], suggesting that, for small basis sets, the MP2-GG0 model recovers less correlation energy than do current R12 models. It is an open question, however, whether this occurs because of the use of the WO rather than SO functional in MP2-GG« theory or through an error cancellation in R12 theory.
CONCLUSION We have presented the mixed GTO-GTG (GG) model for the calculation of correlation energies at the MP2 level of theory, demonstrating that highly accurate energies can be obtained provided the Gaussian correlation functions are introduced in excited pair functions.
ACKNOWLEDGMENTS This work has been supported by the Norwegian Research Council (Grants No. 125903/410 and No. 154011/420), by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), by the Wolfson Foundation through the Royal Society, and by the National Science Foundation (Grant No. CHE-9700627 and Cooperative Agreement No. DACI-96I9020).
TABLE 2. All-electron MP2 correlation energies (niBj,) of some small systems MP2-GG2 He Be Ne H2 LiH HF
37.37729 76.355 388.19 34.252 72.877" 384.41
current best 37.37747* 76.358^^ 388.19 34.252 72.890^^ 384.41
" MP2-GG1 * GTG by Patkowski et al. [11] '^ GTG by Bukowski et al. [12]
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REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
T. Helgaker, P. Jergensen and J. Olsen, Molecular Structure Theory, John Wiley & Sons, Chichester, 2000. R. A. Kendall, T. H. Dunning Jr. and R. J. Harrison, J. Chem. Pliys. 96, 6796 (1992). W. Klopper, F. R. Manby, S. Ten-no and E. F. Valeev, Int Rev. Phys. Chem. 25, 427 (2006). K. Szalewicz, B. Jeziorski, H. J. Monkhorst and J. G. Zabolitzky, Chem. Phys. Lett 91, 169 (1982). K. Szalewicz, B. Jeziorski, H. J. Monkhorst and J. G. Zabolitzky, J. Chem. Phys. 78, 1420 (1983). W. Kutzelnigg, Theoret Chim. Acta 68,445 (1985). W. Klopper and W. Kutzelnigg, Chem. Phys Lett 134, 17 (1986). B. J. Persson and P R. Taylor, J. Chem. Phys. 105, 5915 (1996). P Dahle, T. Helgaker, D. Jonsson and P R. Taylor, Phys Chem. Chem. Phys 9,3112 (2007). GREMLIN, a program for calculating integrals over Gaussian-type geminals. Release LO., P. Dahle and T. Helgaker, 2002. K. Patkowski, W. Cencek, M. Jeziorska, B. Jeziorski and K. Szalewicz, J. Phys. Chem. A (2007), in press. R. Bukowski, B. Jeziorski and K. Szalewicz, J. Chem. Phys. 110,4165 (1999).
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