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Use of Mo”Ller-Plesset Perturbation Theory in Molecular Calculations: Spectroscopic Constants of First Row Diatomic Molecules

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Use of Mo”Ller-Plesset Perturbation Theory in Molecular Calculations: Spectroscopic Constants of First Row Diatomic Molecules JOURNAL OF CHEMICAL PHYSICS VOLUME 108, NUMBER 12 22 MARCH 1998 Use of Mo”ller-Plesset perturbation theory in molecular calculations: Spectroscopic constants of first row diatomic molecules Thom H. Dunning, Jr. Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington 99352 Kirk A. Peterson Department of Chemistry, Washington State University and the Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington 99352 ~Received 9 December 1997; accepted 17 December 1997! The convergence of Mo”ller–Plesset perturbation expansions ~MP2–MP4/MP5! for the spectroscopic constants of a selected set of diatomic molecules ~BH, CH, HF, N2, CO, and F2! has been investigated. It was found that the second-order perturbation contributions to the spectroscopic constants are strongly dependent on basis set, more so for HF and CO than for BH. The MP5 contributions for HF were essentially zero for the cc-pVDZ basis set, but increased significantly with basis set illustrating the difficulty of using small basis sets as benchmarks for correlated calculations. The convergence behavior of the exact Mo”ller–Plesset perturbation expansions were investigated using estimates of the complete basis set limits obtained using large correlation consistent basis sets. For BH and CH, the perturbation expansions of the spectroscopic constants converge monotonically toward the experimental values, while for HF, N2, CO, and F2, the expansions oscillate about the experimental values. The perturbation expansions are, in general, only slowly converging and, for HF, N2, CO, and F2, appear to be far from convergence at MP4. In fact, for HF, N2, and CO, the errors in the calculated spectroscopic constants for the MP4 method are larger than those for the MP2 method ~the only exception is De!. The current study, combined with other recent studies, raises serious doubts about the use of Mo”ller–Plesset perturbation theory to describe electron correlation effects in atomic and molecular calculations. © 1998 American Institute of Physics. @S0021-9606~98!00912-X# I. INTRODUCTION authors found that classical ‘‘single configuration’’ systems such as hydrogen fluoride, with a weight of nearly 95% for The treatment of electron correlation in atomic and mo- the Hartree–Fock wave function in the MPn expansion, can lecular calculations is critical for an accurate prediction of lead to expansions that are either divergent or appear to di- molecular properties. Ignoring correlation effects can lead to verge in low orders of perturbation theory. The reason for errors of 100% or more in bond energies and reaction ener- this behavior has yet to be firmly established. Analyzing getics, 10%–20% in vibrational frequencies, and as much as their perturbation theory calculations on the neon atom, 5% or more in bond lengths. Mo”ller–Plesset perturbation Christiansen et al.11 found that the divergence was due to theory, as implemented in a number of popular quantum avoided crossings between the Hartree–Fock configuration 1 2 3 chemical packages ~GAUSSIAN, CADPAC, MOLPRO, ACES and states dominated by quintuple and higher excitations. 4 II, etc.!, especially second- through fourth-order perturba- Cremer and He5 noted that the convergence behavior of the tion ~MP2, MP3, MP4! theory, is the most widely used tech- perturbation expansion changed depending on the magnitude nique for quantifying the effects of electron correlation on and sign of the triple excitation terms. The pathology in the molecular properties. Techniques and applications software perturbation expansion appears to arise when multiple elec- 1 5 for computing MP5 and MP6 energies and wave functions tron pairs, e.g., the bond and lone pairs in HF, are in close are even coming into use, despite the dramatic increase in proximity. computational demands associated with higher orders of per- In this paper we examine the convergence of Mo”ller– turbation theory. Plesset perturbation theory for molecular properties. The per- Mo”ller–Plesset perturbation theory is based on a single turbation theory expansion of a molecular property, Q, can reference, Hartree–Fock wave function and it has long been be written as recognized that this imposes limits on the radius of conver- gence of the MPn expansions. The numerical implications of Q~MP`!5Q~HF!1DQ~MP2!1DQ~MP3! this limitation have been investigated for a number of ‘‘dif- 1DQ~MP4!1DQ~MP5!1 , ~1! ficult’’ small molecules.6–8 More recently, it has become ap- ¯ parent that convergence problems in the MPn expansion are where not limited to molecules whose zero-order wave function is poorly described by a single configuration.5,9,10 These latter DQ~MPn!5Q~MPn!2Q~MPn21! ~2! 0021-9606/98/108(12)/4761/11/$15.004761 © 1998 American Institute of Physics Downloaded 22 Feb 2001 to 192.101.100.146. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html 4762 J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 T. H. Dunning, Jr. and K. A. Peterson II. THEORETICAL AND COMPUTATIONAL CONSIDERATIONS The correlation consistent basis sets of Dunning and co-workers,12–17 labeled cc-pVnZ with n5D26, form a hi- erarchial family of basis sets that become more and more complete as n increases. As has been demonstrated in nu- merous benchmark studies,18–26 calculations with the corre- lation consistent basis sets exhibit systematic convergence toward their apparent CBS limits. In fact, the basis set con- vergence can often be modeled by a simple exponential function:27 2a~n22! Qn5Q`1DQ2e ~3! to obtain an estimate of the complete basis set limit, Q` ,of property Q. @Other extrapolation procedures have been ex- plored by Wilson and Dunning26 and Martin and co-workers FIG. 1. Definition of apparent and intrinsic errors for the computed quan- tity, Q. Types I and II refer to two distinct behaviors: convergence to a ~see, e.g., Ref. 28!.# In the present work Eq. ~3! was used to value which lies on the same side of the experimental value, or convergence estimate CBS limits whenever possible. Otherwise, the esti- to a value on the other side of the experimental value. Examples for both mates were taken from calculations with the cc-pV6Z basis types of convergence behavior are observed here. sets, which is very close to the CBS limit. We focus here on valence-electron calculations on BH, CH, HF, N2, CO, and F2. For atoms on the left-hand side of is the incremental change in Q from (n21)th order to nth the periodic chart, core and core-valence correlation effects order perturbation theory and Q(MPn) is the value of Q at are significant and their inclusion is essential to obtaining nth order perturbation theory. For the perturbation theory agreement with the experimental results. For example, for expansion, Eq. ~1!, to be a useful approach for describing the BH, all-electron calculations29 show that inclusion of core/ effect of electron correlation on property Q, the series must core-valence correlation effects increases De by 0.2 kcal/ 21 converge, i.e., limn `DQ(MPn)50, and must converge to mol, decreases re by 0.003 Å, and increases ve by 11 cm the exact solution→ of the electronic Schro¨dinger equation, ~at the MP4 level!. Thus, the intrinsic errors associated with i.e., limn `Q(MPn)5Q(exact). If a finite basis set is used, valence-electron calculations include this error in addition to then the→ series must converge to the full configuration inter- any errors arising from the limitations of the electronic struc- action value for Q in the chosen basis set. In this paper, we ture method, as well as those resulting from the breakdown examine the behavior of the perturbation series expansion, of the Born–Oppenheimer approximation and the neglect of Eq. ~1!, for the spectroscopic constants (De , re , ve , vexe , relativistic corrections. However, neglect of core/core- Be , ae) of a set of diatomic molecules ~BH, CH, HF, N2, valence correlation effects has essentially no effect on the 9 CO, and F2! chosen to be representative of a wide range of convergence behavior of the Mo”ller–Plesset perturbation ex- molecules. Computational restrictions limit our studies of pansion, which is the subject of interest here. CH, N2, CO, and F2 to the MP2–MP4 methods and of BH To minimize potential sources of experimental errors, and HF to the MP2–MP5 methods. As we shall see, the we have limited the current study to diatomic molecules for convergence behavior of the Mo”ller–Plesset perturbation ex- which accurate spectroscopic constants have been pansion alluded to above gives rise to significant problems in determined.30 For each molecular species, potential energy computing molecular properties. functions were calculated by fitting nine computed energies In characterizing the behavior of the perturbation expan- that covered a range in Dr(5r2re)of20.4a0<Dr< sions for molecular properties, it is important to distinguish 10.7a0 to seventh- or eighth-order polynomials in Dr. between the apparent and intrinsic accuracies of a MPn Spectroscopic constants were then determined from the fitted model. In Eq. ~1!, Q(MPn) should be obtained from the polynomial coefficients by a Dunham analysis.31 In the cal- exact solution of the MPn equations. This is the only way to culation of dissociation energies, the dissociated limits were establish the intrinsic accuracy of the MPn method. Past obtained from ROHF–MPn32 calculations on the isolated at- 1 studies of the convergence of the Mo”ller–Plesset expansion oms, except in the case of the MP5 calculations, where an were limited by the use of finite basis sets. In this case, one unrestricted Hartree–Fock ~UHF! based method was used. can only determine the apparent accuracy of a MPn method, Unfortunately, software limitations prevented us from carry- which may be very different than the intrinsic accuracy of ing out UHF-MP5 calculations on the boron and carbon at- the method.
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