Towards benchmark second-order correlation energies for large atoms. II. Angular extrapolation problems

Abstract

We have studied the use of the asymptotic expansions (AEs) for the angular momentum extrapolation (to l --> infinity) of atomic second-order Moller-Plesset (MP2) correlation energies of symmetry-adapted pairs (SAPs). The AEs have been defined in terms of partial wave (PW) increments to the SAP correlation energies obtained with the finite element MP2 method (FEM-MP2), as well as with the variational perturbation method in a Slater-type orbital basis. The method employed to obtain AEs from PW increments is general in the sense that it can be applied to methods other than MP2 and, if modified, to molecular systems. Optimal AEs have been determined for all types of SAPs possible in large atoms using very accurate FEM PW increments up to lmax = 45. The impact of the error of the PW increments on the coefficients of the AEs is computed and taken into account in our procedure. The first AE coefficient is determined to a very high accuracy, whereas the second involves much larger errors. The optimum l values (lopt) for starting the extrapolation procedures are determined and their properties, interesting from the practical point of view, are discussed. It is found that the values of the first AE coefficients obey expressions of the type derived by Kutzelnigg and Morgan [J. Chem. Phys. 96, 4484 (1992); 97, 8821(E) (1992)] for He-type systems in the bare-nucleus case provided they are modified by fractional factors in the case of triplet and unnatural singlet SAPs. These expressions give extremely accurate values for the first AE coefficient both for the STO and the FEM Hartree-Fock orbitals. We have compared the performance of our angular momentum extrapolations with those of some of the principal expansion extrapolations performed with correlation consistent basis sets employed in the literature and indicated the main sources of inaccuracy.

Can extrapolation to the basis set limit be an alternative to the counterpoise correction? A study on the helium dimer

“…Complete basis set extrapolations have been inferred from the dependence of the correlation energy on the partial wave quantum number for two-electron atomic systems and second-order pair energies in many-electron atoms [40][41][42]. The most popular two-parameter CBS scheme assumes the form [43] …”

Generalized Uniform Singlet- and Triplet-Pair Extrapolation of the Correlation Energy to the One Electron Basis Set Limit

“…Such results generalize to the Møller-Plesset energy of atoms with any number of electrons. 39,41 Note that the single-term extrapolation formula (2) on the cardinal number X (or L, the maximum partial wave number l max in a partial wave expansion of the correlation energy) finds its justification in the fact that the leading contribution at secondorder of perturbation theory is proportional to (l + 1/2) -4 . Note further that the use of just one term may be accuracy-limiting as the subset of natural singlet-pairs in a MP2 calculation for the Zn 2+ ion has been found to contribute only 54.7% of the total correlation energy.…”

“…Note further that the use of just one term may be accuracy-limiting as the subset of natural singlet-pairs in a MP2 calculation for the Zn 2+ ion has been found to contribute only 54.7% of the total correlation energy. 41 Although Klopper 22 has utilized distinct one-term expansions for the singlet and triplet pairs (thus accounting for the X -3 and X -5 behavior, respectively), we have chosen in our uniform singlet-and triplet-pair extrapolation (USTE 27 ) approach not to decompose the total correlation into such contributions. We have done so, first, because it is unnecessary for accurate results; 27 second, because such a decomposition scheme cannot be implemented for open-shell CCSD calculations 26 (the wave function is not a spin eigenfunction in practical implementations of CCSD theory) nor is it commonly available in most CC codes; and third, because such a decomposition is extraneous to MRCI calculations as a single excitation out of the reference space can be counted either as a singlet-pair or as a triplet-pair depending on the spin coupling of the N -1 part of the determinant.…”

“…Although also utilized for the total energy on the basis of the dominance of the correlation energy lowering (ref 46, and references therein), eq 2 finds its justification in the energy increments of partial-wave expansions of atomic correlation energies [29][30][31][32][33][34][35][36][37][38][39][40][41] or similar expressions derived from the convergence behavior of the principal expansion. 8,32,47 From a MP2 study on arbitrary excited states of He-like atoms, where the first-order wave function ψ behaves for small r 12 as ψ ) (1 + κr 12 )Φ + O(r 12 2 ) with κ ) 1/2 (1/4) for singlet (triplet) states 1,2 and Φ being the HF wave function, Kutzelnigg and Morgan 39 established the following: for natural-parity singlet states, the leading contribution at second-order of perturbation theory is proportional to (l + 1/2) -4 , with no contributing odd-terms proportional either to (l + 1/2) -5 or (l + 1/2) -7 and the term (l + 1/2) -6 being universally -(5/4) that of (l + 1/2) -4 ; for all triplet states, the leading term is proportional to (l + 1/2) -6 ; for unnatural parity singlet states, the coefficient is proportional to (l + 1/2) -8 .…”

“…[29][30][31][32][33][34][35][36][37][38][39][40][41] Specifically, the energy has been shown to vary as an inverse power of the cardinal number, where E X cor is the correlation energy for the basis set of cardinal number X, and E ∞ cor , A cor , and are parameters. 16,17 Asymptotically one expects the value of ) 3 , although Truhlar 17 recommended optimal extrapolation exponents for MP2 42 ( MP2 ) 2.2), CCSD and CCSD(T) [ CCSD ) CCSD(T) ) 2.4] calculations by minimizing the root-mean-square deviation (rmsd) in fits to Halkier et al 16 estimated basis-set limits for Ne, HF, and H 2 O.…”