Application of the overlap model to calculating correlated exchange energies

doi:10.1016/S0009-2614(00)00752-1

Chemical Physics Letters

Volume 326, Issues 3–4, 18 August 2000, Pages 263-268

https://doi.org/10.1016/S0009-2614(00)00752-1 Get rights and content

Abstract

We develop methods for approximating correlated exchange energies which require only Hartree–Fock self-consistent field (SCF) exchange energies and SCF and correlated charge density overlaps. We benchmark the methods using results calculated for the water dimer at the second-order Møller–Plesset (MP2) level using symmetry-adapted perturbation theory. Assuming that the exchange/overlap ratio is transferable between SCF and MP2 calculations gives a weighted RMS error of 3.2% with no fitted parameters. Including a single overall scaling parameter gives an error of 2.3%.

Introduction

Modelling the water dimer accurately has proven to be computationally very demanding, and this task has drawn considerable effort from the scientific community over the last few decades. One of the most sensitive tests of a model potential for this system is in the determination of the vibration–rotation–tunneling spectrum, and only very recently, using state of the art computational techniques, has it been possible to attain near spectroscopic accuracy [1]. The work in Ref. [1]represents a continuation of earlier work on the same system 2, 3and employs symmetry-adapted perturbation theory (SAPT) 4, 5. The success of the SAPT method stems partly from the fact that both intramolecular and intermolecular correlation effects can systematically be considered. However, SAPT is computationally demanding even for systems as small as the water dimer, and more approximate methods are still required for larger molecules.

Interactions between molecules may be divided into long-range (multipolar) and short-range (overlap and exchange–repulsion) contributions. Long-range interactions, including the effects of intramolecular correlation, can be obtained from monomer calculations, but more computationally demanding dimer calculations are required for short-range interactions. Here we concentrate on the exchange–repulsion (called exchange here for brevity) which is a dominant component of the intermolecular potential energy at short range. The exchange energy is relatively straightforward to compute at the uncorrelated (Hartree–Fock, i.e. HF) level. However, correlation corrections are also important as there is a delicate balance between long-range and short-range terms near the minima of the potential energy surface. Correlated exchange energies are difficult to obtain, even approximately. Their magnitude, relative to the HF exchange, is system-dependent, and may vary considerably as a function of the relative orientations of the interacting molecules [6].

The development of accurate ab initio potentials will therefore benefit from methods which can provide high-quality exchange energies for a reasonable computational cost. In this Letter we report the development of methods which can be used to estimate correlated exchange energies from quantities which are easy to evaluate, namely charge density overlaps and HF exchange energies. We report results for the water dimer at the MP2 level, because SAPT results are available for comparison, though the methods are completely general and applicable to higher levels of correlation and to systems too large for SAPT calculations. With reference to the use of the MP2 method, we note a recent study by Heßelmann and Jansen who calculated first-order exchange energies with a reference determinant constructed using a Brueckner doubles (BD) methodology [7]. Six configurations of the water dimer were studied, and the BD and MP2 results were comparable with exchange energies calculated at the CCSD level with an infinite-order summation of the intramolecular correlation corrections 8, 9. This indicates that the MP2 results are a reasonable approximation to highly correlated calculations.

Section snippets

Methods

The charge density overlap between two molecules separated by a distance R, with relative orientations denoted by $Ω$ , is defined as $S_{ρ} (R;Ω)= ∫ ρ_{A} (r)ρ_{B} (r) d^{3} r$ where ρ_A and ρ_B are the ground-state charge densities of isolated monomers A and B, respectively. Note that the charge density only has to be calculated once per molecule at each level of theory being considered, if the internal coordinates remain fixed. The basic premise of the overlap model (see Ref. [10]for previous applications) is that the

Results

The SCF and MP2 exchange energies are plotted against one another in Fig. 1, and it is evident that a simple scaling factor can be used to approximate the MP2 data such that $E_{exch}^{MP2} ≈λ×E_{exch}^{SCF} .$ The optimized scaling parameters λ, and errors Δ, resulting from this approximation are given in Table 1. If we assume that λ=1 then for both weighting schemes the error incurred by using the SCF rather than MP2 exchange energies is about 20%. This is a measure of the significant effect of correlation on

Discussion and conclusions

We believe that this is the first time that correlated exchange energies and charge density overlaps have been compared. The observation that the SCF and MP2 overlap functions are very similar has allowed us to approximate correlated exchange energies to within 2–3% of the exact values. Two strategies for approximating results at the MP2 level, both requiring SCF exchange energies and SCF and MP2 charge density overlaps, have been proposed. The simpler of the two, requiring no fitted

Acknowledgements

We thank Krzysztof Szalewicz and Robert Bukowski for providing us with the SAPT exchange energies and for helpful discussions. M.P.H. thanks the Leverhulme Trust for financial support.

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