A study of the effect of attenuation curvature on molecular correlation energies by introducing an explicit cutoff radius into two-electron integrals

We present a new attenuator function that can be applied to the Coulomb operator. Similar to the popular erf(omegar) attenuator, the function [erf(omega(r + r0)) + erf(omega(r - r0))]/2 divides the Coulomb potential into a singular short-range piece and a non-singular long-range piece. In our attenuator, omega controls the sharpness of the short-range/long-range division at r0. With r0 = 0, this reduces to erf(omegar), but the additional parameter allows more flexible adjustment of the potential, for physical and/or computational reasons. We present some illustrative results for a truncated MP2 method, where mean field effects are handled exactly and correlation is treated locally. This study indicates, somewhat expectedly, that the slope and curvature of the attenuated potential are more important than its value (a trivial constant may always be added to a potential). However, there are some surprising features of the data that suggest what bounds need to be put on the curvature of the attenuated potential in order to achieve reasonable physics. Conveniently, we find that our attenuator form has the ability to preserve the curvature of the Coulomb potential almost exactly at short range, allowing for the truncation of long-range interactions while preserving the local physics very well. The molecular integrals for the resultant operator can be done analytically over Gaussian basis functions, and the extensive algebraic manipulations necessary to evaluate them stably are shown.