9.6 Laplace-transformed SOS-RI-MP2 with O(N4) scaling costs

The ricc2 module contains an implementation of SOS-MP2 which exploits the RI approximation and a Laplace transformation of the orbital energy denominators

                ∫                    n
------1-------    ∞ - (ϵa+ϵb-ϵi-ϵj)t    ∑L    -(ϵa+ ϵb-ϵi- ϵj)tα
ϵa + ϵb - ϵi - ϵj = 0 e           dt ≈ α=1wαe              ,

to achieve an implementation with O(N4) scaling costs, opposed to the conventional O(N5) scaling implementation. In particular for large molecules the Laplace-transformed implementation can reduce a lot the computational costs of SOS-MP2 calculations without loss in accuracy.

The Laplace-transformed implementation for SOS-MP2 calculations is activated with the input


where the parameter conv is a convergence threshold for the numerical integration in Eq. (9.10). A value of conv=5 means that the numerical integration will be converged to a root mean squared error of 10-5 a.u.

Whether the conventional or the Laplace-transformed implementation will be more efficient depends firstly on the system size (the number of occupied orbitals) and secondly on the required accuracy (the number of grid points for the numerical integration in Eq. (9.10)) and can be understood and estimated from the following considerations:

The number of grid points nL depends on the requested accuracy and the spread of the orbital energy denominators in Eq. (9.10). The efficiency of Laplace-transformed SOS-RI-MP2 calculations can therefore (in difference to conventional RI-MP2 calculations) be enhanced significantly by a carefull choice of the thresholds, the basis set, and the orbitals included in the correlation treatment:

Note that for large molecules it is recommended to disable for geometry optimizations (or for gradient or property calculations in general) the preoptimization for the Z vector equations with the nozpreopt option in the $response data group (see Sec. 20.2.19).