7.12.2 Second-Order Perturbative Corrections to RAS-CI

In general, the RAS-CI family of methods within the $hole$ and $particle$ approximation is unable to capture the necessary amounts of dynamic correlation for the computation of relative energies with chemical accuracy. The missed correlation can be added on top of the RAS-CI wave function using multi-reference perturbation theory (MRPT).Casanova:2014 The second order energy correction, i.e. RASCI(2), can be expressed as:

 $E^{(2)}=-\sum_{k}\frac{|\langle k|\hat{H}|0\rangle|^{2}}{E_{k}-E_{0}+\epsilon}$ (7.91)

where $0$ indicates the zero-order space and $\{|k\rangle\}$ is the complementary set of determinants. There is no natural choice for the $\{E_{k}\}$ excited energies in MRPT, and two different models are available within the RASCI(2) approach, that is the Davidson-Kapuy and Epstein-Nesbet partitionings. As it is common practice in many second-order MRPT corrections, the denominator energy differences in Eq. (7.91) can be level shifted with a parameter $\epsilon$.