Efficient Implementation of Approximate Fourth Order <i>N</i>-Electron Valence State Perturbation Theory

In this work, the implementation of a partial fourth order <i>N</i>-electron-valence perturbation theory (NEVPT) is reported and numerically evaluated. The method, termed NEVPT4(SD), includes the internally contracted functions that span the first-order-interacting space (FOIS) and evaluates their contribution to second-order in the wave function and fourth order in the energy. The triple- and quadruple excitations that would additionally enter the second-order-interacting space (SOIS) are not included. As discussed by Grimme [<i>Chem. Phys. Lett.</i> <b>2001,</b> <i>334,</i> 99-106] in order to obtain a size-consistent method, it is necessary to also drop the fourth-order renormalization term if the quadruple excitations are dropped. The NEVPT4(SD) method is demonstrated to be perfectly size consistent. Computationally, the method is still fairly affordable and requires about the same time as a single iteration of the fully internally contracted (FIC) MRCI or MRCEPA(0) and significantly cheaper than the FIC MRCC that serves as the reference for our calculations. The accuracy tests show that NEVPT4(SD) offers significant accuracy improvements over NEVPT2 for transition metal atom/ion multiplets as well as diatomic bond breaking potential energy surfaces. We find that going to fourth order in perturbation theory essentially eliminates the need for a second d-shell, thus showing that the latter primarily serves to capture higher-order dynamic correlation effects that are not present in a second-order treatment. Although it captures fourth-order correlation effects, NEVPT4(SD) is numerically not a large improvement over NEVPT2 for the calculation of Heisenberg exchange couplings as illustrated by test calculations on Cu(II) dimers.