An efficient implementation of the NEVPT2 and CASPT2 methods avoiding higher-order density matrices

A factorization of the matrix elements of the Dyall Hamiltonianin N-electron valence state perturbation theory (NEVPT2) allowing their evaluation with a computational effort comparableto the one needed for the construction of the third-order reduced density matrix at the most is presented.Thus, the computational bottleneck arising from explicit evaluation of the fourth-order densitymatrix is avoided. It is also shown that the residual terms arising in case of an approximate CASCI solutionand containing even the fifth-order density matrix for two excitation classes can be evaluated with little additional effortby choosing again a favorable factorization of the corresponding matrix elements. An analogous argument is provided for avoiding the fourth-order density matrix also in complete active space second-order perturbation theory (CASPT2). Practical calculationsindicate that such an approach leads to a considerable gain in computational efficiency withoutany compromise in numerical accuracy or stability.