Abstract
A factorization of the matrix elements of the Dyall Hamiltonian in N-electron valence state perturbation theory allowing their evaluation with a computational effort comparable to 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 density matrix is avoided. It is also shown that the residual terms arising in the case of an approximate complete active space configuration interaction solution and containing even the fifth-order density matrix for two excitation classes can be evaluated with little additional effort by choosing again a favorable factorization of the corresponding matrix elements. An analogous argument is also provided for avoiding the fourth-order density matrix in complete active space second-order perturbation theory. Practical calculations indicate that such an approach leads to a considerable gain in computational efficiency without any compromise in numerical accuracy or stability.
Highlights
The complete active space self-consistent field (CASSCF) method1 offers a possibility to account for static electron correlation in cases where the electronic state of a molecule cannot even be approximately described by a single Slater determinant
As a consequence of the “rank reduction trick,” there is a residual with matrix elements involving even the fifth-order density matrix, which vanishes in the case of an exact complete active space configuration interaction (CASCI) solution but not for approximate solutions such as the density matrix renormalization group (DMRG),48–50 full configuration interaction Quantum Monte Carlo (FCIQMC),51 the semistochastic heat-bath configuration interaction (SHCI) method,52 selected CI methods (CIPSI: configuration interaction using a perturbative selection carried out iteratively),2,53–60 or the iterative configuration expansion (ICE),61 which are needed for large active spaces where an exact solution is no longer affordable
The sum in Eq (23) or Eq (20) runs over the configuration state functions (CSFs) contained in the truncated CI space, whereas the corresponding sums for the intermediates XuI v, YuIv, and ZI run over the CI space needed for a resolution of the identity (RI), which is larger than the truncated CI space
Summary
The complete active space self-consistent field (CASSCF) method offers a possibility to account for static electron correlation in cases where the electronic state of a molecule cannot even be approximately described by a single Slater determinant. As a consequence of the “rank reduction trick,” there is a residual with matrix elements involving even the fifth-order density matrix, which vanishes in the case of an exact complete active space configuration interaction (CASCI) solution but not for approximate solutions such as the density matrix renormalization group (DMRG), full configuration interaction Quantum Monte Carlo (FCIQMC), the semistochastic heat-bath configuration interaction (SHCI) method, selected CI methods (CIPSI: configuration interaction using a perturbative selection carried out iteratively), or the iterative configuration expansion (ICE), which are needed for large active spaces where an exact solution is no longer affordable Such approximative solutions of the CASCI problem have been used with NEVPT2.62–66 false intruder states have been reported in this context, their origin has been investigated only recently using the full rank NEVPT2 (FR-NEVPT2) method, which takes into account the abovementioned residual. Both the factorization of the Koopmans matrices and the residuals will be considered in the following
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