Size-extensive vibrational self-consistent field method

The vibrational self-consistent field (VSCF) method is a mean-field approach to solve the vibrational Schrdinger equation and serves as a basis of vibrational perturbation and coupled-cluster methods. Together they account for anharmonic effects on vibrational transition frequencies and vibrationally averaged properties. This article reports the definition, programmable equations, and corresponding initial implementation of a diagrammatically size-extensive modification of VSCF, from which numerous terms with nonphysical size dependence in the original VSCF equations have been eliminated. When combined with a quartic force field (QFF), this compact and strictly size-extensive VSCF (XVSCF) method requires only quartic force constants of the ∂⁴V∂Q_i²∂Q_j ² type, where V is the electronic energy and Q_i is the ith normal coordinate. Consequently, the cost of a XVSCF calculation with a QFF increases only quadratically with the number of modes, while that of a VSCF calculation grows quartically. The effective (mean-field) potential of XVSCF felt by each mode is shown to be harmonic, making the XVSCF equations subject to a self-consistent analytical solution without matrix diagonalization or a basis-set expansion, which are necessary in VSCF. Even when the same set of force constants is used, XVSCF is nearly three orders of magnitude faster than VSCF implemented similarly. Yet, the results of XVSCF and VSCF are shown to approach each other as the molecular size is increased, implicating the inclusion of unnecessary, nonphysical terms in VSCF. The diagrams of the XVSCF energy expression and their evaluation rules are also proposed, underscoring their connected structures.