Alternatively, expansion as a Rayleigh-Schroedinger perturbation series, with 642) yields, to first order %(i) = 7f + Km,ge.i From comparison of (Al) and (A3), one may identify (at the level of linear response) the relationship The nature of the convergeance and therefore the relative importance of the neglect of the higher order terms of (Al) and (A3) are not known. In the MFP implementation of Stephens, the wave function, q0, is approximated as the Hartree-Fock determinant, *HF, and the left-hand side of (A4) is evaluated by coupled perturbed H F (CPHF) theory. In VCT, the right-hand side is used. In the implementation of Rauk and Dutler, the ground state is also approximated as *'HF. Trunction of the sum over states necessarily invokes a second approximation, the severity of which depends on the level at which the truncation occurs. Since only singly excited configurations couple directly to qHF, the present implementation uses either SCI correlated or uncorrelated singly excited configurations for *e and the denominator is the correct excitation energy for the type of "excited state" used. Both MFP and VCT implementations incur a basis set error, the severity of which will be different for the two approaches and can only be determined by explicit testing. It should be emphasized that with the adoption of qHF for the ground state, neither the CPHF (MFP) nor the perturbation (VCT) approach takes into account electron correlation in the usual sense. Nakatsuji has shown38 ~~~ ~~ (38) Nakatsuji, H. J. Chem. Phys. 1974,61,3728 and references therein.that the coupled HF wave function may be expressed as a perturbation sum as in (A4) and that the sum will include those doubly excited configurations which are derivable from the singly excited configurations by a further single excitation, with coefficients which are products of the coefficients of the singly excited configurations. Thus, in coupled HF theory, the left-hand side of (A4) through second order may be expressed in the form (*HF/d@B=O i= N(ZCmiSmi+ 9°F + Y~EZCmiCtljsm,+S,+ *HF) (A51 nu tlJ where N is a normalization constant and the excitation operator, S , : , generates a singlet-spin adapted singly excited configuration by substitution of unoccupied orbital, m, for occupied orbital, i. The presence of the "doubly excited" configurations in (AS) does not correspond to inclusion of electron correlation in the usual sense, because the coefficients are not independently determined and not all doubly excited configurations appear.38 However, because doubly excited configurations are present in the right-hand side of (A5), and therefore affect the magnitudes of the coefficients of the singly excited configurations through the normalization constant, N, the MFP treatment of Stephens and the present implementation of VCT are not equivalent even in the HF limit.At the Hartree-Fock limit the CHF procedure, and therefore the MFP method, is clearly superior, since the coefficients of the right-hand side of (A5) are variationally determined and the balance of contribu...
