Characterization of the two-body correlation structures in a molecular many-electron wave function

Abstract

A method has been developed to analyze the two-body correlation structures of a molecular many-electron wavefunction that are involved in the second-order density matrix. It is shown that the densities of singlet and triplet electron pairs and a density representing their interference are the basic quantities from which all information about the two-body density correlation structures can be derived. We define the correlation functions as four kinds: the charge, spin scalar, spin quardrupole, and charge-spin correlation functions. They represent the correlation between electron densities, the scalar product correlation of spin vectors, the anisotropy in the orientational correlation of spin vectors, and the correlation of a spin to the density of other electrons, respectively. Their expressions are derived in terms of the singlet and triplet pair densities and the interference density. The correlation angle of two spin vectors is defined. The densities of the triplet pairs coupled to the total spin S with the composite spin angular momenta S + 1, S, |S − 1|, are expressed in terms of the triplet pair densities using the Racah coefficients. These densities give information about the correlations of free radical electrons. As the standard for the uncorrelated motion of electrons, the unlinked density matrix, which is the antisymmetrized product of the first-order density matrix, is introduced. The linked parts of those two-body quantities subtracted by the unlinked contributions give the quantitative measures for the dynamical correlation effects. The method is applied to the analyses of the correlation structures in the low lying exact eigenstates of the PPP Hamiltonian of benzene, cyclobutadiene, and cyclopentadienyl radical.