Variation-Perturbation Approach to Electron-Cluster Wavefunctions

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Abstract
Sinanoğlu's cluster expansion formula for the exact wavefunction of a many‐electron system is investigated by variation‐perturbation techniques. The perturbation operator is the difference between the actual Hamiltonian and a symmetric sum of arbitrary one‐electron Hamiltonians. Each perturbation function is then expanded in accordance with the cluster formula. When the occupied orbitals are chosen to be eigenfunctions of the one‐electron Hamiltonian the variational equations determining different nth‐order clusters are completely independent. If the orbitals are unitarily transformed to achieve greater localization the variational equations for each cluster are coupled. The perturbation governing first‐order two‐electron clusters has the form of a dipole potential. This result does not depend on the nature of the one‐electron Hamiltonian. It is primarily useful, however, in studying the question of interorbital vs intraorbital correlation, i.e., when the one‐electron Hamiltonian is the Hartree—Fock operator. It is shown that interorbital correlation is small if the Hartree—Fock orbitals are well separated. The advantage of transforming to localized orbitals is shown to be questionable because of the coupling between the clusters. Differential equations governing first‐ and second‐order orbital correction functions are derived and partially interpreted. In accordance with Brillouin's theorem the first‐order corrections vanish as the one‐electron Hamiltonian approaches the form of the Hartree—Fock operator. In the Hartree—Fock case the role of the second‐order orbital correction functions is to relax the charge density subsequent to correlation. An investigation of the derivative of the correlation energy with respect to nuclear charge, a one‐electron property, suggests that the second order correction functions may be important whenever it is necessary to evaluate one‐electron properties to an accuracy beyond the Hartree—Fock estimate. The second order four‐electron cluster is shown to be exactly expressible as a sum of products of first‐order two‐electron clusters. Explicit variational equations are derived also for the second‐order two‐ and three‐electron Hartree—Fock clusters.

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