Variational Principle with Built-In Pure State N-Representability Conditions. The N-Electron Case

Abstract

We consider the problem of obtaining a formal expression for the internal energy functional F^p[γ], where γ is an n-representable reduced first order density operator corresponding to a pure-state. This expression is obtained in the context of a variational procedure where a minimum is sought for the internal energy functional $Tr\left[ {{{\hat H}_O}{\Gamma ^n}} \right]$ , subject to the following conditions: a) that at all stages of variation ${\Gamma ^{\text{n}}}\varepsilon {{\text{P}}_{\text{n}}}$ where P_n is the set of pure-state nth-order density operator, and b) that Γⁿ map into a given approximate γ. It is found that extreme point of variation, ${F^P}\left[ \gamma \right] = T{r_1}\left[ {{{\underline \alpha }_\gamma }\gamma } \right]$ where ${\underline \alpha _\gamma }$ is a Lagrange multiplier matrix. The generaln-electron case is dealt with in this paper and a formal expression for ${\underline \alpha _\gamma }$ is obtained in terms of the vectors ${\vec C_i}$ which are constructed by means of particular combinations of the CI expansion coeffi cients of the many electron wavefunction. Possible applications of the procedure advanced here to the solution of the quantum mechanical n-electron problem, are discussed.