Direct Determination of Pure-State Density Matrices. V. Constrained Eigenvalue Problems

A density-matrix approach to constrained eigenvalue problems is presented. It is shown that all of the linearly independent eigenvectors of an Hermitian matrix can be generated with the idempotency equations (P equations) developed in previous papers of this series. In particular, the method is applied to variational calculations in H₂⁺ and He.Since the local-energy method assumes eigenvalue form, it also can be formulated in terms of the P equations. Various local energies for H₂⁺ and He are calculated. Direct methods of incorporating local energies as constraints are suggested. An orthogonal operator formalism for the P equations is given. Such operators oₖ, Oₗ have the property that TrOₖOₗ=0 for k≠L. The iterative P equations, then, assume the simple form ${P}ₙ₊₁={P}ₙ+{Σ}{}{k}[{({O}ₖ{-}TrP{O}ₖ)}{Tr{{O}ₖ}²}]{O}ₖ,$ where $P{≡}3{P}²{-}2{P}²$ and the constraints, $TrP{O}ₖ={O}ₖ$, are now identically satisfied.