Direct Determination of Pure-State Density Matrices. II. Construction of Constrained Idempotent One-Body Densities

A formalism is developed for the determination of a constrained idempotent one-body density matrix $\mathit{P}$. The method ensures pure-state representability in the Hartree-Fock sense. This is accomplished by minimizing the quantity $\mathrm{Tr}{({\mathit{P}}^2-\mathit{P})}^2$ subject to either empirical or theoretical constraints. The method leads to an iterative matrix equation of the form ${\mathit{P}}_{n+1\mathrm{}}=3\mathrm{}{{\mathit{P}}_n}^2-2\mathrm{}{{\mathit{P}}_n}^3+\Sigma \stackrel{}{k}{{\lambda }_k}^{\mathrm{}\left(n\right)\mathrm{}}{\mathit{O}}_k,$ where $-{\lambda }_k$ is the $k\mathrm{th}$ Lagrangian multiplier pertaining to the constraint $\mathrm{Tr}\mathit{P}{\mathit{O}}_k=O_k$, and $O_k$ is the expectation value of the observable ${\mathit{O}}_k$. Applications to several diatomic molecules are reported using the electrostatic and virial theorems as empirical constraints. Several calculations are carried out in order to corroborate the following result: If the above iterative equations are constrained with a sufficient number of bona fide Hartree-Fock conditions, the solution is the Hartree-Fock $\mathit{P}$ matrix.