Direct Determination of Pure-State Density Matrices. IV. Investigation of Another Constraint and Another Application of thePEquations

An additional theorem of the Hellman-Feynman variety is used in conjunction with the equations derived in paper II for the first-order density matrix. This particular theorem due to Parr relates the energy difference $E\left({\lambda }^{\prime }\right)-E\left(\lambda \right)$ to a "parameter-transition density" $\rho ({\lambda }^{\prime }, \lambda )$. It is shown that $E\left({\lambda }^{\prime }\right)-E\left(\lambda \right)$ can be used as a constraint on either $\rho \left({\lambda }^{\prime }\right)$ or $\rho \left(\lambda \right)$. An additional application of the $\mathit{P}$ equations as a density-fitting technique is investigated. That is, given a set of expectation values in some basis $\mathit{\Psi }$, one can use the $\mathit{P}$ equations to generate the density matrix that corresponds to these expectation values in some other basis $\mathit{\chi }$. Several helium-atom Hartree-Fock densities are fitted to smaller bases with various sets of operators. Density graphs are given for comparison.