Direct Determination of Pure-State Density Matrices. IV. Investigation of Another Constraint and Another Application of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="italic">P</mml:mi></mml:math>Equations

Abstract

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 $E({\ensuremath{\lambda}}^{\ensuremath{'}})\ensuremath{-}E(\ensuremath{\lambda})$ to a "parameter-transition density" $\ensuremath{\rho}({\ensuremath{\lambda}}^{\ensuremath{'}}, \ensuremath{\lambda})$. It is shown that $E({\ensuremath{\lambda}}^{\ensuremath{'}})\ensuremath{-}E(\ensuremath{\lambda})$ can be used as a constraint on either $\ensuremath{\rho}({\ensuremath{\lambda}}^{\ensuremath{'}})$ or $\ensuremath{\rho}(\ensuremath{\lambda})$. 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 $\mathit{\ensuremath{\Psi}}$, one can use the $\mathit{P}$ equations to generate the density matrix that corresponds to these expectation values in some other basis $\mathit{\ensuremath{\chi}}$. Several helium-atom Hartree-Fock densities are fitted to smaller bases with various sets of operators. Density graphs are given for comparison.