Direct Determination of Pure-State Density Matrices. I. Some Simple Introductory Calculations

A method is developed for the semiempirical determination of an electron density in small diatomic molecules. The method involves primarily the electrostatic and virial theorems and empirical potential curve data. The density has been determined, using Slater-type basis functions, for the molecules ${\mathrm{H}}_2^{+}$, ${\mathrm{H}}_2$, ${\mathrm{He}}_2$, ${\mathrm{Li}}_2$, and ${\mathrm{N}}_2$. Expectation values are calculated and compared to exact or Hartree-Fock values. A theoretical discussion of the semiempirical densities via a natural-orbital analysis is given. These analyses show that (a) the electrostatic and virial theorems are effective when used to determine unknown parameters in a first-order density but that (b) some additional means must be provided to ensure $N$ representability.