Orientational invariance and significance of electron densities obtained with approximate all-valence electron wavefunctions

Abstract

The total electron density ρ(R), computed with approximate all-valence electron wavefunctions, at a point defined by the position vector R, should (i) obey the orientational invariance requirement, i.e., be independent of the orientation of the molecule in a general coordinate system provided the position vector R is transformed according to the same “reorientation” matrix as the molecular coordinates, and (ii) satisfy the normalization condition, i.e., ∫ρ(R) dR=N, where N is the total number of valence electrons. It is shown that with these two restrictions in mind, only two approximations are physically significant: (i) the use of NDO-type wave functions (CNDO/2, INDO, etc.) combined with the evaluation of ρ(R) up to an INDO-level; (ii) the use of wavefunctions obtained either by solving the complete Roothaan equations, i.e., where overlap integrals are explicitly taken into account, or by deorthogonalizing NDO-functions, together with the unapproximated expression for ρ(R). A comparison of the density maps for formaldehyde obtained by these two methods, with ab initio plots obtained with wavefunctions giving an energy close to the Hartree–Fock limit, leads to the conclusion that the second possibility is the more significant.