11.6 Spin and Charge Densities at the Nuclei

Gaussian basis sets violate nuclear cusp conditions.^{457, 700, 776} This may lead to large errors in wave function at nuclei, particularly for spin density calculations.¹⁷³ This problem can be alleviated by using an averaging operator that compute wave function density based on constraints that wave function must satisfy near Coulomb singularity.^{777, 778} The derivation of operators is based on hyper virial theorem³⁸³ and presented in Ref. 777. Application to molecular spin densities for spin-polarized⁷⁷⁸ and DFT⁹⁶⁷ wave functions show considerable improvement over traditional delta function operator.

One of the simplest forms of such operators is based on the Gaussian weight function $\exp [-{(Z/r_0)}^2{(𝐫-𝐑)}^2]$ that samples the vicinity of a nucleus of charge $Z$ located at $𝐑$. The parameter $r_0$ has to be small enough to neglect two-electron contributions of the order $𝒪(r_0^4)$ but large enough for meaningful averaging. The range of values between 0.15–0.3 a.u. has been shown to be adequate, with final answer being relatively insensitive to the exact choice of $r_0$.^{777, 778} The value of $r_0$ is chosen by RC_R0 keyword in the units of 0.001 a.u. The averaging operators are implemented for single determinant Hartree-Fock and DFT, and correlated SSG wave functions. Spin and charge densities are printed for all nuclei in a molecule, including ghost atoms.