# 10.6 Spin and Charge Densities at the Nuclei

Gaussian basis sets violate nuclear cusp conditions.Kato:1957, Pack:1966, Rassolov:1996a This may lead to large errors in wave function at nuclei, particularly for spin density calculations.Chipman:1989 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.Rassolov:1996b, Rassolov:1996c The derivation of operators is based on hyper virial theoremHirschfelder:1960 and presented in Ref. Rassolov:1996b. Application to molecular spin densities for spin-polarizedRassolov:1996c and DFTWang:2000b 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 $\exp[-(Z/r_{0})^{2}(\mathbf{r}-\mathbf{R})^{2}]$ that samples the vicinity of a nucleus of charge $Z$ located at $\bf R$. The parameter $r_{0}$ has to be small enough to neglect two-electron contributions of the order ${\cal{O}}({r_{0}^{4}})$ 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 $r_{0}$.Rassolov:1996b, Rassolov:1996c 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.

RC_R0
Determines the parameter in the Gaussian weight function used to smooth the density at the nuclei.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Corresponds the traditional delta function spin and charge densities $n$ corresponding to $n\times 10^{-3}$ a.u.
RECOMMENDATION:
We recommend value of 250 for a typical spit valence basis. For basis sets with increased flexibility in the nuclear vicinity the smaller values of $r_{0}$ also yield adequate spin density.