10.4.1 Spin and Charge Densities at the Nuclei‣ 10.4 Density Analysis ‣ Chapter 10 Molecular Properties and Analysis ‣ Q-Chem 6.3 User’s Manual

Gaussian basis sets violate nuclear cusp conditions. ⁶⁵¹ Kato T.
Commun. Pure Appl. Math.
(1957), 10, pp. 151. Link ^, ⁹⁹² Pack R. T., Brown W. B.
J. Chem. Phys.
(1966), 45, pp. 556. Link ^, ¹¹⁰³ Rassolov V. A., Chipman D. M.
J. Chem. Phys.
(1996), 104, pp. 9908. Link This may lead to large errors in wave function at nuclei, particularly for spin density calculations. ²³⁶ Chipman D. M.
Theor. Chem. Acc.
(1989), 76, pp. 73. Link 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. ¹¹⁰⁴ Rassolov V. A., Chipman D. M.
J. Chem. Phys.
(1996), 105, pp. 1470. Link ^, ¹¹⁰⁵ Rassolov V. A., Chipman D. M.
J. Chem. Phys.
(1996), 105, pp. 1479. Link The derivation of operators is based on hyper virial theorem ⁵⁶¹ Hirschfelder J. O.
J. Chem. Phys.
(1960), 33, pp. 1462. Link and presented in Ref. 1104 Rassolov V. A., Chipman D. M.
J. Chem. Phys.
(1996), 105, pp. 1470. Link . Application to molecular spin densities for spin-polarized ¹¹⁰⁵ Rassolov V. A., Chipman D. M.
J. Chem. Phys.
(1996), 105, pp. 1479. Link and DFT ¹³⁶⁴ Wang B., Baker J., Pulay P.
Phys. Chem. Chem. Phys.
(2000), 2, pp. 2131. Link 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 $r_{0}$ . ¹¹⁰⁴ Rassolov V. A., Chipman D. M.
J. Chem. Phys.
(1996), 105, pp. 1470. Link ^, ¹¹⁰⁵ Rassolov V. A., Chipman D. M.
J. Chem. Phys.
(1996), 105, pp. 1479. Link 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

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.

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10.4.1 Spin and Charge Densities at the Nuclei