On the calculation of the electrostatic potential, electric field and electric field gradient from the aspherical pseudoatom model

Accurate, yet simple and efficient, formulae are presented for calculation of the electrostatic potential (ESP), electric field (EF) and electric field gradient (EFG) from the aspherical Hansen-Coppens pseudoatom model of electron density [Hansen & Coppens (1978). Acta Cryst. A34, 909-921]. They are based on the expansion of |r′-r|^-1 in spherical harmonics and the incomplete gamma function for a Slater-type function of the form R _l(r) = rⁿ exp(-αr). The formulae are valid for 0 ≤r≤∞ and are easily extended to higher values of l. Special treatment of integrals is needed only for functions with n = l and n = l + 1 at r = 0. The method is tested using theoretical pseudoatom parameters of the formamide molecule obtained via reciprocal-space fitting of PBE/6-31G** densities and experimental X-ray data of Fe(CO)₅. The ESP, EF and EFG values at the nuclear positions in formamide are in very good agreement with those directly evaluated from density-functional PBE calculations with 6-31G**, aug-cc-pVDZ and aug-cc-pVTZ basis sets. The small observed discrepancies are attributed to the different behavior of Gaussian- and Slater-type functions near the nuclei and to imperfections of the reciprocal-space fit. An EF map is displayed which allows useful visualization of the lattice EF effects in the crystal structure of formamide. Analysis of experimental 100 K X-ray data of Fe(CO)₅ yields the value of the nuclear quadrupole moment Q(⁵⁷Fe^m) = 0.12 × 10 ^-28 m² after taking into account Sternheimer shielding/antishielding effects of the core. This value is in excellent agreement with that reported by Su & Coppens [Acta Cryst. (1996), A52, 748-756] but slightly smaller than the generally accepted value of 0.16 ± 5% × 10^-28 m² obtained from combined theoretical/spectroscopic studies [Dufek, Blaha & Schwarz (1995). Phys. Rev. Lett. 25, 3545-3548].