The electrostatic potential and its derivatives

Coppens, P.; Su, Z.; Becker, P. J.

doi:10.1107/97809553602060000615

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 718-719

Section 8.7.3.4.2.1. The electrostatic potential and its derivatives

P. Coppens,^a Z. Su^b and P. J. Becker^c

^a 732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA,^bDigital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 01754-2122, USA, and ^cEcole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F-92295 Châtenay Malabry CEDEX, France

8.7.3.4.2.1. The electrostatic potential and its derivatives

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The electrostatic potential Φ(r′) due to the electronic charge distribution is given by the Coulomb equation, $[ \Phi({\bf r}^{\prime }) =-k\int \displaystyle{\rho({\bf r}) \over | {\bf r}-{\bf r}^{\prime }| }{\,{\rm d}}{\bf r},\eqno (8.7.3.50)]$ where the constant k is dependent on the units selected, and will here be taken equal to 1. For an assembly of positive point nuclei and a continuous distribution of negative electronic charge, we obtain $[ \Phi({\bf r}^{\prime }) =\sum _{{M}}\displaystyle {Z_{{M}}\over | {\bf R}_{{M}}-{\bf r}^{\prime } | }-\int \displaystyle {\rho ({\bf r}) \over | {\bf r}-{\bf r}^{\prime } | }{\,{\rm d}}{\bf r},\eqno (8.7.3.51)]$ in which Z_M is the charge of nucleus M located at R_M.

The electric field E at a point in space is the gradient of the electrostatic potential at that point. $[ {\bf E}({\bf r}) =-\nabla \Phi ({\bf r}) =-{\bf i}{\partial \Phi ({\bf r}) \over\partial x}-{\bf j} {\partial \Phi ({\bf r}) \over \partial y}-{\bf k} {\partial \Phi({\bf r}) \over \partial z}.\eqno (8.7.3.52)]$ As E is the negative gradient vector of the potential, the electric force is directed `downhill' and proportional to the slope of the potential function. The explicit expression for E is obtained by differentiation of the operator |r − r′|⁻¹ in (8.7.3.50) towards x, y, z and subsequent addition of the vector components. For the negative slope of the potential in the x direction, one obtains $[ {\bf E}_x({\bf r}^{\prime }) =\int \displaystyle {\rho _{{\rm total}}({\bf r}) \over | {\bf r}^{\prime }-{\bf r} | ^2}\, {({\bf r}^{\prime }-{\bf r}) _x \over | {\bf r}^{\prime }-{\bf r} | }{\,{\rm d}}{\bf r}=\int \displaystyle {\rho _{{\rm total}}({\bf r}) \over | {\bf r}^{\prime }-{\bf r}| ^3}({\bf r}^{\prime }-{\bf r}) _x{\,{\rm d}}{\bf r},\eqno (8.7.3.53)]$ which gives, after addition of the components, $[ {\bf E}({\bf r'}) =-\nabla \Phi({\bf r}^{\prime }) =\int \displaystyle {\rho _{t}({\bf r}) ({\bf r}^{\prime }-{\bf r}) \over \left | {\bf r}-{\bf r}^{\prime }\right | ^3}{\rm d}{\bf r}. \eqno (8.7.3.54)]$

The electric field gradient (EFG) is the tensor product of the gradient operator $[\nabla={\bf i}{\partial\over {\partial x}}+{\bf j}{\partial \over{\partial y}}+{\bf k}{\partial\over {\partial z}}]$ and the electric field vector E; $[ \nabla {\bf E}=\nabla: {\bf E}=-\nabla: \nabla \Phi. \eqno (8.7.3.55)]$ It follows that in a Cartesian system the EFG tensor is a symmetric tensor with elements $[ \nabla {\bf E}_{\alpha \beta }=- {\partial ^2\Phi \over \partial r_\alpha \partial r_\beta }. \eqno (8.7.3.56)]$

The EFG tensor elements can be obtained by differentiation of the operator in (8.7.3.53) for E_α to each of the three directions β. In this way, the traceless result $[\eqalignno{ \nabla {\bf E}_{\alpha \beta }({\bf r}^{\prime }) &= {\partial {\bf E}_\alpha \over\partial (r_\beta -r_\beta ^{\prime }) } \cr &=-\int \displaystyle{1\over | {\bf r}-{\bf r}^{\prime } | ^5}\Big \{ 3(r_\alpha -r_\alpha ^{\prime }) (r_\beta -r_\beta ^{\prime }) \cr &\quad {\,}- | {\bf r}-{\bf r}^{\prime }| ^2\delta _{\alpha \beta }\Big\} \rho _{{\rm total}}({\bf r}) {\,{\rm d}}{\bf r} & (8.7.3.57)}]$ is obtained. We note that according to (8.7.3.57) the electric field gradient can equally well be interpreted as the tensor of the traceless quadrupole moments of the distribution −2ρ_total(r)/|r − r′|⁵ [see equation (8.7.3.17)].

Definition (8.7.3.56) and result (8.7.3.57) differ in that (8.7.3.56) does not correspond to a zero-trace tensor. The situation is analogous to the two definitions of the second moments, discussed above, and is illustrated as follows. The trace of the tensor defined by (8.7.3.56) is given by $[ -\nabla ^2\Phi =-\nabla \cdot \nabla \Phi =-\left ({\partial ^2\Phi \over \partial x^2}+ {\partial ^2\Phi \over \partial y^2}+ {\partial ^2\Phi \over \partial z^2}\right). \eqno (8.7.3.58)]$ Poisson's equation relates the divergence of the gradient of the potential $[\Phi ({\bf r})]$ to the electron density at that point: $[ \nabla ^2\Phi({\bf r}) =-4\pi \left [-\rho _{{e}}({\bf r}) \right] =4\pi \rho _{{e}}({\bf r}). \eqno (8.7.3.59)]$ Thus, the EFG as defined by (8.7.3.56) is not traceless, unless the electron density at r is zero.

The potential and its derivatives are sometimes referred to as inner moments of the charge distribution, since the operators in (8.7.3.50), (8.7.3.52) and (8.7.3.54) contain the negative power of the position vector. In the same terminology, the electrostatic moments discussed in §8.7.3.4.1 are described as the outer moments.

It is of interest to evaluate the electric field gradient at the atomic nuclei, which for several types of nuclei can be measured accurately by nuclear quadrupole resonance and Mössbauer spectroscopy. The contribution of the atomic valence shell centred on the nucleus can be obtained by substitution of the multipolar expansion (8.7.3.7) in (8.7.3.57). The quadrupolar (l = 2) terms in the expansion contribute to the integral. For the radial function $[R_l=\{{\zeta ^{n(l)+3}}/{ [n(l) +2] !}\}r^{n(l) }\exp(-\zeta r)]$ with n(l) = 2, the following expressions are obtained: $[\eqalign{ \nabla {\bf E}_{11} &=+(3/5)\left(\pi P_{22+}-\sqrt {3}P_{20}\right) Q_r, \cr \nabla {\bf E}_{22} &=-(3/5) \left (\pi P_{22+}+\sqrt {3}P_{20}\right) Q_r, \cr \nabla {\bf E}_{33} &=+(6/5) \left (\sqrt {3}P_{20}\right) Q_r, \cr \nabla {\bf E}_{12} &=+(3/5) \left (\pi P_{22-}\right) Q_r, \cr \nabla {\bf E}_{13} &=+(3/5) \left (\pi P_{21+}\right) Q_r, \cr \nabla {\bf E}_{23} &=+(3/5) \left (\pi P_{21-}\right) Q_r,} \eqno(8.7.3.60)]$ with $[\eqalignno{ Q_r &=\langle r^3 \rangle _{3{\,{\rm d}}} \cr &=\textstyle\int\limits _0^\infty [{R}(r) /r] {\,{\rm d}}r \cr &=(\kappa ^{\prime }\zeta) ^3/ [n_2(n_2+1) (n_2+2)] \cr &=(\kappa ^{\prime }\zeta) ^3/120,} ]$ in the case that n₂ = 4 (Stevens, DeLucia & Coppens, 1980).

The contributions of neighbouring atoms can be subdivided into point-charge, point-multipole, and penetration terms, as discussed by Epstein & Swanton (1982) and Su & Coppens (1992, 1994b), where appropriate expressions are given. Such contributions are in particular important when short interatomic distances are involved. For transition-metal atoms in coordination complexes, the contribution of neighbouring atoms is typically much smaller than the valence contribution.

References

Epstein, J. & Swanton, D. J. (1982). Electric field gradients in imidazole at 103 K from X-ray diffraction. J. Chem. Phys. 77, 1048–1060.Google Scholar

Stevens, E. D., DeLucia, M. L. & Coppens, P. (1980). Experimental observation of the effect of crystal field splitting on the electron density distribution of iron pyrite. Inorg. Chem. 19, 813–820.Google Scholar

Su, Z. & Coppens, P. (1992). On the mapping of electrostatic properties from the multipole description of the charge density. Acta Cryst. A48, 188–197.Google Scholar

Su, Z. & Coppens, P. (1994b). On the evaluation of integrals useful for calculating the electrostatic potential and its derivatives from pseudo-atoms. J. Appl. Cryst. 27, 89–91.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 718-719