Nuclear Charge Density and Magnetization Distributions

Abstract

In the study of electronic structure of matter, the atomic nuclei play the role of centers of force to which the electrons are bound. Within this context, almost all of the internal details of nuclear structure can be neglected, and the nuclei can be considered as objects with static extended distributions of charge and magnetic moment. This chapter presents a discussion of nuclear charge density and magnetization distributions. The underlying general principles are discussed, and details are given for model distributions that are widely used in relativistic quantum chemistry. Finally, the principal effects of extended nuclear distributions of charge and magnetic moment are pointed out.

Appendix

This appendix provides definitions for mathematical expressions and special functions used in the main part of this chapter. Unless stated otherwise, further details can be found in standard references [47, 48].

Euler-Mascheroni constant γ _E :

$$\displaystyle\begin{array}{rcl} \gamma _{\mathrm{E}} =\lim \limits _{n\rightarrow \infty }\left (\sum \limits _{k=1}^{n}\frac{1} {k} -\ln (n)\right ) = 0.57721\ldots.& &{}\end{array}$$

(90)

Heaviside step function Θ(x):

$$\displaystyle\begin{array}{rcl} \varTheta (x) = \left \{\begin{array}{ll} 0 &x <0, \\ 1/2&x = 0, \\ 1 &x> 0.\\ \end{array} \right.& &{}\end{array}$$

(91)

Gamma function Γ(a) and its logarithmic derivative:

$$\displaystyle\begin{array}{rcl} \varGamma (a) =\int _{ 0}^{\infty }\mathrm{d}t\,t^{a-1}\mathrm{e}^{-t}\quad (a> 0),\quad \varGamma (a + 1) = a\varGamma (a),\quad \varGamma (1) = 1.& &{}\end{array}$$

(92)

$$\displaystyle\begin{array}{rcl} \psi (a) = \frac{\varGamma '(a)} {\varGamma (a)} = \frac{1} {\varGamma (a)}\int _{0}^{\infty }\mathrm{d}t\,t^{a-1}\ln (t)\mathrm{e}^{-t},\qquad \psi (1) = -\gamma _{\mathrm{ E}}.& &{}\end{array}$$

(93)

Incomplete gamma functions P(a, x) and Q(a, x) and error function erf(x):

$$\displaystyle\begin{array}{rcl} P(a,x) = \frac{1} {\varGamma (a)}\int _{0}^{x}\mathrm{d}t\,t^{a-1}\mathrm{e}^{-t},\quad Q(a,x) = \frac{1} {\varGamma (a)}\int _{x}^{\infty }\mathrm{d}t\,t^{a-1}\mathrm{e}^{-t}\quad (a> 0).& &{}\end{array}$$

(94)

For a equal to a positive integer (a = n + 1, n ≥ 0):

$$\displaystyle\begin{array}{rcl} P(n + 1,x) = 1 - Q(n + 1,x) = 1 -\mathrm{ e}^{-x}\sum \limits _{ k=0}^{n}\frac{x^{k}} {k!}.& &{}\end{array}$$

(95)

For 2a equal to an odd positive integer (2a = 2k + 1, k ≥ 0):

$$\displaystyle\begin{array}{rcl} P(1/2,x^{2})& =\mathrm{ erf}(x) = \frac{2} {\sqrt{\pi }}\int _{0}^{x}\mathrm{d}t\,\exp (-t^{2}).&{}\end{array}$$

(96)

$$\displaystyle\begin{array}{rcl} P((m + 3)/2,x^{2})& =& \frac{2} {\varGamma ((m + 3)/2)}\int _{0}^{x}\mathrm{d}t\,t^{m+2}\exp (-t^{2})\qquad (m = 0,2,4,\ldots ) \\ & =& \mathrm{erf}(x) -\frac{2x} {\sqrt{\pi }}\exp (-x^{2})\sum \limits _{ k=0}^{m/2} \frac{(2x^{2})^{k}} {(2k + 1)!!}. {}\end{array}$$

(97)

Confluent hypergeometric function ₁ F ₁(a; c; x):

$$\displaystyle\begin{array}{rcl} _{1}F_{1}(a;c;x) =\sum \limits _{ k=0}^{\infty }\frac{(a)_{k}} {(c)_{k}} \frac{x^{k}} {k!},\quad (a)_{k} = \frac{\varGamma (a + k)} {\varGamma (a)}.& &{}\end{array}$$

(98)

Spherical Bessel functions of the first kind j _n(x):

$$\displaystyle\begin{array}{rcl} j_{n}(x) = (-x)^{n}\left (\frac{1} {x} \frac{\mathrm{d}} {\mathrm{d}x}\right )^{n}\frac{\sin (x)} {x} \qquad (n \geq 0).& &{}\end{array}$$

(99)

The first two members of this sequence are

$$\displaystyle\begin{array}{rcl} j_{0}(x) = \frac{\sin (x)} {x},\qquad j_{1}(x) = \frac{\sin (x)} {x^{2}} -\frac{\cos (x)} {x}.& &{}\end{array}$$

(100)

Sine integral Si(x):

$$\displaystyle\begin{array}{rcl} \mathrm{Si}(x) =\int _{ 0}^{x}\mathrm{d}t\frac{\sin (t)} {t}.& &{}\end{array}$$

(101)

Fermi-Dirac integrals $\mathcal{F}_{j}(a)$ and $\mathcal{F}_{j}(a,x)$ and related functions:

$$\displaystyle\begin{array}{rcl} \mathcal{F}_{j}(a) = \frac{1} {\varGamma (j + 1)}\int _{0}^{\infty }\mathrm{d}t \frac{t^{j}} {1 +\exp (t - a)},& &{}\end{array}$$

(102)

$$\displaystyle\begin{array}{rcl} \mathcal{F}_{j}(a,x) = \frac{1} {\varGamma (j + 1)}\int _{x}^{\infty }\mathrm{d}t \frac{t^{j}} {1 +\exp (t - a)},& &{}\end{array}$$

(103)

$$\displaystyle\begin{array}{rcl} \mathcal{G}_{j}(a) = \frac{1} {\varGamma (j + 1)}\int _{0}^{\infty }\mathrm{d}t \frac{t^{j}\ln (t)} {1 +\exp (t - a)} =\psi (j + 1)\mathcal{F}_{j}(a) + \frac{\partial } {\partial j}\mathcal{F}_{j}(a).& &{}\end{array}$$

(104)

A general algorithm for the evaluation of complete and incomplete Fermi-Dirac integrals has been published [49, 50].