Coulomb repulsion, point-like nuclear charges, Dirac paradox, soft nuclear charge density and hypermultiplet nuclear repulsion

Abstract

A discussion about the classical Coulomb repulsion via point-like nuclear charges, usually employed within Born–Oppenheimer approximation, leads to the description of Dirac paradox: an inconsistency found when describing nuclear charges by means of Dirac’s distributions and computing with them nuclear Coulomb repulsion integrals. The way of overcoming Dirac paradox is bound to the description of soft Gaussian nuclear charge density and also to adopting a nuclear hypermultiplet Coulomb repulsion formulation. Such theoretical prospect produces a quantum mechanically compliant but simple algorithm in order to compute nuclear repulsion, which also appears to be consistently related to classical Coulomb repulsion energy, while avoiding singularities when nuclei collapse.

Appendix

1.1 Incomplete gamma function

A set of polynomial expressions to accurately compute the incomplete gamma functions needed here was published by Arents [17]. Also, knowing any higher order term, then a unit order inferior function can be easily obtained by means of the descending iteration [11]:

$$\begin{aligned} F_p \left( x \right) =\frac{1}{2p+1}\left( {2xF_{p+1} \left( x \right) +e^{-x}} \right) \end{aligned}$$

Then, one can also write:

$$\begin{aligned} F_0 \left( {\frac{\theta }{2}R_{IJ}^2 } \right) -\exp \left( {-\frac{\theta R_{IJ}^2 }{2}} \right) =\theta R_{IJ}^2 F_1 \left( {\frac{\theta }{2}R_{IJ}^2 } \right) \end{aligned}$$

1.2 Derivatives

Derivatives of the incomplete gamma function and error function are not at all difficult to obtain. They can be useful for general geometry optimization purposes.

1.2.1 A of the incomplete gamma function

The derivatives of the incomplete gamma function of any order can be expressed [12] in terms of the incomplete gamma function of a unit superior order:

$$\begin{aligned} \frac{\partial F_p \left( x \right) }{\partial x}=-F_{p+1} \left( x \right) . \end{aligned}$$

1.2.2 B of the error function

Derivatives of the error function are readily computed in terms of Hermite polynomials [13]:

$$\begin{aligned} \forall n=0,1,2,\ldots : \frac{d^{n+1}}{dx^{n+1}}erf\left( x \right) =\left( {-1} \right) ^{n}\frac{2}{\sqrt{\pi }}H_n \left( x \right) e^{-x^{2}}. \end{aligned}$$

Thus, the first derivative term, owing to the fact that also: $H_0 \left( x \right) =1$ holds, corresponds to:

$$\begin{aligned} \frac{d}{dx}erf\left( x \right) =\frac{2}{\sqrt{\pi }}e^{-x^{2}}. \end{aligned}$$