A Dirac field interacting with point nuclear dynamics

Abstract

The system describing a single Dirac electron field coupled with classically moving point nuclei is presented and studied. The model is a semi-relativistic extension of corresponding time-dependent one-body Hartree-Fock equation coupled with classical nuclear dynamics, already known and studied both in quantum chemistry and in rigorous mathematical literature. We prove local existence of solutions for data in $H^\sigma$ with $\sigma \in [1,\frac{3}{2}[$. In the course of the analysis a second new result of independent interest is discussed and proved, namely the construction of the propagator for the Dirac operator with several moving Coulomb singularities.

Appendix: a remark on regularity of the ground state of Dirac-Coulomb Hamiltonian

Let us consider the Sobolev regularity of Dirac-Coulomb eigenstates when $Z<\sqrt{3}/2$.

In particular, we are interested in the ground state. Its generic component has the form (see [5, 21])

$$\begin{aligned} f(r)=\text {const}\times e^{-a r} r^{b-1} \end{aligned}$$

(4.1)

where

$$\begin{aligned} a=Z\ ,\ \ \ \ \ b=\sqrt{1-(\alpha Z)^2} \equiv \sqrt{1- \nu ^2},\ \ \ \ \nu \in (0,1) \end{aligned}$$

(4.2)

The Fourier transform of the above radial function satisfies

$$\begin{aligned} {\hat{f}}(k)=\frac{2}{\sqrt{2}\pi }\frac{1}{k}\int _0^{\infty }\ rf(r)\sin (kr)\ dr \end{aligned}$$

The above integral is the Fourier sine-transform of $rf(r)=\text {const}\times e^{-a r} r^b$.

According to the Table of Integral Transforms I, Bateman Manuscript Project, formula 2.4 (7) pag 72 in [1], one has

$$\begin{aligned} {\hat{f}}(k)&=\frac{2}{\sqrt{2}\pi }\frac{1}{k} \Gamma (b +1)(a^2+k^2)^{-\frac{1}{2}(b+1)} \sin \left[ (b+1)\tan ^{-1}\left( \frac{k}{a}\right) \right] \end{aligned}$$

(4.3)

$$\begin{aligned}&=\text {const}\times \frac{1}{k} (a^2+k^2)^{-\frac{1}{2}(b+1)} \sin \left[ (b+1)\tan ^{-1}\left( \frac{k}{a}\right) \right] \end{aligned}$$

(4.4)

This is regular at the origin while the asymptotic behavior at infinity is given by

$$\begin{aligned} {\hat{f}} \sim k^{-(b+2)} \end{aligned}$$

(4.5)

Now $f\in H^\sigma \iff |{\hat{f}}|^2(1+k^2)^s$ is integrable in $\mathbb {R}^3$, from which we get the condition $f\in H^\sigma \iff k^{(-2b-4)}(1+k^2)^sk^2$ is integrable at infinity, i.e.

$$\begin{aligned} 2b+2-2\sigma>1 \ \ \ \iff \ \ \ \sqrt{1-\nu ^2}>s-\frac{1}{2}\ \ \end{aligned}$$

(4.6)

This implies that

• 1. $f\in H^1\ \ \forall \,\nu \in (0,\frac{\sqrt{3}}{2})$

• 2. $f\notin H^{\frac{3}{2}}\ \ \text {whatever}\ \ \ \nu \in (0,\frac{\sqrt{3}}{2})$

• 3. $f\in H^\sigma \ \ \ \sigma =\sigma (\nu )=\frac{3}{2}-\epsilon$ with $\nu ^2<2\epsilon -\epsilon ^2$

So the regularity is better and better (but lower than $H^{3/2}$) with the decreasing of the charge Z.