A proposed family of variationally correlated first-order density matrices for spin-polarized three-electron model atoms

Abstract

In early work of March and Young (Phil Mag 4:384, 1959), it was pointed out for spin-free fermions that a first-order density matrix (1DM) for $N-1$ particles could be constructed from a 2DM ($\Gamma$) for $N$ fermions divided by the diagonal of the 1DM, the density $n(\mathbf{r}_1)$, as $2\Gamma (\mathbf{r}_1,\mathbf{r}^{\prime }_2;\mathbf{r}_1,\mathbf{r}_2)/n(\mathbf{r}_1)$ for any arbitrary fixed $\mathbf{r}_1$. Here, we thereby set up a family of variationally valid 1DMS constructed via the above proposal, from an exact 2DM we have recently obtained for four electrons in a quintet state without confining potential, but with pairwise interparticle interactions which are harmonic. As an indication of the utility of this proposal, we apply it first to the two-electron (but spin-compensated) Moshinsky atom, for which the exact 1DM can be calculated. Then the 1DM is found for spin-polarized three-electron model atoms. The equation of motion of this correlated 1DM is exhibited and discussed, together with the correlated kinetic energy density, which is shown explicitly to be determined by the electron density.

Appendix: Some technical details relating to Section 2

The Moshinsky wave function in Eq. (6) is written completely in terms of two quantity $C_1$ and $C_2$ and the normalization constant $C$. These are given below explicitly in terms of the force constant $k$ of the external potential and $K$ for the interparticle interaction $u(r_{12})$:

$$\begin{aligned}&C = \frac{\beta ^{\frac{3}{2}}}{2\sqrt{2}\pi ^{\frac{3}{2}} a^3}, \quad C_1 = \frac{1+\beta ^2}{8a^2}, \quad C_2 = \frac{1-\beta ^2}{4a^2}, \nonumber \\&a = \Bigl (\frac{\hbar }{2 \sqrt{m \,k }}\Bigr )^{\frac{1}{2}} , \quad \beta = \Bigl (\frac{2K + k}{k}\Bigr )^{\frac{1}{4}}. \end{aligned}$$

(38)

Next, We turn to the density $n_1$ in Eq. (12). We summarize below a spherical family of one-level electron densities and the corresponding one-body potential. This family is constructed explicitly by multiplying Eq. (12) by volume element $d\Omega$ and integrate over the angle between $\mathbf{r}_1$ and $\mathbf{r}_2$ to find

$$\begin{aligned} \bar{n}_1 (r_2;r_1) = \frac{d}{C_2 r_1 r_2} \bigl (e^{C_2 r_1 r_2} - e^{-C_2 r_1 r_2}\bigr ) e^{-2C_1 r_2^2}. \end{aligned}$$

(39)

With this density $\bar{n}_1$ inserted for $n$ in the von Weizsäcker Eq. (13) we extract the one-body potential $\bar{V}(\mathbf{r}_2;\mathbf{r}_1)$ to within an additive constant, as

$$\begin{aligned}&\frac{2m}{\hbar ^2} \bar{V}(\mathbf{r}_2;\mathbf{r}_1) = -\frac{1}{4 r_2^2} + 4 C_1^2 r_2^2 -\Bigl (\frac{C_2 r_1}{2} \Bigr )^2 \coth ^2{(C_2 r_1 r_2)} \nonumber \\&\quad + \Bigl ( \frac{C_2 r_1}{2 r_2} - 2C_1 C_2 r_1 r_2 \Bigr ) \coth {(C_2 r_1 r_2)}. \end{aligned}$$

(40)

This means that the Schrödinger equation can be solved exactly for this potential $\bar{V}(\mathbf{r}_2;\mathbf{r}_1)$ for arbitrary $\mathbf{r}_1$, the corresponding wave function $\bar{\Psi }$ being simply the square root of the right hand side of Eq. (39). Figure 2 depicts the general shape of the potential in Eq. (39) for a particular choice of $C_2 r_1$ and $C_1$.