Enhancement models of momentum densities of annihilating electron-positron pairs: The many-body picture of natural geminals

The correlated motion of a positron surrounded by electrons is a fundamental many-body problem. We approach this by modeling the momentum density of annihilating electron-positron pairs using the framework of reduced density matrices, natural orbitals, and natural geminals (electron-positron pair wave functions) of the quantum theory of many-particle systems. We find that an expression based on the natural geminals provides an exact, unique, and compact expression for the momentum density. The natural geminals can be used to define and to determine enhancement factors for enhancement models going beyond the independent-particle model for a better understanding of the results of positron annihilation experiments.

Figure 1

(a) The total momentum density, un-normalized diagonal sums of Eq. (10) for the natural orbitals (NO) and HO orbitals, and the convergence analysis of Eq. (7) for the $\omega =0.2$ system. The inset shows the high-momentum regime and the behavior of the two-shell NO sum. (b) Enhancement factors for the $\omega =0.4$ system. The solid (black) line shows the factor $\sqrt{\gamma \left(\mathbf{r}\right)}$ evaluated with ED. The (blue) dashed lines show the $\sqrt{{\gamma }_j\left(\mathbf{r}\right)}$'s for the $s$ and $p$ shells. The solid (orange) line is the $\sqrt{\gamma \left(\mathbf{r}\right)}$ according to a two-component LDA functional [16], the (red) dashed line is its zero-positron-density limit [5], and the dotted (green) one is the result of the GGA (Ref. [17]), which also is formulated in this limit. The ED results are not converged near the border of the figure because of the finite extent of the basis functions. (c) Two-shell diagonal sums of Eq. (10) for various bases as well as a convergence analysis of Eq. (7) for the $\omega =0.4$ system plotted as a ratio to the accurate ED spectrum. All spectra have been normalized prior to taking the ratio. Also, the position-dependent LDA model employing the LDA $\gamma \left(\mathbf{r}\right)$ (Ref. [8]) has been applied as well as the state-dependent enhancement factor by Alatalo et al. [9].