Quantum Monte Carlo study of correlation energy and pair correlation function at various electron-positron density ratios: Accurate calculation of positron annihilation lifetimes in solids

Positron annihilation lifetime spectroscopy is a unique technique to probe the defect structure of crystalline solids. To distinguish different defect states, accurate calculation of the positron annihilation lifetimes is demanded to compare with the experimental results. We perform quantum Monte Carlo calculations to compute the electron-positron correlation energy and pair correlation function at several electron-positron density ratios. This approach enables us to obtain new fitting parameters under the local density approximation to combine with two-component density functional theory to calculate the positron lifetimes of defect-free bulk and monovacancy defects in various materials. As compared with the experimental results, this method outperformed some previous schemes and provided theoretical support for the previous deduction of certain vacancy defects.

Figure 1

PCF as a function of $r/r_s^e$ for (a) $N_p:N_e=1:64$, (b) $N_p:N_e=30:60$, and (c) $N_p:N_e=40:40$, calculated from DMC simulations with a time step of 0.001 a.u. (d) The $g\left(0\right)$ at three different density ratios derived from the PSN scheme and QMC simulations. The indexes of 0, 1, and 2 represent $R\to 0$ ($N_p:N_e=1:64$), $R=1$ ($N_p:N_e=40:40$), and $R=1/2$ ($N_p:N_e=30:60$), respectively. The dashed lines are the fitted curves of data obtained from the QMC simulations. The solid lines are obtained from the PSN scheme.

Figure 2

The correlation energy obtained from QMC simulations at $R=1$. The red dashed line is the curve fitted by the QMC data points using Eq. (15), and the black solid line is obtained from the PSN scheme. The unit of the energy is ${10}^{-3}$ Rydberg per unit volume.

Figure 3

The comparison between new correlation energy (dashed line) and PSN correlation energy (solid line) as a function of positron density $n_p$ at $r_s^e=2,3,4,5$. The unit of the energy is ${10}^{-3}$ Rydberg per unit volume.

Figure 4

A fully self-consistent TCDFT loop. $V_H^{e-p}$ and $V_{\mathrm{corr}}^{e-p}$ refer to the electron-positron Hartree potential and correlation potential, respectively. $V_{\mathrm{corr}}^{e-p}$ and the enhancement factor used here are obtained from QMC simulations.

Figure 5

The relative error ($\mathrm{\Delta }\tau /{\tau }_{\mathrm{Exp}}$) between the four calculation results and the experimental values for the bulk positron lifetimes of some materials.

Figure 6

The standard deviation of relative errors derived from Fig. 5.

Figure 7

The contact PCF obtained by SJ plane waves (square) and SJB plane waves (circle) at $R=1/2$ (blue) and $R=1$ (red).

Figure 8

The energy and $g\left(0\right)$ obtained by QMC simulations at (a),(d) $R=\frac{1}{2}, r_s^e=4$ and (b),(c) $R=1, r_s^e=4$ with different total particle numbers. The blue and red dashed lines represent the values derived by Eq. (B1).