First-principles method for impurities in quantum fluids: Positron in an electron gas

We propose a first-principles methodology for calculating the behavior of isolated impurities immersed in quantum fluids. To obtain an accurate description of correlation effects between the impurity and the host, we work in the frame of reference in which the impurity is stationary, building on the work of C. H. Leung, M. J. Stott, and C. O. Almbladh [Phys. Lett. 57A, 26 (1976)]. We apply our methodology to the case of a positron immersed in an electron gas. Our positron relaxation energies and annihilation rates are similar to those from the best existing many-body calculations. Our annihilating-pair momentum densities are significantly different from previous data and include a “tail” after the Fermi edge.

Figure 1

(Color online) Kohn-Sham eigenvalues for a HEG in the transformed frame in which the “positron” $\left(m_p=1\text{a}\text{.u}\text{.}\right)$ is stationary (solid line) and in the laboratory (untransformed) frame (dashed line).

Figure 2

(Color online) Dependence of the relaxation energy on the plane-wave cutoff energy $E_{\text{cut}}$ for positrons immersed in HEGs of different system size $N$ at density parameter (a) $r_s=2$ and (b) $r_s=5$.

Figure 3

(Color online) Relaxation energy against inverse system size $N^{-1}$ for positrons immersed in HEGs at different density parameters $r_s$. The relaxation energies have been extrapolated to basis-set completeness in each case.

Figure 4

(Color online) Relaxation energy against $r_s$ for a positron immersed in a 114-electron HEG.

Figure 5

Dependence of the annihilating-pair MD at particular momenta on the plane-wave cutoff energy $E_{\text{cut}}$ for positrons immersed in (a) a 114-electron HEG of density parameter $r_s=2$, (b) a 162-electron HEG at $r_s=2$, (c) a 114-electron HEG at $r_s=5$, and (d) a 54-electron HEG at $r_s=5$. The solid line shows the fit to Eq. (59) in each case.

Figure 6

(Color online) Annihilating-pair MDs for positrons immersed in HEGs of density parameter (a) $r_s=2$ and (b) $r_s=5$. The MDs have been extrapolated to basis-set completeness. The curves labeled “Nonint.” show the MD of a free-electron gas.

Figure 7

(Color online) PCFs for positrons immersed in 114-electron HEGs of density parameter (a) $r_s=2$ and (b) $r_s=5$. The solid lines show the uncorrected PCFs and the dashed lines show the cusp-corrected PCFs.

Figure 8

(Color online) Dependence of the contact PCF $g\left(0\right)$ on the plane-wave cutoff energy $E_{\text{cut}}$ for positrons immersed in HEGs of density parameter (a) $r_s=2$ and (b) $r_s=5$. The data have been multiplied by $N/\left(N-1\right)$, as discussed in Sec. 7C2, and the data labeled “CC” have been cusp-corrected using the scheme described in Appendix .

Figure 9

(Color online) Cusp-corrected PCFs for positrons immersed in HEGs of density parameter (a) $r_s=2$ and (b) $r_s=5$. The data have been multiplied by $N/\left(N-1\right)$. The plane-wave cutoff energy was 50 and 10 a.u. at $r_s=2$ and $r_s=5$, respectively. The insets show the PCFs relative to that obtained with $N=246$ electrons.

Figure 10

(Color online) Relaxation energy against density parameter. Results obtained with $N=114$, 162, and 246 electrons for $1\le r_s\le 3.5$, with $N=54$, 114, and 162 electrons for $4\le r_s\le 6.5$, and with $N=14$, 54, and 114 electrons for $7\le r_s\le 8$ were averaged to obtain the final results. Also shown are the relaxation energies obtained by other authors (Refs. 25, 26, 27, 28). The dashed line following the data of Arponen and Pajanne (Ref. 25) is the widely used fit of Boroński and Nieminen (Ref. 7). The energy of the ${\text{Ps}}^{-}$ ion is taken from Ref. 24. The curve labeled “No extra” was obtained without the extra interaction term in the transformed Hamiltonian.

Figure 11

(Color online) Annihilating-pair MDs $\rho \left(\overline{p}\right)$ for different densities. The solid lines show the MDs calculated with the extra interaction included; the dotted line shows the MD calculated at $r_s=8$ without the extra interaction. Model functions have been fitted to the results obtained with $N=406$ electrons at $r_s=1$, $N=246$ electrons for $2\le r_s\le 7$, and $N=162$ electrons at $r_s=8$. MD data calculated by Kahana (Ref. 19) and Stachowiak (Ref. 30) using other approaches are also shown. The dotted curve labeled No extra was obtained without the extra interaction term in the transformed Hamiltonian.

Figure 12

(Color online) Electron-positron PCF for positrons immersed in HEGs of different density parameter $r_s$. The solid line shows results obtained with the extra interaction terms present; the dotted lines show results obtained without the extra interaction. The data are for 54-electron HEGs and the finite-size correction discussed in Sec. 7C2 has been applied. The cusp correction described in Appendix has been applied. The plane-wave cutoff energies are 400, 100, 100, 50, 50, 20, 20, and 20 a.u. at $r_s=1$, 2, 3, 4, 5, 6, 7, and 8, respectively. The dotted curves labeled No extra were obtained without the extra interaction term in the transformed Hamiltonian.

Figure 13

(Color online) Electron-positron contact PCF $g\left(0\right)$ against density parameter $r_s$. The results were obtained by averaging the $g\left(0\right)$ values obtained in cells with $N=114$, 162, and 246 electrons for $1\le r_s\le 7$ and $N=114$ and 162 at $r_s=8$. Various contact PCF results from the literature are also shown. The results of Leung et al. (Ref. 9) were obtained by rescaling the data for a proton in a HEG obtained by Almbladh et al. (Ref. 32). The curve labeled No extra was obtained without the extra interaction term in the transformed Hamiltonian.