Two-component density functional theory study of quantized muons in solids

The quantum effect of light nuclei in materials is usually considered as lattice vibration and zero-point motion from an ab initio perspective. Here we start from full-quantized particles and take the muon as an example, considering the two-body quantized system of the muon and the electrons within two-component density functional theory, which can calculate the related two-body wave functions directly and automatically taking into account the quantum effect of the muon. We first simulate the two-body correlation energy and the pair-correlation function by quantum Monte Carlo, then construct a two-body self-consistent loop to solve the two-body density and then the hyperfine couplings are estimated. This is an attempt to reveal the quantum effect of light nuclei in materials from a different perspective. We finally find this fundamental method agrees better with experiments and shows good potential for further such calculations.

Figure 1

Some $g\left(r\right)$ results. $r_s^{-}={(3/4\pi n^{-})}^{1/3}$, $r_s^{+}$ also. (a) PCF calculated from 54 electrons and 2 muons, as a function of distance r and $r_s^{-}$, (b) PCF calculated from 54 muons and 2 electrons, as a function of distance $r$ and $r_s^{+}$, (c)PCF calculated from 28 muons and 28 electrons, as a function of distance $r$ and muon or electrons density. All $g\left(0\right)$ are estimated from the extrapolation of $g\left(r\right)$.

Figure 2

$g\left(0\right)$ for $n^{+}\ll n^{-}$, calculated from 54 electrons and 2 muons system. The fitting curve is the $g_3$ in Eq. (12).

Figure 3

$g\left(0\right)$ for $n^{+}\gg n^{-}$, calculated from 54 muons and 2 electrons system. This corresponds to $g_2$ in Eq. (11).

Figure 4

Correlation potential of electrons calculated from 54 muons and 2 electrons. This corresponds to $V_2$ in Eq. (16).

Figure 5

TCDFT loop. $+$ stands for muon, $-$ stands for electrons. At present, the lattice optimization calculation using muon density (the arrow of lattice relax) is not implanted due to the calculation efficiency, the point-like muon optimization structure is simply applied in this work. Further study about the full relax calculation should be improved in the future.

Figure 6

Convergent density of vacuum for $\alpha =1$.

Figure 7

Convergent density for (a) NaF, plotted from (0,0,0) to (2,2,2), muon site is (${\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}}$). (b) ${\mathrm{CaF}}_2$, from (0,0,0) to (2,2,2), muon site is (1,1,1). (c) Diamond, from (0,0,0) to (2,2,2), muon site is (${\displaystyle \frac{2}{3}},{\displaystyle \frac{2}{3}},{\displaystyle \frac{2}{3}}$). (d) Fe-bcc, from (0,0,0) to (2,1,0), muon site is (${\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{4}}$,0). (e) Co-fcc, from (0,${\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}$) to (3,${\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}$), muon site is (${\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}$). (f) Co-hcp, from (0,1,${\displaystyle \frac{3}{4}}$) to (3,$-2,{\displaystyle \frac{3}{4}}$), muon site is (${\displaystyle \frac{1}{3}},{\displaystyle \frac{2}{3}},{\displaystyle \frac{3}{4}}$). Crystal structures with the muon can be found in Refs. [1, 5, 29].

Figure 8

Relative errors of different methods. Dashed curves are a guide to the eye.