Quantum Monte Carlo study of three-dimensional Coulomb complexes: Trions and biexcitons, hydrogen molecules and ions, helium hydride cations, and positronic and muonic complexes

Three-dimensional (3D) excitonic complexes influence the optoelectronic properties of bulk semiconductors. More generally, correlated few-particle molecules and ions, held together by pairwise Coulomb potentials, play a fundamental role in a variety of fields in physics and chemistry. Based on statistically exact diffusion quantum Monte Carlo calculations, we have studied excitonic three- and four-body complexes (trions and biexcitons) in bulk 3D semiconductors, as well as a range of small molecules and ions in which the nuclei are treated as quantum particles on an equal footing with the electrons. We present interpolation formulas that predict the binding energies of these complexes, either in bulk semiconductors or in free space. By evaluating pair distribution functions within quantum Monte Carlo simulations, we examine the importance of harmonic and anharmonic vibrational effects in small molecules.

Figure 1

(a) An exciton X is a bound electron-hole pair. (b) A negative trion is a negatively charged exciton ${\mathrm{X}}^{-}$. (c) A positive trion is a positively charged exciton ${\mathrm{X}}^{+}$.

Figure 2

DMC binding energies of negative trions against rescaled mass ratio $x=\sigma /(1+\sigma )$. The solid line shows the fit of Eq. (17).

Figure 3

DMC BO potential energy, i.e., DMC total energy of a positive trion in the infinite mass limit of two heavy holes and one light electron ($\sigma =0$) against the hole-hole distance. The solid curve shows a fit of Eq. (18) to the DMC data.

Figure 4

Nucleus-nucleus RDF $4\pi r^2{g}_{\text{hh}}\left(r\right)$ for three dihydrogen cations ($\mathrm{H}{{}_2}^{+}, \mathrm{D}{{}_2}^{+}$, and $\mathrm{T}{{}_2}^{+}$) vs the nucleus-nucleus distance.

Figure 5

Electron-nucleus PDF $g_{\text{eh}}\left(r\right)$ and the corresponding RDF $4\pi r^2{g}_{\text{eh}}\left(r\right)$ in three dihydrogen cations (${{\mathrm{H}}_2}^{+}, {{\mathrm{D}}_2}^{+}$, and ${{\mathrm{T}}_2}^{+}$) vs the electron-nucleus distance. The curves for the different ions are almost on top of each other.

Figure 6

Proton-proton RDF in a hydrogen molecular cation obtained using the three different approaches listed in the caption of Table 7.

Figure 7

DMC binding energies of biexcitons against rescaled mass ratio $x=\sigma /(1+\sigma )$. The solid line shows the fit of Eq. (28).

Figure 8

DMC BO potential energy, i.e., DMC total energy of a biexciton in the infinite mass limit of two heavy holes and two light electrons ($\sigma =0$) against the hole-hole distance. The solid curve shows a fit of Eq. (29) to the DMC data.

Figure 9

Nucleus-nucleus RDF $4\pi r^2{g}_{\text{hh}}\left(r\right)$ for three dihydrogen molecules (${\mathrm{H}}_2, {\mathrm{D}}_2$, and ${\mathrm{T}}_2$) vs the nucleus-nucleus distance.

Figure 10

Electron-nucleus PDF $g_{\text{eh}}\left(r\right)$ and RDF $4\pi r^2{g}_{\text{eh}}\left(r\right)$ in three dihydrogen molecules (${\mathrm{H}}_2, {\mathrm{D}}_2$, and ${\mathrm{T}}_2$) vs the electron-nucleus distance.

Figure 11

Proton-proton RDF in a hydrogen molecule ${\mathrm{H}}_2$ from three different approaches: BO approximation, $\mathrm{BO}+\text{harmonic}$ approximation, and the DMC solution to the four-particle problem.

Figure 12

DMC total energy against time step for an $\mathrm{H}{{}_2}^{+}$ ion (a positive trion with a very small electron-hole mass ratio of $\sigma \approx 0.0005446$). A quadratic fit to the DMC energies at time steps of 0.0025, 0.005, 0.008, 0.01, 0.08, and 0.16 e.u. is shown as a dashed blue line. A linear fit to the DMC energies at two small time steps of 0.0025 and 0.01 e.u. is shown as a solid red line. The DMC energies extrapolated to zero time step using the quadratic and linear fits are $-0.5974637\left(4\right)$ and $-0.5974636\left(7\right)$ e.u., respectively.

Figure 13

As Fig. 12, but for an ${\mathrm{H}}_2$ molecule. The DMC energies extrapolated to zero time step using quadratic and linear fits are $-1.164653\left(5\right)$ and $-1.164649\left(8\right)$ e.u., respectively.

Figure 14

Fits of a Morse potential [Eq. (C1), raw DMC data offset by $E_{\text{X}}$] and Eq. (18) to the DMC BO potential of a positive trion. The Morse fitting parameters are $D_{\text{eq}}=0.102553\left(8\right)$ a.u., $R_{\text{eq}}=2.0181\left(1\right)$ a.u., and $a=0.7003\left(2\right)$ a.u. The SSE and RMSE are $6.451\times {10}^{-6}$ a.u. and $0.0006788$ a.u., respectively. The value of $R_{\text{eq}}$ is slightly larger than the value predicted by fitting Eq. (18). However, the Morse potential shows much more reasonable behavior at small and large hole-hole separations. The fitted function given by Eq. (18) increases unphysically at large separations beyond 2.6 a.u. However, Eq. (18) fits the DMC data very well in the vicinity of the equilibrium point, as seen in the inset; consequently, spectroscopic constants predicted by Eq. (18), shown in Table 5, are more accurate than those predicted by the Morse potential, shown in Table 19.

Figure 15

Fits of a Morse potential [Eq. (C1), raw DMC data offset by $2E_{\text{X}}$] and Eq. (29) to the DMC BO potential of a biexciton. To obtain a better SSE and RMSE using the Morse model, we only included separations in the range 1 a.u.$\le r_{\text{hh}}\le 1.8$ a.u. in the fit. The Morse fitting parameters are $D_{\text{eq}}=0.17459\left(1\right)$ a.u., $R_{\text{eq}}=1.4054\left(1\right)$ a.u., and $a=1.0402\left(6\right)$ a.u. The SSE and RMSE are $1.018\times {10}^{-5}$ a.u. and $\mathrm{RMSE}=0.001128$ a.u., respectively. The $R_{\text{eq}}$ is close to the value obtained by fitting Eq. (29). In the large separation limit, contrary to the Morse model, Eq. (29) becomes unphysical. In the vicinity of the equilibrium point, Eq. (29) fits the DMC data better and produces more accurate spectroscopic constants, as seen by comparing Tables 10 and 20.