Quantum Monte Carlo calculations of energy gaps from first principles

We review the use of continuum quantum Monte Carlo (QMC) methods for the calculation of energy gaps from first principles, and present a broad set of excited-state calculations carried out with the variational and fixed-node diffusion QMC methods on atoms, molecules, and solids. We propose a finite-size-error correction scheme for bulk energy gaps calculated in finite cells subject to periodic boundary conditions. We show that finite-size effects are qualitatively different in two-dimensional materials, demonstrating the effect in a QMC calculation of the band gap and exciton binding energy of monolayer phosphorene. We investigate the fixed-node errors in diffusion Monte Carlo gaps evaluated with Slater-Jastrow trial wave functions by examining the effects of backflow transformations, and also by considering the formation of restricted multideterminant expansions for excited-state wave functions. For several molecules, we examine the importance of structural relaxation in the excited state in determining excited-state energies. We study the feasibility of using variational Monte Carlo with backflow correlations to obtain accurate excited-state energies at reduced computational cost, finding that this approach can be valid. We find that diffusion Monte Carlo gap calculations can be performed with much larger time steps than are typically required to converge the total energy, at significantly diminished computational expense, but that in order to alleviate fixed-node errors in calculations on solids the inclusion of backflow correlations is sometimes necessary.

Figure 1

Scaled difference of exciton binding energy $E_{\mathrm{B}}^{\mathrm{X}}$ and the exciton Rydberg against the lattice parameter $L$ in an effective-mass model of a three-dimensional exciton confined in a periodic FCC cell. $R_{\mathrm{y}}^{*}=m_{\mathrm{e}}^{*}{m}_{\mathrm{h}}^{*}/\left[2{\epsilon }^2(m_{\mathrm{e}}^{*}+m_{\mathrm{h}}^{*})\right]$ and $a_0^{*}=\epsilon (m_{\mathrm{e}}^{*}+m_{\mathrm{h}}^{*})/\left(m_{\mathrm{e}}^{*}{m}_{\mathrm{h}}^{*}\right)$ are the exciton Rydberg and the exciton Bohr radius, respectively, where $m_{\mathrm{e}}^{*}$ and $m_{\mathrm{h}}^{*}$ are the electron and hole masses and $\epsilon$ is the permittivity. The exciton Rydberg is the binding energy of an exciton in a cell of infinite extent. The gradient on this log-log plot gives the scaling exponent of the finite-size error in the exciton binding energy.

Figure 2

Approximations to the first-excited-state energy of an H atom using the ${\psi }^{\gamma }$ excited-state trial wave function of Eq. (14) as a function of $\gamma$, obtained by various means. DMC errors are smaller than the thickness of the lines. The pocket eigenvalues outside and inside the nodal surface, $E_{\mathrm{outside}}^{\mathrm{pocket}}$ and $E_{\mathrm{inside}}^{\mathrm{pocket}}$, were determined by numerical integration of the Schrödinger equation with fixed-node boundary conditions, and $\langle \widehat{H}\rangle =\langle {\psi }^{\gamma }|\widehat{H}|{\psi }^{\gamma }\rangle$, where $\widehat{H}$ is the Hamiltonian.

Figure 3

DMC first-excited-state energies of an H atom with the trial wave function ${\psi }_L^{\alpha ,\beta }$ [see Eq. (15)] for $L=2$ and 4 at various amplitudes $\alpha$ of wrinkling of the nodal surface [see Eq. (15)]. DMC error bars are of order the size of the symbols.

Figure 4

Nodal surface of the SJB trial wave function ${\psi }_{L=4}^{\alpha =0.1,\beta }$ [see Eq. (15)] for the first excited state of an H atom. The wave function is optimized by VMC energy minimization. SJB-$n$ labels the nodal surface of the SJB wave function after the $n^{\text{th}}$ cycle of energy minimization. The $n=3$ and 4 cases are indistinguishable from each other, and correspond to the termination of the optimization process.

Figure 5

Nondimer molecules whose energy gaps we have calculated. From left to right: anthracene $\left({\mathrm{C}}_{14}{\mathrm{H}}_{10}\right)$, tetracyanoethylene $\left({\mathrm{C}}_6{\mathrm{N}}_4\right)$, benzothiazole $\left({\mathrm{C}}_7{\mathrm{H}}_5\mathrm{NS}\right)$, and boron trifluoride $\left({\mathrm{BF}}_3\right)$.

Figure 6

Finite-size errors in uncorrected SJ-DMC quasiparticle and excitonic gaps ${\mathrm{\Delta }}_{\text{QP}}$ and ${\mathrm{\Delta }}_{\text{Ex}}$ of Si as a function of the number of primitive cells $N_{\mathrm{P}}$ in the supercell. The dashed lines show the infinite-system gaps estimated by subtracting the supercell Madelung constant from the gaps obtained in finite cells and averaging over the different cells.

Figure 7

Uncorrected SJ-DMC quasiparticle and excitonic energy gaps of Si at $\mathrm{\Gamma }$, plotted against DFT analog “quasiparticle” (AQP) gaps, obtained using the defect formation energies for positive and negative charged defects. The results were obtained in different sizes and shapes of periodic cell. The straight lines are linear fits of SJ-DMC gap against DFT AQP gaps.

Figure 8

Time-step bias in (a) SJ-DMC energy gaps and (b) SJ-DMC total energies for ground $\left({\mathrm{\Gamma }}^0\right)$, excitonic $\left(\mathrm{\Gamma }\to \mathrm{\Gamma }\right)$, cationic $\left({\mathrm{\Gamma }}^{-}\right)$, and anionic $\left({\mathrm{\Gamma }}^{+}\right)$ states of Si. All calculations have been performed in a $2\times 2\times 2$ supercell with a target population of 256 walkers. The Madelung correction is not included (and would only offset the gaps by a constant).

Figure 9

Geometry of a phosphorene layer: (a) tilted view, (b) top view, and (c) front view.

Figure 10

DMC quasiparticle gaps ${\mathrm{\Delta }}_{\text{QP}}$, excitonic gaps ${\mathrm{\Delta }}_{\text{Ex}}$, and exciton binding energies $E_{\text{B}}^{\text{X}}$ at $\mathrm{\Gamma }$ against the inverse of the number $N_{\mathrm{P}}$ of primitive cells in the supercell for a free-standing phosphorene monolayer. The Keldysh Madelung constant correction has been applied to the quasiparticle gaps; no finite-size correction has been applied to the excitonic gaps. The nondiagonal supercell results (filled symbols) have been slightly shifted relative to the $3\times 2$ supercell result for readability.

Figure 11

Exciton binding energy in phosphorene within the effective-mass approximation with the Keldysh interaction between charges as a function of the permittivity of the surrounding medium $\epsilon$. Results were obtained using the fitting formula from Ref. [74].