Phaseless auxiliary-field quantum Monte Carlo calculations with plane waves and pseudopotentials: Applications to atoms and molecules

The phaseless auxiliary-field quantum Monte Carlo (AF QMC) method [S. Zhang and H. Krakauer, Phys. Rev. Lett. 90, 136401 (2003)] is used to carry out a systematic study of the dissociation and ionization energies of second-row group 3A–7A atoms and dimers: Al, Si, P, S, and Cl. In addition, the ${\mathrm{P}}_2$ dimer is compared to the third-row ${\mathrm{As}}_2$ dimer, which is also triply bonded. This method projects the many-body ground state by means of importance-sampled random walks in the space of Slater determinants. The Monte Carlo phase problem, due to the electron-electron Coulomb interaction, is controlled via the phaseless approximation, with a trial wave function $\mid {\Psi }_T⟩$. As in previous calculations, a mean-field single Slater determinant is used as $\mid {\Psi }_T⟩$. The method is formulated in the Hilbert space defined by any chosen one-particle basis. The present calculations use a plane wave basis under periodic boundary conditions with norm-conserving pseudopotentials. Computational details of the plane wave AF QMC method are presented. The isolated systems chosen here allow a systematic study of the various algorithmic issues. We show the accuracy of the plane wave method and discuss its convergence with respect to parameters such as the supercell size and plane wave cutoff. The use of standard norm-conserving pseudopotentials in the many-body AF QMC framework is examined.

Figure 1

Convergence of the total energy of a phosphorus atom in a $14\times 14\times 14a_0^3$ unit cell. The OPIUM pseudopotential is used here. The convergence behavior is similar in both LDA and AF QMC methods. A constant shift is added to each data set for convenience, such that the converged energies are approximately 0 and $0.5\mathrm{eV}$ for LDA and AF QMC, respectively.

Figure 2

Convergence of the phosphorus total energy with respect to the inverse of the simulation cell volume $\Omega$. The triangles denote the results of LDA calculations, while the solid circles denote those of the AF QMC. The OPIUM pseudopotential is used here.

Figure 3

Convergence of the AF QMC total energy with simulation cell size for sulfur, plotted as a function of the inverse volume, from box sizes $12\times 12\times 12a_0^3$ through $19\times 19\times 19a_0^3$. The OAL pseudopotential is used here. The dotted line shows the fitting of the total energy as a linear function of $1∕\Omega$. The data point shown at $1∕\Omega =0$ shows the extrapolated energy at infinite box size.

Figure 4

Phosphorus dissociation energy (top panel) and the total energies of P (times two) and ${\mathrm{P}}_2$ (bottom panel) for different supercell sizes, computed using the OPIUM pseudopotential. The dissociation energy obtained with the largest simulation cell, $18\times 18\times 18a_0^3$, is within $0.1\mathrm{eV}$ of the experimental value of $5.08\mathrm{eV}$ (indicated by the horizontal dotted line in the top panel). $E_{\mathrm{conv}}$ is the energy extrapolated to $1∕\Omega \to 0$ (using cubic cells only).

Figure 5

An illustration of the size effect in the calculation of charged atoms. The top panel shows the corrected total energies for P, ${\mathrm{P}}^{+}$, and ${\mathrm{P}}^{++}$, while the bottom panel shows the uncorrected ones. Cell sizes used in this study are the same as in Fig. 4. The error bars are smaller than the point size.

Figure 6

Trotter errors for P, ${\mathrm{P}}^{+}$, and ${\mathrm{P}}^{++}$ energies (shown in the top, middle, and bottom panels, respectively) in a $14\times 14\times 14a_0^3$ simulation box. The data points at $\Delta \tau =0$ represent the Trotter extrapolation.