Trail-Needs pseudopotentials in quantum Monte Carlo calculations with plane-wave/blip basis sets

We report a systematic analysis of the performance of a widely used set of Dirac-Fock pseudopotentials for quantum Monte Carlo (QMC) calculations. We study each atom in the periodic table from hydrogen ($Z=1$) to mercury ($Z=80$), with the exception of the $4f$ elements ($57\le Z\le 70$). We demonstrate that ghost states are a potentially serious problem when plane-wave basis sets are used in density functional theory (DFT) orbital-generation calculations, but that this problem can be almost entirely eliminated by choosing the $s$ channel to be local in the DFT calculation; the $d$ channel can then be chosen to be local in subsequent QMC calculations, which generally leads to more accurate results. We investigate the achievable energy variance per electron with different levels of trial wave function and we determine appropriate plane-wave cutoff energies for DFT calculations for each pseudopotential. We demonstrate that the so-called “T-move” scheme in diffusion Monte Carlo is essential for many elements. We investigate the optimal choice of spherical integration rule for pseudopotential projectors in QMC calculations. The information reported here will prove crucial in the planning and execution of QMC projects involving beyond-first-row elements.

Figure 1

Magnitude of difference between plane-wave and Gaussian DFT total energies $E_{\mathrm{DFT}\left(\mathrm{PW}\right)}$ and $E_{\mathrm{DFT}\left(\mathrm{Gauss}\right)}$ for each TN pseudopotential, for different choices of local channel. No results are given for lutetium because of the lack of availability of an accurate Gaussian basis set; this does not imply that lutetium is affected by ghost states. The DFT calculations for niobium with $s$ and $p$ local, vanadium, iron, and nickel with $p$ local, and manganese with $d$ local persistently failed to converge and are not shown in the graph.

Figure 2

TN pseudopotentials that are affected by ghost states for different choices of local channel in plane-wave DFT calculations. For elements in white cells, there are no ghost states; for elements in blue cells, choosing $p$ (but not $s$ or $d$) to be local results in ghost states; for elements in yellow cells, choosing $d$ (but not $s$ or $p$) to be local results in ghost states; for elements in red cells, choosing $p$ or $d$ (but not $s$) to be local leads to ghost states; and elements in gray cells are haunted by ghost states for any choice of local channel. Gaussian DFT results are not available for lutetium ($Z=71$), which is therefore omitted.

Figure 3

TN pseudopotentials that are affected by ghost states or similar difficulties for different choices of local channel in QMC calculations using plane-wave DFT orbitals. For elements in white cells, there are no ghost-state-like symptoms; for elements in blue cells, choosing $p$ (but not $s$ or $d$) to be local results in ghost-state-like symptoms; for elements in yellow cells, choosing $d$ (but not $s$ or $p$) to be local results in ghost-state-like symptoms; for elements in red cells, choosing $p$ or $d$ (but not $s$) to be local leads to ghost-state-like symptoms; and elements in gray cells are affected by ghost-state-like symptoms for any choice of local channel.

Figure 4

Plane-wave cutoff energy required to achieve a given level of convergence in the DFT energy per atom for each TN pseudopotential. The $s$ channel is chosen to be local in each case. 120 Ha is the cutoff energy used for the tests reported in Fig. 7.

Figure 5

Top panel: DFT, VMC, and DMC energies as a function of plane-wave cutoff energy $E_{\mathrm{cut}}$ for an isolated oxygen atom. Bottom panel: VMC energy variance against plane-wave cutoff energy for an isolated oxygen atom. Two different levels of correlated wave function were used in the VMC calculations, and energy minimization (“Emin”) and unreweighted variance minimization (“Varmin”) were used to optimize the wave functions. The dashed and dash-dotted vertical lines show the cutoff energies at which the DFT total energy is converged to within chemical accuracy and to within 0.1 mHa, respectively. The DMC calculations used the T-move scheme. Note that unreweighted variance minimization does not actually minimize the true variance of the energy; hence higher variances can be obtained with unreweighted variance minimization than with energy minimization [33].

Figure 6

As Fig. 5, but for an isolated oxygen molecule.

Figure 7

Top panel: VMC energy variance per electron achieved with different angular-momentum channels chosen to be local (“$s\to d$” indicates that $s$ was chosen to be local in the DFT orbital-generation calculation, while $d$ was chosen to be local in the VMC calculation). We used the spin-polarized ground-state electronic configuration in each case. Energy minimization was used to optimize the trial wave function. Bottom panel: VMC energy variance per electron with the $s$ channel local in the DFT calculation and the $d$ channel local in the VMC calculation, using different levels of Slater-Jastrow trial wave function. Energy minimization (“Emin”) and unreweighted variance minimization (“Varmin”) were used to optimize the Jastrow factor in each case.

Figure 8

DMC energy of a small-core copper pseudoatom against time step with different trial wave functions, with and without the use of T-moves. The trial wave functions were optimized by energy minimization (Emin) and unreweighted variance minimization (Varmin). The solid and dashed lines show fits to the data without and with T-moves, respectively. The orbitals were generated using a plane-wave cutoff energy of 120 Ha, and the $s$ channel was chosen to be local to avoid ghost-state problems.