Fixed-node DQMC

QMC methods were first applied to the case of the electron gas by Ceper-ley in the late 1970s,and the results have been widely used in density functional theory. Only recently have these early calculations been extended by others to provide greater detail. Pickett and Broughton carried out VQMC calculations for the spin-polarized gas. Ortiz and Ballone used both VQMC and fixed-node DQMC for the spin-polarized gas in the density range most important to density functional theory. Kenny et al. performed VQMC and DQMC calculations for the nonpolarized homogeneous electron gas, incorporating relativistic effects via first-order perturbation theory. 

Methane was one of the first molecules used to illustrate the effectiveness of fixed-node DQMC calculations relative to standard ab initio methods. The first fixed-node DQMC calculations for methane recovered 97% of the correlation energy and gave a total electronic energy 30 kcal/mol below the lowest energy variational result (at the time) and only 3 kcal/mol above the experimental value. Since then many more calculations for a large variety of carbon and hydrocarbon systems have shown similarly impressive results. These systems ranged from methylene to graphite and diamond structures and were treated with and without pseudopotentials. 

Fixed-node DQMC calculations by Grossman and Mitas give excellent... 

Because of the Pauli principle antisymmetry requirement, the ground-state wave function has nodal surfaces in 3n-dimensional space, and to ensure that the walkers converge to the ground-state wave function, one must know the locations of these nodes and must eliminate any walker that crosses a nodal surface in the simulation. In the fixed-node (FN) DQMC method, the nodes are fixed at the locations of the nodes in a known approximate wave function for the system, such as found firom a large basis-set Hartree-Fock calculation. This approximation introduces some error, but FN-DQMC calculations are variational. (In practice, the accuracy of FN-DQMC calculations is improved by a procedure called importance sampling. Here, instead of simulating the evolution of with t, one simulates the evolution off, where / = where is a known accurate trial variation function for the ground state.)... 

Figure 1 A family tree of quantum chemistry DFT, density functional theory QMC, quantum Monte Carlo RRV, Rayleigh-Ritz variational theory X-a, X-alpha method KS, Kohn-Sham approach LDA, BP, B3LYP, density functional approximations VQMC, variational QMC DQMC, diffusion QMC FNQMC, fixed-node QMC PIQMC, path integral QMC EQMC, exact QMC HF, Hartree-Fock EC, explicitly correlated functions P, perturbational MP2, MP4, Maller-Plesset perturbational Cl, configuration interaction MRCI, multireference Cl FCI, full Cl CC, CCSD(T), coupled-cluster approaches. Other acronyms are defined in the text.

We reserve a description of the quantum Monte Carlo (QMC) branch until the next section. We note, however, the names variational (VQMC), diffusion (DQMC), fixed-node (FNQMC), path integral (PIQMC), and EQMC (exact quantum Monte Carlo) in the third branch of the family tree of Figure 1. 

DQMC calculations for atoms and molecules such as H2, H4, Be, H2O, and HF made by means of fixed-node structures obtained from optimized single-determinant SCF calculations typically recover more than 90% of the correlation energies of these species and yield total electronic energies lower than the lowest energy analytic variational calculations. These results suggest that optimized single-determinant wavefunctions have node structures that are reasonably correct. 

The fixed-node method was first applied in DQMC calculations for the systems H P, H2 H4 and Be The results indicated that good energies could be obtained with node locations of relatively poor quality. Because the nodal surfaces of ground state systems are typically located in regions of low electron density (i.e., according to hAq), one might expect the calculated energies to be insensitive to small departures in node locations from those of the true wavefunctions. 

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