Fixed-node quantum Monte Carlo

P.J. Reynolds et al., Fixed-node quantum Monte Carlo for molecules. J. Chem. Phys. 77, 5593-5603 (1982)... 

J. B. Anderson, Int. Rev. Phys. Chem., 14, 85 (1995). Fixed-Node Quantum Monte Carlo. 

The quality of a variational quantum Monte Carlo calculation is determined by the choice of the many-body wavefunction. The many-body wavefunction we use is of the parameterized Slater-Jastrow type which has been shown to yield accurate results both for the homogeneous electron gas and for solid silicon (14) (In the case of silicon, for example, 85% of the fixed-node diffusion Monte Carlo correlation energy is recovered). At a given coupling A, 4>A is written as... 

Keywords Electronic structure theory ab initio quantum chemistry Many-body methods Quantum Monte Carlo Fixed-node diffusion Monte Carlo Variational Monte Carlo Electron correlation Massively parallel Linear... 

Today, the most important QMC method for molecules is the diffusion quantum Monte Carlo method (DMC). It has been presented in the review articles mentioned above and in detail in the monograph by Hammond et al Here only an overview is given without mathematical rigor. A mathematical analysis of the DMC method, and in particular of its fixed-node approximation, has recently been published by Cances et al. ... 

This method is known as fixed-node diffusion quantum Monte Carlo (FN-DMC). 

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. 

He ), the node problem can be overcome by exact cancellation methods (described below), and exact solutions can be obtained. For systems of as many as 10 electrons,released-node or transient estimate methods (also described below) can provide excellent approximate solutions. But, in general, the method of choice for systems of more than about 10 electrons is the fixed-node method. Although the fixed-node method is variational and does not yield exact results, it is the only choice available for quantum Monte Carlo calculations on many larger systems. The fixed-node method is remarkably accurate and generally yields energies well below those of the best available analytic variational calculations. 

A Systematic Study of Fixed-Node and Localization Error in Quantum Monte Carlo Calculations with Pseudopotentials for Group III Elements. 

Quantum Monte Carlo Without Fixed Nodes... 

Reptation quantum Monte Carlo (RQMC) [15,16] allows pure sampling to be done directly, albeit in common with DMC, with a bias introduced by the time-step (large, but controllable in DMC e.g. [17]) and the fixed-node approach (small, but not controllable e.g. [18]). Property estimation in this manner is free from population-control bias that plagues calculation of properties in diffusion Monte Carlo (e.g. [19]). Inverse Laplace transforms of the imaginary time correlation functions allow simulation of dynamic structure factors and other properties of physical interest. 

For both bosonic systems and fermionic systems in the fixed-node approximation, G has only nonnegative elements. This is essential for the Monte Carlo methods discussed here. A problem specific to quantum mechanical systems is that G is known only asymptotically for short times, so that the finite-time Green function has to be constructed by the application of the generalized Trotter formula [6,7], G(r) = limm 00 G(z/m)m, where the position variables of G have been suppressed. 

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