Green’s functions Monte Carlo

Other quantum simulations involve simulations with effective Hamiltonians [261-263] or the simulation of ground state wave properties by Green s function Monte Carlo or diffusion Monte Carlo for reviews and further references on these methods see Refs. 162, 264-268. 

We now turn to approaches that begin with an arbitrary initial function and can, in principle, be iterated to an exact or accurate solution of the SE. The earliest approach is Green s function Monte Carlo (GFMC) in which the time-independent Schrodinger equation is employed [24] DMC was developed later and follows from the time-dependent SE (TDSE) in imaginary time. 

D.M. Ceperley, The statistical error of green s function Monte Carlo. J. Stat. Phys. 43, 815-826 (1996)... 

B. Diffusion Monte Carlo and Green s Function Monte Carlo... 

This review is a brief update of the recent progress in the attempt to calculate properties of atoms and molecules by stochastic methods which go under the general name of quantum Monte Carlo (QMC). Below we distinguish between basic variants of QMC variational Monte Carlo (VMC), diffusion Monte Carlo (DMC), Green s function Monte Carlo (GFMC), and path-integral Monte Carlo (PIMC). 

Green s Function Monte Carlo (GFMC) [3, 26, 62] relies on the standard resolvent operator of Schroedinger equation... 

Schmidt KE (1986) Variational and green s function Monte Carlo calculations of few body systems. Conference on models and methods in few body physics, Lisbon... 

Note, that the isospin symmetry of nuclear forces is not exact. The modem ab initio Green s function Monte Carlo calculations show that isospin-breaking terms are necessary in the strong interaction to reproduce the experimental data (Pieper et al. 2001), see later in O Sect. 2.3.7.2. The charge symmetry breaking in the strong interaction occurs because the up and down quarks have different masses and quark electromagnetic effects present themselves (Miller et al. 2006). 

Variational Monte Carlo (VMC), and a similar Green s function Monte Carlo (GFMC) method. 

The Green s function Monte Carlo calculation is very similar it starts with a trial function P obtained as a result of a variational optimization. 

Experimental (right-hand side for each nucleus) and Green s function Monte Carlo theoretical energies for the ground and narrow excited states of light nuclei (Pieper et al. 2001). The Argonne Vi8 two-nucleon and Urbana (UIX), Illinois (IL2) and Illinois (IL4) three-nucleon potentials were used in the calculations. The dashed lines indicate the breakup thresholds... 

B. H. Wells, in Electron Correlation in Atoms and Molecules, S. Wilson, Ed., Plenum Press, New York, 1987, pp. 311-350. Green s Function Monte Carlo Methods. 

D. W. Skinner, J. W. Moskowitz, M. A. Lee, P. A. Whitlock, and K. E. Schmidt, /. Chem. Phys., 83, 4668 (1985). The Solution of the Schrodinger Equation in Imaginary Time by Green s Function Monte Carlo. The Rigorous Sampling of the Attractive Coulomb Potential. 

M. Caffarel and D. M. Ceperley, /. Chem. Phys., 97, 8415 (1992). A Bayesian Analysis of Green s Function Monte Carlo Correlation Functions. 

DMC = diffusion Monte Carlo GFMC = Green s function Monte Carlo QMC = quantum Monte Carlo VMC = variational Monte Carlo. 

We would like to emphasize here some additional methodological developments and their results. The first is the variational treatment of fully antisymmetrized trial functions (5). The second is the Green s function Monte Carlo algorithm ( , ) which has, in effect, made possible the numerical integration of the Schrodinger equation. 

The fermion and Green s Function Monte Carlo are important in themselves and interesting as hints to the richness of methodology that can be brought to bear on the computation of quantum systems. In the short term we expect to broaden the specific trial functions used in fermion Monte Carlo to permit the more accurate study of He-3 and the treatment of more realistic models of nuclear and neutron matter. We expect also to try a variant in Quantum Chemistry problems. 

The most immediate research we plan with the Green s Function Monte Carlo is the exploration of the equation of state of He-4 liquids and crystals with two body potentials which fit more data than the... 

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