Variational Monte Carlo

Rick SW, Lynch DL, Doll JD (1991) A variational Monte Carlo study of argon, neon, and helium clusters. J Chem Phys 95 3506-3520. 

The strategy in Variational Monte Carlo (VMC) is therefore to pick a proper form for a trial wave function based on physical insight for the particular system under study. In general, a number of parameters (oi,..., a ) will appear in the wave function to be treated as variational parameters. For any given set of a the Metropolis algorithm is used to sample the distribution... 

The variational Monte Carlo method is reviewed here. It is in essence a... 

Variational Monte Carlo (or VMC, as it is now commonly called) is a method that allows one to calculate quantum expectation values given a trial wavefunction [1,2]. The actual Monte Carlo methodology used for this is almost identical to the usual classical Monte Carlo methods, particularly those of statistical mechanics. Nevertheless, quantum behavior can be studied with this technique. The key idea, as in classical statistical mechanics, is the ability to write the desired property <0> of a system as an average over an ensemble... 

The simplest method is the variational Monte Carlo method, discussed in the next section. Here an approximate expectation value is computed by employing an approximate eigenvector of G. Typically, this is an optimized trial state, say Mr>, in which case variational Monte Carlo yields X f° which is simply the expectation value of X in the trial state. Clearly, variational Monte Carlo estimates of X0 have both systematic and statistical errors. 

The required matrix elements can be computed using the variational Monte Carlo method discussed in the previous section. Furthermore, the power method can be used to reduce the variational bias. Formally, one simply defines new basis states... 

If we stopped here, we would have an exact and efficient variational Monte Carlo algorithm to sample from . Now, we reintroduce the branching term to convert the steady-state distribution from to 0. This is accomplished by reweighting the walkers with the branching factor [see Eq. (6.14)]... 

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). 

In variational Monte Carlo (VMC), one samples, using the Metropolis rejection method, the square of an assumed trial wavefunction, j where I = r, are the coordinates of all the particles... 

There have been a few studies that do not seem to fit into any of the categories thus far identified. Moss [70] has made a number of high-precision nonadiabatic calculations on hJ both by variational methods that specifically include vibrational wavefunctions and by methods based on scattering theory. The same system has also been studied by Ackermann and Shertzer [71] using finite-element methods. Finally, we mention the very precise variational Monte Carlo studies of Ps2 and its generalization to four-charge systems of type M" m M m by Bressanini et al. [72]. 

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... 

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