Trial wavefunctions, Monte Carlo

Umrigar C J, Wilson K G and Wilkins J W 1988 Optimized trial wavefunctions for quantum Monte Carlo calculations Phys. Rev. Lett. 60 1719-22... 

An overview of quantum Monte Carlo electronic structure studies in the context of recent effective potential implementations is given. New results for three electron systems are presented. As long as care is taken in the selection of trial wavefunctions, and appropriate frozen core corrections are included, agreement with experiment is excellent (errors less than 0.1 eV). This approach offers promise as a means of avoiding the excessive configuration expansions that have plagued more conventional transition metal studies. 

Now let us look at the paper. Eqn. (1) gives the form of the transcorrelated wave function C0, where C = li >jfiri,rj) is a Jastrow factor, and is a determinant. This compact wave function includes the effects of electron correlation through the introduction of r in C. The form C0 is taken as the trial wave function in quantum Monte Carlo (QMC) molecular computations today. Indeed the explicit form for/(r r,j is most often used by the QMC community. The transcorrelated wave function was obtained by solving (C //C - W) = 0, which Boys called the transcorrelation wave equation. Because C //C is a non-Hermitian operator, it was important to devise independent assessments of the accuracy of the wavefunction C0. 

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... 

A typical VMC computation to estimate the energy or other expectation values for a given 4/x(R) might involve the calculation of the wavefunction value, gradient, and Laplacian at several millions points distributed in configuration space. Computationally this is the most expensive part. So a desirable feature of TVR), from the point of view of Monte Carlo, is its compactness. It would be highly impractical to use a trial wavefunction represented, for example, as a Cl expansion of thousands (or more) of Slater determinants. 

C. Path-Integral Monte Carlo Trial Wavefunctions Treatment of Atomic Cores in QMC... 

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... 

One of the main advantages of the Monte Carlo method of integration is that one can use any computable trial function, including those going beyond the traditional sum of one-body orbital products (i.e., linear combination of Slater determinants). Even the exponential ansatz of the coupled cluster (CC) method [27, 28], which includes an infinite number of terms, is not very efficient because its convergence in the basis set remains very slow. In this section we review recent progress in construction and optimization of the trial wavefunctions. 

In this short review we have pointed out only very few of the basic issues involving the simulation of chemical systems with Quantum Monte Carlo. What has been achieved in the last few years is remarkable very precise calculations of small molecules, the most accurate calculations of the electron gas, silicon and carbon clusters, solids, and simulations of hydrogen at temperatures when bonds are forming. New methods have been developed as well high-accuracy trial wavefunctions for atoms, molecules, and solids, treatment of atomic cores, and the generalization of path-integral Monte Carlo to treat many-electron systems at positive temperatures. 

To obtain the importance sampling version of diffusion quantum Monte Carlo, we first multiply the basic equation, Eq. [9] by a trial wavefunction /q and define a new function f - x A /o, which is the product of the true wavefunction and the trial wavefunction. After several pages of rearrangement, one may obtain the basic equation for DQMC with importance sampling, ... 

The problem of node locations—the sign problem in quanmm Monte Carlo —remains one of the major obstacles to obtaining exact solutions for systems of more than a few electrons. In analytic variational calculations and in VQMC, the locations of the nodal smfaces of a trial wavefunction may be and usually are optimized along with the rest of the wavefunction in the attempt to reach a minimum in the expectation value of the energy. In DQMC and GF-QMC, the node locations are not so easily varied. 

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