Quantum Monte Carlo procedure

We hope to have convinced the reader that the VMC approach to obtaining quantum expectation values of interest in both chemical and physical problems is a very powerful one. We believe that it, in combination with fully quantum Monte Carlo procedures, will be the preferred choice in the near future, for many of the calculations performed these days by more traditional nonstochastic means. VMC is the first, and a necessary step, toward a complete quantum simulation of a system. It has the very desirable feature (often a rare one) that it can be learned, implemented, and tested in a short period of time. (It is now part of the folklore in the quantum Monte Carlo community, that the original code, written by J. B. Anderson and used for his first QMC paper, was only 77 lines of FORTRAN code.) We hope our readers will be inspired to write their own (toy or otherwise) VMC code possibly thereby contributing to and enlarging the growing Monte Carlo community. 

Most of the present discussion has been concerned with applications of REPs within the framework of otherwise essentially orbital-based calculations. On the other hand, a recent application 110) involved a quantum Monte Carlo (QMC) procedure. [A useful overview of Monte Carlo electronic structure work has been given by Ceperly and Alder 111). ] Currently, QMC offers little, if any, competition for conventional calculations in that the computer time required to reduce statistical errors to acceptable limits increases rapidly as a function of atomic number and is excessive for all but the smallest systems. Recent fluorine calculations required nearly 100 hours of supercomputer time 112). Although, on the surface, it would appear totally impractical, the appeal of this approach in the context of heavy-element work is its avoidance of extensive basis sets and enormous configuration expansions that plague present studies. 

Another technique which has gained prominence in recent years is the Quantum Monte Carlo (QMC) technique. This technique maps a d-dimensional quantum model onto a d - - 1 dimensional classical model via a Trotter decomposition of the partition function or the ground state projection operator [56, 57]. The quantum model is then studied by performing a Monte Carlo sampling procedure on the classical model in higher dimension. For fermions, the mapping of the interacting quantum model system to the classical system could... 

So far, we have discussed quantum Monte Carlo approaches to quantum phase transitions in boson and spin systems. In these systems, the statistical weight in the Monte Carlo procedure is generally positive definite, so there is no sign problem. Note that for spin systems, this is only true if there is no frustration. Frustrated spin systems in general do have a sign problem. 

In the real time domain, the sampling of quantum paths involves oscillating positive and negative weights which are problematical for any currently known Monte Carlo procedure. Thus we must await new developments in time-dependent path integral quantum Monte Carlo. 

Figure 8 Rate constants k T) for the reaction OH + H2 — H2O + H. Results are shown for diffusion quantum Monte Carlo calculations (DMC) and for the more approximate Morse quadratic-quartic (MQQ) procedure. (From Ref. 132.)...

Beyond the clusters, to microscopically model a reaction in solution, we need to include a very big number of solvent molecules in the system to represent the bulk. The problem stems from the fact that it is computationally impossible, with our current capabilities, to locate the transition state structure of the reaction on the complete quantum mechanical potential energy hypersurface, if all the degrees of freedom are explicitly included. Moreover, the effect of thermal statistical averaging should be incorporated. Then, classical mechanical computer simulation techniques (Monte Carlo or Molecular Dynamics) appear to be the most suitable procedures to attack the above problems. In short, and applied to the computer simulation of chemical reactions in solution, the Monte Carlo [18-21] technique is a numerical method in the frame of the classical Statistical Mechanics, which allows to generate a set of system configurations... 

The quantum/classical procedures recover the nuclear fluctuation properties of the surrounding medium via the Monte Carlo statistical approach or by using molecular dynamics simulations. In the following section we examine the problem of energy exchange between solute and solvent from a quantum dynamical viewpoint. 

With the development of powerful computers, these methods have been restricted in practice to the application in specific problems where no other calculations are available. For non-relativistic atomic systems, they have been replaced by quantum mechanical calculations like Monte Carlo or multiconfigurational Hartree-Fock ones. Nevertheless, Thomas-Fermi estimates can be easily evaluated by non-specialists in theoretical calculations and in some problems they provide a starting point for more sophisticated procedures. Moreover, they are interesting for theoretical purposes such as finding relationships among different average quantities [4]. 

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