Variational Monte Carlo optimization

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. 

Variation and selection turns out to be an enormously potent tool for improvement also in vitro. Why this is so, does not trivially follow from the nature of random searches. The efficiency of Monte-Carlo methods may work very poorly as we know from other optimization problems. The intrinsic regularities of genotype-phenotype mappings with high degrees of neutrality and very wide scatter of the points in sequence space, which lead to the same or very similar solutions, are the clues to evolutionary success. 

The naive application of the variational principle to the optimization problem is limited by the statistical uncertainty inherent in every Monte Carlo calculation. The magnitude of this statistical error has a great impact on the convergence of the optimization, and on the ability to find the optimal parameter set as well. 

An important step for getting high-quality trial function is the optimization process. One usually takes a set of configurations (Monte Carlo samples of electron positions) from previous runs and minimizes the variational energy or the fluctuations of the local energy [16] ... 

There are two ways of constructing the Fock matrix for solving the DSCF equations one is based on the variational optimization of the energy of Eq. 2.4, and the other, which was first used in Monte Carlo simulations where anal3ftic forces are not required, is written by assuming that each monomer is embedded in the fixed electrostatic field of the rest of the system. The two approaches are discussed next. 

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