Energy optimization, variational Monte

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. 

Minimization of the Monte Carlo energy estimate minimizes the sum of the true value and the error due to the finite sample. Although the variational principle provides a lower bound for the energy, there is no lower bound for the error of an energy estimate. Fixed sample energy minimization is therefore notoriously unstable [140, 146], Optimization algorithms based on Newton s, linear and perturbative methods have been proposed [43,48, 140,147-151]. 

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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