Chapter 16
Improved Monte Carlo Approaches to Systems of
Fermions
AbstractThis chapter develops the equations and formalism that are necessary to study
many-particle systems of fermions. The crucial part of any variational or diffusion Monte
Carlo code for many particles is the Metropolis evaluation of the ratios between wave func-
tions and the computation of the local energy. In particular, we develop efficient ways of
computing the ratios between Slater determinants
16.1 Introduction
For fermions we need to pay particular attention to the way wetreat the Slater determinant.
The trial wave function, as discussed in chapter 14, consists of separate factors that incorpo-
rate different mathematical properties of the total wave function. There are three types that
will be of concern to us: The Slater determinant, the productstate, and the correlation factor.
The two first are direct functions of the spatial coordinatesof the particles, while the last one
typically depends on the relative distance between particles.
The trial wave function plays a central role in quantum variational Monte Carlo simula-
tions. Its importance lies in the fact that all the observables are computed with respect to the
probability distribution function defined from the trial wave function. Moreover, it is needed
in the Metropolis algorithm and in the evaluation of the quantum force term when impor-
tance sampling is applied. Computing a determinant of anN×Nmatrix by standard Gaussian
elimination is of the order ofO(N^3 )calculations. As there areN·dindependent coordinates
we need to evaluateNdSlater determinants for the gradient (quantum force) andN·dfor the
Laplacian (kinetic energy). Therefore, it is imperative tofind alternative ways of computating
quantities related to the trial wave function such that the computational perfomance can be
improved.
16.2 Splitting the Slater Determinant
Following for example Ref. [128], assume that we wish to compute the expectation value of
a spin-independent quantum mechanical operatorÔ(r)using the spin-dependent stateΨ(x),
wherex= (r,σ)represents the space-spin coordinate par. Then,
〈Ô〉=〈Ψ(x)|
Ô(r)|Ψ(x)〉
〈Ψ(x)|Ψ(x)〉.
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