Trial wavefunctions Metropolis sampling

The key problem is how to create and sample the distribution (R) (from now on, for simplicity, we consider only real trial wavefunctions). This is readily done in a number of ways, possibly familiar from statistical mechanics. Probably the most common method is simple Metropolis sampling... 

Specifically, this involves generating a Markov chain of steps by box sampling R = R + qA, with A the box size, and q a 3M-dimensional vector of uniformly distributed random numbers q e [— 1, +1]. This is followed by the classic Metropolis accept/reject step, in which ( PX(R )/T,X(R))2 is compared to a uniformly distributed random number between zero and unity. The new coordinate R is accepted only if this ratio of trial wavefunctions squared exceeds the random number. Otherwise the new coordinate remains at R. This completes one step of the Markov chain (or random walk). Under very general conditions, such a Markov chain results in an asymptotic equilibrium distribution proportional to, FX(R). Once established, the properties of interest can be measured at each point R in the Markov chain (which we refer to as a configuration) using Eqs. (1.2) and... 

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... 

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