St r J dRGR R irR r5

where G(R -» K',St) = (R' | e~SrH | R) is the imaginary time Green's function. Substituting Eq. 5 into Eq. 4, and remembering that if 6t = 0 then + r) = ^(R',r), it is easy to show that the

Green's function is a solution of the Schrodinger equation with the initial condition G(R -> R',0) = 5(R' - R) [77]. The general solution of both Eq. 4 and Eq. 5 is given by

where Viand £jare the eigenfunctions and eigenvalues of the Schrodinger equation, and c» = (if), | i(R,0)). This equation shows that acting with the Green's function in imaginary time r over a starting wave function i(R, 0) projects out all the excited states, leaving only the ground state solution for r oo. If the exact analytical form of the Green's function were known, this task could be accomplished by means of the Monte Carlo method [78]. Unfortunately, the exact Green's function is known only for very simple model Hamiltonians. For systems whose ground state wave function has no nodes, the DMC theory relies completely on the ability to find a short time approximation (STA) to the imaginary time Green's function [72, 74, 76, 79]. Once this approximation is known, equation 5 can be iterated to produce an asymptotic wave function ^st/i(R) t oo)[51]. Using a STA of the exact Green's function introduces a systematic difference between ^sr^R, r -» oo) and the exact 4>o for any finite St, an error that is usually called "time step bias" [80]. Nevertheless, using small time steps, this error can be reduced so that the difference between the computed values and the exact ones are smaller than any chosen threshold [81].

However, it is customary to improve the efficiency and the accuracy of the DMC algorithms by means of the Importance Sampling (IS) procedure [55], Here, an approximate trial wave function $is introduced to guide the displacement of the configurations [82], or to modify the reaction rate by which configurations are deleted or created [78]. The IS procedure is usually carried out simulating a modified form of Equation 4 that reads [82]

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