Variance as calculated in “Zero Monte Carlo Error or Quantum Mechanics is Easier”

The Monte Carlo Method[1] 

An estimate for E_t for a lithium atom could be formed by randomly choosing N sets of values for r_i = (r₁,r₂,r₃)_i.   This would yield

(1)

In practice, however, y_t² varies over an enormous range of values so that the two sums would be completely dominated by the relatively few choices for r_i which correspond to electronic positions close to the nucleus at the origin. To avoid this, the nuclear-electronic correlation function, g_ne(r), can be used as a guide in selecting the values for r_i.

            To be specific, random values for |r₁|, |r₂|, and |r₃| are selected with probability r²g(r) where r²g(r) is approximately g_ne(0.l5a_B)/0.l5a_B for r less than 0.15, g_ne(r)/r for 0.l5a <r <a_B, the larger of g_ne(r) and 0.0l/r² for a_B <r <9.6a_B and zero elsewhere. Approximately refers to the fact that g(r) was held fixed at a reasonable approximation to g_ne(r) and furthermore that r²g(r) was rigorously taken to be the straight lines connecting 65 uniformly spaced values of r²g(r) in the interval 0 to 9.6a_B. This is importance sampling[2] which concentrates the random values in the region where y_t² is large and introduces a weight, w = (g(|r_1i|) g(|r_2i|) g(|r_3i|))^-1, for the ith choice of r_i = (r₁,r₂,r₃) which tends to make the random values w_i relatively constant. Then with H_t,i = (Hy_t(r_i))/ y_t(r_i) we have

(2)

In order to form an estimate of the random error resulting from the use of a finite N in E_N,t, we notice that the difference between it and E_t is due to the fact that the N points individually predict somewhat different values for E_t In particular, the difference between E and E_N,t can be written as

(3)

(4)

If we square this difference and average over a large number of sets of N points we have

(5)

The left-hand side of Eq. (5) is the standard deviation of E_N,t. The second term on the right-hand side averages to zero from the statistical independence of points j and k, while the first term can be approximated by a single sum over N points using E_N,t as the best available estimate for E This yields

(6)