Probability that a sample falls outside a centralized interval of size for , a Normal distribution, and for an example Stable distribution, (Ref. 5) that is symmetric with power law tails and mean 0 (the mean exists for this stable distribution). This form of stable distribution is typical of the distributions that occur for an invalid CLT in standard sampling QMC.
Evolution of total energy estimates with computational time. (a) shows the evolution of confidence intervals for efficient, standard, and optimum samplings. No confidence interval is available for SD sampling, and only the estimated value is shown. An arbitrary constant offset has been added for each sampling strategy to aid clarity. (b) shows the evolution of standard errors for efficient, standard, and optimum samplings. For SD sampling the quantity shown arises from evaluating the standard error estimate when it is incorrect to do so, and is unrelated to confidence intervals.
Evolution of total energy estimates with optimization for all-electron C and using efficient sampling. Four optimization processes are shown using 384, 1480, 14 800, and 80 000 samples in order of decreasing . All total energy estimates are generated using 2 200 000 samples.
Evolution of total energy estimates with optimization for O, for both standard and efficient sampling. (a) shows the evolution of estimates using standard sampling and (b) shows the evolution of estimates using efficient sampling, with the same total computational cost of each. The gray regions are the confidence interval for the final total energy estimate constructed using an average of converged parameters and equal computational cost.
Total energy estimates for fixed computational time and optimum, standard, efficient, and SD samplings. No standard error is available for SD sampling and the number of significant figures is unrelated to random error.
Summary of parameters for the isolated atoms and diatomic molecules. Experimental bond lengths used are from Ref. 31 for the homogeneous diatomic molecules and from Ref. 32 for the rest.
Total electronic energies for isolated atoms and diatomic molecules. The table presents VMC estimates obtained as described in this paper . Approximate exact total energies for isolated atoms are from Ref. 33 for Li and Ref. 34 otherwise. Approximate exact total energies are from Ref. 31 for , , , and , and from Ref. 35 otherwise.
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