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44848-AC6
Benchmark Quantum Monte Carlo Calculations for Carbon and Hydrocarbon Species
We have explored a new scheme for quantum Monte Carlo (QMC) calculations intended to improve the accuracies of electronic structures and energies obtained for large molecular systems. The method is based on an earlier procedure for the direct Monte Carlo determination of corrections to trial wavefunctions and their energies in combination with correlated sampling for accurate determinations of energy differences. The new scheme extends correlated sampling beyond the previously available calculations for variational QMC to fixed-node diffusion QMC to allow differences in energy to be calculated with a minimal statistical error for very large molecules.
Fixed-node diffusion QMC provides some of the most accurate predictions of molecular properties for both very large and very small systems. Although such calculations are computationally very expensive they scale with the third power of system size (typically as the number of atoms cubed) while most other accurate methods scale as the fifth or sixth power of system size. The QMC method is the method of choice for very large systems, those beyond the range of other accurate methods.
Our targets are the carbon systems C10, C20, and C60 and hydrocarbons in the range of C10H22 to C20H42. Our investigations thus far have been carried out mainly with much smaller systems which require lower computation efforts and may have known values for their energies. Typical of these are H2, Ne, and H2O, as well as special cases without electron correlation, but we have also obtained results for C10. For each of these systems we have succeeded in calculations of energy differences for similar systems with very high accuracies.
For H2 the test case was H2 with an internuclear distance of 1.4000 bohr vs. H2 with an internuclear distance of 1.4011 bohr. The total electronic energies determined were -1.174 470 644 hartrees and -1.174 470 901 hartrees respectively with statistical uncertainties of about +/- 0.000 005 000 hartrees. The difference in energies determined from correlated sampling at the fixed-node diffusion level was 0.000 000 257 hartrees with an uncertainty of about +/- 0.000 000 040 hartrees. This represents a gain in accuracy of more than a factor of 100 relative to that from uncorrelated or independent calculations. The essentially exact value of the energy difference, from other sources, is 0.000 000 210 hartrees.
The next larger test case investigated was that of the neon atom. In this case we compared energies of two independent neon atoms placed 0.02 bohr apart for purposes of the calculations. The known difference in energies is zero, but the calculation makes no use of that information. Correlated sampling at the fixed-node level gave an energy difference of 0.000 115 +/- 0.000 077 hartrees (1 standard deviation) out of the total energy of -128.937 hartrees per atom. The uncertainty in the energy for each of the atoms was a factor of ten higher.
In the case of C10 we again carried out correlated sampling at the fixed-node diffusion level to determine energy differences for C10 molecules in circular geometries of different radii. For these we obtained energies of about -380.450 hartrees with uncertainties of about 0.013 hartrees and energy differences with uncertainties of about 0.0013 hartrees. Since this is unknown territory at these accuracies, exact results are not available for comparison. The minimum in energy was found for a radius of 2.08 bohr.
We have much more to do, but we have successfully demonstrated the feasibility of the new correlated sampling scheme. A number of possible improvements remain to be investigated and a few technical questions remain unanswered. We plan to continue to the still larger target systems in the coming year. We also hope to explore the use of this type of correlated sampling in optimizing node locations in trial wavefunctions.
Two doctoral students, Matthew C. Wilson and Patrick D. O'Connor, have participated in this work. Matthew Wilson finished his doctoral dissertation with calculations for C10 molecules and was awarded his Ph.D. degree in May 2007. Patrick O'Connor will complete his doctoral thesis work later this year. The research made possible by this grant provides an important extension to available QMC methods and it furthers the career of the principal investigator in this area.
