7.4 Tests of orbitals

7.4.1 LDA and natural orbitals compared

Fixed node DMC calculations were performed using LDA and natural orbitals to form the determinants, in the expectation that a determinant of natural orbitals would have lower nodal error than a determinant of LDA orbitals. Re-optimization of the Jastrow and $\chi$ functions to improve sampling efficiency in the DMC calculation was found to be unnecessary in the two cases. VMC calculations were also performed to test for the possibility that the variational energy had been improved.

The energies obtained were $-107.59$ eV (LDA), $-107.69(1)$ eV (VMC with LDA orbitals), $-107.71(1)$ eV (VMC with natural orbitals), $-108.10(1)$ eV (DMC with LDA orbitals), and $-108.09(1)$ eV (DMC with natural orbitals). The VMC wave function appears to show a very slight improvement with natural orbitals compared with LDA orbitals. However, to within statistical accuracy, the DMC energies obtained with LDA and natural orbitals are the same. This indicates that the nodal surfaces given by the LDA and natural orbitals are of the same quality.

7.4.2 LDA and HF orbitals compared

In light of the above results it is interesting to compare the quality of the nodal surfaces obtained with LDA and HF orbitals, which are both commonly used in the determinantal parts of trial wave functions for QMC calculations.

This was investigated by performing DMC calculations in silicon with an fcc simulation cell containing 16 atoms. The smaller simulation cell enabled a large number of independent configurations to be obtained rapidly. Wave functions expanded in a basis of atom-centered Gaussians were obtained from the HF and DFT code CRYSTAL95. [130] Special care was taken to ensure that the LDA and HF calculations were done in equivalent ways to try and eliminate any bias in the comparison. A basis set of four uncontracted $sp$ functions and one $d$ polarization function per pseudo-atom was optimized separately for each calculation. The quality of the basis set is high - to obtain the same energy within a plane wave calculation would require a basis set cutoff of 12.5 Ry. The same non-local LDA pseudopotential was used as in the earlier calculations. The Ewald interaction was used in the many-body Hamiltonian to avoid any charge-density dependence on the interaction. Approximately $6.7\times 10^5$ walker moves obtained DMC total energies and statistical accuracies of -107.488(3) eV per atom and -107.464(3) eV per atom for the LDA and HF guiding wave functions respectively.

The walker energies were approximately normally distributed. Using a conventional students t-test, [19] the 95% confidence interval on the difference in energies obtained was $0.002 - 0.046$ eV per atom, showing that for this system it is very likely that the DMC energy from a determinant of LDA orbitals is lower than that from a determinant of HF orbitals. Therefore, for this system, a determinant of LDA orbitals has a marginally better nodal structure than a determinant of HF orbitals.