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Tests on Jellium

Wavefunctions were optimised for the HEG at a range of densities from tex2html_wrap_inline7459 to tex2html_wrap_inline7461 . Excellent results were obtained at all densities, but for brevity only the results for tex2html_wrap_inline7463 =1 are presented here. A wavefunction of Slater-Jastrow type (cf. Eq.(gif)) was used, where the determinants, tex2html_wrap_inline7465 , were constructed from the lowest energy plane waves at zero wavevector within the simulation cell Brillouin zone. The one-body tex2html_wrap_inline6237 function was set to zero and the u function of Eqs.(gif-gif) was used. Separate u functions for parallel and antiparallel spins were used for fcc simulation cells containing N=30, 54, 178 and 338 electrons. In each case the numbers of up- and down-spin electrons were equal. Typically 10,000 electron configurations were sampled from a VMC run of sufficient length to ensure that the chosen configurations are statistically independent as described in section gif.

The variance minimisation procedure is stable for small N, but gradually becomes unstable as N increases. The technique described in section gif, was used to control this instability by fixing the reweighting factors to unity and regenerating configurations several times. This proved to be completely successful for all the system sizes studied. All calculations used 9 Chebyshev polynomials to represent f(r), which tests show to give essentially complete convergence for the systems studied. The minimisation problem then has 20 parameters.

Table gif shows the energy and standard deviation, tex2html_wrap_inline7481 , of the energy as a function of system size[2], comparing our u function Eqs.(gif-gif) with that of Eq.(gif), which includes a sum over simulation cells, and with DMC results. For the DMC calculations we used a time step of 0.01 au and an average population of 640 configurations. After equilibration the averages were collected over 5000 moves of all the electrons. The results obtained using our new u function are of similar quality to those obtained with the u function of Eqs.(gif-gif), but the new u function is much faster to evaluate. In figure gif we show the optimised spin-parallel u function for N=338 , together with the u function of Eqs.(gif-gif) which is plotted in the [100] and [110] directions (for all other directions the u function lies between the values in these directions). In figure gif the derivatives of the functions are shown. These figures show that the two functions are similar, but the optimised u function exhibits a slightly smaller derivative at intermediate distances. The optimised spin-antiparallel u function shows similar behaviour.

N tex2html_wrap_inline6909 tex2html_wrap_inline7481 tex2html_wrap_inline6909 tex2html_wrap_inline7481 tex2html_wrap_inline7513
Eqs.(gif-gif) Eqs.(gif-gif) Eqs.(gif-gif) Eqs.(gif-gif)
30 0.4657 0.2 0.46979 0.22
54 0.6085 0.24 0.6110 0.22 0.6069
178 0.6161 0.17 0.61679 0.17 0.6141
338 0.5772 0.14 0.57707 0.17
Table: Energies, E, and standard deviations of the energy, tex2html_wrap_inline7481 , for the HEG at a density of tex2html_wrap_inline7521 as a function of the number of electrons in the simulation cell, N. All entries are in Hartree atomic units per electron. The VMC energies are calculated with the Yukawa form of the Jastrow factor (Eqs.(gif-gif)) and our optimised spherically symmetric form (Eqs.(gif-gif)). The DMC energies do not depend on which of the two Jastrow factors is used.

Figure: Comparison of spin-parallel u functions for the HEG at tex2html_wrap_inline7521 . The optimised function (black line) is shown along with the Ewald summed Yukawa form along the [100] direction (red line) and the [110] direction (blue line). Fig. gifa shows the u functions themselves while Fig. gifb shows the first derivatives.

The reduction in computing cost from using the new u function is very significant. It is particularly effective when combined with our recently developed technique for evaluating the expectation value of Coulomb interactions in homogeneous systems [3], (see chapter gif). This combination of techniques entirely eliminates the need for time-consuming sums over simulation cells, and the resulting algorithm is extremely fast, with the most costly remaining operation being the calculation of determinants which are evaluated for each electron move using the standard Sherman-Morrison[18] formula to update the matrix of cofactors.

next up previous contents
Next: Applying the New u Up: A New u function Previous: Form of the New

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996