Form of the New u function

The new u function is similar to one used earlier for the HEG by Ortiz and Ballone [68, 69]. In common with Ortiz and Ballone a spherically symmetric u function is chosen, which is short ranged so that it need not be summed over simulation cells. This u function folds in the long range behaviour of the Jastrow factor in an approximate manner, and therefore it depends on the size of the simulation cell as well as on the electron density of the system. For each electron pair the separation vector is reduced to its minimum length (by subtraction of supercell lattice vectors) giving the vector between electron i and the nearest periodic image of electron j. This reduction procedure is illustrated in figure .

  
Figure: Reduction of the vector to its minimum length. The figure contains a square simulation cell and just one of the periodic images in each direction. The blue vector shows the original vector. The red vector has been been reduced to its minimum length by subtraction of a vertical and a horizontal lattice vector.

The precise form of the new u is different from that used by Ortiz and Ballone. It has certain advantages which will be described below. We demand that u obeys the following conditions:

1. u(r) satisfies the cusp conditions as ;
2. u(r) is continuous and has a continuous first derivative for all r > 0;
3. u(r) is linear in the variable parameters.

The only condition that u(r) must satisfy for our QMC procedures to work is condition (ii) given above. If this condition is not obeyed then the kinetic energy estimator, , will have -functions at the discontinuities, which will be missed by the sampling procedure. To ensure continuity of the first derivative of u(r) for r>0 it is required that goes (almost exactly) to zero at the surface of the sphere of radius inscribed within the Wigner-Seitz cell of the simulation cell. For , u(r) and are set to zero. The cusp conditions are imposed on the first derivative of u at because this is a property of the exact wavefunction. In contrast to Ortiz and Ballone, continuity of the second derivative of u is not imposed. We write u(r) as

 

where is a fixed function and f contains the variable parameters. f is expanded as a linear sum of some basis functions, :

For the fixed part of u, the following form was chosen,

where F is chosen so that the cusp condition is obeyed and is chosen so that is effectively zero . Typically and A is fixed by the plasma frequency[25]. The function is chosen to give a good description of the correlation so that the variable part of u is small. For the variable part we choose

 

where B and the are variational coefficients, is the lth Chebyshev polynomial, and

 

so that the range is mapped into the orthogonality interval of the Chebyshev polynomials, [-1,1]. The use of Chebyshev polynomials rather than a simple polynomial expression improves the numerical stability of the fitting procedure. The function f is the most general polynomial expression containing powers up to which satisfies the following conditions:

1. ;
2. ;
3. .

Condition (i) ensures that u(r) obeys the cusp conditions, which are incorporated in . Addition of a constant to u(r) changes the normalisation of the wavefunction but not its functional form, and condition (ii) eliminates this unimportant degree of freedom. Condition (iii) ensures continuity of the first derivative of u at .

To start the optimisation process we perform a VMC run to produce the electron configuration data for the initial distribution as described in section . For each electron configuration, u(r) is summed over all distinct pairs of electron coordinates i and j in the simulation cell (with the separation vector reduced into the Wigner-Seitz simulation cell). For each configuration the following summation is performed

 

Instead of storing the individual electron coordinates in each configuration we store the , which is sufficient because the functional form for u is linear in the variable parameters. This reduces the storage and CPU time needed for the minimisation procedure, which requires no further summations over the electron coordinates when the values of the parameters, , are altered. The first and second derivatives of u, which enter the expression for the energy, are dealt with in a similar manner. These savings are very significant when dealing with a large number of electrons in the simulation cell, and for the HEG we have performed full minimisations with up to 338 electrons.