Variational quantum Monte Carlo calculation of electronic and structural properties of crystals Page: 6 of 14
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nonlocal energy involves the many-electron wavefunction on a sphere
about each atom. For an ion core at the origin, the contribution from
the angular momentum potential Vj(P) to the energy of the ith electron
for a given configuration R - t^i*^*****^
El(ri'•••,ri ™ r,...,rN)
* r N )
where fl* is the angular coordinate of r* with r pointed along the
polar axis. In principle, the expression for the nonlocal energy due
to one atom in Eq. (11) should be summed over all the atoms in the
crystal to give the total nonlocal energy for the ith electron. In
practice, one needs only to sum the potentials of at most two neigh-
boring atoms because the Vj(r)’s are, in general, very short range (- 2
a.u.). However, even with this simplification, it is impractical to
evaluate the nonlocal terms using standard fixed-grid methods. We find
that the integral in Eq. (11) can be evaluated accurately in a statisti-
cal fashion using a special point scheme for . The scheme involves
choosing a set of values for at random but at fixed relative posi-
tions and uses the sumnation over the values of i|»(R) at these special
points (with appropriate weighting factors) to obtain an unbiased esti-
mator of the integral. The procedure for generating special points for
different angular momenta is straightforward and is discussed in Ref.
16. With this scheme, the computational effort involved in the nonlo-
cal energy calculation is quite manageable and is comparable to that
for the kinetic energy.
3. Application to Solids and Atoms
We present in this section several applications of the present
approach to atoms and solids. Results on binding energies and struc-
tural properties as well as those on the single-particle properties are
3.1. Binding Energies and Structural Properties
Atoms. The total energy, ionization potential, and electron
affinity of atoms have been determined. These results were obtained by
carrying out calculations for the ground-state energy of the neutral,
positively, and negatively charged atoms. In each case, we used the
cups condition to fix the parameter a in the expression (Eq. (4)) for
the two-body term u(r-jj) in the Jastrow factor and searched the b, a
parameter space to determine the optimal u and X functions to minimize
the total energy. Note that since the atoms are spin-polarized (ne-
glecting spin-orbit interactions), Eq. (5) gives a different X function
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Louie, S.G. Variational quantum Monte Carlo calculation of electronic and structural properties of crystals, article, September 1, 1989; [Berkeley,] California. (https://digital.library.unt.edu/ark:/67531/metadc1093861/m1/6/: accessed April 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.