Variational quantum Monte Carlo calculation of electronic and structural properties of crystals Page: 4 of 14
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where A and F are spin-dependent variational parameters. The variation-
al QMC approach has been successfully applied to the uniform electron
gasl-5 using Eq. (2) for ¥ with X - 0 and a two-body term u of the above
form. The obtained results are shown to be highly accurate as compared
to those of the more exact Green's function QMC calculations^. The
form of u in Eq. (3) has the expected behavior: u is large and posi-
tive for r - 0 and decreases with increasing r. The general asymptotic
form of u is, in fact, constrained by physical considerations of the
Coulomb interaction. As discussed in previous work on the uniform
electron gas, at large r, u is dominated by the zero-point motions of
the plasmons leading to a 1/r dependence with coefficient given by
e^/Hup. There is also a "cusp" condition on u(r), owing to the singu-
larity of the Coulomb interaction as r + 0. These two conditions give
some guidance in the search for the values of A and F. In the calcula-
tions, we find that the optimal values of A and F are, indeed, very
closed to the values given by the physical considerations for the
crystals examined. For atoms, in addition to the form of u(r) for the
solid [Eq. (3)], we also have used a form of
“frl --TrfbFT <4)
and obtained identical energies within statistical noises.
The one-particle term X(f) in the Jastrow factor serves to allow a
variational relaxation of the electron density in the presence of the
two-body u(r-jj) term which tends to make the electron density overly
diffuse. We rind that, although the one-body term is irrelevant in
homogeneous systems such as liquids or the uniform electron gas, it is
quite important for atoms and solids. There are several possible im-
plementations of X 16. For simplicity of calculation, we have either
where p(r) is the electron density and o is a variational parameter, or
in the case where the LDA electron density might be significantly
different from the X-0 QMC density, X(r) is iteratively obtained by
X(r) - I X.(r) (6)
where X^ is given by Eq. (5) and X..+1(r) - a/2 ln[px u_g(r)/pXi(r)].
For the systems considered, we find that the optimal value for a is
very close to 1. This is not unexpected since the LDA charge density
is generally in excellent agreement with experiment^.
2.3. The Hamiltonian and Total Energy
For a given many-electron wavefunction <|»(R), we obtain the expec-
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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/m1/4/: accessed March 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.