Variational quantum Monte Carlo calculation of electronic and structural properties of crystals Page: 5 of 14
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tation value of an operator F by evaluating the multi-dimensional
<1f\?\V> - f F(R)|i|»(R)|2 dR (7)
where R - 1s a P°int in configuration space specified by
the coordinates of all the electrons. For the total energy, the func-
tion F(R) is then taken to be [H\Jj](R)/ij>(R) where H is the Hamiltonian
operator. Since typically several hundred electrons in a box (corre-
sponding to tens of atoms) with periodic boundary conditions are re-
quired to simulate accurately the properties of solids, the only prac-
tical way of evaluating many-body integrals of the form in Eq. (7) is
by the Metropolis Monte Carlo algorithm^ for importance sampling with
the importance function given by |i|j(R)|2.
The many-electron Hamiltonian for the crystal
is consisted of the usual three terms: the kinetic energy of the va-
lence electrons, the external potential due to the ion cores, and the
Coulomb repulsion between the valence electrons. In the Metropolis
scheme, E(R) • [H\J/](R)/i|>(R) is evaluated along a random walk in config-
uration space so as to visit points R with probability density equal to
H»(R)|Z. The average of E(R) over this walk is then an unbiased esti-
mator of the total energy:
E - <i|>|H|\|j> - i 2 E(R-j) . (9)
'As in the electron gas case, the evaluation of the electron-electron
energy at each step of the walk may be carried out straightforwardly
using Ewald summation techniques provided some care is given to the
periodic boundary conditions imposed on the finite simulation region.
Similarly, although the single-particle orbitals in the Slater deter-
minant are no longer plane waves, the form of i|> in Eq. (2) allows the
kinetic eneray to be calculated using techniques developed for the
The evaluation of the external potential energy is more involved
because of the nonlocality of the pseudopotential. The local part is
straightforward since it is diagonal in the coordinate representation
of the electrons given by
WM- I C^-^on) <10>
where §-jnn are the positions of the ions in the crystal. The value of
the local potential at each configuration on the random walk is also
evaluated using Ewald summation techniques. The nonlocal part (second
term in Eq. (1)) is a more complicated form, and the evaluation of the
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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/m1/5/: accessed April 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.