Variational quantum Monte Carlo calculation of electronic and structural properties of crystals Page: 8 of 14
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for different spin components.
The computed values for the ionization energy and electron affin-
ity of atomic carbon and silicon are presented in Table I together with
the experimental values. The agreement between theory and experiment
is quite good (both C“ and Si“ are unbounded in the LDA). We find that
both the one-body and two-body terms in the Jastrow factor are import-
ant to obtain quantitative results in the present approach. Without
inclusion of the one-body term X, the presence of a nonzero u(r-jj)
significantly alters the charge density from that of the Slate** deter-
minant alone. (See Fig. 1.) Because u(rjj) is a two-body correlation
term, it has the effect of reducing the electron density in the high-
density regions and increasing it in the low-density regions. The
resulting electron density is, in fact, too diffuse as compared to
experiment. The inclusion of a X term as given in Eqs. (5) and (6)
relaxes the electron distribution to one very similar to that of the
LDA and lowers the total energy. For neutral carbon, the one-body term
further lowered the total energy by 1.8 eV.
We note that the form of the wavefunction in Eq. (2) neglects
three-body and higher order terms in the Jastrow factor. Since the
number cf three-body interactions is very different for C and Si in the
three different charge states, the results in Table I suggest that
three-body terms in the Jastrow factor appear to be not very
significant in this case. Furthermore, the variational QMC results not
only gives the relative energies for the various ionization states
correctly. It also gives the absolute energies quite accurately at
least for the case of the Si atom where our results may be compared to
a recent Green's function QMC calculation using a pseudo-Hamiltonian
formal ism.17 The Green's function QMC result for the total valence
electron energy is -103.57(3) eV, which is only - 0.1 eV lower than our
result of -103.42(5) eV for the neutral Si atom.
Sol ids. The approach has been applied to study carbon- and
silicon-based crystals. Simulation cells with periodic boundary condi-
tion containing up to N - 216 electrons (or 54 atoms) were used. We
find that, with this size simulation cell, the many-electron part of
the energy is well converged. Finite size scaling to the final N + ®
limit is primarily dominated by the one-electron terms which are de-
pendent on the k-point sampling in the Brillouin zone. (A fine grid in
k-space is equivalent to a large simulation cell in real space.)
The result for diamond is summarized in Fig. 2. The total energy
per carbon atom in the diamond crystal structure is calculated as a
function of the lattice constant and fitted with a Murnaghan equation
of state. We obtained a calculated equilibrium lattice constant of
3.54 ± 0.03 A and a bulk modulus of 420 ± 50 GPa in good agreement with
experimental values of 3.567 A and 443 GPa, respectively^. Similarly
accurate results for these structural parameters have been obtained for
the case of silicon.
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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/m1/8/: accessed April 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.