6.4 Finite size correction and extrapolation formulae

The basic idea behind finite size correction formulae is to write the energy for the infinite system as

where the subscript denotes the system size. A highly
accurate many-body calculation for the -particle system is then
performed to obtain an approximation for , and the correction
term in brackets is approximated using a much less expensive scheme
which can be applied to very large systems:

where the prime again indicates that a less expensive scheme is used and is an extrapolation function. Clearly, the optimal form of depends on the method used to calculate the correction term. The practical difference between correction and extrapolation is that correction requires a single calculation of using the accurate (and normally very costly) many-body technique, while extrapolation requires several such calculations for different values of and a subsequent fit to the chosen functional form . The extrapolation procedure is costly because it involves more calculations and is prone to inaccuracy because one has to perform a fit with only a few data points. In designing a correction/extrapolation procedure one therefore tries to make the extrapolation term as small as possible.

Candidate methods for evaluating the correction term include
Kohn-Sham DFT and HF theory. The most convenient methods are the
local density approximation (LDA) to DFT, or extensions such as the
Generalized Gradient Approximation. These methods are very widely
applied in periodic boundary conditions calculations and are
computationally inexpensive, while retaining a realistic description
of the system. Within an independent particle theory, such as
Kohn-Sham DFT, calculations for periodic systems are normally
performed by solving within the primitive unit cell.^{6.1} To obtain
the correct result for the infinite system it is necessary to
integrate over k-space, and the integral is normally approximated by
a sum over a finite set of k-points. A determinant formed from the
occupied orbitals at a single k-point in the first Brillouin zone
(BZ) of the primitive unit cell is a many-body wave function for a
simulation cell of the size of the primitive unit cell. Adding a
second k-point doubles the size of the determinant and is equivalent
to doubling the size of the simulation cell, etc.

An important subtlety arises when finite size corrections derived from
an independent particle theory such as LDA-DFT are used to correct
the results of a true many-body method such as QMC. Suppose the
many-body calculation is performed using the Ewald interaction, so
that the Coulomb energy is that of an infinite periodic array of
copies of the simulation cell. Given that the solid we are trying to
model is crystalline, and hence the charge density is truly periodic,
the Ewald interaction gives a good description of the classical
Coulomb or Hartree energy. However, because the electronic positions
are mirrored exactly in every copy of the simulation cell, the
*electronic correlations* are also forced to be periodic, and the
exchange-correlation (XC) energy corresponds to a system with a
*periodic* XC hole. This unphysical approximation is particularly
inaccurate when the simulation cell is small. In Kohn-Sham DFT
calculations the XC energy is evaluated using a standard functional
such as the LDA of Perdew and Zunger [88]. This was
obtained by fitting to the results of DMC calculations for
jellium, [13] which were extrapolated to the
infinite-system-size limit *prior* to performing the fit.

The consequences of this difference may be illustrated by considering many-body and LDA calculations for jellium, in which the charge density is uniform. The LDA gives the XC energy for the infinite system irrespective of the size of the simulation cell, but the XC energy obtained from a many-body simulation using the Hamiltonian of equation 6.1 with the Ewald interaction gives an XC energy which depends on the size and shape of the simulation cell. Consequently, evaluating the correction term in equation 6.4 within the LDA does not give a good approximation to the finite size correction in the many-body simulation and a significant extrapolation term remains.

The issue of finite size corrections to both the kinetic and interaction energies has been addressed by Ceperley and coworkers. [89,90,91,33] Their approach is to add separate extrapolation terms for the kinetic and interaction energies. In their work on hydrogen solids, Ceperley and Alder [90] performed DMC calculations for a number of different system sizes and fitted to the formula

where and are parameters, and is the kinetic
energy of the non-interacting electron gas. The term accounts
for the finite size effects arising from the interaction energy and
the difference of the parameter from unity accounts for the
difference between the kinetic energies of the interacting and
non-interacting systems. Normally because the interacting
kinetic energy is larger than the non-interacting kinetic energy.
Engel * et al.* [92] used the following formula for
inhomogeneous systems:

(6.6) |

which reduces to equation 6.5 for a homogeneous system.