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Subsections
4.5 Pseudopotentials
The inclusion of core
electrons poses special problems in QMC calculations distinct from
those encountered in other electronic structure methods. The differing
timescale (or equivalently distance scale) on which the electrons move
compared with valence electrons requires the use of special
modified sampling schemes or the use of very small timesteps which reduce
the efficiency of the simulation. Large ionic and kinetic
energies in the core regions are a further problem. Although
accurate wavefunctions may be constructed for low
atoms,^{4.7} for high atoms accurate
wavefunctions have yet to be designed and obtained.
These two difficulties conspire to focus computational effort on the
core electrons, forcing attention away from the chemically interacting
and physically interesting valence and bonding regions. Fortunately
for many properties of interest, and for many materials, the core
remains almost independent of its environment and may be substituted
by a pseudopotential with negligible loss of accuracy.^{4.8} This
replacement serves to reduce the effective of the atoms that must
be dealt with. The dependence in the scaling of QMC calculations
is then determined by the number of valence electrons that must be
included in order to obtain satisfactory results.
Pseudopotentials are used routinely in DFT and quantum chemical
applications and consequently there exists a large literature and
continued research effort on the
subject. [54,55,56,57,58,59,60,61]
Improvements to the pseudopotential approximation are not an aim of
this work: accurate potentials exist for carbon and silicon, the main
atomic species used. The pseudopotentials user were generated within
LDADFT. Evaluation of the pseudopotentials requires techniques
particular to QMC, and these are described in the remainder of the
section.
The pseudopotentials modify the manybody Hamiltonian, replacing the
electronion coulomb terms with

(4.34) 
where signifies the electronion dependent terms of ion
, and
and
are respectively a
longranged local potential and a shortranged nonlocal
potential. The two potentials are chosen within a pseudopotential
construction scheme to accurately model the valence electrons. At
longrange, typically of order 23 a.u., the local potential returns
to the Coulomb tail.
The nonlocal potentials are written in terms of several shortranged
angular momentum dependent potentials. The operator
is
given by these potentials multiplied by the appropriate projection
operators, written as integrals over the angular interval :

(4.35) 
The local component of the pseudopotential may be directly evaluated
during a Monte Carlo simulation. In supercell simulations, Ewald's
method (see section 4.6.1) is required to deal with the
longrange of the potential, but this this not a significant
complication. In VMC, the nonlocal potentials must be projected onto
the trial function which requires statistical evaluation of the
projection operators.
4.5.1.1 Evaluation of the nonlocal energy
Evaluation of the nonlocal projection operators, and hence
the nonlocal energy cannot be performed analytically. For general
correlated wavefunctions, the projection operators depend on all
interparticle distances and are therefore not amenable to analytic or
numerical integration. Fahy [38,26]
developed an exact scheme for evaluating the nonlocal energy in VMC and
first applied the technique in simulations of bulk carbon and
silicon in the diamond structure. This scheme is now used almost
exclusively, approximate evaluation methods having been discarded due
to insufficient accuracy.
In the Fahy scheme, evaluation of
is
performed stochastically during a VMC simulation. The projection is
written in terms of ratios of the trial function,

(4.36) 
where the sum runs over all electrons at distances
from the ion. The orientation of the coordinate system (which is
arbitrary) is chosen for each electron so that the vector lies along the direction, thus eliminating the
dependence of the integrals. [26]
The angular integral over
is performed
stochastically during the simulation. Optimised integration grids have
been developed to exactly integrate functions of up to a certain
angular momentum. [62] The variance of the
estimator for the nonlocal energy is chosen to optimise the balance of
work spent evaluating the nonlocal pseudopotential with the work
performing other parts of the simulation. The grids are very
efficient, and in calculations on bulk silicon and carbon, taking
multiple samples of a grid was always found to be less efficient than
using a single larger grid. This result is expected to be general; the
integrals are effectively of low dimensionality and gridbased
integration methods should converge more rapidly than Monte Carlo
methods. In practice, 612 points give sufficient accuracy for first
and second row elements, [26,63] where the
character of the wavefunctions is dominated by and
angularmomenta. 6 point grids are sufficient to exactly integrate
momenta, and 12 points grids, momenta.
The nonlocality of accurate pseudopotentials is a significant problem
in DMC calculations. An explicit form for the manybody wavefunction
is not available in DMC and it has been shown
(e.g. Ref. [64]) that the matrix elements of the
nonlocal operator for imaginary time diffusion are negative. This
creates a sign problem similar to the sign problem of fermion
DMC. This problem has been avoided by replacing the nonlocal
potential operator by an approximate local potential determined
by evaluating the full nonlocal operator on the trial wavefunction,
as in VMC. [65,66] This approximation
has been shown to converge quadratically to the exact energy as the
trial wavefunction improves. [63] In practice this
approximation is very good and is not overly sensitive to details of
the trial function. The ``locality approximation'' results in a
nonvariational DMC energy.
Core polarisation potentials (CPPs) are a refinement of
pseudopotential theory particularly designed for use in manybody
calculations. The pseudopotentials most commonly used in manybody
calculations are derived from mean field calculations. This represents
an approximation at some level, and to consistently achieve an
accuracy better than eV on atomic energy levels, some
manybody effects must be incorporated into the pseudopotential.
Direct methods of generating pseudopotentials within QMC have met with
little success due to the difficulties in performing accurate
allelectron calculations of sufficient statistical
accuracy. [67]. Methods based on configuration
interaction calculations and quasiparticle calculations have met with
most
success. [60,68,69,70]
The accuracy of pseudopotentials may be increased by taking into
account the polarisability of the core, which is one of the most
important effects excluded by a conventional rigid core approach. It
is has been shown that this incorporates the leading terms of ``core
relaxation'', the change that a core undergoes in different chemical
environments (Ref. [71] and references within). The
formulation of CPPs considers the polarisation of the atomic core by
the valence electrons, within a pointdipole picture. To avoid
divergences in the potential, the effect fields experienced by valence
electrons are truncated close to the core. By parameterising the
truncation, the correct binding energies of valence electrons is
ensured. The scheme includes some of the corecore, corevalence and
valencevalence correlation effects of an allelectron, manybody
calculation. [71,68]
The manybody Hamiltonian is initially modified to include additional
CPP terms resulting from the polarisation induced electric
fields[68]
where is the core polarisability, and the sums run over
all ions, , and electrons. The terms are due to ionion,
ionionelectron, electronion and electronelectronion interactions
respectively. The electron dependent terms are further parameterised,
by means of a cutoff function, removing the divergences:

(4.38) 
where is a rescaled coordinate. The Hamiltonian terms become
where is the projection operator for angular momentum
, and the and
are parameters obtained
during construction of the CPP.[68]
These terms may be directly evaluated within VMC and DMC. The
nonlocal projection operator, , is evaluated at the same
time as nonlocal component of the full pseudopotential, avoiding
costly evaluations of the manybody wavefunction.
Sample data
comparing DMC results for the ionisation potentials of a Ti atom for
different pseudopotentials [61] is given in Table
4.1.
Table 4.1:
Ionisation potentials of Ti
for several pseudopotentials computed using DMC. Energies are in eV
and statistical error bars are given in brackets. Experimental data
from [72]. RelTM denotes a relativistic
pseudopotential constructed in the TroullierMartins
scheme, [58] HF and HF+CPP denote a bare HF
pseudopotential and an HF pseudopotential with corepolarisation
potential terms respectively. [68] The
pseudopotentials were generated in the configuration suggested by
G. B. Bachelet et al.[56] Data courtesy
Y. Lee [61]
IP 
Expt 
RelTM 
HF 
HF+CPP 



1st 
6.82 
6.862(15) 
6.693(13) 
6.767(16) 



2nd 
13.58 
12.347(8) 
12.218(8) 
13.639(9) 



3rd 
27.48 
27.666(4) 
28.288(4) 
28.413(5) 



4th 
43.24 
44.336(0) 
44.774(0) 
46.228(0) 




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© Paul Kent