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40
A possibility to treat exchange exactly in DFT is offered by the OEP method discussed
in Sec. 5.3.
41
This is due to the integral operator in the HF equations.
42
The GW approximation [68, 69, 70], mentioned in footnote 39, is one such approxi-
mation for £, but in actual implementations of it one usually takes DFT-KS results as an
input.
38
4.3 Basis functions
In practice, numerical solution of the KS differential equation (71) typically
proceeds by expanding the KS orbitals in a suitable set of basis functions and
solving the resulting secular equation for the coefficients in this expansion
and/or for the eigenvalues for which it has a solution. The construction
of suitable basis functions is a major enterprise within electronic-structure
theory (with relevance far beyond DFT), and the following lines do little
more than explaining some acronyms often used in this field.
In physics much is known about the construction of basis functions for
solids due to decades of experience with band-structure calculations. This in-
cludes many calculations that predate the widespread use of DFT in physics.
There is a fundamental dichotomy between methods that work with fixed ba-
sis functions that do not depend on energy, and methods that employ energy-
dependent basis functions. Fixed basis functions are used e.g., in plane-wave
expansions, tight-binding or LCAO (linear combination of atomic orbitals)
approximations, or the OPW (orthogonalized plane wave) method. Exam-
ples for methods using energy-dependent functions are the APW (augmented
plane wave) or KKR (Korringa-Kohn-Rostoker) approaches. This distinction
became less clear-cut with the introduction of linear methods [75], in which
energy-dependent basis functions are linearized (Taylor expanded) around
some fixed reference energy. The most widely used methods for solving the
Kohn-Sham equation in solid-state physics, LMTO (linear muffin tin or-
bitals) and LAPW (linear augmented plane waves), are of this latter type
[76]. Development of better basis functions is an ongoing enterprise [77, 78].
The situation is quite similar in chemistry. Due to decades of experience
with Hartree-Fock and CI calculations much is known about the construction
of basis functions that are suitable for molecules. Almost all of this continues
to hold in DFT a fact that has greatly contributed to the recent popularity
of DFT in chemistry. Chemical basis functions are classified with respect
to their behaviour as a function of the radial coordinate into Slater type
orbitals (STOs), which decay exponentially far from the origin, and Gaussian
type orbitals (GTOs), which have a gaussian behaviour. STOs more closely
resemble the true behaviour of atomic wave functions [in particular the cusp
condition of Eq. (19)], but GTOs are easier to handle numerically because
the product of two GTOs located at different atoms is another GTO located
in between, whereas the product of two STOs is not an STO. The so-called
contracted basis functions , in which STO basis functions are reexpanded in
39
a small number of GTOs, represent a compromise between the accuracy of
STOs and the convenience of GTOs. The most common methods for solving
the Kohn-Sham equations in quantum chemistry are of this type [4, 49].
Very accurate basis functions for chemical purposes have been constructed
by Dunning [79] and, more recently, by da Silva and collaborators [80, 81].
More details on the development of suitable basis functions can be found,
e.g., in these references and Ref. [49].
A very popular approach to larger systems in DFT, in particular solids,
is based on the concept of a pseudopotential (PP). The idea behind the PP
is that chemical binding in molecules and solids is dominated by the outer
(valence) electrons of each atom. The inner (core) electrons retain, to a good
approximation, an atomic-like configuration, and their orbitals do not change
much if the atom is put in a different environment. Hence, it is possible to
approximately account for the core electrons in a solid or a large molecule
by means of an atomic calculation, leaving only the valence density to be
determined self-consistently for the system of interest.
In the original Kohn-Sham equation the effective potential vs[n] = vext +
vH[n] + vxc[n] is determined by the full electronic density n(r), and the self-
consistent solutions are single-particle orbitals reproducing this density. In
the PP approach the Hartree and xc terms in vs[n] are evaluated only for
the valence density nv, and the core electrons are accounted for by replac-
P P P
ing the external potential vext by a pseudopotential vext . Hence vs P [nv] =
P P P P
vext + vH[nv] + vxc[nv].43 The PP vext is determined in two steps. First, one
P
determines, in an auxiliary atomic calculation, an effective PP, vs P , such
that for a suitably chosen atomic reference configuration the single-particle
P
orbitals resulting from vs P agree outside a cut-off radius rc separating the
core from the valence region with the valence orbitals obtained from the
all-electron KS equation for the same atom. As a consequence, the valence
densities nat obtained from the atomic KS and the atomic PP equation are
v
the same. Next, one subtracts the atomic valence contributions vH[nat] and
v
P P P
vxc[nat] from vs P [nat] to obtain the external PP vext ,44 which is then used
v v
43
Note that the effective potential vs is a way to deal with the electron-electron inter-
action. The pseudopotential is a way to deal with the density of the core electrons. Both
potentials can be profitably used together, but are conceptually different.
44
This external PP is also called the unscreened PP, and the subtraction of vH[nat] and
v
P
vxc[nat] from vs P [nat] is called the unscreening of the atomic PP . It can only be done
v v
exactly for the Hartree term, because the contributions of valence and core densities are
not additive in the xc potential (which is a nonlinear functional of the total density).
40
in the molecular or solid-state calculation, together with vH[nv] and vxc[nv]
taken at the proper valence densities for these systems.
P
The way vs P is generated from the atomic calculation is not unique.
Common pseudopotentials are generated following the prescription of, e.g.,
Bachelet, Hamann and Schlüter [82], Kleinman and Bylander [83], Vanderbilt
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