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Subsections
Density functional theory (DFT) is a powerful, formally
exact theory (see
Refs. [8,9,3] and
references within). It is distinct from quantum chemical methods in
that it is a noninteracting theory and does not yield a correlated body
wavefunction. In the KohnSham[10] DFT, the theory is a
oneelectron theory and shares many similarities with HartreeFock.
DFT has come to prominence over the last decade as a method
potentially capable of very accurate results at low cost. In
practice, approximations are required to implement the theory, and a
significantly variable accuracy results. Calibration studies are
therefore required to establish the likely accuracy in a given class
of systems.
The HohenbergKohn theorem[11] states that if
interacting electrons move in an external potential
, the groundstate electron density
minimises
the functional

(2.17) 
where is a universal functional of and the minimum value of
the functional is the exact groundstate electronic energy.
The proof of equation 2.17 is straightforward. It is a
proof only of existence; additional theory is required before a
method can be implemented.
Levy[12] gave a particularly simple proof of the
HohenbergKohn theorem which is as follows:
A functional is defined as

(2.18) 
where the expectation value is found by searching over all
wavefunctions, , giving the density and
selecting the wavefunction which minimises the expectation value of
.
is defined by

(2.19) 
so that

(2.20) 
Considering an electron ground state wavefunction which
yields a density and minimises
, then from the definition of the functional

(2.21) 
The Hamiltonian is given by
, and so
must obey the variational principle,

(2.22) 
This completes the first part of the proof, which places a lower bound
on .
From the definition of equation 2.19 we obtain

(2.23) 
since is a trial wavefunction
yielding . Combining
with the above equation gives

(2.24) 
which in combination with equation 2.22 yields the key result

(2.25) 
completing the proof.
Kohn and Sham[10] derived
a coupled set of differential equations enabling the ground state
density to be found.
Kohn and Sham separated into three distinct parts,
so that the functional becomes

(2.26) 
where
is defined as the kinetic energy
of a noninteracting electron gas with density ,

(2.27) 
and not the kinetic energy of the real
system. Equation 2.26 also defines the exchangecorrelation
energy functional . Introducing a normalisation constraint
on the electron density,
, we obtain
Equation 2.28 may now be rewritten in terms of an effective
potential,
,

(2.29) 
where

(2.30) 
and

(2.31) 
Crucially, noninteracting electrons moving in an
external potential
would result in the same
equation 2.29. To find the ground state energy, , and
the ground state density, , the one electron Schrödinger
equation

(2.32) 
should be solved selfconsistently with

(2.33) 
and equations 2.29 and 2.30. A selfconsistent
solution is required due to the dependence of
on
.
The above equations provide a theoretically exact method for finding
the ground state energy of an interacting system provided the form of
is known. Unfortunately, the form of is in general
unknown and its exact value has been calculated for only a
few very simple systems. In electronic structure calculations
is most commonly approximated within the local density approximation
or generalisedgradient approximation.
In the local density approximation (LDA), the value of
is approximated by the exchangecorrelation
energy of an electron in an homogeneous electron gas of the same
density , i.e.

(2.34) 
The most accurate data for
is from Quantum
Monte Carlo calculations.[13]^{2.3} The LDA is often
surprisingly accurate and for systems with slowly varying charge
densities generally gives very good results. The failings of the LDA
are now well established: it has a tendency to favour more homogeneous
systems and overbinds molecules and solids. In weakly bonded systems
these errors are exaggerated and bond lengths are too short.
In good systems where the LDA works well, often those mostly
consisting of bonds, geometries are good and bond lengths and angles
are accurate to within a few percent. Quantities such as the
dielectric and piezoelectric constant are approximately 10% too
large.
The principle advantage of LDADFT over methods such as HartreeFock
is that where the LDA works well (correlation effects are well
accounted for) many experimentally relevant physical
properties can be determined to a useful level of
accuracy. Difficulties arise where it is not clear whether the LDA is
applicable. For example, although the LDA performs well in bulk
groupIV semiconductors it is not immediately clear how well it
performs at surfaces of these materials.
Despite the remarkable success of the LDA, its limitations mean that
care must be taken in its application. For systems where the density
varies slowly, the LDA tends to perform well, and chemical trends are
well reproduced. In strongly correlated systems where an independent
particle picture breaks down, the LDA is very inaccurate. The
transition metal oxides XO (X=Fe,Mn,Ni) are all Mott insulators, but
the LDA predicts that they are either semiconductors or metals. The
LDA has been applied to high superconductors, but finds
several to be metallic, when in reality they are insulating at
0K. [3]
The LDA finds the wrong ground state for in many simpler cases. For
example, the LDA finds the wrong ground state for the titanium
atom. The LDA does not account for van der Waals bonding, and gives a
very poor description of hydrogen bonding. These phenomena are essential
for most of biochemistry: the structure of DNA of depends critically
on hydrogen bonding, as do the changes in the structure of most
molecules on solvation.
The success of the LDA has been shown by QMC calculations to result
from a realspace cancellation of errors in the LDA exchange and
correlation energies. [14,15] This is
illustrated in figure 2.1, where the exchange and
correlation energy densities of silicon are compared with an accurate
QMC calculation. The cancellation represents a difficulty when
improvements to the LDA are attempted, as an improvement in only the
exchange or correlation contributions may give worse results.
Figure 2.1:
Contour plots of the LDA exchange and correlation energy
densities compared with VMC calculations for Si in the diamond
structure in the (110) plane: (a)
; (b)
. The chain of atoms and bonds are
represented schematically. The contours are in atomic units. Note
that in the bonding regions between silicon atoms, the error in the
exchange energy density tends to cancel with the error in the
correlation energy density. Figure courtesy
R. Q. Hood. [15]

An obvious approach to improving the LDA is to include gradient
corrections, by making a functional of the density and its gradient:

(2.35) 
where is a correction chosen to satisfy one or several known
limits for .
Clearly, there is no unique recipe for , and several dozen
functionals have been proposed in the literature. They do not always
represent a systematic improvement over the LDA and results must be
carefully compared against experiment.^{2.4}The development of improved functionals is currently a very active
area of research and although incremental improvements are likely, it
is far from clear whether the research will be successful in providing
the substantial increase in accuracy desired.
Next: 2.6 Summary
Up: 2. Electronic structure methods
Previous: 2.4 Post HartreeFock techniques
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© Paul Kent