Chemistry Reference
In-Depth Information
This expansion is substituted into the KS equations which lead to matrix form
like the HF Roothaan equations
HC
ΒΌ
SCE
:
(16)
The Kohn-Sham iteration procedure is presented in Fig.
3b
.
Obviously the procedure is very similar to the HF scheme. Details of the DFT
approach are given in [
2
,
3
].
A variety of DFT-based codes is commercially available, for example, VASP
[
12
,
65
], Cambridge Serial Total Energy Package (CASTEP) [
66
], TURBOMOLE
[
67
], CRYSTAL [
68
], Jaguar [
69
], GAUSSIAN [
70
], General Atomic and Molecu-
lar Electronic Structure System (GAMESS) [
71
] and QChem [
13
], amongst others.
A rather complete list of DFT codes is available from an internet address
1
which
also contains programs available for free. Some of these programs also comprise
many other QM approaches.
There is a problem with DFT: the conventional functionals cannot treat disper-
sive forces. GGA, meta-GGA and hybrid functionals are unreliable for systems
where vdW interactions are important. The most rigorous description of dispersion
interactions is provided by explicitly non-local correlation functionals. However,
these methods are computationally demanding and far more complicated than
standard DFT. Although vdW interactions are often considered to be weak, they
dominate the behaviour of all neutral physical systems at separations of order
0.5 nm or larger. vdW interactions are crucial for the chemistry and physics of
weakly bound systems, for example bio-molecules, layered materials, organic
crystals or adsorbing of neutral molecules on surfaces. To overcome this deficiency
of DFT, two strategies have been adopted: semi-empirical approaches have been
developed where an approximately derived R
6
term, multiplied by a suitable short-
range damping function, is explicitly introduced. The R
6
term describes the corre-
lated instantaneous dipole fluctuations together with higher order terms. The second
strategy is the introduction of new density functionals and/or complex schemes
that allow for a first-principles treatment of the vdW interactions. For example, a
seamless vdW density functional (vdW-DF), valid for all interatomic distances, has
been developed by Langreth and co-workers [
72
,
73
]. An example of this approach
is given by Rudenko et al. [
74
], which describes adsorption of halogen molecules on
graphene. Implementation of vdW-DF is non-trivial. An implementation of vdW-
DF with Gaussian basis functions has been presented by Vydrov et al. [
75
]. The
semi-empirical so-called DFT plus dispersion approaches (DFT-D) employ addi-
tion of empirical, pair-wise atomic dispersion corrections of the form -C
6
R
6
,
which are used in the force field methods. To avoid double-counting electron
correlation effects at short range, these contributions are damped for small inter-
nuclear distances. This approach has been refined by Grimme et al. [
76
] by intro-
ducing atom-pair-wise specific dispersion coefficients and cut-off radii that are both
1
http://dft.sandia.gov/Quest/DFT-codes.html.
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