E
B3LYP
XC
¼ a 0 E
HF
X þ 1 À a 0
ð
ÞE
LDA
X
þ a x E
Becke
X
þ 1 À a c
ð
ÞE
VVN
C
þ a c E
LYP
C
ð1:64Þ
and includes additionally LDA exchange and correlation functionals.
The last group of XC functionals is fully non-local functionals such as ADA [80]
(average density approximation) or WDA [80, 81] (weighted density approximation), proposed in mid-70s of the twentieth century, where the starting point is the
exact functional of the density defined as:
E XC ½nðrÞ ¼
1
2
Z
nðrÞdr
Z n XC ðr; r
0
Þ
r À r 0
j
j
dr
0
ð1:65Þ
where n XC ðr; r
0
Þ is the electron density of the exchange–correlation hole (density
depletion) resulting directly from the exchange–correlation effects related to
electron interactions and modeling of the exchange–correlation hole using
some analytic function. Such a fully non-local functional has finally the following
form:
E
NL
XC ½nðrÞ ¼
1
2
Z
nðrÞdr
Z n
model
XC ðr; r
0
Þ
r À r 0
j
j
dr
0
ð1:66Þ
Thus, the crucial point of this approach is the best possible selection of the
analytical function, modeling the electron density n
model
XC ðr; r
0
Þ of the exchange–
correlation hole. Non-local functionals have a number of advantages over previous
approximations (the minimal number of simplifications), but they are very computationally demanding and hence less frequently used.
Density Functional Theory quickly gained popularity in calculations in
solid-state physics, but in chemistry, it began to be applied to a broader range of
systems only in the 90s of the twentieth century, when better approximations of XC
potentials were proposed. Nowadays, due to significantly lower computational
power requirements compared to HF and post-HF methods, DFT has dominated the
electronic structure calculations for both solids and molecules/clusters. However,
due to the difficulty of using this formalism, for example, in the description of
intermolecular interactions (of key importance for the understanding of chemical
reactions), especially dispersive as well as of transition states and defects containing or highly correlated systems, it is still being intensively developed.
One of the fundamental problems associated with DFT in Kohn–Sham’s formulation, resulting from the lack of knowledge of the exact form of the exchange–
correlation functional and hence the necessity to use of approximate ones, is the
need to select the appropriate functional for the problem studied (no universal XC
functional); hence, a wide variety of functionals have been proposed exhibiting
different efficiency in calculations for a given type of system and/or physical
property. In the case of solid-state calculations, the simplest LDA and GGA
potentials are still quite often used, but in the case of strongly correlated systems,
e.g., 3d-block transition metal oxides, modified versions of these potentials are
1 Computational Methods in Spectroscopy
21
B3LYP
XC
¼ a 0 E
HF
X þ 1 À a 0
ð
ÞE
LDA
X
þ a x E
Becke
X
þ 1 À a c
ð
ÞE
VVN
C
þ a c E
LYP
C
ð1:64Þ
and includes additionally LDA exchange and correlation functionals.
The last group of XC functionals is fully non-local functionals such as ADA [80]
(average density approximation) or WDA [80, 81] (weighted density approximation), proposed in mid-70s of the twentieth century, where the starting point is the
exact functional of the density defined as:
E XC ½nðrÞ ¼
1
2
Z
nðrÞdr
Z n XC ðr; r
0
Þ
r À r 0
j
j
dr
0
ð1:65Þ
where n XC ðr; r
0
Þ is the electron density of the exchange–correlation hole (density
depletion) resulting directly from the exchange–correlation effects related to
electron interactions and modeling of the exchange–correlation hole using
some analytic function. Such a fully non-local functional has finally the following
form:
E
NL
XC ½nðrÞ ¼
1
2
Z
nðrÞdr
Z n
model
XC ðr; r
0
Þ
r À r 0
j
j
dr
0
ð1:66Þ
Thus, the crucial point of this approach is the best possible selection of the
analytical function, modeling the electron density n
model
XC ðr; r
0
Þ of the exchange–
correlation hole. Non-local functionals have a number of advantages over previous
approximations (the minimal number of simplifications), but they are very computationally demanding and hence less frequently used.
Density Functional Theory quickly gained popularity in calculations in
solid-state physics, but in chemistry, it began to be applied to a broader range of
systems only in the 90s of the twentieth century, when better approximations of XC
potentials were proposed. Nowadays, due to significantly lower computational
power requirements compared to HF and post-HF methods, DFT has dominated the
electronic structure calculations for both solids and molecules/clusters. However,
due to the difficulty of using this formalism, for example, in the description of
intermolecular interactions (of key importance for the understanding of chemical
reactions), especially dispersive as well as of transition states and defects containing or highly correlated systems, it is still being intensively developed.
One of the fundamental problems associated with DFT in Kohn–Sham’s formulation, resulting from the lack of knowledge of the exact form of the exchange–
correlation functional and hence the necessity to use of approximate ones, is the
need to select the appropriate functional for the problem studied (no universal XC
functional); hence, a wide variety of functionals have been proposed exhibiting
different efficiency in calculations for a given type of system and/or physical
property. In the case of solid-state calculations, the simplest LDA and GGA
potentials are still quite often used, but in the case of strongly correlated systems,
e.g., 3d-block transition metal oxides, modified versions of these potentials are
1 Computational Methods in Spectroscopy
21