2.1 Computational Methods
41
E XC [ρ] = (T [ρ] − T S [ρ]) + (E ee [ρ] − J [ρ])
(2.22)
Hence, the exchange-correlation energy contains all parts of the total energy
equation that are unknown, including effects of self-interaction, exchange, correlation
and components of the kinetic energy. The total energy therefore becomes
E
ρ( r )
= T S [ρ] + J [ρ] + E XC [ρ] + E Ne [ρ]
(2.23)
The final difficulty was therefore to establish a means to define a unique set of KS
orbitals that represent the non-interacting system. It turns out that this can be done by
application of the variational principle [5], and leads to the so-called (one-electron)
Kohn-Sham equation,
−
1
2
∇
2
+
ρ( − → r 2 )
r 2
d − → r 2 + V XC ( − → r 1 ) −
M
A
Z A
r 1 A
ϕ i
=
−
1
2
∇
2
+ V eff ( − → r 1 )
ϕ i = ε i ϕ i
(2.24)
Similar to the HF one-electron equations, this must be solved iteratively. It is
also worth noting the similarity of the Kohn-Sham equation with the HF equation
(Eq. 2.8), with the main difference being the nature of the spin-orbitals and the
exchange-correlation terms. In Eq. 2.24, the only term that remains unknown is V XC
(the exchange-correlation potential), which is defined as,
V XC =
δ E XC
δ ρ
(2.25)
Thus, if the form of V XC were known, the KS approach would lead to an exact
solution of the Schrödinger equation.
2.1.4.3 Exchange-Correlation Functionals
The most active area of research in DFT development surrounds developing approximate forms for V XC . A number of approaches have been made, with varying complexities. A full discussion of these approaches is outside the scope of this thesis, but an
excellent introduction is provided in Refs. [16] and [17]. The most basic form of V XC
is based on the work of Thomas and Fermi [18, 19] and known as the Local Density
Approximation (LDA) [15]. It makes the assumption that the electron density can
be treated as a uniform electron gas and thus the exchange-correlation at a point r
with density ρ(r) should be the same as that of a uniform gas of the same density
[20]. Physically, this is similar to the electronic structure of solid metals, for which
LDA works quite well. However, when molecular solids are considered, this approximation becomes rather poor. Electrons in such systems are not delocalised, but are
41
E XC [ρ] = (T [ρ] − T S [ρ]) + (E ee [ρ] − J [ρ])
(2.22)
Hence, the exchange-correlation energy contains all parts of the total energy
equation that are unknown, including effects of self-interaction, exchange, correlation
and components of the kinetic energy. The total energy therefore becomes
E
ρ( r )
= T S [ρ] + J [ρ] + E XC [ρ] + E Ne [ρ]
(2.23)
The final difficulty was therefore to establish a means to define a unique set of KS
orbitals that represent the non-interacting system. It turns out that this can be done by
application of the variational principle [5], and leads to the so-called (one-electron)
Kohn-Sham equation,
−
1
2
∇
2
+
ρ( − → r 2 )
r 2
d − → r 2 + V XC ( − → r 1 ) −
M
A
Z A
r 1 A
ϕ i
=
−
1
2
∇
2
+ V eff ( − → r 1 )
ϕ i = ε i ϕ i
(2.24)
Similar to the HF one-electron equations, this must be solved iteratively. It is
also worth noting the similarity of the Kohn-Sham equation with the HF equation
(Eq. 2.8), with the main difference being the nature of the spin-orbitals and the
exchange-correlation terms. In Eq. 2.24, the only term that remains unknown is V XC
(the exchange-correlation potential), which is defined as,
V XC =
δ E XC
δ ρ
(2.25)
Thus, if the form of V XC were known, the KS approach would lead to an exact
solution of the Schrödinger equation.
2.1.4.3 Exchange-Correlation Functionals
The most active area of research in DFT development surrounds developing approximate forms for V XC . A number of approaches have been made, with varying complexities. A full discussion of these approaches is outside the scope of this thesis, but an
excellent introduction is provided in Refs. [16] and [17]. The most basic form of V XC
is based on the work of Thomas and Fermi [18, 19] and known as the Local Density
Approximation (LDA) [15]. It makes the assumption that the electron density can
be treated as a uniform electron gas and thus the exchange-correlation at a point r
with density ρ(r) should be the same as that of a uniform gas of the same density
[20]. Physically, this is similar to the electronic structure of solid metals, for which
LDA works quite well. However, when molecular solids are considered, this approximation becomes rather poor. Electrons in such systems are not delocalised, but are