2.1 Computational Methods
39
This is a particularly attractive approach, since the electron density is experimentally
observable, while a wavefunction is not. An excellent introduction to DFT can be
found in References [11] and [12], and more detailed derivations can be found in
dedicated texts [5].
2.1.4.1 Hohenberg-Kohn Theorems
Rooted in the BOA Hamiltonian, Eq. 2.3, the development of DFT began with the
seminal theories proposed by Hohenberg and Kohn: the Hohenberg-Kohn Theorems
(HK). According to the HK theorems: [13]
1. The external potential, V ext ( r) is uniquely defined by ρ( r)
Since the Hamiltonian is fixed by this term, Eq. 2.3, this suggests that the energy of
the system is uniquely defined by ρ( r ). In addition, this posits that the 3N spatial
coordinates required to define a system in the HF equations can be reduced to only
3 spatial coordinates.
Conceptually, this theorem stems from the fact that the electron density (whose
integral is the total number of electrons in the system) depends on the number, charge
and position of nuclei. It can be shown [14] that because the total energy is a function
of the electron density, so too must its components, and thus Eq. 2.3 can be recast as,
E 0 [ρ 0 ] = T [ρ 0 ] + E ee [ρ 0 ] + E Ne [ρ 0 ]
(2.16)
This equation is conveniently separated into the system-dependent term
(E Ne [ρ 0 ] =
ρ 0 ( r )V Ne d
r ) and the terms which are universal (i.e. their form is
independent of N, R A and Z A ),
F H K [ρ] = T [ρ] + E ee [ρ] = T [ρ] + J [ρ] + E ncl [ρ]
(2.17)
where the electron-electron energy is decomposed into the Coulombic component
(J ) and a non-classical component, E ncl , which includes correlation and exchange
effects. And thus the total energy is defined by
E 0 [ρ 0 ] = F H K [ρ] + ∫ ρ 0 ( r )V Ne d
r
(2.18)
Hence, it appears that the first HK theorem offers a direct link between density
and energy. Despite the immense simplicity of these equations, the problem again
arises that, due to the effects of electron correlation and exchange, no explicit form
for T [ρ] or E ee [ρ] are known.
2. Variational Principle
This theorem states that the functional that returns the ground state energy of a system
will deliver the lowest energy only if the input electron density is the true ground
39
This is a particularly attractive approach, since the electron density is experimentally
observable, while a wavefunction is not. An excellent introduction to DFT can be
found in References [11] and [12], and more detailed derivations can be found in
dedicated texts [5].
2.1.4.1 Hohenberg-Kohn Theorems
Rooted in the BOA Hamiltonian, Eq. 2.3, the development of DFT began with the
seminal theories proposed by Hohenberg and Kohn: the Hohenberg-Kohn Theorems
(HK). According to the HK theorems: [13]
1. The external potential, V ext ( r) is uniquely defined by ρ( r)
Since the Hamiltonian is fixed by this term, Eq. 2.3, this suggests that the energy of
the system is uniquely defined by ρ( r ). In addition, this posits that the 3N spatial
coordinates required to define a system in the HF equations can be reduced to only
3 spatial coordinates.
Conceptually, this theorem stems from the fact that the electron density (whose
integral is the total number of electrons in the system) depends on the number, charge
and position of nuclei. It can be shown [14] that because the total energy is a function
of the electron density, so too must its components, and thus Eq. 2.3 can be recast as,
E 0 [ρ 0 ] = T [ρ 0 ] + E ee [ρ 0 ] + E Ne [ρ 0 ]
(2.16)
This equation is conveniently separated into the system-dependent term
(E Ne [ρ 0 ] =
ρ 0 ( r )V Ne d
r ) and the terms which are universal (i.e. their form is
independent of N, R A and Z A ),
F H K [ρ] = T [ρ] + E ee [ρ] = T [ρ] + J [ρ] + E ncl [ρ]
(2.17)
where the electron-electron energy is decomposed into the Coulombic component
(J ) and a non-classical component, E ncl , which includes correlation and exchange
effects. And thus the total energy is defined by
E 0 [ρ 0 ] = F H K [ρ] + ∫ ρ 0 ( r )V Ne d
r
(2.18)
Hence, it appears that the first HK theorem offers a direct link between density
and energy. Despite the immense simplicity of these equations, the problem again
arises that, due to the effects of electron correlation and exchange, no explicit form
for T [ρ] or E ee [ρ] are known.
2. Variational Principle
This theorem states that the functional that returns the ground state energy of a system
will deliver the lowest energy only if the input electron density is the true ground