Biomedical Engineering Reference
In-Depth Information
20
Density Functional Theory and the
Kohn-Sham LCAO Equations
The 1998 Nobel Prize for Chemistry was awarded jointly to John A. Pople and Walter
Kohn, and the texts of their addresses on the occasion of the awards are reported in Pople
(1999) and Kohn (1999). Pople's name should be familiar to you if you have assiduously
studied this text. Kohn opened his address with the following remarks.
The citation for my share of the 1998 Nobel Prize in Chemistry refers to the 'Development
of Density Functional Theory'. The initial work on Density Functional Theory (DFT) was
reported in two publications: the first with Pierre Hohenberg (P Hohenberg and W Kohn,
Phys Rev 136 (1964) B864) and the next with Lu J Sham (W Kohn and L J Sham, Phys Rev
140 (1965) A1133). This was almost 40 years after E Schrödinger (1926) published his first
epoch-making paper marking the beginning of wave mechanics. The Thomas-Fermi theory,
the most rudimentary form of DFT, was put forward shortly afterwards (E Fermi, Attl Accad
Naz Lincei, Cl Sci Fis Mat Nat Rend 6 (1927) 602, L H Thomas, Proc Camb Phil Soc 23
(1927) 542) and received only modest attention.
It is worth spending a little time on the historical aspects before launching into modern
DFT, and the
exchange potential
plays a key role in our discussion. If we consider a closed
shell system with electron configuration
(ψ
1
)
2
(ψ
2
)
2
...(ψ
M
)
2
then in Hartree's original theory we would write the total wavefunction as a simple orbital
product for which the electronic energy is
M
ψ
R
(
r
1
)
h
(
r
1
) ψ
R
(
r
1
) dτ
1
ε
el
=
2
R
=
1
(20.1)
2
ψ
R
(
r
1
) ψ
R
(
r
1
)
M
M
+
g
(
r
1
,
r
2
)ψ
S
(
r
2
) ψ
S
(
r
2
) dτ
1
dτ
2
ˆ
R
=
1
S
=
1
