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��
�
E �
NR
DFT
NI �R� � Ek
� Een
� Eee
Exc
With the
22
Coul
(2–6)
NI
Ek term accounting for the kinetic energy of the non-interacting model
system of the same density, and the remaining interacting kinetic energy being accounted
for within the Exc term.
With this ingenious and ground-breaking step, Kohn and Sham introduced the
new concept of using a non-interacting reference system, with the exact same density as
the real system, to account for almost all of the kinetic energy, instead of the exact
amount accounted for by the Schrödinger equation. The remaining small amount of
kinetic energy is then merged into the only unknown term in equation (2–6), Exc. In order
to solve for this term an appropriate method of approximation must be chosen.
The most commonly used approach to determining approximations for Exc is to
separate it into an exchange term Ex and a correlation term Ec, and approximate each of
these terms separately.
E �
xc
� Ex
Ec
(2–7)
Given an expression for Exc one can formulate a one-electron Schrödinger
equation (2–8) analogous to the HF equation (2–4). This allows us to use the well-
established HF SCF technique.
ˆ
f KS � i � � i� i (2–8)