6
In the above ansatz, the dependence of electronic wavefunction on the nuclear
coordinates gives a correction for the electronic eigenvalues of the order
m/M(which comes from applying the kinetic energy operator of the nuclei on
the electronic wavefunctions). The small correction of the order m/M when
included results to the adiabatic approximation and when neglected gives the
so called Born-Oppenheimer approximation. The Born-Oppenheimer approximation
results in an electronic Schrödinger equation Hamiltonian which can
be written as 6:
ˆh
e = −
Xn
i=1
~2
2mir2i
+ e2
2
Xn
i=1
Xn
j6=i
1
ri − rj − e2
XN
I=1
Xn
i=1
ZI
|RI − ri|
.
The above approximation simplifies the electronic problem significantly but not
sufficiently to make it tractable for complex systems. The complexity mainly
arises from the electron-electron interaction term in the electronic Hamiltonian.
Density functional theory (DFT) which is discussed in the next section
provides an elegant way to solve the electronic structure problem of complex
electronic systems.
2.2 Density Functional Theory: An Introduction
The density functional theory came into being from the two theorems by Hohenberg
and Kohn which are 7:
Theorem 1: The electronic density uniquely determines the external potential
up to a trivial additive constant.
Theorem 2: The ground state energy of an electron system is a universal
functional of the ground state electronic density.
Above theorems make it possible to map an interacting system to a noninteracting
electron system with the same electronic density leading to so called
Kohn-Sham equations. The non-interacting electron system is much easier to
solve since the wavefunction of the system factorizes. The mapping significantly
simplifies the electronic structure problem since the electronic density
which is dependendent only on three coordinates becomes the central object as
opposed to the wavefunction in the Schrödinger equation which has 3N degrees
of freedom. The potential which enters the independent particle Hamiltonian
can be derived from the total energy of the system if one knows how the energy
depends on the electronic density. Kohn-Sham equation for independent