5
in most of the cases, it is relevant to consider the time independent Schrödinger
equation. The time independent Schrödinger equation in the position basis can
be written as 6:
H (r, R) = E(r, R) (r, R), (2.2)
where E(r, R) is the eigenvalue of the Hamiltonian of the system and r and
R represent the electronic and nuclear coordinates. In an expanded form the
Hamiltonian can be written as:
H = −
XN
I=1
~2
2MI r2I
−
Xn
i=1
~2
2mir2i
+ e2
2
XN
I=1
XN
J6=I
ZIZJ
|RI − RJ |
(2.3)
+e2
2
Xn
i=1
Xn
j6=i
1
ri − rj − e2
XN
I=1
Xn
i=1
ZI
|RI − ri|
,
where first and second term on the right hand side represent the kinetic
energy of the nuclei and electrons respectively, third term corresponds to
the nuclear-nuclear Coulomb interaction, fourth term represents the electronelectron
Coulomb interaction and the last term is Coulomb interaction between
the electrons and nuclei.
Unfortunately, the eigenvalues and eigenfunctions of the full Hamiltonian
with coupled electronic and nuclear degrees of freedom can only be obtained
for very few simple systems. Therefore, approximations are needed to make
the electronic structure problem tractable.
2.1.1 Adiabatic and Born-Oppenheimer approximation
One of the commonly used approximation to decouple the nuclear and electronic
degrees of freedom is the adiabatic approximation. It is based on the
fact that the ratio of of the mass of the electrons and nuclei is very small,
therefore, the electrons instantaneously adjust their wavefunctions if there is a
dynamical evolution of the nuclear wavefunctions. In other words, due to the
sluggish dynamics of the nuclear wavefunction the electrons are always in a
stationary state of the Hamiltonian with the instantaneous nuclear potential.
The wavefunction of the system within the adiabatic approximation can be
written as 6:
(R, r, t) = n(R, t)n(R, r), (2.4)
where n denotes the nth adiabatic state of the electrons, n(R, t) represents
the nuclear wavefunction and n(R, r) denotes the electronic wavefunction.