Atomic units; Molecular Hamiltonian; Born-Oppenheimer ...

Molecular Hamiltonian
The non-relativistic time-independent Schrödinger equation:
H|Ψ>=E|Ψ>
H – Hamiltonian operator for a system of nuclei and electrons
N
riA
| riA
| |
ri
RA
|
rij
| rij
| |
ri
rj
|
RAB
|
RAB
| |
RA
RB
|
1 2 1 2 ZA
1 ZAZB
i
A
M
A A
r
2 2
i i
A iA r
i j i ij R
1
1
1 1
1 A
1 B
A
AB
T
e
M
T
N
N
M
V
eN
N1
V
N
ee
M1
H
2
i
A molecular coordinate system
2
xi
2
2
y
2
i
2
z
2
i
- the major challenge in solving the time-independent Schrodinger
equation
MA - the ratio of the mass of nucleus A to the mass of an electron
ZA – the atomic number of nucleus A
Te – the operator for the kinetic energy of the electrons
TN – the operator for the kinetic energy of the nuclei
VeN – the operator for the Coulomb attraction between electrons and nuclei
Vee – the operator for the repulsion between electrons
VNN – the operator for the repulsion between nuclei
(1)– represents the general problem
to be separated in two parts: electronic and nuclear problems
Exact solution for systems containing more than one electron is unknown!
approximations, approximations
i, j – electrons (N)
A, B – nuclei (M)
M
V
NN
(1)
3