that since nuclei are much heavier then electrons (approximately 10
3
–10
5 times,
depending on the number of protons and neutrons making up particular nucleus), they
move much slower than electrons which have a response time of the order of femtoseconds and can adjust themselves almost instantaneously to the new atomic configuration (thus—from electrons’ point of view—nuclei seem to be clamped and the
idea goes, we can treat the motions of electrons and nuclei as independent). In this
case, the effective interactions between the atoms take place via electrons, and we
carry out the calculations looking for a solution for the ground state of the electron
cloud, for the assumed, fixed spatial configuration of motionless atoms. The
assumption of the independent movement of the atomic nuclei and electrons allows
separation of the variables describing their positions and writing the multi-electron
wave function as a product of two functions: one dependent only on nuclei positions
(nuclei wave function w N R a ; R b ; . . .; R m
À
Á
) and the other on positions of electrons
(electronic wave function w el r 1 ; r 2 ; . . .; r n ; R a ; R b ; . . .; R m
À
Á
) and current geometry
(nuclei positions):
w r 1 ; r 2 ; :::; r n ; R a ; R b ; :::; R m
À
Á ¼ w el r 1 ; r 2 ; :::; r n ; R a ; R b ; :::; R m
À
Á
w N R a ; R b ; :::; R m
À
Á
ð1:12Þ
This enables us to solve the electron and nuclear Schrödinger equation separately, starting with the one for electrons:
^
H el w el r; R
ð Þ ¼ E el w el r; R
ð Þ
ð1:13Þ
where
^
H el ¼ À
h
2
2m e
X
i
r
2
i À
X
a
X
i
Z a e
2
r ia
þ
X
j
X
i [ j
e
2
r ij
ð1:14Þ
The Born–Oppenheimer approximation leads to a very important concept—the
potential energy hypersurface U(R):
UðRÞ ¼ E el þ V NN ¼ E el þ
X
a
X
a [ b
Z a Z b e
2
r ab
ð1:15Þ
Once we have the potential energy surface (PES), we can solve the nuclear
Schrödinger equation:
^
H N w N R
ð Þ ¼ E N w N R
ð Þ
ð1:16Þ
where
^
H N ¼ À
h
2
2
X
a
1
m a
r
2
a þ U R
ð Þ
ð1:17Þ
1 Computational Methods in Spectroscopy
9
3
–10
5 times,
depending on the number of protons and neutrons making up particular nucleus), they
move much slower than electrons which have a response time of the order of femtoseconds and can adjust themselves almost instantaneously to the new atomic configuration (thus—from electrons’ point of view—nuclei seem to be clamped and the
idea goes, we can treat the motions of electrons and nuclei as independent). In this
case, the effective interactions between the atoms take place via electrons, and we
carry out the calculations looking for a solution for the ground state of the electron
cloud, for the assumed, fixed spatial configuration of motionless atoms. The
assumption of the independent movement of the atomic nuclei and electrons allows
separation of the variables describing their positions and writing the multi-electron
wave function as a product of two functions: one dependent only on nuclei positions
(nuclei wave function w N R a ; R b ; . . .; R m
À
Á
) and the other on positions of electrons
(electronic wave function w el r 1 ; r 2 ; . . .; r n ; R a ; R b ; . . .; R m
À
Á
) and current geometry
(nuclei positions):
w r 1 ; r 2 ; :::; r n ; R a ; R b ; :::; R m
À
Á ¼ w el r 1 ; r 2 ; :::; r n ; R a ; R b ; :::; R m
À
Á
w N R a ; R b ; :::; R m
À
Á
ð1:12Þ
This enables us to solve the electron and nuclear Schrödinger equation separately, starting with the one for electrons:
^
H el w el r; R
ð Þ ¼ E el w el r; R
ð Þ
ð1:13Þ
where
^
H el ¼ À
h
2
2m e
X
i
r
2
i À
X
a
X
i
Z a e
2
r ia
þ
X
j
X
i [ j
e
2
r ij
ð1:14Þ
The Born–Oppenheimer approximation leads to a very important concept—the
potential energy hypersurface U(R):
UðRÞ ¼ E el þ V NN ¼ E el þ
X
a
X
a [ b
Z a Z b e
2
r ab
ð1:15Þ
Once we have the potential energy surface (PES), we can solve the nuclear
Schrödinger equation:
^
H N w N R
ð Þ ¼ E N w N R
ð Þ
ð1:16Þ
where
^
H N ¼ À
h
2
2
X
a
1
m a
r
2
a þ U R
ð Þ
ð1:17Þ
1 Computational Methods in Spectroscopy
9