Chemistry Reference
In-Depth Information
Let us start by considering the general many-electron problem of
N
e
valence
electrons, which contribute to chemical bonding, and
N
ion
ions, which contain the
nuclei and the tightly bound core electrons. The positions of the electrons and ions
are given by
r
i
and
R
j
, respectively, referred to the same arbitrary origin. This
problem can be described quantum-mechanically, in the absence of external fields,
by the Hamilton operator
H
0
:
H
0
=
H
ee
+
H
ion
−
ion
+
H
e
−
ion
,
(1.3)
where
H
ee
,
H
ion
−
ion
and
H
e
−
ion
correspond to the Hamilton operators concerning
electron-electron, ion-ion and electron-ion interactions, respectively, which are
given by the expressions:
N
e
¯
h
2
2
m
e
∇
e
2
2
i
H
ee
=−
+
|
,
(1.4a)
|
r
i
−
r
j
i
=
1
i
>
j
N
ion
¯
h
2
2
M
j
∇
2
j
H
ion
−
ion
=−
+
V
ion
−
ion
(
R
i
−
R
j
)
,
(1.4b)
j
=
1
i
>
j
N
e
N
ion
H
e
−
ion
=
V
e
−
ion
(
r
i
−
R
j
)
.
(1.4c)
i
=
1
j
=
1
In Eqs. (1.4a), (1.4b) and (1.4c)
m
e
and
M
j
represent the electron and ion masses,
respectively, and
2
2
x
2
2
y
2
∇
i
the Laplacian operator (
∇
≡∇·∇=
∂
/∂
+
∂
/∂
+
z
2
). In Eq. (1.4a) we have inserted a Coulomb term for the electron-electron
repulsive interactions and for
V
ion
−
ion
and
V
e
−
ion
, the ion-ion and electron-ion
interaction potentials, we leave open their explicit form but we assume that they
can be described as sums over two-particle interactions.
Since electrons are much faster than nuclei, owing to
m
e
2
∂
/∂
M
j
, ions can be
considered as fixed and one can thus neglect the
H
ion
−
ion
contribution (formally
H
ion
−
ion
H
ee
, where
V
ion
−
ion
is a constant). This first approximation, as formu-
lated by N. E. Born and J. R. Oppenheimer, reflects the instantaneous adapta-
tion of electrons to atomic vibrations thus discarding any electron-phonon effects.
Electron-phonon interactions can be a-posteriori included as a perturbation of the
zero-order Hamiltonian
H
0
. This is particularly evident in the photoemission spec-
tra of molecules in the gas phase, as already discussed in Section 1.1 for N
2
, where
the
π
u
state exhibits several lines separated by a constant quantized energy.
Equation (1.3) simply transforms to:
H
0
H
ee
+
H
e
−
ion
.
(1.5)
