Atoms and Molecules
135
5.2 SHELLS AND SUBSHELLS IN ATOMS
The Hamiltonian for the electrons in an atom with the nucleus placed at the
origin, can be written as:
H = H 0 + H 1 + H 2 + H 3
(5.14)
where
H 0 =
2
2
0
1
( )
2
4
i
i
i
i
Ze
p
Vr
m
e
−
+
π ε
∑
(5.15)
H 1 =
2
0
1
( )
2
4
i
ij
i j
i
e
V r
r
≠
−
πε
∑
∑
(5.16)
H 2 =
2 2
1
1
.
2
2
∑
i i
i
i
i
dV
dr
m c r
l s
(5.17)
and H 3 includes the corrections due to spin-spin interaction, relativisitic
corrections, etc. The term H 0 contains the potential due to the nucleus and V(r i )
which represents some averge potential due to the other electrons (the other
electrons screen the nuclear charge). The term V(r i ) is not important for small
r i but will become increasingly important as r i increases. In a simple model, the
screening contribution is assumed to be of the form
V(r) =
2
/
0
(
1) (1
)
4
r b
Z
e
e
r
−
−
−
πε
(5.18)
which has the correct behaviour for r → 0 and r → ∞. Since b is expected to be
large compared to the radius of the inner orbits, the potential can be expanded in
powers of r to obtain an approximate expression for V(r i ) as
V(r i ) ≈
2
2
0
(
1)
1
1
...
4
2
i
Z
e
r
b
b
−
−
+
πε
(5.19)
The second term H 1 represents the deviations of the actual repulsive potential
from the average potential V(r i ). The term H 2 describes the spin-orbit interaction
of the electrons. It is of the same form as Eq. (4.51) except that Ze
2
/4πε 0 r
2
has
been replaced by the more general expression dV/dr. Of these three terms, the
general structure of the atoms is determined mainly by H 0 . The terms H 1 and
H 2 , however, play an important role in the determination of the energy levels of
the atom, in particular the fine structure of the levels.
For obtaining the structure of the atoms, one starts with only the kinetic
energy of the electrons and the electrostatic interaction of the electrons with the
nucleus, i.e. the first two terms in H 0 [Eq. (5.15)]. The energy levels of the
electrons are then those of a one-electron atom, i.e.
135
5.2 SHELLS AND SUBSHELLS IN ATOMS
The Hamiltonian for the electrons in an atom with the nucleus placed at the
origin, can be written as:
H = H 0 + H 1 + H 2 + H 3
(5.14)
where
H 0 =
2
2
0
1
( )
2
4
i
i
i
i
Ze
p
Vr
m
e
−
+
π ε
∑
(5.15)
H 1 =
2
0
1
( )
2
4
i
ij
i j
i
e
V r
r
≠
−
πε
∑
∑
(5.16)
H 2 =
2 2
1
1
.
2
2
∑
i i
i
i
i
dV
dr
m c r
l s
(5.17)
and H 3 includes the corrections due to spin-spin interaction, relativisitic
corrections, etc. The term H 0 contains the potential due to the nucleus and V(r i )
which represents some averge potential due to the other electrons (the other
electrons screen the nuclear charge). The term V(r i ) is not important for small
r i but will become increasingly important as r i increases. In a simple model, the
screening contribution is assumed to be of the form
V(r) =
2
/
0
(
1) (1
)
4
r b
Z
e
e
r
−
−
−
πε
(5.18)
which has the correct behaviour for r → 0 and r → ∞. Since b is expected to be
large compared to the radius of the inner orbits, the potential can be expanded in
powers of r to obtain an approximate expression for V(r i ) as
V(r i ) ≈
2
2
0
(
1)
1
1
...
4
2
i
Z
e
r
b
b
−
−
+
πε
(5.19)
The second term H 1 represents the deviations of the actual repulsive potential
from the average potential V(r i ). The term H 2 describes the spin-orbit interaction
of the electrons. It is of the same form as Eq. (4.51) except that Ze
2
/4πε 0 r
2
has
been replaced by the more general expression dV/dr. Of these three terms, the
general structure of the atoms is determined mainly by H 0 . The terms H 1 and
H 2 , however, play an important role in the determination of the energy levels of
the atom, in particular the fine structure of the levels.
For obtaining the structure of the atoms, one starts with only the kinetic
energy of the electrons and the electrostatic interaction of the electrons with the
nucleus, i.e. the first two terms in H 0 [Eq. (5.15)]. The energy levels of the
electrons are then those of a one-electron atom, i.e.