CHEM 356, Lecture 26, Fall 2009
1
Atomic Structure and Atomic Spectroscopy.
We shall assume that the wavefunctions for a multielectron atom are determined in
zeroth order by a central–field Hamiltonian,
ℋ
0
: we shall study the effect of adding
systematically inter–electron repulsion, followed by spin–orbit coupling.
Russell–Saunders Coupling Scheme:
The most common coupling scheme for angular momenta is the Russell–Saunders scheme,
which corresponds to building up the atomic Hamiltonian in three stages,
ℋ
=
ℋ
0
,
ℋ
=
ℋ
0
+
ℋ
ee
,
ℋ
=
ℋ
0
+
ℋ
ee
+
ℋ
so
.
Notation:
lower case letters are used to designate the angular momenta of the individual
electrons in a multielectron atom:
ˆ
ℓ
𝑖
,
ˆ
s
𝑖
,
ˆ
j
𝑖
, etc.
Procedure:
makes extensive use of the fact that mutually commuting operators can have si
multaneous eigenfunctions. We shall
require
at each step that the
operators
used
to label our energy eigenfunctions
all commute with
ℋ
.
1.
Central–field Hamiltonian
:
ℋ
0
All
ˆ
ℓ
𝑖
,
ˆ
s
𝑖
commute with
ℋ
0
and with one another: we say that
ˆ
ℓ
2
𝑖
,
ˆ
ℓ
𝑖𝑧
,
ˆ
𝑠
2
𝑖
, and
ˆ
𝑠
𝑖𝑧
are
SHARP operators, which means that we can obtain simultaneous eigenfunctions
of the full set of operators
{
ˆ
ℓ
2
𝑖
,
ˆ
ℓ
𝑖𝑧
,
ˆ
𝑠
2
𝑖
,
ˆ
𝑠
𝑖𝑧
,
ℋ
0
}
.
∙
The form for the Hamiltonian provides a description of the atom at the level of
electronic CONFIGURATIONS.
2.
Inclusion of Inter–electronic repulsions:
ℋ
=
ℋ
0
+
ℋ
ee
The individual vector operators
ˆ
ℓ
𝑖
no longer commute with
ℋ
(because they do
not commute with
ℋ
ee
); however, two new angular momentum operators
ˆ
L
=
∑
𝑖
ˆ
ℓ
𝑖
and
ˆ
S
=
∑
𝑖
ˆ
s
𝑖
do.
The corresponding set of sharp operators is then
{
ˆ
𝐿
2
,
ˆ
𝑆
2
,
ˆ
𝐿
𝑧
,
ˆ
𝑆
𝑧
}
.
∙
This form for the Hamiltonian provides a description of the atom at the level of
electronic TERMS.
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CHEM 356, Lecture 26, Fall 2009
2
3.
Inclusion of Spin–Orbit Coupling:
ℋ
=
ℋ
0
+
ℋ
ee
+
ℋ
so
.
ˆ
L
and
ˆ
S
do not commute with
ℋ
so
=
𝐴
ˆ
S
⋅
ˆ
L
, and hence they do not commute with
ℋ
. The vector operator
ˆ
J
, with
ˆ
J
=
ˆ
L
+
ˆ
S
, which represents the total electronic
angular momentum of the multielectron atom does commute with
ℋ
, however.
The corresponding set of sharp operators is now
ˆ
𝐽
2
, and
ˆ
𝐽
𝑧
.
∙
This form for the Hamiltonian provides a description of the atom at the level of
electronic LEVELS.
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 Fall '09
 Prof.Iaskjd
 Atom, Electron, Angular Momentum, coupling scheme, russell–saunders coupling scheme

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