32  1
5.73
Lecture
#32
updated September 19,
Last time:
Matrix elements of Slater determinantal wavefunctions
Normalization:
(N!)
–1/2
F
(i):
selection rule (∆so ≤ 1), sign depending on order
G
(i,j):
selection rule (∆so ≤ 2), two terms with opposite signs
TODAY:
Configuration
∅
which LS terms?
∅
LS basis states
∅
matrix elements
Method of crossing out
boxes
ladders plus orthogonality
Many worked out examples will not be covered in lecture.
MM
LS
,
Longer term goals:
represent “electronic structure” in terms of
properties of atomic orbitals
1. Configuration
→
L,S terms
2. Correct linear combination of Slater determinants for each
L,S term:
several methods
3. 1/r
ij
matrix elements
→
F
k
, G
k
SlaterCondon parameters,
Slater sum rule trick
4.
H
SO
*
ζ
(NLS) — coupling constant for each LS term in an
electronic configuration
*
ζ
(NLS)
↔
ζ
n
l
one spinorbit orbital integral for entire
configuration
* full H
SO
matrix in terms of
ζ
n
l
5. Stark, Zeeman, optical transitions
6. transition strengths
There are a vastly smaller number of orbital parameters than the
number of electronic states.
The periodic table provides a basis for
rationalization of orbital parameters (dependence on atomic number
and on number of electrons.)
matrix elements of
, g  values
r
r
()
nrn
ll
′
+
1
KEY IDEAS:
*1
/r
ij
destroys spinorbital labels as good quantum numbers.
* Configuration splits into widely spaced LSJ “terms.”
*
is a scalar operator with respect to
L, S,
and
J
thus matrix elements
are independent of M
L
, M
S
, and M
J
.
* Configuration generates all possible M
L
, M
S
components of each LS term.
* It can’t matter which M
L
, M
S
component we use to evaluate the 1/r
ij
matrix
elements
* Method of microstates and boxes: Bookkeeping which LS states are present,
organize the algebra to find eigenstates of L
2
and S
2
, basis for “sum rule”
method (next lecture).
1/r
ij
ij
>
∑
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View Full Document32  2
5.73
Lecture
#32
updated September 19,
Which LS terms belong to (nf)
2
* shorthand notation for spin  orbitals
n m
/
e.g.
4f3
,
could suppress 4 and f
( main diagonal for Slater determinant,
for simple product
of spin  orbitals)
* standard order (to get signs internally consistent)
3 3 2 2
 3  3 is my standard order for f
spin orbitals
* which Slater determinants are nonzero and distinct (i.e., not identical when
spin  orbitals are permuted to a different ordering)?
l
l
l
αβ
α
αβαβ
α
β
…
…
+
()
+
=
=
21
7
4
s
f
2
 take any 2 so’s and list in standard order
How many nonzero and distinct Slater determinants are there for f
2
?
14
14 13
2
spin  orbitals
2 identical electrons
⋅
=
91
general
n
p
p
p
n
l
l
l
l
+
[]
+
−
∏
:
!
!
!
22
1
1
1
subshell : one such factor for each subshell
How to generate all 91 linear combinations of Slater determinants that correspond
to the 91 possible
LM
L
SM
S
⟩
basis states that arise from f
2
?
Next lecture.
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 Spring '04
 RobertField
 Determinant, ml, Slater, Slater determinants

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