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5.80 Small-Molecule Spectroscopy and Dynamics
Fall 2008
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5.80 Lecture #14
Fall, 2008
Page 1 of 7 pages
Lecture #14: Definition of Angular Momenta and |A α MA〉.
 ROT
Evaluation of H
We want to be able to set up effective Hamiltonian models for rotation-vibration-electronic structure of
diatomic molecules with non-zero electronic angular momenta.
 are
The important terms in H
 ROT = B(R)R
2
H
 SO =
H
∑ a (r )̂ ·sˆ
i
i
i
i
electrons
 ROT and H
 SO — HUND’S CASE A
* a convenient basis set for evaluating matrix elements of H
 ,A
 , and A
 ,A
 ± and A
 2 operators and |A α M 〉 basis functions
* A
z
±
Z
A
* Heff and van Vleck Corrections to Heff
* Limiting cases where Heff is approximately diagonal and energy levels are expressed in terms of
a pattern-forming rotational quantum number like J(J + 1), N(N + 1), R(R + 1). (Example in
next lecture for 2∏ and 2∑+ states.)
* effects of accidental degeneracies — perturbations.
Angular Momenta

R

L
S
J

N
J
a
nuclear rotation
e– orbital angular momentum
e– spin

 
+L
 + S
R
total angular momentum
+L

angular momentum. exclusive of spin J − S = R
 + S
total electron angular momentum
= L
See H. Lefebvre-Brion/R. W. Field, pages 72-81.
All angular momenta can be defined by their commutation rules.
5.80 Lecture #14
Fall, 2008
Page 2 of 7 pages
⎡⎣A
 ,A
 ⎤⎦ = +i ε A
 k
∑ ijk
i
j
k
εijk
+1
for
x, y, z in cyclic order
–1
for
non-cyclic
0
repeated coordinator
 , S , R
,
Above is a “normal” Commutation Rule which is applicable for all SPACE components of J , L
 , J a and all body components of L
 , S (but not J , R
, N
 ).
N

involve rotation
of body
 2, A
 , A
 from
Trivial matter to derive properties of eigenbasis |A α MA〉 under operation by A
i
±
commutation rule.
2

A AαM A =  2 A(A + 1) AαM A
 z AαM = α AαM
A
A
A
 Z AαM = M AαM
A
A
A
A
±




 X ± iA
 Y (up for upper case)
A ± ≡ A x ± iA y and A ≡ A
 ± AαM =  [ A(A +1) − α(α ±1)]1/2 Aα ±1M
A
A
A
±
 “raises” α
A
+
 “lowers” α
A
–
1/2

A AαM A =  [ A(A +1) − M A (M A ±1)] AαM A ±1
 + “raises” Μ
A
Α
5.80 Lecture #14
Fall, 2008
Page 3 of 7 pages
This is all you need to know for rotation of diatomic molecules
EXCEPT
 i, A
 j ⎤⎦ = −i ε A
Anomalous commutation rule ⎡⎣A
∑ ijk  k applies only to BODY components of J , R , N .
k
The only difference is
 ± AαM = + [ A(A + 1) − α(α  1)]1/2 A α  1 M
A
A
A
 acts as a “lowering” operator rather than as raising operator.
A
+
 do to |A α M 〉 basis functions.
Now, suppose we want to evaluate what other angular momenta than A
A
 by similar commutation rules.
We classify these other operators as vectors or scalars with respect to A
 (some other operator
The Wigner-Eckart Theorem will eventually tell us how to evaluate the effect of B
 ) on |A α M 〉.
classified by its commutation rule with respect to A
A
 i ⎤⎦ = ⎡⎣S , A
 ± ⎤⎦ = 0 all i, I and S |A α M 〉 = s |A α M 〉.
A scalar is defined as ⎡⎣S , A
A
A
A
 ) is defined as
A normal vector (with respect to A
 ,V
 j ⎤ = +i
⎡⎣ A
i
⎦
∑ ε ijk Vk
k
It happens that
 ,S ⎤ = ⎡ L
 
⎡⎣ L
i
j ⎦
⎣ I ,S J ⎤⎦ = 0
 , S operate on different coordinates and are
L
scalar operators with respect to each other
and all angular momenta obey normal vector operator commutation rules with respect to J for space
fixed components.
J generates rotations in lab frame
 ⎤ = i ε A
⎡⎣ J I , A
∑ IJK K
J ⎦
k
 for body
and all angular momenta obey anomalous vector operator commutation rules with respect to R
fixed components.
 generates rotations in body frame.
R
 ,A
 ⎤ = –i ε A
⎡⎣ R
∑ ijk k
i
j ⎦
k
+L
 + S . See this in example.
 j ⎤⎦ = ⎡⎣J i ,S j ⎤⎦ = 0 because J = R
This has convenient effect that ⎡⎣J i , L
5.80 Lecture #14
Fall, 2008
Page 4 of 7 pages
⎡⎣J i , L
 j ⎤⎦ = ⎡⎣R
 i, L
 j ⎤⎦+ ⎡⎣L
 i, L
 j ⎤⎦+ ⎡⎣S i , L
 j ⎤⎦
E.g.
–iεijkLk
+iεijkLk
=0
which means that J acts as a scalar operator with respect to |n L Λ S ∑〉 so we can factor ψ into
electronic ⊗ vibration ⊗ rotation factors!
So we can write a convenient basis set.
Case (a)
fully “uncoupled” basis set in body
configuration
or index
|v〉
|nLΛ〉|S∑〉
|ΩJM〉
electronic
rotation
very convenient for body fixed matrix elements
vibration
 2 in this case(a) basis set.
Now we can work out matrix elements of B(R) R
v B(R) v ≡ Bv 
to cancel the 2 from all
angular momentum matrix
elements
2
(Note 〈v|B(R)|v′〉 = Bvv′ ≠ 0)
We will see these again when we use J = 0
potential energy curve to derive centrifugal
distortion effects.
   
 = J – L

– S

R
R
Evidently
z ≡ 0
So R
5.80 Lecture #14
Fall, 2008
Page 5 of 7 pages
2
2
2





 x − S x ) + ( J y − L
 y − S y )
R = R ·R = R x + R y = ( J x − L
2
2
= ( J 2x + J 2y ) + ( L2x + L2y ) + (S2x + S2y )
–2(JxLx + JyLy)
No need to be careful of order of
operators because body components of
 and S commute with each other and
L
with those of J .
–2(JxSx + JySy)
+2(LxSx + LySy)
Now for some convenient simplifications.
A 2x + A 2y = A 2 − A 2z
2 ( A x Bx + A y By ) = ( A + B− + A − B+ )
confirm for yourself
 2 = ( J 2 − J 2 ) + ( L2 − L2 ) + (S2 − S2 )
Thus R
z
z
z
plus off-diagonal terms below.
L–uncoupling term
S–uncoupling term
Rotation-electronic term
–(J+L– + J–L+)
–(J+S– + J–S+)
+(L+S– + L–S+)
diagonal part
selection rules
∆Ω = ∆Λ = ±1
∆Ω = ∆Σ = ±1
∆Λ = –∆Σ = ±1
(∆Ω = 0)
So we are almost ready to set up Heff. However L is not a well defined quantity.
Non-Lecture: Stark effect in atoms
∆  = ±1
+↔ –
∆ ML = ∆ λ = 0
Electric field (axially symmetric) of atom B mixes L’s in atom A
5.80 Lecture #14
Fall, 2008
Page 6 of 7 pages
typically 105 cm–1
for 1e– at 1Å
ψ A = ∑ a nL nLA M LA
nL
a nL



 n′ L′
nL H
~ o
E nL − E on ′L′

typically 104 cm–1 for
different nL states of atom
* LA destroyed
* M LA preserved
* L2 and L± matrix elements not explicitly defined (become perturbation parameter)
* L2 and L± and Lz selection rules on Λ are preserved!
* ( L2 − L2z ) ≡ L2⊥
treated as a constant
*
 + v′, n′ Λ −1 S ∑ ≡ B β or β perturbation parameter.
v, n Λ S ∑ BL
vv′
v′v
 ROT = B(R)R
2
Matrix elements of H
1.
Diagonal part (in v, n, Λ, S, ∑, J, Ω, M)
B(R)⎣⎡( J 2 − J 2z ) + (L2⊥ ) + (S2 − S2z )⎤⎦
Bv ⎡⎣J(J + 1) − Ω 2 + L2⊥ + S(S +1) − ∑2 ⎤⎦
include BvL2⊥
in Te + G(v)
2.
Within a 2S+1Λ multiplet state.
The splitting of Ω-components within an S ≠ 0 state is
 SO and is usually small relative to splittings
due to H
between different nΛ states. HSO gives “fine structure”.
o
Spin-uncoupling term — will destroy ∑, Ω provided that E Ωo − E Ω±1
is small.
5.80 Lecture #14
Fall, 2008
Page 7 of 7 pages
−B(R)[ J + S− + J − S+ ]

− ( BV /  2 ) ∑ ±1 Ω ± 1 J S H
ROT
∑ΩJS
= −Bv [ J(J + 1) − (Ω ± 1)Ω ]
1/2
[S(S + 1) − (∑ ±1)∑ ]1/2
notice Ω′Ω, ∑′∑
 ROT will always overwhelm
Above matrix element is ≈ proportional to J, so at high J H
o
∑ and Ω are destroyed, thus spin “uncouples from body frame”.
∆ E Ω,Ω±1
3.
Between two electronic states
∆S = 0, ∆∑ = 0, ∆Λ = ∆Ω = ±1
“L-Uncoupling” (even though L is already destroyed) destroys Λ and Ω.
 − + J − L
+ ]
− ( B(R)  2 )[ J + L
− ⎡⎣ vΛ ±1 B(R) vΛ
 − + J − L
 + ) n Λ S ∑ J Ω
 2 ⎤⎦ n′ Λ ± 1 S ∑ J Ω ± 1 ( J + L
−BvΛ±1vΛ [ J(J + 1) − (Ω ± 1)Ω ]
1/2
4.
n′Λ ± 1 L ± nΛ 



β ( n′Λ′, nΛ )
−BvΛ±1vΛ β ( n′Λ′, nΛ ) ≡ β
an unknown perturbation parameter to be determined directly from spectrum
Between two electronic states ∆Ω = 0
∆Λ = –∆∑ = ±1
+β[S(S + 1) – ∑(∑ ± 1)]1/2
same β as above in #3 for L-uncoupling