 OH, a prominent flame emitter, absorber.
Useful for T, XOH measurements.
2
1. Introduction
Selected region of
A2Σ+←X2Π(0,0)
band at 2000K
1517
 Term energies
4
2. Energy levels
 Separation of terms: Born-Oppenheimer approximation
 G(v) = ωe(v + 1/2) – ωexe(v + 1/2)2
 Sources of Te, ωe, ωexe  Herzberg
 Overall system : A2Σ+←X2Π
       JFvGnTJvnE e ,,
elec. q. no.
vib. q. no.
ang. mom. q. no.
Electronic
energy
Vibrational
energy
Angular momentum energy (nuclei + electrons)
A2Σ+
Te ωe ωexe
X2Π
Te ωe ωexe
32682.0 3184.28 97.84 0.0 3735.21 82.21
in [cm-1]
Let’s first look at the upper state  Hund’s case b!
 Hund’s case b (Λ=0, S≠0) – more standard, especially for hydrides
5
2. Energy levels
Recall:
 Σ, Ω not rigorously defined
 N = angular momentum without spin
 S = 1/2-integer values
 J = N+S, N+S-1, …, |N-S|
 i = 1, 2, …
Fi(N) = rotational term energy
Now, specifically, for OH?
 The upper state is A2Σ+
6
2. Energy levels
For OH:
 Λ = 0, ∴	Σ not defined  use Hund’s case b
 N = 0, 1, 2, …
 S = 1/2
 J = N ± 1/2
 F1 denotes J = N + 1/2
F2 denotes J = N – 1/2
Common to write either F1(N) or F1(J)
 The upper state: A2Σ+
7
2. Energy levels
 for pure case b
 ∴ the spin-splitting is γv(2N+1)  function of v; increases with N
      
        111
11
2
2
2
1


NNNDNNBNF
NNNDNNBNF
vvv
vvv


 
 21
OHforcm1.0constantsplitting Av
 Notes:
 Progression for A2Σ+
 “+” denotes positive
“parity” for even N [wave
function symmetry]
 Importance? Selection
rules require parity
change in transition
γv(2N+1) ~ 0.1(5) ~ 0.5cm-1 for N2
Compare with ∆νD(1800K) = 0.23cm-1
 The ground state: X2Π (Λ=1, S=1/2)
8
2. Energy levels
 Note:
1. Rules less strong for hydrides
2. OH behaves like Hund’s a @ low N
like Hund’s b @ large N
 at large N, couples more to N, Λ is less defined, S
decouples from A-axis
3. Result? OH X2Π is termed “intermediate case”
L

Hund’s case a Hund’s case b
Λ ≠ 0, S ≠ 0, Σ defined Λ = 0, S ≠ 0, Σ not defined
 The ground state: X2Π
9
2. Energy levels
 Notes:
3. For “intermediate/transition cases”
where Yv ≡ A/Bv (< 0 for OH); A is effectively the moment of inertia
Note: F1(N) < F2(N)
           
       22/12222
2
22/12222
1
144
2
1
1414
2
1
1
















NNDYYNNBNF
NNDYYNNBNF
vvvv
vvvv
For large N
    
 NNBF
NNBF
v
v


22
2
22
1 11
     0121 22
21  NNNBFF v
For small N
Behaves like Hund’s a, i.e., symmetric top, with spin splitting ΛA
Behaves like Hund’s b, with small (declining) effect from spin
 The ground state: X2Π
10
2. Energy levels
 Notes:
4. Some similarity to symmetric top
Showed earlier that F1 < F2
N
3
2
1
J
5/2
3/2
1/2
J
7/2
5/2
3/2
F1: J = N + 1/2 F2: J = N – 1/2
Ω = 3/2 Ω = 1/2
Te = T0 + AΛΣ
For OH, A = -140 cm-1
 Te = T0 + (-140)(1)(1/2), Σ = 1/2
+ (-140)(1)(-1/2), Σ = -1/2
 ∆Te = 140 cm-1
Not too far off the 130 cm-1 spacing
for minimum J
130
Hund’s a → 2|(A-Bv)|
Recall: Hund’s case a has
constant difference of 2(A-Bv) for same J
F(J) = BJ(J+1) + (A-B)Ω2
(A–B)Ω2 ≈ -158.5Ω2
(A for OH~ -140, B ~ 18.5), Ω = 3/2, 1/2
 Ω = 3/2 state lower by 316 cm-1
Actual spacing is only 188 cm-1, reflects
that hydrides quickly go to Hund’s case b
Hund˚uv typ a
 The ground state: X2Π
11
2. Energy levels
 Notes:
5. Role of Λ-doubling
Showed earlier that F1 < F2
   
    icid
diid
ciic
FF
JJJFF
JJJFF






1
1


 Fic(J) – Fid(J) ≈ 0.04 cm-1 for
typical J in OH
 c and d have different parity
(p)
 Splitting decreases with
increasing N
N
3
2
1
J
5/2
3/2
1/2
J
7/2
5/2
3/2
F1: J = N + 1/2 F2: J = N – 1/2
Ω = 3/2 Ω = 1/2
p
+
–
–
+
+
–
p
–
+
+
–
+
–
Now let’s proceed to draw transitions, but
first let’s give a primer on transition notation.
 Transition notations
 General selection rules
 Parity must change + → – or – → +
 ∆J = 0, 1
 No Q (J = 0) transitions, J = 0 → J = 0 not allowed
12
3. Allowed radiative transitions
Full description: A2Σ+ (v')←X2Π (v") YXαβ(N" or J")
where Y – ∆N (O, P, Q, R, S for ∆N = -2 to +2)
X – ∆J (P, Q, R for ∆J = -1, 0, +1)
α = i in Fi'; i.e., 1 for F1, 2 for F2
β = i in Fi"; i.e., 1 for F1, 2 for F2
 "or" JNXY

 Notes:
1. Y suppressed when ∆N = ∆J
2. β suppressed when α = β
3. Both N" and J" are used
Strongest trans.
e.g., R1(7) or R17
Example: SR21:
∆J = +1, ∆N = +2
F' = F2(N')
F" = F1(N")
1521
 Allowed transitions
13
3. Allowed radiative transitions
Allowed rotational transitions from N"=13 in the A2Σ+←X2Π system
 12 bands possible (3 originating from each lambda-doubled, spin-split X state)
 Main branches: α = β; Cross-branches: α ≠ β
 Cross-branches weaken as N increases
F1(13)
F1c(13)
F1d(13)
State or level
a specific
v",J",N",and
Λ-coupling
 Allowed transitions
14
3. Allowed radiative transitions
Allowed rotational transitions from N"=13 in the A2Σ+←X2Π system
 Notes:
 A given J" (or N") has12 branches (6 are strong; ∆J = ∆N)
 + ↔ – rule on parity
 F1c–F1d ≈ 0.04N(N+1) for OH  for N~10, Λ-doubling is ~ 4cm-1, giving clear separation
 If upper state has Λ-doubling, we get twice as many lines!
 Allowed transitions
15
3. Allowed radiative transitions
Allowed rotational transitions from N"=13 in the A2Σ+←X2Σ+ system
 Note:
1. The effect of the parity selection rule in reducing the number of
allowed main branches to 4
2. The simplification when Λ=0 in lower state, i.e., no Λ-doubling
 Absorption oscillator strength
17
4.1. Oscillator strengths
   
1"2
notationshorthandinor
1"2
'"
'"'"'"
'"
'"'"',',',',',",",",","




J
S
qff
J
S
qff
JJ
vvnnJJ
JJ
vvnnJvnJvn
elec. vib. spin ang. mom. Λ-doubling
elec. osc. strength F-C factor H-L factor
'"vvf = band oscillator strength
(v',v") fv'v"
(0,0) 0.00096
(1,0) 0.00028
 Notes: qv"v' and SJ"J' are normalized


1
'
'" v
vvq
  


2
Xfor4"
'
'" 121"2
elg
J
JJ SJS 
this sum includes the S values for all states with J"
1 for Λ = 0 (Σ state), 2 otherwise
For OH A2Σ+–X2Π
 Is SJ"J' = SJ'J"?  Yes, for our normalization scheme!
 From g1f12 = g2f21, and recognizing that 2J+1 is the ultimate (non removable) degeneracy at
the state level, we can write, for a specific transition between single states
In this way, there are no remaining electronic degeneracy and we require, for detailed
balance, that and
 Do we always enforce for a state?  No!
 But note we do enforce (14.17)
and (14.19)
where, for OH A2Σ←X2Π, (2S+1) = 2 and δ = 2.
 When is there a problem?
 Everything is okay for Σ-Σ and Π-Π, where there are equal “elec. degeneracies”, i,e., g"el =
g'el. But for Σ-Π (as in OH), we have an issue. In the X2Π state, gel = 4 (2 for spin and 2 for Λdoubling),
meaning each J is split into 4 states. Inspection of our H-L tables for SJ"J' for OH
A2Σ←X2Π (absorption) confirms ΣSJ"J' from each state is 2J"+1. All is well. But, in the upper
state, 2Σ, we have a degeneracy g'el of 2 (for spin), not 4, and now we will find that the sum
of is twice 2J'+1 for a single J' when we use the H-L values for SJ"J' for SJ'J". However,
as there are 2 states with J', the overall sum as required by (14.19)
18
4.1. Oscillator strengths
   
1'2
'1'2
1"2
"1"2 "'
"'
'"
'"




J
S
qfJ
J
S
qfJ JJ
vvel
JJ
vvel
 1"2
'
'"  JS
J
JJ
"''" JJJJ SS 
"
"'
J
JJS
"''",'" vvvvelel qqff 
  121"2
'
'"  SJS
J
JJ
  121'2
"
'"  SJS
J
JJ
 41'2
"
"'  JS
J
JJ
 Absorption oscillator strength for f00 in OH A2Σ+–X2Π
19
4.1. Oscillator strengths
Source f00
Oldenberg, et al. (1938) 0.00095 ± 0.00014
Dyne (1958) 0.00054 ± 0.0001
Carrington (1959) 0.00107 ± 0.00043
Lapp (1961) 0.00100 ± 0.0006
Bennett, et al. (1963) 0.00078 ± 0.00008
Golden, et al. (1963) 0.00071 ± 0.00011
Engleman, et al. (1973) 0.00096
Bennett, et al. (1964) 0.0008 ± 0.00008
Anketell, et al. (1967) 0.00148 ± 0.00013
 Absorption oscillator strength
20
4.1. Oscillator strengths
Hönl-London factors for selected OH transitions
Transition SJ"J'/(2J"+1) ΣF1(J) ΣF2(J) Σ[F1(J)+F2(J)]
Q12(0.5) 0.667 0 2 2
Q2(0.5) 0.667
R12(0.5) 0.333
R2(0.5) 0.333
P1(1.5) 0.588 2 2 4
P12(1.5) 0.078
P21(1.5) 0.392
P2(1.5) 0.275
Q1(1.5) 0.562
Q12(1.5) 0.372
Q21(1.5) 0.246
Q2(1.5) 0.678
R1(1.5) 0.165
R12(1.5) 0.235
R21(1.5) 0.047
R2(1.5) 0.353
P1(2.5) 0.530 2 2 4
P12(2.5) 0.070
P21(2.5) 0.242
P2(2.5) 0.358
Q1(2.5) 0.708
Q12(2.5) 0.263
Q21(2.5) 0.214
Q2(2.5) 0.757
R1(2.5) 0.256
R12(2.5) 0.173
R21(2.5) 0.050
R2(2.5) 0.379
Transition SJ"J'/(2J"+1) ΣF1(J) ΣF2(J) Σ[F1(J)+F2(J)]
P1(3.5) 0.515 2 2 4
P12(3.5) 0.056
P21(3.5) 0.167
P2(3.5) 0.405
Q1(3.5) 0.790
Q12(3.5) 0.195
Q21(3.5) 0.170
Q2(3.5) 0.814
R1(3.5) 0.316
R12(3.5) 0.131
R21(3.5) 0.044
R2(3.5) 0.402
P1(9.5) 0.511 2 2 4
P12(9.5) 0.016
P21(9.5) 0.038
P2(9.5) 0.488
Q1(9.5) 0.947
Q12(9.5) 0.050
Q21(9.5) 0.048
Q2(9.5) 0.950
R1(9.5) 0.441
R12(9.5) 0.035
R21(9.5) 0.014
R2(9.5) 0.462