1. Electronic Spectroscopy
Outlines
Introduction
Molecular Term Symbols
Transitions between Electronic States of Diatomic Molecules
Vibrational Fine Structure of Electronic Transitions in Diatomic Molecules : Franck-Condon Principle
- UV-Visible Light Absorption in Polyatomic Molecules
Transitions Between Ground and Excited States
Jablonski diagram, Fluorescence, Phosphorescence

2. Introduction
- Excitations between molecular electronic states (the electron is excited from an initial low energy state to a higher state by absorbing photon energy).
- Molecular electronic transitions are induced by UV/VIS radiation (Eelec >> Evib >> Erot)

3. 2S + 1L Molecular Term Symbols: diatomic molecules
Describe the electronic states of molecules.
Molecular electronic configuration relied on the orbital and spin angular momentums.
“Spin multiplicity”
(S : the total spin quanmtum number)
Principle z-axis
2S + 1L
The “z-component” orbital momentum quanmtum number
 = |ML|
(similar to atomic term symbol) L 1 2 3 4 S P D F G
(or 2S + 1 L g/u for homonuclear diatomic molecules indicate by inversion symmetry of MO’s functions) ,

4. Determine molecular term symbol
Only unfilled subshells contribute to the term orbital and spin angular momentum.
In the first or second row diatomic molecules, MOs are either  or  type. MO s p d f g ml 1 2 3 4 H2 CO

5. 3. Calculate ML and MS
where mli, mls = z components of orbital and spin angular momentum for the i th electron in its molecular orbital.
For MO's with  symmetry, ml = 0
For MO's with  symmetry, ml = +1, -1
Note that there is no ml = 0 for  MO's since the Pz atomic orbital is associated with the  MO
-L  ML  L, -S  MS  S
4. Determine L value from |ML| to assign  symbol, and S value from |MS| to calculate 2S+1
5. Generate term symbols
2S + 1L

6.  L= 0 ; L   1S 1S g Molecular term symbol for Ground state of H2
Ground-state Electronic configuration of H2 : (1g)2 MO
 msi  mli
= 0
1 = 0
Singlet
ML = 0
MS = 0
|ML | = 0
|MS| =  S = 0  2S+1 = 1
 L= 0 ; L  
Term symbol 1S
Note that H2 is homonuclear diatomic molecules. Inversion symmetry of g MO’s wave functions is also considered. Using multiplication rules
(g)(g) = g
(g)(u) = (u)(g) = u
(u)(u) = g
1S g

7.  L = 0; L   2S 2S u Ground state of 𝐻 2 − 1*u 1* MO  msi  mli
Gr. state configuration (1g)2 (1*u)2
1*u
1* MO
 msi  mli 1
* = 0
Fully occupied, not need to be considered
1*g
ML = 0
MS = 1/2, -1/2
|ML | = 0
|MS | = 1/2  S = 1/2  2S+1 = 2
 L = 0; L  
Term symbol 2S
An electron fills in *u orbital
2S u

8. Molecular term symbol for Ground state of O2
2 e- are in * MO

9. MO  msi  mli S (|MS|) L L 1 -1 1+1 = 2 1-1 = 0 -1+-1 = -2 1 2   
2 e- in * MO
Possible combination = 4C2 = 6 MO
Term symbol
 msi  mli
S (|MS|) L L 1 -1
1+1 = 2
1-1 = 0
-1+-1 = -2 1 2
   
1 3 
1 
3 
1 
Three terms : 1, 3 , 1 

10. Gerade = symmetric with inversion
Parity :Gerade (g) and ungerade (u)
Gerade = symmetry with respect to a center of inversion
represents center of inversion
Gerade = symmetric with inversion
inversion
Ungerade = antisymmetric with inversion
inversion

11. σg σg σu σu πu πg πu πg Gerade and ungerade Antibonding MO Bonding MO
From s orbs
From pz orbs
From s orbs
From p orbs πu πg πu πg
From px orbs
From py orbs
From px orbs
From py orbs

12. Energy diagram and electronic configuration of O2
1s
*1s
2s
*2s
2p
2px, 2py
*2px , *2py
*2p
1g
1*u
2g
2*u
3g
1u
1*g
3*u
Energy
Ground state electronic configuration of O2
(1g)2 (1*u)2 (2g)2 (2*u)2 (3g)2 (1u)2 (1u)2 (1*g)1 (1*g)1

13. Because 2 e- are in * MO so we use the notation g
Term symbol
1  g
3  g
1  g
1  g
1  g
*g*g = *g
Energy
3  g
Hund’s rule
3  g
Electrons of the same spins separated in two orbitals (lowest energy)
1  g
magnetic attraction between electrons of opposite spins
Electrons of opposite spins paired in a single orbital
1  g
Electrons of opposite spins separated in two orbitals

14. 3  g 1  g 1  g Electronic states and molecular term symbol of O2
Ground state configuration of O2
(1g)2 (1*u)2 (2g)2 (2*u)2 (3g)2 (1u)2 (1u)2 (1*g)1 (1*g)1
Ground state
Excited states
3  g
1  g
1  g
lowest energy due to greatest spin multiplicity (Hund’s rules)
Singlet state O2
Triplet state O2

15. the +/- superscript applies only to  states, and indicate whether the wavefunction is symmetric or antisymmetric with respect to reflection through a plane containing the two nuclei.

16. Assigning + and – sign to  terms of diatomic molecules
+ and - refer to the change in sign of the molecular wave function on reflection in a plane that contains the molecular axis.
“+” refers to no change in sign of .
“-” refers to  does change sign.
General rule for “+”:
- All MOs are filled,
- Unpaired electron in  MOs
“+” superscript because  MOs has plan of reflection. Therefore  unchanges the sign.

17.   - + - + + - - +
“-” superscript because  MOs does not have plan of reflection. Therefore  changes the sign.
“+” superscript because  MOs does not have plan of reflection. Therefore  changes the sign.

18. For the case of O2, electronic configuration is
(1g)2 (1*u)2 (2g)2 (2*u)2 (3g)2 (1u)2 (1u)2 (1*g)1 (1*g)1
Consider only 
(1*g)1 (1*g)1
There are six possible combinations of wave functions for ml = 1 and for ms = 1

19. Spatial (orbital) function Symmetric orb function
Spin function 
|S|
symbol 1
+1+1
symmetry
12 - 12 2
1g 2
-1-1 3
+1-1 + -1+1
1 +g 4
+1-1 - -1+1
anti-symmetry
12 1
3-g 5
12 + 12 6
12

20. Ex. Predict molecular term symbol of the ground state for Li2.
the electronic configuration for Li2
(1g)2 (1*u)2 (2g)2
each electron is in the same orbitals
 = 0
For the total orbital angular momentum:
 mli = = 0   = 
For the total spin angular momentum
 msi = 1/2 + -1/2 = 0  2S+1 = 1
Two electron fills in *g whose the orbital is assigned as gerade. According to the multiplication rule (g) (g) = g
Molecular term symbol of the ground state for Li2 is
1Sg or 1S+g

21. Electronic states of excited state configuration of O2
Ground state
Excited state
Ground state configuration of O2
(1g)2 (1*u)2 (2g)2 (2*u)2 (3g)2 (1u)2 (1u)2 (1*g)1 (1*g)1
Excited state configuration of O2
(1g)2 (1*u)2 (2g)2 (2*u)2 (3g)2 (1u)1 (1u)2 (1*g)1 (1*g)2
1u
1*g
Possible combination (4C3 )= 4  = 16 states

22. 1u
1*g

23. 1u 1*g  msi  mli Term symbol +1 -1 +1 -1 +1 2 3u 0 2 1u 1 0 3u
 msi  mli u u u u

24. 1u 1*g  msi  mli Term symbol +1 -1 +1 -1 0 2 1u -1 2 3u 0 0 3u
 msi  mli u u u u

25. 1u
1*g
Term symbol
 msi  mli

26. 1u
1*g
Term symbol
 msi  mli

27. Description for potential energy curves of electronic states
- As the energy states become higher in energy, the bond length increases.
- Excited states have more anti-bonding character (less bonding order, less bonding energy and longer bond).
the four lowest excited states of O2
X refers to the ground state.
A, B, … refers to the higher energy states with the same multiplicity as the ground state.
a, b, … refers to the higher energy states with the different multiplicity as the ground state.
Ground state of O2

28. Allowed transition Examples
Transitions Between Electronic States of Diatomic Molecules
Selection rules:
Allowed transition
Examples
 = 0, 1
    
S = 0
11 33 11 33
g  u
1g1u 1u1g
+ + or -  -
1+g1+u 3-u3-g
Note
- The rule is applicable for the atomic number < 40.
- All of these selection rules can be derived by calculating the transition dipole element.

29. transitions forbidden
Ex. Which of the following electronic transitions are allowed or forbidden? 1)
1+u1-g
3) 3u3g
5) 1g1 -g 2)
1+u1g
4) 1-u3-g
1) 1+u1-g
transitions forbidden
 = 0, S = 0, u  g, +  -
2) 1+u1g
 = 2, S = 0, u  g
transitions forbidden
3) 3u3g
 = 1, S = 0, u  g
transitions allowed
transitions forbidden
4) 1-u3-g
 = 0, S = 2, u  g, -  -
transitions forbidden
5) 1g1 -g
 = 1, S = 0, g  g

30. X3-ga1g forbidden (S0) X3-gb 1+g forbidden (S0)
Ex. Using molecular term symbols of the ground and excited states of O2 to indicate which transition is allowed or forbidden.
X3-ga1g forbidden (S0)
X3-gb 1+g forbidden (S0)
X3-g A3+u forbidden (- +)
X3-g B3-u allowed
(the lowest allowed transition)

31. The X3-g B3-u transition is allowed
Note that the selection rule n = ±1 does not apply to vibrational transitions between different electronic states.
Absorption from the ground state into various vibrational levels of the B3-u excited states is possible (vibronic coupling).

32. Ex. The energy difference between the two lowest vibrational states in the electronic ground state (X3-g) and in the first excited state (B3-u) of O2 is cm-1. The observed spectral line of cm-1 corresponds to the transition of the ground vibrational state of X3-g to the n vibrational state of B3-u . Find the n value of this transition? Given the vibrational wavenumber for an allowed vibrational transition in B3-u is 700 cm-1 (Ignore any rotational structure or anharmonicity.)
Ignore
B3-u
X3-g
n'=?
n'=3
n'=2
n'=1
700 cm-1
n'=0
52,500 cm-1
49,000 cm-1
n=3
n=2
n=1
For transition between different electronic states, the selection rule for vibrational transitions n = ±1 does not need to consider.
n=0

34. Pure electronic transition vs vibrational-electronic transition
sometimes energy (a photon) can excite a molecule to an excited electronic and vibrational state

35. electrons (associated with the motion of electrons)
Vibrational Fine Structure of Electronic Transitions in Diatomic Molecules
Vibrational and rotational quantum numbers can change during electronic excitation.
Born-Oppenheimer approximation can be used to determine vibrational transition between electronic states.
Nuclei are much more massive than electrons
Wave functions
Born-Oppenheimer approximation
electrons (associated with the motion of electrons)
nuclei (associated with vibration of the molecule)

36. The total wave functions is a product of vibrational (nuclei) and electronic parts within the Born-Oppenheimer approximation
where R1,…,Rm depends on position of the nuclei
r1,…rn depends on the position of electrons
The spectral line of an electronic transition (initial final) has a measurable intensity if the transition of electric dipole moment is not zero:
the dipole moment operator is given by

37. For simplification, we consider the short notation :
For initial state:
For final state:
the dipole moment operator :

38. a measure of the expected intensity of an electronic transition
0 (orthogonal if)
Represents the overlap between the vibrational wave functions in the initial and final states.
Franck-Condon factor
a measure of the expected intensity of an electronic transition

39. If the overlap between the vibrational wave functions in the initial and final states is not zero (S  0),
the spectral line of the corresponding transition will be observed.

40. Franck-Condon principle
States that transitions between electronic states correspond to vertical lines on an energy versus inter-nuclear distance diagram.
Electronic transitions occur at a much faster rate than the nuclei’ motion (The atoms do not move during the transition).
The electronic transition occur at the initial state that have the ground (n=0) vibrational state (equilibrium bond distance).
Morse potential

41. Franck-Condon principle
1) Separation distance remains constant during electronic transitions
2) Later moves to new equilibrium position
Separation distance does
NOT change during transition
3) An electronic transition can go to any number of different vibrational levels in the excited electronic state depending on the energy.
No longer have selection rule for vibrations (n=1)

42. The Franck-Condon principle determine the n values in the excited state that give the most intense spectral lines.
The electronic transition will lift the highest populated molecules in the n=0 vibrational state, therefore the n excited state can be from the peak with the highest intensity.
nearly all of the molecules in the ground vibrational state

43. Ex. Consider the refined structure of UV/VIS absorption spectrum for a diatomic molecule and sketch the energy diagram with the corresponding state of electronic transition

44. UV-Visible Light Absorption in Polyatomic Molecules
Absorption lines in condense phases are usually board and obscure fine structure.
Polyatomic molecules exhibit many rotational and vibrational transitions. Because their spectral lines overlap, absorption lines are board and featureless.
atom
diatomic molecule
polyatomic molecule
This makes it difficult to extract information on the initial and final states involved in an electronic transition in polyatomic molecule.

45. Chromophores
a chemical entity embedded within a molecule that absorbs radiation at the same wavelength in different molecules.
Common chromophores: C=C, C=O, C=S, CN
Electronic excitation from HOMO to LUMO with the configurations : nπ*, π π*, and σσ*

46. The energy increases in the sequence nπ*, π π*, and σσ*.

47. Transitions Between Ground and Excited States
Types of transitions
1) Radiative Transitions: photons absorbed or emitted
1.1 Fluorescence: S = 0 singlet-singlet transition
1.2 Phosphorescence: S  0 singlet-triplet transition
2) Nonradiative Transitions: energy transfer between internal degrees of freedom of a molecule or to surroundings:
2.1 Internal Conversion: transition without a change in energy between states of the same multiplicity (S = 0, i.e. singlet-singlet transition)
2.2 Intersystem Crossing: transition without a change in energy between states of the different multiplicity (S  0, i.e. singlet-triplet)

48. Jablonski diagram IC Fluorescence: singlet-singlet transition (S = 0)
ISC
Phosphorescence
Fluorescence
Fluorescence: singlet-singlet transition (S = 0)
Phosphorescence: singlet-triplet transition (S  0)

49. Singlet-Singlet Transitions: Absorption and Fluorescence
Fluorescence is a radiative transition from the lowest vibrational state of excited states back to the ground state.
The fluorescence process involves:
Absorption from the lowest vibrational level of the ground state to the various vibrational levels of the excited (singlet state)
Internal conversion of energy (non-radiative): molecules in the excited vibrational levels of the excited state collide with other molecules. (Note: Non-radiative transitions occur much more rapidly compared to radiative transitions from excited vibrational levels of the excited state)

50. The fluorescence process involves:
3) Once in the ground vibrational level of the excited state, the molecule undergoes radiative transition to any vibrational level in the ground state

51. Intersystem Crossing and Phosphorescence
Singlet-triplet transition (S  0).
Although intersystem crossing between singlet and triplet electronic states is forbidden, the transition probability is enhanced by two factors:
- very similar molecular geometry in the excited singlet and triplet states,
- a strong spin-orbit coupling

53. Transitions Between Ground and Excited States
2.1
2.2 
1.1
1.2

54. Ex. Consider the transition from one electronic state to another, their bond lengths being Re and R’e and their force constants unchanged. Calculate the Franck–Condon factor for the 0–0 transition and show that the transition is most intense when the bond lengths are equal.
We need to calculate S(0,0), the overlap integral of the two ground-state vibrational wavefunctions,
We use

55. Singlet-Singlet Transitions:
Transitions Between Ground and Excited States
For transitions from ground singlet states to excited singlet or triplet states,
three types of transitions are possible:
Radiative Transitions: photons absorbed or emitted (Fluorescence)
Nonradiative Transitions: energy transfer between internal degrees of freedom of a molecule or to surrondings.
Intersystem Crossing (singlet-triplet). (Phosphorescence)

58. From the previous slide
Rearranging the differential equation separating the θ-dependent terms from the -dependent terms:
Only 
Only 
the Schrödinger equation can be solved using separation of variables.

59. Use the substitution method (similar to the previous one)
For the J=0 → J=2 transition,
Consider
Use the substitution method (similar to the previous one)
Replace x with  and integrate from 0 to , we get:
Do the same for

60. From the previous derivation:
For the J=0 → J=2 transition,
Thus:
From the previous derivation:
Therefore:
Thus, the J=0 → J=2 transition is forbidden.

61. Home Work 2
1. Spectral line spacing of rotational microwave spectrum of OH radical is 37.8 cm-1 . Determine the OH bond length (in pm unit) and moment of inertia (in kg m2)
mO = amu
Spectrum (cm-1)
mH = amu
37.8 cm-1
2. Use the bond length of diatomic molecules in Table to predict line spacing (in cm-1 unit) of rotational microwave spectrum.
molecule
bond length (pm) HF
91.7 HI
161
HCl
128
HBr
141

62. 3. Determine the bond length of these diatomic gases in Table and arrange them in order of increasing the bond length.
Bond length (pm) OH
37.80
ICl
0.11
ClF
1.03
AlH
12.60
4. Using the information in the Table to calculate the ratio between the transition energy of rotation from J=0 to J =1 and vibration from n=0 to n=1 for H2
Atomic mass
1.008 amu
Bond length of H2
74.14 pm
Force constant of H2
575 N/m