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

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)

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) ,

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

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

 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 0 + 0 = 0 1 = 0 Singlet ML = 0 MS = 0 |ML | = 0 |MS| = 0  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

 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

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

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 +1 -1  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 

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

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

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

Because 2 e- are in * MO so we use the notation g Term symbol +1 -1 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

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

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.

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.

  - + - + + - - + “-” 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.

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

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

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 + 0 = 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

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  4 = 16 states

1u 1*g +1 -1 +1 -1

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

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

1u 1*g Term symbol +1 -1  msi  mli +1 -1

1u 1*g Term symbol +1 -1  msi  mli +1 -1

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

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.

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

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)

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).

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 49000 cm-1. The observed spectral line of 52500 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

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

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)

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

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

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

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.

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

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)

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

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

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.

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*

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

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)

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

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)

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

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

Transitions Between Ground and Excited States 2.1 2.2  1.1 1.2

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

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)

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.

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

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.

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 = 15.994 amu Spectrum (cm-1) mH = 1.008 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

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