1. Laser Molecular Spectroscopy CHE466 Fall 2009 David L. Cedeño, Ph.D. Illinois State University Department of Chemistry Electronic Spectroscopy

2. Electron Motion: Orbital and Spin Angular Momentum Consider one electron in the hydrogen atom. Depending on its energy, electron motion is restricted to certain regions of space in a probabilistic manner. The waveunction associated to such probability is called an orbital. The particular “position” of an electron is dependent on its orbital angular momentum, which is a quantized property (with quantum number l, which depends on principal quantum number n). Magnitude of orbital angular momentum: for l = 0, 1, 2, …(n-1) Since an electron is charged, then its motion also generate a magnetic momentum called the spin angular momentum. The spin angular momentum is also quantized. Magnitude of spin angular momentum: for s = ±1/2 Thus the total angular momentum of an electron is the sum of both orbital and spin angular momentum. Magnitude of total angular momentum: for j = l+s, l+s-1, …|l-s| Electronic Motion

3. Electron Motion: Orbital and Spin Angular Momentum For multiple electrons in an atom, the angular momentum is defined by the sum of the angular momentum of the individual electrons, but bear in mind that angular momentum are vectors. Example: Find the angular momentum of an excited state of Helium corresponding to electron configuration 1s 1 2p 1 The total orbital momentum L = l 1 + l 2, l 1 + l 2 – 1, …, |l 1 – l 2 | = 1 + 0 = 1 The total spin momentum S = s 1 + s 2, s 1 + s 2 – 1, …, |s 1 – s 2 | = ½ + ½, ½ - ½ = 1, 0 The total angular momentum J = L + S, L + S – 1, …, |L – S| = 2, 1, 0 for L = 1 and S = 1; and 1 for L = 1, S = 0. Electron Motion: Spectroscopic Term Symbols A term symbol is used to identify a given electronic state with a given electron configuration in terms of the angular momentum. For an atom it corresponds to: 2S+1 L J For the example above there are 4 electronic states associated to 1s 1 2p 1 configuration: 3 P 2, 3 P 1, 3 P 0 corresponding to the S = 1 (called triplet states) and 1 P 1 for that with S = 0 (a singlet state) Electronic Motion

4. Electron Motion: Angular Momentum and Energy Hund rules determine the energy ordering of the states of a given electron configuration. These state that for an atom: 1.If the states have the same orbital angular momentum (L), then the one with the greatest spin angular momentum (S) has the lowest energy 2.If two states have the same spin angular momentum (S), then the one with the largest orbital angular momentum (L) has the lowest energy 3.The ordering in terms of J for a given term depends on whether an electron shell is less or more than half-full. For a less than half-full, the one with the lowest J is the lowest energy, for more than half-full, the one with the highest J is the lowest energy. In the previous example: 3 P 0 < 3 P 1 < 3 P 2 < 1 P 1 Electronic Motion

5. Diatomic Molecules: Molecular Orbital Theory The calculation of the energy of a molecule using the Schrodinger equation is complicated by the fact that a molecule contains more than one electron that is somehow shared at different extents by a set of nuclei. In addition, the interaction between electrons is difficult to represent. These problems translate in our inability to solve the Schrodinger equation exactly for a molecule. Furthermore, suitable molecular wavefunctions are not easy to find. Molecular orbital theory is a successful approach to obtain the energy of molecular electronic states which utilizes the variational principle to obtain the energy of molecular wavefunctions (i.e. orbitals) that are built as linear combinations of atomic orbitals (i.e. hydrogen-like wavefunctions). Let us illustrate this methodology with the H 2 molecule. Each atom contributes one 1s orbitals to the molecular wavefunctions. As a rule of thumb, the number of molecular orbitals (MOs) obtained is equal to the number of atomic orbitals (AOs) combined. These AOs are combined algebraically: For H 2 :  =  c i (1s) i Therefore there will be two MOs resulting from the combination of 2 1s AOs. Electronic Motion in Molecules

6. Diatomic Molecules: Molecular Orbital Theory The variational principle is used to obtain the energy: in this case the expectation value of the energy is obtained on the basis of the optimization of the coefficients (c i ) in a way that the minimum possible energy is obtained. This process is usually known as energy minimization. The expectation value of the energy E is: Since AO’s are normalized and the Hamiltonian is a Hermitian operator then: And the quantity Electronic Motion in Molecules

7. Diatomic Molecules: Molecular Orbital Theory The expectation value of the energy E can be written as: Using the variational principle: It is possible to have a system of equations that allow us to obtain the energy and coefficients. For this example: Electronic Motion in Molecules

8. Diatomic Molecules: Molecular Orbital Theory This determinant yields a quadratic equation on E with solutions:  is called the resonance integral and has a negative value.  is the Coulomb integral. Thus, the state with E – = (  –  )/(1 – S) is higher in energy than the other one with energy E + = (  +  )/(1 + S). Note that For the hydrogen molecule a is equal to the energy of a hydrogen atom (  = E H ) The coefficients can be obtained from the equations (called secular) and the fact that: If we assume S = 0, the wavefunction of the MO with the lower energy has c 1 = c 2 = √2/2 and the wavefunction for the highest energy has c 1 = –c 2 = √2/2. Electronic Motion in Molecules

9. Diatomic Molecules: Molecular Orbital Theory The molecular orbital energy diagram for H 2 is shown below: Electronic Motion in Molecules  g (1s)  u *(1s) E   ––

10. The molecular orbital energy diagram for valence shell of N 2 is shown here. Electronic Motion in Molecules  g (2s)  u *(2s) E  u (2p)  g *(2p)  g (2p)  u *(2p)

11. Diatomic Molecules: Molecular Orbital Theory The order of MOs as a function of energy for Li 2, Be 2, C 2 and N 2 is:  g (2s) <  u *(2s) <  u (2p) <  g (2p) <  g *(2p) <  u *(2p) But this is changed for O 2, F 2 and Ne 2 :  g (2s) <  u *(2s) <  g (2p) <  u (2p) <  g *(2p) <  u *(2p) The bond order is related to the strength of the molecule and their relative stability: Electronic Motion in Molecules

12. Diatomic Molecules: States and Term Symbols Consider the ground state configuration of N 2 :  He 2  g (2s) 2  u *(2s) 2  u (2p) 4  g (2p) 2 The total spin (S) is zero, so it is a singlet (2S + 1 = 1). The total orbital angular momentum is also zero. In order to assign the term symbol we utilize the symmetry symbols of the HOMO via the cross product:  g x  g =  g. The term symbol is 1  g +. Consider the excited state:  He 2  g (2s) 2  u *(2s) 2  u (2p) 4  g (2p) 1  g *(2p) 1 :  g x  u =  u. However, there are two possible spin multiplicities 1 and 3 (for S = 0 and 1 respectively). There is also the possibility of adding the spin and orbital angular momentum, which carries out a split of the energy of the possible states for such an electron configuration. The sum of these angular momentum defines a total angular momentum J (called  ). For this configuration J = L + S, L + S -1, …, |L – S|. = 1. For the triplet state: J = 2, 1, and 0; for the singlet state: J = 1. The term symbols are: 1  u, 3  u,2, 3  u,1, 3  u,0. Thus four excited states are linked to the electron configuration above. Hund’s rule still follows. Electronic Motion in Molecules

13. Diatomic Molecules: States and Term Symbols Consider the following electron configuration of O 2 :  He 2  g (2s) 2  u *(2s) 2  g (2p) 2  u (2p) 4  u *(2p) 2 In this case there are two electrons in two degenerate MOs. The cross product is  g x  g =  g + +  g - +  g The spin multiplicity is singlet or triplet, however Pauli’s exclusion principle forbids two of the possible triplet state. The only allowed triplet state is the ground state (Hund’s rules) with a term symbol is 3  g -. The negative parity is a consequence of the antisymmetric nature of the electronic wavefuction. Pauli’s exclusion principle excludes the states with symbols: 3  g + and 3  g. The antisymmetric nature of the electronic wavefunction also excludes the singlet with term symbol 1  g -. This, there are three states for the configuration above: 3  g - (ground state), 1  g, and 1  g +. The 1  g state is 7882.39 cm -1 above the ground state, while the 1  g + is 13120.91 cm -1 above. Electronic Motion in Molecules g*g* g*g* 3g-3g- g*g* g*g* 1g+1g+ g*g* g*g* 1g1g

14. Diatomic Molecules: States and Term Symbols Another example, NO: Ground state  He 2  (2s) 2  *(2s) 2  (2p) 2  (2p) 4  *(2p) 1 In this case there is one unpaired electrons in two degenerate MOs. There is no need for cross product, the ground state has a  term symbol The total spin is ½, thus the state is a doublet. The total angular momentum (spin + orbital) could be (1 + ½) or (1 – ½), that is  = 3/2 and ½. The ground state of NO is the one with the lowest W, thus it has the symbol: 2  1/2. The other state ( 2  3/2 ) is only 119.73 cm -1 above the ground state, indicating a small coupling of the spin and orbital momentum. With such a small separation both states are populated at room temperature (see selection rules later). Electronic Motion in Molecules

15. Diatomic Molecules: Selection Rules Whether an electronic transition is allowed depends on the effect of changing the total angular momentum of the molecule as the induced dipole couples with the photons: 1.  = 0, ±1 is allowed 2.  S = 0 is allowed (exception to the rule in heavy diatomics like I 2, where  S = 1 is allowed) 3.  = 0, ±1 is allowed 4. For  to  transitions: + ↔ + and – ↔ – are allowed 5. For homonuclear diatomics: g ↔ u is allowed 6. For molecules containing heavy atoms: regarding the states with W = 0, 0 + ↔ 0 + and 0 – ↔ 0 – are allowed. Example: Would the transition from O 2 ground state ( 3  g - ) to its first excited state ( 1  g ) be allowed? No because  = 2 (  to  transition), and  S = 1. This is a doubly forbidden transition. Also it violates the inversion selection rule because g ↔ g. Example: Would the transition from I 2 ground state ( 1  g + ) to its second excited state ( 3  u ) be allowed? Yes because  = 1 (  to  transition) and  S = 1 is now allowed (strong spin-orbital coupling). The inversion rule is also allowed g ↔ u. The last rule could also be considered here, the complete term symbol for the excited state is 3   u +, thus 0 + ↔ 0 + also holds. Electronic Motion in Molecules

16. Diatomic Molecules: Vibrational coupling The following shows the visible spectrum of I 2 at a resolution of 0.02 nm. The transition corresponds to the ( 1  g + → 3  u ) transition. The bands observed are the result of changes in the vibrational motion that are concomitant to the changes in electronic motion. Overall there are three band progressions, with transitions originating at v” = 0, 1, and 2. Electronic Motion

17. Quantum mechanically, the probability is associated to: The square of the quantity is the Franck-Condon factor of the transition The Franck-Condon Principle The probability of a vibrational transition is dictated by the relative overlap of the potential energy surfaces and the overlap of the vibrational wavefunctions at a particular internuclear distance in the lower electronic state. In essence, the electronic transition occurs before the nuclei move (in other words, from the perspective of electron motion, nuclei are stationary). Electronic Motion

18. Franck-Condon Principle and Band Shape The effect of the electronic transition on the internuclear distance is important in terms of the probability of the vibrational transitions and band shapes. Electronic Motion

19. Diatomic Molecules: Molecular Constants and Dissociation Energies from Spectra It is possible to obtain the dissociation energy from the spectrum by carrying out a combination of differences within a given band progression (i.e. constant v” or v’). In this case the difference between two consecutive bands (with vibrational numbers v’ and v’+1) belonging to a given progression (say v” = 0) is plotted against the value of v’+1. It can be shown that such difference is: G(v’+1)-G(v’) =  e ’ – 2  e ’x e ’(v’+1) Similarly for two bands with vibrational number v” and v”+1 that belong to the same progression (same v’): G(v”)-G(v”+1) =  e ” – 2  e ”x e ”(v”+1) The dissociation energy relative to the v’= 0 (zpe), D 0, is the area under the line from v’+1 = 1 to the max v’+1 Electronic Motion in Molecules

20. Diatomic Molecules: Bond Dissociation Energies from Spectra Electronic Motion in Molecules

21. Polyatomic Molecules: Molecular Orbital Theory It is possible to extend the treatment of the LCAO and variational principle to any molecule. Indeed most computational software in the market does this to provide the energy of the orbitals and the contribution of atomic orbitals. Application: AH 2 molecules: This can be bent (C 2v ) or linear (D ∞h ). In any case the valence MOs are built from the combination of the 1s orbitals of the H atoms and the ns and np orbitals of the A atom. If the molecule were linear the MOs would be labeled according to the symmetry representation (using the greek symbols), but once the molecule is bent the MOs are named using the a 1, a 2, b 1 or b 2 symmetry symbols from the C 2v group. The MOs in both geometries are correlated via a Walsh diagram (see next page) Electronic Motion in Molecules

22. Example: BeH 2 is though to have a linear ground state, with 4 valence electrons, the electron configuration is (2  g ) 2 (1  u ) 2. Term symbol is 1  g +. The first excited state is bent with electron configuration (2a 1 ) 2 (1b 2 ) 1 (3a 1 ) 1. The possible term symbols are 3 B 2 and 1 B 2.

23. Electronic Motion in Molecules Example: Formaldehyde. A MO treatment yields the frontier MOs shown. The MOs are labeled according to the symmetry representation (C 2v ). The energy ordering is 1b 1 < 2b 2 < 2b 1. The ground state is 1 A 1 (b 2 x b 2 ). The lowest energy transition corresponds to a HOMO to LUMO electronic motion. This transition is common in carbonyl containing compounds and is called a n → p* transition. This transition results in two excited states (triplet and singlet) with the same term symbol: A 2, thus the 3 A 2 and 1 A 2 excited states. 1b 1 : HOMO-1 (a  MO) 2b 2 : HOMO (non bonding) 2b 1 : LUMO (non bonding) (A  * MO)

24. Electronic Motion in Molecules The Huckel MO Method For conjugated molecules involving valence MOs that originate from pure p AOs, the Huckel method provides a simplified, yet satisfactory modeling ot the MOs. The Huckel method considers a LCAO of p z orbitals from atoms that participate in the conjugation. The secular determinant is given by: The following approximations apply: 1.For any S mn, if m ≠ n, then S mn = 0, if m = n, then S nm = 1 2.Any H nn = a (Coulomb integral) 3.Any H mn = b (resonance integral) only if m and n are adjacent (i.e. atoms are bonded), while H mn = 0 for atoms that are not bonded.

25. Electronic Motion in Molecules Example: Benzene The secular determinant for benzene is: Solutions are x = ±2, ±1, ±1, or E 1 =  + 2 , E 2 = E 3 =  + , E 4 = E 5 =  – , E 6 =  – 2 .

26. Example: Benzene The MOs are represented by a LCAO:  i =  c i  i (p z ). Six pz AOs produce six MOs. Electronic Motion in Molecules Images courtesy of http://chemical-quantum-images.blogspot.com/2007/01/benzenes-mos.html  1 (a 2u )  2 (e 1g )  3 (e 1g )  5 (e 2u )  4 (e 2u )  6 (b 2g )

27. Selection Rules in Polyatomic Molecules Electronic motion is highly allowed in polyatomic molecules. The one rule that hold is the one concerning to the change in the spin angular momentum:  S = 0 (for low spin-orbital coupling) The selection rules of a polyatomic molecule depends on symmetry properties of the electronic wavefunctions. In terms of transition that depend on the coupling of light with the electric dipole moment (  ), the rules is: Г(  el ’) x Г(  ) x Г(  el ”) ⊇ A or Г(  el ’) x Г(  x ) x Г(  el ”) ⊇ A Г(  el ’) x Г(  y ) x Г(  el ”) ⊇ A Г(  el ’) x Г(  z ) x Г(  el ”) ⊇ A Which is then written as:Г(  el ’) x Г(  el ”) ⊇ Г(  x ) Г(  el ’) x Г(  el ”) ⊇ Г(  y ) Г(  el ’) x Г(  el ”) ⊇ Г(  z ) Electronic Motion in Molecules

28. Selection Rules in Polyatomic Molecules Example: The ground state of benzene has a term symbol 1 A 1g. Determine if the transition to the excited state 1 B 1u is allowed. Cross product: A 1g x B 1u = B 1u. For D 6h, Г(  z ) = A 2u and Г(  x,T y ) = E 1u. Since the cross product of the symmetry labels of the excited state does not contain any of the symmetry representations for translation, the transition is NOT allowed. Note that in order to have a transition from the ground state in benzene, the excited state must be a 1 A 2u, or 1 E 1u. Electronic Motion in Molecules

29. Diffuse Spectra and Photophysical Processes The UV-Vis spectra of polyatomic molecules (even in the gas phase) is mostly composed of very broad bands. The reason being that many vibrational (and rotational) transitions concur in a relatively small range of frequencies that correspond to the electronic transition. Electronic Motion in Molecules In cyclohexane solution, Du et al, Photochem. Photobiol. 1998, 68, 141 This spectrum of pyrene shows 3 distinctive bands, in which some vibrational progressions are evident. The ground state of pyrene (D 2h ) is 1 A g, with allowed transitions into 1 B 1u and 1 B 2u and 1 B 3u excited states.

30. Photophysical Processes Electronic Motion in Molecules Absorption IC Fluorescence Phosphorescence ISC 1 M 0 1 M 1 3 M 1 Absorption IC Fluorescence Phosphorescence ISC 1 M 0 1 M 1 3 M 1 Radiative Processes Fluorescence: Light emission  S = 0 Phosphorescence: Light emission  S =1 Non-Radiative Processes Internal Conversion: heat release via vibrational relaxation induced by collisions (  S = 0) Intersystem Crossing: heat release as a result of a spin-orbital change induced by internal or external factors (  S = 1)

31. Photophysical Processes Quantum Yields: In any process initiated by light, the quantum yield is the efficiency of the process per every 100 photons absorbed. For example, a 0.12 fluorescence quantum yield means that out of 100 photons absorbed by the compound, 12 of them are converted into emitted photons. The quantum yield of a photophysical process depends on the relative rates of competing processes. For example the fluorescence quantum yield is defined as: Where the k f, k ISC and k IC are the rate constants for fluorescence, intersystem crossing and internal conversion respectively. Note that the sum of these rate constants represent the rate constant for decay of the excited singlet state. The rates of photophysical processes and therefore the quantum yields are thus dependent on the structure of the compound and the molecular environment around it. Electronic Motion in Molecules

32. Photophysical Processes: Effects of excitation on molecular structure From N. Turro, Modern Molecular Photochemistry, 1 st Ed, 1978

33. Photophysical Processes: Chromophores and Orbital Configurations (Typical) From N. Turro, Modern Molecular Photochemistry, 1 st Ed, 1978

34. Photophysical Processes: Effect of Environmental Conditions From N. Turro, Modern Molecular Photochemistry, 1 st Ed, 1978

35. Photophysical Processes: Effect of Environmental Conditions From N. Turro, Modern Molecular Photochemistry, 1 st Ed, 1978

36. Photophysical Processes: Effect of Environmental Conditions From N. Turro, Modern Molecular Photochemistry, 1 st Ed, 1978