1. MODULE 15 (701) Selection Rules for Electronic Transitions We start with transitions between atomic states, then generalize to molecular states. Selection rules for electric dipole transitions may be defined as: “a set of conditions that apply to the quantum numbers of the eigenfunctions of the initial and final states. If a pair of eigenfunctions possesses quantum numbers that do not conform to the conditions, then the matrix element of the electric dipole moment becomes zero.” We can use a symmetry argument to elaborate this.

2. MODULE 15 (701) We first revisit parity. We met parity earlier when we looked at the symmetry properties of the two orbitals that we obtained for the hydrogen molecule- ion, 1  g and 2  u, g = “gerade” and u = “ungerade”. These characteristics were obtained by performing a parity operation on the orbitals, which means looking at the sign of the orbital when the coordinates of a point are inverted through the center of symmetry. In a Cartesian system, an eigenfunction having even parity would satisfy the equality

3. MODULE 15 (701) An orbital with odd parity satisfies the equality EVENODD All eigenfunctions that are bound-state solutions to time- independent Schrödinger equations for a potential that can be written as V(r) have a definite parity, either g or u.

4. MODULE 15 (701) Such eigenfunctions are for systems that are constrained by a centro-symmetric Coulomb potential. Examples are hydrogenic atoms, homonuclear diatomic molecules, and some polyatomics. These all exhibit parity. Let us examine this for the hydrogenic (one-electron) atom wavefunctions we have worked with earlier. To see this we need to transform our coordinates into spherical polar (r, ,  ) 

5. MODULE 15 (701) When we do this If we apply these parity transformations to some of the hydrogenic wavefunctions we find that

6. MODULE 15 (701) We see that the OAM quantum number, l determines the parity of the wavefunction. If l is even, the parity is even. If l odd, the parity is odd (the sign of the wavefunction changes). What about the parity of the dipole moment operator in the matrix element? The position vector (r) changes into its negative when the signs of the Cartesian coordinates are changed. Therefore the parity of r, and hence is odd.

7. MODULE 15 (701) Recall the transition moment dipole: Since the operator is odd, the whole integrand will be odd if the two wavefunctions have the same parity (both even; both odd). Then the integral will vanish and the transition rate will be zero. Considering the schematic electric dipole transition: For it to be effective, the two states involved must be of different parity.

8. MODULE 15 (701) According to our analysis of the behavior of the hydrogenic wavefunctions the parities of wavefunctions are determined by the factor (-1) l. Thus l must change by 1 for the transition to have a non-zero rate (to be allowed). This generates one part of the Laporte selection rule “For a transition to be allowed,  l = ±1.”  l = 0, or  l = ±2 changes do not change the parity and are therefore not allowed.

9. MODULE 15 (701) Theory states that photons emitted in electric dipole transitions have angular momentum of 1 in atomic units (ħ). Thus to conserve angular momentum during an electric dipole transition, the total angular momentum of the atom must change according to the rule  j = 0, ±1. The  j = 0 case is understood by allowing for a change in the orientation in space of the total angular momentum vector when the transition occurs. The  j selection rule forbids transitions  l = ±3, ±5, (acceptable by parity) which would lead to too large changes in total AM.

10. MODULE 15 (701) These rules forbid some ED transitions that have favorable energy correspondence. Formally forbidden transitions can have non-zero rates (10 4 to 10 6 reduction) because of the second order influences of temporal fluctuations in magnetic dipoles/electric quadrupoles Electric dipole transitions between states of different multiplicity are forbidden because spin is not conserved during such events. This is because the perturbing influence is an electric field, which has no influence on magnetic moments due to electron spin.

11. MODULE 15 (701) The selection rules for diatomic and larger molecules bear some similarities to those for atomic species, but complications arise due to nuclear rotation and nuclear spin. The techniques of Group Theory are usefully employed in determined the selection rules for diatomics and higher. Where parity is definable (molecules with inversion symmetry) g- to-g and u-to-u transitions are forbidden, as are singlet-to-triplet and similar intersystem crossing processes. Polyatomic organic and organometallic molecules are discussed in terms of their light-absorbing residues (chromophores). In such molecules the transitions (absorption and emission) can be identified as arising from a particular grouping of atoms.

12. MODULE 15 (701) Examples of chromophores are carbonyl, nitro, phenyl, naphthyl, The wavelength regions in which transitions occur are typical for the chromophores, irrespective of the molecular environment. The MOs of such systems have designations such as HOMO, HOMO-1, LUMO, LUMO+1,... Labels such as , , , n,  *,  *  *, … occur which inform about the bonding/anti-bonding nature of the orbital and about its electron density distribution. The lowest energy (longest wavelength) transition is HOMO-to- LUMO, but which orbital type is HOMO and which is LUMO can vary from compound to compound, and solvent to solvent.

13. MODULE 15 (701) The carbonyl chromophore is a good example. The plots show the results of a HF 6-31G** calculation for H 2 CO. LUMO HOMO-1 HOMO n   The HOMO-to-LUMO transition is  *<-n, and the HOMO-1 to LUMO is  *<- . The  *<-n (or n,  *) transition that occurs in simple carbonyls near 290 nm thus involves the transfer of electron density from the O atom to the C atom (the  * orbital spreads over both atoms whereas the n-orbital is localized on O).

14. MODULE 15 (701) The HOMO-1 to LUMO absorption transition will occur at shorter wavelengths (higher energies) than the above, and its character will be  *<-  (or  This affects the electron density at C and O much less, and the state generated by this transition will be less polar than that generated by the n,  * transition.

15. MODULE 15 (701) This electron density redistribution during  *<-n transitions causes interesting solvent effects. In H-bonding and polar solvents the lone pair on O will either be involved in H-bonding, or will induce specific solvent orientations. These interactions stabilize the ground state (lower energy). During a transition the electron redistribution occurs much faster than nuclei can respond, thus the instantaneous excited state is in an unfavorable solvent environment. This raises the transition energy. Such influences are not effective in the  transition, and the energy is hardly influenced by the nature of the solvent. The solvent-induced shift in the transition energy can be large enough to cause a change in the state ordering, i.e. the  can become the lowest energy transition.

16. MODULE 15 (701) Another major difference between these two types of transitions is that  is allowed while the  n  is forbidden for symmetry reasons. To a good approximation the n-orbital confined on the O atom is O2p y (the z-direction by convention is along the bond). The  * orbital has a node in the yz plane and it can be approximately described as an LCAO-MO of the type:. The n (2p) and  * orbitals are orthogonally disposed and the integral in the matrix element will vanish. The transition dipole moment is thus zero and the transition is forbidden.

17. MODULE 15 (701) Forbidden transitions are rarely completely forbidden. Magnetic dipole and electric quadrupole oscillations can assist, as can coupling between electronic and vibrational motions. The  transition is fully allowed since it involves orbitals of the same symmetry. Its transition dipole is directed along the inter-nuclear axis. As a result of this transition the  * orbital becomes occupied, the bond order is lowered and the bond weakens. The resulting  framework acquires torsional freedom. Thus the excited state of such species may be perpendicular, whereas the ground state was planar--photoisomerization.