MODULE 26 (701) RADIATIONLESS DEACTIVATION OF EXCITED STATES We have used terms such as "internal conversion" and "intersystem crossing" without thinking very much about what governs such processes. It is useful to think of radiative and non-radiative transitions as competing horizontal and vertical crossings between the zeroth vibrational energy level of an upper electronic state (e.g. S 1 ) to a set of closely spaced vibrational levels on a lower electronic surface (e.g. S 0 ) S1S1 S0S0

MODULE 25 (701) In radiationless transitions, the sum of vibrational and electronic energy is constant. In radiative transitions, the photon removes (in the case of emission) most of the energy from the system. The Fermi Golden Rule is applicable to radiative and non-radiative transitions between states. For radiative transitions the rate constant for the transition is We employed the BO approximation to separate out the electronic and nuclear wavefunctions

MODULE 25 (701) where   ’,  is the (approximately constant) electronic matrix element for the transition and S v’v is the overlap integral between the nuclear wavefunctions. The Franck-Condon factor was defined as The treatment of radiationless transitions is similar except that the transition dipole moment becomes replaced by a matrix element involving the nuclear kinetic energy operator.

MODULE 25 (701) A widely-used concept is that the rate of radiationless deactivations is determined by intramolecular interactions. The role of the solvent is restricted to that of a thermal bath, i.e. a source or sink of energy. In Siebrand’s theory the radiationless transition probability per unit time (the transition rate) is given by where  (E) is the density of vibrational states in the final electronic state.

MODULE 25 (701) In polyatomic molecules in fluid solutions there are a large number of states that are broadened by solvent interactions and they merge into a continuum. The hamiltonian inside the matrix element in the Siebrand equation can be separated into its electronic and vibrational components (Born-Oppenheimer) where J N is an operator for nuclear kinetic energy. The  quantities are the electronic wavefunctions of the final and initial states and the v quantities are the nuclear (vibrational) wavefunctions.

MODULE 25 (701) P =2  E /ħ is a density of states factor. C fi = J fi 2 is an electronic factor involving the wavefunctions of the initial and final electronic states. Thus radiationless transitions are subject to the same multiplicity selection rules as are radiative transitions. A radiative transition involves an electric dipole moment operator and occurs between vibronic states that differ in energy. A radiationless transition involves the nuclear kinetic energy operator and occurs between vibronic states of the same energy

MODULE 25 (701) The plot shows the rate constants for ISC for a series of condensed aromatic molecules. All the molecules have  orbitals only, and we see that the rate constants for the ISC process vary with energy gap.

MODULE 25 (701) The rate constants for IC in the same molecules are much larger because of spin allowedness. They show similar energy gap dependence. In general log k NR is inversely proportional to the energy gap between the participating states. This ENERGY GAP LAW results from differences in the overlap integral and their effect on the Franck-Condon factor. The magnitude of the overlap integral can be imagined by mentally superimposing the vibrational wavefunctions (or their squares -- probability distributions) of the initial and final states.

MODULE 25 (701) The red, brown, and blue nested parabolas represent three electronic envelopes. Consider the blue and red states. Their energy separation is large and the v = 0 energy of the blue state is close to the v = n energy level of the red state. The energy separation is large and there is very little overlap of the iso-energetic vibrational functions v blue = 0 and v red = n. Thus F is small and the radiationless rate constant is small.

MODULE 25 (701) Consider the red and the brown states. The electronic energy difference is smaller and the v’ = 0 level of the brown state is isoenergetic with the v = 3 vibrational level of the red curve (the ground state). Now the wavefunctions show effective overlap, and F is large, as is the rate constant for the radiationless transition.

MODULE 25 (701) Consider the case where the energy gap is large and the upper and lower states have very different geometries (Figure). Even though the energy gap is large, the fact that the upper state is displaced with respect to the lower means that the v’ = 0 wavefunction has significant overlap with v = n. Thus F is large, as is the radiationless rate constant.

MODULE 25 (701) In rigid molecules having nested curves the radiative process (fluorescence) is favored because the F-C factors are small and internal conversion and intersystem crossing are weak. In non-rigid molecules, significant geometry differences between the initial and final states favor ISC and IC and weaken the fluorescence. Siebrand obtained a functional form for F N C and N H are the number of carbon and hydrogen atoms respectively in the molecule and where E-E 0 is the energy of the transition, is the wave number of the vibration(s) in the final state.

MODULE 25 (701) Thus F (and hence the rate) decreases as (E-E 0 ) increases for both ISC and IC, in agreement with the empirical observation. In general So IC from upper states is more rapid than from S 1 and IC competes effectively with the radiative transitions from S 2 and S 3 (Kasha's rule). Thus, a molecule's fluorescence spectrum and lifetime are determined only by the S 1 state independently of which state is excited (Vavilov law).

MODULE 25 (701) Siebrand identified anharmonic C-H stretching modes as the dominant vibrations in the radiationless deactivation of upper states (hence the N C and N H terms). Thus, deuterium substitution should lower k ISC and k IC since the vibration wave number ( ) will drop from ca 3000 cm -1 to ca 2250 cm -1. compound benzene 17.6 benzene – d benzene – d benzene – d 6 3.7

MODULE 25 (701) The Presence of Hetero-atoms We have focused on IC and ISC for aromatic hydrocarbons (  *). Hetero-atoms (O, N, etc.) introduce complications. The carbonyls are archetypal and we have thought of molecules as being composed of a single electronic orbital configuration and specific multiplicity. These may be thought of as Zero Order states, e.g. pure n,  * and pure  *, and pure singlet or triplet. To achieve transitions between states, we must achieve mixing and processes that can do this include vibrations, collisions, spin- orbit couplings and electron-electron interactions.

MODULE 25 (701) Consider the n,  *   * transition in a ketone. To achieve this an electron in a  -orbital moves to the n-orbital (assuming that the  * electron remains unchanged). The switch of orbital location must be iso-energetic and for this to happen vibrational motion within the molecule causes the nuclei to reach a point where the potential energy curves intersect. The switch of the electron then causes the molecule to deactivate along the  surface. Collisions with solvent molecules can also bring about the necessary mixing. The most effective vibrations are those that are out of the molecular plane, because the n and  orbitals are orthogonal to each other.

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