Robinson-Frosch theory

The Robinson-Frosch theory considers a resonance interaction between an initial electronic state and an electronic final state with weak coupling between the two states. The rate process then becomes ... 

In their discussion of the rates of non-radiative energy transfer in molecules (Robinson and Frosch, 1%3) pointed out that large isotope effects may arise in solution where significant amounts of electronic energy are converted into vibrational modes in the system. This motivated some of the earlier investigations of lanthanides in H2O-D2O systems. More recently Siebrand (1%7) carried out an extensive investigation of the Franck-Condon factors that are part of the Robinson and Frosch theory and obtained the expression ... 

Robinson and Frosch<84,133> have developed a theory in which the molecular environment is considered to provide many energy levels which can be in near resonance with the excited molecules. The environment can also serve as a perturbation, coupling with the electronic system of the excited molecule and providing a means of energy dissipation. This perturbation can mix the excited states through spin-orbit interaction. Their expression for the intercombinational radiationless transition probability is... 

In Chapters 4 and 5 we made use of the theory of radiationless transitions developed by Robinson and Frosch.(7) In this theory the transition is considered to be due to a time-dependent intramolecular perturbation on non-stationary Bom-Oppenheimer states. Henry and Kasha(8) and Jortner and co-workers(9-12) have pointed out that the Bom-Oppenheimer (BO) approximation is only valid if the energy difference between the BO states is large relative to the vibronic matrix element connecting these states. When there are near-degenerate or degenerate zeroth-order vibronic states belonging to different configurations the BO approximation fails. 

Internal conversion refers to radiationless transition between states of the same multiplicity, whereas intersystem crossing refers to such transitions between states of different multiplicities. The difference between the electronic energies is vested as the vibrational energy of the lower state. In the liquid phase, the vibrational energy may be quickly degraded into heat by collision, and in any phase, the differential energy is shared in a polyatomic molecule among various modes of vibration. The theory of radiationless transitions developed by Robinson and Frosch (1963) stresses the Franck-Condon factor. Jortner et al. (1969) have extensively reviewed the situation from the photochemical viewpoint. 

The interest aroused by the field of radiationless transitions in recent years has been enormous, and several reviews have been published 72-74) Basically, the ideas of Robinson and Frosch 75) who used the concepts on non-stationary molecular states and time-dependent perturbation theory to calculate the rate of transitions between Born-Oppenheimer states, are still valid, although they have been extended and refined. The nuclear kinetic energy leads to an interaction between different Born-Oppenheimer states and the rate of radiationless transitions is given by... 

Various theories have been proposed for horizontal transfer at the isoenergetic point. Gouterman considered a condensed system and tried to explain it in the same way as the radiative mechanism. In the radiative transfer, the two energy states are coupled by the photon or the radiation field. In the nonradiative transfer, the coupling is brought about by the phonon field of the crystalline matrix. But this theory is inconsistent with the observation that internal conversion occurs also in individual polyatomic molecules such as benzene. In such cases the medium does not actively participate except as a heat sink. This was taken into consideration in theories proposed by Robinson and Frosch, and Siebrand and has been further improved by Bixon and Jortner for isolated molecules, but the subject is still imperfectly understood. 

These three factors are included in the expression derived by Robinson and Frosch from time-dependent perturbation theory, for the nonradiative energy transfer or radiationless transition probability kNR per unit time,... 

The theory of radiationless transition considers the transition to occur in two steps (i) horizontal transition from one energy state to the other at the isocnergetic point, for the two combining states and (ii) vibrational relaxation of the lower energy state. The step (i) is the rate determining step. The rate constant is given by the theory of Robinson and Frosch as,... 

Because the analysis we have presented is based on the concept of compound states of the molecular system it is more general than the theory, based on the use of zero-order states, proposed by Robinson and Frosch, Lin and Bersohn, and Siebrand. The particular advantages of our approach are the following ... 

As for the previous two theories, we are interested in W(t)/t — k as a function of the energy separation Eai — Es. Robinson and Frosch have empirically correlated the square of the overlap integral [as in eq. (7)] with the energy separation for a singlet-triplet transition. From experimentally determined rate constants and an arbitrary but reasonable value for pel (0.01 cm-1) an expression relating < 0>2>tal to Eti ESo was deduced,... 

Robinson, G.W., Frosch, R.P. (1962), Theory of Electronic Energy Relaxation in the Solid Phase. J. Client. Pliy.s. 37, 1962. 

Theory of electronic energy relaxation in the solid phas. J. Chem. Phys., 37, 1962 Robinson, G.W. and Frosch, R.P. (1963) Electronic excitation transfer and relaxation. J. Chem. Phys., 38, 1187. 

---