Coupling of nuclear and electronic motion

The electronic contributions to the g factors arise in second-order perturbation theory from the perturbation of the electronic motion by the vibrational or rotational motion of the nuclei [19,26]. This non-adiabatic coupling of nuclear and electronic motion, which exemplifies a breakdown of the Born-Oppenheimer approximation, leads to a mixing of the electronic ground state with excited electronic states of appropriate symmetry. The electronic contribution to the vibrational g factor of a diatomic molecule is then given as a sum-over-excited-states expression... 

The coupling of nuclear and electronic motions in electronic transitions may then happen through Z //<(Q) if induced by light, or through Hjf) and Gw if induced by nuclear displacements. The nuclear motion functions can be obtained by additional expansion in a basis of functions of nuclear coordinates, or by numerical solutions on a grid of points in the space of nuclear positions. The second approach is specially suitable for non-stationary states, and is briefly described. 

Sutcliffe, B.T. The coupling of nuclear and electronic motions in molecules, J. Chem. Soc. Faraday Trans. 1993,89,2321-35. 

Born and Oppenheiraer s 1927 paper justifying the Born-Oppenheimer approximation is seriously lacking in rigor. Subsequent work has better justified the Bom-Oppenheimer approximation, but significant questions still remain the problem of the coupling of nuclear and electronic motions is, at the moment, without a sensible solution and... is an area where much future work can and must be done [B.T. Sutcliffe,/. Chem. Soc. Faraday Trans.,99,2321 (1993)]. 

So far, we have restricted ourselves to the situation that the nuclei are fixed in space, i.e. we have considered molecular properties or contributions to the molecular properties that can be obtained from the electronic Schrodinger equation, Eq. (2.10), alone. In this chapter, and the following chapter we will finally lift this restriction and allow the nuclei to move again. In this chapter, we will look at properties that arise or at least have contributions due to a breakdown of the Born-Oppenheimer approximation. This means that in order to derive quantmn mechanical expressions for these experimentally observable properties we have to take into account the coupling of nuclear and electronic motion, i.e. some of the terms that are neglected in the Born-Oppenheimer approximation. 

With tlie development of femtosecond laser teclmology it has become possible to observe in resonance energy transfer some apparent manifestations of tire coupling between nuclear and electronic motions. For example in photosyntlietic preparations such as light-harvesting antennae and reaction centres [32, 46, 47 and 49] such observations are believed to result eitlier from oscillations between tire coupled excitonic levels of dimers (generally multimers), or tire nuclear motions of tire cliromophores. This is a subject tliat is still very much open to debate, and for extensive discussion we refer tire reader for example to [46, 47, 50, 51 and 55]. A simplified view of tire subject can nonetlieless be obtained from tire following semiclassical picture. 

In this chapter we present in detail the separation of the nuclear and electronic motions, the nuclear motion within a molecule, and the coupling between nuclear and electronic motion. 

In discussing molecular systems which must be described in terms of more than one potential surface, it is desirable to have a clear definition of the variously used term crossing. It is also important to distinguish between (a) interaction of potential surfaces, and (b) transitions from one adiabatic surface to another induced by coupling between nuclear and electronic motions (failure of the Bom-Oppenheimer approximation). 

The general treatment of the theory of chemical reactions presented in this book is based on the usual adiabatic separation of nuclear and electronic motions which permits a definition of the potential energy as a function of internuclear distances This approach proves to be very useful for the study of electronically adiabatic reactions, provided a separation of the rotation of the reacting system, treated as a supermolecule, is possible. In general, such a separation seems to be a bad approximation /10/. A consideration of the coupling of the overall rotation with the internal motions of the system means taking into account the possibility of non-adiabatic transitions from one to another potential energy surface. This is still an unsolved problem of theoretical chemistry which is open for discussion. 

In the previous sections we have studied Born-Oppenheimer-breakdown corrections to two molecular properties, the rotational g tensor and the nuclear spin-rotation constant, i.e. the effect of the coupling between nuclear and electronic motion on the electronic energies. In this and the following sections we will now turn our attention to the effect of this coupling on the motion of the nuclei and will discuss Born-Oppenheimer-breakdown corrections to the rotational and vibrational energies. For the sake of a simpler presentation we will illustrate it for a diatomic molecule AB, where there is only one vibrational mode that involves changes in the internuclear... 

Non-adiabatic coupling is also termed vibronic coupling as the resulting breakdown of the adiabatic picture is due to coupling between the nuclear and electronic motion. A well-known special case of vibronic coupling is the Jahn-Teller effect [14,164—168], in which a symmetrical molecule in a doubly degenerate electronic state will spontaneously distort so as to break the symmetry and remove the degeneracy. 

These are the equations on which a number of approximations are carried out to obtain approximate model solutions. In particular, the Bom-Oppenheimer (BO) frame allows for a useful separation between nuclear and electronic motion [49-52]. See also Park s book where some interesting elementary examples are analyzed concerning electro-nuclear coupling effects[53]. 

If the Born-Oppenheimer approximation is not valid—for example, in the vicinity of surface crossings—nonadiabatic coupling effects (that couple nuclear and electronic motion) need to be taken in account to correctly describe the motion of the molecular system. This is done, for instance, when one needs to describe a jump between two different PESs. In this case, one uses semiclassi-cal theories and the surface-hopping method, which we discuss subsequently. We now discuss in some detail how the region in which nonadiabatic effects become important can be characterized topologically. 

In the non-CT radiationless transition the change in electronic charge interacts with the nuclei in a similar maimer both before and after the transition. Two types of processes can be identified internal conversion processes in which the transition is between spin states of the same multiplicity and intersystem crossing process in which the transition is between states of different spin multiplicity. For non-CT internal conversion processes the full BO (Bom—Oppenheimer) adiabatic wave-functions for the supramolecular complex are used as the zero-order basis [42-44]. The perturbations that cause the transition are the vibronic coupling between the nuclear and electron motions. These are just the terms that are neglected in the BO approximation [45]. The terms are expanded (normally to first order) in the normal vibrational coordinates of the nuclei as is customarily done for optical vibronic transitions. Thus one obtains Eq. 61b for cases when only one normal mode couples the two states... 

---