Non-adiabatic terms

It will be recalled that our use of the Bom adiabatic approximation in section 2.6 enabled us to separate the nuclear and electronic parts of the total wave function. This separation led to wave equations for the rotational and vibrational motions of the nuclei. We now briefly reconsider this approximation, with the promise that we shall study it at greater length in chapters 6 and 7. 

We could return to the exact equation (2.131) and examine the matrix elements of the Cn, n terms in the same manner as we dealt with the diagonal f,, terms. It is, however, easier to turn to the form of the exact Hamiltonian given in equations (2.113) and (2.120). The terms in this Hamiltonian which cause a breakdown of the adiabatic separation of nuclear and electronic motion are 

This term has matrix elements off-diagonal in PI, i.e. between different electronic states. 

We shall show in chapter 7 that these matrix elements are of the form 

It is important to note that the Hamiltonian (2.120) contains the terms which produce both the adiabatic and non-adiabatic effects. In chapter 7 we shall show how the total Hamiltonian can be reduced to an effective Hamiltonian which operates only in the rotational subspace of a single vibronic state, the non-adiabatic effects being treated by perturbation theory and incorporated into the molecular parameters which define the effective Hamiltonian. Almost for the first time in this book, this introduces an extremely important concept and tool, outlined in chapter 1, the effective Hamiltonian. Observed spectra are analysed in terms of an appropriate effective Hamiltonian, and this process leads to the determination of the values of what are best called molecular parameters . An alternative terminology of molecular constants , often used, seems less appropriate. The quantitative interpretation of the molecular parameters is the link between experiment and electronic structure. 

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As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

In this section, the adiabatic picture will be extended to include the non-adiabatic terms that couple the states. After this has been done, a diabatic picture will be developed that enables the basic topology of the coupled surfaces to be investigated. Of particular interest are the intersection regions, which may form what are called conical intersections. These are a multimode phenomena, that is, they do not occur in ID systems, and the name comes from their shape— in a special 2D space it has the form of a double cone. Finally, a model Hamiltonian will be introduced that can describe the coupled surfaces. This enables a global description of the surfaces, and gives both insight and predictive power to the formation of conical intersections. More detailed review on conical intersections and their properties can be found in [1,14,65,176-178]. 

Two - particle energy correction correction to electron - electron correlation energy due to the phonon field. This non-adiabatic term represents full attractive contribution, and can be compared to the reduced form of Frohlich effective Hamiltonian which maximizes attractive contribution of electron - electron interaction and that can be either attractive or repulsive (interaction term of the BCS theory). For superconducting state transition at the non-adiabatic conditions, the two-particle correction is unimportant - see [2],... 

The right-hand side of (11.109) contains terms off-diagonal in the electronic state (i.e. the non-adiabatic terms). If the right-hand side is set equal to zero, we obtain the adiabatic eigenvalue problem,... 

Recently though, Launay and LeDorneuf and Rbmelt have shown that if one uses the radial Delves coordinate system and solves the diagonal part of the Schrbdinger equation including all Miagonal non-adiabatic terms, then one finds quantitative agreement with exact quantal computations. ... 

If we neglect the so-called non-adiabatic terms on the right-hand side, we arrive at an eigenvalue equation. 

Equation (2.13) constitutes the Born-Oppenheimer approximation. When the electronic adiabatic wavefunctions are chosen real, the non-adiabatic term simply reads A = G = ( 6 T ). Usually, A is very small and Eq.(2.13) is only used if very high accuracy is sought. Neglecting A leads to the so-called adiabatic approximation... 

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