Adiabatic eigenfunction

The effective nuclear kinetic energy operator due to the vector potential is formulated by multiplying the adiabatic eigenfunction of the system, t t(/ , r) with the HLH phase exp(i/2ai ctan(r/R)), and operating with T R,r), as defined in Eq. fl), on the product function and after little algebraic simplification, one can obtain the following effective kinetic energy operator. 

The Born-Oppenheimer approximation may then be thought of as keeping the electronic eigenfunctions independent and not allowing them to mix under the nuclear coordinates. This may be seen by expanding the total molecular wavefunction using the adiabatic eigenfunctions as a basis... 

Equation (19) is the sufficient condition for equation (18) to have a solution. The question to be asked is under what circumstances is equation (19) fulfilled. It can be shown that if the BO adiabatic eigenfunctions form a sub-Hilbert space equation (19) is satisfied [19,25]. In other words the diabatization can be carried out only for a group of states which form a sub-Hilbert space. 

Given that the total hamiltonian may be written as Hw = P2/2M + hw(R), the adiabatic eigenfunctions a R) are the solutions of the eigenvalue problem, hw(R) ot R) = Ea(R) a R). In this adiabatic basis the quantum-classical Liouville operator has matrix elements [12],... 

Here, the matrix of adiabatic eigenfunctions (p ) and the diagonal matrix of adiabatic potential energy curves e(p) are solutions of the eigenvalue problem... 

Instead of working in a stable subspace of (i.e. in a basis of lOp determinants), one may work in a basis of a rather limited number of configurations, those which play a major role in the adiabatic eigenfunctions of interest, and build an effective electronic Hamiltonian in this model space. For instance, for the curve crossing between the ionic and neutral configuration in NaCl, one may define as a model space the two leading configurations... 

If now the nuclear coordinates are regarded as dynamical variables, rather than parameters, then in the vicinity of the intersection point, the energy eigenfunction, which is a combined electronic-nuclear wave function, will contain a superposition of the two adiabatic, superposition states, with nuclear... 

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrddinger equation is then written... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

The fact that the electronic eigenfunctions aie modified as presented in Eq. (A.5) has a direct effect on the non-adiabatic coupling terms as introduced in Eqs. (8a) and (8b). In particular, we consider the term rJi (which for the case of real eigenfunctions is identically zero) for the case presented in Eq. (A.5) ... 

There would, however, be a certain probability, dependent on the nature of the eigenfunctions, that actual non-adiabatic dissociation would give ions rather than atoms, and this might be nearly unity, in case the two potential curves come very close to one another at some point. See I. v. Neumann and E. Wigner, Physik. Z., 30, 467 (1929). 

The time-dependent wave function, t /(r, R, t e(t)), can be expanded in terms of the field-dependent adiabatic electronic eigenfunctions of Eq. (25) ... 

The electronically adiabatic wave functions v(/f ad(r q ) are defined as eigenfunctions of the electronic Hamiltonian Hel with electronically adiabatic potential energies ad(qjJ as their eigenvalues ... 

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