Adiabatic basis

Setting the diabatic basis equal to the adiabatic basis at the degenerate point, Ro, the expansions can be written in vector notation as... 

A final point to be made concerns the symmetry of the molecular system. The branching space vectors in Eqs. (75) and (76) can be obtained by evaluating the derivatives of matrix elements in the adiabatic basis... 

If more than one electronic state is involved, then the electronic wave function is free to contain components from all states. For example, for non-adiabatic systems the elecbonic wave function can be expanded in the adiabatic basis set at the nuclear geometry R t)... 

Robb, Bemaidi, and Olivucci (RBO) [37] developed a method based on the idea that a conical intersection can be found if one moves in a plane defined by two vectors xi and X2, defined in the adiabatic basis of the molecular Hamiltonian H. The direction of Xi corresponds to the gradient difference... 

In Eq. (11b), we observed that since the crude adiabatic basis is used S = 0, for kzQ. Therefore the degeneracy is lifted at first order in the Q-space only, which is therefore used to identified the branching space. The first-order result is... 

At this stage, we would like to emphasize that the same transformation has to be applied to the electronic adiabatic basis set in order not to affect the total wave function of both the elecbons and the nuclei. Thus if is the electronic basis set that is attached to 4> then and are related to each other as... 

The components of the two vectors ( 1 i 2X when multiplied by the electronic (diabatic) basis set ( cj>i), 14b)), form the corresponding electronic adiabatic basis... 

As with STIRAP, we introduce an adiabatic basis [2, 3] for the Stokes vector. 

B. A. Hess Prof. Jungen, in your talk you emphasized that you don t have to calculate matrix elements of d/dQ or Coriolis coupling. My impression is that this is due to your most appropriate choice of a diabatic basis, which is generally what ab initio quantum chemists do when they want to avoid singularities in the adiabatic basis. On the other hand, the absence of explicit Coriolis coupling matrix elements is due to the transformation to a space-fixed coordinate system. 

M. Lombardi What is not needed is the validity of the adiabatic approximation, that is, that there is no transition between adiabatic states. But the geometric phase is defined by following states along a path in parameter space (here nuclear coordinates) with some continuity condition. In the diabatic representation, there is no change of basis at all and thus the geometric phase is identically zero. Do not confuse adiabatic basis (which is required) and adiabatic approximation (which may not be valid). 

Figure 18 Occupancy of the states (i.e., projection on the adiabatic basis states of aft) given in Eq. [22]) for a molecular system wavefunction on the Sj excited (VF1) or S0 ground (Tq ) states.

In contrast to the subsystem representation, the adiabatic basis depends on the environmental coordinates. As such, one obtains a physically intuitive description in terms of classical trajectories along Born-Oppenheimer surfaces. A variety of systems have been studied using QCL dynamics in this basis. These include the reaction rate and the kinetic isotope effect of proton transfer in a polar condensed phase solvent and a cluster [29-33], vibrational energy relaxation of a hydrogen bonded complex in a polar liquid [34], photodissociation of F2 [35], dynamical analysis of vibrational frequency shifts in a Xe fluid [36], and the spin-boson model [37,38], which is of particular importance as exact quantum results are available for comparison. 

Shi and Geva [15] have also derived the QCLE in the adiabatic basis starting from the full path integral expression for the quantum mechanical problem. In this representation the derivation starts with the partial Wigner transform of the environmental degrees of freedom in contrast to what is done... 

When the quantum-classical Liouville equation is expressed in the adiabatic basis, the most difficult terms to simulate come from the off-diagonal force matrix elements, which give rise to the nonadiabatic coupling matrix elements. As described above, contributions coming from this term were computed using the momentum-jump approximation in the context of a surface-hopping scheme. 

The surface-hopping trajectories obtained in the adiabatic representation of the QCLE contain nonadiabatic transitions between potential surfaces including both single adiabatic potential surfaces and the mean of two adiabatic surfaces. This picture is qualitatively different from surface-hopping schemes [2,56] which make the ansatz that classical coordinates follow some trajectory, R(t), while the quantum subsystem wave function, expanded in the adiabatic basis, is evolved according to the time dependent Schrodinger equation. The potential surfaces that the classical trajectories evolve along correspond to one of the adiabatic surfaces used in the expansion of the subsystem wavefunction, while the subsystem evolution is carried out coherently and may develop into linear combinations of these states. In such schemes, the environment does not experience the force associated with the true quantum state of the subsystem and decoherence by the environment is not automatically taken into account. Nonetheless, these methods have provided com-... 

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],... 

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