Off-diagonal coupling

Tbe model is that tbe ground-state PES is first altered by the electronic excitations (on-diagonal coupling leads to a change in equilibrium geomeby and frequency), and these smooth diabatic states are then further altered by vibronic (off-diagonal) coupling. 

Fig. 3. McConnell s idea of the effective 2x2 Hamiltonian is based on matching the two lowest eigenvalues of the n x n state space with an effective 2 x 2 matrix. Note that the true nxn Hamiltonian can have zero direct off-diagonal coupling between reactant and product, but non-zero values in the shaded off-diagonal region. The reactant and product states may be considered trap states embedded in the higher energy medium, and the resulting lowest eigenvalues (must) correspond principally to linear combinations of the trap states...

A one-level system e) that can exchange its population with the bath states [/) represents the case of autoionization or photoionization. However, the above Hamiltonian describes also a qubit, which can undergo transitions between the excited and ground states e) and g), respectively, due to its off-diagonal coupling to the bath. The bath may consist of quantum oscillators (modes) or two-level systems (spins) with different eigenfrequencies. Typical examples are spontaneous emission into photon or phonon continua. In the RWA, which is alleviated in Section 4.4, the present formalism applies to a relaxing qubit, under the substitutions... 

This coupling means that there is no population decay to the bath and the decoherence is only via the phase. In other pioneering works [39-41], off-diagonal coupling to a Lorentzian bath was considered within the RWA. The Hamiltonians in all of these works should be contrasted with the more general Hamiltonians (4.18), which account for both dephasing and decay into arbitrary baths, without the RWA. 

By neglecting the off-diagonal coupling potentials in the coupled equations [Eq. (95)], we obtain a single-channel approximation... 

This Hamiltonian is similar to the usual electron-phonon Hamiltonian, but the vibrations are like localized phonons and q is an index labeling them, not the wave-vector. We include both diagonal coupling, which describes a change of the electrostatic energy with the distance between atoms, and the off-diagonal coupling, which describes the dependence of the matrix elements tap over the distance between atoms. 

This relationship scales the diagonal force constants by the full value of C, where i is the matrix index corresponding to F-, and the off-diagonal coupling constants... 

When the off-diagonal coupling, the last term in the left-hand-side of (28), can be neglected... 

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