On diabatic representation

The nuclear derivative couplings emerge from a connection matrix for the orthogonal electronic basis states h/), which are generated along a trajectory that is characterized in terms of an adiabatic parameter vector R. Suppose that R depends smoothly on time t and the electron basis set is determined at each point in R-space, namely, R(t). The wavefunction 

It should obey the Schrodinger equation with a system Hamiltonian operator, which also depends on R(t), as shown below 

Multiplication of ( h/(R(t)) in this equation of motion from the left side yields 

We first consider a formal equation for such a transformation. (See Ref. [27, 28] for an extensive review of the transformation by Baer.) We start with an adiabatic basis set 4 /(r R), each term of which is associated with the eigen-energy Ej. If we expand the total wavefunction in 

The diabatic basis Ta(r R) is such a basis set that eliminates the derivative coupling (only locally). If we write the transformation matrix U, such that J2j jUja = Fa, then the associated nuclear wavefunctions Ca are related by xj = For an X in the adiabatic representation, a 

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