Coupled dynamics in electronic excitation

The time evolution of a molecular system T (r, R, t) under (possibly several) external fields is given by the Schrodinger equation, 

The total wavefunction of our system, T(r,i2, t), may be decomposed as in the usual Born-Huang representation into several neutral electronic states 4 n(r R), 

employing the dipole form of the external field interaction [Eq. (3.43)], the Schrodinger equation for the nuclear wavefunctions in the diabatic representation is given in the matrix form. 

Chemical Theory Beyond the Born-Oppenheimer Paradigm 

Within the split-operator scheme, for example, numerical short-time propagation would involve diagonalizing the interaction matrix (either together with the potential energy matrix or as a separate term) at each time step. We will see several other time-propagation methods in Sec. 5.2.1. 

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