The time dependent Schroeodinger equation (TDSE) reads
where H is the complete electronic Hamiltonian:
where Z are the charges of the nuclei A, B
Then the wavefunction is expanded over a suitable basis set of functions
The choice of the set of basis functions is the main task of the physicist approaching the problem. The set must be small enough to allow numerical solution with reasonable computational effort, while it must be large enough to represent the wave function accurately, at least in the region of space important for the studied processes.
Multiplying of complex conjugate state, and evaluating the matrix elements, we obtained a general set of coupled equations for the expansion coefficients.
This equation is valid generally, but simplifies for special cases.
The general coupling matrix contains two parts:
For a fixed (in time) base, the second part vanishes. For the molecular orbitals, or the instantaneous eigenstates basis set, the first term vanishes.