Is it something to do with breakdown of the Born-Oppenheimer approximation?
In molecular spectroscopy you occasionally hear this term thrown around. Google scholar yields more than 3000 hits. But I have found its precise meaning and the relevant physics hard to pin down. Quantum mechanics in chemistry by Schatz and Ratner is an excellent book, but the discussion on page 204 did not help me. "Herzberg-Teller" never appears in Atkins' Molecular quantum mechanics.
So here is my limited understanding.
Herzberg and Teller wanted to understand why one observed certain vibronic (combined electronic and vibrational) transitions that were not expected, particularly some that were expected to be forbidden on symmetry grounds. "Intensity borrowing" occurred.
Herzberg and Teller pointed out that his could be understood if the dipole transition moment for the electronic transition depended on the nuclear co-ordinate associated with the vibration. In the Franck-Condon approximation one assumes that there is no such dependence.
There is a nice clear discussion of this in Section 2.2 of
Spectroscopic effects of conical intersections of molecular potential energy surfaces
by Domcke, Koppel, and Cederbaum
They start with a simple Hamiltonian involving two diabatic states coupled to two vibrational modes. The diabatic states, by definition, do not depend on the nuclear co-ordinates.
They show how in the adiabatic approximation [which I would equate with Born-Oppenheimer] one neglects the nuclear kinetic energy operator and diagonalises the Hamiltonian to produce adiabatic states. But, the diagonalisation matrix depends on the nuclear co-ordinates. Hence, the adiabatic eigenstates depend on the nuclear co-ordinates. In the crude adiabatic approximation one ignores this dependence.
The photoelectron and optical absorption spectra depend on calculated the dipole transition
elements between electronic eigenstates. These depend on the nuclear co-ordinates via the diagonalisation matrix. In Franck-Condon (FC) one ignores this dependence. This dependence is the origin of the Herzberg-Teller coupling.
The figure below, taken from the paper, shows spectra for a model calculation for the butatriene cation. The curves from top to bottom are for Franck-Condon approximation, adiabatic approximation, and the exact result. (Note: the vertical scales are different). Comparing the top two curves on can clearly see intensity borrowing for the high energy transitions. Comparing to the bottom curve shows the importance of non-adiabatic effects; these are amplified by the presence of a conical intersection in the model.