quantum mechanics - Non Adiabatic Coupling Term in Born Oppenheimer Approximation

I am attaching a section from a text book (Conical Intersections Electronic Structure, dynamics and spectroscopy: David R Yarkony & Horst Koppel).

Here I am not understanding the so called 'Non Adiabatic Coupling Term"... From eqn (7.b), this term depends only on nuclear kinetic energy and nuclear coordinates. It doesn't involve anything from electronic coordinates!!! Then how come it is a coupling term? How this term describes the dynamical interaction between the electronic and nuclear motion? Also, why it is called non adiabatic?

Answer

Equation (7a) describes a system of differential equations: you have one such equation for each value of $j$, i.e. for each nuclear state $\chi_j$. And I don't like the way author(s?) write the first term on the left side, I would better write it as $[T_{n} + V_{nn}] \, \chi_j$.

Anyway, the point is that equations in the system (7a) are as we say coupled in a sense that solution of $j$-th equation ($\chi_j$) enters all other equations through this $\Lambda_{ji} \, \chi_i$ terms. That is why we call equations (7a) coupled channel equations and terms $\Lambda_{ji} \, \chi_i$ coupling terms.

Later to simplify the system we decouple the equations by introducing what is called the adiabatic approximation, in which we neglect all off-diagonal coupling terms. And thus, we call these terms non-adiabatic.

Why do we call the approximation in which we neglect $\Lambda_{ji} \, \chi_i$ for $j \neq i$ adiabatic? The term "adiabatic" from Ancient Greek αδιαβατος (α - "not", δια - "through", βατος - "passable") literally means the situation when something "is not passing through" something else. In thermodynamics adiabatic process is a process occurring without exchange of heat of a system with its environment, i.e. a process in which heat is not passing through system enclosure. In quantum mechanics adiabatic refers to a process in which no abrupt transition from one state to another with respect to continuous changes of some parameters happens and thus energy of a system changes continuously with respect to continuous changes of these parameters. And that exactly the picture we have here because if you take a look at (7b) then $\Lambda_{ji} \, \chi_i = 0$ for $j \neq i$ can be interpreted as follows: for some particular $i$-th nuclear state when varying nuclear coordinates no transition from corresponding $i$-th electronic state to any other $j$-th electronic states can happen.

P. S. You're reading quite advanced book on the subject. Since you do not know the meaning of the word "non-adiabatic", you would better start from something simpler.