and |\chi_k( \mathbf{r\mid q} ) \rangle;\ k= 1,M,are the electronic adiabatic eigenfunctions which depend explicitly on the electronic coordinates \mathbf{r} and parametrically on the nuclear coordinates \mathbf{q} .
To derive the matrix \mathbf{A} one has to solve the above given first order differential equation along a specified contour \Gamma .