Nonadiabatic coupling second derivative

Lengsfield III, B.H. and Yarkony, D.R. (1986). On the evaluation of nonadiabatic coupling matrix elements using SA-MCSCF/CI wavefunctions and analytic gradient methods. Ill Second derivative terms, J. Chem. Phys. 84, 348-353. 

In this subsection the determination of the second derivative nonadiabatic coupling matrix elements is considered. Real-valued wave functions... 

The second derivative terms 3 2 of the nonadiabatic coupling are mostly much smaller than the first derivative terms and hence are usually believed to be negligible. However, their omission will lead to a non-hermitian Hamiltonian due to a non-hermitian coupling... 

The coupling terms Ay,- are made of two contributions, the nonadiabatic derivative couplings, Fji Q) = (Oy-(i, 2) V 0,-(r, Q)), which are vectors in the nuclear space, and the scalar nonadiabatic couphngs [second term on the right-hand side (RHS) of Eq. 8.7]. By differentiating Eq. 8.4 with respect to the normal coordinates Q, one easily obtains the following expression for the derivative couplings [17] ... 

As usual, the bracket notation ( ) implies integration over the electronic coordinates only, while F/j(R ) and G/j(R ) are the first- and second-order nonadiabatic matrix coupling elements, respectively. By using a derivation similar to that employed for the Hellmann-Feynman theorem, it may be shown that... 

---