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It is shown that to obtain the (2n)th and (2n+1)th energy gradients, it is only necessary to solve equations of the difficulty of the nth order coupled perturbed equations for the orbital and configuration interaction (CI) parameters. For example, to find analytic second and third energy derivatives for CI wave functions, it is only necessary to solve the first order coupled perturbed equations and some related equations, for the effects of orbital rotations. Similar results apply for gradients of energies derived using perturbation theory.
Handy et al. (Sat,) studied this question.