  The analytic energy gradients in the atomic orbital representation have
recently been published (J. Chem. Phys. 146, 014102, 2017) within the framework
of the natural orbital functional theory (NOFT). We provide here an alternative
expression for them in terms of natural orbitals, and use it to derive the
analytic second-order energy derivatives with respect to nuclear displacements
in the NOFT. The computational burden is shifted to the calculation of
perturbed natural orbitals and occupancies, since a set of linear
coupled-perturbed equations obtained from the variational Euler equations must
be solved to attain the analytic Hessian at the perturbed geometry. The linear
response of both natural orbitals and occupation numbers to nuclear geometry
displacements need only specify the reconstruction of the second-order reduced
density matrix in terms of occupation numbers.
