### Abstract

In this work, the derivatives of molecular potential energy surfaces*V*({**R**}) with respect to nuclear coordinates **R** ^{ K } are related to derivatives of the electronic charge density with respect to applied electric fields. New equations are obtained for second, third, and fourth derivatives of *V*({**R**}) in terms of the charge density, the nonlocal polarizability density α(**r**,**r**′), and the hyperpolarizability densities β(**r**,**r**′,**r**″) and γ(**r**,**r**′,**r**″,**r**‴). In general, the *n*th derivative of the potential *V*({**R**}) depends on electrical susceptibility densities through (*n*−1)st order. The results hold for arbitrary nuclear coordinates {**R**}, not restricted to the equilibrium configuration {**R** _{ e }}. Specialization to {**R** _{ e }} leads to a new result for harmonic frequencies in terms of α(**r**,**r**′), and to new results for vibration–rotation coupling constants and anharmonicities in terms of α(**r**,**r**′), β(**r**,**r**′,**r**″) and higher‐order hyperpolarizability densities. This work provides a simple physical interpretation for force derivatives obtained by use of analytic energy differentiation techniques in *ab* *initio* work, or in density functional theory: The charge reorganization terms in harmonic force constants give the electronic induction energy in the change of field δ**F** due to an infinitesimal shift in nuclear positions.

Cubic anharmonicity constants depend on the hyperpolarization energy of the electrons in the field δ**F**, on the induction energy bilinear in δ**F** and the second variation of the field δ^{2} **F**, and on the gradients of the field from the unperturbed charge distribution. The results are derived by use of the Hohenberg–Kohn theorem*or* the electrostatic Hellmann–Feynman theorem, together with a chain of relations that connects the derivative of an electrical property of order *n* to the susceptibility density of order *n*+1. These derivatives are taken *with* *respect* *to* *the* *nuclear* *coordinates* **R** ^{ K }, in contrast to the well known relations for derivatives with respect to an applied electric field. Analytic expressions are compared for the property derivatives that depend on susceptibility densities through γ(**r**,**r**′,**r**″,**r**‴). This includes the derivatives of *V*({**R**}) listed above; first, second, and third derivatives of the dipole moment; first and second derivatives of the polarizability; and the first derivative of the β hyperpolarizability with respect to the nuclear coordinates **R** ^{ K }.

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