Vibrational force constants and anharmonicities: Relation to polarizability and hyperpolarizability densities

Abstract

In this work, the derivatives of molecular potential energy surfaces V({R}) with respect to nuclear coordinates RK 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 nth 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 {Re}. Specialization to {Re} 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 δF due to an infinitesimal shift in nuclear positions. Cubic anharmonicity constants depend on the hyperpolarization energy of the electrons in the field δF, on the induction energy bilinear in δF and the second variation of the field δ2F, 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 RK, 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 RK.