Anharmonic springs, nonlinear

Anharmonic oscillator-molecular orbital theory connection anharmonic energy profile, 97,98/ two-orbital calculation, 96-97,98/ Anharmonic springs, nonlinear polarizabilities, 90 Anionic group theory assumptions, 364-365... 

Classical anharmonic spring models with or without damping [9], and the corresponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequently described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polarizability [6,11,12]. 

The nonlinear polarizabilities in the classical spring problem arise from anharmonic contributions to the spring constant. Resolution of eq. 3 into harmonics of frequency nu using trigonometric identities provides an understanding of how specific orders of anharmonicity in V(x) lead to anharmonic polarizations at frequencies different from that of the applied field S(t). In the classical problem, the coefficients an are determined by the anharmonicity constants in V(x) [10]. 

We have shown the molecular orbital theory origin of structure - function relationships for electronic hyperpolarizability. Yet, much of the common language of nonlinear optics is phrased in terms of anharmonic oscillators. How are the molecular orbital and oscillator models reconciled with one another The potential energy function of a spring maps the distortion energy as a function of its displacement. A connection can indeed be drawn between the molecular orbitals of a molecule and its corresponding effective oscillator . 

---