Anharmonic springs, nonlinear polarizabilities

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... 

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]. 

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]. 

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