Anharmonic oscillator-molecular orbital

The anharmonic oscillator - molecular orbital theory connection... 

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 chemical structure dependence of electronic hyperpolarizability is discussed. Strategies for developing structure-function relationships for nonlinear optical chromo-phores are presented. Some of the important parameters in these relationships, including the relative ionization potential of reduced donor and acceptor and the chain length, are discussed. The correspondence between molecular orbital and classical anharmonic oscillator models for nonlinear polarizability is described. 

The emphasis of the theoretical discussion is (1) derivation and interpretation of the sum on states perturbation theory for charge polarization (2) development of physical models for the hyperpolarizability to assist molecular design (e.g., reduction of molecular orbital representations to the corresponding anharmonic oscillator description for hyperpolarizability). 

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]. 

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 . 

Since the relevant dimensional parameter is 1/D, the pseudoclas-sical large-Z) limit is closer to D = 3 than is the hyperquantum low-D limit. As in Fig. 3, for D finite but very large, equivalent to a very heavy electronic mass, the electrons are confined to harmonic oscillations about the fixed positions attained in the D oo limit. We call these motions Langmuir vibrations, to acknowledge his prescient suggestion 70 years ago [89] that the electrons could...rotate, revolve, or oscillate about definite positions in the atom. In a dimensional perturbation expansion the first-order term, proportional to 1/D, corresponds to these harmonic vibrations, whereas higher-order terms correspond to anharmonic contributions. Standard methods for analysis of molecular vibrations [90] thus become directly applicable to electronic structure. These methods are semiclassical in form and far simpler, both conceptually and computationally, than the conventional orbital formulation. 

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