Anharmonic oscillator models, nonlinear

Amino-5-nitropyrimidine, cocrystallization, 455 Amorphous polymers, criteria for use in second harmonic generation, 250-251 Amphiphilic molecules polar Z-type Langmuir-Blodgett films, formation, 473-479 structures, transfer behavior, and contact angles, 474,476-477r Anharmonic oscillator models, nonlinear optical effect-microstructure relationship, 361... 

The interest in efficient optical frequency doubling has stimulated a search for new nonlinear materials. Kurtz 316) has reported a systematic approach for finding nonlinear crystalline solids, based on the use of the anharmonic oscillator model in conjunction with Miller s rule to estimate the SHG and electro optic coefficients of a material. This empirical rule states that the ratio of the nonlinear optical susceptibility to the product of the linear susceptibilities is a parameter which is nearly constant for a wide variety of inorganic solids. Using this empirical fact, one can arrive at an expression for the nonlinear coefficients that involves only the linear susceptibilities and known material constants. 

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. 

In order to illustrate some of the basie aspeets of the nonlinear optieal response of materials, we first discuss the anharmonic oscillator model. This treatment may be viewed as the extension of the classical Lorentz model of the response of an atom or molecule to include nonlinear effects. In such models, the medium is treated as a collection of electrons bound about ion cores. Under the influence of the electric field associated with an optical wave, the ion cores move in the direction of the applied field, while the electrons are displaced in the opposite direction. These motions induce an oscillating dipole moment, which then couples back to the radiation fields. Since the ions are significantly more massive than the electrons, their motion is of secondary importance for optical frequencies and is neglected. 

While the Lorentz model only allows for a restoring force that is linear in the displacement of an electron from its equilibrium position, the anharmonic oscillator model includes the more general case of a force that varies in a nonlinear fashion with displacement. This is relevant when the displacement of the electron becomes significant under strong driving fields, the regime of nonlinear optics. Treating this problem in one dimension, we may write an appropriate classical equation of motion for the displacement, x, of the electron from equilibrium as... 

Figure Bl.5.3 Magnitude of the second-order nonlinear susceptibility x versus frequency co, obtained from the anharmonic oscillator model, in the vicinity of the single- and two-photon resonances at frequencies coq and cOq/2, respectively.

Takahashi, A., and S. Mukamel. 1994. Anharmonic oscillator modeling of nonlinear susceptibilities and its application to conjugated polymers. Journal of Chemical Physics... 

P. N. Prasad, E. Perrin, and M. Samoc, A coupled anharmonic oscillator model for optical nonlinearities of conjugated organic structures, J. Chem. Ph s. 91 2360 (1989). 

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 . 

Nonlinear oscillators (NLOs) have been extensively used as realistic models for chemical bonds (Merzbacher 1970 Morse 1929), especially for describing the bondbreaking process (dissociation), simulating vibrational spectra of molecules (Child and Lawton 1981 Lehmann 1992), and modeling the nonlinear optical responses of several classes of molecules (Kirtman 1992 Takahashi and Mukamel 1994), catalytic bond activation, and dissociation processes (McCoy 1984). The nonlinear (anharmonic) oscillator, defined by the equation... 

These derivatives are evaluated using so-called coupled Hartree-Fock techniques and either static or oscillating fields (117). A second approach is to model the nonlinear media as a set of coupled anharmonic oscillators, with resonant frequencies corresponding to excited state transition frequencies. The strength of the coupling is a fimction of the proximity of the external field frequency to the resonant frequency of the oscillator. 

It should be stressed that the above calculations refer to a perhaps overly simplified model. When anharmonicities in the oscillator potential and processes like (n + m) + (n — m) 2n with m / 1 are taken into account, both the linear and the nonlinear term in the equation changes drastically. For this case one can, however, still obtain a simple analytical expression for the steady-state distribution, as discussed by Treanor9 and Fisher and Kummler.8 Our main concern here has been the dynamic... 

The limitation to second derivatives in computing the force constants is called the harmonic approximation. The calculation does not take into account either the zero-point oscillation or the anharmonicity of the potentials. It is, strictly speaking, valid only at T = 0. Thermal expansion, thermal conductivity and other nonlinear effects are thus not contained in this model. 

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