Perturbation theory anharmonic oscillator

In the estimation of Acon(t), only the first two terms are considered, neglecting the higher-order terms. (Q - Goo) and (Q m - Goo) 810 die quantum mechanical expectation values of the anharmonic oscillator. They can be calculated using perturbation theory and is given by... 

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]. 

In the lowest order of perturbation theory, the energy levels of the three-dimensional anharmonic oscillator are... 

While the classical model of an anharmonic oscillator describes the effects of non-linearity, it cannot provide information on molecular properties. Calculation of molecular properties requires a quantum mechanical model. Application of perturbation theory (Boyd, 2003) leads to the following expression ... 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

Problem 40-3. Using first-order perturbation theory, find perturbed wave functions for the anharmonic oscillator with V = + ax2,... 

The Mie solution given here assumes a linear response if the classical oscillator is driven hard, anharmonic terms appear which give rise to a nonlinear Mie scattering, similar to effects considered in chapter 9. The perturbative theory of nonlinear Mie scattering has been given for spherical metal clusters, and gives rise to second and third harmonic generation [699]. 

Fernandez, F. M. and Ogilvie, J. F. (1993), Perturbation Theory by the Moment Method Applied to Coupled Anharmonic Oscillators , Phys. Lett. A 178, 11. 

The basic theory of vibration-rotation of polyatomic molecules has been worked out for many years, including the form of the vibration-rotation Hamiltonian, the quantum mechanical solution of certain anharmonic oscillators," and the relationship, in terms of perturbation theory,between experimental spectroscopic data and higher than quadratic terms in the potential function. Still, prediction of harmonic... 

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