Harmonic-oscillation equation

Bond stretching is most often described by a harmonic oscillator equation. It is sometimes described by a Morse potential. In rare cases, bond stretching will be described by a Leonard-Jones or quartic potential. Cubic equations have been used for describing bond stretching, but suffer from becoming completely repulsive once the bond has been stretched past a certain point. 

Thus, we have demonstrated that a small motion of a mass is described by the equation of a harmonic oscillator, Equation (3.25), and, as is well known, its solution is... 

The j-th harmonic bath mode is characterized by the mass mj, coordinate Xj, momentum pxj and frequency coj. The exact equation of motion for each of the bath oscillators is mjxj + mj(0 Xj = Cj q and has the form of a forced harmonic oscillator equation of motion, ft may be solved in terms of the time dependence of the reaction coordinate and the initial value of the oscillator coordinate and momentum. This solution is then placed into the exact equation of motion for the reaction coordinate and after an integration by parts, one obtains a GLE whose... 

The return to equilibrium of a polarized region is quite different in the Debye and Lorentz models. Suppose that a material composed of Lorentz oscillators is electrically polarized and the static electric field is suddenly removed. The charges equilibrate by executing damped harmonic motion about their equilibrium positions. This can be seen by setting the right side of (9.3) equal to zero and solving the homogeneous differential equation with the initial conditions x = x0 and x = 0 at t = 0 the result is the damped harmonic oscillator equation ... 

The selection rules for the QM harmonic oscillator pennit transitions only for An = 1 (see Section 14.5). As Eq. (9.47) indicates diat the energy separation between any two adjacent levels is always hm, the predicted frequency for die = 0 to n = 1 absorption (or indeed any allowed absorption) is simply v = o). So, in order to predict die stretching frequency within the harmonic oscillator equation, all diat is needed is the second derivative of the energy with respect to bond stretching computed at die equilibrium geometry, i.e., k. The importance of k has led to considerable effort to derive analytical expressions for second derivatives, and they are now available for HF, MP2, DFT, QCISD, CCSD, MCSCF and select other levels of theory, although they can be quite expensive at some of the more highly correlated levels of theoiy. 

Eqn (9-3.3) is the same as the well known one-dimensional harmonic oscillator equation and has as its solutions... 

The Lie algebraic indices are implied. The Higgs field is described by the harmonic oscillator equation where the field has the mass Mh — 1.0 TeV/c2. On the physical vacuum the gauge fields are... 

As was shown by Bohm and Pines,90 under certain conditions the equations of motion for a Fourier component of the electron density can be reduced to harmonic-oscillation equations with the frequency depending only on the density of electrons ne ... 

This form of the harmonic oscillator equation is particularly convenient for solution by the methods of matrix mechanics, based on the commutation relationships ... 

In section 6.8.2 we described and solved the Schrodinger equation for a harmonic oscillator, equation (6.178). The potential energy was expressed in terms of a vibrational coordinate q which was equal to R - Re, Re being the equilibrium bond length. The dependence of the electric dipole moment on the internuclear distance may be expressed as a Taylor series,... 

In the quantum mechanical treatment of vibrational normal modes, the vibrational Schrddinger equation is separated into individual harmonic oscillator equations by exactly the same transformation of variables [28]. 

Instead of the quantum mechanical partition function for a harmonic oscillator, equation (28) may be replaced by the classical partition function... 

Since F( ) also equals RTj2 (a classical Boltzmann distribution) and also equals K < )j2 (a harmonic oscillator. Equation 13.5), it follows, using Equation 13.12, that ... 

FIGURE 20.7 The potential energy for a diatomic molecule has its minimum at the equilibrium bond length Z e. The displacement coordinate Q = / - Z e represents stretching the bond (Q > 0) or compressing the bond (Q < 0). For small Q, the potential energy is approximated by the harmonic oscillator equation /(Q) = (V2)kQ. The bond dissociation energy is defined from the bottom of the potential well at / =. 

Note that U and Fb are calculated from the intermolecular potential energies < > k and separation distances ak which are calculated in the static theory for the phase of lowest free energy as a function of T, P, system composition, and molecule chemical structures. Thus, U and Fb are determined by the details of the molecule chemical structures and the orientational and positional ordering of the molecules. < >b xb / b From t le harmonic oscillator equation. 

As to the model of particle production, let us take the matter-inflaton coupling of the form, g spatially homogeneous, one may Fourier-decompose the quantum field ip that couples to . The mode equation for the held harmonic oscillator equation,... 

These equations have the form of harmonic oscillator equations of motion for a coordinate q and momentum /). Indeed, Eqs (3.11) can be derived from the Hamiltonian... 

Such correlation functions are often encountered in treatments of systems coupled to their thennal environment, where the mode 1 for the system-bath interaction is taken as a product of A or B with a system variable. In such treatments the coefficients Cj reflect the distribution of the system-bath coupling among the different modes. In classical mechanics these functions can be easily evaluated explicitly from the definition (6.6) by using the general solution of the harmonic oscillator equations of motion... 

These relationships are based on a pure harmonic oscillator, but real molecular vibrations are often anharmonic. Consequently the solution for the quantized harmonic oscillator (Equation (3.18)) is replaced by... 

Each equation is now a total differential equation in one variable, Q, This is the linear harmonic oscillator equation in terms of the normal coordinate Q. The solution is then expandable as the product of harmonic oscillator functions, one for each normal mode, and the total energy corresponds to the sum of the energies of the 3A atom 6 oscillators. 

There are a number of other methods which may be used to obtain approximate wave functions and energy levels. Five of these, a generalized perturbation method, the Wentzel-Kramers-Brillouin method, the method of numerical integration, the method of difference equations, and an approximate second-order perturbation treatment, are discussed in the following sections. Another method which has been of some importance is based on the polynomial method used in Section 11a to solve the harmonic oscillator equation. Only under special circumstances does the substitution of a series for 4 lead to a two-term recursion formula for the coefficients, but a technique has been developed which permits the computation of approximate energy levels for low-lying states even when a three-term recursion formula is obtained. We shall discuss this method briefly in Section 42c. 

To obtain the standard form of Onsager s theory [37,38], we next linearize the thermodynamic forces in eqs. (A. 15) and (A.28). This linearization reduces these equations to coupled damped harmonic oscillator equations of motion. 

Referring to the original harmonic-oscillator equation (12.107) leads to the general formula for energy eigenvalues... 

These are simply harmonic oscillator equations with solutions... 

Another important task of the vibration spectroscopy is the determination of force constants if). For diatomic molecules, these can be obtained directly from the vibration frequencies (co) and the reduced mass (in) of oscillating atoms by the harmonic oscillator equation... 

The solutions of the harmonic oscillator equation, Eq. (6), Sec. 3-1, have been described in many places. They are called the Hermite orthogonal functions and are of the form... 

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