Harmonic oscillator kinetic energy

We make use of the assumption which is conventional in kinetic theory of the harmonic oscillator [193] as well as in energy-corrected IOS [194]. All the transition rates from top to bottom in the rotational spectrum are supposed to remain the same as in EFA. Only transition rates from bottom upwards must be corrected to meet the demands of detailed balance. In the same way the more general requirements expressed in Eq. (5.21) may be met ... 

To solve this equation, an appropriate basis set ( >.,( / ) is required for the nuclear functions. These could be a set of harmonic oscillator functions if the motion to be described takes place in a potential well. For general problems, a discrete variable representation (DVR) [100,101] is more suited. These functions have mathematical properties that allow both the kinetic and potential energy... 

Explicit forms of the coefficients 7j and A depend on the coordinate system employed, the level of approximation applied, and so on. They can be chosen, for example, such that a part of the coupling with other degrees of freedom (typically stretching vibrations) is accounted for. In the space-fixed coordinate system at the infinitesimal bending vibrations, Ty, + T f. reduces to the kinetic energy operator of a two-dimensional (2D) isotropic harmonic oscillator,... 

Pj and p2 represent the displacement vectors of the nuclei A and D (the corresponding polar coordinates are p1 cji, and p2, < )2, respectively) p, and pc are the displacement vectors and pT, r and pc, <[)f the corresponding polar coordinates of the terminal nuclei at the (collective) trans-bending and cis-bending vibrations, respectively. As a consequence of the use of these symmetry coordinates the nuclear kinetic energy operator for small-amplitude bending vibrations represents the kinetic energy of two uncoupled 2D harmonic oscillators ... 

The first term is the intrinsic electronic energy of the adsorbate eo is the energy required to take away an electron from the atom. The second term is the potential energy part of the ensemble of harmonic oscillators we do not need the kinetic part since we are interested in static properties only. The last term denotes the interaction of the adsorbate with the solvent the are the coupling constants. By a transformation of coordinates the last two terms can be combined into the same form that was used in Chapter 6 in the theory of electron-transfer reactions. 

The statistical treatment of the vibrational degrees of freedom of crystals is far more difficult compared to gases. Let us initially consider a monoatomic crystal. An atom in a crystal vibrates about its equilibrium lattice position. In the simplest approach, three non-interacting superimposed linear harmonic oscillators represent the vibrations of each atom. The total energy, given by the sum of the kinetic and potential energies for the harmonic oscillators, is... 

What is the lowest possible energy for the harmonic oscillator defined in Eq. (5.10) Using classical mechanics, the answer is quite simple it is the equilibrium state with x 0, zero kinetic energy and potential energy E0. The quantum mechanical answer cannot be quite so simple because of the Heisenberg uncertainty principle, which says (roughly) that the position and momentum of a particle cannot both be known with arbitrary precision. Because the classical minimum energy state specifies both the momentum and position of the oscillator exactly (as zero), it is not a valid quantum... 

The initial exploration in this unit requires the students to compare the trajectories calculated for several different energies for both Morse oscillator and harmonic oscillator approximations of a specific diatomic molecule. Each pair of students is given parameters for a different molecule. The students explore the influence of initial conditions and of the parameters of the potential on the vibrational motion. The differences are visualized in several ways. The velocity and position as a function of time are plotted in Figure 2 for an energy approximately 50% of the Morse Oscillator dissociation energy. The potential, kinetic and total energy as a function of time are plotted for the same parameters in Figure 3. 

Figure 3. Potential energy (solid line), kinetic energy (dashed line), and total energy (dot-dashed line), for an anharmonic (top panel) and harmonic (bottom panel) oscillator with the same ground state vibrational frequency. The parameters are the same as in Figure I.

A complete treatment of this derivation can be found in Ref. [19]. The first three terms in the kinetic energy operator indicates the presence of a 3D harmonic oscillator, and the final two terms indicate the presence of a 2D rotator (as for the hydrogen atom). A similar conclusion was made by Auberbach et al. [22] where they use a semi-classical quantisation method and the molecule is said to undergo a unimodal distortion and then the semi-classical Hamiltonian is found to be separated into two parts - a harmonic oscillator part with three vibrational coordinates and a rotational part with two rotational coordinates. However, more progress in terms of specifying the wavefunctions of the system can be made by following a different approach. 

Thus, with this result for the kinetic energy and Eq. (E.10) for the potential energy, we conclude that the quantum dynamics of the normal modes is just the dynamics of n uncoupled harmonic oscillators that is,... 

Cf. V 23, Section 6. In the case of sinusoidal oscillations the time average of the potential energy is equal to that of the kinetic energy. The theorem of equipartition of kinetic energy therefore determines also the total energy content for harmonic oscillators. 

Equation (6.169) gives the kinetic energy, but we have still to obtain a classical expression for the potential energy. A harmonic oscillator can be defined as a mass point m which is acted upon by a force F proportional to the distance x from the equilibrium position and directed towards the equilibrium position. Since force equals mass times acceleration, we have... 

In -> Butler-Volmer equation describing the charge transfer kinetics, the transfer coefficient a (or sometimes symbol jS is also used) can range from 0 to 1. The symmetrical energy barrier results in a = 0.5. Typically, a is in the range of 0.3 to 0.7. In general, a is a potential-dependent factor (which is a consequence of the harmonic oscillator approximation, see also - Marcus theory) but, in practice, one can assume that a is potential-independent, as the potential window usually available for determination of kinetic parameters is rather narrow (usually not more than 200 mV). 

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