Harmonic oscillator total energy

Fig. 3. Resonance Raman spectra and excitation profiles for a displaced (B = 2) harmonic oscillator (totally symmetric normal mode). The spectra plotted as line spectra perpendicular to the plane of the paper include the Rayleigh line u = 0, the Raman fundamental = 1, and the first two overtones u = 2 and u = 3. The panels parallel to the plane of the paper represent the corresponding excitation profile. The front (i.e., Rayleigh) excitation profile resembles the squared absorption spectrum. Unless otherwise stated, all figures are based on unit vibrational frequencies and transition moments, and values , = 15 for the energy of the excited state considered and F = 0.2 for its vibronic bandwidth.

The problem of estimating densities of vibrational states p is a large one that we will only touch on here. For single harmonic oscillator with uniform energy spacing hv, p is of course l/hv. The number of ways of placing n vibrational quanta in q identical oscillators (total energy E = nhv) is [12]... 

Fig. 1. Total energy (in kj/mol) versus time (in fs) for different integrators for a collinear collision of a classical particle with a harmonic quantum oscillator (for details see [2]). Dashed line Nonsymplectic scheme. Dotted Symplectic integrator of first order. Solid PICKABACK (symplectic, second order).

Most spectroscopic properties are related to second derivatives of the total energy. As a simple illustrative example, vibrational modes, which arise from the harmonic oscillations of atoms around their equilibrium positions, are characterized by the quadratic variation of the total energy as a function of the atomic displacements SRy... 

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... 

The total vibrational energy is a sum of energies of 3N-6 distinct harmonic oscillators. Indeed, 3N-6 is the final number of coordinates in Equation 8. Namely, the number of 3N+3 coordinates of the initial equation has been reduced by three through the elimination of internal rotations. Furthermore, the equation of nuclear motion (mainly its potential) has to be invariant under rotations and translations of a molecule as a whole (which is equivalent to the momentum and angular momentum preservation laws). The latter requirement leads to a further reduction of the number of coordinates by six (five in the case of linear molecules for which there are only two possible rotations). 

The harmonic oscillator energies and wavefunctions comprise the simplest reasonable model for vibrational motion. Vibrations of a polyatomic molecule are often characterized in terms of individual bond-stretching and angle-bending motions each of which is, in turn, approximated harmonically. This results in a total vibrational wavefunction that is written as a product of functions one for each of the vibrational coordinates. 

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 2. Internuclear separation (top panel) and velocity (bottom panel) as a function of time for a Morse (dashed line) and harmonic (solid line) oscillator having the same total energy, ca. 50% of the dissociation energy of the Morse potential. Note the rapid change in velocity at the inner turning point and slow change in velocity at the outer turning point for an anharmonic oscillator.

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.

Hinshelwood assumed that the molecule C is composed of s identical harmonic oscillators of frequency v. Let n be the total number of vibrational quanta that the molecule must possess for an internal energy e. That is,... 

The addition of the spin-orbit term to the nuclear harmonic oscillator potential causes a separation or removal of the degeneracy of the energy levels according to their total angular momentum (j = l + s). In the nuclear case, the states with... 

The normal coordinates are required to be combinations of the qt such that the total energy, when expressed in terms of these coordinates, becomes the sum of the energies of individual harmonic oscillators. This means that no cross terms should appear in V and T when written in normal coordinates. For an oscillator of mass 1, wc would therefore have... 

The first part of the Hamiltonian (16), Hc.o.m, describes the center-of-mass contribution, as in the quasi-one-dimensional cases, and contributes the eigenenergy of a two-dimensional isotropic harmonic oscillator to the total energy. The second part of the Hamiltonian, Hint, depends on the antisymmetric coordinates xa and ya, and represents the contribution to the total energy due to the internal degrees of freedom. 

The assumption about a uniform probability for any distribution of the energy between the harmonic oscillators may now be used to determine the probability Pet >e (E). It can be expressed as the ratio between the density of states corresponding to the situation where the energy exceeds the threshold energy in the reaction coordinate and the total density of states at energy E, that is, N(E) of Eq. (7.36). 

The total wave function can, accordingly, be written as a product of wave functions corresponding to each mode. The energy eigenfunctions corresponding to each mode are, in particular, just the well-known eigenfunctions for a harmonic oscillator. 

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. 

Exact calculations of AE have been carried out by Kelley and Wolfsberg [19] for colinear collisions between an atom and a diatomic molecule. The oscillator potential was considered to be both harmonic and Morse-type, and the interaction between the colliding pair was taken both as an exponential repulsion and as a Lennard-Jones 6 12 potential. Two important conclusions were reached First, when the initial energy of the oscillator increases, the total energy transferred from translation to vibration, AE, decreases. Second, the effect of using a Morse-oscillator potential in place of the harmonic oscillator was generally to decrease AE, often by more than a factor of 10. 

We have demonstrated here that for the one-dimensional harmonic oscillator the integral required in the Euler equation, involving the functional derivative 8t/8p, can be exactly expressed in terms of the total kinetic energy. Indeed, the relation, involving a factor of 3, is exactly that given by the TF statistical theory. This latter theory gives for the density in d dimensions... 

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