Total harmonic oscillator

Using the wave functions of the harmonic oscillator in each potential well of the proton, we can estimate the total effect of the inertia on the transition probability in the high-temperature approximation for the medium67 ... 

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... 

In the harmonic approximation the functions Xi and Xf are products of harmonic oscillator functions. We therefore specify the initial state by a set of quantum numbers n — (ni, ri2,..., n/v), and those for the final state by m = (mi,m2,..., tun)- So the nuclear wavefunctions are henceforth denoted by Xi,n and Xf,m- Equation (19.21) tells us how to calculate the rate of transition from one particular initial quantum mode n to a final quantum state m. This is more than we want to know. All we are interested in is the total rate from any initial state to any final state. The ensemble of reactants is in thermal equilibrium therefore... 

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 heat capacity models described so far were all based on a harmonic oscillator approximation. This implies that the volume of the simple crystals considered does not vary with temperature and Cy m is derived as a function of temperature for a crystal having a fixed volume. Anharmonic lattice vibrations give rise to a finite isobaric thermal expansivity. These vibrations contribute both directly and indirectly to the total heat capacity directly since the anharmonic vibrations themselves contribute, and indirectly since the volume of a real crystal increases with increasing temperature, changing all frequencies. The constant volume heat capacity derived from experimental heat capacity data is different from that for a fixed volume. The difference in heat capacity at constant volume for a crystal that is allowed to relax at each temperature and the heat capacity at constant volume for a crystal where the volume is fixed to correspond to that at the Debye temperature represents a considerable part of Cp m - Cv m. This is shown for Mo and W [6] in Figure 8.15. 

It is easy to show, analogously to the classical system, the quantum problem also has solutions corresponding to 3N uncoupled harmonic oscillators. The total wave... 

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. 

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,... 

Let the total system of harmonic oscillator plus radiation field come to equilibrium. Then we know from equilibrium statistical mechanics... 

Exercise. Two harmonic oscillators are linearly coupled with total Hamiltonian... 

If in addition to this harmonic oscillator approximation for nuclear motion we simply approximate the electronic Hamiltonian as //sp(Qeq) we obtain the harmonic approximation12,13 for the total wave function... 

The contribution to the total electron transfer rate from a single vibrational distribution of the reactants, j, is given by (1) summing over the transition rates from j to each of the product vibrational distributions k, I. Jc,k, and (2) multiplying I. kkjk by the fraction of reactant pairs which are actually in distribution,/, p. The result is p,Xkkkh which is the fraction of total electron transfer events that occur through distribution j. Recall that for a harmonic oscillator normal mode the fractional population in a specific vibrational level j is given as a function of temperature by... 

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... 

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