Harmonic oscillator normalization

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... 

Consider three harmonic oscillators (normal modes) with frequencies 1600 cm-1, 3650 cm-1, and 3750 cm-1. We want to calculate the sum of vibrational states... 

C.6. Small amplitude harmonic vibrations of a molecule (JV atoms) are described by 3iV—6 independent harmonic oscillators (normal modes). Each normal mode is characterized by an irreducible representatioiL A scheme of the vibrational energy levels of three normal modes corresponding to the irreducible representations Fj, T2, r3. The modes have different frequencies, so the interlevel separations are different for all of them (but equal for a given mode due to the harmonic potential). On the right side, all these levels are shown together. 

The atoms in a Debye solid are treated as a system of weakly coupled harmonic oscillators. Normal modes with wavelengths that are large compared to the atomic spacing do not depend on the discrete nature of the crystal lattice, and consequently these normal modes can be obtained by treating the crystal as an isotropic elastic continuum. In the Debye treatment of a solid all of the normal modes are treated as elastic waves. The partition function for a Debye solid cannot be obtained In closed form, but the thermodynamic functions for a Debsre solid have been tabulated as a function of 9p/T- For the pair of Isotopic metals Li(s)... 

To establish the vibrational election itilco, let us first define the vimational states of 3A — 6 harmonic oscillators (normal modes). The ground state of the system is no doubt the state in which every normal mode i is in its ground state. The ground-state wave function of the i-th normal mode reads as (p. 166)... 

Since the very beginning, almost a century ago, spectroscopy has revealed that vibrational dynamics are quantal in nature the energy is quantized and dynamics are described in terms of eigenstates. The quantum theory has developed predominantly within the harmonic approximation, that is the simplest expansion, to quadratic terms, of the potential hypersurface around the equilibrium position. Vibrational dynamics can be thus represented with harmonic oscillators (normal modes) corresponding to coherent oscillations of all degrees... 

In an approximation which is analogous to that which we have used for a diatomic molecule, each of the vibrations of a polyatomic molecule can be regarded as harmonic. Quantum mechanical treatment in the harmonic oscillator approximation shows that the vibrational term values G(v ) associated with each normal vibration i, all taken to be nondegenerate, are given by... 

This relation may be interpreted as the mean-square amplitude of a quantum harmonic oscillator 3 o ) = 2mco) h coth( /iLorentzian distribution of the system s normal modes. In the absence of friction (2.27) describes thermally activated as well as tunneling processes when < 1, or fhcoo > 1, respectively. At first glance it may seem surprising... 

We proceed now to the calculation of B, following [Benderskii et al. 1992a]. The denominator in (4.11) (apart from normalization) is equal to the harmonic-oscillator partition function [2 sinh(ico+ )] The numerator is the product of the s satisfying an equation of the Shrodinger type... 

If we further assume that the vibrational wavefunctions associated with normal mode i are the usual harmonic oscillator ones, and r = u + 1, then the integrated intensity of the infrared absorption band becomes... 

In this approximation the nuclear wavefunctions are a product of N harmonic oscillator functions, one for each normal mode ... 

In Chapter 3, a formula was presented which connects the normal vibrational frequencies of two rigid-rotor-harmonic-oscillator isotopomers with their respective atomic masses m , molecular masses Mi and moments of inertia (the Teller-Redlich product rule). If this identity is substituted into Equation 4.77, one obtains... 

In a first model, these motions are represented by harmonic vibrations, and the functions (Q) and Xbw (Q) are then replaced by products of harmonic oscillator-like wavefunctions. The solutions of Eqs. (9) take this particular form when the T jJ are negligible and when and H b can be expanded in terms of normal coordinates ... 

Epsilon notation is defined similarly, although the deviations are in parts per 10,000. Comparison between these numbers is straightforward as shown in Figure 2, which plots the co-variations in Fe/ Fe and Te/ Fe of layers from BIFs in both 5 and e notation. The mass-dependent fractionation line, based on a simple harmonic oscillator approximation (Criss 1999), lies close to a line of 1.5 1 for 5 Fe-5 Te variations. For example, point A in Figure 2 has an 5 Te value of+15.0, which would be approximately equal to a 5 Fe value of+1.00, as defined here, assuming normalization to an identical reference reservoir. 

The first attempts (G. Klein and I. Prigogine, 1953, MSN.5,6,7) were very timid and not very conclusive. They were devoted to a chain of harmonic oscillators. In spite of a tendency to homogenization of the phases, there was no intrinsic irreversibility here, because an essential ingredient is lacking in this model the interaction among normal modes. The latter were introduced as a small perturbation in the fourth paper of the series (MSN.8). 

In this approach, the diffusion constant, Di, is related to the corresponding characteristic time, x, describing the distortions of the normal coordinate, Westlund et al. (85) used the framework of the general slow-motion theory to incorporate the classical vibrational dynamics of the ZFS tensor, governed by the Smoluchowski equation with a harmonic oscillator potential. They introduced an appropriate Liouville superoperator ... 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the imstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

---