Partition function harmonic oscillators

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... 

Fig. 4.1 The zero point energy or low temperature approximation As temperature drops and u increases above u 4 the harmonic oscillator partition function Q (Harm. Osc.) is better and better approximated by the zero point energy term, exp(—u/2). For a typical CH stretching frequency, v = 3000 cm-1, u 4 at 1050 K and it is reasonable to use the ZPE approximation for that frequency at temperatures below 1000 k...

Using harmonic oscillator partition functions to describe both internal and external modes, the logarithmic Q ratios introduced above, ln(Qg/Qc/QgQc/) = ln(Qc/QcO + ln(Qg7Qg), become... 

Ascribe to q the form of a harmonic-oscillator partition function so that the preexponential factor becomes ... 

The partition function q b of HCl or DCl is well approximated by the harmonic-oscillator partition function (f°. Since the energy levels of a harmonic oscillator are given by (v + j)tiv, one obtains ... 

The free energy of the system also includes entropic contributions arising from the internal fluctuations, which are expected to be different for the separate species and for the liganded complex. These can be estimated from normal-mode analyses by standard techniques,136,164 or by quasi-harmonic calculations that introduce approximate corrections for anharmonic effects 140,141 such approaches have been described in Chapt. IV.F. From the vibrational frequencies, the harmonic contribution to the thermodynamic properties can be calculated by using the multimode harmonic oscillator partition function and its derivatives. The expressions for the Helmholtz free energy, A, the energy, E, the heat capacity at constant volume, C , and the entropy are (without the zero-point correction)164... 

Comparing the form (2.51) with the consecrated result for harmonic oscilators partition function (see Volume I/Chapter 4 of the present five-volume book) one remains with the condition ... 

For his calculations, Burton chose the simplest possible material—a cluster of atoms interacting with nearest-neighbor harmonic forces and with the atoms packed onto lattice positions of a close-packed cubic material. He then calculated the partition functions in Eq. (43) and ultimately the cluster concentrations and nucleation rates. The major problem in this calculation was the internal partition function Zmt> which was calculated by diagonalizing the 3i X 3i dynamical matrices of i atom clusters to obtain normal mode vibrational frequencies and ultimately harmonic oscillator partition functions. This calculation was very expensive and could not be done for i larger than about 100. 

The low-frequency motions of 5 MHz and 3 kHz lower the free energy of the folded and assembled structure of the elastic protein-based polymers. A coarse sense of the amount each mechanical resonance could be contributing to the stability of the structure of these elastic protein-based polymers is obtained from a plot of the logarithm of the frequency as a function of the entropy contribution of resonance frequency using the harmonic oscillator partition function. It is not expected that the harmonic oscillator would give an accurate measurement of the magnitude of the entropy contribution of such low-frequency motions. However, the use of the harmonic oscillator partition function provides a sense of direction and magnitude of the contribution of low-frequency motions to... 

The harmonic oscillator partition function (fv) provides a relatively simple and yet informative representation of internal chain dynamics. 

Figure 4. Plot, based on the harmonic oscillator partition function, of the entropy of the oscillator as a function of log(oscillator frequency). On the left-hand ordinate is plotted the entropy in cal/mol deg (EU, entropy units), and on the right-hand ordinate is plotted the entropic contribution to the Gibbs free energy in kcal/mol pentamer at 298°K. This illustrates the increasing contribution of low frequency oscillations to the entropy and free energy of a protein chain segment. Adapted with permission from Urry et al. (1988).

The uni-particle (electronic) density (4.127) is then used for computing the harmonic oscillator partition function ... 

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