Harmonic oscillator entropy

Molecular structure enters into the rotational entropy component, and vibrational frequencies into the vibrational entropy component. The translational entropy component cancels in a (mass) balanced reaction, and the electronic component is most commonly zero. Note that the vibrational contribution to the entropy goes to oo as v goes to 0. This is a consequence of the linear harmonic oscillator approximation used to derive equation 7, and is inappropriate. Vibrational entropy contributions from frequencies below 300 cm should be treated with caution. 

Note that there is nothing wrong widi Eq. (10.45). The entropy of a quantum mechanical harmonic oscillator really does go to infinity as the frequency goes to zero. What is wrong is that one usually should not apply the harmonic oscillator approximation to describe those modes exhibiting the smallest frequencies. More typically than not, such modes are torsions about single bonds characterized by very small or vanishing barriers. Such situations are known as hindered and free rotors, respectively. 

Now, consider the normalized density operator pa of a system of equivalent quantum harmonic oscillators embedded in a thermal bath at temperature T owing to the fact that the average values of the Hamiltonian //, of the coordinate Q and of the conjugate momentum P, of these oscillators (with [Q, P] = ih) are known. The equations governing the statistical entropy S,... 

Rigid rotor-harmonic oscillator-ideal gas approximation. The AMBER 4.1 free energy values are summarized in Table 7. The entropy term is important and compensates for the interaction energy (enthalpy) term. A similar type of compensation has also been found in the case of DNA base pairs [40]. FI-bonded structure 4 remains the most stable and also HB6 and HBl structures remain as the second and third most stable ones. The following order of stability is however, changed. The H-bonded structure 7 and the T-shaped structure are surprisingly more stable than H-bonded structures 2, 3 and 5. Analyzing veirious... 

It follows from the preceding discussion that the equilibrium constant for complex formation evaluated using the rigid rotor-harmonic oscillator approximation, with molecular constants derived from ab initio SCF calculations with a medium basis set (of DZ quality), is not very accurate. Comparison of the AG° values calculated using extended and medium basis sets indicates that the major uncertainty in AG is derived from AH . TASP is not as dependent on the basis set used. Furthermore, it is evident that the entropy term plays an extremely important rote in complex formation neglecting it may result not only in quantitative, but even in qualitative failure. 

As indicated briefly above, the calculated entropy of Cg gas is higher than the experimental value even when a summation is performed only over the observed first six levels of the bending frequency. Treatment of the vibration as a harmonic oscillator or as a anharmonic oscillator gives values considerably higher. Strauss and Thiele (1 ) have also calculated functions based on a quartlc potential function whlich still yields values several units higher than the experiments. In order to reduce the entropy to the approximate range of the measurements we have made the assumption that the potential function is... 

Functions were calculated from the constants given above using the rigid rotator harmonic oscillator method. The entropy was increased by R tn 2 because two rotameric forms are implied by use of the torsional frequency. Small but arbitrary adjustments were made in the assignment of the bending mode frequencies in order to reproduce the vapor pressure data of Scott et al. (1 ) as closely as possible. Calculated values of S (298.15 K) = 57.03 and S (340 K) = 58.70 cal k" raol" may be compared with 56.99 and 58.69, respectively, derived from the data of Scott et al. Internal rotation calculations would require a complex potential function in order to fit the data. The barrier to inversion (990 cm 2.8 kcal mol ) is slightly less than the barrier to... 

Statistical Mechanics of the Harmonic Oscillator. As has already been argued in this chapter, the harmonic oscillator often serves as the basis for the construction of various phenomena in materials. For example, it will serve as the basis for our analysis of vibrations in solids, and, in turn, of our analysis of the vibrational entropy which will be seen to dictate the onset of certain structural phase transformations in solids. We will also see that the harmonic oscillator provides the foundation for consideration of the jumps between adjacent sites that are the microscopic basis of the process of diffusion. 

In the text we sketched the treatment of the statistical mechanics of the harmonic oscillator. Flesh out that discussion by explicitly obtaining the partition function, the free energy, the average energy, the entropy and specific heat for such an oscillator. Plot these quantities as a function of temperature. 

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... 

Paralleling the harmonic oscillator expansion of the potential function of a mechanical system, we next approximate the equilibrium entropy function 5(Tp A) by its quadratic order Taylor series expansion about A (the point at which 5 has its maximum). That is, we assume... 

Microscopically, to understand the concept of thermal entropy, or heat capacity for that matter, one needs to appreciate that the vibrational energy levels of atoms in a crystal are quantized. If the atoms are assumed to behave as simple harmonic oscillators, i.e., miniature springs, it can be shown that their energy will be quantized with a spacing between energy levels given by... 

The details of lattice vibrations will not be discussed here. But for the sake of discussion, the main results of one of the simpler models, namely, the Einstein solid, are given below without proof. By assuming the solid to consist of Avogadro s number jVav of independent harmonic oscillators, all oscillating with the same frequency Einstein showed that the thermal entropy per mole is given by... 

Figure 8. Vibration-rotation contributions to the entropy for H20 as a function of temperature. The lower curve (O, solid line) shows the entropy predicted by the classical rigid rotator-quantum harmonic oscillator (RR/HO) model, the intermediate curve ( , dashed line) shows data taken from the JANAF Thermochemical Tables, and the upper curve (0> solid line) is obtained from differentiating the fourth-order fit to the AOSS-U free energy calculations shown in Figure 6. Note that (as expected) all three curves coincide at low temperatures and that the JANAF and AOSS-U results are in dose agreement throughout the entire temperature range.

In order to account for some of the differences in thermodynamic properties of H2O and D2O, theoretical studies have been applied. Swain and Bader first calculated the differences in heat content, entropy, and free energy by treating the librational motion of each water molecule as a three-dimensional isotopic harmonic oscillator. Van Hook demonstrated that the vapor pressure of H2O and D2O on liquid water and ice could be understood quantitatively within the framework of the theory of isotope effects in condensed systems. Nemethy and Scheraga showed that in a model based on the flickering cluster concept, the mean number of hydrogen bonds formed by each water molecule is about 5% larger in D2O than in H2O at 25 °C. 

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