Vibrational contributions

Within the Born-Oppenheimer approximation, excitations occur between two roto-vibrational levels of two electronic states. The excitation energy is thus  

Concerning the transition probability, we saw in Eq. (4.6) that it is proportional to the square of the transition dipole 

When the nuclear motion is taken into account the electric dipole moment operator must include nuclear contributions and it is written as  

The quantity /itnmCB) is the electronic transition dipole between the electronic states n and m, as a function of the normal nuclear coordinate R (which can be represented also with the normal coordinated or q ). 

For a practical evaluation of integrals of the form of Eq. (4.17), two more issues must be still resolved. First of all, the normal coordinates of the state n must be transformed into the normal 

The sum runs over 3N-6 normal coordinates. The order of the displacements in the presence of the field determines the corresponding order of the FIC. Depending 

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Thus the entropy of localized adsorption can range widely, depending on whether the site is viewed as equivalent to a strong adsorption bond of negligible entropy or as a potential box plus a weak bond (see Ref. 12). In addition, estimates of AS ds should include possible surface vibrational contributions in the case of mobile adsorption, and all calculations are faced with possible contributions from a loss in rotational entropy on adsorption as well as from change in the adsorbent structure following adsorption (see Section XVI-4B). These uncertainties make it virtually impossible to affirm what the state of an adsorbed film is from entropy measurements alone for this, additional independent information about surface mobility and vibrational surface states is needed. (However, see Ref. 15 for a somewhat more optimistic conclusion.)... 

E. Rotational, Translational, and Vibrational Contributions to the Correlation Funetion... 

The classical values of each of drese components can be calculated by ascribing a contribution of R/2 for each degree of freedom. Thus tire U ansla-tional and tire rotational components are 3/27 each, for drree spatial components of translational and rotational movement, and (3 — 6)7 for die vibrational contribution in a non-linear polyatomic molecule containing n atoms and (3 — 5)7 for a linear molecule. For a diatomic molecule, the contributions ate 3/27 ti.a s -f 7 [.ot + 7 vib-... 

The rotational terms are slightly different for a linear molecule, and the vibrational terms will contain one more vibrational contribution. 

It is clear that nonconfigurational factors are of great importance in the formation of solid and liquid metal solutions. Leaving aside the problem of magnetic contributions, the vibrational contributions are not understood in such a way that they may be embodied in a statistical treatment of metallic solutions. It would be helpful to have measurements both of ACP and A a. (where a is the thermal expansion coefficient) for the solution process as a function of temperature in order to have an idea of the relative importance of changes in the harmonic and the anharmonic terms in the potential energy of the lattice. 

The thermodynamic functions of primary interest in chemistry are Cp.m, Sm, and Gm-Ho.m- The translational, rotational, and vibrational contributions are summarized in Table 10.4.u We will not attempt to derive all the equations in this table but will do enough to show how it is done. 

Vibrational Contribution to the Gibbs Free Energy For a Linear Diatomic Molecule From equations (10.84) and (10.101)... 

Calculation of Thermodynamic Properties We note that the translational contributions to the thermodynamic properties depend on the mass or molecular weight of the molecule, the rotational contributions on the moments of inertia, the vibrational contributions on the fundamental vibrational frequencies, and the electronic contributions on the energies and statistical weight factors for the electronic states. With the aid of this information, as summarized in Tables 10.1 to 10.3 for a number of molecules, and the thermodynamic relationships summarized in Table 10.4, we can calculate a... 

Vibration The vibrational contribution is calculated from the fundamental vibrational frequencies and the relationship in Table 10.4. CO is a linear molecule with (3r/ - 5) = 4 fundamentals. The values of Jj are obtained from... 

Equation (10.148) does not correctly predict CV.m at low temperatures because it assumes all the atoms are vibrating with the same frequency. In the ideal gas, this is a good assumption, and in the previous section we used an equation similar to (10.148) to calculate the vibrational contribution to the heat capacity of an ideal gas. 

Table A4.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the various degrees of freedom, including translation, rotation, and vibration (see Section 10.7). For most monatomic gases, only the translational contribution is used. For molecules, the contributions from rotations and vibrations must be included. If unpaired electrons are present in either the atomic or molecular species, so that degenerate electronic energy levels occur, electronic contributions may also be significant see Example 10.2. In molecules where internal rotation is present, such as those containing a methyl group, the internal rotation contribution replaces a vibrational contribution. The internal rotation contributions to the thermodynamic properties are summarized in Table A4.6.

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

We have verified the applicability of the assumption concerning the additivity of the widths for methane [163], The vibrational contribution, known in advance, was subtracted not from the experimental contour width, but from that calculated for given magnitudes of aE and fTdp with the following formulae... 

Use the estimates of molar constant-volume heat capacities given in the text (as multiples of R) to estimate the change in reaction enthalpy of N2(g) + 3 H,(g) —> 2 NH.(g) when the temperature is increased from 300. K to 500. K. Ignore the vibrational contributions to heat capacity. Is the reaction more or less exothermic at the higher temperature ... 

The CCSD model gives for static and frequency-dependent hyperpolarizabilities usually results close to the experimental values, provided that the effects of vibrational averaging and the pure vibrational contributions have been accounted for. Zero point vibrational corrections for the static and the electric field induced second harmonic generation (ESHG) hyperpolarizability of methane have recently been calculated by Bishop and Sauer using SCF and MCSCF wavefunctions [51]. 

In the following we consider nitrogen atoms adsorbed on a ruthenium surface that is not completely flat but has an atomic step for each one hundred terrace atoms in a specific direction. The nitrogen atoms bond stronger to the steps than to the terrace sites by 20 kj mok. The vibrational contributions of the adsorbed atoms can be assumed to be equal for the two types of sites. (Is that a good assumption ) Determine how the coverage of the step sites varies with terrace coverage. 

The mean and mean square values of the LA coordinate s represent the principal anharmonic and harmonic vibrational contributions, respectively [3]. 

The entropy difference A5tot between the HS and the LS states of an iron(II) SCO complex is the driving force for thermally induced spin transition [97], About one quarter of AStot is due to the multiplicity of the HS state, whereas the remaining three quarters are due to a shift of vibrational frequencies upon SCO. The part that arises from the spin multiplicity can easily be calculated. However, the vibrational contribution AS ib is less readily accessible, either experimentally or theoretically, because the vibrational spectrum of a SCO complex, such as [Fe(phen)2(NCS)2] (with 147 normal modes for the free molecule) is rather complex. Therefore, a reasonably complete assignment of modes can be achieved only by a combination of complementary spectroscopic techniques in conjunction with appropriate calculations. 

The so-called Boson peak is visible as a hump in the reduced DOS, g(E)IE (Fig. 9.39b), and is a measure of structural disorder, i.e., any deviation from the symmetry of the perfectly ordered crystal will lead to an excess vibrational contribution with respect to Debye behavior. The reduced DOS appears to be temperature-independent at low temperatures, becomes less pronounced with increasing temperature, and disappears at the glass-liquid transition. Thus, the significant part of modes constituting the Boson peak is clearly nonlocalized on FC. Instead, they represent the delocalized collective motions of the glasses with a correlation length of more than 20 A. 

Table 13-1. Computed reaction barriers and isomer stabilities [kcal/mol] for the electrocyclic ring opening of cyclobutene (relative to cyclobutene 1, including zero-point vibrational contributions). Except for G2, the results were obtained using the 6-311+G(d,p) basis set.

Table 13-2. Computed activation (AEa) and reaction energies (AEr) for the concerted gas-phase cycloaddition of ethylene to Irans-butadiene [kcal/mol]. The HF and DFT calculations were performed with the 6-311+G(d,p) basis set and include zero-point vibrational contributions.

If only the spins of the nuclei are altered in a given transition, the translational and vibrational contributions to the partition function are identical. Thus, for tiie reaction ortho-Hz - para-H2, the partial-pressure ratio at equilibrium is given by... 

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