Energy levels small molecules

Vibrational energy states are too well separated to contribute much to the entropy or the energy of small molecules at ordinary temperatures, but for higher temperatures this may not be so, and both internal entropy and energy changes may occur due to changes in vibrational levels on adsoiption. From a somewhat different point of view, it is clear that even in physical adsorption, adsorbate molecules should be polarized on the surface (see Section VI-8), and in chemisorption more drastic perturbations should occur. Thus internal bond energies of adsorbed molecules may be affected. 

Of course, using the additive scheme rules out the possibility of calculating of the contribution of electron correlation component in bond energy. In fact, the difference between estimated from the Morse potential and experimental bond energies [40] decreases in HI, HBr, HC1, HF, H2 92.8 (31.4%), 81.8 (22.5 %), 59.5 (13.8 %), -25.8 (4.5 %), 30.4 (7.0 %) kJ/mol, respectively. Apparently, the accuracy of the calculation of bond energy in small molecules is usually low-level. Exceptions can be found with molecules containing atoms of inert gases. The enforced dissociation of NeOH was shown to be described by Morse-like potential... 

The fortuitous near-degeneracy of the levels poses severe difficulties to a complete quantitative description of the process in the too-many level small-molecule limit, and even in the small-molecule limit itself. The interaction potentials are poorly known. It is still unclear why collisional processes proceed on a much more rapid time scale in electronically excited states than in ground electronic states. The intuitive explanation, that the excited states are larger, is insufficient. An analysis of the excited potential energy surfaces should prove enlightening in this regard. 

Another recent development is the implementation of DK Hamiltonians which include spin-orbit interaction. An early implementation shared the restriction of the relativistic transformation to the kinetic energy and the nuclear potential with the efficient scalar relativistic variant electron-electron interaction terms were treated in nonrelativistic fashion. Further development of the DKH approach succeeded in including also the Hartree potential in the relativistic treatment. This resulted in considerable improvements for spin-orbit splitting, g tensors and molecular binding energies of small molecules of heavy main group and transition elements. Application of Hamiltonians which include spin-orbit interaction is still computationally demanding. On the other hand, the SNSO method is an approximation which seems to afford a satisfactory level of accuracy for a rather limited computational effort. 

Brunner, K., Van Dijken, A., Bomer, H. et al. 2004. Carbazole compounds as host materials for triplet emitters in organic light-emitting diodes tuning the HOMO level without influencing the triplet energy in small molecules. /. Am. Chem. Soc. 126 6035. 

The mathematical foundation for statistical thermodynamics was set by Ludwig Boltzmann in the late 1800s and has been developed to a useful technology for small molecules. Other aspects of thermodynamics still need use of empirically fitted polynomials, but as computers continue to improve, statistical thermodynamics offers a detailed treatment to connect quantized energy levels of molecules to macroscopic thermal properties. There are extensive treatises on statistical thermodynamics but here we have tried to give just the basic essentials as a way to use some of the quantized energy-level formulas we have obtained from quantum mechanics. 

The rotational energy of a rigid molecule is given by 7(7 + l)h /S-n- IkT, where 7 is the quantum number and 7 is the moment of inertia, but if the energy level spacing is small compared to kT, integration can replace summation in the evaluation of Q t, which becomes... 

Translational energy, which may be directly calculated from the classical kinetic theory of gases since the spacings of these quantized energy levels are so small as to be negligible. The Maxwell-Boltzmann disuibution for die kinetic energies of molecules in a gas, which is based on die assumption diat die velocity specuum is continuous is, in differential form. 

Vibrational energy, which is associated with the alternate extension and compression of die chemical bonds. For small displacements from the low-temperature equilibrium distance, the vibrational properties are those of simple harmonic motion, but at higher levels of vibrational energy, an anharmonic effect appears which plays an important role in the way in which atoms separate from tire molecule. The vibrational energy of a molecule is described in tire quantum theory by the equation... 

Step 1 of the parametrization process is the selection of the appropriate model compounds. In the case of small molecules, such as compounds of pharmaceutical interest, the model compound may be the desired molecule itself. In other cases it is desirable to select several small model compounds that can then be connected to create the final, desired molecule. Model compounds should be selected for which adequate experimental data exist, as listed in Table 1. Since in almost all cases QM data can be substimted when experimental data are absent (see comments on the use of QM data, above), the model compounds should be of a size that is accessible to QM calculations using a level of theory no lower than HE/6-31G. This ensures that geometries, vibrational spectra, conformational energetics, and model compound-water interaction energies can all be performed at a level of theory such that the data obtained are of high enough quality to accurately replace and... 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

This procedure assumes that the translational, rotational, vibrational, and electronic energy levels are independent. This is not completely so. In the instance of diatomic molecules, we will see how to correct for the interaction. For more complicated molecules we will ignore the correction since it is usually a small effect. 

FIGURE 3.41 In large molecules, there are many closely spaced energy levels and the HOMO-LUMO gap is quite small. Such molecules are often colored because photons of visible light can be absorbed when electrons are excited from the HOMO to the LUMO. 

We can also use the Boltzmann formula to interpret the increase in entropy of a substance as its temperature is raised (Eq. 2 and Table 7.2). We use the same par-ticle-in-a-box model of a gas, but this reasoning also applies to liquids and solids, even though their energy levels are much more complicated. At low temperatures, the molecules of a gas can occupy only a few of the energy levels so W is small and the entropy is low. As the temperature is raised, the molecules have access to larger numbers of energy levels (Fig. 7.10) so W rises and the entropy increases, too. 

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