Vibrational-rotational partition functions applications

This method is applicable only to radicals that show sufficient saturation (>10%) of a vibrational transition by the radiation field, The prerequisites for successful detection of the saturation are radiation Intensity in the cavity of the CO2 laser of >100 W cm , rather a small rotational partition function (<10, i.e. nonlinear radicals with low moments of inertia, or linear radicals), and moderate transition dipole moment ( 0.1 D). 

A further complication associated with the application of molecular mechanics calculations to relative stabilities is that strain energy differences correspond to A (AH) between conformers with similar chromophores (electronic effects) and an innocent environment (counter ions and solvent molecules), whereas relative stabilities are based on A (AG). The entropy term, TAS, can be calculated by partition functions, and the individual terms of AS include vibrational (5vib), translational (5 trans) and rotational (Arot) components, and in addition to these classical terms, a statistical contribution (5stat). These terms can be calculated using Eqs. 3.40-3.43tl21]. 

In addition the reader may find tables with selection rules for the Resonance Raman and Hyper Raman Effect in the book of Weidlein et al. (1982). Special discussions about the basics of the application of group theory to molecular vibrations are given in the books of Herzberg (1945), Michl and Thulstrup (1986), Colthup et al. (1990) and Ferraro and Nakamoto (1994). Herzberg (1945) and Brandmiiller and Moser (1962) describe the calculation of thermodynamical functions (see also textbooks of physical chemistry). For the calculation of the rotational contribution of the partition function a symmetry number has to be taken into account. The following tables give this number in Q-... 

The partition functions and numbers of accessible states can all be calculated from the reaction path data, (j) and F(s), assuming harmonic vibrations and separation of vibration and rigid-body rotation. The vibrational partition function may be improved in accuracy by accounting for anharmonicity in some modes. This has been done simply in a separable mode approximation (e.g., Morse stretches and quartic terms in the bending potentials) [129-131]. There are now a number of examples of applications of various forms of canonical variational transition-state theory using ab initio reaction path calculations [15,106,108,112,131-142]. 

We have seen how statistical thermodynamics can be applied to systems composed of particles that are more than just a single atom. By applying the partition function concept to electronic, nuclear, vibrational, and rotational energy levels, we were able to determine expressions for the thermodynamic properties of molecules in the gas phase. We were also able to see how statistical thermodynamics applies to chemical reactions, and we found that the concept of an equilibrium constant presents itself in a natural way. Finally, we saw how some statistical thermodynamics is applied to solid systems. Two similar applications of statistical thermodynamics to crystals were presented. Of the two, Einstein s might be easier to follow and introduced some new concepts (like the law of corresponding states), but Debye s agrees better with experimental data. 

Except for floppy molecules, thermal contributions at room temperature can be quite accurately evaluated using the familiar rigid rotor-harmonic oscillator (RRHO) approach. If data at high temperatures are required, this approach is no longer sufficient, and an anharmonic force field and analysis, combined with a procedure for obtaining the rotation-vibration partition function therefrom, are required. Two practical procedures have been proposed. The first one, due to Martin and co-workers " is based on asymptotic expansions for the nonrigid rotor partition function inside an explicit loop over vibration. It yields excellent results in the medium temperature range but suffers from vibrational level series collapse above 2000 K or more. A representative application (to FNO and CINO) is found in Ref. 42. 

For a molecular ion with charge number Q a transformation between isotopic variants becomes complicated in that the g factors are related directly to the electric dipolar moment and irreducible quantities for only one particular isotopic variant taken as standard for this species these factors become partitioned into contributions for atomic centres A and B separately. For another isotopic variant the same parameters independent of mass are still applicable, but an extra term must be taken into account to obtain the g factor and electric dipolar moment of that variant [19]. The effective atomic mass of each isotopic variant other than that taken as standard includes another term [19]. In this way the relations between rotational and vibrational g factors and and its derivative, equations (9) and (10), are maintained as for neutral molecules. Apart from the qualification mentioned below, each of these formulae applies individually to each particular isotopic variant, but, because the electric dipolar moment, referred to the centre of molecular mass of each variant, varies from one cationic variant to another because the dipolar moment depends upon the origin of coordinates, the coefficients in the radial function apply rigorously to only the standard isotopic species for any isotopic variant the extra term is required to yield the correct value of either g factor from the value for that standard species [19]. 

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