Partitioning anharmonicity

Except for the nonlocal last term in the exponent, this expression is recognized as the average of the one-dimensional quantum partition function over the static configurations of the bath. This formula without the last term has been used by Dakhnovskii and Nefedova [1991] to handle a bath of classical anharmonic oscillators. The integral over q was evaluated with the method of steepest descents leading to the most favorable bath configuration. 

C) The error in AE" /AEq is 0.1 kcal/mol. Corrections from vibrations, rotations and translation are clearly necessary. Explicit calculation of the partition functions for anharmonic vibrations and internal rotations may be considered. However, at this point other factors also become important for the activation energy. These include for example ... 

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

The ZPVE may be partitioned into harmonic and anharmonic contributions... 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

For diatomic molecules, corrections can be made for the assumption used in the derivation of the rotational partition function that the rotational energy levels are so closely spaced that they can be considered to be continuous. The equations to be used in making these corrections are given in Appendix 6. Also given are the equations to use in correcting for vibrational anharmonicity and nonrigid rotator effects. These corrections are usually small.22... 

In our discussion the usual Born-Oppenheimer (BO) approximation will be employed. This means that we assume a standard partition of the effective Hamiltonian into an electronic and a nuclear part, as well as the factorization of the solute wavefunction into an electronic and a nuclear component. As will be clear soon, the corresponding electronic problem is the main source of specificities of QM continuum models, due to the nonlinearity of the effective electronic Hamiltonian of the solute. The QM nuclear problem, whose solution gives information on solvent effects on the nuclear structure (geometry) and properties, has less specific aspects, with respect the case of the isolated molecules. In fact, once the proper potential energy surfaces are obtained from the solution of the electronic problem, such a problem can be solved using the standard methods and approximations (mechanical harmonicity, and anharmonicity of various order) used for isolated molecules. The QM nuclear problem is mainly connected with the vibrational properties of the nuclei and the corresponding spectroscopic observables, and it will be considered in more detail in the contributions in the book dedicated to the vibrational spectroscopies (IR/Raman). This contribution will be focused on the QM electronic problem. 

The Adiabatic Approximation [51] Dealing With the Strong Anharmonic Coupling Theory of Weak H-Bonds Partition of the Full Hamiltonian into Diabatic and Adiabatic Parts Weakness of the Diabatic Hamiltonian [122]... 

Analyses of linear nonrigid rotators, anharmonic oscillators, and vibrating rotators, yielding first-order corrections for nonrigidity, anharmon-icity, and vibration-rotation interaction (nonseparability of vibrational and rotational modes), respectively, have also been completed and are conventionally used in obtaining corrections (which are most important at elevated temperatures) to the simple product form of the molecular partition... 

In some instances, a quantitative understanding of anharmonic effects may be required to acheive a priori accuracies of better than a factor of two. Procedures for incorporating one-dimensional corrections, particularly for hindered rotors are well developed and commonly employed. Increased quantum chemical and computational capabilities should now allow for studies of the fully coupled nonrigid anharmonic state densities and/or partition functions via direct Monte Carlo sampling. Such accurate state density studies are a necessary prerequisite to furthering our understanding of the accuracy limits of both quantum chemical estimates and of RRKM theory itself. 

The thermal functions are calculated from the partition function Q Q Eg Q Q exp (-CgGj /T) in which 0 and 0 contain first order corrections for anharmonicity. The electronic and molecular constants are taken from the compilation of Suchard (6). 

The vibrational and rotational constants of the respective electronic levels were taken from Rosen (2 ). The thermodynamic functions are calculated using first-order anharmonic corrections to and 0 in the partition function Q = Q,j.EQ Q gj exp(-... 

An atomistic approach, which has relevance to the current work, is the previously discussed normal-mode method. In the normal-mode method the constituent monomer units in the cluster are assumed to interact with a reasonable model potential in a fixed structure. From the assumed structure and model potential a normal-mode analysis is jjerformed to determine a vibrational partition function. Rotational and translational partition functions are then included classically. The normal-mode method treats the cluster as a polyatomic molecule and is most appropriate at very low temperatures where anharmonic contributions to the intermolecular forces can be ignored. As we shall show by numerical example, as the temperature is increased, the... 

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