Partition function, potential energy surfaces

Equation 4.117 makes complete sense. One of the first things one learns in dealing with phase space integrals is to be careful and not over-count the phase space volume as has already been repeatedly pointed out. In quantum mechanics equivalent particles are indistinguishable. The factor n ni is exactly the number of indistinguishable permutations, while A accounts for multiple minima in the BO surface. It is proper that this factor be included in the symmetry number. Since the BO potential energy surface is independent of isotopic substitution it follows that A is also independent of isotope substitution and cannot affect the isotopic partition function ratio. From Equation 4.116 it follows... 

Early in the development of VTST calculations on simple three atom systems compared rates obtained by exact classical dynamics with conventional TST and VTST, the same potential energy surface and classical partition functions being used throughout. These calculations confirmed the importance of eliminating the recrossing phenomenon in VTST. While TST yielded very much larger rate constants than the exact classical calculations, the VTST calculations yielded smaller rate constants, but never smaller than the exact classical values. 

It is important to point out once again that explanations (rationalizations) of isotope effects which employ arguments invoking hyperconjugation and/or steric effects are completely equivalent to the standard interpretation of KIE s in terms of isotope independent force constant differences, reactant to transition state. In turn, these force constant differences describe isotope dependent vibrational frequencies and frequency differences which are not the same in reactant and transition states. The vibrational frequencies determine the partition functions and partition function ratios in the two states and thus define KIE. The entire process occurs on an isotope independent potential energy surface. This is not to claim that the... 

Since potential energy surfaces of isotopic molecules are nearly identical, equilibrium isotope effects can only arise from the effect of isotopic mass on the nuclear motions of the reactants and products. Thus the ratio can be expressed in terms of partition functions for nuclear... 

The most accurate theories of reaction rates come from statistical mechanics. These theories allow one to write the partition function for molecules and thus to formulate a quantitative description of rates. Rate expressions for many homogeneous elementary reaction steps come from these calculations, which use quantum mechanics to calculate the energy levels of molecules and potential energy surfaces over which molecules travel in the transition between reactants and products. These theories give... 

These are only rough guides to the true entropy change on activation. Detailed calculations using spectroscopic data for the reactants and a calculated potential energy surface for the activated complex will yield accurate partition functions for the translational, rotational and vibrational terms involved. Since the quantities contributing to the partition functions for each molecule will be different, then accurate calculations will be able to differentiate between such reactions as... 

In this approach properties of potential energy surfaces are investigated from the point of view of all possible monomolecular transformations of the given reactants. A plausible suggestion concerning the mechanism of the reaction under study is usually made on the basis of reaction barriers or activation energies. Moreover, in some studies, partition functions are evaluated and rate constants are obtained within the framework of the absolute rate theory. 

To study CO2 on clean Pd(lll), two different clusters Pdio(7,3) and Pd 15(10,5) were selected to represent mono-coordinated and bi-coordinated adsorption modes respectively. The local/outer separation described above was employed, pseudopotentials and basis sets chosen according to this partition. The hybrid B3LYP density functional method was used to explore the potential energy surface. The different optimizations converged to three unique species corresponding to two coordination models only. For theri -C coordination two different species were found, one being a physisorbed and... 

TlntermedtaleTiomJalizetfwave lncfl6n7 t2 Internal energy, 298, 374 Internal rotation, partition function for, 306 Intersecting potential energy surfaces model, 48... 

The proper evaluation of the quantized energy levels within the SACM requires a separable reaction coordinate and thus numerical implementations have implicitly assumed a center-of-mass separation distance for the reaction coordinate, as in flexible RRKM theory. Under certain reasonable limits the underlying adiabatic channel approximation can be shown to be equivalent to the variational RRKM approximations. Thus, the key difference between flexible RRKM theory and the SACM is in the focus on the underlying potential energy surface in flexible RRKM theory as opposed to empirical interpolation schemes in the SACM. Forst s recent implementation of micro-variational RRKM theory [210], which is based on interpolations of product and reactant canonical partition functions, provides what might be considered as an intermediate between these two theories. 

In the case of isotropic potential energy surfaces, such as appropriate (approximately) for ion A - induced dipole B systems, the situation is even simpler. In this case, the external rotational levels of A and B transfer unchanged into W(E,T) and g becomes a centrifugal partition function... 

The simple stochastic model bypasses the need for an evaluation of cluster partition functions and dynamics that have previously been employed ° to study the density dejjendence of kf. The density dependence of the recombination rate is the simplest example of the rollover predicted by the Kramers theory for passage over a barrier. In the theory of this chapter the barrier arises from entropy considerations in a free energy surface rather than from a potential energy surface. 

The initial assumption is made — xa. This assumption will be discussed further subsequently. The problems of calculating ratios of rate constants become then a matter of calculation of ratios of isotopic partition functions. The latter is simplified by the fact that the potential energy surfaces for isotopic molecules are the same to a very high degree of approximation. 

Q is the usual partition function of the activated complex referred to the minimum in the potential of the normal molecule as the zero of energy, Q is the partition function qf the three rotations and three translations of the normal molecule, Ea IS the activation energy of the reaction as measured from the minimum of the normal molecule potential energy surface to the minimum of the activated complex, 0 is the zero-point energy of the activated complex, and the v( s are the vibrational frequencies, of the normal molecule. Moreover, A the rate of deactivation of active molecules to normal molecules, has been set equal to the collision number Z times an efficiency factor y, assumed to be isotope independent. 

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