Partition function factorization

The Horiuti group treats the temperature coefficient of the rate differently from the way it is usually treated in TST. They clearly identify E as the experimentally observed activation energy, but according to TST [cf. Eq. (5)] the (E — RT) quantity of Eq. (52) is the enthalpy of activation. The RT term in Eq. (5) arises because the assumption that the Arrhenius plot is linear is equivalent to the assumption that the preexponential factor A of the Arrhenius equation is constant, whereas, according to TST, A always contains the factor (kT/h). In addition, the partition function factors of Table I are also part of A, and most of them are functions of T. Since the Horiuti group takes this temperature dependency of the preexponential factor into account, the factor exp[(5/2)(vi -I- V2)] (where 5/2 is replaced by 3 for nonlinear molecules) arises. 

Since Ti=K. + V, the canonical ensemble partition function factorizes into ideal gas and excess parts, and as a consequence most averages of interest may be split into corresponding ideal and excess components, which sum to give the total. In MC simulations, we frequently calculate just the excess or configurational parts in this case, y consists just of the atomic coordinates, not the momenta, and the appropriate expressions are obtained from equation b3.3.2 by replacing Efby the potential energy V. The ideal gas contributions are usually easily calculated from exact... 

TST predicts the trend of decreasing Arrhenius pre-exponential factor with increasing reactant size and molecular complexity that is revealed by experimental measurements of rate coefficients, and that SCT explained away by invoking the steric factor. This trend arises in TST through the internal degrees of freedom, which are accounted for in the partition functions, and which are not present in the structureless point masses of SCT. If electronic, vibrational, rotational, and translational dfs are independent, the molecular partition function factors into electronic, vibrational, rotational, and translational contributions, i.e., Q = e v r t-The orders of magnitude of the partition functions per df are v l-10 per vibrational or internal rotational df, per overall... 

Within the harmonic approximation, one can easily obtain the total vibrational zero point energy, e t(s), and the vibrational partition function factor, Q (T,s), contained in Q (T,s) from the generalized normal mode frequencies discussed above. Specifically, can be written as the sum of the harmonic zero point energies in each generalized bound mode m, e9jin(s), while Q (T,s) can be written as the product of the harmonic vibrational partition functions for each generalized bound mode m, Qvib.m(T,s), where e j (s) and Qvib.m given by standard formulas found in almost any physical chemistry text. However, the... 

To study the effects of incorporating the anharmonic nature of the generalized normal modes transverse to the MEP on the vibrational partition function factor, Q° (T,s), in the generalized transition state partition function, Q (T,s), in eq. (4), we computed at the saddle point of surface 5SP from sets of either harmonic or anharmonic bound vibrational energy levels E /hc (in wave numbers) [176], where Vj,...,V5 are the vibrational quantum numbers and the energy is measured relative to the saddle point (i.e., from the bottom of the vibrational well). That is, we take... 

The canonical ensemble is the name given to an ensemble for constant temperature, number of particles and volume. For our purposes Jf can be considered the same as the total energy, (p r ), which equals the sum of the kinetic energy (jT(p )) of the system, which depends upon the momenta of the particles, and the potential energy (T (r )), which depends upon tlie positions. The factor N arises from the indistinguishability of the particles and the factor is required to ensure that the partition function is equal to the quantum mechanical result for a particle in a box. A short discussion of some of the key results of statistical mechanics is provided in Appendix 6.1 and further details can be found in standard textbooks. 

The factor of 2 in the denominator of the H2 molecule s rotational partition function is the "symmetry number" that must be inserted because of the identity of the two H nuclei. 

This factor can be obtained from the vibration partition function which was omitted from the expression for the equilibrium constant stated above and is, for one degree of vibrational freedom where vq is the vibrational frequency in the lowest energy state. 

This is die fonn diat chemists and physicists are most accustomed to. The probabilities are calculated from the Boltzmann equation and the energy difference between state t and state it — 1. Because we are using a ratio of probabilities, the normalization factor, i.e., the partition function, drops out of the equation. Another variant when 6 is multidimensional (which it usually is) is to update one component at a time. We define 6, = 6, i,... 

At high temperatures (/S -r 0) the centroid (3.53) collapses to a point so that the centroid partition function (3.52) becomes a classical one (3.49b), and the velocity (3.63) should approach the classical value Uci- In particular, it can be directly shown [Voth et al. 1989b] that the centroid approximation provides the correct Wigner formula (2.11) for a parabolic barrier at T > T, if one uses the classical velocity factor u i. A. direct calculation of Ax for a parabolic barrier at T > Tc gives... 

This is our principal result for the rate of desorption from an adsorbate that remains in quasi-equihbrium throughout desorption. Noteworthy is the clear separation into a dynamic factor, the sticking coefficient S 6, T), and a thermodynamic factor involving single-particle partition functions and the chemical potential of the adsorbate. The sticking coefficient is a measure of the efficiency of energy transfer in adsorption. Since energy supply from the... 

We assume that exploring all possible forms for the fields corresponds to exploring the overall usual phase space. To determine the partition function Z the contributions from all the p+ r) and P- r) distributions are summed up with a statistical weight, dependent on p+ r) and p (r), put in the form analogous to the Boltzmann factor exp[—p (F)]], where the effective Hamiltonian p (F)] is a functional of the fields. The... 

Thus far we have specified the energy level relative to some arbitrary zero level. It is common, however, to represent energies as the difference between the ith level and the zeroth level for the molecules. We can write e, = (e, — eo) + eq, thus obtaining exp( — E,) = exp( —Eo)-exp[-(E, - Eo)]. Therefore, the quantity exp(-Eo) can be factored out of each term in the partition function, giving... 

Q is the partition function of the transition state with this special vibrational mode factored out. 

The partition function may alternatively be written as a sum over all distinct energy levels, times a degeneracy factor g,-. 

For each of the partition functions the sum over allowed quantum states runs to infinity however, since the energies become larger, the partition functions are finite. Let us examine each of the -factors in a little more detail. 

In the infinite sum each successive term is smaller than the previous by a constant factor ( -hujVT which is <1), and can therefore be expressed in a closed form. Only the vibrational frequency is needed for calculating the vibrational partition function for a harmonic oscillator, i.e. only the force constant and the atomic masses are required. 

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. 

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 ... 

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