Sum of states

The tenn (E-E ) is tire sum of states at the transition state for energies from 0 to E-E. Equation (A3.12.15) is the RRKM expression for the imimolecular rate constant. 

The intennolecular Hamiltonian of the product fragments is used to calculate the sum of states of the transitional modes, when they are treated as rotations. The resulting model [28] is nearly identical to phase space theory [29],... 

The classical mechanical RRKM k(E) takes a very simple fonn, if the internal degrees of freedom for the reactant and transition state are assumed to be hamionic oscillators. The classical sum of states for s harmonic oscillators is [16]... 

The reactant density of states m equation (A3.12.15) is given by the above expression for p( ). The transition state s sum of states is... 

For a RRKM calculation without any approximations, the complete vibrational/rotational Flamiltonian for the imimolecular system is used to calculate the reactant density and transition state s sum of states. No approximations are made regarding the coupling between vibration and rotation. Flowever, for many molecules the exact nature of the coupling between vibration and rotation is uncertain, particularly at high energies, and a model in which rotation and vibration are assumed separable is widely used to calculate the quantum RRKM k(E,J) [4,16]. To illustrate this model, first consider a linear polyatomic molecule which decomposes via a linear transition state. The rotational energy for tire reactant is assumed to be that for a rigid rotor, i.e. 

If K is adiabatic, a molecule containing total vibrational-rotational energy E and, in a particular J, K level, has a vibrational density of states p[E - EjiJ,K). Similarly, the transition state s sum of states for the same E,J, and Kis [ -Eq-Ef(J,K)]. The RRKM rate constant for the Kadiabatic model is... 

The bulk of the infomiation about anhannonicity has come from classical mechanical calculations. As described above, the aidiannonic RRKM rate constant for an analytic potential energy fiinction may be detemiined from either equation (A3.12.4) [13] or equation (A3.12.24) [46] by sampling a microcanonical ensemble. This rate constant and the one calculated from the hamionic frequencies for the analytic potential give the aidiannonic correctiony j ( , J) in equation (A3.12.41). The transition state s aidiannonic classical sum of states is found from the phase space integral... 

The above expressions are empirical approaches, with m and D. as parameters, for including an anliamionic correction in the RRKM rate constant. The utility of these equations is that they provide an analytic fomi for the anliamionic correction. Clearly, other analytic fomis are possible and may be more appropriate. For example, classical sums of states for Fl-C-C, F1-C=C, and F1-C=C hydrocarbon fragments with Morse stretching and bend-stretch coupling anhamionicity [M ] are fit accurately by the exponential... 

Modifying equation (A3.12.45) to represent the transition state s sum of states, the aniiamionic correction to the RRKM rate constant becomes... 

Now we do one of the standard quantum mechanical tricks, inserting the identity operator as a complete sum of states in the coordinate representation ... 

The term partition function conveys the idea of distribution over states the German word is Zustandsumme, sum of states. From the above relationships, we have N = KQ and p. = ArT In (N/Q). 

It is possible, however, to eliminate this drawback [56] by enlarging the above Markov chain through a combination of several of its states into a single one. Such an enlargement is attainable in two ways. Following the first of them it is necessary as a transient state (j) of the enlarged chain to choose the sum of states lj + 2j +...+ mj, whereas the second way suggests that as such a state (i) the... 

I. The theory of molecular dislocations used to describe deformation and relaxation is based on the assumption of a distribution of the thermal vibration energy similar to that applicable to gas molecules. In general we consider the superposition of thermal motion in one direction to be given by the geometric position of two possible conformations. In this single dimension the phase space elements are dx (space coordinates) and dp (momentum coordinates). The sum of states... 

RRKM theory, an approach to the calculation of the rate constant of indirect reactions that, essentially, is equivalent to transition-state theory. The reaction coordinate is identified as being the coordinate associated with the decay of an activated complex. It is a statistical theory based on the assumption that every state, within a narrow energy range of the activated complex, is populated with the same probability prior to the unimolecular reaction. The microcanonical rate constant k(E) is given by an expression that contains the ratio of the sum of states for the activated complex (with the reaction coordinate omitted) and the total density of states of the reactant. The canonical k(T) unimolecular rate constant is given by an expression that is similar to the transition-state theory expression of bimolecular reactions. 

Motion along the reaction coordinate was limited to classical mechanics, whereas the sum and density (or, to be precise, the degeneracy) of states should be evaluated according to quantum mechanics. The integral in Eq. (7.49) should really be replaced by a sum N (E) is not a continuous function of the energy, but due to the quantization of energy, it is only defined at the allowed quantum levels of the activated complex. That is, the sum of states G (E ) should be calculated exactly by a direct count of the number of states ... 

An additional consequence of the omission of rotation is that symmetry numbers (see Appendix A. 1.1) were ignored in calculating the sum and density of states. Thus, the sum of states of the activated complex and the density of states of the reactant should be divided by their symmetry numbers, rA and a, respectively. 

To that end, variational transition-state theory has been introduced, which is based on Wigner s variational theorem, Eq. (5.10). When a saddle point exists, it represents a bottleneck between reactant and products. It is the point along the reaction coordinate where we have the smallest rate of transformation from the reactant to products. This can be seen from Eq. (7.50), where it should be noted that only the sum of states G (E ) changes as the reaction proceeds along the reaction coordinate. We have the smallest sum of states of the activated complex G (E ) on top of the barrier because at this point the available energy E is at a minimum see Fig. 7.3.3. 

From Appendix A.2, we have classical expressions for the sum and density of states of s uncoupled harmonic oscillators. Thus, the sum of states is... 

We consider the sum of states, density of states, and energies of an ideal gas in a box of volume V. The Hamiltonian for a free particle of mass m is... 

We first consider the sum of states. Now, in Eq. (A.33) the integration over coordinates gives the volume of the container, and the integral over the momenta is the momentum-space volume for H having values between 0 and E. Equation (A.41) is the equation for a sphere in momentum space with radius j2rn, 11. Thus, the volume of the sphere is 4Tt(y/2mH)3/3 and... 

Finally, since Eq. (A.48) is based on a classical evaluation of the sum of states, the fact that according to quantum mechanics no vibrational states exist at energies below the zero-point energy Ez is clearly violated. Thus, we can anticipate that a better estimate of the sum of states at the vibrational energy E, defined as the energy in excess of the lowest possible vibrational energy, is G(E) = G(E + Ez) — G(EZ). 

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