Generalized transition state partition function

The classical phase-space averages for bound modes in Eq. (11) are replaced by quantum mechanical sums over states. If one assumes separable rotation and uses an independent normal mode approximation, the potential becomes decoupled, and onedimensional energy levels for the bound modes may be conveniently computed. In this case, the quantized partition function is given by the product of partition functions for each mode. Within the harmonic approximation the independent-mode partition functions are given by an analytical expression, and the vibrational generalized transition state partition function reduces to... 

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... 

Generalized Transition State Partition Functions in Rectilinear Coordinates... 

The minimization of the canonical transition state partition function as in Eq. (2.13) is generally termed canonical variational RRKM theory. This approach provides an upper bound to the more proper E/J resolved minimization, but is still commonly employed since it simplifies both the numerical evaluation and the overall physical description. It typically provides a rate coefficient that is only 10 to 20% greater than the E/J resolved result of Eq. (2.11). 

Finally, the generalization of the partition function q m transition state theory (equation (A3.4.96)) is given by... 

In this expression, Qgt and Qr are the partition functions of the generalized transition state and of the reactants, and Vmep(s) is the potential energy of the MEP at s. 

This approach has already been shown to provide accurate results for the vibrational partition functions of the bound molecules H2O and S02, and eq. (53) should be equally applicable for generalized transition states. The harmonic partition functions are given bySl... 

The superscript GT denotes the generalized transition state theory, ff is l/ksT, kg is the Boltzmann constant, h is Planck s constant, is the value of s at which is minimum (that is, the location of the canonical variational transition state), <7 is the symmetry factor, and and are partition functions for the generahzed transition state (GTS) and reactants, respectively. To include the tunneUng effect, the calculated rate constant is multi-... 

Calculation of the rate constant involves the ratio of partition functions for the generalized transition state and for reactants. The three degrees of freedom corresponding to translation of the center of mass of the system are the same in the reactants and transition state, and they are therefore removed in both the numerator and the denominator of Eq. [15]. The reactant partition function per unit volume for bimolecular reactions is expressed as the product of partition functions for the two reactant species and their relative translational motion... 

The generalized transition state number of states needed for microcano-nical variational theory calculations counts the number of states in the transition state dividing surface at s that are energetically accessible below an energy E. Consistent with approximations used in calculations of the partition functions, we assume that rotations and vibrations are separable to give... 

Because the transition state geometry optimized in solution and the solution-path reacton path may be very different from the gas-phase saddle point and the gas-phase reaction path, it is better to follow the reaction path given by the steepest-descents-path computed from the potential of mean force. This approach is called the equilibrium solvation path (ESP) approximation. In the ESP method, one also substitutes W for V in computing the partition functions. In the ESP approximation, the solvent coordinates are not involved in the definition of the generalized-transition-state dividing surface, and hence, they are not involved in the definition of the reaction coordinate, which is normal to that surface. One says physically that the solvent does not participate in the reaction coordinate. That is the hallmark of equilibrium solvation. 

These equations lead to fomis for the thermal rate constants that are perfectly similar to transition state theory, although the computations of the partition functions are different in detail. As described in figrne A3.4.7 various levels of the theory can be derived by successive approximations in this general state-selected fomr of the transition state theory in the framework of the statistical adiabatic chaimel model. We refer to the literature cited in the diagram for details. 

However, since and -5 asymptote to the same function, one might approximate (U) = S dJ) in (3.57) so that the acceptance probability is a constant.3 The procedure allows trial swaps to be accepted with 100% probability. This general parallel processing scheme, in which the macrostate range is divided into windows and configuration swaps are permitted, is not limited to density-of-states simulations or the WL algorithm in particular. Alternate partition functions can be calculated in this way, such as from previous discussions, and the parallel implementation is also feasible for the multicanonical approach [34] and transition-matrix calculations [35],... 

Following Fey nman s original work, several authors pmsued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g., Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed, which gave rise to path integral quantum transition state theory and even more general approaches. The Centroid Molecular Dynamics (CMD) method for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the efiective centroid potential. Various analysis and numerical tests for realistic systems have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

This entropy of activation is determined by the ratio of partition functions, which generally has a slight temperature dependence. The typical value of kc = 10-I-10 5 s 1 corresponds to a drop in AS of 65-85 cal/mol-K. Since only vibrational degrees of freedom are involved in a solid-state reaction, the sole reason for this change may be the increase in their frequencies in the transition state ... 

Computing the partition function for an /V-statc chain requires enumerating all possible states. The clever trick associated with the helix-coil transition theory is to generalize this calculation using the statistical-weight matrix ... 

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