Computing the Partition Function

This section is split in three parts dealing with  

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To compute the r.h.s. of Eq. (15.35), we need to define how we compute the partition function (and the ZPVE) for the non-stationary point 5. In this case, we simply continue to take advantage of our decision to treat the activated complex as a species having 3N — 7 bound degrees of freedom. In order to define this space for an arbitrary point on the MEP,... 

To compute the partition function. Using the above hybrid treatment for the ensemble of ion-binding modes, we calculate the partition function from the sum of the modes (Eq. (22.11)). For each mode, the free energies A Q, and Ac ° for the TBIs are direcdy given by the pretabulated potentials of the mean force (Eqs. (22.7) and (22.9)). 

In Eq. (2.9), we have a general expression for mean square fluctuations, if only we can express E — Ts as a function of x. This ordinarily can be done conveniently for the internal energy E. We shall now show that, to a very good approximation, equals the entropy S, so that it also can be expressed in terms of the parameter x, by ordinary thermodynamic means. To do this, we shall compute the partition function Z... 

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 ... 

Apart from prediction of MFE structures RNAalifold also implements an algorithm to compute the partition function over all possible (consensus) structures and the thermodynamic equilibrium probability for each possible pair. These basepairing probabilities are useful to see structural alternatives, and to distinguish well-defined regions, where the predicted structure is most likely correct, from ambiguous regions. 

Although the energy differences (E —E for 1,2-difluoroethane obtained by the electron diffraction method (Table 23) must be regarded as a rough estimate and little credence should be placed on the quantitative magnitudes, the sign of (Eg-E ) appears to be correct. To deduce from diffraction data accurate energy differences between rotational isomers one must (i) compute the partition function of the isomers (ii) estimate... 

The strength of the path integral (PI) formalism [23-27] lies in the fact that it allows computing the partition function without finding the eigenstates of the Hamiltonian. Let us therefore consider a molecular system with D degrees of freedom with masses m Starting from the exact expression Q P) = Tr(e ), one obtains PI... 

Before we compute the partition function, let s complete the expression for the wavefunction Probabilities must integrate to 1, Jg dx = 1. So... 

C)ur aim is to compute the partition function, based on the combinatorics for counting the numbers of arrangements of H and C units in the chain. We will consider three different models for the partition function. To show the nature of cooperativity, it is useful to start with a model that has no cooperativity. 

Because each chain of length N can have any possible configuration—any sequence of H s and C s—the partition function is the sum over all the possible chain configurations. You can compute the partition function using a simple matrix method. First, express the statistical weights for the two possible states of the first monomer as a statistical weight vector qi ... 

EXAMPLE 26.3 Compute the partition function for a trimer. Substitute N = 3 into Equation (26.21) ... 

To compute the partition function of a freely moving atom, the summation over / in (4.34) has to be replaced by an integration over the variables defining the energy of a free atom. The translational energy of an atom (or center of mass motion of a molecule) is... 

The introduction of coupled cluster theory into quantum chemistry is attributed to Cfzek who, in 1966, published a seminal paper [61] in the Journal of Chemical Physics entitled On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods. The theory uses an exponential ansatz which was originally employed in statistical mechanics to compute the partition function of a non... 

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. 

The strategy now consists in estimating the partition function from below by considering only the trajectories that visit the lower half plane exactly at all the atypical stretches. Notice that computing the partition function restricted to these trajectories is particularly easy, because this... 

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