Partition functions, calculation

For strongly shocked Argon (see Table 1), calculations based on Eq 8 give a s that are 30 to 40% higher than the more accurate (partition function) calculations of Reynolds... 

The problem of reaction mechanism has been attacked from the standpoint of entropies of activation on two somewhat different fronts. On the one hand, mechanistic significance has sometimes been attached to the actual value of AS observed for a given reaction. The value can be compared with that expected for a given transition state structure on the basis of partition function calculations, empirical considerations or thermodynamic data for corresponding equilibria. On the other hand, a change in mechanism between two similar reactants is sometimes indicated by very dissimilar entropies of activation. Both types of information are of a qualified nature. 

The partition function calculation for the activated complex is more tricky. If the surface is known the quantities quoted above can be found from the dimensions and the curvature of the surface around the critical configuration. If this is not possible, then estimates of Qf can be made by analogy with a molecule of similar structure. The vital feature is that the free translation along the reaction coordinate has already been accounted for and must not be included in the calculation, and so the activated complex has one degree of vibrational freedom less than that for a molecule with the same number of atoms. 

Detailed calculations using partition functions, calculated from spectroscopic data for reactants and the PE surface for the activated complex, allows K, AG ", AH, AS " and AV to be found. Experimental data can give experimental values to compare direct with the theoretical ones. Equation (4.41) gives... 

Let us remind that the problem of the partition function calculation for a closed polymer chain with topological constraints is usually formulated as integration over the set fl of closed paths from a fixed topological class, or with fixed value of the topological invariant ... 

Here U N) is the interaction potential energy for the complete system at a specific configuration, uniformly the same quantity that has been discussed above. U(TV) is an ejfective potential designed to be used in classical-limit partition function calculations, e.g. Eq. (3.17), p. 40, in order to include quantum mechanical effects approximately. We will call this /(TV) the quadratic Feynman-Hibbs (QFH) model. In Eq. (3.67), Mj is the mass of atom j, and V is the Laplacian of the... 

The thermal functions for the five alkali metal monatomic gases are calculated by the same procedure. Oberved and estimated atomic energy levels are included in the partition function calculation, using an ionization potential lowering (IP-kT) technique... 

In tabulations of partition function calculations (e.g. Richet et al. 1977), reduced partition ratios are commonly reported as/ = P values, or 1000 In P values. In such cases, the fractionation factor between two substances is simply ... 

For isotope exchange reactions written as above involving only isotopically pure molecules (e g. pure C 0 or C 0), the symmetry number of a molecule and its isotopic derivative are identical. Therefore, o/a = 1, and the term need not be included in the calculations. The vibrational frequencies used in our calculations are the same as those on which Urey s (1947) calculations are based. However, Urey corrected these frequencies for anharmonicity (zero-order frequencies), whereas we used observed (measured) fundamental frequencies with no anharmonicity correction (see discussions in Bottinga 1969a, p. 52 McMillan 1985, p. 15 and Polyakov and Kharlashina 1995, p. 2568). Vibrational frequencies are generally reported in wave numbers (co), which have units of cm . For partition function calculations, wave numbers must be converted to units of sec by multiplying by c, the velocity of light (v = cco). 

The ortho-para ratio is determined by the statistical calculation of the availability of states given by the partition function. Calculated ortho-para equilibria for H2, D2, and T2 are shown in Figure 6 at the temperature range from 0 to 300K. When the two nuclear spins are parallel, the resultant nuclear spin quantum number is 1 (i.e. 1/2 -E 1 /2) and the state is threefold degenerate. When the two spins are opposed, however, the resultant nuclear spin is zero and the state is nondegenerate. Therefore para-H2 has the lower energy and this state is favored at lower temperatures. The equilibrium concentration of H2 approaches three parts ortho to one part para at room temperature. As the nuclear spin quantum number of the deuteron is 1 rather than 1/2 for the proton (see Table 1), the D2 system is described by Bose-Einstein... 

B. Partition Function Calculation Using the PFRG Method... 

For more complicated reactions, RRKM theory can be applied, using the statistical mechanical partition functions calculated within the BAC-MP4 method. ... 

Nickel atoms have a low-lying excited state at about 200 cm (3.97 X 10- J). Assuming that both electronic states have a degeneracy of 3 and that no additional low-lying excited states contribute significantly to the electronic partition function, calculate at 1000 K. 

Here is the rotational partition function calculated at the transition state geometry and kj=o is the thermal rate calculated only for J=0. Thus, within the J-shifting approximation, the thermal rate constant can be obtained from a dynamical simulation for J=0 only. 

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