The Direct Counting Method

If all identical atoms in the reactants of a given reaction are labelled, a number / of sets of products can be obtained differing only for the labels of the identical atoms. By the same talk, if the reaction is reversible, a number r of different sets of reactants can be obtained by labeUing all identical atoms of the products. After evaluation of the number of microspedes of products and reactants, / and r, respectively, the statistical factor of the generic equilibrium in Eq. [34] can be calculated by Eq. [36], where the factorial of the stoichiometric coefficients is required to correct for the redundancy of the generated microspecies 

From the latter example, it appears that the direct counting method is more suitable than the symmetry number method to obtain statistical factors 

---

It is evident that the combined sum of states, N,2, is given by a convolution between the sum and the density P2, or by Pi A 2 (not the convolution between two sum functions). This property is essential in understanding the principles underlying the direct count method for the sum of states. 

The most accurate procedure for determining the density of harmonic vibrational states is by the direct count method. A particularly clever scheme for doing this was proposed by Beyer and Swinehart (1973). As demonstrated by Gilbert and Smith (1990), this approach is based on the convolution of state densities. Suppose that the system consists of s harmonic oscillators with vibrational frequencies, co, = v,/c (cm" )- Each will have a series of equally spaced states located at , = nco, (n = 0, 1,. . . ). We choose the zero of energy at the molecule s zero point energy, and divide the energy into bins. The vibrational frequencies must be expressed as integral numbers of bin sizes, for example, as multiples of 10 cm for a 10 bin size. A convenient bin size is 1 cm so that the s frequencies can be simply rounded off to the nearest wave-number. 

Thus, at very low energies, the direct count method is recommended. At higher energies, the faster steepest descent approach is preferred, especially when the densities or sums of states are required at a high resolution. It must be recalled that both of these approaches are only approximate because they do not include anharmonicities. 

An accurate and consistent evaluation of statistical factors in self-assembly processes is of crucial importance to predict the expected stability constant in the absence of cooperative effects and, therefore, to spodight the emergence of either positive or negative cooperativity as a marked deviation firom statistical behaviour. However, the evaluation of statistical factors can be controversial and doubtful sometimes. A critical re-examination of the methods to assess statistical factors in self-assembly processes has been published in 2007. " Two methods appear the most useful, namely, the symmetry number method and the direct counting method. The two methods if properly appHed give the same results however, the symmetry... 

The direct count methods are dependent on the ability of the investigator to recover bacteria from the sample, to observe stained particles, and to identify such particles as bacteria. Each one of these steps could limit the accuracy of the method, depending on the circumstances. 

The CO stretch corresponds to the mode of vibration leading to the product CO vibrational excitation, and the CF stretch corresponds to the reaction coordinate. Equation (I) can be evaluated by either the direct-count method or the approximation of Whitten and Rabinovitch (27) ... 

The direct count method based on Eq. (l) predicted exactly the same distribution. 

Because physiological deterioration is generally accompanied by an increase in bacterial population, as pointed out by Nielsen, Wolford, and Campbell (33), estimation of bacterial numbers might serve as the basis of a test for condition. Obviously, the plate count method is not adaptable because of the time limitations imposed. Direct microscopic count would be much more appropriate, especially if a positive field count were substituted for cell count as suggested by Wolford (39). 

Methods for calculating these quantities are discussed in detail in the book by Holbrook, Pilling and Robertson [29]. These methods fall into several basic categories - classical approximations, inversion of the partition function, direct count methods and Monte Carlo methods, each of which is introduced briefly. 

Comparison with direct count methods for systems of harmonic oscillators indicates that the classical density of states is generally an underestimate. However, it is so convenient to have an explicit formula for the density of states that more accurate semiclassical approximations have been developed. 

Gordon and Davison136 initiated the path counting method. In the path counting method the number of Kekule structures Kof a benzenoid B is equal to the number of the mutually self-avoiding directed peak-to-valley paths 138,139... 

Figure 21. Vibrational density of states (pv b) versus vibrational energy (Evib) in anthracene calculated using a direct count method and the frequencies of Refs. 64. Given in the Figure are characteristic times for IVR in the absent, restricted, and dissipative regimes.

Surprisingly, one of the major difficulties encountered when applying this model in coordination chemistry is cormected with the calculation of reliable statistical factors. Two parallel methods have been developed by using either the symmetry numbers of the molecules or the direct count of the microspecies formed in the reactants and products (Ercolani et al.,... 

---