Direct-count method

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. 

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.

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. 

One type of direct counting method which has been standardised is the Howard mould count which is widely used by the Food and Drug Administration in the USA. Although widely used, the method is inaccurate, subjective, and shows poor reproducibility. 

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... 

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

Direct counting method, 46—48 Dispersion interactions, 205 DL. See Dynamic library (DL)... 

A counting chamber can be used to determine the number of bacterial cells in water. In such a chamber, the number of bacterial cells in a particular, small quantity of water (e.g. a cube with 0.1 mm sides = 0.001 mm ) can be counted under a microscope. Each bacterial cell determined in such a cube represents 10 germs in 1 ml. Thus, in order to determine a bacterial count at all with this method, the germ content of the water must be very high. This direct counting method cannot be used for water containing low levels of germs. 

The IVR rates in the aniline(Ar)i case are rather sensitive to the densities of states. In this case, the values of pi (=N(E used in the construction of the IVR transition probability matrix are calculated with a direct count method, Vibrational frequencies of 45 cm l (stretch), and IS cm l (both bends) and anharmonicities of 3% (i.e., Avi, i+i /Avi-i,i =. 97) are assumed. 

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. 

The quantum mechanical trace in equation (17) is most readily carried out for separable Hamiltonians. This became feasible for collections of harmonic oscillators with the advent in 1973 of the discrete convolution algorithm of Beyer and Swinehart, which was generalized to the case of arbitrary separable Hamiltonians by Stein and Rabinovitch. These direct-count methods are exceedingly fast and exact for separable Hamiltonians. Direct counts can also be implemented for energy levels derived from perturbative expansions of the Hamiltonian. 

Direct counting Method of inferring " C concentration by directly easuring ions by accelerator mass spectrometry. 

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