Conformational space discrete approximations

It is not uncommon for a single molecule to have multiple populations. At non-zero temperatures, the population of different conformations will be dictated by Boltzmann statistics. If we make the approximation that we may neglect the continuous character of conformational space and simply work with discrete potential energy minima, we can replace a statistical mechanical probability integral with a discrete sum, and the equilibrium fraction F of any given conformer A at temperature T may be computed as... 

A more efficient approach to the discrete approximation of the conformational space is based on decomposing the molecule into a number of building blocks or structural elements, each defined as a group of connected atoms whose conformation is known or can be separately investigated. In organic molecules, typical conformational units include rings... 

The CMF approach can offer an alternative solution to this problem. Instead of using discrete sets of representative conformations, one can consider for each molecule an infinite number of conformations organized into a continuous manifold, so-called corrformational space . This provides the ability to apply functional data analysis not only to molecular fields but also to molecular geometry in a consistent way. Such corrformational space can be described by means of some probability density function pdf) in 3N-dimensional Euclidean space, where N is the number of atoms in the molecule under study. Having applied several approximations from the arsenal of statistical physics, one can obtain the following expression for calculating atomic kernels instead of Eq. (13.6) ... 

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