Boltzmann distribution, from conformational

The structures of most molecules in crystals are quite similar to their structures when the molecules are in isolation. (The principle exception is that crystals usually contain a single conformation, while fluid phases are likely to contain a Boltzmann distribution of conformations.) So for the most part, MM4 studies on crystals have led to rather few surprises. There are, however, things that can be learned by molecular mechanics from a smdy of molecules packed into crystals that cannot necessarily be easily learned in other ways. There is some introductory discussion of the usefulness and limitations of X-ray crystallography in determining molecular structures in Chapter 2. If the reader is unfamiliar with the technique, it may be helpful to refer to that discussion and the references cited therein. 

It will be noted that the first three terms of Eq. (11.3) are the same as Eq. (11.2), with the additional last four terms from statistical mechanics added. Taking these last four terms in order, POP is the additional energy that comes from the Boltzmann distribution of conformations, TOR is needed to account for extra torsional energy that results when there are low barriers to rotation in the molecule, T/R is the energy from the translational and rotational degrees of freedom of the whole molecule plus R (the gas constant) to convert energy to enthalpy, and E a, is the vibrational energy of the molecule. These will be discussed in turn. 

There are two additive terms to the energy, POP and TORS, that have not been mentioned yet because they are zero in minimal ethylene. The POP term comes from higher-energy conformers. If the energy at the global minimum is not too far removed from one or more higher conformational minima, molecules will be distributed over the conformers according to the Boltzmann distribution... 

Figure 1.12 Rotational level distributions in HC1 (v = 2) from the H + Cl2 and Cl + HI reactions [264], The abscissa records both J, the rotational quantum number, and Fj, the fraction of available energy present as rotation. The arrows indicate the limits of excitation determined by the fact that E (v = 2) + Zi(./)cannot exceed Q, the total available energy. The broken lines indicate best fit Boltzmann distributions and show that the majority of the rotators are in a highly non-Boltzmann distribution. The subsidiary peaks at low J conform to a low-temperature Boltzmann distribution.

Another procedure to overcome the inefficiency of Metropolis Monte Carlo is adaptive importance sampling.194-196 In this technique, the partition function (and quantities derived from it, such as the probability of a given conformation) is evaluated by continually upgrading the distribution function (ultimately to the Boltzmann distribution) to concentrate the sampling in the region (s) where the probabilities are highest. 

Palm et al. [3] took into account the flexibility of molecules by using molecular mechanics to calculate an averaged PSA according to a Boltzmann distribution. Later Clark [4,5] found that the use of a representative conformation was sufficient for the calculation of reliable PSA values. Ertl [6] developed a method to calculate PSA as the sum of fragment contributions and proposed a topological PSA (TPSA). The advantage of TPSA is that it can be directly calculated from the 2D chemical structure, which makes the calculation rapid and reproducible. 

Hence, in the light of our both accounts of causality, the molecular dynamics model represents causal processes or chains of events. But is the derivation of a molecule s structure by a molecular dynamics simulation a causal explanation Here the answer is no. The molecular dynamics model alone is not used to explain a causal story elucidating the time evolution of the molecule s conformations. It is used to find the equilibrium conformation situation that comes about a theoretically infinite time interval. The calculation of a molecule s trajectory is only the first step in deriving any observable structural property of this molecule. After a molecular dynamics search we have to screen its trajectory for the energetic minima. We apply the Boltzmann distribution principle to infer the most probable conformation of this molecule.17 It is not a causal principle at work here. This principle is derived from thermodynamics, and hence is statistical. For example, to derive the expression for the Boltzmann distribution, one crucial step is to determine the number of possible realizations there are for each specific distribution of items over a number of energy levels. There is no existing explanation for something like the molecular partition function for a system in thermodynamic equilibrium solely by means of causal processes or causal stories based on considerations on closest possible worlds. 

While it is true that large, high-energy deformations are less likely to occur (and be observed) than small, low-energy ones, there is a serious flaw in these arguments. An ensemble of structural parameters obtained from chemically different compounds in a variety of crystal structures does not even remotely resemble a closed system at thermal equilibrium and does not therefore conform to the conditions necessary for the application of the Boltzmann distribution. It is thus misleading to draw an analogy between this distribution and those derived empirically from statistical analysis of observed deformations in crystals [20]. 

Figure 1. Temperature-dependent macrostate dissection of a two-dimensional potential-energy landscape, (a) Potential V as a function of two coordinates, (b) Gibbs-Boltzmann distribution p at low (left), medium (middle), and high (right) temperatures, (c) Corresponding p at each temperature constructed from solutions to the characteristic packet equations, (d) Characteristic packet solution parameters R° and 0 for each macrostate (labeled with indices a, / , 6, y, and s). (e) Trajectory diagram of macrostate conformational free energies Fa as a function of temperature. (Reproduced from Church et al. [17] with permission obtained.)...

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