Partitioning the Free Energy

The Hamiltonian can be written as a sum of contributions from bond stretching, angle bending, and so on  

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Since E = - 2 iU)tn., a given spin excess value 2m. implies a given energy partition. The free energy for such a specific energy partition is... 

A major drawback of MD and MC techniques is that they calculate average properties. The free energy and entropy fiinctions caimot be expressed as simple averages of fimctions of the state point y. They are directly coimected to the logaritlun of the partition fiinction, and our methods do not give us the partition fiinction itself Nonetheless, calculating free energies is important, especially when we wish to detennine the relative thenuodynamic stability of different phases. How can we approach this problem ... 

The solubilization of diverse solutes in micelles is most often examined in tenns of partitioning equilibria, where an equilibrium constant K defines the ratio of the mole fraction of solute in the micelle (X and the mole fraction of solute in the aqueous pseudophase. This ratio serves to define the free energy of solubilization -RT In K). 

For these sequences the value of Gj, is less than a certain small value g. For such sequences the folding occurs directly from the ensemble of unfolded states to the NBA. The free energy surface is dominated by the NBA (or a funnel) and the volume associated with NBA is very large. The partition factor <6 is near unify so that these sequences reach the native state by two-state kinetics. The amplitudes in (C2.5.7) are nearly zero. There are no intennediates in the pathways from the denatured state to the native state. Fast folders reach the native state by a nucleation-collapse mechanism which means that once a certain number of contacts (folding nuclei) are fonned then the native state is reached very rapidly [25, 26]. The time scale for reaching the native state for fast folders (which are nonnally associated with those sequences for which topological fmstration is minimal) is found to be... 

The HYBOT descriptors were successfully applied to the prediction of the partition coefficient log P (>i--octanol/water) for small organic componnds with one acceptor group from their calculated polarizabilities and the free energy acceptor factor C, as well as properties like solubility log S, the permeability of drugs (Caco-2, human skin), and for the modeling of biological activities. 

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

LS now consider the problem of calculating the Helmholtz free energy of a molecular 1. Our aim is to express the free energy in the same functional form as the internal that is as an integral which incorporates the probability of a given state. First, we itute for the partition function in Equation (6.21) ... 

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

Calculations of relative partition coefficients have been reported using the free energy perturbation method with the molecular dynamics and Monte Carlo simulation methods. For example, Essex, Reynolds and Richards calculated the difference in partition coefficients of methanol and ethanol partitioned between water and carbon tetrachloride with molecular dynamics sampling [Essex et al. 1989]. The results agreed remarkably well with experiment... 

The overall free energy can be partitioned into individual contributions if the thermo-lynamic integration method is used [Boresch et al. 1994 Boresch and Karplus 1995]. The itarting point is the thermodynamic integration formula for the free energy ... 

The last equation follows from the definition of the partition function, eq. (16.2). Analogously to eq. (16.10) the free energy difference can be evaluated as an ensemble average. 

Given an energy function as in eq. (16.21), the partition function and thereby also the free energy are functions of A. 

The free-energy increment 6G(Am— Am.) associated with the change of Um to Um, can be obtained by examining the corresponding change in the partition function of the system. That is,... 

The assessment of k is of some importance since it relates to the question as to how much if any of the free energy of activation barrier is due to the spin-forbidden character of the transition. From the experimental point of view, Eq. (49) shows that the transmission coefficient k and the activation entropy AS appear in the temperature-independent part of the rate constant and thus cannot be separated without additional assumptions. Possible approaches to the partition of — TAS have been discussed in Sect. 4 for spin transition complexes of iron(II) and iron(III). If the assumption is made that the entropy of activation is completely due to k, minimum values between 10 and 10 are obtained for iron(II) and values between 10 and 10 for iron(III). There is an increase of entropy for the transition LS -+ HS and thus the above assumption implies that the transition state resembles the HS state. On the other hand, volumes of activation indicate that the transition state should be about midway between the LS and HS state. This appears indeed more reasonable and has the... 

The net retention volume and the specific retention volume, defined in Table 1.1, are important parameters for determining physicochemical constants from gas chromatographic data [9,10,32]. The free energy, enthalpy, and. entropy of nixing or solution, and the infinite dilution solute activity coefficients can be determined from retention measurements. Measurements are usually made at infinite dilution (Henry s law region) in which the value of the activity coefficient (also the gas-liquid partition coefficient) can be assumed to have a constant value. At infinite dilution the solute molecules are not sufficiently close to exert any mutual attractions, and the environment of each may be considered to consist entirely of solvent molecules. The activity... 

As suggested previously, the density of states has a direct connection to the entropy, and, hence, to thermodynamics, via Boltzmann s equation. Alternately, we can consider the free energy analogue, using the Laplace transform of the density of states - the canonical partition function ... 

To illustrate how stratification works in the context of free energy calculations, let us consider the transformation of state 0 into state 1 described by the parameter A. We further assume that these two states are separated by a high-energy barrier that corresponds to a value of A between Ao and Ai. Transitions between 0 and 1 are then rare and the free energy estimated from unstratified computer simulations would converge very slowly to its limiting value, irrespective of the initial conditions. If, however, the full range of A is partitioned into a number of smaller intervals, and... 

To understand better the difficulties connected with the partitioning of free energy into contributions, let us break down the potential energy into a sum of two terms,... 

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