Exact Free Energy Calculations

There now exist several methods for predicting the free energy associated with a compositional or conformational change.7 These can be crudely classified into two types "exact" and "approximate" free energy calculations. The former type, which we shall discuss in the following sections, is based directly on rigorous equations from classical statistical mechanics. The latter type, to be discussed later in this chapter, starts with statistical mechanics, but then combines these equations with assumptions and approximations to allow simulations to be carried out more rapidly. 

Equations (42)-(44) are all calculated via the moment free energy. The three corresponding quantities obtained from the exact free energy (38) are, first, the chemical potentials conjugate to p([Pg.287]

The simulations (molecular dynamics [MD] and free energy calculations) have identified two important water chains in the enzyme (Figure 4.5), which connect Glu242, a known proton donor, both for chemical and pumped protons, to the binuclear Fe-Cu center (BNC), and to the region above heme a3, via its Prop D. The exact number of water molecules in this region, however, is not known. 

Other publications, however, report more accurate values of B3LYP gas phase Gibbs free energy calculations on aliphatic amines, diamines, and aminoamines. In 2007 Bryantsev et al. reported that B3LYP calculations with the basis set 6-31-h-G had a mean absolute error of 0.78 kcal/mol from experimental values of the gas phase basicity (AGg s) of the reverse reaction of equation 1 reported in the NIST database [58]. This accuracy is comparable to that of expensive, high level model chemistries, but because the experimental values have uncertainties of 2 kcal/mol, it is difficult to discern exactly how accurate the calculations are in comparison to values in the other publications [81]. The take-home message remains the same always benchmark DFT calculations for the systems you are interested in computing [52]. 

For a system at a nonideal state, one method is to find a reversible path from the current state of interest to a reference state (Frenkel and Smit 1996). The ideal gas is a good choice for the reference state because the exact free energy can be easily calculated. The usual approach to find this particular path is thermodynamic integration and is readily explained from a thermodynamic relation as follows ... 

Parameter A can be chosen to be any pointer of variation within the system for example, a torsional rotation to be followed in small angular steps or, more radically, the change of a solute molecule into a solvent molecule, of a hydroxyl group into a methyl group, or the actual transmutation of a reactant into a product in a chemical reaction, provided a suitable hamiltonian is available. In principle one could chose the starting state as the ideal gas, use equation 9.25 to calculate the exact free energy by statistical mechanics, and then use 9.26 or 9.27 to turn on the intermolecular potential and obtain the value of the free energy of a condensed phase. 

We have, so far, discussed the treatment of the free energy d/i due to the electron density fluctuation in solution. As a result a simple and exact functional has been formulated in terms of the energy distribution functions which describes Sfi. By virtue of the versatile form of the fiinctional it can also be applied to the perturbation approach for the QM/MM-ER method. The major purpose of this subsection is to introduce our recent development which combines QM/MM-ER with a second-order perturbation theory (PT2) [76] and demonstrate its efficiency. With this method the free energy calculation can be substantially expedited since the... 

The most conunon choice for a reference system is one with hard cores (e.g. hard spheres or hard spheroidal particles) whose equilibrium properties are necessarily independent of temperature. Although exact results are lacking in tluee dimensions, excellent approximations for the free energy and pair correlation fiinctions of hard spheres are now available to make the calculations feasible. 

It is possible to calculate derivatives of the free energy directly in a simulation, and thereby detennine free energy differences by thenuodynamic integration over a range of state points between die state of interest and one for which we know A exactly (the ideal gas, or hanuonic crystal for example) ... 

In the present case, each endpoint involves—in addition to the fully interacting solute—an intact side chain fragment without any interactions with its environment. This fragment is equivalent to a molecule in the gas phase (acetamide or acetate) and contributes an additional term to the overall free energy that is easily calculated from ideal gas statistical mechanics [18]. This contribution is similar but not identical at the two endpoints. However, the corresponding contributions are the same for the transfonnation in solution and in complex with the protein therefore, they cancel exactly when the upper and lower legs of the thermodynamic cycle are subtracted (Fig. 3a). 

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