Relative free energy

Estimating Relative Free Energies from a Single Simulation of the Initial State... 

Liu, H., Mark, A. E., van Gunsteren, W. F. Estimating the relative free energy of different molecular states with respect to a single reference state. J. Phys. Chem. 100 (1996) 9485-9494... 

Many different approaches have been suggested as possible approaches to this problem, from the 1960s onwards [Verwer and Leusen 1998]. What is obvious from all of these ellorts is that this is an extremely difficult problem. Both thermodynamics and kinetics can be important in determining which crystalline form is obtained under a certain se1 of experimental conditions. Kinetic effects are particularly difficult to take into accouni and so are usually ignored. A proper treatment of the thermodynamic factors would lequire one to deal with the relative free energies of the different possible polymorphs... 

Schafer H, W F van Gunsterten and A E Mark 1999. Estimating Relative Free Energies from a Sing] Ensemble Hydration Free Energies, Journal of Computational Chemistry 20 1604-1617. 

The free energy of activation at the QCISD(T)/6-31 H-- -G(d,p) level amounts to 21.1 kcal/mol. According to the authors, the large electron density redistribution arising upon cyclization makes it necessary to use extended basis sets and high-order electron correlation methods to describe the gas-phase thermodynamics, which indicates clearly the gas-phase preference of the azido species. However, the equilibrium is shifted toward the tetrazole as the polarity of a solvent is increased. For instance, SCRF calculations (e = 78.4) yield a relative free energy of solvation with respect to the cw-azido isomer of —2.4 kcal/mol for the tmns-zziAo compound and of —6.8 kcal/mol for the tetrazole isomer. At a much lower level, the... 

In summary for non-metal transfer situations chemical thermodynamics is a useful guide to probable behaviour. The transfer of a non-metal, X, dissolved in a molten metal, M to another metal M", will depend on the relative free energies of formation of M X and M X (see Section 7.6). Thus sodium will give up oxygen to Zr, Nb, Ti and U, as the free energy of oxide formation of these metals is greater than that for sodium on the other hand, sodium will remove oxygen from oxides of Fe, Mo and Cu unless double oxides are formed. 

There have, however, been attempts to correlate Q-e values and hence reactivity ratios to, for example, c NMR chemical shifts 50 or the results of MO calculations 51153 and to provide a better theoretical basis for the parameters. Most recently, Zhan and Dixon153 applied density functional theory to demonstrate that Q values could be correlated to calculated values of the relative free energy for the radical monomer reaction (PA + Mn — PA ). The e values were correlated to values of the electronegativities of monomer and radical. 

Various statistical treatments of reaction kinetics provide a physical picture for the underlying molecular basis for Arrhenius temperature dependence. One of the most common approaches is Eyring transition state theory, which postulates a thermal equilibrium between reactants and the transition state. Applying statistical mechanical methods to this equilibrium and to the inherent rate of activated molecules transiting the barrier leads to the Eyring equation (Eq. 10.3), where k is the Boltzmann constant, h is the Planck s constant, and AG is the relative free energy of the transition state [note Eq. (10.3) ignores a transmission factor, which is normally 1, in the preexponential term]. 

Relative free energy changes along the reaction coordinate may be calculated via MD simulations. Subsequently, they may be combined with the QM total energies of the reacting QM subsystem to obtain the overall free energy profiles associated with the reaction steps in the following way... 

Initially, we have applied the modified NEB method to the calculation of both steps of the 40T catalyzed reaction. The free energy profiles and relative free energies obtained with this method were compared to our previously determined profiles [33], As we had previously shown, the calculated MEPs for Ref. [33] determined with the reaction coordinate driving method, and the MEPs for Ref. [25] calculated with the parallel path optimizer method, agree in the overall shape and relative potential energies. This provides a good starting point for our comparison. 

Figure 14-4. Intrinsic reaction coordinate for the transesterification of the dinucleotide model with B3LYP and M06-2X functionals. Relative free energies of reaction and activation are provided in kcal/mol...

Wang, C. X. Liu, H. Y. Shi, Y. Y. Huang, F. H., Calculations of relative free energy surfaces in configuration space using an integration method, Chem. Phys. Lett. 1991, 179, 475 178... 

Fig. 2.5. Possible applications of a coupling parameter, A, in free energy calculations, (a) and (b) correspond, respectively, to simple and coupled modifications of torsional degrees of freedom, involved in the study of conformational equilibria (c) represents an intramolecular, end-to-end reaction coordinate that may be used, for instance, to model the folding of a short peptide (d) symbolizes the alteration of selected nonbonded interactions to estimate relative free energies, in the spirit of site-directed mutagenesis experiments (e) is a simple distance separating chemical species that can be employed in potential of mean force (PMF) calculations and (f) corresponds to the annihilation of selected nonbonded interactions for the estimation of e.g., free energies of solvation. In the examples (a), (b), and (e), the coupling parameter, A, is not independent of the Cartesian coordinates, x. Appropriate metric tensor correction should be considered through a relevant transformation into generalized coordinates...

Nanda, H. Lu, N. Kofke, D. A., Using non-Gaussian density functional fits to improve relative free energy calculations, J. Chem. Phys. 2005,122, 134110 1-8... 

Ota, N. Stroupe, C. Ferreira da Silva, I.M.S. Shah, S.A. Mares-Guia, M. Brunger, A.T., Non-Boltzmann thermodynamic integration (NBTI) for macromolecular systems relative free energy of binding of trypsin to benzamidine and benzylamine, Proteins 1999, 37, 641-653... 

Two concepts are helpful in understanding and characterizing relative free energy computational errors phase space relationships and probability distribution functions of perturbations. 

Fig. 6.5. Graphical illustration of the inaccuracy model and the relative free energy error in forward and reverse free energy calculations. A limit-perturbation Xf is adopted to (effectively) describe the sampling of the distribution the regions above x/ are assumed to be perfectly sampled while regions below it shaded area) are never sampled. We may also put a similar upper limit x f for the high-rr tail, where there is no sampling for regions above it. However, this region (in a forward calculation) makes almost zero contribution to the free energy calculation and its error. Thus for simplicity we do not apply such an upper limit here...

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