Free energy differences

This type of procedure is referred to as a kinetic resolution since the enantiomers of the racemic substrate exhibit different rates of reaction with the optically active compound, i.e. the diastereomeric transition states that arise from differences in e.g. non-bonded interactions have different free energies. Horeau and Nouaille (1966) estimate that a rate difference corresponding to A AG of the order of 0 2 cal mol at 25°C could in principle be revealed by this method. 

In this chapter, we focus on the method of constraints and on ABF. Generalized coordinates are first described and some background material is provided to introduce the different free energy techniques properly. The central formula for practical calculations of the derivative of the free energy is given. Then the method of constraints and ABF are presented. A newly derived formula, which is simpler to implement in a molecular dynamics code, is given. A discussion of some alternative approaches (steered force molecular dynamics [35-37] and metadynamics [30-34]) is provided. Numerical examples illustrate some of the applications of these techniques. We finish with a discussion of parameterized Hamiltonian functions in the context of alchemical transformations. 

Fig. 5.3. Comparison of different free energy estimators. Plotted are distributions of estimated free energies using sample sizes (i.e., number of independent simulation runs) of N = 100 simulations (solid lines), as well as N = 1, 000 (long dashed) and N = 10,000 simulations short dashed lines), (a) Exponential estimator, (5.44). (b) Cumulant estimator using averages from forward and backward paths, (5.47). (c) Cumulant estimator using averages and variances from forward and backward paths, (5.48). (d) Bennett s optimal estimator, (5.50)...

Fig. 5.4. Comparison of different free energy estimators for asymmetric perturbation from A = 0 to 2 within t = 1. Shown are distributions of free energies estimated using the direct exponential average, (5.44), averaged over forward and backward perturbations (solid line), averages (5.47) from forward and backward paths (long dashed line)-, averages and variances (5.48) from forward and backward paths (short dashed line)-, and Bennett s optimal estimator, (5.50), (dotted line). In all cases, free energies were estimated from N 1,000 simulations. The vertical arrow indicates the actual free energy difference of (3AA —6.6...

The degree of stereoselectivity is usually not large in these reactions and appears to be due to transition-state rather than initial-state interactions. In other words the diastereomeric transition states derived from the enantiomeric substrates have different free energies in the micelle. To this extent the situation is essentially no different to the stereoselectivity which is often observed in non-micellar reactions involving reactions of enantiomeric substrates with a chiral reagent. In some cases it is possible to identify the noncovalent interactions which are responsible for the stereoselectivity (Brown et al., 1981). 

The different free energy contributions lead, upon minimization with respect to the two length scales h and d, to different behaviors. Let us first consider the weak charging limit, i.e. the situation where the counterions leave the brush, d>h.ln this case, minimization of Fion + Fint with respect to the counterion height d in the limiting case of vanishing brush height (h = 0) and monomer volume = 0) leads to... 

Irrespective of the mechanism of resolution, HPLC CSPs work by providing a chiral environment for analyte stereoisomers to interact with. Resolution relies upon the formation of reversible, transient diastereomers on the CSP that have different free energies of interaction and therefore stability. The stereoisomer forming the most stable diastereomer with the CSP will be the most retained and vice versa. Free energy differences are typically small in such systems but may be large enough to produce useable resolutions provided the column efficiency is sufficient [41]. If column efficiency is insufficient to allow complete separation of the stereoisomers, inaccurate integrations can result in erroneous... 

All the above transitions are accompanied by changes in Seebeck coefficient, structural parameters, heat capacity and other characteristics. The V2O3 system has been explained in terms of a thermodynamic model which uses different free energy expressions for electrons in the itinerant and localized regimes (Honig Spalek, 1986). 

The slope of greater than 1 in Eq. 7-8 indicates that structural differences in the solutes have a somewhat greater impact on their partitioning behaviors in the hexadecane-water, as compared to the octanol-water system. This can be rationalized as arising from the different free-energy costs related to the cavity formation in the two solvents, which is larger in the bipolar octanol (see discussion in Chapter 6). 

If steric interactions are the main factors in determining the different free energies for the four possible transition states, the substituent would occupy preferentially that quadrant in which more space is available. If the quadrant preferred by the substituent is known, i.e., if the relative positions of L, S, Z, and H are known, we can predict the enantiomer that is formed prevailingly. The same model also allows us to predict which of the two unsaturated carbon atoms of the substrate will be bound to the metal and carbonylated eventually. For instance, in the case of a monosubstituted olefin (see Scheme VII), Isomer b will prevail over the sum of the antipodes of the other isomer (a + c) and Antipode a will prevail over Antipode c. 

The prevalence of one antipode in the chiral product obtained by hydroformylation of an olefin must be connected with the different free energy of the transition states in the step (or steps) in which asymmetric induction occurs. 

Is there a function which can represent the balance between these two opposing factors, and thus be a measure of the tendency for a reaction to take place Such a function does exist and is known as the free energy Junction or simply free energy. We now derive two different free energies, namely, Helmholtz free energy and Gibbs free energy. 

Different phases, say, a and P, in a given substance in general have different free energies and entropies at a specific temperature and pressure. First, let s look at the a phase. 

Stereoselectivity axiom 2 If two or more stereoisomers of different free energy can enter into or result from similar elementary processes, the rate of one will be preferred. 

Diastereoisomers have different free energies (internal energies) so reactions of this kind are similar to ordinary competitive reactions between two different substances, and need not be exemplified. 

While the preceding statements apply to the bulk of the crystal, the surface of the crystal is subject to the same constraints with some additional complexity. Specifically, because different point defects are likely to have different free energies of formation, there is an enhanced concentration of one of the two defects in the near surface region and a compensating layer of charge on the... 

A simplified version of this process is presented in Fig, 1. In this example, the two solute/CSP complexes, d-solute-CSP and i-solute-CSP, have different free energies, with the d-soIute-CSP complex being more stable, that is, the one with the lower free energy. As a result, the d isomer will remain on the column longer than the I isomer, and the two enantiomers will be resolved. 

Stability of Microemulsions. The first attempt to describe the microemulsion stability in terms of different free energy components was made by Ruckenstein and Chi (55) who evaluated the enthalpic (Van der Waals potential, interfacial free energy and the potential due to the compression of the diffuse double layer) and entropic... 

---