Free energy approximation

Specifically, when A and B are salt solutions and medium m is a nonconducting dielectric without the high dielectric constant of water, then the A s go to 1. The leading term in the n = 0 contribution to the interaction free energy approximation (LI. 5) becomes (see Fig. LI.46)... 

Relative free energies were also calculated for clusters of low-energy conformers. This analysis is useful because it is difficult to capture the tme accessibiUty of individual structures based on a pointwise approximation of entropic effects. That is, the harmonic free energy approximation does not provide a continuous free energy landscape. By clustering structures into larger... 

The linear free energy approximation states that the free energy of folding is linearly dependent on the concentration of the denaturant, a statement which is found empirically to be true... 

Harkins then estimated Tc for diamond to be about 6700 K and, using Eq. HI-10, found the entropy correction at 25°C to be negligible so that the preceding values also approximate the room temperature surface free energies. These... 

For the steady-state case, Z should also give the forward rate of formation or flux of critical nuclei, except that the positive free energy of their formation amounts to a free energy of activation. If one correspondingly modifies the rate Z by the term an approximate value for I results ... 

The effect is to write the adsorption free energy or, approximately, the energy of adsoiption Q as a sum of electrostatic and chemical contributions. A review is provided by Ref. 156. 

This expression contains the contribution of the short-range potential included earlier in, so that the excess free energy, to this level of approximation, is... 

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. 

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

By integrating over the hard cores in the SL expansion and collecting tenns it is easily shown this expansion may be viewed as a correction to the MS approximation which still lacks the complete second virial coefficient. Since the MS approximation has a simple analytic fomi within an accuracy comparable to the Pade (SL6(P)) approximation it may be more convenient to consider the union of the MS approximation with Mayer theory. Systematic improvements to the MS approxunation for the free energy were used to detemiine... 

In two classic papers [18, 46], Calm and Flilliard developed a field theoretic extension of early theories of micleation by considering a spatially inliomogeneous system. Their free energy fiinctional, equations (A3.3.52). has already been discussed at length in section A3.3.3. They considered a two-component incompressible fluid. The square gradient approximation implied a slow variation of the concentration on the... 

It may be iisefiil to mention here one currently widely applied approximation for barrierless reactions, which is now frequently called microcanonical and canonical variational transition state theory (equivalent to the minimum density of states and maximum free energy transition state theory in figure A3,4,7. This type of theory can be understood by considering the partition fiinctions Q r ) as fiinctions of r similar to equation (A3,4.108) but with F (r ) instead of V Obviously 2(r J > Q so that the best possible choice for a... 

Equation (ASA. 110) represents the canonical fonn T= constant) of the variational theory. Minimization at constant energy yields the analogous microcanonical version. It is clear that, in general, this is only an approximation to the general theory, although this point has sometimes been overlooked. One may also define a free energy... 

Within this contimiiim approach Calm and Flilliard [48] have studied the universal properties of interfaces. While their elegant scheme is applicable to arbitrary free-energy fiinctionals with a square gradient fomi we illustrate it here for the important special case of the Ginzburg-Landau fomi. For an ideally planar mterface the profile depends only on the distance z from the interfacial plane. In mean field approximation, the profile m(z) minimizes the free-energy fiinctional (B3.6.11). This yields the Euler-Lagrange equation... 

Such linear free energy relationships are available for alkyl sulphates and for tire C4 to C9 homologues of tire dialkanoyl lecitliins (see table C2.3.3 for stmcture). Most of tire naturally occurring phospholipids are too insoluble to fonn micelles, but tire lower alkanoyl lecitliins, also known as phosphotidylcholines, do fonn micelles. The ernes for tliese homologues are listed in table C2.3.6. The approximately linear free energy relationship between tire alkyl chain iengtli and log cmc is given by ... 

Van Gunsteren, W.F., Beutler, T.C., Praternali, F., King, P.M., Mark, A.E., Smith, P.E. Computation of free energy in practice Choice of approximations and accuracy limiting factors, in Computer Simulations of Biomolecular Systems, Vol 2, W.F. van Gunsteren, P.K. Weiner and A.J. Wilkinson, eds. Escom, Leiden (1993) 315-348. 

A linear dependence approximately describes the results in a range of extraction times between 1 ps and 50 ps, and this extrapolates to a value of Ws not far from that observed for the 100 ps extractions. However, for the simulations with extraction times, tg > 50 ps, the work decreases more rapidly with l/tg, which indicates that the 100 ps extractions still have a significant frictional contribution. As additional evidence for this, we cite the statistical error in the set of extractions from different starting points (Fig. 2). As was shown by one of us in the context of free energy calculations[12], and more recently again by others specifically for the extraction process [1], the statistical error in the work and the frictional component of the work, Wp are related. For a simple system obeying the Fokker-Planck equation, both friction and mean square deviation are proportional to the rate, and... 

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