Statistical artifact? Free energy

These arguments do not invalidate the immense array of linear free energy relationships gathered over the past 70 years or so. Class II systems such as Hammett or Taft relationships can be excluded from these considerations because the two variables are independent. The following arguments can be employed to demonstrate experimentally that a Class I correlation such as a Bronsted or a Leffler Equation does not arise from a statistical artifact in a system under investigation. Rearranging the Leffler Equation (Equation 27) yields Equation (29). 

At the outset we emphasize, however, that F c) is not a well-defined quantity thermodynamic potentials are well defined for thermal equilibrium states only states with (d F (c)ldc )T < 0 violate the basis laws of statistical thermodynamics. For Ccoex < c < CcoeJ- the only well-defined free energy is the free energy which corresponds to the lever rule, i.e., F(c) = F(cil, T))X + F(c ,(7))(l - X), i.e., a linear function of the concentration. The double-well shape of F(c) obtained from mean-field theories, such as the Bragg-Williams approximation of binary mixtures, is an artifact of an uncontrolled approximation. So the singular behavior resulting at the spinodal should not be taken seriously. 

As with atomic EAs, comparisons to higher-level calculations suggest that correlation effects on VDEs beyond the CCSD(T) level are quite small for molecular anions.Consider, for example, the notoriously challenging HNC and HCN anions, " whose binding energies are only 0.004 eV and w 0.002 eV, respectively, with the VDE for HCN arising almost entirely from electron correlation effects. For these two species, VDEs computed at the CCSD(T) level and the CCSDT level agree to within 0.001 eV. The (H20) anion provides another example here, the VDE computed at the CCSD(T) leveP lies within the statistical error bars of a quantum Monte Carlo (QMC) calculation, the latter of which is free of basis-set artifacts and does not require truncation of the excitation level. 

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