Free energy inner shell

The right-most term is similar to the familiar PDT formula except that the indicator function combinations forbid binding of solution molecules to the defined inner shell. That last factor is recognized as the Boltzmann factor of the hydration free energy that would result if inner-shell binding were prohibited. The Km are recognizable ratios of equilibrium concentrations - equilibrium constants - that are discussed... 

The identifier HC means that this is the solvation free energy for the case of a hardcore solute with the excluded-volume region established by the inner-shell definition. 

For ionic as for molecular solutes (Section III.3), some studies have applied the discrete molecular model to the solvent in the immediate environment of the solute, and treated the remainder as a continuum. This can in principle help to deal with the problem of inner-shell structure as well as that of long-range effects. Thus Straatsma and Berendsen used the Bom equation to correct simulation-obtained free energies of hydration for six monatomic ions.174 This helped in some instances but not in others. 

In semiclassical ET theory, three parameters govern the reaction rates the electronic couphng between the donor and acceptor (%) the free-energy change for the reaction (AG°) and a parameter (X.) related to the extent of inner-shell and solvent nuclear reorganization accompanying the ET reaction [29]. Additionally, when intrinsic ET barriers are small, the dynamics of nuclear motion can limit ET rates through the frequency factor v. These parameters describe the rate of electron transfer between a donor and acceptor held at a fixed distance and orientation (Eq. 1),... 

The free energy required to reorganize the solvent molecules around the reactants (the outer coordination shell) and to reorganize the inner coordination shell of the reactants. These are termed and X, respectively. 

Equation (3) incorporates relativistic effects, effects of target density, and corrections to account for binding of inner-shell electrons, as well as the mean excitation energy C/Z is determined from the shell corrections, S/2 is the density correction, Ifj accounts for the maximum energy that can be transferred in a single collision with a free electron, m/M is the ratio of the electron mass to the projectile mass, and mc is the electron rest energy. If the value in the bracket in Eq. (4) is set to unity, the maximum energy transfer for protons... 

The electron, upon excitation, is ejected from an inner shell into vacuum and the energy of the free electron is then measured. This technique is called X-ray photoelectron spectroscopy. If the electron is ejected from the valence band by ultraviolet radiation, the technique is called ultraviolet photoelectron spectroscopy. Excitation energies not greater than those provided by ultraviolet radiation are necessary for electron excitation from the valence band or for electrons from the valence shell of adsorbed molecules. 

Calculations by Gryzinski and Kowalski (1993) for inner shell ionization by positrons also confirmed the general trend. Theirs was essentially a classical formulation based upon the binary-encounter approximation and a so-called atomic free-fall model, the latter representing the internal structure of the atom. The model allowed for the change in kinetic energy experienced by the positrons and electrons during their interactions with the screened field of the nucleus. 

The different emission products which are possible after photoionization with free atoms lead to different experimental methods being used for example, electron spectrometry, fluorescence spectrometry, ion spectrometry and combinations of these methods are used in coincidence measurements. Here only electron spectrometry will be considered. (See Section 6.2 for some reference data relevant to electron spectrometry.) Its importance stems from the rich structure of electron spectra observed for photoprocesses in the outermost shells of atoms which is due to strong electron correlation effects, including the dominance of non-radiative decay paths. (For deep inner-shell ionizations, radiative decay dominates (see Section 2.3).) In addition, the kinetic energy of the emitted electrons allows the selection of a specific photoprocess or subsequent Auger or autoionizing transition for study. 

The above discussion shows that the electrostatic free energy of solvation can be divided into an coordination shell or inner solvation sphere in which eA is close to 1, where the Pu interaction depends only on - xAa, and an outer solvation sphere where the PA interaction depends to a good approximation on Eqs. (55)-(57), but in which the electrostatic Gibbs energy may be approximated by the integral in Eq. (58), which resembles the Bom charging equation, but it is obtained in a different way with a more definite physical meaning. 

Figure 7. Calculated two-dimensional energy surfaces for heterogeneous non-Franck-Condon transition of the three-dimensional 6-oscillator trikisoctahedral Inner Shells of 3+ and 2+ ions at (A) equal potential energies after ground state energy correction (B) at equal ground state free energies. Barrier height in B +22.9kT at 298 K (+56.9 kJ/mole, +0.59 eV, experimentally +0.59 eV177).

It also is important to note that the aforementioned treatments of free-energy barriers refer only to weak-overlap reactions. This corresponds to the curve PAS in Fig. 1, where the transition-state energy is essentially unaffected (at least in a specific manner) by the proximity of the metal surface. When these reactant-electrode interactions become sufficiently strong and specific, marked decreases in both the inner- and outer-shell intrinsic barriers can be anticipated. This is discussed further in Sects. 3.5.1 and 4.6. 

Figure 7.2 Quasi-chemical contributions of the hydration free energy of Be (aq). Cluster geometries were optimized using the B3LYP hybrid density functional (Becke, 1993) and the 6-31- -G(d, p) basis set. Frequency calculations confirmed a true minimum, and the zero point energies were computed at the same level of theory. Single-point energies were calculated using the 6-311- -G(2d, p) basis set. A purely inner-shell n = 5 cluster was not found the optimization gave structures with four (4) inner-sphere water molecules and one (1) outer-sphere water molecule. For n = 6 both a purely inner-shell configuration, and a structure with four (4) inner-shell and two (2) outer-shell water molecules were obtained. The quasi-chemical theory here utilizes only the inner-shell structure. O - rin [/ff -f (left ordinate) vs. n. A ...

We focus first on the outer-shell contribution of Eq. (7.8), p. 145. That contribution is the hydration free energy in liquid water for a distinguished water molecule under the constraint that no inner-shell neighbors are permitted. We will adopt a van der Waals model for that quantity, as in Section 4.1. Thus, we treat first the packing issue implied by the constraint Oy [1 i>a (7)] of Eq. (7.8) then we append a contribution due to dispersion interactions, Eq. (4.6), p. 62. Einally, we include a contribution due to classic electrostatic interactions on the basis of a dielectric continuum model. Section 4.2, p. 67. 

The second basis set is the split-valence (or extended) 4-31G basis.m In this basis set, inner shell orbitals are written as the sum of four gaus-sian functions while valence orbitals are split into inner and outer parts consisting of three gaussians and one gaussian, respectively. Because the ratio of the inner and outer contributions is free to be determined by the SCF procedure, this basis set provides a more flexible description of the electronic distribution than ST0-3G. It has proved more reliable in energy comparisons than STO-3G. T2-i4) Wg therefore carry out for this purpose, single 4-31G calculations at the ST0-3G optimized geometries for each molecule. 

---