Exchange non-additivity

What about the exchange contribution This contribution does not exist in the polarization approximation. It appears only in symmetry-adapted perturbation theory, in pure form in the first-order energy correction and coupled to other effects in higher order energy corrections. The exchange interaction is difficult to interpret, because it appears as a result of the antisymmetry of the wave fimction (Pauli exclusion principle). The antisymmetry is forced by one of the postulates of quantum mechanics (see Chapter 1) and its immediate consequence is that the probability density of finding two electrons with the same spin and space coordinates is equal to zero. 

A CONSEQUENCE OF THE PAULI EXCLUSION PRINaPLE In an atom or molecule, the Pauli exclusion principle results in a shell-like electronic structure (electrons with the same spin coordinates hate each other and try to occupy different regions in space). The valence repulsion may be seen as the same effect manifesting itself in the intermolecular interaction. Any attempt to make the molecular charge distributions overlap or occupy the same space ( pushing ) leads to a violent increase in the energy. 

The Pauli exclusion principle leads to a deformation of the wave functions describing the two molecules (by projecting the product-like wave function by the antisymmetrizer A) with respect to the product-like wave function. The Pauli deformation (cf. Appendix Y) appears in the zeroth order of perturbation theory, whereas in the polarization approximation, the deformation of the wave function appears in the first order and is not related to the Pauli exclusion principle. 

The antisyiiimetrizc portakuS to the pcimutation o yiinneti of the wave function with respect to the coordinates of all electrons and therefore is different for a pair of molecules and for a system of three molecules. 

The expression for the three-bo(ty non-additivity of the valence repulsion [given by formula (f3.39), based on definition (13.37) of the first-order correction in symmetry-adapted perturbation theoiy and from definition (13.52) of the three-body contribution] is  

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First,we write down the exact expression for the first-order exchange non-additivity. 

Next, we see that the exchange non-additivity expression contains terms of the order of and higher, where S stands for the overlap integrals between the orbitals of the interacting molecules. 

Fig. 13.11. A scheme of the SE and TE exchange non-additivities. Rgs. (a), (b), (c) show the single exchange mechanism (SE). (a) Three non-interacting molecules (schematic representation of electron densities), (b) Pauli deformation of molecules A and B. (c) Electrostatic interaction of the Pauli deformation (resulting from exchange of electrons 1 and 2 between A and B) with the dipole moment of C. (d) The TE mechanism molecules A and B exchange an electron with the mediation of molecule C.

Additivity of the Electrostatic Interaction Exchange Non-additivity Induction Non-additivity... 

Implemented as outlined above, the PCM seems to correctly account for the main non-additive effects for cations in water. Except for cations like NH4 where exchange seems the principal source of non additivity [133], they are basically polarization of water in the electric field of the cation and electron transfer from water to the cation. A second water molecule nearby reduces both these effects, giving a less deep potential well in the effective two-body potential compared to the strictly two-body one. In the PCM picture, a distribution of negative charge on the cavity, due to the polarization of the dielectric continuum induced by the cation, decreases the electric field of the cation and hence both water polarization and electron transfer from water to the cation. 

A similar linearized formula can be written using approximate quantities Es, pA, and pb- Es and Es differ only in the exchange-correlation and non-additive kinetic energy parts ... 

The matrix Kemb relates the linear changes in the Coulomb, the exchange-correlation, and the non-additive kinetic energy components of the KSCED effective potential with the changes of in the electron density in the subsystem A ... 

Figure 4-3. Electrochemical techniques and the redox-linked chemistries of an enzyme film on an electrode. Cyclic voltammetry provides an intuitive map of enzyme activities. A. The non-turnover signal at low scan rates (solid lines) provides thermodynamic information, while raising the scan rate leads to a peak separation (broken lines) the analysis of which gives the rate of interfacial electron exchange and additional information on how this is coupled to chemical reactions. B. Catalysis leads to a continual flow of electrons that amphfles the response and correlates activity with driving force under steady-state conditions here the catalytic current reports on the reduction of an enzyme substrate (sohd hne). Chronoamperometry ahows deconvolution of the potenhal and hme domains here an oxidoreductase is reversibly inactivated by apphcation of the most positive potential, an example is NiFe]-hydrogenase, and inhibition by agent X is shown to be essentially instantaneous.

The interaction energy of N molecules is not pairwise additive i.e., it is not the sum of the interactions of all possible pairs of molecules. Among the energy corrections up to the second order, the exchange and, first of all, the induction terms contribute to the non-additivity. The electrostatic and dispersion (in the second order) contributions ate pairwise additive. 

Single-Exchange A contribution to the exchange interaction (valence repulsion of molecules) non-additivity effect coming from the interaction of the Pauli deformation of the electron cloud due to two interacting molecules with the electric field created by the third molecule. 

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