Equations of the Multiplet Theory

Of the energy factors involved in heterogeneous catalysis the reaction heat (—AH), the activation energy of the reaction (e), and the catalyst sublimation energy (A) will be of primary importance to us. As will be shown later, owing to the existence of a far-going additivity, AH and e values may be derived from the more fundamental ones, i.e., the bond energy Qab.  

The bond energy in turn is nothing else than the dissociation energy or AH of the reaction 

Graphically AH is represented by the depth of the potential pit (minus the zero energy) on the potential curve (Fig. 40). The energy factors 

we shall consider the idealized case of a catalyst or an adsorbent with smooth surface normal to the plane of the drawing and of an isolated chemisorbed atom A. Let us denote as KK the section of this surface with the plane of the paper and the distance between the surface KK and the center of the atom A as x (Fig. 40a). Then the potential curve showing the mutual potential energy W of the system as function of x will have the appearance shown in Fig. 40b where x is plotted against If. The coordinate axis If, Fig. 40b, is normal to the plane of the plot. Fig. 40a, thus the planes of the plots of Figs. 40a and 40b are mutually perpendicular. The case of the smooth surface is approximately realized when the atom A is at sufficiently great distance from the surface. 

However, the real surface of the catalyst or of the adsorbent is not smooth. It consists of atoms and therefore, even when the centers of the surface atoms of the catalyst lie in the same plane, the catalyst surface itself has depressions V and projections P of atomic size. Then the line ZZ will represent the section of the catalyst surface by the plane of the drawing Fig. 40a. The position of the atom A in one of the surface depressions V will show its extreme stable position on the surface along the section ZZ. 

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We have already pointed out (see Section I) the similarity of the multiplet theory and the theory of catalysis by Polanyi. The equations of the multiplet theory [(1I.9)-(1I.11)] were given by the author as early as in 1929 Polanyi suggested similar equations in 1934 (35), how ever, for one reaction only—the para-ortho-conversion of hydrogen. In 1957 Temkin (366) developed the Polanyi theory and also obtained Eqs. (II.9) and (II.10), the same as in the multiplet theory. 

Equations (II.9)-(II.12) are the fundamental energy equations of the multiplet theory. It is of extreme importance that these equations allow estimation of the respective ease of the reaction sequence if the bond energies Q are known. The latter depend on the nature of mo-... 

To apply the equations of the multiplet theory one should know the bond energies with the catalysts, Qak The bond energies may be found experimentally by thermochemical, comparative, and kinetic methods. 

According to this method, put forward by the author (7, 381), the relative rates of reactions of one and the same type involving different bonds in the same molecule on a given catalyst are compared. Having a set of such molecules and bonds and using the equations of the multiplet theory (II.9) and (II. 10) for them one can form a number of inequalities. 

Equations (II.9) and (11.12) of the multiplet theory for the first stage (adsorption) applied to reactions (11.40) give the system of equations... 

The symmetry-adapted functions themselves may then be constructed directly from equations (59), (60), and (61). The same procedure can be applied without any difficulty to the other types of ionic configuration. The results show that there are altogether a total of 268 multiplets of which 22 correspond to xAig states. A similar result, of course, would be obtained from MO theory with full configuration interaction.33... 

The same defect thermodynamics and diffusion theory can be applied to ionic crystals with one important proviso, which is the need to account for the charges on the ions (and hence effective charges on the defects), and that the crystal must remain electrically neutral overall. This means that the defects will occur as multiplets to satisfy this later condition. For example, in a MX crystal they will occur as pairs the Schottky pair- a cation vacancy and an anion vacancy the cation-Prenkel pair- a cation vacancy and an interstitial cation and the anion-Frenkel pair - an anion vacancy and an interstitial anion. The concentrations of the defects in the pair are related by a solubility product equation, which for Schottky pairs in an MX equation takes the form ... 

The application of the principle of energetic correspondence to the theory of the selection of catalysts consists of use of the equations of multiplet theory (II.9-II.14) and the resultant volcano-shaped curves. 

Alternative approaches have been presented using a more general field theory with all the J multiplets of the ground state serving as a basis. This leads to deviations of the dependence of Ap (except for Eu + and Sm +). However, McGarvey has shown that equation (6) is adequate for estimating shifts to an accuracy of 10-20%. ... 

LFT originated as a purely electrostatic model - crystal-field theory (CFT) [18], in which d-electronic multiplets of transition metals are perturbed by ligands as point charges or point dipoles. The CF operator (Equation 1) acts within the space of Slater determinants (SD) composed of purely d-spin-orbitals in which two-electron energies are taken into account with the Coulomb operator and one-electron energies with a crystal-field potential (vcp), the first and second terms in Equation 1, respectively. 

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