LEPS equation

By making a number of simplifying assumptions, the quantum mechanical problem is reduced to one that is solvable in analytical form. The valence-bond result, with these approximations, for H3 gives the LEPS equation ... 

The semiempirical methods combine experimental data with theory as a way to circumvent the calculational difficulties of pure theory. The first of these methods leads to what are called London-Eyring-Polanyi (LEP) potential energy surfaces. Consider the triatomic ABC system. For any pair of atoms the energy as a function of intermolecular distance r is represented by the Morse equation, Eq. (5-16),... 

The lEP for aminoacid can be calculated from the ionization constant according to the equation ... 

On the basis of London equation (9.31), Eyring and Polanyi calculated the potential energy surface, which is known as London-Eyring-Polanyi (LEP) surface. In this treatment, the coulombic energy A and exchange energy a for a diatomic molecule have been assumed to be the constant fractions of the... 

Although the theoretical roots of this technique are very well established, it is more often used as a flexible surface which can be adjusted to lit either exprimental data or data established by better electronic-structure methods. The LEPS formalism has also been extensively used to explore the relationships between the potential energy surface and the details of chemical dynamics . Because of the widespread use of this potential for studying gas-phase reactions, the specific form of the equations will not be discussed here. The interested reader is instead referred to references which discuss this approach in more detail . ... 

The modification of theoretical gas-phase reaction techniques to study gas-surface reactions continues to hold promise. In particular, the LEPS formalism appears to capture a sufficient amount of realistic bonding characteristics that it will continue to be used to model gas-surface reactions. One computational drawback of the LEPS-style potentials is the need to diagonalize a matrix at each timestep in the numerical integration of the classical equations of motion. The size of the matrix increases dramatically as the number of atoms increases. Many reactions of more direct practical interest, such as the decomposition of hydrocarbons on metal surfaces, are still too complicated to be realistically modeled at the present time. This situation will certainly change in the near future as advances in both dynamics techniques and potential energy surfaces continue. 

A semi-empirical extension of the London equation—the LEPS method—allows for a simple but somewhat crude construction of potential energy surfaces. 

The London-Eyring-Polanyi-Sato (LEPS) method is a semi-empirical method.8 It is based on the London equation, but the calculated Coulombic and exchange integrals are replaced by experimental data. That is, some experimental input is used in the construction of the potential energy surface. The LEPS approach can, partly, be justified for H + H2 and other reactions involving three atoms, as long as the basic approximations behind the London equation are reasonable. 

The LEPS method is also used for general triatomic systems. Here equations that are equivalent to Eqs (3.37) and (3.38) are written down and solved for the other diatomic pairs (AC and BC). The LEPS method provides a quick but somewhat crude estimate of potential energy surfaces. 

For their calculations, Polanyi and co-workers and Karplus, Porter, and Sharma (20) have employed versions of the LEPS (London-Eyring-Polanyi-Sato) potential, which has some connection with formal theory since it is based on the London equation for a system of three atoms [320] ... 

The acidic surfaces of quartz and tungsten oxide are noted in this table, as well as, the basic surfaces of alumina and magnesium oxide. It should also be noted that the method of powder fabrication is important in establishing the structure of the surface and, therefore, the isoelectric point of the powder surfiice. The lEP for a simple oxide is inversely proportional to the ratio of the valence, z, to the radius, Rcaom > of the metal cation making up the oxide. The regression equation for lEP data from Parks [53] is as follows ... 

The pressure variable can be used in defining wettability in terms of LEP, as represented by the Laplace (Cantor) equation... 

Equations for mutual relationship between different zero points have been derived. The equality of PZC and lEP of pure oxides was also challenged by Charmas et al. [13] who wrote ... 

Several models allow differences between the PZC, CIP, and lEP, and the equality of these quantities has been challenged in a few publications [638-641], even for sparingly soluble metal oxides. Rejection of the assumption that CIP = PZC implies that the correction for an acid or base associated with solid particles illustrated in Eigure 2.7 is not applicable, and the correction term in Equation 2.12 has to be determined by another means. 

London Equation and its Generalizations (LEPS) Ionic (Rittner type) Models Convenient Analytic Formulae... 

In the London-Eyring-Polanyi-Sato (LEPS) method224 the original London equation is multiplied by an empirical factor which is supposed to account for the effect of overlap. 

Shift in lEP. The intrinsic constants may be used to estimate the composition of the oxide surface as a function of solution variables (pH, concentration of specifically adsorbable cations and anions). Without correcting for coulombic attraction or repulsion such calculations should give reasonable predictions only for surfaces that have a fixed surface charge of zero (or nearly zero). Hence, it should be possible to use intrinsic constants to predict shifts in lEP caused by specific cation and anion binding. Equation 20 gives the condition for zero fixed charge (lEP). 

By using the intrinsic equilibrium constants given in Appendix II, and mass-balance equations for MgT and the available surface groups (a, and =A10t ), the shift in lEP as a function of Mg " or MgT can be predicted. Generally, for the specific binding of a cation the following relationship can be derived ... 

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