Mott-Littleton

Grimes R W, Catlow C R A and Stoneham A M 1989 A oomparlson of defeot energies In MgO using Mott-LIttleton and quantum meohanloal prooedures J. Phys. Condens Matter 1 7367-84... 

The summation is over the different types of ion in the unit cell. The summation ca written as an analytical expression, depending upon the lattice structure (the orij Mott-Littleton paper considered the alkali halides, which form simple cubic lattices) evaluated in a manner similar to the Ewald summation this typically involves a summc over the complete lattice from which the explicit sum for the inner region is subtractec... 

Table I LI gives the results of Mott-Littleton calculations on some simple ionic sj st [Mackrodt 1982]. By comparing the relative energies of the various tj pes of defect.

Grimes R W, C R A Catlow and A M Stoneham 1989. Quantum-mechanical Cluster Calculations anc the Mott-Littleton Methodology. Journal of the Chemical Society, Faraday Transactions 85 485-495. 

Sometimes, the system of interest is not the inhnite crystal, but an anomaly in the crystal, such as an extra atom adsorbed in the crystal. In this case, the inhnite symmetry of the crystal is not rigorously correct. The most widely used means for modeling defects is the Mott-Littleton defect method. It is a means for performing an energy minimization in a localized region of the lattice. The method incorporates a continuum description of the polarization for the remainder of the crystal. 

The chemistry of interest is often not merely the inhnite crystal, but rather how some other species will interact with that crystal. As such, it is necessary to model a system that is an inhnite crystal except for a particular site where something is diherent. The same techniques for doing this can be used, regardless of whether it refers to a defect within the crystal or something binding to the surface. The most common technique is a Mott-Littleton defect calculation. This technique embeds a defect in an inhnite crystal, which can be considered a local perturbation to the band structure. 

Figure 2.9 Mott-Littleton method of dividing a crystal matrix for the purpose of calculating defect energies. Region I contains the defect and surrounding atoms, which are treated explicitly. Region lib is treated as a continuum reaching to infinity. Region Ha is a transition region interposed between regions I and lib.

In these calculations, the isolated defect or defect cluster is embedded in the crystal, which extends to infinity, and the contrast between this approach and that used in the supercell methods is illustrated diagrammatically in Figure 1. The normal procedure in a Mott-Littleton calculation is to relax all the atoms in a region of crystal surrounding the defect, containing typically 100-300 atoms, until all are at zero force. Newton-Raphson minimization methods are generally used. The relaxation of the remainder of the crystal is then described by more approximate methods in which the polarization, P at a point r, is calculated for crystals that have dielectric isotropy, from the expression ... 

Figure 1 Schematic representation ofthe Mott-Littleton technique for treating relaxations around defects. (An interface region 11a is included between the inner and outer regions)...

The location of extra framework cations is a major problem in characterising zeolites. Simulation is becoming an increasingly powerful tool for the exploration and rationalisation of cation positions, since it not only allows atomic level models to be compared to bulk experimental behaviour, but can also make predictions about the behaviour of systems not readily accessible to experimental probing. In the first part of this work we use the Mott-Littleton method in conjunction with empirical potential energy functions to predict and explore the locations of calcium cations in chabazite. Subsequently, we have used periodic non-local density functional calculations to validate these results for some cases. 

Defect modelling with the Mott-Littleton method... 

The Mott-Littleton calculations revealed three sites for calcium in the high-silica chabazite, as show in Figure 3 ... 

Site Idealised Mott-Littleton Defect centre ... 

Configurations 14 and 110 were examined with sites A, A, N and B1 and compared to the defect calculations. Both the periodic empirical potential calculations and the periodic DFT calculations gave mainly the same sites as the Mott-Littleton study. With the surprising exception that the DFT calculation found no minimum at site N for 14 or 1 10. 

The Mott-Littleton method has been employed to model Ca-Chabazite in the limit as the... 

Figure 4. Comparison of Mott-Littleton, periodic empirical potential functions and periodic DFT quantum mechanics. The horizontal scale is not significant. Transition state energies are shown as open symbols...

Fig. 2.2. Space partitioning in EPE embedded cluster calculations. I - internal region treated at a QM level II - shell model enviromnent of the QM cluster subdivided into regions of explicit optimization (Ila), of the effective (Mott-Littleton) polarization (lib) and of the external area (lie). The sphere indicates an auxiliary surface charge distribution which represents the Madelung field acting on the QM cluster (dashed line).

Most calculations of the internal energy of a defect process (hereafter referred to simply as the energy) use classical potential models. Two kinds of approach have been used to calculate defect processes first, that based on the Mott Littleton approximation and second, the supercell method (closely related to the lattice minimization methods discussed in Chapter 3). 

A computer code, CHAOS (Duffy and Tasker, 1983), has been written to calculate the energy of defects at interfaces. It uses the structure of the relaxed interface as a starting point and then applies the Mott-Littleton method discussed above. The presence of the interface complicates the calculation. Details are given in the reference above the essential difference is that part of the polarization calculation must be done as a sum over planes. 

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