Empirical shell model potential

It is instructive to compare the performance of empirically derived force fields with those potentials derived from quantum mechanical calculations. Sierka and Sauer <> compared results from Jackson and Catlow s empirical shell model potential with Schroder and Sauer s Hartree-Fock based and their own density functional based shell model potentials. The mean deviation between computed and observed unit cell parameters was found to be 0.7%, 1.9%, and 1.4%, respectively, for the three potentials. This means that the empirical shell model potential is twice as accurate as the best quantum chemically derived force field for unit cell predictions. However, the calculated vibrational spectra of silicalite are in good agreement with experiment for both of the quantum chemically derived potentials,whereas agreement is not as satisfactory for the empirical force field. ... 

The derivation of potential parameters is a vast and important topic, which cannot possibly be covered comprehensively here. Hence, the focus will be on two topics concerning the particular approaches used for empirical shell model potential derivation for ionic materials. However, it is noted that derivation of parameters from ab initio energy surfaces will increasingly become the method of choice for more complex materials due to the lack of suitable experimental data. 

De Boer and coworkers ° °" parameterized the shell model for silica polymorphs on the results of ab initio calculations of the potential energy surfaces, polarizabilities, and dipole moments of Si(OH)4 and (0H)3Si-0(H)-Si(OH)3 clusters. The structural characteristics and elastic moduli calculated with this set of parameters for three structures compared well with results computed with the use of both the rigid ion and the empirical shell models. ... 

The highly ionic nature of MgO means that quite accurate empirical potentials can be constmcted. The polarizable shell model potential is the most widely used for MgO and also for a wide range of other ionic materials. It is instmctive to discuss the main elements of this potential in order to understand the nature of interactions between the ions. The dominating contribution to the interaction is electrostatic and in the simplest approximation can be represented by associating a point charge (usually the formal charge) with each ion. In addition there is a short-range repulsive term due to the overlap of electron density between the ions (Born-Mayer) and a weakly attractive... 

Tables 8-11 list results for some commonly studied zeolites. Faujasite and silicalite are industrially important catalysts when they contain aluminum. The structural features of sodalite are predicted best by both Jackson and Catlow s potentiaPS and Schroder s empirical force field. a pgr faujasite, all force fields are similar in their ability to predict the observed unit cell dimensions, whereas the best for mordenite and silicalite is Sierka and Sauer s potential, which consistently reproduces unit cells both for dense and microporous silica. In general, shell model potentials provide slightly better structural agreement with experiment for zeolites than do ion pair potentials and molecular mechanics force fields.

I have not described the calculation of the eigenvalues, which requires the solution of the equations of motion and therefore a knowledge of the force constants. The shell model for ionic crystals, introduced by Dick and Overhauser (1958), has proved to be extremely useful in the development of empirical crystal potentials for the calculation of phonon dispersion and other physical properties of perfect and imperfect ionic crystals. There is now a considerable literature in this field, and the following references will provide an introduction Catlow etal. (1977), Gale (1997), Grimes etal. (1996), Jackson et al. (1995), Sangster and Attwood (1978). The shell model can also be used for polar and covalent crystals and has been applied to silicon and germanium (Cochran (1965)). 

Future directions in the development of polarizable models and simulation algorithms are sure to include the combination of classical or semiempir-ical polarizable models with fully quantum mechanical simulations, and with empirical reactive potentials. The increasingly frequent application of Car-Parrinello ab initio simulations methods " may also influence the development of potential models by providing additional data for the validation of models, perhaps most importantly in terms of the importance of various interactions (e.g., polarizability, charge transfer, partially covalent hydrogen bonds, lone-pair-type interactions). It is also likely that we will see continued work toward better coupling of charge-transfer models (i.e., EE and semiem-pirical models) with purely local models of polarization (polarizable dipole and shell models). 

This is undertaken by two procedures first, empirical methods, in which variable parameters are adjusted, generally via a least squares fitting procedure to observed crystal properties. The latter must include the crystal structure (and the procedure of fitting to the structure has normally been achieved by minimizing the calculated forces acting on the atoms at their observed positions in the unit cell). Elastic constants should, where available, be included and dielectric properties are required to parameterize the shell model constants. Phonon dispersion curves provide valuable information on interatomic forces and force constant models (in which the variable parameters are first and second derivatives of the potential) are commonly fitted to lattice dynamical data. This has been less common in the fitting of parameters in potential models, which are onr present concern as they are required for subsequent use in simulations. However, empirically derived potential models should always be tested against phonon dispersion curves when the latter are available. 

The variable retention of diatomic differential overlap (VRDDO) method of Kaufman and collaborators also uses semiorthogonalization. In addition, VRDDO uses a model potential to eliminate consideration of the inner-shell electrons. Both Kashiwaga s method and the VRDDO method approximate ab initio calculations, and neither introduces empirically based parameters. 

Figure 7 Schematic representation of the empirical method for deriving short-range potential energy function parameters p, are the computed properties for the system, observed properties, and S the sum of squared deviations from target values p and C,y are potential parameters of Eq. [13] K and fCy are shell model parameters, and qi and qj the charges of component species.

Fisler DK, Gale JD, Cygan RT (2000) A shell model for the simulation of rhombohedtal carbonate minerals and their point defects. Am Miner 85 217-224 Gale JD (1996) Empirical potential derivation for ionic materials. Phil Mag B 73 3-19 Gale JD (1997) GULP - A computer program for the syrmnetry adapted simulation of solids. J Chem Soc... 

For supported metal and metal oxide systems, one typically has to resort to using two different QM methods owing to the lack of accurate force fields or empirical potentials to describe these systems. Both Whitten and Yangl l and Govind et al.1 1 have developed schemes which embed more accurate Cl wavefunction methods into lower level QM methods in order to provide for more accurate descriptions than DFT. Sauer s group has used standard ab initio methods along with shell models to describe the oxide environment for zeolite systemsl 1. 

Fig. 4.4 VDOS calculated from a a two-body potential (VB) derived by Kramer et al. on a quantum-chemical calculation of an H4Si04 cluster [55], b a two-body potential (TS) derived by Tsuneyuki et al. using a Hartree-Fock calculation on SiOJ cluster [9], c a three-body (3B) potential by Sander, Leslie, and Catlow with a shell-model description and with three-body interactions [48] and d a two-body potential (KR) proposed by Kramer et al. using a mixed self-consistent field and empirical procedures [10], Note the KR potential here is the BKS potential. Figure taken from [56]...

---