Molecular cluster model

FIGURE 1. Molecular clusters models for compounds 1-6. Reproduced with permission from... 

Experimental trends in Si shielding observed experimentally arise from variations in the coordination number (i.e. the number of atoms in the 1st coordination sphere), the extent of polymerization of the silicate tetrahedra, the degree of replacement of one net-work forming cation by another (e.g. coupled Na+, Al+3 for Si+4 substitution), the size of the rings of tetrahedra present and the Si-O-Si angles (1,2). Similar trends are seen in gas-phase molecules, species in aqueous solution and in both crystalline and amorphous solids. Polarized double-zeta basis set Hartree-Fock level calculations using small molecular cluster models reproduce these trends semiquantitatively, as we will show. 

Whereas selective diffusion can be better investigated using classical dynamic or Monte Carlo simulations, or experimental techniques, quantum chemical calculations are required to analyze molecular reactivity. Quantum chemical dynamic simulations provide with information with a too limited time scale range (of the order of several himdreds of ps) to be of use in diffusion studies which require time scale of the order of ns to s. However, they constitute good tools to study the behavior of reactants and products adsorbed in the proximity of the active site, prior to the reaction. Concerning reaction pathways analysis, static quantum chemistry calculations with molecular cluster models, allowing estimates of transition states geometries and properties, have been used for years. The application to solids is more recent. 

The MS-A"a method has been very successful for the calculation of ionization energies for both gas-phase molecules and molecular cluster models of solids. DeAlti et al. (1982) have employed both the MS-Aa method and the LCAO-Aa scheme of Sambe and Felton (1975) in the transition-state procedure for calculating ionization potentials for a number of molecules where the Koopmans approximation within Hartree-Fock theory gives the wrong ordering. They found that the LCAO-Aa ionization potentials were more accurate than the Hartree-Fock Koopmans values, while the MS-Aa results typically showed errors of one or two electron volts. Representative results are given for ozone (O3) in Table 3.8. 

These results for Si(OH)4 are consistent with the observation that average bond distances in solids can be reproduced quite well using molecular cluster models and ab initio Hartree-Fock techniques, at least for three- to six-coordinate central metal atoms from the first through third rows of the Periodic Table. Gibbs et al. (1987) have shown excellent agreement between calculated equilibrium bond distances in tetrahedral molecules and average experimental bond distances for tetrahedrally coordinated metal atoms. For the third-row elements, the addition of flexible d polarization functions was required to match experiment closely (in... 

Some fundamental aspects of the nucleation process have been investigated by molecular dynamics (MD) methods. In a recent review [44] the advantages and limitations of molecular cluster models in simulating the dynamics of nucleation and phase changes have been discussed. In this approach, molecular dynamic simulations are correlated with experimental nucleation rates extracted from electron diffraction patterns of molecular supersonic jets. The dynamics of freezing of ammonia, CCI4 and water, and the phase transformations of t-butyl chloride have been analysed. A useful feature of the MD computational... 

The Q coefficients (and derived Cg) in the D3 method have been computed using a modified form of this relation, where the a(ia>) are computed nonempirically by TDDFT and A and B are reference molecules from which atomic values are derived [42]. Because the reference system can also be a molecular cluster modeling a solid environment, special coefficients for atoms in the bulk can be derived [68]. The final form for the DFT-D3 two-body part of the dispersion energy employs the so-called Becke-Johnson (BJ) damping [61, 69] and truncates the expansion at Cg... 

In 2009, Wu and co-workers compared the QCC results obtained with the molecular cluster model and the periodic crystal lattice approach [110]. In particular, authors computed the QC and CS tensors for the carbonyl... 

Kubicki JD, Blake GA, Apitz SE (1996b) Ab initio calculations on Si and Al species Implications for atomic structure of mineral surfaces. Am Mineral 81 789-799 Kubicki JD, Itoh MJ, Schroeter LM, Apitz SE (1997) The bonding mechanisms of salicyUc acid adsoibed onto illite clay An ATR-FTIR and MO study. Env Sci Tech 31 1151-1156 Kubicki JD, Apitz SE (1998) Molecular cluster models of aluminum oxide and aluniinum hydroxide surfaces. AmMiner 83 1054-1066... 

Martins JBL, Andres J, Longo E, Taft CA (1995) A theoretical-study of (1010) and (0001) ZnO surfaces—molecular cluster model, basis-set and effective core potential dependence. Theochem-J Mol Struc 330 301-306... 

As compared to the molecules the wavefunction-based correlation methods for periodic systems are practically reliable only when the molecular cluster model is used. Unfortunately, the well-known problems of the cluster choice and the influence of the dangling bonds on the numerical results restricts the application range of the molecular cluster model to the essentially ionic systems. 

In the next sections we briefly discuss the basic ideas of Cl, MCSCF and CC post-HF methods for molecules as they are directly extended to the crystalline solids in the framework of the molecular cluster model. In more detail, the local correlation and MP2 methods are considered both for the molecules and the periodic systems. 

The idea to use relatively small cyclic clusters for comparative perfect-crystal and point-defect calculations appeared as an alternative to the molecular-cluster model in an attempt to handle explicitly the immediate environment of the chemisorbed atom on a crystalline surface [285] and the point defects in layered solids [286,287] or in a bulk crystal [288,289,292,293]. The cluster is formed by a manageable group of atoms around the defect and the difference between the molecular-cluster model (MCM) and the cychc cluster model (COM) is due to the choice of boundary conditions for the one-electron wavefunctions (MOs). Different notations of COM appeared in the literature molecular vmit ceU approach [288], small periodic cluster [286], large rmit cell [289,290]. We use here the cychc cluster notation. 

In MCM (as in molecules) MOs are supposed to be orthonormahzed in the whole space this condition is also fulfilled for the molecular clusters embedded by some or another means into the crystaUine environment. Recent apphcations of the molecular-cluster model and MO calculations for materials design in the Hartree-Fock-Slater method are discussed in [291]. 

The electronic structure of the molecular cluster can be calculated using any of the methods developed for the molecules. However, the problem arises at the molecular cluster surface, i.e. with the representation of the rest of the crystal. As is shown schematically in Fig. 10.1, there are different possibilities for this representation. In molecular-cluster models the group of atoms chosen is embedded into the crystaUine environment embedded-cluster model) or the cluster surface atoms are saturated by hydrogen or other species atoms saturated-cluster model). In the most crude model the free cluster (neutral or charged) is chosen. 

LCAO calculations of the charged point defects in metal oxides are made mainly in the molecular-cluster model, considered in the next section. As we already noted PW molecular-cluster calculations are impossible as use of the PW basis requires the periodicity of the structure under consideration. 

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