Point cyclic cluster

The trimer of hydrogen fluoride, (HF)3, does not have an open-chain structure. All experimental evidence points to a symmetric cyclic cluster... 

In the HF LCAO method, (4.57) for the periodic systems replaces (4.33) written for the molecular systems. In principle, the above equation should be solved at each SCF procedure step for all the (infinite) fe-points of the Brillouin zone. Usually, a finite set kj j = 1, 2,..., L) of fe-points is taken (this means the replacing the infinite crystal by the cyclic cluster of L primitive cells). The convergence of the results relative to the increase of the fe-points set is examined in real calculations, for the convergent results the interpolation techniques are used for eigenvalues and eigenvectors as these are both continuous functions of k [84]. The convergence of the SCF calculation results is connected with the density matrix properties considered in Sect. 4.3... 

We note that the procedure described above for interpolating the DM in the BZ is not uniquely determined, because the LUC (le. the set of vectors Ji° ) can be variously chosen for the same superlattice. Furthermore, the LUC can be chosen differently for the pairs of DM indices r, r and p, v. In this book, the LUC is taken to be the Wigner-Seitz ceU, because only this cell has a symmetry identical to the point symmetry of the superlattice in all cases. In order to correlate the LUC with a cyclic cluster, we choose the LUC to be dependent on the pair of DM indices as follows. In the coordinate representation, the LUC (VA-region) is centered at the point (r — r ) therefore, we have... 

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. 

The CCM model allows real-space calculations (formaUy corresponding to the BZ center for the infinite crystal composed of the supercells). From this point of view the cyclic cluster was termed a quasimolecular large unit ceb [289] or a molecular unit ceb [288]. 

What happens when the cychc cluster is increased Depending on its shape and size different sets of fe-points are reproduced, but in the EHT matrix elements the number of interactions included (interaction radius) increases as the periodically reproduced atomic sites distance is defined by the translation vector of a cyclic cluster as a whole. It is important to reproduce in the cyclic-cluster calculations the states defining the bandgap. As the overlap matrix elements decay exponentially with the interatomic distance one obtains the convergence of results with increasing cyclic cluster. Of course, this convergence is slower the more diffuse are the AOs in the basis. Prom band-structure calculations it is known that for BNhex in the one-layer model the top of the valence band and the bottom of the conduction band are at the point P of the BZ reproduced in the cyclic cluster considered. 

The EHT method is noniterative so that the results of COM apphcation depend only on the overlap interaction radius. The more complicated situation takes place in iterative Mulliken-Riidenberg and self-consistent ZDO methods. In these methods for crystals, the atomic charges or the whole of the density matrix are calculated by summation over k points in the BZ and recalculated at each iteration step. The direct lattice summations have to be made in the surviving integrals calculation before the iteration procedure. However, when the nonlocal exchange is taken into account (as is done in the ZDO methods) the balance between direct lattice and BZ summations has to be ensured. This balance is automatically ensured in cychc-cluster calculations as was shown in Chap. 4. Therefore, in iterative MR and self-consistent ZDO methods the increase of the cyclic cluster ensures increasing accuracy in the direct lattice and BZ summation simultaneously. This advantage of COM is in many cases underestimated. 

A calculation performed solely at the F k = 0) point of the supercell BZ would become entirely equivalent to special ff-points (6.59) calculation if all the direct lattice summations were be made over the whole crystal. However, in CCM the interaction range depends on the cyclic-cluster size. To determine the fc = 0 Fock matrix elements we need the full k dependence of the superceU density matrices Pfi k ), see (6.61) and (6.62). Meanwhile, each iteration in a cychc-cluster calculation only provides the eigenvector coefficients C jp(O) at k = 0, and hence only Pfu 0). Therefore, it is necessary to relate the reduced BZ integrals over the fully fc -dependent density matrices, namely J2PN k ) and J])Ppj,(fe )exp(—ife i2 ) to Pn 0) and P i/(0), re-fe fe ... 

Table 6.3. Cyclic clusters used for the representation of the rutile bulk special point (SP)-set and its accuracy J... 

SCM is used as a rule for the neutral-point-defects calculations (for the charged point defects the field of periodically repeated charge has to be suppressed in one or another way). The molecular- and cyclic-cluster models are more universal as they can be apphed both for the neutral and charged point defects. 

The point symmetry of the cyclic-cluster model coincides with point groups F and Fd for the host and defective crystal. 

Supercell and Cyclic-cluster Models of Neutral and Charged Point Defects... 

The use of localized orbitals for the cluster calculation is an efficient approach for defective crystals. To connect the perfect crystal localized orbitals and molecular cluster one-electron states the molecular cluster having the shape of a superceU was considered [699]. Such a cluster differs from the cyclic cluster by the absence of PBC introduction for the one-electron states. Evidently, the molecular cluster chosen is neutral and stoichiometric but its point symmetry can be lower than that of the cyclic cluster. Let the locaUzed orthogonal crystalline orbitals (Wannier functions) be defined for the infinite crystal composed of snpercells. The corresponding BZ is L-times reduced (the snpercell is supposed to consist of L primitive unit cells). The Wannier functions W r — Ai) are now introdnced for the supercells with the translation vectors and satisfy the following equation ... 

Table 10.13. Convergence of results for pure SrTiOs (a=3.905 A) obtained for pure HP (a) and DFT-PWGGA (b) LCAO band calculations corresponding to cyclic clusters of an increasing size [680]. All energies in eV, total energies are presented with respect to the reference point of 80 a.u.=2176.80 eV. q and V are effective atomic charges and valencies (in e), respectively. Rm and M are explained in the text...

This review will restrict itself to boron-carbon multiple bonding in carbon-rich systems, as encountered in organic chemistry, and leave the clusters of carboranes rich in boron to the proper purview of the inorganic chemist. Insofar as such three-dimensional clusters are considered at all in these review, interest will focus on the carbon-rich carboranes and the effect of ring size and substituents, both on boron and carbon, in determining the point of equilibrium between the cyclic organoborane and the isomeric carborane cluster. A typical significant example would be the potential interconversion of the l,4-dibora-2,5-cyclohexadiene system (7) and the 2,3,4,5-tetracarbahexaborane(6) system (8) as a function of substituents R (Eq. 2). 

Why do we believe that a Cu monolayer is inserted between SAM and gold substrate The 2D-deposit grows and dissolves extremely slowly. Another indication is that the 2D deposit is very stable and shows no displacement by the scanning tip. Cu clusters on top of an alkanethiol-SAM would be only weakly bound and should be easily pushed away by the tip at higher tunnel currents, very much like metal clusters on a hydrogen-terminated Si(lll) surface, which for that very reason are difficult to image by STM (or AFM [122]). And finally, the cyclic voltammograms (Fig. 33) point to the formation of a buried monolayer . 

The actual natural charge in the cyclic pentamer is qnat = 0.570, which makes the Coulombic point-charge estimate entirely unrealistic.) Thus, no matter how q is chosen, a simple Coulombic point-charge model will give >10% errors for one or the other of these clusters. 

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