Density cyclic cluster

In a recent thermodynamic and spectroscopic study Redington66 -68) evaluated AH and AS for the formation of cyclic and open-chain oligomers of hydrogen fluoride. His best fit to vapor density, heat capacity, excess entropy, excess enthalpy, and infra red absorption data shows a monotonous increase in AH per hydrogen bond with increasing number of HF molecules both for open-chain and cyclic clusters, using the relation... 

In the theory of electronic structure two symmetric models of a boundless crystal are used or it is supposed that the crystal fills aU the space (model of an infinite crystal), or the fragment of a crystal of finite size (for example, in the form of a parallelepiped) with the identified opposite sides is considered. In the second case we say, that the crystal is modeled by a cyclic cluster which translations as a whole are equivalent to zero translation (Born-von Karman Periodic Boundary Conditions -PBC). Between these two models of a boundless crystal there exists a connection the infinite crystal can be considered as a limit of the sequence of cychc clusters with increasing volume. In a molecule, the number of electrons is fixed as the number of atoms is fixed. In the cyclic model of a crystal the number of atoms ( and thus the number of electrons) depends on the cyclic-cluster size and becomes infinite in the model of an infinite crystal. It makes changes, in comparison with molecules, to a one-electron density matrix of a crystal that now depends on the sizes of the cyclic cluster chosen (see Chap. 4). As a consequence, in calculations of the electronic structure of crystals it is necessary to investigate convergence of results with an increase of the cyclic cluster that models the crystal. For this purpose, the features of the symmetry of the crystal, connected with the presence of translations also are used. 

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... 

The study of the approximate density matrix properties allowed the implementation of the cyclic cluster model in the Hartree- Fock LCAO calculations of crystalUne systems [100] based on the idempotency relations of the density matrix. The results... 

Hybrid approaches combining ab-initio or DFT and semiempirical approaches have become popular. As an example, we can refer to LEDO (hmited expansion of differential overlap) densities application to the density-functional theory of molecules [262]. This LEDO-DFT method should be well suited to the electronic-structure calculations of large molecules and in the anthors opinion its extension to Bloch states for periodic structures is straightforward. In the next sections we discuss the extension of CNDO and INDO methods to periodic stmctures - models of an infinite crystal and a cyclic cluster. 

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 ... 

Analysis of the difference electron-density plots, calculated for the band and the 80-atom cyclic-cluster calculations confirms that the latter well reproduces the electron-density distribution in a perfect crystal. Lastly, the total and projected density of states for a perfect crystal show that the upper valence band consists of O 2p atomic orbitals with admixture of Ti 3d orbitals, whereas the Sr states contribute mainly to the energies close to the conduction-band bottom, in agreement with previous studies. 

Fig. 10.7. (a) The electronic-density plots for the (010) cross section of Fe and nearest ions in SrTiOs as calculated by means of the HF method for the cyclic cluster of 160 atoms. Isodensity curves are drawn from 20.8 to 0.8 e a.u. with an increment of 0.0022 e a.u. , b) the same as (a) for the (001) section, (c) the same for the (110) section. Left panels ate HF difference electron densities, right panels are spin densities. 

Janetzko, F., Koster, A. M., 8c Salahub, D. R. (2008). Development of the cyclic cluster model formalism for Kohn-Sham auxiliary density functional theory methods. Journal of Chemical Physics, 128, 024102. 

Xantheas, S. 1995. Ab initio studies of cyclic water clusters (HzO)n, n = 1-3. III. Comparison of density functional with MP2 results. J. Chem. Phys. 102, 4505. 

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