Site Symmetry in Crystals

Ėvarestov, R. A.; Smirnov, V. P.

doi:10.1007/978-3-642-60488-1

Cited by 105 publications

(40 citation statements)

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“…The matrix elements of the Fourier transform of a function on a lattice motion group can be defined as: (10) Using the properties of the delta function, (10) can be rewritten as (11) where k is drawn from the reciprocal lattice (see the appendix). By setting , (11) becomes (12) Equation (12) can be identified as a d-dimensional space DFT-transform, where d is the dimension of the lattice.…”

Section: Analysis: Fourier Transformmentioning

confidence: 99%

Position and Orientation Distributions for Non-Reversal Random Walks using Space-Group Fourier Transforms

2010

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This paper presents an efficient group-theoretic approach for computing the statistics of non-reversal random walks (NRRW) on lattices. These framed walks evolve on proper crystallographic space groups. In a previous paper we introduced a convolution method for computing the statistics of NRRWs in which the convolution product is defined relative to the space-group operation. Here we use the corresponding concept of the fast Fourier transform for functions on crystallographic space groups together with a non-Abelian version of the convolution theorem. We develop the theory behind this technique and present numerical results for two-dimensional and three-dimensional lattices (square, cubic and diamond). In order to verify our results, the statistics of the end-to-end distance and the probability of ring closure are calculated and compared with results obtained in the literature for the random walks for which closed-form expressions exist.

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Section: Analysis: Fourier Transformmentioning

confidence: 99%

Position and Orientation Distributions for Non-Reversal Random Walks using Space-Group Fourier Transforms

2010

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“…of G is in a one-to-one correspondence with the small irrep [G k ] γ of G k ⊂ G and is obtained from the latter by induction procedure [7,8] [G]…”

Section: Connection Between Small Representations Of Space Groups Andmentioning

confidence: 99%

A point group approach to selection rules in crystals

Smirnov¹,

Ėvarestov

Tronc³

2003

Phys. Solid State

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The problem of generation of the selection rules for a transition between Bloch states at any point of the Brillouin zone in crystals is equivalent to the problem of the decomposition of Kronecker products of two representations (reps) of a space group into irreducible components (the full group method). This problem can be solved also by the subgroup method where small reps of little groups are used. In this article, we propose the third method of the selection rules' generation which is formulated in terms of projective reps of crystal point groups. It is based on a well known relation between small irreducible reps (irreps) of little space groups and projective irreps of corresponding little co-groups. The proposed procedure is illustrated by calculations of the Kronecker products for different irreps at the W point of the Brillouin zone for the nonsymmorphic space group O 7 h being one of the most complicated space groups for the selection rules' generation. As an example, the general procedure suggested is applied to obtain the selection rules for direct and phonon-assisted electrical dipole transitions between some states in crystals with the space group O 7 h . One of the authors (V.P.S.) acknowledges the support of Ministére de la Recherche (France).

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“…(6), are found, the induced representations (IR) of DG space groups have to be used to find the symmetry of CC electron states. To construct the IR table of a given DG, the table of the IR for the corresponding triperiodic group may be used [5]. In particular, to obtain the table of IR for DG61 (symmetry group of cyclic clusters of MgO (001) surface) the table of IR is for the related space group G123(D 1 4h ) has to be used.…”

Section: Finite Models Of Surfacesmentioning

confidence: 99%

“…We used the semi-empirical SCF MO method MSINDO [40], the successor of the SINDO1 method that was successfully applied in studies of metal oxide surfaces [23][24][25]28,29]. Water was adsorbed in molecular and dissociated form on the clusters Ti 5 (Fig. 7) chosen in accordance with the rules suggested here.…”

Section: Finite Models Of Surfacesmentioning

confidence: 99%

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Connection between slab and cluster models for crystalline surfaces

2001

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Different models for the theoretical description of bare crystalline surfaces are compared and discussed in terms of stoichiometry and conservation of the point symmetry. While infinite models such as the semi-infinite model or the slab model in general preserve the symmetry of the perfect crystal surface, special care has to be taken when finite cluster models are considered. The connection between molecular cluster choice and surface unit cell of the slab model is demonstrated for metal oxides such as MgO, TiO 2 , V 2 O 5 , and Al 2 O 3 , analyzing how atoms of the primitive unit cell of the parent three-dimensional crystal are distributed in different planes of slab and cluster models. General rules for the construction of finite cluster models based on stoichiometry and symmetry considerations are given and illustrated with calculations on water adsorption at rutile (110).Crystalline surfaces are of high importance in many chemical and physical processes [1,2]. They are therefore subject of an increasing number of studies, both experimental and theoretical [3]. In the last decades, quantum-chemical calculations have become an important tool for investigations of structural, electronic and catalytic properties of surfaces. Methodological developments and the rapid improvement of computer hardware enabled theorists to treat systems of increasing complexity. Thereby it was possible not only to reproduce experimental findings with increasing accuracy, but also to aid in the interpretation of experimental results.For simulations of the electronic structure of surfaces of crystalline solids three basic approaches are used: cluster, slab, and semi-infinite crystal models [4]. The latter is the most appropriate, because it takes into account an infinite number of atoms of the crystal below the surface [5]. Slab and cluster models are nevertheless by far more popular, since they are more feasible from the computational point of view. The cyclic cluster model [4] is intermediate between slab and molecular cluster models. It takes into account the translational symmetry of the surface but considers only a finite number of interatomic interactions within a strictly defined region.The choice of the cluster model (both molecular and cyclic) for a surface allow to employ all those quantum chemical techniques that have been developed for molecular systems. It is also possible to study defects or adsorption reactions in the limit of low coverage. Care has to be taken in the selection of cluster size and shape due to the unavoidable presence of boundary effects. Different schemes have been developed to reduce boundary effects in the cluster model, either by embedding procedures or by the introduction of cyclic boundary conditions. Slab models eliminate twodimensional boundary effects and are widely used for the study of periodic surface structures.In the present work, the connection between slab and cluster models based on a symmetry analysis of the crystalline surface is considered. As far as we know, this connection has not...

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Site Symmetry in Crystals

Abstract: The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Cited by 105 publications

References 0 publications

Position and Orientation Distributions for Non-Reversal Random Walks using Space-Group Fourier Transforms

Position and Orientation Distributions for Non-Reversal Random Walks using Space-Group Fourier Transforms

A point group approach to selection rules in crystals

Connection between slab and cluster models for crystalline surfaces

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