Real Space Properties

The coordination numbers hap were obtained by using an integration range of 0—2.7 A where the upper limit corresponds to the first minimum in the total pair distribution fimctions for the GeX4 models 

To provide a more complete description of the network we calculated the average percentages of the individual a-l structural units where an atom of species a (Ge, X = S or Se) is Z-fold coordinated to other atoms. To clarify this notation, Ge-GeS3 represents a Ge atom that is connected to 1 other Ge atom and 3 S atoms while Ge-S4 represents a Ge atom that is connected to 4 S atoms. Bonds are deemed to be formed when the interatomic distance for a given pair of atoms is smaller than 2.7 A, this value corresponds to the first minimum in the total pair distribution functions for both systems. The proportion of units (/) are summarized in Table 12.4. 

First-principles molecular dynamics studies of glassy GeS4 and glassy GeSe4 have shown than both systems are characterized by tetrahedral connections interlinked with X-X (X=S or Se) homopolar bonds. From the methodological point of view. 

We also provide the number of Ge atom involved in homopolar bond(s) iVoeCe and the fraction of X atom (S or Se) involved in a homopolar bond(s) iVxx- These quantities have been calculated including neighbors separated by less than 2.7 A 

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The performances of the BLYP XC functional have been also assessed through the study of the amorphous GeSe2 [26, 27]. In this case, a clear improvement has been found on both the real space properties and the reciprocal space properties with respect to the PW scheme. This was substantiated by the good agreement between the proportion of homopolar bonds calculated for theFPMD model and the experiment. 

The LMTO method [58, 79] can be considered to be the linear version of the KKR teclmique. According to official LMTO historians, the method has now reached its third generation [79] the first starting with Andersen in 1975 [58], the second connnonly known as TB-LMTO. In the LMTO approach, the wavefimction is expanded in a basis of so-called muffin-tin orbitals. These orbitals are adapted to the potential by constmcting them from solutions of the radial Scln-ddinger equation so as to fomi a minimal basis set. Interstitial properties are represented by Hankel fiinctions, which means that, in contrast to the LAPW teclmique, the orbitals are localized in real space. The small basis set makes the method fast computationally, yet at the same time it restricts the accuracy. The localization of the basis fiinctions diminishes the quality of the description of the wavefimction in die interstitial region. 

It is a known property of Fourier transforms that given a convolution product in the reciprocal space, it becomes a simple product of the Fourier transforms of each term in the real space. Then, as the peak broadening is due to the convolution of size and strains (and instrumental) effects, the Fourier transform A 1) of the peak profile I s) is [36] ... 

S(r) will be treated in real space but needs to be applied only in a small neighborhood around the reference charge distribution. L(r), which represents the bulk of the electrostatic interactions (in terms of their number, not in the contribution to the energy), can be treated using various approximations with favorable scaling properties. For example, in the origi-... 

Since the electrostatic potential is closely related to the electronic density, it may be useful to discuss how the information that can be obtained from V(r) differs from that provided by the p(r). Both are real physical properties, related by Eqs. (3.1) and (3.4). An important difference between V(r) and p(r) is that the electrostatic potential explicitly reflects the net effect of all of the nuclei and electrons at each point in space, whereas the electron density directly represents only the concentration of electrons at each point. A molecule s interactions with another chemical system is affected by its total charge distribution, both positive and negative, and thus can be better understood in terms of its electrostatic potential than its electronic density alone. Examples illustrating this point have been discussed elsewhere (Politzer and Daiker 1981 Politzer and Murray 1991). 

The late 1980s saw the introduction into electrochemistry of a major new technique, scanning tunnelling microscopy (STM), which allows real-space (atomic) imaging of the structural and electronic properties of both bare and adsorbate-covered surfaces. The technique had originally been exploited at the gas/so id interface, but it was later realised that it could be employed in liquids. As a result, it has rapidly found application in electrochemistry. 

Order and polydispersity are key parameters that characterize many self-assembled systems. However, accurate measurement of particle sizes in concentrated solution-phase systems, and determination of crystallinity for thin-film systems, remain problematic. While inverse methods such as scattering and diffraction provide measures of these properties, often the physical information derived from such data is ambiguous and model dependent. Hence development of improved theory and data analysis methods for extracting real-space information from inverse methods is a priority. 

The atoms defined in the quantum theory of atoms in molecules (QTAIM) satisfy these requirements [1], The atoms of theory are regions of real space bounded by a particular surface defined by the topology of the electron density and they have all the properties essential to their role as building blocks ... 

Two objects are similar and have similar properties to the extent that they have similar distributions of charge in real space. Thus chemical similarity should be defined and determined using the atoms of QTAIM whose properties are directly determined by their spatial charge distributions [32]. Current measures of molecular similarity are couched in terms of Carbo s molecular quantum similarity measure (MQSM) [33-35], a procedure that requires maximization of the spatial integration of the overlap of the density distributions of two molecules the similarity of which is to be determined, and where the product of the density distributions can be weighted by some operator [36]. The MQSM method has several difficulties associated with its implementation [31] ... 

Abstract Chemists may find it difficult to admit that their concepts and opportunities have always been strongly influenced by the available methods for characterization and analysis. Physics, has, of course, the lead when it comes to the visualization of single molecules in real space and to the detection of their specific, not ensemble-averaged properties. The challenge for chemistry is to provide molecules as objects of study which really disclose new concepts of structure and function. This chapter presents a chemical approach toward nanosciences which comprises (i) design and synthesis, (ii) immobilization, often using principles of self-assembly, (iii) visualization, e.g. by scanning probe... 

A modem description of a conventional hydrogen bond as well as its older, more accurate definition are based on Bader s theory of atoms in molecules (AIM theory) [4]. Bader considers matter a distribution of charge in real space of point-like nuclei embedded in the diffuse density of electron charge, p(r). All the properties of matter are manifested in the charge distribution and the topology... 

When the electrostatic properties are evaluated by AF summation, the effect of the spherical-atom molecule must be evaluated separately. According to electrostatic theory, on the surface of any spherical charge distribution, the distribution acts as if concentrated at its center. Thus, outside the spherical-atom molecule s density, the potential due to this density is zero. At a point inside the distribution the nuclei are incompletely screened, and the potential will be repulsive, that is, positive. Since the spherical atom potential converges rapidly, it can be evaluated in real space, while the deformation potential A(r) is evaluated in reciprocal space. When the promolecule density, rather than the superposition of rc-modified non-neutral spherical-atom densities advocated by Hansen (1993), is evaluated in direct space, the pertinent expressions are given by (Destro et al. 1989)... 

In addition, many of the ferroelectric solids are mixed ions systems, or alloys, for which local disorder influences the properties. The effect of disorder is most pronounced in the relaxor ferroelectrics, which show glassy ferroelectric behavior with diffuse phase transition [1]. In this chapter we focus on the effect of local disorder on the ferroelectric solids including the relaxor ferroelectrics. As the means of studying the local structure and dynamics we rely mainly on neutron scattering methods coupled with the real-space pair-density function (PDF) analysis. 

Albeit a number of conclusions can be gained from parity considerations, chemistry is described in terms of real space variables and particle ideology. Reaction coordinate, molecular species, molecular structure and properties are to be related to the present approach. This cannot be made rigorously because quantum mechanics is about quantum states and not objects in real space. This confusion has been fatal to a correct understanding of molecular phenomena in spite of the effort made by Primas [15]. 

Just as a reminder The dots between the vectors denote the scalar (inner) product and the crosses denote the cross (outer) product of the vectors. These vectors 6 are in units of nr, which is proportional to the inverse of the lattice constants of the real space crystal lattice. This is why one calls the three-dimensional space spanned by these vectors the reciprocal space and the lattice defined by these primitive vectors is called the reciprocal lattice. These primitive reciprocal vectors have the following properties ... 

The last condition is identical to Eq. (A.10). Therefore, g must be a reciprocal lattice vector. This shows us another important property of the reciprocal lattice it is the Fourier transform of the real space lattice. 

It is evident that in representing energy levels in solids extensive use is made of momentum (reciprocal- or k-) space rather than the real-space representations which theoretical chemists frequently employ for the description of isolated molecules. One of the obvious advantages in so doing is that optical and spectroscopic properties are concisely illustrated and the various symmetry-allowed transitions clearly identified with reference to such E/k plots. 

Therefore we propose here an alternative route to inspect the local dielectric and polarization properties using non-destructive and non-invasive methods based on scanning force microscopy (sfm). Simultaneously, these techniques offer a high resolution in real space being extended down to the atomic scale when inspecting ferroelectric systems under ultra-high... 

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