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Real-Space

For example, the lowest order rule in one dimension, the three point rule reads [Pg.229]

In our opinion, the main advantage of real-space methods is the simplicity and intuitiveness of the whole procedure. First of all, quantities like the density or the wave-functions are very simple to visualize in real space. Furthermore, the method is fairly simple to implement numerically for 1-, 2-, or 3-dimensional systems, and for a variety of different boundary conditions. For example, one can study a finite system, a molecule, or a cluster without the need of a super-cell, simply by imposing that the wave-functions are zero at a surface far enough from the system. In the same way, an infinite system, a polymer, a surface, or bulk material can be studied by imposing the appropriate cyclic boundary conditions. Note also that in the real-space method there is only one convergence parameter, namely the grid-spacing. [Pg.229]


Flamers R J, Tromp R M and Demuth J M 1986 Surface electronic structure of Si(111)-7 7 resolved in real space Phys. Rev. Lett. 56 1972... [Pg.316]

Figure Bl.9.5. Geometrical relations between the Cartesian coordmates in real space, the spherical polar coordinates and the cylindrical polar coordinates. Figure Bl.9.5. Geometrical relations between the Cartesian coordmates in real space, the spherical polar coordinates and the cylindrical polar coordinates.
This expression holds true only in the large g region in reciprocal space (or small r in real space). Since... [Pg.1405]

The integral describes the spatial amplitude modulation of the excited magnetization. It represents the excitation or slice profile, g(z), of the pulse in real space. As drops to zero for t outside the pulse, the integration limits can be extended to infinity whereupon it is seen that the excitation profile is the Fourier transfonn of the pulse shape envelope ... [Pg.1523]

STM found one of its earliest applications as a tool for probing the atomic-level structure of semiconductors. In 1983, the 7x7 reconstructed surface of Si(l 11) was observed for the first time [17] in real space all previous observations had been carried out using diffraction methods, the 7x7 structure having, in fact, only been hypothesized. By capitalizing on the spectroscopic capabilities of the technique it was also proven [18] that STM could be used to probe the electronic structure of this surface (figure B1.19.3). [Pg.1679]

Roberts C J, Williams P M, Davies J, Dawkes A C, Sefton J, Edwards J C, Haymes A G, Bestwick C, Davies M C and Tendler S J B 1995 Real-space differentiation of IgG and IgM antibodies deposited on microtiter wells by scanning force microscopy Langmuir 1822... [Pg.1724]

The major role of TOF-SARS and SARIS is as surface structure analysis teclmiques which are capable of probing the positions of all elements with an accuracy of <0.1 A. They are sensitive to short-range order, i.e. individual interatomic spacings that are <10 A. They provide a direct measure of the interatomic distances in the first and subsurface layers and a measure of surface periodicity in real space. One of its most important applications is the direct determination of hydrogen adsorption sites by recoiling spectrometry [12, 4T ]. Most other surface structure teclmiques do not detect hydrogen, with the possible exception of He atom scattering and vibrational spectroscopy. [Pg.1823]

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. [Pg.2213]

One of the most efficient algorithms known for evaluating the Ewald sum is the Particle-mesh Ewald (PME) method of Darden et al. [8, 9]. The use of Ewald s trick of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fourier space can be adjusted within certain limits via a free parameter in the method, but one is still left with two distinct calculations. PME performs the real-space calculation in the conventional manner, evaluating the complementary error function within a cutoff... [Pg.464]

The real-space sum is particularly easy to parallelize as it simply involves a spatial decomposition with appropriate attention to the overlap between adjacent regions due to the cut-off radius. Performance for the real-space contribution alone is given in Fig. 4. [Pg.465]

The Fourier sum, involving the three dimensional FFT, does not currently run efficiently on more than perhaps eight processors in a network-of-workstations environment. On a more tightly coupled machine such as the Cray T3D/T3E, we obtain reasonable efficiency on 16 processors, as shown in Fig. 5. Our initial production implementation was targeted for a small workstation cluster, so we only parallelized the real-space part, relegating the Fourier component to serial evaluation on the master processor. By Amdahl s principle, the 16% of the work attributable to the serially computed Fourier sum limits our potential speedup on 8 processors to 6.25, a number we are able to approach quite closely. [Pg.465]

Fig. 4. Performance (top) and scaling behavior (bottom) of the real space part of PME on the Cray T3D. Fig. 4. Performance (top) and scaling behavior (bottom) of the real space part of PME on the Cray T3D.
In the Ewald summation method the initial set of charges are surrounded by a Gaussian distribution lated in real space) to which a cancelling change distribution must be added (calculated in reciprocal space). [Pg.350]

The unitary transform does the same thing as a similarity transform, except that it operates in a complex space rather than a real space. Thinking in terms of an added imaginary dimension for each real dimension, the space of the unitary matrix is a 2m-dimensionaI space. The unitary transform is introduced here because atomic or molecular wave functions may be complex. [Pg.44]

The X-ray and neutron scattering processes provide relatively direct spatial information on atomic motions via detennination of the wave vector transferred between the photon/neutron and the sample this is a Fourier transfonn relationship between wave vectors in reciprocal space and position vectors in real space. Neutrons, by virtue of the possibility of resolving their energy transfers, can also give infonnation on the time dependence of the motions involved. [Pg.238]

Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships, (a) Scattering m real space (h) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample t and the phonon wave vector, q. Heavy dots represent Bragg reflections. Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships, (a) Scattering m real space (h) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample t and the phonon wave vector, q. Heavy dots represent Bragg reflections.
Real-space three-dimensional imaging in air, vacuum, or solution with unsurpassed resolution high-resolution profilometry imaging of nonconductors (SFM). [Pg.9]

One of the major advantages of SEXAFS over other surface structutal techniques is that, provided that single scattering applies (see below), one can go direcdy from the experimental spectrum, via Fourier transformation, to a value for bond length. The Fourier transform gives a real space distribudon with peaks in at dis-... [Pg.232]

In a difiraction experiment one observes the location and shapes of the diffracted beams (the diffraction pattern), which can be related to the real-space structure using kinematic diffraction theory. Here, the theory is summarized as a set of rules relating the symmetry and the separation of diffracted beams to the symmetry and separation of the scatterers. [Pg.267]


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Advection real space

Analysis in Real Space

Angles, real-reciprocal space

Basic Formalism Describing the Relation between Real-Space Structure and Scattering Intensity in a SAXS Experiment

Chains with two-body interactions real space expansions

Coordinates real space

Crystal real-space crystallographic methods

Diffusion real space

Direct-space techniques real structures

Distance, real-reciprocal space relationships

Electron density maps real space refinement

Electronic structure real-space analysis

Electronic structure real-space methods

Electrostatic potential real-space

Existing Real-Space and Multigrid Codes

Force real-space

General description of renormalization techniques in real space

Imaging real-space

Nanostructure in Real Space (CDF)

Real Space Properties

Real Space Refinement Difference Fourier Syntheses

Real Space Tight-Binding Methods

Real space grid

Real space lattice vector

Real space refinement, of electron density maps

Real space renormalization

Real space structures, Fourier transform

Real space wave functions

Real space, surface structure

Real world, chemical space

Real-Space Basics

Real-Space Calculations

Real-Space Measurement of 3D Structure

Real-space R-factor

Real-space averaging

Real-space correlation coefficient

Real-space crystallographic methods

Real-space distribution, electronic states

Real-space information

Real-space method

Real-space refinement

Real-space renormalization group method

Real-space representation

Real-space structure

Reiteration of Why Real Space

Renormalization group real space

The Core-Valence Separation in Real Space

The real-space representation

Valence electrons real-space energy

What Are the Limitations of Real-Space Methods on a Single Fine Grid

Why Real Space

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