Real-space representation

Fig. 8 LEED pattern as observed during preparation of a MgCL-film. a Pd(lll), b 1 ML MgCl2(001)/Pd(lll), c multilayer MgCl2(001)/Pd(lll). d Schematic real space representation of b the mesh represents the Cl lattice and spots the underlying Pd lattice...

We now consider the first order correction to the average potential, i.e. Vn>. In real space representation, substituting Equations (11) and (13) into (10) gives [36]... 

Abraham, N.L. and Probert, N.J., A periodic genetic algorithm with real space representation for crystal structure and polymorph prediction, Phys. Rev. B., 73, 224106,2006. 

Algorithm with Real-Space Representation for Crystal Structure and Polymorph Prediction. 

Table 6.3 Contributions to the binding energy (in Ry per atom) of sodium, magnesium, and aluminium within the second order real-space representation, eqn (6.73), using Ashcroft empty-core pseudopotentials. L/gf is defined by eqn (6.75). The numbers in brackets correspond to the simple expression, eqn (6.77), for = 0) and to the experimental values of the binding energy and negative cohesive energy respectively.

This can be written within the real-space representation as... 

The beauty of the real-space representation is that it separates out the very small structure-dependent contribution to the total binding energy as a single sum over pair potentials. As illustrated in Fig. 6.7, these metallic pair... 

Figure 4. Real space representations of CO chemisorbed on a Rh(lll) surface (a) (y/ 3 X y/ 3)R30° overlayer structure visible at low CO exposures (b) (2 X 2) structure seen at high surface coverage...

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. 

An important feature that affects the numerical solution strategy is that these equations are written in the spectral space, either in the three dimensional space of wave-vectors (/-propagated UPPE) or in a two-dimensional space of transverse wave-vectors plus a one dimensional angular-frequency space (z-propagated UPPE). At the same time, the nonlinear material response must be calculated in the real-space representation. Consequently, a good implementation of Fast Fourier Transform is essential for a UPPE solver. 

Step three consist in transforming the equation from the spectral- to the real-space representation. Mathematically, this is nothing but a Fourier transform that results in the following standard rules for differential operators ... 

Finally, transforming into the real-space representation, we arrive at NEE... 

This step is meant to make it easy to implement a numerical solver in the real space, as it results in the equation that only contains simple differential operators in the real-space representation ... 

The latest data has been analysed by a novel use of a cluster form-factor to fit the data in the high-Q region. The inter-molecular correlations within the tetrahedral network are evaluated up to a range of second nearest neighbour molecules. The transformation to a real-space representation in Fig 4, shows how well the local stmcture is represented by this approach and confirms that the four-bonded network gives excellent results. 

A simple estimate of the computational difficulties involved with the customary quantum mechanical approach to the many-electron problem illustrates vividly the point [255]. Consider a real-space representation of ( ii 2, , at) on a mesh in which each coordinate is discretized by using 20 mesh points (which is not very much). For N electrons, becomes a variable of 3N coordinates (ignoring spin), and 20 values are required to describe on the mesh. The density n(r) is a function of three coordinates and requires only 20 values on the same mesh. Cl and the Kohn-Sham formulation of DFT (see below) additionally employ sets of single-particle orbitals. N such orbitals, used to build the density, require 20 values on the same mesh. (A Cl calculation employs in addition unoccupied orbitals and requires more values.) For = 10 electrons, the many-body wave function thus requires 20 °/20 10 times more storage space than the density and sets of single-particle orbitals 20 °/10x 20 10 times more. Clever use of symmetries can reduce these ratios, but the full many-body wave function remains inaccessible for real systems with more than a few electrons. 

Fig. 2. Dominant fluctuation modes in the HEX phase, (a) Real-space representation according to the work of Laradji et al. (1997). (b) Location of the fluctuation peaks, shown as gray dots, in reciprocal space proposed by Qi and Wang (1998). The black dots are the Bragg peaks of the HEX phase. Reproduced with permission from C. Y. Ryu, M. S. Lee, D. A. Hajduk, and T. P. Lodge. J. Polym. Sci. B Polym. Phys., 1998 81 5345-5357.)...

This represents an initial concentration field in which all of the diffusing species is concentrated at the origin. To find Co of eqn (7.21), we require c k, 0). If we Fourier tranform the initial concentration profile given in eqn (7.22), the result is c(k,0) = 1. We are now prepared to invert the transformed concentration profile to find its real space representation given by... 

Here ET is a shift in energy such that E0 — ET 0, where E0 is the ground-state energy. In the real-space representation we have... 

The scheme would be to construct each term in the effect of the Hartree-Fock operator on a real-space representation of an MO, using either that spatial representation or the linear expansion representation, whichever is appropriate, easier or more tractable, and transform the expansion-method terms to the spatial representation by means of or its inverse and solve the resulting equation on a chosen grid of points. 

Davidson s (Appendix E) algorithms. A two-dimensional real space representation of the resulting transition density matrices is convenient for an analysis and visualization of each electronic transition and the molecular optical response in terms of excited-state charge distribution and motions of electrons and holes (Section IIC). Finally, the computed vertical excitation energies and transition densities may be used to calculate molecular spectroscopic observables such as transition dipoles, oscillator strengths, linear absorption, and static and frequency-dependent nonlinear response (Appendix F). The overall scaling of these computations does not exceed X in time and in memory (A being the... 

In this section we derive an effective Hamiltonian that describes the high energy physics associated with particle-hole (or ionic) excitations across the charge gap. The Hamiltonian will describe a hole in the lower Hubbard band and a particle in the upper Hubbard band, interacting with an attractive potential. This attractive potential leads to bound, excitonic states. In the next chapter we derive an effective-particle model for these excitons. A real-space representation of an ionic state is illustrated in Fig. 5.5(b). 

Figure 5-34. a) Left total and Gd projected (shaded) DOS for GdioCli8C4, center C projected DOS, right MO s for interacting C2 units (a, and a omitted). The dashed line indicates the Fermi level, b) Real space representation of the split-off d band shown by an arrow in a). Gd and Cl atoms in the base of the bioctahedron are connected by lines. 

The principles of X-ray and neutron diffraction studies of molecular liquids are presented. The measured liquid structure factor S (Q) provides detailed information on the conformation of the molecular unit which is interpreted through the molecular form-factor fx(Q). The structural characteristics of the liquid are contained in the term D / (Q) which may be transformed to give a real-space representation dL(r) which contains a weighted-sum of the partial pair-correlation functions, gotg(r). 

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