The real-space representation

The total binding energy can be rewritten as a sum over real space rather than q-space vectors by changing the order of their summation in . From eqn (6.57) the band-structure energy is given by 

This allows us to define the band-structure contribution to a pair potential, namely 

The Madelung energy, eqn (6.61), can also be expressed as a pairwise sum over coulomb interactions between the point ions, plus a q - 0 contribution arising from the electron-ion and electron-electron interactions. Grouping this together with the band-structure contribution we have 

In 1974 Finnis showed that the q - 0 limit could be evaluated directly by using the compressibility sum rule of the free-electron gas, which relates the long wavelength behaviour of the dielectric constant to its compressibility. He found that 

the total binding energy per atom of a NFE metal can be expressed in a physically transparent form, as the sum of a volume-dependent contribution and a pair-potential contribution in a manner that is reminiscent of the semi-empirical embedded atom potential of eqn (5.68). It follows from eqs (6.59M6.72) that 

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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... 

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 ... 

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... 

We summarize this chapter by reviewing the major results. In Sect. 6.2.1 we provided the outline of the real-space grid Kohn-Sham DFT (RS-DFT) which aims at massively parallel implementation. Since the Hamiltonian matrix of the KS-DFT is dominated by the diagonal part in the real-space representation the parallelization can be achieved only by local communications between neighboring nodes in contrast to the LCAO approach. In Sect. 6.2.2 we assessed a parallel performance of our RS-DFT code on a modern parallel machine equipped with 512 cores for a water cluster with an ice structure. To avoid the global communications associated with FFT we instead employ the Poisson equation to construct the Hartree potential. Then, we achieved a high parallelization ratio of 99.8 % measured on this... 

To facilitate the discussion, we couch DFT in the language of p, the first-order reduced density operator of the noninteracting reference system. Consider an N electron system in a spin-compensated state and in an external potential Wext(r) (extension to spin-polarized state is trivial). The real space representation of p is the density matrix... 

The first step in structure parameter determination is the interpretation of the obvious features in the real-space representation of the nanostructure topology. This interpretation leads to a qualitative model describing the nanostructure. Based on this description, a mathematical model may be set up and fitted to the data. Such fitting can be sometimes replaced by the direct determination of parameters of physical meaning. An example concerning the analysis of the transverse structure is presented in the sequel. 

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