Real space methods

D. Van Dyck and W. Coene, The real space method for dynamical electron diffraction calculations in HREM... 

Similarly, expanding the KS potential in an LCAO expansion makes molecular density-functional calculations practical [9]. For metals and similar crystalline solids, it is best to expand the Kohn-Sham potential in momentum space via Fourier coefficients. For molecular solids various real-space method are under investigation. For molecules studied with the big, well-chosen Gaussian basis sets of quantum chemistry, it is undoubtedly best to expand the KS potential in linear-combination-of-Gaussian-type-orbital (LCGTO) form [10]. 

Electronic structure methods for studies of nanostructures can be divided broadly into supercell methods and real-space methods. Supercell methods use standard k-space electronic structure techniques separating periodically repeated nanostructures by distances large enough to neglect their interactions. Direct space methods do not need to use periodic boundary conditions. Various electronic structure methods are developed and applied using both approaches. In this section we will shortly discuss few popular but powerful electronic structure methods the pseudopotential method, linear muffin-tin orbital and related methods, and tight-binding methods. 

While the supercell approach works well for localized systems, it is typically necessary to consider a very large supercell. This results in a plane-wave basis replicating not only the relevant electronic states but also vacuum regions imposed by the supercell. A much more efficient method to implement for investigating the electronic structures of localized systems is to use real space methods such as the recursion methods [27] and the moments methods [28], These methods do not require symmetry and their cost grows linearly with the number of inequivalent atoms being considered. For these reasons, real space methods are very useful for a description of the electronic properties of complex systems, for which the usual k-space methods are either inapplicable or extremely costly. 

Within the real-space method, the kinetic energy operator is expressed by the finite-difference scheme. Here, we derive the matrix elements for the kinetic energy operator of one dimension in the first-order finite difference. By the Taylor expansion of a wavefunction i/r (/) at the grid point Z we obtain the equations,... 

In the implementation of the QM/MM approach with the real-space method, the QM cell that contains the real-space grids is embedded in the MM cell. One should take care for the evaluation of the potential upc(r) defined as Eq. (17-20). When a point charge in MM region goes inside the QM cell, it makes a singularity in the effective Kohn-Sham Hamiltonian, which may give rise to a numerical instability. To circumvent the problem, we replace a point charge distribution... 

S. Fitzwater and H. A. Scheraga, Proc. Natl. Acad. Sci. USA, 79, 2133 (1982). Combined-Information Protein Structure Refinement Potential Energy-Constrained Real-Space Method for Refinement with Limited Diffraction Data. 

Structure solution in powder diffraction is approached by two different methodologies. One is using the conventional reciprocal space methods. The second is by real space methods where all the known details about the sample (say, molecular details such as bond distances, angles, etc., for an organic molecule, and coordination spheres such as octahedral, tetrahedral etc., in case of inorganic compounds) in question are exploited to solve the structure. 

The choice of whether to use the conventional reciprocal space approach or the modem real space method depends on the quality of the powder diffraction and the complexity of the problem in hand. 

We focus on a different approach based on real space methods, which are basis free. Real space methods have gained ground in recent years [10-13] owing in great part to their great simplicity and ease of implementation. In particular, these methods are readily implemented in parallel computing environments. A second advantage is that, in contrast with a plane wave approach, real space methods do not impose artificial periodicity in non-periodic systems. While plane wave basis... 

Therefore, real space methods have received increased interest in recent years [38]. In real space, the number of degrees of freedom m can be increased more easily (the computing time typically scales like m or mlnm). However, the time integration of the diffusion equation can no longer be performed exactly. 

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. 

The results show that both the LDA and the GGA are over-estimating the CH bond length and the vibrational frequency. These calculations were repeated using a real-space method (see next section). 

To illustrate the use of real-space methods, we again chose to study methane (CH4). For all calculations, we used the program octopus [90] (see also http //www.tddft.org/programs/octopus), which was written by some of the authors, and is freely available under an open source license. Furthermore, we employed the Troullier-Martins pseudo-potentials which are distributed with the code, and the GGA in the parameterization of Perdew, Burke and Ernzerhof. 

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