Reciprocal space sum

The sums in Eqs. (1) and (2) run, respectively, over the reciprocal space lattice vectors, g, and the real space lattice vectors, r and Vc= a is the unit cell volume. The value of the parameter 11 affects the convergence of both the series (1) and (2). Roughly speaking, increasing ii makes slower the convergence of Eq. (1) and faster that of Eq. (2), and vice versa. Our purpose, here, is to find out, for an arbitrary lattice and a given accuracy, the optimal choice, iiopt > tbal minimises the CPU time needed for the evaluation of the KKR structure constants. This choice turns out to depend on the Bravais lattice and the lattice parameters expressed in dimensionless units, on the... 

In order to And rjopt, we minimise an expression of the total CPU time, T, required for the Ewald sums. We assume that T is proportional to the number of vectors used in both the reciprocal and direct space sums, G /(2j ) and R /a . and to t, and t<, the CPU times required for the evaluation of a single term in each series. In formulae. 

This sum over a// reciprocal space vectors of the form (IV.2) should be carefully distinguished from the expansion (III.4) of the density of a periodic crystal. If the density has the "little period", the e>mansion (IV.3) reduces to a sum over all reciprocal lattice vectors. The general case (IV.3) and the periodic case (III.4) actually represent two extreme cases. The presence of "more and more symmetry" in the density can be gauged... 

N.2 Computational speedup for the direct and reciprocal sums Computational speedups can be obtained for both the direct and reciprocal contributions. In the direct space sum, the issue is the efficient evaluation of the erfc function. One method proposed by Sagui et al. [64] relies on the McMurchie-Davidson [57] recursion to calculate the required erfc and higher derivatives for the multipoles. This same approach has been used by the authors for GEM [15]. This approach has been shown to be applicable not only for the Coulomb operator but to other types of operators such as overlap [15, 62],... 

Since Ap is the Fourier transform of AF, Eq. (5.12) implies that minimization of J (Fobs - Pcaic )2 dr and of J (Fobs - Fcalc)2 dS are equivalent. Thus, the structure factor least-squares method also minimizes the features in the residual density. Since the least-squares method minimizes the sum of the squares of the discrepancies in reciprocal space, it also minimizes the features in the difference density. The flatness of residual maps, which in the past was erroneously interpreted as the insensitivity of X-ray scattering to bonding effects, is an intrinsic result of the least-squares technique. If an inadequate model is used, the resulting parameters will be biased such as to produce a flat Ap(r). 

The rates at which the direct-space and the reciprocal-space parts of the lattice sums converge are a function of the value of K. According to Williams (1981), the choice of K = 0.3/a minimizes the total computation time in the case of NaCl. With a lower K value of %0.2/a, the reciprocal sum can be neglected completely because of the rapid decay of the exponential factor in the Fourier summation. Generally, K can be chosen to be of the order of 0.1 A". ... 

The first Brillouin zone, formed by the bisector lines between the center and the nearest lattice points in reciprocal space, is a square bounded with lines h = ula and ky - nia. The lowest Fourier components of the sum of the local density of states (LDOS) over a range of energy A should have the form ... 

In infinite periodic systems, an attractive alternative to the use of a cut-off distance is the Ewald sum technique, first described for chemical systems by York, Darden and Pedersen (1993). By using a reciprocal-space technique to evaluate long-range contributions, the total electrostatic interaction can be calculated to a pre-selected level of accuracy (i.e., the Ewald sum limit is exact) witli a scaling that, in tlie most favorable case (called Particle-mesh Ewald , or PME), is AlogA. Prior to the introduction of Ewald sums, the modeling of polyelectrolytes (e.g., DNA) was rarely successful because of the instabilities introduced... 

In infinite periodic systems, an attractive alternative to the use of a cut-off distance is the Ewald sum technique, first described for chemical systems by York, Darden and Pedersen (1993). By using a reciprocal-space technique to evaluate long-range contributions, the total... 

The field of the gaussian distribution is summed in the reciprocal space. 

The convergence of the Ewald sum is independent of i], but seems to be optimal in both direct and reciprocal spaces if t] V1/3. Using 64-binary bit precision in computer programs, EM and a values precise to eight decimal figures can be obtained. 

First, the irreducible part of the Brillouin zone now varies from k = 0 to k = Tr/d = tt/2d. Indeed, doubling the parameter of the unit cell in real space halves the size of the Brillouin zone (or the reciprocal-space unit cell). Second, recall that orbital interactions are additive and that the final MO diagram (or band structure) is just the result of the sum of all the orbital interactions. Within each individual H2 unit the interactions simply correspond to the bonding (a) and antibonding (a ) MOs of each individual H2 unit. There are three types of interactions involving the MOs of different H2 units interactions between all the a orbitals interactions between all the a orbitals and interactions between the a and the a orbitals. Since all the an orbitals are equivalent by translational symmetry, their interaction is described by the Bloch function ... 

The sums may be carried out with respect to the atomic positions in direct (real) space or to lattice planes in reciprocal space, an approach introduced in 1913 by Paul Peter Ewald (1888-1985), a doctoral student under Arnold Sommerfeld (Ewald, 1913). In reciprocal space, the structures of crystals are described using vectors that are defined as the reciprocals of the interplanar perpendicular distances between sets of lattice planes with Miller indices (hkl). In 1918, Erwin Rudolf Madelung (1881-1972) invoked both types of summations for calculating the electrostatic energy of NaCl (Madelung, 1918). 

Electrostatic Terms. The electrostatic energy of a lattice of atoms of zero polarizability may be calculated exactly by the method of Bertaut, provided the position and charge of each atom in the structure are known, This method involves the infinite sum in reciprocal space... 

Here Mj is the Madelung constant based on I as unit distance, n is the number of molecules in the unit cell, zy is the charge number of atom j, V is the volume of the unit cell and h is the magnitude of the vector (hi, h2, ha) in reciprocal space or the reciprocal of the spacing of the planes (hihjha). The coordinates of atom j are a i/, X2, Xay. The sums over j are taken over all the atoms in the imit cell. F(h) is the Fourier transform of the Patterson function and (h) is the Fourier transform of the charge distribution /(r). F h) is given by... 

The Fourier transform equations show that the electron density is the Fourier transform of the structure factor and the structure factor is the Fourier transform of the electron density. Examples are worked out in Figures 6.14 and 6.15. If the electron density can be expressed as the sum of cosine waves, then its Fourier transform corresponds to the sum of the Fourier transforms of the individual cosine waves (Figure 6.16). The inversion theorem states that the Fourier transform of the Fourier transform of an object is the original object, hence the opposite signs in Equations 6.12.1 and 6.12.2. This theorem provides the possibility of using a mathematical expression to go back and forth between reciprocal space (structure factors) and real space (electron density), so that the phrase and vice versa is applicable here. 

We have seen that the diffracted waves Fhki, from a particular family of planes hkl, when Bragg s law is satisfied, depends only on the perpendicular distances of all of the atoms from those hkl planes, which are h xj for all atoms j. Therefore each Fhki carries information regarding atomic positions with respect to a particular family hkl, and the collection of Fhki for all families of planes hkl constitutes the diffraction pattern, or Fourier transform of the crystal. If we calculate the Fourier transform of the diffraction pattern (each of whose components Fhki contain information about the spatial distribution of the atoms), we should see an image of the atomic structure (spatial distribution of electron density in the crystal). What, then, is the mathematical expression that we must use to sum and transform the diffraction pattern (reciprocal space) back into the electron density in the crystal (real space) ... 

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