Direct lattice sum

The subscript labels a, b,... (i, j,...) correspond to unoccupied (occupied) bands. The Mulliken notation has been chosen to define the two-electron integrals between crystalline orbitals. Two recent studies demonstrate the nice converging behaviour of the different direct lattice sums involved in the evaluation of these two-electron integrals between crystalline orbitals [30]. According to Blount s procedure [31], the z-dipole matrix elements are defined by the following integration which is only non zero for k=k ... 

The Direct Lattice Sum. Dispersion forces between two atoms can be described by a potential function expressed in terms containing inverse powers of the internuclear separations, s. The simplest function of this sort includes a potential energy of attraction proportional to the inverse sixth power of the separation and a repulsion that is zero at distances of separation greater than a particular value se and infinite at separations less than sc. This is the so-called hard sphere or van der Waals model. Such an approximate potential function can be improved in two respects. Investigations of the second virial coefficient have revealed that the potential energy of repulsion is best described as proportional to the inverse twelfth power of the separation and the term in sr9, which accounts for the greater part of the total attraction potential, due to the attraction of mutually induced dipoles, should have added to it the dipole-quadrupole and quadrupole-quadru-pole attractions, expressed as terms in sr8 and s-10, respectively. The complete potential function for the forces between two atoms is, therefore ... 

The method described above has been used by Avgul et al. (I) to calculate the dispersion interactions between a number of adsorbates and the cleavage surface of graphite. A prime disadvantage of the method of direct lattice sums, apart... 

The concept, incorporated in this approximation, that matter is distributed continuously in each layer plane, while it might not be permissible for a localized adsorbed film, is actually close to the truth for a mobile adsorbed film, in which the rapid translation of the molecules along the surface prevents their responding to its fine structure. This approximatioi i produces just the sort of average that is desired and is so difficult to obtain from the direct lattice sum. 

Direct lattice-sum calculations show that the lattice contributions to e qQ of the two sites are of opposite sign [30]. The signs of the coupling constants have been confirmed as positive by the magnetic perturbation method [32], and the electronic excited states estimated assuming only one type of site [5, 19]. 

Computer simulations of bulk liquids are usually performed by employing periodic boundary conditions in all three directions of space, in order to eliminate artificial surface effects due to the small number of molecules. Most simulations of interfaces employ parallel planar interfaces. In such simulations, periodic boundary conditions in three dimensions can still be used. The two phases of interest occupy different parts of the simulation cell and two equivalent interfaces are formed. The simulation cell consists of an infinite stack of alternating phases. Care needs to be taken that the two phases are thick enough to allow the neglect of interaction between an interface and its images. An alternative is to use periodic boundary conditions in two dimensions only. The first approach allows the use of readily available programs for three-dimensional lattice sums if, for typical systems, the distance between equivalent interfaces is at least equal to three to five times the width of the cell parallel to the interfaces. The second approach prevents possible interactions between interfaces and their periodic images. 

In this expression SA is summed over the direct lattice, R( SB is summed over the reciprocal lattice, b,. The parameter r is chosen so as to obtain equally rapid convergence in the sums over the direct and reciprocal lattices. The 6 functions are defined by... 

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". ... 

Owing to the high symmetry of the lattice for the LiA and NaA zeolites, an analysis was made only of the section comprising 1/48 of the total cavity volume (Figure 9). The volume of the corresponding section for CaA zeolite was 1/24 of the total volume of the cavity. The potential energy was calculated for the inner cavity of the selected section for 16 different directions in the case of LiA and NaA zeolites and for 31 directions in the case of the CaA zeolite. For each of these directions, the (fi) was calculated for 40 different positions of the molecule. The lattice sums... 

A crystalline solid can be described by three vectors a, b and c, so that the crystal structure remains invariant under translation through any vector that is the sum of integral multiples of these vectors. Accordingly, the direct lattice sites can be defined by the set... 

The lattice sums in Eq. (6.65) reflect the fact that the Coulombic system between conducting walls has, in a way, three-dimensional periodicity. The basic cell of this three-dimensional array contains the original cell with the N particles plus the first set of images, that is, the N images resulting from the presence of just the lower wall [six Fig. 6.9(b)]. In fact, as we show explicitly in Appendix F.3.3, the energy of the extended system with a total of 2N charges, is directly linked to Uq by the relation... 

Obtain an analytic expression for the zone edge phonons in fee Cu using the Morse potential derived in the previous chapter. To do so, begin by deriving eqn (5.37) and then carry out the appropriate lattice sums explicitly for the Morse potential to obtain the force constant matrix. In addition, obtain a numerical solution for the phonon dispersion relation along the (100) direction. 

In detail, the solution of the equations of motion requires strict observance of the crystal symmetry and precise equilibrium coordinates of the molecules and their atoms in dependence on the temperature (and the pressure), since only then can the atom-atom potentials be computed correctly. Natkaniec et al. [7] solved the equations of motion numerically at T = 0 K for the perdeuterated naphthalene crystal N-ds (and later for many other molecular crystals). The lattice sum in Eq. (5.14) was limited to 24 neighbouring molecules j, after it had been found that inclusion of additional neighbour shells had no influence on the results. This is a direct consequence of the short range of the van der Waals interaction. 

Comparing this approximation with the actual lattice stuns (as determined by direct summation), one finds that it is accurate to within 1% for all densities. Since this error is smaller than that introduced by the dimensional scaling approximation, we will use the approximate form for the Coulomb sum in order to simplify calculations. It should be noted that these lattice sums will be corrected systematically when electron correlation is introduced in Sec. 5. 

M (r k) is verified to be periodic throughout the direct lattice (the equivalence of the sum over lattice vectors m = g -I-1 and the sum over g originates from translation invariance and the periodic boundary conditions). 

Representing S and F matrices in the Bloch function basis set at every k point of the sampling set. In this basis, the expression of the matrix elements contains a double sum over the direct lattice vectors. For example, a generic element of the Fock matrix represented in the reciprocal space is given by... 

The total energy of an infinite crystal is obviously infinite and has no physical meaning, but the total energy per cell, which includes the interaction of the nuclei and electrons in the 0-cell with all nuclei and electrons in the crystal, is finite. In this expression, a new sum over the infinite direct lattice vectors appears. 

Again, steps 2-6 are iterated to self-consistency. Basically, two aspects are specific for the application of this method to solids the calculation of matrices in direct space, which involve multiple sums over all the infinite direct lattice vectors, and the integration in reciprocal space. This latter aspect will be discussed with reference to a few specific examples in the next sections. 

The lattice sums in parameters Bpq absolutely converge and may be calculated by direct summation. Special care is needed only when calculating conditionally convergent sums in parameters of quadrupole components of the point charge field, which are proportional to the parameters Qap( ) of the Lorentz field ... 

The phonon frequencies are thus obtained by diagonalizing a 3i/ X 3i/ matrix, where v is the number of atoms per primitive cell. In some symmetry directions, this diagonalization can be performed analytically, and the calculation of the phonon frequencies then only requires some lattice sums. For the ionic contribution, these lattice sums can often even be done analytically. But the numerical evaluation in general is quite simple, so that... 

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