Infinite lattice sums

Ewald s formalism reduces the infinite lattice sum to a serial complexity of in the number of particles n, which has been reduced to n logn in more recent formulations. A review of variants on Ewald summation methods which includes a more complete derivation of the basic method is in [3]. 

Calculation of the infinite lattice sums remains the most difficult step in ab initio polymer calculations. They can be evaluated in configuration space as well as in momentum space (or the two procedures can be also combined). There is not enough experience accumulated in the literature to decide which approach... 

A more specific problem for simulations of liquid water, and one that took some time to understand, is the question of properly incorporating long-range dipole-dipole interactions into the calculations. Initially, such interactions were either ignored [7,8] or included as a reaction field associated with a point dipole contained in a spherical cavity surrounded by a dielectric [33]. Next, infinite lattice sums over the dipole-dipole interactions associated with the periodic images... 

It follows from these considerations that, for a physically correct truncation of the infinite lattice sums, the following criteria have to be fulfilled ... 

Summarizing the above considerations for symmetry-adapted and electrostatically balanced truncation of the infinite lattice sums, the following requirements must be satisfied in actual calculations (using again the convention that indices 0, q, Qx, qi refer to cells, and r, s, u, v and a, refer to orbitals and atoms, respectively). ... 

Table S. 1 presents the HF energy per H atom in the infinite chain at the optimized lattice constant as well as values of the Fermi level (cp), the width of the half-filled valence band (Se), and the cohesion energy Ecoh (defined as the difference between the free atomic energy and the lattice energy per cell in the infinite chain) at the HF level. The computations involved 64 atomic neighbors in the crystal, so that all infinite lattice sums convei ed properly for each basis set. We can also see that the cohesion enei y is saturated with respect to the size of the atomic basis and lies somewhat above 1 eV/H atom. The equilibrium lattice constant is larger than the interatomic distance in the hydrogen molecule, as predicted earlier by cluster calculations. ...

Ewald summation was invented in 1921 [7] to permit the efl5.cient computation of lattice sums arising in solid state physics. PBCs applied to the unit cell of a crystal yield an infinite crystal of the appropriate. symmetry performing... 

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. 

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 ... 

In Eqs. (4), (6) and (11), there are infinite summations in the overlap matrix Sk, the Fock matrix Fk, and the total energy per unit cell E JF. There are also infinite summations in T, V, Jlpq, and Kln. The infinite lattice summations in Sk, Tl and K converge themselves while V , J and the internuclear interactions have to be summed together to get converged results [20-22], In real calculations, cutoffs for the lattice summations have been imposed and multipole expansion techniques have been applied to hasten the convergence [22],... 

The crystal truncation rod (CTR). We now calculate the scattering intensity for a semi-infinite lattice, i.e., which has only one reflecting interface. This sum is nearly... 

The CTR shape is sensitive to the termination of the crystal surface. Calculations show (Fig. 5B) that CTR data are sensitive not only to the presence of the crystal termination but also to the detailed termination of the lattice. Here, we compare the scattering intensity for a semi-infinite lattice in which the outermost surface layer is ideally terminated with a bulk-like termination, or has been modified, either by its position, ds, or its scattering strength, fs. The total structure factor of the crystal with a modified surface is just the sum of individual structure factors for all atoms in the crystal (a table of commonly used structure factors is given in Appendix 3). Conceptually, this quantity can be broken into two parts, consisting of contributions from the modified surface layer, Fsurf, and from the semi-infinite substrate, Fsub -... 

A perfect crystal is completely defined by the contents of its unit cell. This is a region of the material that when repeated indefinitely in three dimensions, generates an infinite lattice. The lattice energy, Eiat is calculated by summing the interactions described in Section 1.2 for all atoms in the unit cell interacting with each other (in distinct pairs, triads, etc.) and with atoms in other parts of the lattice. The summed interactions between the cell and the rest of the lattice are then halved so that the total energy of a portion of lattice consisting of n cells is X Elat- Failure to do this would include multiple accumulations of the same interactions. 

For any finite system there is no problem the results are always finite. The only danger, therefore, is the summation to infinity ( lattice sums ), which always ends with the interaction of a part or whole unit cell with an infinite number of distant cells. Let us take such an example in the simplest case of a single atom per cell. Let us assume that the atoms interact by the Lennard-Jones pairwise potential (p. 284) ... 

Here e is the electronic charge magnitude, Z is the valence, is the free space static electric permittivity, a is the nearest neighbor lattice constant and C is the bulk Madelung constant computed from the lattice sum of an infinite extent. The bulk bandgap Egi is then given by... 

The atomic multipole expansion of the BI electrostatic potential is extremely useful, when the long-range, purely point-multipolar part of the potential yields an important contribution. This is the case in crystals, where the multipolar sums (up to the quadrupolar potential) are conditionally convergent lattice sums. Special techniques, like Ewald summation method [130, 131] and its generalizations [132] are needed to handle properly these infinite sums. Recently we applied the multipolar BI method, coupled with Ewald summation for the evaluation of electrostatic potentials and fields in zeolite cavities [133] and for the prediction of the IR frequency sequence of the different acidic sites in H-faujasite [134]. 

The difference of (6.44) from its molecular analog is defined by the lattice summation over all atoms N (including M = N) in the reference cell and in different unit cells of a crystal. In the model of an infinite crystal these lattice sums are infinite. All the simplifications mentioned are introduced in the Fock and overlap matrices (4.56) and HF LCAO equations (4.57) for periodic systems so that one obtains instead of (4.56) and (4.57) ... 

The first term is the lattice sum effected by the Ewald formula (28), the second is a surface dipole term which depends on the symmetry of the infinite lattice and e. The last term in (30) can be set equal to zero if z ... 

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