Lattice sum calculations

Potzel and Kalvius were the first to investigate zinc compounds with Zn Mossbauer spectroscopy. The first measurements were carried out with Znp2 powder at 4.2 K [80]. The observed quadrupole splitting could not be explained by a simple lattice sum calculation. More detailed measurements were carried out at... 

The equilibrium configuration of the surface region comprising n layers is determined by solving simultaneously the 4n equations obtained by equating to zero the partial derivatives of AU with respect to each of the variables. The equations so obtained are nonlinear and are solved by an iterative Newton-Raphson procedure (12), which necessitates calculating the second partial derivatives of AU with respect to all possible pairs of variables. A Bendix G15D computer was used for all numerical computations—i.e., evaluation of the various lattice sums, calculation of the derivatives of AU, and solution of the linearized forms in the Newton-Raphson treatment. 

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]. 

The many attempts at lattice-sum calculations of the electric field gradient tensor have already been listed in Section 5.4 in connection with the evaluation of the iron nuclear quadrupole moment. 

High-temperature measurements of the paramagnetic quadrupole splittings have been made in Y, Sm, Gd, Dy, Yb, and Lu garnets, but comparison with estimates from point-charge lattice-sum calculations gives only poor agreement [136]. 

Surface Stresses and Edge Energies. Some surface tension values, that is, values of the surface stress t, are included in Table VII-2. These are obtained by applying Eq. Vll-5 to the appropriate lattice sums. The calculation is very sensitive to the form of the lattice potential. Earlier calculations have given widely different results, including negative r [43, 51, 52]. 

B. A. Luty, I. G. Tironi, and W. F. van Gunsteren. Lattice-sum methods for calculating electrostatic interactions in molecular simulations. J. Chem. Phys., 103 3014-3021, 1995. 

U. Essmann, L. Perera, M. L. Berkowitz, T. Darden, H. Lee, and L. G. Pedersen. The smooth particle mesh ewald method. J. Chem. Phys., 103 8577, 1995. Brock A. Luty, Ilario G. Tironi, and Wilfried F. van Gunsteren. Lattice-sum methods for calculating electrostatic interactions in molecular simulations. J. Chem. Phys., 103 3014-3021, 1995. 

The packing energy of an organic crystal can be easily calculated by a lattice sum over pairwise interactions. The potential parameters for these calculations are summarized in Table 15. The packing energy is usually a quite accurate estimate of the crystal sublimation energy. 

Kashiwagi et al.10) determined the second moment anisotropy for the one-way drawn polyethylene terephthalate sheets discussed above. The three lattice sums S00, S2q and S4o were calculated from the crystal structure determination of Daubeny et al., the proton positions being calculated on the basis of known bond angles and lengths. The isotropic lattice sum S00 was adjusted to a value consistent with the measured isotropic second moment of 10.3G2. The values for P200, P220 etc. were then used to predict the optical anisotropy. The predicted refractive indices for the sheets of draw ratio 2 1 and 2.5 1 are shown in Fig. 10, together with the experimental... 

Yh fyT = X)( z) > as cubic and completely disordered lattices (incidentally, the type of arrangements for which in the classical Lorentz cavity-field calculation the contribution of the dipoles inside the small sphere vanishes) these lattice sums vanish for large spherical samples. The sums and... 

For a polyatomic adsorbate it may be necessary to sum the interactions over each atom of the adsorbate molecule as well. The various values of sy in Equation 2 can, for a particular adsorbent of known crystal lattice, be calculated in terms of a single distance, z, which is defined as the distance between the center of an adsorbate atom and the mathematical plane in which lie the centers of the surface atoms of the adsorbent. 

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... 

We have developed general expressions in the preceding sections for the calculation of the dispersion potential contribution to the potential energy for adsorption and for the vibrational frequency of the adsorbed molecule, using various expressions for the lattice sums. We now propose to apply these equations, for the purpose of comparison with experiment, to the adsorption of the inert gases by graphite. 

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... 

In Section 2 we briefly summarize the basic mathematical expressions of the LCAO Hartree-Fock crystal orbital method both in its closed-shell and DODS (different orbitals for different spin) forms and describe the difficulties encountered in evaluating lattice sums in configuration space. Various possibilities for calculating optimally localised Wannier functions are also presented. They can be efficiently used in the calculation of excited states and correlation effects discussed in Section 3. 

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