Lattice sum

It looks as if the hep lattice has slightly lower energy and would be favored. Interestingly, most van der Waals-bonded substances, primarily the noble gases actually form fee structures. The only explanation is that it is not a perfect theory. Remember, the assumed form for the potential repulsive potential was arbitrary and has no theoretical basis. 

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Only even values of Wi -t- m2 -t- m3 are used for the FCC lattice. The numerical values of these lattice sums are dependent on the exponents used for U(r), and Eq. VII-11 may be written... 

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]. 

A related approach carries out lattice sums using a suitable interatomic potential, much as has been done for rare gas crystals [82]. One may also obtain the dispersion component to E by estimating the Hamaker constant A by means of the Lifshitz theory (Eq. VI-30), but again using lattice sums [83]. Thus for a FCC crystal the dispersion contributions are... 

If the simulated system uses periodic boundary conditions, the logical long-range interaction includes a lattice sum over all particles with all their images. Apart from some obvious and resolvable corrections for self-energy and for image interaction between excluded pairs, the question has been raised if one really wishes to enhance the effect of the artificial boundary conditions by including lattice sums. The effect of the periodic conditions should at least be evaluated by simulation with different box sizes or by continuum corrections, if applicable (see below). 

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. 

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... 

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]. 

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 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. 

FIGURE 21. Packing energy, PE (kcal mol l at 10 A cutoff in the lattice sums), as a function of total molecular free surface, SM, in compounds 1-8. 

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... 

Table 6. Total Lattice Sums for Trans and Gauche Conformations in PET and Contributions from Intramolecular Interactions...

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 ... 

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... 

In the case of a square lattice, only two values of the lattice sums are necessary ... 

From the point of view of the NMR spectrum, it appears that y-AUOj is a highly disordered structure, and one takes as a model for quantitative interpretation the MgO type structure, AljgO, with the aluminum atoms randomly occupying % of the octahedral sites. This is an extremely simple type of lattice for which the necessary lattice sums of 1/R (second order interaction) are known. 

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". ... 

An essential element of the expressions derived for %, c, and the dipolar fields are the following lattice sums ... 

The long range of the dipole-dipole interaction leads to a sample shape dependence of the physical quantities dependent on an external magnetic field [32,33]. In the expressions obtained for the susceptibility, this sample shape dependence is home by the slowly convergent lattice sums 5, and If we consider sufficiently isotropic lattices, in the sense of fulfilhng =... 

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... 

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