Organic lattice sums

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. 

We will consider that each molecule of a liquid or an amorphotis organic polymer is centered on a normal site in a face centered cubic lattice and that all interactions may be described in terms of van der Waals interactions. The total potential of a molec ile will be calculated by summing over a large number of nei bors. Similarly, calcxilations of various properties of such a system will be peiv formed in terms of the fee lattice array. There is, of course, no physical basis for the selection of any one of the isotropic crystalline models except from the standpoint of simplifying the subsequent calculations. One cotild also perform calculations using a radial distribution function and integrating rather than performing lattice sums. Clearly these calculations are of an approximate na-ttare and are used only to demonstrate that several properties of liquids and polymers may be described in teims of van der Waals interactions. ... 

In typical organic crystals, molecular pairs are easily sorted out and ab initio methods that work for gas-phase dimers can be applied to the analysis of molecular dimers in the crystal coordination sphere. The entire lattice energy can then be approximated as a sum of pairwise molecule-molecule interactions examples are crystals of benzene [40], alloxan [41], and of more complex aziridine molecules [42]. This obviously neglects cooperative and, in general, many-body effects, which seem less important in hard closed-shell systems. The positive side of this approach is that molecular coordination spheres in crystals can be dissected and bonding factors can be better analyzed, as examples in the next few sections will show. 

In organic metals, the nature of the molecular tt orbitals that form the conduction bands leads to a dipolar hyperfine interaction that may be nonnegligible when compared with the contact contribution discussed above [23]. The various terms in the dipolar interaction modify K and Tx 1 in different ways. The dipolar component [3] can be written as a sum of terms, some of which produce anisotropic Knight shifts (or line broadening in powder samples) and contribute to the spin-lattice relaxation rate. 

Yalkowsky proposed that the entropy of fusion of an organic compound is the sum of translational, rotational, and internal entropy changes when it is released from the crystal lattice (Yalkowsky, 1979) ... 

The quantum mechanical approach cannot be used for the calculation of complete lattice energies of organic crystals, because of intrinsic limitations in the treatment of correlation energies. The classical approach is widely applicable, but is entirely parametric and does not adequately represent the implied physics. An intermediate approach, which allows a breakdown of the total intermolecular cohesion energy into recognizable coulombic, polarization, dispersion and repulsion contributions, and is based on numerical integrations over molecular electron densities, is called semi-dassical density sums (SCDS) or more briefly Pixel method. [12-14]... 

In the early days, optical nonlinearity of organic materials was measured usually with powder samples, mainly because it is very difficult to isolate organic compounds in the form of molecular crystals. In the case of centrosymmetric crystal lattices, macroscopic second-order nonlinear optical characteristics are not detected. Molecular crystals are organized assemblies of individual molecules held together by intermolecular forces. Their macroscopic nonlinear optical constants are estimated as the sum of the molecular polarizability of individual molecules. Thus, neglecting intermolecular interactions in the crystal, the nonlinear optical constant, dtlK, is expressed by... 

The lattices in which covalencies link atoms through many unit cells are themselves of various kinds. The simplest is perhaps that of diamond, in which each carbon is joined by tetrahedral oovalenoies to four others. The interatomic distances correspond to those between carbon atoms finked by single bonds in the molecules of organic compounds, and the total energy of the lattice is the sum of all the bond energies. 

Two general forms have been used for the pair potential v. The first was introduced by Walmsley and Pople (1964) in their treatment of the <1 = 0 lattice frequencies of solid COj. It consists of a 6-12 Lennard-Jones term (between molecular centers) and an orientation-dependent term in the form of a quadrupole-quadrupole interaetion. The seeond form, which has found wide application, consists of a sum over atom-atom interactions, summed over the nonbonded atoms of the two molecules. This type of pair potential function was developed for organic molecules and was used to account for the crystal structures of these systems. Kitaigorodskii (1966) determined the parameters for such potentials in this way. Dows (1962) first applied such a potential to the calculation of the librational lattice modes frequencies of solid ethylene using hydrogen-hydrogen repulsion terms as given by de Boer (1942). The usual form of this type of potential contains 6-exponential atom-atom terms ... 

The key concept in organic crystal modelling, after the crystal structure, is the static lattice energy (or packing energy), which literally sums the intermo-lecular interaction energy between all the rigid molecules in the crystal ... 

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