Polar crystals, Coulomb energy

Present-day diffraction facilities provide easy access to very low-temperature data collection and hence to an accurate determination of electron densities in crystals. Application of standard theorems of classical physics then provides an evaluation of the Coulombic interaction energies in crystal lattices [27]. These calculations are parameter-less and hence are as accurate as the electron density is. Moreover, for highly polar compounds, typically aminoacid zwitterions and the like, a fortunate coincidence cancels out all other attractive and repulsive contributions, and the Coulombic term almost coincides with the total interaction energy. 

If the charge distribution is described by a set of distributed multipoles, as described in Section 4.2.3, the coulombic contributions to the intermolecular potential energy are calculated as multipole-multipole terms. The main disadvantage of even a rigorous distributed multipole model is that such a representation is still very localized, so that coulombic energies miss a large part of the penetration contribution. For use in a complete representation of intermolecular interactions, the dispersion, polarization, and repulsion terms must be evaluated separately by some semi-empirical or fiilly empirical method, for example the approximate atom-atom formulations of equations 4.38. 39. This approach has been extensively exploited by S. L. Price and coworkers over the years, in applications to molecular crystals [48]. 

The problem of convergence of coulombic sums can be acute in crystals with polar space groups, when molecular dipoles are parallel or nearly parallel to the polar axis. Indeed, in such cases even the physical meaning of a coulombic energy is questionable. The problem can be approximately solved by the estimation of an additional term for the coulombic energy, derived by an integration over a uniform distribution of dipolar unit cells [22] ... 

DFT for the pair energies in the coordination shell of the nitroguanidine crystal. The picture is instructive because the molecular pairs where uncorrected DFT gives the worst errors (pairs E and L) are dispersion-dominated stacked pairs. Not only are total energies nearly identical in DFT/D and PIXEL, but also the dispersion contributions are nearly identical, lending mutual support to the evaluation of the sum of Coulombic-polarization and repulsion terms in the two methods, as well as further validation to the PIXEL parameterization. 

Here a and d are the number of atoms in the acceptor and the donor, respectively, Ry is the distance between atoms i and j and and are the van der Waals and electrostatic potentials, respectively. The van der Waals potential is often represented by a Lennard-Jones potential (Eq. 8) or by a Buckingham potential (Eq. 9). The parameters a, fi, y and o are obtained from solid-state crystal data. The leading term in the electrostatic potential is the Coulomb interaction (first term in Eq. 10), where D is the effective dielectric constant (usually < D <2). Other terms may be added to represent induced polarization, etc. [40]. The geometries of the two components of the cluster are obtained from microwave or electron diffraction data or from quantum chemical calculations. It is assumed that these geometries do not change upon adduct formation. An initial guess is made for the structure of the adduct, and then the relative positions of the two (or more) components are varied until a local energy minimum is obtained. 

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]... 

Nonbonded terms typically include steric (e.g., van der Waals) and electrostatic (e.g., Coulombic) terms but may also include polarization contributions. Force field parameters for each bonded or nonbrmded term are obtained by fitting potential energy terms to ab initio (e.g., HF/6-31G ) or DFT calculations of small molecules or by fitting to experimental data such as crystal structure and the heat of vaporization (A/fy) for low-molecular-weight compounds. The form of specific terms used by different commercial, public domain, and customized force fields for polymer simulations are given in the sections that follow. 

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