Semi-classical Approaches the SCDS-Pixel Method

5 Semi-classical Approaches the SCDS-Pixel Method 

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] 

For the calculation of Coulombic energies, the electrostatic potential J generated by molecule A at point (i) of the charge density of molecule B, and the potential m generated by molecule A at nucleus m of molecule B, are calculated, and the total electrostatic potential energy between the two molecules is the sum of the electrostatic energies at points (i), Ek = q j, and at points (m), Em = Zmd m. 

Calling E the total electric field exerted by surrounding molecules at pixel i, ak the polarizability at pixel i, and g the dipole induced at pixel i by that field, the linear polarization energy is  

In order to assign pixel polarizabilities, each pixel has to be first assigned to a particular atom in the molecule. Let p be the number of atoms for which the nucleus-pixel distance is smaller than the atomic radius. If p=l, the charge pixel is within one atomic sphere only, and it is assigned to that atom. Ifp 1, the pixel is assigned to the atom from which the distance is the smallest fraction of the atomic radius. If p = 0, the pixel is assigned to the atom whose atomic surface is nearest. 

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