Crystal Potential Formula for the Calculation of Crystal Lattice Sums

A new formula is derived for the determination of the potential energy of the central unit cell of a finite crystal; this formula is called the crystal potential formula. The crystal potential formula is based on a two-center Cartesian multipole expansion. The key feature of the crystal potential formula is that it achieves a separation of the lattice geometry and multipole moments. This feature allows one to compute in a highly effective way the crystal energies as a function of the molecules contained in the unit cell. Numerical implementation is discussed, and a numerical example is given. The example mimics an organic crystal composed of dipolar molecules. It is found that the crystal environment may create local minima and that the effects of the environment might be well-approximated by consideration of the proximate unit cells only. Both of these effects have important implications for the crystal engineering of molecular organic materials