Ewald gradient

The Ewald series for the three-dimensional crystal can also be differentiated. The first derivative yields expressions for the Madelung electric field Fm (due to local charges). The second derivative yields the Madelung field gradient, or, equivalently, the internal or dipolar or Lorentz field FD (due to local dipoles) [68-71],This second derivative can also generates the dimensionless 3x3 Lorentz factor tensor L with its nine components Lv/t ... 

The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

By having written a point-multipole as the spherical tensor gradients passing through a point, one easily derives the particle mesh Ewald method for point multipoles. The main differences occur in the calculation of the structure factor, which requires spherical tensor gradients of the Cardinal B-spline weight, and the calculation of the short-range real-space correction (see Section 1.6.3). 

The intramolecular van der Waal interactions were calculated only between atoms that were located at distances greater than their fourth nearest neighbors. A modified Ewald summation method (Karasawa and Goddard 1989) was used for calculating the nonbonded coulomb interactions, and an atom-based direct cutoff method was used for van der Waal interactions. Smart minimizer, as implemented in Cerius /Material Studio, was used for geometry optimization. The optimization was considered to be converged when a gradient of 0.0001 kcal/mole was reached. 

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