Coulomb interactions in a computer simulation of a system periodic in two directions

The integral representation of the gamma function and the Poisson summation formula are used to calculate the interaction energy of charged particles in a 3-dimensional system periodic in two directions. A parallelogram shape simulation box is considered. Calculations are carried out for interactions described by any inverse power, and analytical continuation of the energy function leads to the final expression for the Coulomb interaction energy. Summation over the simulation box replica along one or the other side of the box base is replaced by summation in reciprocal space. Therefore there are two equivalent formulas for the potential energy that offer the possibility of avoiding slowly convergent series. The energy expressions are identical to those obtained from the Lekner method. The special case is considered where the functions defining the energy are infinite, i.e. when two charges lie on a line parallel to the simulation box side that was chosen to convert real space summation into reciprocal space.     

Diminished reactivity of ortho-substituted phenacyl bromides toward nucleophilic displacement

A systematic increase of substitution rates by tert-butylamine on alpha-bromopropiophenones is observed with meta or para substituents with increasing electron-withdrawing ability (k x 10(3) L M(-1) min(-1) = 12.7 (p-CH(3)), 15.7 (o-F), 20.5 (H), 20.0 (p-Cl), 23.6 (m-Cl), 27.3 (p-CF(3))). Within an ortho-substituted series, the reactivities decrease (k x 10(3) L M(-1) min(-1) = 7.64 (o-OCH(3)), 5.31 (o-CH(3)), 2.85 (o-Cl), 2.40 ( o-CF(3))). Ortho-substitution results occur from rotational barrier effects and an Adelta(sigma)+ Bdelta(sigma)+ repulsion. The major bonding contribution between reaction and alpha-substituent centers (A-B) is only the sigma bond. When pi bonding is allowed between A and B (meta/para-substitution), delocalization and stabilization of the reacting center occurs.