Atomic multipolar tensors

In the same way that atomic charges are definable by the atomic polar tensor, higher order atomic multipoles can be defined in terms of higher-order atomic multipolar tensors.Specifically, atomic dipoles and quadrupoles, and for diatomics and linear molecules even atomic octupoles, can be calculated. For example, atomic dipoles are given by... 

Atomic Charges and Charge Flux Parameters from the Atomic Multipolar Tensors. The Planar Case... 

AMT = atomic multipolar tensor APT = atomic polar tensor FR = force related. 

ATOMIC CHARGES AND CHARGE FLUX PARAMETERS FROM THE ATOMIC MULTIPOLAR TENSORS. THE PLANAR CASE... 

Equation (1) specifically refers to EFP-EFP interactions. EFP-QM interactions are discussed later. Ecoui refers fo the Coulomb portion of fhe electrostatic interaction. This term is obtained using the distributed multipolar expansion introduced by Stone, with the expansion carried out through octopoles. The expansion centers are taken to be the atom centers and the bond midpoints. So, for water, there are five expansion poinfs (fhree at the atom centers and two at the O-H bond midpoints), while in benzene there are 24 expansion points. Ejn is the induction or polarization part of fhe electrostatic interaction. This term is represented by the interaction of fhe induced dipole on one fragment with the permanent dipole on another fragment, expressed in terms of the dipole polarizability. Although this is just the first term of fhe polarizability expansion, it is robust, because the molecular polarizability is expressed as a tensor sum of localized molecular orbital (LMO) polarizabilities. That is, the number of polarizability points is equal to the number of bonds and lone pairs in fhe molecule. This dipole-induced dipole term is iterated to self-consistency, so some many body effects are included. 

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. 

---