Atom-centered multipolar functions

The nature of the charge density parameters to be added to those of the structure refinement follows from the charge density formalisms discussed in chapter 3. For the atom-centered multipole formalism as defined in Eq. (3.35), they are the valence shell populations, PLval, and the populations PUmp of the multipolar density functions on each of the atoms, and the k expansion-contraction parameters for... 

To evaluate this expression for distributions expressed in terms of their multipolar density functions, the potential <1> and its derivatives must be expressed in terms of the multipole moments. The expression for charge distribution has been given in chapter 8 [Eq. (8.54)]. Since the potential and its derivatives are additive, a sum over the contributions of the atom-centered multipoles is again used. The resulting equation contains all pairwise interactions between the moments of the distributions A and B, and is listed in appendix J. 

The Ce dispersion coefficients for dipole dipole dispersion between pairs of interacting species, the coefficients for terms involving higher multipolar dispersion, and coefficients for three-body dispersion terms can be and have been evaluated by ab initio techniques [114 119] as well as through relations to experimental optical data based on moments of the dipole oscillator strength [120 122]. These are parameters of the interaction, not properties. However, as noted in Section IVA, values for Ce coefficients of like pairs (e.g., A-A), and possibly for other dispersion coefficients, can be used in simple [Eq. (4)] or in more complete forms [Eq. (2)] as an intrinsic property of a molecule. The basis set and correlation requirements for adequate evaluation show, in part, the same requirements for describing polarizabilities however, there are further needs and other than atom-centered functions are seen as being suited [49 52]. 

The key is that a single-center expansion of the transition density, implicit in a multipolar expansion of the Coulombic interaction potential, cannot capture the complicated spatial patterns of phased electron density that arise because molecules have shape. The reason is obvious if one considers that, according to the LCAO method, the basis set for calculating molecular wavefunctions is the set of atomic orbital basis functions localized at atomic centers a set of basis functions localized at one point in a molecule is unsatisfactory. 

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. 

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