Atomic multipolar models

The core and valence monopole populations used for the MaxEnt calculation were the ones of the reference density (electrons in the asymmetric unit iw = 12.44 and nvalence = 35.56). The phases and amplitudes for this spherical-atom structure, union of the core fragment and the NUP, are already very close to those of the full multipolar model density to estimate the initial phase error, we computed the phase statistics recently described in a multipolar charge density study on 0.5 A noise-free data [56],... 

For a number of 1907 acentric reflexions up to 0.463 A resolution, the mean and rms phase angle differences between the noise-free structure factors for the full multipolar model density and the structure factors for the spherical-atom structure (in parentheses we give the figures for 509 acentric reflexions up to 0.700A resolution only) were (Acp) = 1.012(2.152)°, rms(A( >) = 2.986(5.432)° while... 

For the multipolar modeling [11, 12] of the X-ray diffraction data the program XD [13] was used. The atomic density contributions are parametrized into a core term, p re, a spherical valence term, p,leiKe, and a set of multipolar functions ... 

All the above methods are somehow based on an orbital hypothesis. In fact, in the multipolar model, the core is typically frozen to the isolated atom orbital expansion, taken from Roothan Hartree Fock calculations (or similar [80]). Although the higher multipoles are not constrained to an orbital model, the radial functions are typically taken from best single C exponents used to describe the valence orbitals of a given atom [81]. Even tighter is the link to the orbital approach in XRCW, XAO, or VOM as described above. Obviously, an orbital assumption is not at all mandatory and other methods have been developed, for example those based on the Maximum Entropy Method (MEM) [82-86] where the constraints/ restraints come from statistical considerations. 

Fig. 17. (a) Orientation of molecular dipole moments in 3-methyl-4-nitropyridine. Y-oxide fih molecular dipole moment from direct integration methods fi2, from multipolar model and Hj from semi-empirical calculation, (b) Electrostatic potential around the molecule in the plane of ring atoms. Contours at 0.2kcal/mol (reproduced with permission from Hamazaoui et al. [79]). 

Purely mathematical models can be criticized as curve fitting exercises with very little appeal to chemical intuition and very little transferability. But models that seem reasonable chemically can be inaccurate, and maybe they will not be transferable either. To set some stakes on this rugged landscape, let us first examine the atomic multipole model, In this model each atom is a multipolar center which makes a contribution to the electric potential. 

We have described in this paper the first implementation of this Bayesian approach to charge density studies, making joint use of structural models for the atomic cores substructure, and MaxEnt distributions of scatterers for the valence part. Used in this way, the MaxEnt method is safe and can usefully complement the traditional modelling based on finite multipolar expansions. This supports our initial proposal that accurate charge density studies should be viewed as the late stages of the structure determination process. 

The least-squares Molly program based on the Hansen-Coppens model [10] was used to determine atomic coordinates, thermal parameters and multipolar density coefficients in scolecite. In the Hansen-Coppens model, the electron density of unit cell is considered as the superposition of the pseudo-atomic densities. The pseudoatom electron density is given by... 

The parameters Pim , Pcore, and k can be refined within a least square procedure, together with positional and thermal parameters of a normal refinement to obtain a crystal structure. In the Hansen and Coppens model, the valence shell is allowed to contract or expand and to assume an aspherical form [last term in (11)], as it is conceivable when the atomic density is deformed by the chemical bonding. This is possible by refining the k and k radial scaling parameters and population coefficients Pim of the multipolar expansion. Spherical harmonics functions yim are used to describe the deformation part. Several software packages [68-71] are available for multipolar refinement of the electron density and some of them [68, 70, 72] also compute properties from the refined multipolar coefficients. 

While exchange- and dispersion-induced dipole components are of a quantum nature, the multipole-induced dipole components can be modeled by classical relationships, if the quantum effects are small. For many systems of practical interest, multipolar induction generates the dominant dipole components. The classical multipole induction approximation has been very successful, except for the weakly polarizable partners (e.g., He atoms) [193]. It models the dipole induced in the collisional partner by polarization in the molecular multipole fields. 

One other aspect of nonprimitive electric double layer theories which is particularly relevant to the inner Stern region are the models for the water molecule and the ions. The simplest models for a water molecule and an ion are a hard-sphere point dipole and point charge, respectively. A more realistic model of the hard-sphere water molecule would include quadrupoles and octupoles and also polarizability. However the hard-sphere property is best avoided and replaced, for example, by a Lennard-Jones potential. An alternative to a multipolar water model are three point charge sites associated with the atoms within the water molecule. 

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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