Multipolar density function

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 matrix M 1 is given in appendix I. In all but triclinic point groups, site-symmetry restrictions limit the allowed functions beyond the / even requirement. The symmetry-allowed multipolar density functions are given by the index-picking rules of appendix D, section D.3, and are listed in Table 10.1. 

These imperfections have occasioned to review the spherical DFT approach with respect to a more correct description for fluids which consists of non-spherical particles. The paper applies a statistic thermodynamic approach [7, 8] which uses density functional formulation to describe the adsorption of nitrogen molecules in the spatial inhomogeneous field of an adsorbens. It considers all anisotropic interactions using asymmetric potentials in dependence both on particular distances and on the relative orientations of the interacting particles. The adsorbens consists of slit-like or cylinder pores whose widths can range from few particle diameters up to macropores. The molecular DFT approach includes anisotropic overlap, dispersion and multipolar interactions via asymmetric potentials which depend on distances and current orientations of the interacting sites. The molecules adjust in a spatially inhomogeneous external field their localization and additionally their orientations. The approach uses orientation distributions to take the latter into account. 

Kusaka et al. (1995b) also present a density functional theory for ion-induced nucleation of polarizable multipolar molecules. For a fixed orientation of a molecule, the... 

Kusaka et al. (1995b) also present a density functional theory for ion-induced nucle-ation of polarizable multipolar molecules. For a fixed orientation of a molecule, the ion-molecule interaction through the molecular polarizability is independent of the sign of the ion charge, while that through the permanent multipole moments is not. As a result of this asymmetry, the reversible work acquires a dependence on the sign of the ion charge. 

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. 

Aubert E, Leb gue S, Marsman M, Bui TT, Jelsch C, Dahaoui S, Espinosa E, Angyan JG (2011) Periodic projector augmented wave density functional calculations on the hexachlorobenzene crystal and comparison with the experimental multipolar charge density model. J Phys Chem A 115 14484... 

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 ... 

We recall that in the multipolar expansion, the 3d density is expressed in terms of the density-normalized spherical harmonic functions dlmp as... 

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. 

As anticipated, the multipolar model is not the only technique available to refine electron density from a set of measured X-ray diffracted intensities. Alternative methods are possible, for example the direct refinement of reduced density matrix elements [73, 74] or even a wave function constrained to X-ray structure factor (XRCW) [75, 76]. Of course, in all these models an increasing amount of physical information is used from theoretical chemistry methods and of course one should carefully consider how experimental is the information obtained. 

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 evidence available suggests that the two approaches are about equally accurate, although the approach based on site-site correlation functions is more readily generalized to the treatment of multipolar interactions as well as to the effect of the attractive forces upon the structure and free energy at moderate and low density. In addition to the efforts made at extending the WCA theory to interaction site fluids, the Barker-Henderson theory has also been extended to these systems by Lombardero, Abascal, Lago and their co-workers. ... 

Accurate calculation of molecular electrostatic potentials in an algorithm which scales at most with the cube of molecule size has long remained a challenge. The established method for molecules is to fit the molecular density calculated from the orbital densities with multipolar functions attached to the atoms [39]. This method leads to a term which scales like the cube of the molecule size. It requires the introduction of an auxiliary basis set for the multipolar functions. The choice of such a basis set requires expertise and can easily lead to uncertain results. 

N, is the normalization factor, n, and are parameters depending on the atomic type. Pi are the multipolar population parameters and k and k are the contraction-expansion coefficients [11] for, respectively, spherical and multipolar valence densities. We have chosen orthogonal reference axes which respect the tetrahedral (23) T point group for Si and A1 atoms of the scolecite in order to reduce the number of multipolar parameters only the cubic harmonic multipoles (one octupole / = 3 and two hexadecapoles / = 4) have been refined for these two atoms. The pseudo-atom expansion was extended to the octupoles (/ = 3) for 0 including oxygen of water, and to the dipoles (/ = 1) for H. The best radial functions of Si and A1 atoms were obtained by inspection of the residual maps [12], ( / = 4,4,4,4 (1 = 1-4)) s were taken from Clement and Raimondi [13] i si = 3.05 bohr, = 2.72 bohr. For 0 atoms, = 4.5 bohr and the multipole exponents were respectively n = 2, 3, 4 up to the octupole level. 

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