The Multipole Density Formalism (also referred to as Hansen-Coppens Formalism) is an X-ray crystallography method of electron density modelling proposed by Niels K. Hansen and Philip Coppens in 1978. Unlike the commonly used Independent Atom Model, the Hansen-Coppens Formalism presents an aspherical approach, allowing one to model the electron distribution around a nucleus separately in different directions and therefore describe numerous chemical features of a molecule inside the unit cell of an examined crystal in detail.
Theory
Independent Atom Model The Independent Atom Model (abbreviated to IAM), upon which the Multipole Model is based, is a method of charge density modelling. It relies on an assumption that electron distribution around the atom is isotropic, and that therefore charge density is dependent only on the distance from a nucleus. The choice of the radial function used to describe this electron density is arbitrary, granted that its value at the origin is finite. In practice either Gaussian- or Slater-type 1s-orbital functions are used. Due to its simplistic approach, this method provides a straightforward model that requires no additional parameters (other than positional and Debye–Waller factors) to be refined. This allows the IAM to perform satisfactorily while a relatively low amount of data from the diffraction experiment is available. However, the fixed shape of the singular basis function prevents any detailed description of aspherical atomic features.
Kappa Formalism In order to adjust some valence shell parameters, the Kappa formalism was proposed. It introduces two additional refineable parameters: an outer shell population (denoted as P v a l {\displaystyle P_{val}} ) and its expansion/contraction ( κ {\displaystyle \kappa } ). Therefore, the electron density is formulated as:
ρ a t o m = ρ c o r e + ρ v a l e n c e ′ ( κ r ) = ρ c o r e + P v a l κ 3 ρ v a l ( κ r ) {\displaystyle \rho _{atom}=\rho _{core}+\rho '_{valence}(\kappa r)=\rho _{core}+P_{val}\kappa ^{3}\rho _{val}(\kappa r)}
While P v a l {\displaystyle P_{val}} , being responsible for the charge flow part, is linearly coupled with partial charge, the normalised κ {\displaystyle \kappa } parameter scales radial coordinate r {\displaystyle r} . Therefore, lowering the κ {\displaystyle \kappa } parameter results in expansion of the outer shell and, conversely, raising it results in contraction. Although the Kappa formalism is still, strictly speaking, a spherical method, it is an important step towards understanding modern approaches as it allows one to distinguish chemically different atoms of the same element.
Multipole description In the multipole model description, the charge density around a nucleus is given by the following equation:
ρ a t o m ( r , θ , ϕ ) = P c o r e ρ c o r e ( r ) + P v a l κ 3 ρ v a l ( κ r ) + ∑ l = 0 l m a x κ ′ 3 R l ( κ ′ r ) ∑ m = − l l P l m ± Y l m ± ( θ , ϕ ) {\displaystyle \rho _{atom}(r,\theta ,\phi )=P_{core}\rho _{core}(r)+P_{val}\kappa ^{3}\rho _{val}(\kappa r)+\sum _{l=0}^{l_{max}}\kappa '^{3}R_{l}(\kappa 'r)\sum _{m=-l}^{l}P_{lm\pm }Y_{lm\pm }(\theta ,\phi )}
… excerpt ends here. Continue reading the full article.
