Isosurface of static electron density for doxycycline resulting from the Multipole Model refinement at level.Equivalent isosurface resulting from the Independent Atom Model. Notice the lack of covalent bonds' electrons.
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.
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The MultipoleDensityFormalism (also referred to as Hansen-Coppens Formalism) is an X-ray crystallography method of electron density modelling proposed...
higher multipoles. Evidently, inclusion of higher multipoles signifies that the polarization density P no longer is determined by a dipole moment density p...
length uniform density thin rod. In weak field general relativity, it is convenient to treat isolated sources using another type of multipole, which generalize...
the CMB temperature maps at higher multipoles, or ℓ ≥ 2, is considered to be the result of perturbations of the density in the early Universe, before the...
sphere. The solution takes the form of an infinite series of spherical multipole partial waves. It is named after German physicist Gustav Mie. The term...
series can be viewed as a potential arising from the 2n-moments (see multipole expansion). For many purposes only the monopole, dipole, and quadrupole...
interactions. Advanced algorithms, such as the Ewald summation or Fast Multipole Method, reduce this to O ( N log N ) {\displaystyle {\mathcal {O}}(N\log...
regarding the reproducibility of published results. While models based on the multipole expansion of the charge distribution of a molecule are limited to small...
We will use the density matrix formalism, and the optical Bloch equations for this. The main idea here is that the non-diagonal density matrix elements...
non-relativistic background, a few of them (e.g. the Dirac or path-integral formalism) also work with special relativity. Key features common to all RQMs include:...
1103/PhysRevD.59.083506. S2CID 119444449. Thorne, Kip S. (1980-04-01). "Multipole expansions of gravitational radiation" (PDF). Reviews of Modern Physics...
derivative of the quadrupole moment (or the l-th time derivative of the l-th multipole moment) of an isolated system's stress–energy tensor must be non-zero...
Self-Consistent Tight-Binding Quantum Chemical Method with Multipole Electrostatics and Density-Dependent Dispersion Contributions". Journal of Chemical...
stability of its solutions, 1974 – R. O. Hansen introduces Hansen–Geroch multipole moments, 1974 – Stephen Hawking discovers Hawking radiation. 1974 – Stephen...
Charges, magnetic moments, and orbital moments at individual atoms Electric multipole moments and magnetic dipole moments Heisenberg interaction parameters...
high-energy accelerator (the Berkeley HILAC) and to observe relativistic and multipole atomic transitions in those ions. This work led to the new field of "high-energy...
uses the computed solution to refine the mesh only where necessary Fast multipole method — hierarchical method for evaluating particle-particle interactions...
distribution of charge within the atom can allow dipole (or more generally multipole) moments to form, with which is associated a dipole field. Separating...
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