On the Expansion Formula of Atomic Orbitals in Spherical Harmonics around Another Center

1956 ◽  
Vol 11 (9) ◽  
pp. 980-986 ◽  
Author(s):  
Yukito Tanabe
Keyword(s):  
Spherical Harmonics ◽  
Expansion Formula ◽  
Atomic Orbitals

On the expansion of a Coulomb potential in spherical harmonics

1950 ◽  
Vol 46 (4) ◽  
pp. 626-633 ◽  
Author(s):  
B. C. Carlson ◽  
G. S. Rushbrooke
Keyword(s):  
Spherical Harmonics ◽  
Coulomb Potential ◽  
Addition Theorem ◽  
Electrostatic Potentials ◽  
Legendre Functions ◽  
Expansion Formula ◽  
Image Position

The addition theorem for Legendre functions leads, as is well known, to a useful expansion formula of importance in the theory of electrostatic potentials,or, in an alternative notation,

Crystal-field theory for the Rydberg states of polyatomic molecules

1986 ◽  
Vol 64 (7) ◽  
pp. 782-795 ◽  
Author(s):  
Ying-Nan Chiu
Keyword(s):  
Crystal Field ◽  
Spherical Harmonics ◽  
Rydberg States ◽  
Second Order ◽  
Crystal Field Theory ◽  
Linear Combinations ◽  
Atomic Orbitals ◽  
Square Planar ◽  
Rydberg Electron ◽  
D Orbitals

The potential on a Rydberg electron due to the cluster of atoms near the center of a polyatomic molecule is expanded in powers of spherical harmonics. Nonvanishing potentials in totally symmetric irreducible representations are obtained using the crystal field of the cluster of atoms in D3h, C3v, D4v, C4v, Td, and D2d symmetries. Odd as well as the usual even powers of spherical harmonics are included up to [Formula: see text]. Spectroscopically observable differences in potentials between a planar versus a nonplanar XY3 molecule and among a square planar, pyramidal, tetrahedral, and dihedral XY4 molecule are exhibited. First-order energies are given for a Rydberg [Formula: see text] state showing λ dependence. Second-order energies due to mixing of Rydberg states by odd and even power potentials and splitting of ±λ degeneracies are shown analytically for an nd as well as an nf Rydberg electron. The formalism is applicable to nonpenetrating Rydberg orbitals. Approximate radial integrals are obtained. Exact angular integrals for the first- and second-order energies are given. Symmetry-adapted combinations of the separated Y3 and Y4 ligand atomic orbitals are derived up to d orbitals. The correlations between these linear combinations of atomic orbitals as molecular configurations change are shown, e.g., as an XY4 molecule distorts from (D4h, C4v) to (D2d, Td) and vice versa.

Spin-resolved charge density and wavefunction refinements using MOLLYNX: a review

2021 ◽  
Vol 77 (5) ◽  
Author(s):  
Mohamed Souhassou ◽  
Iurii Kiblin ◽  
Maxime Deutsch ◽  
Ariste Bolivar Voufack ◽  
Claude Lecomte ◽  
...  
Keyword(s):  
High Resolution ◽  
Neutron Diffraction ◽  
Spherical Harmonics ◽  
Experimental Information ◽  
Theoretical Ground ◽  
X Ray Diffraction ◽  
Precise Description ◽  
New Approach ◽  
X Ray ◽  
Atomic Orbitals

MOLLYNX is a new crystallographic tool developed to access a more precise description of the spin-dependent electron density of magnetic crystals, taking advantage of the richness of experimental information from high-resolution X-ray diffraction (XRD), unpolarized neutron (UND) and polarized neutron diffraction (PND). This new program is based either on the well known Hansen–Coppens multipolar model (MOLLYNX-mult) or on a new expansion over a set of atomic orbitals (MOLLYNX-orb). The main difference between the two models is the basis of the expansion: in MOLLYNX-mult the expansion is over atom centered real spherical harmonics, in MOLLYNX-orb the expansion is over a set of atomic orbitals with which mono and bicentric contributions are calculated. This new approach of MOLLYNX-orb can also be applied to nonmagnetic crystals. This paper summarizes the theoretical ground of two models and describes the first applications to organic, organometallic and inorganic magnetic materials

Zonal and tesseral harmonic perturbations of an artificial satellite

1966 ◽  
Vol 25 ◽  
pp. 323-325 ◽  
Author(s):  
B. Garfinkel
Keyword(s):  
Angular Velocity ◽  
Spherical Harmonics ◽  
Artificial Satellite ◽  
Compact Form ◽  
Earth’S Rotation ◽  
Earth's Rotation ◽  
Secular Variations ◽  
Tesseral Harmonics

The paper extends the known solution of the Main Problem to include the effects of the higher spherical harmonics of the geopotential. The von Zeipel method is used to calculate the secular variations of orderJmand the long-periodic variations of ordersJm/J2andnJm,λ/ω. HereJmandJm,λare the coefficients of the zonal and the tesseral harmonics respectively, withJm,0=Jm, andωis the angular velocity of the Earth's rotation. With the aid of the theory of spherical harmonics the results are expressed in a most compact form.

Ab initio band theory approach to electron energy loss near edge structures

1988 ◽  
Vol 46 ◽  
pp. 506-507
Author(s):  
Xudong Weng ◽  
O.F. Sankey ◽  
Peter Rez
Keyword(s):  
Ab Initio ◽  
Local Density ◽  
Interpolation Method ◽  
Atomic Orbital ◽  
Plane Waves ◽  
Large Set ◽  
Theory Approach ◽  
Band Theory ◽  
Atomic Orbitals ◽  
Pseudopotential Calculations

Single electron band structure techniques have been applied successfully to the interpretation of the near edge structures of metals and other materials. Among various band theories, the linear combination of atomic orbital (LCAO) method is especially simple and interpretable. The commonly used empirical LCAO method is mainly an interpolation method, where the energies and wave functions of atomic orbitals are adjusted in order to fit experimental or more accurately determined electron states. To achieve better accuracy, the size of calculation has to be expanded, for example, to include excited states and more-distant-neighboring atoms. This tends to sacrifice the simplicity and interpretability of the method.In this paper. we adopt an ab initio scheme which incorporates the conceptual advantage of the LCAO method with the accuracy of ab initio pseudopotential calculations. The so called pscudo-atomic-orbitals (PAO's), computed from a free atom within the local-density approximation and the pseudopotential approximation, are used as the basis of expansion, replacing the usually very large set of plane waves in the conventional pseudopotential method. These PAO's however, do not consist of a rigorously complete set of orthonormal states.

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