Numerical Calculation of Energy Eigen-values of the Hydrogen Negative Ion in the 2p^2 Configuration by Using the Variational Method

Calculation of energy eigen value of hydrogen negative ion (H − ) in 2p^2 configuration using the method of variation functions has been done. A work on H − can lead to calculations of electric multipole moments of a hydrogen molecule. The trial function is a linear combination of 8 expansion terms each of which is related to the Chandrasekhar’s basis. This work produces a series of 7 energy eigen values which converges to a value of −0.2468 whereas the value of this convergence is expected to be −0.2523. This deviation from the expected value is mainly due to the elimination of interelectronic distance (u) coordinate. The values of the exponent parameters used in this work contribute also to this deviation. This variational method will be applied to the construction of some energy eigen functions of Hv2 .

Quantized electric multipole insulators

The Berry phase provides a modern formulation of electric polarization in crystals. We extend this concept to higher electric multipole moments and determine the necessary conditions and minimal models for which the quadrupole and octupole moments are topologically quantized electromagnetic observables. Such systems exhibit gapped boundaries that are themselves lower-dimensional topological phases. Furthermore, they host topologically protected corner states carrying fractional charge, exhibiting fractionalization at the boundary of the boundary. To characterize these insulating phases of matter, we introduce a paradigm in which “nested” Wilson loops give rise to topological invariants that have been overlooked. We propose three realistic experimental implementations of this topological behavior that can be immediately tested. Our work opens a venue for the expansion of the classification of topological phases of matter.

Crystal structure, atomic net charges and electric moments in pyroelectric LiNaSO4at 296 K

The crystal structure and electric charge distribution in LiNaSO4have been studied at 296 K by X-ray diffraction using a spherical crystal. LiNaSO4is pyroelectric, nonferroelectric and an optically uniaxial insulator which crystallizes in the space groupP31c. Least-squares refinement (MOLLY) was based on 13 026 reflections. The asymmetric unit contains Li+, Na+and three SO42−ions, where one O and S lie on a threefold axis about which three O atoms are related with a threefold symmetry in each sulfate ion. Two of the O—S—O groups suffer from disorder. The net charges of the atoms in three independent sulfate ions were determined under ionic charge constraints. The S atoms have positive net electric charges and O atoms are negative. The components of the significant electric multipole moments in the principal axis directions are determined from the distribution of net atomic charges in each sulfate ion. Electric moments in the unit cell generate macroscopic electric moments in the crystal which interact with light. This interaction results in two axial vectors of second rank associated with an optical indicatrix. The ratio of the calculated axial vector components in the principal axis directions originating from the asymmetric unit is 1.0061 (1), which compares well with the ratio of 1.006 for the corresponding optical refractive indices of LiNaSO4for λNa= 589.29 nm.