Closed Shell States in Periodic Compounds

Journal “Scientific Israel ‒ Technological Advantages", 2019

It is shown that the basis of the periodic system of elements is a cyclical change in the moment of distribution of valence electrons as one of the parameters of the inhomogeneity of the system. An approximate expression of this moment is found through the degree of filling of the outer shell of an atom and on its basis a graphical representation of a periodic law is proposed, taking into account its number, charge, atomic radius and distribution of valence electrons. The graph reveals the existence of additional periods ending in inert metals, not only in known elements but also in the group of lanthanides and actinides, allowing them to be organically incorporated into the periodic system. The consistency of such a representation of the classical and quantum models of atoms is shown.

Physical Review B, 1993

We present a scheme for calculating the (spin-unrestricted) Hartree-Fock (HF) band structure of periodic solids that uses the accurate linearized-augmented-plane-wave basis set. In contrast with linear-combination-of-atomic-orbitals-like schemes, the convergence of the variational HF results can be easily monitored, and the cumbersome evaluation of multicenter integrals is avoided. Potentials and charge densities are evaluated without any shape approximation, and the singularity due to the long-range nature of the Coulomb potential is handled in reciprocal space. All the elements of the Periodic Table can be equally treated, and the relativistic effects for heavy elements are included as in the standard local-density-approximation case. The method is tested on silicon and diamond, where recent HF calculations are available for comparison. The diagonal Coulomb-holeplus-screened-exchange approximation can also be implemented with little additional computational effort.

Symmetry

The group theoretical description of the periodic system of elements in the framework of the Rumer–Fet model is considered. We introduce the concept of a single quantum system, the generating core of which is an abstract C*-algebra. It is shown that various concrete implementations of the operator algebra depend on the structure of the generators of the fundamental symmetry group attached to the energy operator. In the case of the generators of the complex shell of a group algebra of a conformal group, the spectrum of states of a single quantum system is given in the framework of the basic representation of the Rumer–Fet group, which leads to a group-theoretic interpretation of the Mendeleev’s periodic system of elements. A mass formula is introduced that allows giving the termwise mass splitting for the main multiplet of the Rumer–Fet group. The masses of elements of the Seaborg table (eight-periodic extension of the Mendeleev table) are calculated starting from the atomic number Z...

Physical Review B, 1997

The theory of space-group representations is extended to aperiodic crystals by reformulating it as the theory of symmetry-required degeneracies of electronic levels that emerges from the Fourier-space approach to crystal symmetry. As an illustration it is shown that the nonvanishing of a simple linear combination of phase functions belonging to commuting elements from the little group of q requires the degeneracy of all levels with generalized Bloch wave vector q. This condition is applied to all cubic and icosahedral centrosymmetric nonsymmorphic space groups, and to the two nonsymmorphic space groups of periodic crystals that have no systematic extinctions. ͓S0163-1829͑97͒00146-X͔

1994

This paper is a review of research on molecular periodic systems, a developed field of research.

Physical Review Letters, 1999

The group theoretical treatment of bound and scattering state problems is extended to include band structure. We show that one can realize Hamiltonians with periodic potentials as dynamical symmetries, where representation theory provides analytic solutions, or which can be treated with more general spectrum generating algebraic methods. We find dynamical symmetries for which we derive the transfer matrices and dispersion relations. Both compact and non-compact groups are found to play a role.

Recent Trends in Theory of Physical Phenomena in High Magnetic Fields, 2003

The novel meso-nucleo-spinics effect of the nuclear spin polarization induced periodic structure creation in a low-dimensional electron system is studied theoretically. It is shown that the periodically distributed nuclear magnetization results in the periodic hyperfine magnetic field which, in turn, creates a periodic electron structure. The electron wave functions and energy spectrum for such a structure are evaluated.

2005

We present a simple and general method for construction of localized orbitals to describe electronic structure of extended periodic metals and insulators as well as confined systems. Spatial decay of these orbitals is found to exhibit exponential behavior for insulators and power law for metals. While these orbitals provide a clear description of bonding, they can be also used to determine polarization of insulators. Within density functional theory, we illustrate applications of this method to crystalline Aluminium, Copper, Silicon, PbTiO$_3$ and molecules such as ethane and diborane.

Journal of Mathematical Physics, 2000

We show that dynamical symmetry methods can be applied to Hamiltonians with periodic potentials. We construct dynamical symmetry Hamiltonians for the Scarf potential and its extensions using representations of su(1, 1) and so(2, 2). Energy bands and gaps are readily understood in terms of representation theory. We compute the transfer matrices and dispersion relations for these systems, and find that the complementary series plays a central role as well as non-unitary representations.

The Journal of Chemical Physics, 2021

The localization spread gives a criterion to decide between metallic and insulating behavior of a material. It is defined as the second moment cumulant of the many-body position operator, divided by the number of electrons. Different operators are used for systems treated with open or periodic boundary conditions. In particular, in the case of periodic systems, we use the complex position definition, which was already used in similar contexts for the treatment of both classical and quantum situations. In this study, we show that the localization spread evaluated on a finite ring system of radius R with open boundary conditions leads, in the large R limit, to the same formula derived by Resta and co-workers [C. Sgiarovello, M. Peressi, and R. Resta, Phys. Rev. B 64, 115202 (2001)] for 1D systems with periodic Born-von Kármán boundary conditions. A second formula, alternative to Resta's, is also given based on the sum-overstate formalism, allowing for an interesting generalization to polarizability and other similar quantities.