scholarly journals Understanding Topological Insulators in Real Space

Molecules ◽  
2021 ◽  
Vol 26 (10) ◽  
pp. 2965
Author(s):  
Angel Martín Pendás ◽  
Francisco Muñoz ◽  
Carlos Cardenas ◽  
Julia Contreras-García
Keyword(s):  
Quantum Theory ◽  
Chiral Symmetry ◽  
Topological Insulator ◽  
Real Space ◽  
Electron Delocalization ◽  
Electron Localization ◽  
Visualization Tool ◽  
Atom Pairs ◽  
Chemical Concepts ◽  
Bulk Behavior

A real space understanding of the Su–Schrieffer–Heeger model of polyacetylene is introduced thanks to delocalization indices defined within the quantum theory of atoms in molecules. This approach enables to go beyond the analysis of electron localization usually enabled by topological insulator indices—such as IPR—enabling to differentiate between trivial and topological insulator phases. The approach is based on analyzing the electron delocalization between second neighbors, thus highlighting the relevance of the sublattices induced by chiral symmetry. Moreover, the second neighbor delocalization index, δi,i+2, also enables to identify the presence of chirality and when it is broken by doping or by eliminating atom pairs (as in the case of odd number of atoms chains). Hints to identify bulk behavior thanks to δ1,3 are also provided. Overall, we present a very simple, orbital invariant visualization tool that should help the analysis of chirality (independently of the crystallinity of the system) as well as spreading the concepts of topological behavior thanks to its relationship with well-known chemical concepts.

Large-scale real-space density-functional calculations: Moiré-induced electron localization in graphene

2015 ◽  
Vol 117 (11) ◽  
pp. 112811 ◽  
Author(s):  
Atsushi Oshiyama ◽  
Jun-Ichi Iwata ◽  
Kazuyuki Uchida ◽  
Yu-Ichiro Matsushita
Keyword(s):  
Density Functional Calculations ◽  
Density Functional ◽  
Large Scale ◽  
Real Space ◽  
Electron Localization ◽  
Space Density

How do electron localization functions describe π-electron delocalization?

2011 ◽  
Vol 13 (46) ◽  
pp. 20584 ◽  
Author(s):  
Stephan N. Steinmann ◽  
Yirong Mo ◽  
Clemence Corminboeuf
Keyword(s):  
Electron Delocalization ◽  
Electron Localization

scholarly journals Real Space Delocalization, Resonance and Aromaticity

2020 ◽  
Author(s):  
Leonard Reuter ◽  
Arne Lüchow
Keyword(s):  
Electronic Structure ◽  
Probability Density ◽  
Saddle Points ◽  
Valence Bond ◽  
Special Kind ◽  
Real Space ◽  
Structure Framework ◽  
Chemical Concepts ◽  
Bond Theory ◽  
The Many

<div>When chemists want to explain a molecule’s stability and reactivity, they often refer to the concepts of delocalization, resonance, and aromaticity. Resonance is commonly discussed within the electronic structure framework of valence bond theory as the stabilizing effect of mixing different Lewis structures. Yet, most computational chemists work with delocalized molecular orbitals, which are also usually employed to explain the concept of aromaticity, a special kind of ring delocalization that shows up in cyclic planar systems which abide certain number rules. As an intuitive picture for aromaticity, an electronic ring current has been hypothesized. However, all three concepts lack a real space definition, that is not reliant on orbitals or specific wave function expansions. Here, we outline a redefinition from first principles: the concepts are of kinetic nature and related to saddle points of the all-electron probability density |Ψ|². Delocalization means that likely electron arrangements are connected via paths of high probability density in the many-electron real space. In this picture, resonance is the consideration of additional electron arrangements, which offer alternative paths of higher probability. Most notably, the concept of aromatic ring currents in absence of a magnetic field is rejected and the famous 4n+2 Hückel rule is derived from nothing but the antisymmetry of fermionic wave functions. The analysis developed in this work allows for a quantitative discussion of important chemical concepts that were previously only accessible qualitatively or restricted to specific electronic structure frameworks.</div>

scholarly journals Twisted bulk-boundary correspondence of fragile topology

Science ◽  
2020 ◽  
Vol 367 (6479) ◽  
pp. 794-797 ◽  
Author(s):  
Zhi-Da Song ◽  
Luis Elcoro ◽  
B. Andrei Bernevig
Keyword(s):  
Boundary Conditions ◽  
Topological Insulator ◽  
Experimental Verification ◽  
Real Space ◽  
Two Dimensional ◽  
Edge States ◽  
Quantum Numbers ◽  
Boundary Correspondence ◽  
Topological States ◽  
Twisted Boundary Conditions

A topological insulator reveals its nontrivial bulk through the presence of gapless edge states: This is called the bulk-boundary correspondence. However, the recent discovery of “fragile” topological states with no gapless edges casts doubt on this concept. We propose a generalization of the bulk-boundary correspondence: a transformation under which the gap between the fragile phase and other bands must close. We derive specific twisted boundary conditions (TBCs) that can detect all the two-dimensional eigenvalue fragile phases. We develop the concept of real-space invariants, local good quantum numbers in real space, which fully characterize these phases and determine the number of gap closings under the TBCs. Realizations of the TBCs in metamaterials are proposed, thereby providing a route to their experimental verification.

scholarly journals Quantum theory of atoms in molecules, electron localization function, and localized-orbital locator investigations on trans-(NHC)PtI2(para-NC5H4X) complexes

2020 ◽  
Vol 44 (7-8) ◽  
pp. 482-486
Author(s):  
Sarvin Hossien Saraf ◽  
Reza Ghiasi
Keyword(s):  
Quantum Theory ◽  
Substituent Effect ◽  
Atoms In Molecules ◽  
Electron Localization Function ◽  
Electron Localization ◽  
Localization Function ◽  
Hammett Constants ◽  
Localized Orbital ◽  
In Trans ◽  
Electron Donating Groups

In this study, the MPW1PW91 method is applied to analyze the quantum theory of atoms in molecules, the electron localization function, and the localized-orbital locator in trans-(NHC)PtI2( para-NC5H4X) (X = H, F, COOH, CN, NO2, Me, OH, NH2) complexes. The substituent effect is assessed in the presence of electron-withdrawing groups and electron-donating groups and their influence on the Pt–C and Pt–N bonds of the molecules is analyzed using quantum theory of atoms in molecules, electron localization function, and localized-orbital locator methods. In addition, the eta index (η) is used to evaluate the Pt–C and Pt–N bonds in the studied complexes. The correlations between electron localization function, localized-orbital locator, and the η index values of Pt–C and Pt–N bonds with Hammett constants (σp) and dual parameters (σI and σR) are given.

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