Asymmetric topological pumping in nonparaxial photonics

Topological photonics was initially inspired by the quantum-optical analogy between the Schrödinger equation for an electron wavefunction and the paraxial equation for a light beam. Here, we reveal an unexpected phenomenon in topological pumping observed in arrays of nonparaxial optical waveguides where the quantum-optical analogy becomes invalid. We predict theoretically and demonstrate experimentally an asymmetric topological pumping when the injected field transfers from one side of the waveguide array to the other side whereas the reverse process is unexpectedly forbidden. Our finding could open an avenue for exploring topological photonics that enables nontrivial topological phenomena and designs in photonics driven by nonparaxiality.

Many-electron Systems and the Pauli Principle

It is shown how the quantum Hamiltonian for a general molecule is set up, using the ‘quantum recipe’ of chapter 3. In the most restrictive Born Oppenheimer approximation, the nuclei are held fixed and the Schrodinger equation (SE) is set up for the electrons only. The wavefunction depends on the positions and spin projections of all electrons. The electron spin projection is introduced heuristically as another two-valued electron degree of freedom. The electronic SE cannot be solved exactly, and (spin-) orbitals are introduced to construct an approximate wavefunction. The Pauli principle demands that a many-electron wavefunction is antisymmetric upon the exchange of electron labels, which leads to the construction of the approximate orbital-model wavefunction as a Slater determinant rather than a simple Hartree product. The orbital model wavefunction does not describe the Coulomb electron correlation, but it incorporates the (Fermi) correlation leading to the Pauli exclusion.

Embedding-theory-based simulations using experimental electron densities for the environment

The basic idea of frozen-density embedding theory (FDET) is the constrained minimization of the Hohenberg–Kohn density functional E HK[ρ] performed using the auxiliary functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B], where Ψ A is the embedded N A -electron wavefunction and ρ B (r) is a non-negative function in real space integrating to a given number of electrons N B . This choice of independent variables in the total energy functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B] makes it possible to treat the corresponding two components of the total density using different methods in multi-level simulations. The application of FDET using ρ B (r) reconstructed from X-ray diffraction data for a molecular crystal is demonstrated for the first time. For eight hydrogen-bonded clusters involving a chromophore (represented as Ψ A ) and the glycylglycine molecule [represented as ρ B (r)], FDET is used to derive excitation energies. It is shown that experimental densities are suitable for use as ρ B (r) in FDET-based simulations.

Electronic Wavefunction Tiles

We review the pre-quantum theories of electronic structure of Lewis and Langmuir, and how this relates to the post-quantum double-quartet theory of Linnett. Linnett’s ideas are put on a firm theoretical footing through the emergence of the wavefunction tile: The 3N-dimensional repeating structure of the N-electron wavefunction. Wavefunction tiles calculated by the dynamic Voronoi Metropolis sampling method are reviewed, and new results are presented for bent bonds of cyclopropane, and electron correlation in Be-O-Be.

Electronic Transitions of Molecules: Vibrating Lewis Structures

In this work we demonstrate a simple and intuitive description of electronic resonances in terms of localized electron vibrations. By partitioning the 3N-dimensional space of a many-electron wavefunction into hyper-regions related by permutation symmetry, chemical structures naturally result which correspond closely to Lewis structures, with identifiable single and double bonds, and lone pairs. Here we demonstrate how this picture of electronic structure develops upon the admixture of electronic wavefunctions, in the spirit of coherent electronic transitions. We show that pi-pi* transitions correspond to double-bonding electrons oscillating along the bond axis, and n-pi* transitions reveal lone-pairs vibrating out of plane. In butadiene and hexatriene, the double-bond oscillations combine with in- and out-of-phase combinations, revealing the correspondence between electronic transitions, molecular normal mode vibrations, and molecular plasmonics. This analysis allows electronic excitations to be described by building upon ground state electronic structures, without the need for molecular orbitals.

Electronic Transitions of Molecules: Vibrating Lewis Structures

In this work we demonstrate a simple and intuitive description of electronic resonances in terms of localized electron vibrations. By partitioning the 3N-dimensional space of a many-electron wavefunction into hyper-regions related by permutation symmetry, chemical structures naturally result which correspond closely to Lewis structures, with identifiable single and double bonds, and lone pairs. Here we demonstrate how this picture of electronic structure develops upon the admixture of electronic wavefunctions, in the spirit of coherent electronic transitions. We show that pi-pi* transitions correspond to double-bonding electrons oscillating along the bond axis, and n-pi* transitions reveal lone-pairs vibrating out of plane. In butadiene and hexatriene, the double-bond oscillations combine with in- and out-of-phase combinations, revealing the correspondence between electronic transitions, molecular normal mode vibrations, and molecular plasmonics. This analysis allows electronic excitations to be described by building upon ground state electronic structures, without the need for molecular orbitals.