Symmetry-adapted Linear Combinations Research Articles

Semiconductor Band Structure, Symmetry, and Molecular Interface Hybridization for the Chemist.

Merging molecular bonding concepts with semiconductor- and materials-based concepts of band structure is challenging due to the mutually exclusive historical development and notations used in those respective fields: symmetry adapted linear combinations (SALCs) and Mulliken terms for molecules, versus k space and Bloch sums for materials. This lack of commonality brings the issue of hybridization (aka electronic coupling) between molecules and materials to the forefront in many aspects of modern chemical research─including nanocrystal properties, solar energy conversion, and molecular computing. It is thus critical to establish a holistic approach to hybridizing orbital (molecule) and plane-wave (semiconductor/material) systems to better describe symmetry-based molecule|material bonding and the corresponding symmetry-adapted molecular orbital (MO) diagrams. Such a unified approach would enable the construction of testable hypotheses about the role of symmetry and electronic structure in determining the extent of electronic coupling between molecular orbitals and semiconductor band structure. This Perspective provides an analysis and compendium of "translations" between the physics and chemistry language of group theory. In this vein, this approach describes the symmetries─and corresponding point groups─that occur in k space along the available descent in symmetry pathways (k space vectors). As a result, chemists may arrive at a more intuitive understanding of the band symmetries of semiconductors, as well as insights into the corresponding algebraic formulations. This analysis can ultimately generate MO diagrams for hybrid molecule|material systems. Lastly, an Outlook provides some context to the application of this analysis to modern problems at the interface of molecular and materials chemistry.

Dressed jeff-1/2 objects in mixed-valence lacunar spinel molybdates

The lacunar-spinel chalcogenides exhibit magnetic centers in the form of transition-metal tetrahedra. On the basis of density-functional computations, the electronic ground state of an Mo413+ tetrahedron has been postulated as single-configuration a12 e4 t25, where a1, e, and t2 are symmetry-adapted linear combinations of single-site Mo t2g atomic orbitals. Here we unveil the many-body tetramer wave-function: we show that sizable correlations yield a weight of only 62% for the a12 e4 t25 configuration. While spin–orbit coupling within the peculiar valence orbital manifold is still effective, the expectation value of the spin–orbit operator and the g factors deviate from figures describing nominal t5jeff = 1/2 moments. As such, our data documents the dressing of a spin–orbit jeff = 1/2 object with intra-tetramer excitations. Our results on the internal degrees of freedom of these magnetic moments provide a solid theoretical starting point in addressing the intriguing phase transitions observed at low temperatures in these materials.

Analysis of Symmetric Electromagnetic Components Using Magnetic Group Theory

We discuss a method of analysis of symmetric electromagnetic components with magnetic media based on magnetic group theory. In this description, some of the irreducible corepresentations assume complex values exp(iθ) with the real parameter θ. A possible physical interpretation of this parameter is given. We demonstrate the application of the symmetry-adapted linear combination method combined with the corepresentation theory to the problem of current modes in an array of magnetized graphene elements where Faraday and Kerr effects can exist. The elements are described by the magnetic symmetry C4 or C4v(C4). The scattering matrix of the array and its eigensolutions are defined and analyzed and some numerical simulations are presented as well. An example of a waveguide described by symmetry C4v(C2v) with a specific type of degeneracy is also discussed.

Symmetry analysis of trimer-based all-dielectric metasurfaces with toroidal dipole modes

Herein, we discuss the conditions for excitation of symmetry-protected toroidal dipole modes in all-dielectric metasurfaces composed of trimer or twin-trimer clusters of dielectric disks. Such metasurfaces permit enhanced light–matter interaction due to spatially confined light in resonant systems with a high-quality factor. To describe characteristics of toroidal modes existing in the clusters, we use the magnetic dipole moments approximation, group-theoretical methods, group representation theory, symmetry-adapted linear combination method, and circuit theory. To validate the obtained theoretical results, we fulfill both full-wave numerical simulations and microwave experiments. In particular, we have shown that the toroidal dipole mode appears as a quasi-dark state of the trimer. It can be excited in the metasurface by the field of a linearly polarized wave, providing the symmetry of the trimer is properly reduced. In the metasurface, the properties of the toroidal dipole mode are determined primarily by the parameters of a single trimer and are not a consequence of the periodicity of the array. The coupling of the toroidal dipole modes in the twin-trimers can appear in both bonding and anti-bonding fashion resulting in different orders of the net toroidal dipole moment. Due to the unique field configuration of these modes, the proposed metasurfaces can be considered as a platform for efficient light–matter interaction for enhanced absorption, non-linear switching, and sensing.

Characterization of bonding modes in metal complexes through electron density cross-sections.

Qualitative inspection of molecular orbitals (MOs) remains one of the most popular analysis tools used to describe the electronic structure and bonding properties of transition metal complexes. In symmetric coordination complexes, the use of group theory and the symmetry-adapted linear combination (SALC) of fragment orbitals allows for a very accurate and informative interpretation of MOs, but the same procedure cannot be performed for asymmetric complexes, such as Schrock and Fischer carbenes. In this work, we present a straight-forward approach for classifying and quantifying MO contributions to a particular metal-ligand interaction. Our approach utilizes the topology of MO density contributions to a cross-section of an inter-nuclear region, and is computationally inexpensive and applicable to symmetric and asymmetric complexes alike. We also apply the same approach with similar decompositions using Natural Bond Orbitals (NBO) and the recently developed Fragment, Atomic, Localized, Delocalized and Interatomic (FALDI) density decomposition scheme. In particular, FALDI analysis provides additional insights regarding the multi-centric nature of metal-carbene bonds without resorting to expensive multi-reference calculations.

Automatic procedure for generating symmetry adapted wavefunctions

Automatic detection of point groups as well as symmetrisation of molecular geometry and wavefunctions are useful tools in computational quantum chemistry. Algorithms for developing these tools as well as an implementation are presented. The symmetry detection algorithm is a clustering algorithm for symmetry invariant properties, combined with logical deduction of possible symmetry elements using the geometry of sets of symmetrically equivalent atoms. An algorithm for determining the symmetry adapted linear combinations (SALCs) of atomic orbitals is also presented. The SALCs are constructed with the use of projection operators for the irreducible representations, as well as subgroups for determining splitting fields for a canonical basis. The character tables for the point groups are auto generated, and the algorithm is described. Symmetrisation of molecules use a projection into the totally symmetric space, whereas for wavefunctions projection as well and partner function determination and averaging is used. The software has been released as a stand-alone, open source library under the MIT license and integrated into both computational and molecular modelling software.Graphical abstract.

Symmetry-adapted linear combinations for the eigenvalues and eigenvectors of reciprocal graphs

ABSTRACTThree classes of reciprocal graphs, viz. monocycle (GCn), linear chain (GLn) and star (GKn) with reciprocal pairs of eigenvalues (λ, 1/λ), are well known. Reciprocal graphs of monocycle (GCn) and linear chain (GLn) are obtained by putting a pendant vertex to each vertex of simple monocycle (Cn) and simple linear chain (Ln), respectively. A star graph of such kind is obtained by attaching a pendant vertex to the central vertex and to each of the (n − 1) peripheral vertices of the star graph (K1, (n−1)). An n-fold rotational axis of symmetry for GCn and (n − 1)-fold rotational axis of symmetry for GKn have been exploited for obtaining their respective condensed graphs. The condensed graph for GLn has been generated from that of GCn incorporating proper boundary conditions. Condensed graphs are lower dimensional graphs and are capable of keeping all eigeninformation in condensed form. Thus the eigensolutions (i.e. the eigenvalues and the eigenvectors) in analytical forms for such graphs are obtained by solving 2 × 2 or 4 × 4 determinants that in turn result in the charge densities and bond orders of the corresponding molecules in analytical forms. Some mathematical properties of the eigenvalues of such graphs have also been explored.

Theoretical model for optical properties of symmetric trimer nanoholes in a gold film

In this study, optical properties of a symmetric nanohole trimer milled in a gold film are studied using theoretical calculations and numerical simulations. The theoretical model is based on a closed-form relation of the electric and magnetic polarizabilities of a circular nanohole. By modeling each nanohole as a magnetic point-dipole and extending the symmetry-adapted linear combination to the in-plane nanoholes, the mutual effects of the nanoholes on the scattering spectrum are demonstrated. The numerical approach is based on the finite-difference time-domain method, and the numerical results support the theoretical approach. The results indicate that two distinct resonant peaks can be seen in the scattering spectrum of the nanohole trimer, which are due to antibonding and bonding plasmon resonant modes, respectively. The magnetic fields and the corresponding magnetic point-dipoles of the lower nanoholes are in-phase and out-of-phase of the perpendicular and parallel magnetic field components of the incident field, respectively. The upper nanohole at the antibonding resonant wavelength has a phase difference of approximately 30° with the lower nanoholes.

A theoretical study on the stability difference of the borane BnHn2− and carborane C2Bn−2Hn(5 ≤ n ≤ 7) clusters

In order to study the electronic structure and structural stability of borane and carborane C2Bn−2Hn (5 ≤ n ≤ 7) clusters, especially the stability difference between the borane and carborane C2B3H5. The frontier orbital energy levels of the borane and carborane C2Bn−2Hn (5 ≤ n ≤ 7) clusters are calculated at CCSD(T)/aug-cc-pVXZ//B3LYP/def2-TZVPP level. The results are further analyzed by qualitative frontier orbital method based on the cap–ring interaction. The results reveal that: (1) the larger Egap(HOMO-LUMO energy gap) of carborane C2Bn−2Hn (5 ≤ n ≤ 7) clusters than borane (5 ≤ n ≤ 7) clusters originates from the more effective cap–ring orbital overlap of carborane C2Bn−2Hn (5 ≤ n ≤ 7) clusters than that of borane (5 ≤ n ≤ 7) clusters; (2) the smallest Egap of the borane results from the highest energy level of the ring symmetry-adapted linear combination orbital of cluster; and (3) the largest Egap of the carborane C2B3H5 is induced by the most effective cap–ring orbital interaction of C2B3H5 cluster. © 2014 Wiley Periodicals, Inc.

Spectroscopic Detection of Medium Range Order in Hydrogenated Amorphous Silicon, a-Si(H): Applications in Photovolatics, Thin Film Transistors and Si-based Microelectronics

Abstract Hydrogenated amorphous silicon, a-Si1-xHx, with ~ at.10% or x ~0.1±0.02, is used in photovoltaics (PV), and thin film transistors (TFT's). Amorphous Si (a-Si) thin films, thought to be free of H, are used as precursors for polycrystalline gate electrodes in microelectronics. PV and TFT alloys has been deposited by the glow discharge method (GD), remote plasma-enhanced chemical vapor deposition (RPECVD, and reactive magnetron sputtering (RMS) with bonded-H determined by deposition precursors and substrate temperatures. Two conditions are required for low Si dangling bond densities ~0.5 to 1x1016 cm−3: (i) a bonded monohydride, Si-H, concentration of ~10 at. % H, and (ii) a deposition, and/or a post-deposition anneal at ~240 °C to 300 °C. These reduce strain by introducing medium range order (MRO) as nano-meter scale aperiodically-assembled clusters. Si L2,3 X-ray absorption spectroscopy (XAS) has been used to confirm MRO extending beyond short-range order (SRO) of continuous random networks (CRN) through observation of ligand-field split d-state features. These are associated with symmetry-adapted linear combinations (SALCs) of atomic states forming molecular orbital valence bands. XAS and photoelectron spectroscopy (PES) studies have confirmed that a-Si used for microelectronic applications also has MRO as well. This is associated with a H-transfer reaction from an Si-H bond of the cluster into a Si-H-Si bonding arrangement on near-neighbor sites. MRO has also identified by other spectroscopic techniques: spectroscopic ellipsomentry (SE), and ultra-violet or X-ray photoemission spectroscopy (UPS or XPS). The same H-atom motion induced by absorption of sun-light is responsible for the Staebler-Wronski Effect (SWE); the generation of Si-atom dangling bonds that degrade PV performance.

Construction of Ligand Group Orbitals for Polyatomics and Transition-Metal Complexes Using an Intuitive Symmetry-Based Approach

A molecular orbital (MO) diagram, especially its frontier orbitals, explains the bonding and reactivity for a chemical compound. It is therefore important for students to learn how to construct one. The traditional methods used to derive these diagrams rely on linear algebra techniques to combine ligand orbitals into symmetry-adapted linear combinations or ligand group orbitals (LGOs). Over the past 10 years, I have developed and refined a simple graphical method for generating ligand group orbitals. The method is an extension of concepts that the students already know from general chemistry: VSEPR and valence bond theory, which are then used to create a generator function. LGOs are prepared by matching symmetry (nodes and phase behavior) of the generator function. The LGOs generated by this technique are qualitatively correct and sufficient for the “back-of-the-envelope” MO diagrams for which they are intended. Through a series of in-class group work and out-of-class problem sets, students learn to derive LGOs for main-group, organometallic, and coordination complexes quickly and correctly.

Symmetry effects on the optical coupling between plasmonic nanoparticles with applications to hierarchical structures

We present theoretical studies on the collective optical response of symmetric configurations of metallic nanoparticles. We show that within the electrostatic approximation, the surface plasmon resonance of these symmetric multiparticle systems can be expressed as symmetry-adapted linear combinations of the plasmon modes of each particle of the ensemble, closely resembling the situation encountered in molecular systems. By making use of group theoretical arguments, we show that such linear combinations can be written down by simple geometrical considerations through the use of point group character tables, without using extensive numerical computations. Furthermore, we apply this formalism to study the coupling of hierarchical arrays containing a large number of nanoparticles. This theory thus provides an intuitive and formal approach for the rational design of plasmonic nanostructures.

Theoretical Investigations of Mode Mixing in Vibrationally Excited States of CH5+

Using diffusion Monte Carlo, five vibrational excited states of CH(5)(+) and CD(5)(+) are evaluated and analyzed. Here, we focus on the fundamentals in the five modes that are generated by requiring that the wave functions change sign at specified values of the five symmetry-adapted linear combinations (SALCs) of the CH or CD bond lengths. Even though the definitions of these modes are based on displacements of the CH or CD bond lengths, the frequencies are found to be low compared to previously calculated CH vibrational frequencies in this molecule. The totally symmetric mode, with A(1)(+) symmetry, has a calculated frequency of 2164 and 1551 cm(-1) for CH(5)(+) and CD(5)(+). The frequencies of the four fundamentals that arise from excitation of the four SALCs that transform under G(1)(+) symmetry have frequencies that range from 1039 to 1383 cm(-1) and from 628 to 893 cm(-1) in CH(5)(+) and CD(5)(+), respectively. The origins of the broken degeneracy are investigated and explained to reflect extensive coupling to the two low-frequency modes that lead to isomerization of CH(5)(+).

Length scale discontinuities between non-crystalline and nano-crystalline thin films: Chemical bonding self-organization, broken constraints and reductions of macroscopic strain

This paper identifies different length scales, λs, for strain-reducing chemical bonding self-organizations in non-crystalline and nano-crystalline thin films. Length scales have been identified through spectroscopic studies, thermal heat flow measurements, and are analyzed by semi-empirical bond-constraint theory (SE-BCT) and symmetry adapted linear combinations (SALC) of atomic states. In both instances, strain-reducing self-organizations result in reduced defect densities that are minimized and enabling for device applications. The length scale for non-crystalline solids extends to at most 1nm, and more generally to 0.5–0.8nm; however, there are two different length scales for nano-crystalline films: one is <2.5nm and is characterized by suppression of longer range ordering required for complex unit cells based on more than one primitive unit cell and the second is >3–3.5nm and defines a regime where complex unit cells, comprised of two or more primitive unit cells are stabilized and the electronic structure is changed.

Bond Orders between Molecular Fragments

An extension of the Mayer bond order for the interaction between molecular fragments is presented. This approach allows the classical chemical concepts of bond order and valence to be utilised for fragments and the interactions between the fragments and symmetry-adapted linear combinations to be analysed. For high-symmetry systems, the approach allows the contribution from each irreducible representation to be assessed and provides a semiquantitative measure of the role of each bonding mode to interfragment interactions. The utility of this tool has been examined by a study of the bonding in symmetrical sandwich complexes. The validity of the frontier-orbital approach and the contributions from each frontier-orbital interaction can also be assessed within this model. As demonstrated by a study of a number of mixed-sandwich complexes, the model proves to be especially useful for low-symmetry systems in which separation of the sigma, pi and delta roles in bonding of the ligand is difficult to assess. The fragment bond order describes the interaction between preoptimized fragment orbitals and is independent of the charges that are placed on these fragments. Although the method allows the chemist to define fragments in any way they choose, most insight is gained by using the same frontier orbitals employed so successfully in perturbational molecular-orbital approaches. The results are free from the influence of the electron-counting method used to describe fragments, such as the rings and metals in sandwich complexes.

Moving Beyond the Single Center-Ways To Reinforce Molecular Orbital Theory in an Inorganic Course

Rather than ending the discussion of molecular orbital (MO) theory in an inorganic chemistry course with molecules such as octahedral ML6 or square planar ML4, we suggest moving beyond the single-atom center to include the MO diagram of ethene (C2H4). Embracing the use of group theory and symmetry adapted linear combinations (SALCs) at the level already taught, we extend the coverage to include this simple organic dimer as a means of teaching skills that enable students to tackle significantly more complicated molecules. Ethene provides a bridge to the fragment approach pioneered by Roald Hoffmann and used in some high-level treatises. Larger molecules are split into component fragments poised to form bonds. The MO diagram is constructed for each fragment and then combined to obtain the MO diagram of the full molecule. In this way, more sophisticated examples can be addressed if the process is reduced to smaller, more chemically intuitive steps. To bring the fragment approach to the undergraduate curriculum, we present the MO diagram of C2H4 first using SALCs of atomic orbitals and then contrasting this with a two-step fragment approach that first creates the MO diagram of the CH2 diradical.

Orbital symmetry as a tool for understanding the bonding in Krossing's cation.

The geometric and electronic structure of Krossing's cation, Ag(eta(2)-P(4))(2)(+), which shows an unexpected planar coordination environment at the metal center and D(2)(h) symmetry both in solution and in the solid state, have been investigated using density functional theory and orbital-symmetry-based energy decomposition. The analysis reveals that the contribution from electrostatic interactions to the bond energy is greater than that of orbital interactions. Partitioning of the latter term into the irreducible representations shows that, in addition to the 5s orbital, 5p orbitals of silver act as acceptor orbitals for electron donation from sigma(P-P) orbitals (a(1)(g), b(1)(u)) and n(P) orbitals (b(3)(u)). Back-donation from the 4d(10) closed shell of Ag into sigma orbitals of the pnictogen cages (b(2)(g)) is also important. However, this contribution is shown not to determine the D(2)(h) structure, contradicting conclusions from the pioneering study of the title cation (J. Am. Chem.Soc. 2001, 123, 4603). The contributions from the irreducible representations to the stabilizing orbital interactions in the D(2)(h) structure and in its D(2)(d)-symmetric conformer are analogous, indicating that the planar coordination environment at the metal center in Ag(eta(2)-P(4))(2)(+) is induced by intermolecular rather than by intramolecular interactions. Because ethylene coordination to a metal ion is an elementary reaction step in industrial processes, the bonding in Ag(C(2)H(4))(2)(+) has been analyzed as well and compared to that in Krossing's cation. Surprisingly, similar contributions to the bond energies and an involvement of metal 4d and 5p orbitals have been found, whereas a recent atoms in molecules analysis suggested that the metal-ligand interactions in silver(I) olefin complexes fundamentally differ from those in tetrahedro P(4) complexes. The only qualitative difference between the bonding patterns in Ag(eta(2)-P(4))(2)(+) and Ag(C(2)H(4))(2)(+) is the negligible energy contribution from the b(3)(u) irreducible representation in the ethylene complex because a respective symmetry-adapted linear combination of ligand orbitals is not available.

Theoretical Studies of Diiron(II) Complexes that Model Features of the Dioxygen-Activating Centers in Non-Heme Diiron Enzymes

We have applied high-level Density Functional Theory to investigate the properties of recently characterized carboxylate-bridged diiron(II) complexes supported by 2,6-di(p-tolyl)benzoate (ArTolCO2-) ligands. These compounds, prepared as synthetic models for the reduced non-heme diiron centers in the enzymes MMO, RNR-R2, and Δ9D, reproduce the composition of the first coordination sphere ligands as well as the core geometry. The experimentally observed flexibility of the diiron cores in the model compounds, a main design target, was confirmed computationally. Details of a possible interconversion mechanism that transforms quadruply and doubly carboxylate-bridged isomers of [Fe2(ArTolCO2)4L2], L = pyridine or related ligand, were examined. The orientation of the pyridine ligands plays a major role and promotes an initial carboxylate shift of the bridging carboxylate ligand that is orthogonal to the pyridine ring plane. Alternative mechanisms were explored and evaluated. Structural features of the strongly coupled diiron centers could only be reproduced reliably by using the experimentally determined antiferromagnetic spin-coupling properties of the high-spin d6 iron(II) centers. Use of the ferromagnetic-coupling scheme gave rise to a poor correlation of the computed structure with the experiment. The broken-symmetry orbitals required to describe the antiferromagnetic coupling are compared to the MOs as classical symmetry-adapted linear combinations of atomic orbitals that form the basis for the magnetic coupling scheme. The molecular orbitals responsible for the dependence of the structural results on spin coupling were identified and used to evolve an intuitive explanation for the structural differences observed.

Factorization of the Hückel Hamiltonian matrix for C 540

Symmetry-adapted linear combinations (SALCs), which are constructed via the character table of the I h group, are employed as the starting states for a recursion treatment for C 540 to attain its group-theoretical factorization within the tight-binding (Hückel) framework. Factorized characteristic polynomials and related eigenvalues referring to each irreducible representation (IR) are presented.

Symmetry adaptation of correlated states in the valence bond basis

Symmetry-adapted linear combinations of valence-bond (VB) diagrams are constructed for arbitrary point groups and total spin S using diagrammatic VB methods. VB diagrams are related uniquely to invariant subspaces whose size reflects the number of group elements; their nonorthogonality leads to sparser matrices and is fully incorporated into a binary integer representation. Symmetry-adapated linear combinations of VB diagrams are constructed for the 1764 singlets of a half-filled cube of eight sites, the 2.8 million π-electron singlets of anthracene, and for illustrative S≥0 systems.