Huckel MO Research Articles

Hückel molecular orbital theory on a quantum computer: A scalable system-agnostic variational implementation with compact encoding.

Hückel molecular orbital (HMO) theory provides a semi-empirical treatment of the electronic structure in conjugated π-electronic systems. A scalable system-agnostic execution of HMO theory on a quantum computer is reported here based on a variational quantum deflation (VQD) algorithm for excited state quantum simulation. A compact encoding scheme is proposed here that provides an exponential advantage over the direct mapping and allows for quantum simulation of the HMO model for systems with up to 2n conjugated centers with n qubits. The transformation of the Hückel Hamiltonian to qubit space is achieved by two different strategies: an iterative refinement transformation and the Frobenius-inner-product-based transformation. These methods are tested on a series of linear, cyclic, and hetero-nuclear conjugated π-electronic systems. The molecular orbital energy levels and wavefunctions from the quantum simulation are in excellent agreement with the exact classical results. However, the higher excited states of large systems are found to suffer from error accumulation in the VQD simulation. This is mitigated by formulating a variant of VQD that exploits the symmetry of the Hamiltonian. This strategy has been successfully demonstrated for the quantum simulation of C60 fullerene containing 680 Pauli strings encoded on six qubits. The methods developed in this work are easily adaptable to similar problems of different complexity in other fields of research.

Graph derivative indices interpretation from the quantum mechanics perspective

This contribution examines the interpretation of the Graph Derivative Indices (GDIs) from the Quantum Mechanics (QM) perspective and its relation with the Hückel molecular orbital (HMO) method. The different elements used for calculating Graph Discrete Derivatives over atom-pairs are related to the QM integrals appearing in the Hückel Determinant. The relation between the Coulomb and Resonance Integrals was estimated by the topological way and quantitative values for resonance energies were calculated. Some GDI calculations were performed to the topological interpretation of the aromaticity, expressing the frequencies as probabilities. Starting from topological interpretation for aromaticity, GDI calculations were performed. There is the possibility of expressing the frequencies as probabilities. It allows explaining the atypical formation of cyclobutadiene from the entropic point of view. Considering this hypothesis, experimental resonance energies for 14 molecules were correlated with GDI-topological resonance energies by the same structures. Taking into consideration the regularity and coherence stablished in experiments performed with the GDIs, it is possible to assure that GDIs have interpretations in QM terms.

Digraph Energy of Directed Polygons

Background: The energy E(G) of G is defined as the sum the absolute values of the eigenvalues of its adjacency matrix. In theoretical chemistry, within the Hu ̈ckel molecular orbital (HMO) approximation, the energy levels of the π-electrons in molecules of conjugated hydrocarbons are related to the energy of the molecular graphs. Objective: Generally, the energy to digraphs was proposed. Methodology: Let Δ_n be the set consisting of digraphs with n vertices and each cycle having length≡2 mod(4). The set of all the n-order directed hollow k-polygons in Δ_n based on a k-polygon G is denoted by H_k (G). Results: In this research, by using the quasi-order relation over Δ_n and the characteristic polynomials of digraphs, we describe the directed hollow k-polygon with the maximum digraph energy in H_k (G). Conclusion: The n-order oriented hollow k-polygon with the maximum digraph energy among Hk(G) only contains a cycle. Moreover, such a cycle is the longest one produced in G.

Theoretical Study on the Conduction Properties of π-Stacked Single-Molecule Junctions with Stacked Aromaticity

π-stacked single-molecule junctions stacked with π-conjugated molecules have the potential to be used as building blocks for single-molecule scale three-dimensional integrated circuits. In this study, we investigate the relationship between π-stacking distance and conductance in face-to-face π-stacked single-molecule junctions with benzene as the monomer unit using the non-equilibrium Green's function (NEGF), which combines the Hückel molecular orbital (HMO) and density functional theory (DFT) methods. As the π-stack distance between two benzene molecules decreases, the pseudo-para junction, which is insulating, turns conductive. Furthermore, it was found that the cause of this change can be explained by orbital interactions.

Color Energy of Some Cluster Graphs

Let $G$ be a simple connected graph. The energy of a graph $G$ is defined as sum of the absolute eigenvalues of an adjacency matrix of the graph $G$. It represents a proper generalization of a formula valid for the total $\pi$-electron energy of a conjugated hydrocarbon as calculated by the Huckel molecular orbital (HMO) method in quantum chemistry. A coloring of a graph $G$ is a coloring of its vertices such that no two adjacent vertices share the same color. The minimum number of colors needed for the coloring of a graph $G$ is called the chromatic number of $G$ and is denoted by $\chi(G)$. The color energy of a graph $G$ is defined as the sum of absolute values of the color eigenvalues of $G$. The graphs with large number of edges are referred as cluster graphs. Cluster graphs are graphs obtained from complete graphs by deleting few edges according to some criteria. It can be obtained on deleting some edges incident on a vertex, deletion of independent edges/triangles/cliques/path P3 etc. Bipartite cluster graphs are obtained by deleting few edges from complete bipartite graphs according to some rule. In this paper, the color energy of cluster graphs and bipartite cluster graphs are studied.

Augmented Hückel molecular orbital model of π‐electron systems: from topology to metric. I. General theory

AbstractThe π‐electron molecule is planar (but the strict planarity requirement may be relaxed), with a set of electrons residing on the (valence) molecular orbitals of the π‐symmetry (πMOs). The quantum state of these electrons influences the geometry of the molecule, most notably the bond lengths. The Hückel molecular orbital (HMO) model provides the simplest (yet quite effective) description of the πMOs. Longuet‐Higgins and Salem, and their followers, made efforts to augment this model by making its empirical parameters dependent on the bond lengths; this approach was later often referred to as the SCβ HMO model. In the present paper, we introduce the AugHMO model that is a generalization of these earlier efforts. We show that the mathematical formulation of this model necessarily implies the principle of a bond‐order–bond‐length (BO‐BL) relationship that is universal for a given type of bond (C–C, C–N, etc.) in all π‐electron molecules. A plausible assumption that the BO‐BL relationship is a linear one leads to a family of workable computational methods, such as the Hückel–Su–Schrieffer–Heeger method and the Hückel–Longuet–Higgins–Salem method. Our AugHMO model is developed with the following goals in mind: (i) to offer a predictive tool for calculating the bond lengths in π‐electron molecules, capable of dealing with very large systems; (ii) to provide insights into the problem of coupling between the π‐electrons and the bond lengths; and (iii) to offer a simple educational model illustrating the principles of geometry optimization in quantum chemistry (involving the gradient and Hessian of the molecular energy).

Graphical Representation of Hückel Molecular Orbitals

In this paper, we develop a general but very simple mathematical foundation for the predefined coefficient graphical method of Huckel molecular orbital theory (HMO). We first present the general solution for the recurrence relation of the coefficients of Huckel molecular orbitals (MOs). Subsequently, for all the three unbranched hydrocarbons, i.e., open-chain, cyclic Huckel and Mobius polyenes, different boundary conditions are explored for obtaining the MOs and their energy levels. The analytic continuation of the recurrence relation, in which one extends the domain from integral to real, allows us to analyze the symmetric properties of Huckel MOs in an elegant fashion without even knowing the actual expressions. In fact, we can use the symmetric properties to derive the Huckel MOs of the unbranched hydrocarbons and some branched hydrocarbons such as naphthalene. Consequently, this work also provides a pedagogical alternative to present the HMO model for students in an advanced physical chemistry course. Finally, the graphical approach could be a good mnemonic device for students’ comprehension of the HMO theory.

Influence of long-range interactions on quantum interference in molecular conduction. A tight-binding (Hückel) approach.

When certain pairs of atoms in a π-conjugated molecule are connected with nanometer-scale source and drain electrodes, the remarkable quantum interference (QI) effect may arise. In this case, the electron transmission probability is significantly suppressed due to the QI effect. Tight-binding approaches, such as the Hückel molecular orbital (HMO) model, have revealed important features of this quantum phenomenon. However, important deviations from experiments and from more sophisticated calculations are known for a variety of cases. Here, we propose an extension of the HMO method to include non-nearest-neighbor interactions. Such long-range interactions (LRIs) are implemented in the HMO model in the form of a damping function that decays as the topological distance-the number of bonds separating two atoms-gets larger. The proposed model is further developed so that a geometric modification, i.e., the rotation around a single bond, can be taken into account. Our results show that LRI affects both the location of the antiresonance peak due to QI and the intensity of QI, even suppressing it in some cases. These results agree well with what was observed in a Density Functional based Tight-Binding (DFTB) study reported in the literature. These properties can be interpreted on the basis of a graph-theoretic path-counting model as well as the molecular orbital theory. In addition, the geometric LRI model is shown to reproduce the change of transmission as a function of rotation around the single bond separating two benzene rings in biphenyl, in agreement with what was observed in both experiment and DFTB calculation.

Practical Training in Simple Hückel Theory: Matrix Diagonalization for Highly Symmetric Molecules and Visualization of Molecular Orbitals

At an advanced stage of learning quantum chemistry, undergraduate students usually encounter simple Huckel-molecular-orbital (HMO) theory, whose primitive approach gives very useful insight into the electronic structure of π-conjugated molecules. However, on one hand, computational HMO software, when programmed without using molecular symmetry, does not necessarily output chemically reasonable shapes of degenerate molecular-orbitals in benzene. On the other hand, separately inputting the molecular symmetry into the HMO calculation is mathematically redundant (i.e., duplicate information input). To solve this problem, in this study, a Microsoft Excel macro for obtaining reasonable HMOs of highly symmetric molecules such as benzene was developed. Furthermore, two other Excel macros for readily drawing quantitative contour plots of HMOs were created as student-friendly tools. The practical training in simple HMO theory proposed in the present report using these three macros will be very useful for undergradu...

Extended Hückel Calculations on Solids Using the Avogadro Molecular Editor and Visualizer

The “Yet Another extended Huckel Molecular Orbital Package” (YAeHMOP) has been merged with theAvogadro open-source molecular editor and visualizer. It is now possible to perform YAeHMOP calculations directly from the Avogadro graphical user interface for materials that are periodic in one, two, or three dimensions, and to visualize band structures, total and projected density of states, and crystal orbital overlap/Hamilton populations (COOPs/COHPs). Calculations on graphite, silicon, sodium, and a one-dimensional hydrogen chain are provided to illustrate the functionality. Similar exercises have been carried out in an upper-level undergraduate quantum theory course.

Genealogy of Conjugated Acyclic Polyenes.

Based on the total π-electron energies Eπs of Hückel Molecular Orbital (HMO) method for all the possible isomers of conjugated acyclic polyenes (C2nH2n+2) up to n = 7, the structure–stability relation of the possible isomers was analyzed. It was shown that the mean length of conjugation L can roughly predict the ordering of stability among isomers, while the Z-index, or Hosoya-index, can almost perfectly reproduce their stability. Further, the genealogy of the conjugated acyclic polyene family was obtained by drawing systematic diagrams connecting these isomers of different n, and governed by several simple rules. Namely, the stability change of a given isomer in the genealogy connecting n and n + 1 polyenes can be classified into three different modes of vinyl addition (elongation, inner and outer branching) and horn growing, i.e., substitution of –HC=CH– moiety with –HC(=CH2)–C(=CH2)H–. By using the Z-index, we can extend this type of discussion to polyene radicals and even to “cross-conjugated” cyclic polyenes containing only one odd-membered cycle, such as radialene and fulvene.

Energy gaps of graphene clusters: the first-principles calculations based on high-throughput screening

With the enumeration of the triangular lattice fragments, we have systematically investigated the graphene clusters (CnHm, n = 14 – 24) with various sizes and shapes, whose structural stabilities and electronic properties are studied by the Hückel molecular orbital (HMO) method and the first-principles calculation. According to the formation energies, we show the structural stabilities of the clusters are closely related to the shape and size, as well as the chemical potential of hydrogen. The energy gaps obtained from the HMO method are in the same trend with the ones calculated by the first-principles calculations, indicating the effective screening of the gap minimum and maximum in a fast speed. There is a general decreasing of the energy gaps with the size increment due to the quantum confinement, meanwhile, the gaps are also highly dependent on the shape of the clusters for those with the same number of carbon atom.

Hückel Theory + Reorganization Energy = Marcus–Hush Theory: Breakdown of the 1/n Trend in π-Conjugated Poly-p-phenylene Cation Radicals Is Explained

Among the π-conjugated poly-p-phenylene wires, fluorene-based poly-p-phenylene (FPPn) wires have been extensively explored for their potential as charge-transfer materials in functional photovoltaic devices. Herein, we undertake a systematic study of the redox and optical properties of a set of FPPn (n = 2–16) wires. We find that, while their absorption maxima (νabs) follow a linear trend against cos[π/(n + 1)] up to the polymeric limit, redox potentials (Eox) show an abrupt breakdown from linearity beginning at n ∼ 8. These observations prompted the development of a generalized model to describe the unusual evolution of redox and optical properties of poly-p-phenylene wires. We show that the cos[π/(n + 1)], commonly expressed as 1/n, dependence of the properties of various π-conjugated wires has its origin in Huckel molecular orbital (HMO) theory, which however, fails to predict the evolution of the redox potentials of these wires, as the oxidation-induced structural/solvent reorganization is unaccounted...

Systematic Construction and Calculation of Electronic Properties of Fullerene Series Related by Rotational Symmetry: From Fullerenes to Bicapped Nanotubes.

The results herein demonstrate that the methods of circumscribing and the facile calculation of Hückel molecular orbital (HMO) eigenvalues by mirror-plane fragmentation have a broad application in the construction of carbon cluster series and the systematic study of trends in their electronic properties. In comparing open-ended nanotubes and their isomeric elongated fullerenes (bicapped nanotubes), we show that the former are more aromatic but the latter are more conjugated and that progressive elongation increases aromaticity and conjugation in both. Recursion equations that will allow one to obtain the eigenvalues to all 5-endcapped nanotubes are given.

THE EIGEN-COMPLETE DIFFERENCE RATIO OF CLASSES OF GRAPHS- DOMINATION, ASYMPTOTES AND AREA

The energy of a graph is related to the sum of -electron energy in a molecule represented by a molecular graph, and originated by the HMO (Huckel molecular orbital) theory. Advances to this theory have taken place which includes the difference of the energy of graphs and the energy formation difference between a graph and its decomposable parts. Although the complete graph does not have the highest energy of all graphs, it is significant in terms of its easily accessible graph theoretical properties, and has a high level of connectivity and robustness, for example. In this paper we introduce a ratio, the eigen-complete difference ratio, involving the difference in energy between the complete graph and any other connected graph G, which allows for the investigation of the effect of energy of G with respect to the complete graph when a large number of vertices are involved. This is referred to as the eigen-complete difference domination effect. This domination effect is greatest negatively (positively), for a strongly regular graph (star graphs with rays of length one), respectively, and zero for the lollipop graph. When this ratio is a function f(n), of the order of a graph, we attach the average degree of G to the Riemann integral to investigate the eigen-complete difference area aspect of classes of graphs. We applied these eigen-complete aspects to complements of classes of graphs.

Diels–Alder Reactions of Graphene: Computational Predictions of Products and Sites of Reaction

The cycloaddition reactions and noncovalent π interactions of 2,3-dimethoxybutadiene (DMBD), 9-methylanthracene (MeA), tetracyanoethylene (TCNE), and maleic anhydride (MA) with graphene models have been investigated using density functional theory (DFT) calculations. Reaction enthalpies have been obtained to assess the reactivity and selectivity of covalent and noncovalent functionalization. Results indicate that graphene edges may be functionalized by the four reagents through cycloaddition reactions, while the interior regions cannot react. Noncovalent complexation is much more favorable than cycloaddition reactions on interior bonds of graphene. The relative reactivities of different sites in graphene are related to loss of aromaticity and can be predicted using Hückel molecular orbital (HMO) localization energy calculations.

On the determinant of bipartite graphs

The nullity of a graph G, denoted by η(G), is the multiplicity of 0 in the spectrum of G. Nullity of a (molecular) graph (e.g., a bipartite graph corresponding to an alternant hydrocarbon) has important applications in quantum chemistry and Hückel molecular orbital (HMO) theory. A famous problem, posed by Collatz and Sinogowitz in 1957, asks to characterize all graphs with positive nullity. Clearly, detA(G)=0 if and only if η(G)>0. So, examining the determinant of a graph is a way to attack this problem. For a graph G, we define the matching core of G to be the graph obtained from G by successively deleting each pendant vertex along with its neighbour. In this paper, we show that the determinant of a graph G with all cycle lengths divisible by four (e.g., the 1-subdivision of a bipartite graph), is 0 or (−1)|V(G)|/2. Furthermore, the determinant is 0 if and only if the matching core of G is nonempty.

English

Extended Huckel Molecular Orbital (EHMO) calculations were carried out on octahedral transition metal clusters with general formula (M6X8 iY6 a) (M = Molybdenum; X and Y = π-Donor Ligands) in order to rationalize their electronic structure. In these species the opti- mal metallic electron (ME) count is of 24 but for many clusters experimentally synthesized, the ME count can vary from 20 to 48 without dramatically altering the architecture of the octahedral cluster. Herein are reported the geometrical parameters and electronic properties, of a series of clusters with 20 to 24 valence electrons per cluster. The calculated characteristics for all the considered structures are in excel- lent agreement with the experimental ones and requires to learn more about the relationships that exist between their structural arrangement and electronic properties especially the number of electrons available for metal-metal bonding in M6 octahedral clusters. Keywords- Cluster Compounds, Molybdenum, Electron counting, Extended Huckel Molecular Orbital (EHMO) calculations, Electronic Struc- ture, Frontier orbitals.

Nickel poly-acetylide carbonyl clusters: structural features, bonding and electrochemical behaviour

The reactions of [NEt(4)](2)[Ni(6)(CO)(12)] with miscellaneous carbon halides (e.g. CCl(4), C(4)Cl(6)) in CH(2)Cl(2) have been extensively investigated particularly focusing on the stoichiometric ratio of the reagents and reaction temperature. This allowed the preparation of the previously known acetylide clusters [Ni(16)(C(2))(2)(CO)(23)](4-), [HNi(25)(C(2))(4)(CO)(32)](3-) and [Ni(22)(C(2))(4)(CO)(28)Cl](3-), as well as isolation and full characterisation of the closely related [Ni(17)(C(2))(2)(CO)(24)](4-) and [Ni(25)(C(2))(4)(CO)(32)](4-) tetraanions. From a structural point of view, all these clusters are based on a Ni(16) square orthobicupola which contain interstitial C(2), Ni(η(2)-C(2))(4) or Ni(2)(μ-η(2)-C(2))(4) moieties, displaying rather short C-C bonds. Electrochemical studies reveal that all these species undergo different redox processes, even if their stability is rather limited. This is corroborated by an extensive analysis of the interaction between interstitial C(2) acetylide units and the metal cluster cage by Extended Huckel Molecular Orbital (EHMO) calculations, which indicates that tightly bonded C-C units are less effective than isolated C-atoms in stabilising the cluster cage.

A DFT Study of the Reactivity Indexes of Ionic [4 + 2+] Diels-Alder Cycloaddition to Nitrilium and Immonium Ions

The global electrophilicity index, defined within the conceptual density functional theory (DFT), was used to classify the dienes and dienophiles currently used in Diels-Alder reactions on a unique scale of electrophilicity. The index, obtained within the Kohn-Sham scheme, is based on the HOMO and LUMO energies. A systematic study of the global reactivity indexes of the reagents involved in formal [4 + 2+] Diels-Alder cycloaddition reactions of nitrilium and immonium ions with isoprene is presented. Keywords: Diels-Alder, reactivity indexes, nitrilium, immonium, electrophilicity, nucleophilicity, Huckel molecular orbital (HMO) theory, electrofugality and nucleofugality, Berny analytical gradient, isoprene