On the calculation of the spectrum of large Hückel matrices, representing carbon nanotubes, using fast Hadamard and symplectic transforms

Cóllado, J. R. Alvarez

doi:10.1080/00268970601005235

Cited by 7 publications

(4 citation statements)

References 27 publications

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“…The Hückel spectra of two non-helical (zigzag) nanotubes, having 2 ny cycles of length 2 nx atoms, were presented in the previous paper. 15 These results have been ratified by the present code to be much more reliable, since the correct behavior of the several kinds of eigenvector have been checked by suitable subroutines. Two recent works, 33,34 which have been found by using the Google searching engine, 35 confirm the fine structure of the density of states, for infinite (cyclic) nanotubes.…”

Section: Analysis Of Resultssupporting

confidence: 54%

“…10 Some of these papers describe, on a theoretical basis, how electronic structure depends on the length of the tube itself. [11][12][13][14] The present study is a continuation of a previous paper, 15 where large finite length (N ≈ 16.000) nanotubes were studied. The spectra of their Hückel matrices was obtained by applying SVD Hadamard-Symplectic transforms on them.…”

Section: Introductionmentioning

confidence: 78%

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On the Diagonalization of Hermitian Matrices, and Its Use to Calculate the Hückel Electronic Structure of Large Carbon Nanotubes

Cóllado

2007

J. Theor. Comput. Chem.

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In this paper, a method for calculating all the Hückel molecular orbitals (MO) of large (16.000 atoms) carbon zigzag nanotubes is presented. These MO have been obtained by combining the singular value decomposition (SVD), the Sylvester-Hadamard transform, and the theory of Hamiltonian-Symplectic matrices. Numerical diagonalization of hermitian matrices is reviewed and improved. A new, more advantageous, (tri-diagonal) algorithm is proposed and analyzed. The reactivity of the atoms is described by calculating their free valence indices.

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Section: Analysis Of Resultssupporting

confidence: 54%

Section: Introductionmentioning

confidence: 78%

On the Diagonalization of Hermitian Matrices, and Its Use to Calculate the Hückel Electronic Structure of Large Carbon Nanotubes

Cóllado

2007

J. Theor. Comput. Chem.

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“…The PPP-weighted graphs are expected to grow rapidly in size as a function of tessellation parameters. Even for very large tessellations, it has been shown that Hadamard-transform-based orthogonal matrices provide all the eigenvalues and eigenvectors quite efficiently as shown in previous works. − Consequently, graph-theory-based PPP techniques offer a viable alternative for energy computations of large tessellations of kekulenes with ab initio -derived parameters compared to DFT or other techniques, as the graph-theory-based PPP method focuses on π electrons of these systems. Moreover, computations of TREs of larger tessellations appear to be intractable due to combinatorial explosions in the coefficients of their matching polynomials.…”

Section: Results and Discussionmentioning

confidence: 90%

Topological Characterization and Graph Entropies of Tessellations of Kekulene Structures: Existence of Isentropic Structures and Applications to Thermochemistry, Nuclear Magnetic Resonance, and Electron Spin Resonance

Kavitha¹,

Abraham²,

Arockiaraj³

et al. 2021

J. Phys. Chem. A

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Tessellations of kekulenes and cycloarenes are of considerable interest as nanomolecular belts in trapping and transportation of heavy metal ions and chloride ions, as they possess optimal electronic features and pore sizes. A class of cycloarenes called kekulenes have been the focus of several experimental and theoretical studies from the stand point of aromaticity, superaromaticity, chirality, and novel electrical and magnetic properties. In the present study, we investigate the entropies and topological characterization of different tessellations of kekulenes through topological computations of superaromatic structures with pores. We introduce the self-powered vertex degree-based topological indices and then derive the graph entropy measures for three different tessellations (zigzag, armchair, and rectangular) via various molecular descriptors that we derive here. Several applications to computing the molecular properties are pointed out. We demonstrate the existence of isentropic and yet nonisomorphic tessellations of kekulenes for the first time. The two tessellations are predicted to be quite close in energy with comparable energy gaps. Graph theory-based PPP methods with parameters derived from higher levels of theory are proposed to be promising tools for the predictions of relative stabilities of kekulene tessellations. We show that the developed techniques can be applied in the general context of artificial intelligence for the machine generation of nuclear magnetic resonance and electron spin resonance spectroscopic patterns as well as in robust computations of thermochemistry of a large combinatorial libraries of tessellations of kekulenes through the generation of bond-equivalence classes.

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“…The subject matters of hypercubes, polycubes and wreath product groups have been dealt with by several researchers over the years , including techniques to obtain topological Symmetry 2023, 15, 557 2 of 20 indices [41][42][43][44][45][46][47][48][49][50] by cut methods which involve subgraphs of hypercubes [41,42]. Furthermore, graph spectra, matching polynomials, distance degree sequences, topological indices and other properties of graphs pertinent to chemical applications have been the subject matter of several studies [51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67].…”

Section: Introductionmentioning

confidence: 99%

Topological Indices, Graph Spectra, Entropies, Laplacians, and Matching Polynomials of n-Dimensional Hypercubes

Balasubramanian

2023

Symmetry

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We obtain a large number of degree and distance-based topological indices, graph and Laplacian spectra and the corresponding polynomials, entropies and matching polynomials of n-dimensional hypercubes through the use of Hadamard symmetry and recursive dynamic computational techniques. Moreover, computations are used to provide independent numerical values for the topological indices of the 11- and 12-cubes. We invoke symmetry-based recursive Hadamard transforms to obtain the graph and Laplacian spectra of nD-hypercubes and the computed numerical results are constructed for up to 23-dimensional hypercubes. The symmetries of these hypercubes constitute the hyperoctahedral wreath product groups which also pave the way for the symmetry-based elegant computations. These results are used to independently validate the exact analytical expressions that we have obtained for the topological indices as well as graph, Laplacian spectra and their polynomials. We invoke a robust dynamic programming technique to handle the computationally intensive generation of matching polynomials of hypercubes and compute all matching polynomials up to the 6-cube. The distance degree sequence vectors have been obtained numerically for up to 108-dimensional cubes and their frequencies are found to be in binomial distributions akin to the spectra of n-cubes.

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On the calculation of the spectrum of large Hückel matrices, representing carbon nanotubes, using fast Hadamard and symplectic transforms

Cited by 7 publications

References 27 publications

On the Diagonalization of Hermitian Matrices, and Its Use to Calculate the Hückel Electronic Structure of Large Carbon Nanotubes

On the Diagonalization of Hermitian Matrices, and Its Use to Calculate the Hückel Electronic Structure of Large Carbon Nanotubes

Topological Characterization and Graph Entropies of Tessellations of Kekulene Structures: Existence of Isentropic Structures and Applications to Thermochemistry, Nuclear Magnetic Resonance, and Electron Spin Resonance

Topological Indices, Graph Spectra, Entropies, Laplacians, and Matching Polynomials of n-Dimensional Hypercubes

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