A formula for Eigenpairs of certain symmetric tridiagonal matrices

Shin, Byeong Chun

doi:10.1017/s0004972700033918

Cited by 23 publications

(28 citation statements)

References 4 publications

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“…If the oxygen atom is removed, the remaining π system is that of a 2M+1 polyene. For such a system, in the Hückel approximation, the LUMO is singly occupied in the excited state and has coefficients that are exactly vanishing on the even‐number centers, with alternating signs on the odd‐number atoms (i.e., this orbital presents oscillations of period equal to four) 32. In the excited state of the molecule, the delocalized electron occupying this orbital interacts with the hole in the strongly localized n O orbital.…”

Section: Resultsmentioning

confidence: 99%

Localized molecular orbitals for excited states of polyenals, polyendials, and polyenones

Pitarch‐Ruiz

Evangelisti

Maynau

2003

Int J of Quantum Chemistry

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ABSTRACT:The work is focused on the generation of localized molecular orbitals for excited states. A recently developed a priori method based in a CAS-SCF-type algorithm is applied. The method generates directly localized orbitals and can be applied to multireference wavefunctions. A detailed description of the performance of the method as well as the locality of the MOs for the example of the singlet n3* (CO) excited state is given. It is in general possible to obtain local orbitals for the doubly occupied and virtual valence orbitals. The partial delocalization of the * (CO) orbital is discussed, as is the effect of the use of different CAS spaces. The systems under study are polyenals, polyendials, and polyenones where the * (CO) orbital interacts with the rest of the system.

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Section: Resultsmentioning

confidence: 99%

Localized molecular orbitals for excited states of polyenals, polyendials, and polyenones

Pitarch‐Ruiz

Evangelisti

Maynau

2003

Int J of Quantum Chemistry

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“…22 The odd-order case, N =2M + 1, admits an explicit expression for the eigenvalues. 22 The odd-order case, N =2M + 1, admits an explicit expression for the eigenvalues.…”

Section: ͑13͒mentioning

confidence: 99%

The metal-insulator transition in dimerized Hückel chains

Monari

Bendazzoli

Evangelisti

2008

The Journal of Chemical Physics

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The metal-insulator transition is investigated in the case of linear chains described by a one-electron Hückel Hamiltonian. In these systems, the transition is a consequence of a dimerization of the chain bond length, which induces a similar dimerization of the hopping integral. Three indicators of the chain character are considered: The highest occupied molecular orbital-lowest unoccupied molecular orbital gap, the polarizability, and the localization tensor. In the case of even open chains, the behavior of the large chains depends in a crucial way on the alternating structure of the hopping integrals. If the ending atoms of the chain are weakly bonded to their neighbors, the energy spectrum of the Hamiltonian shows two quasidegenerated eigenvalues, and all the indicators would predict a (spurious) metallic behavior. It is shown that if the corresponding eigenvectors are removed from the Hamiltonian, the ordinary insulating behavior of alternating chains is recovered.

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“…The use of symmetry splits the secular matrices of the graphs into two non-interacting blocks, each of which represents a single path (or cycle) with alternating weighted vertices and edges. It is these backbone graphs that possess the analytical solutions derived previously by Gover [9] and Shin [18]. An active avenue of research is exploration of the influence of molecular topology in conduction behaviour.…”

Section: Graph Theoretical Backgroundmentioning

confidence: 79%

“…We consider paths, P M (a, b | c, d), with alternating vertex weights a, b, and edge weights c, d. Eigenvalues and eigenvectors for such weighted paths have been deduced by Gover [9] and Shin [18]. Gover used recursion to show that the spectrum of the odd-vertex chain, P 2N +1 , could be expressed in terms of two sets of polynomials.…”

Section: Planmentioning

confidence: 99%

Spectra and structural polynomials of graphs of relevance to the theory of molecular conduction

et al. 2017

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In chemistry and physics, distortivity of π-systems (stabilisation of bond-alternated structures) is an important factor in the calculation of geometric, energetic, and electronic properties of molecules via graph theoretical methods. We use the spectra of paths and cycles with alternating vertex and edge weights to obtain the eigenvalues and eigenvectors for a class of linear and cyclic ladders with alternating rung and backbone edge weights. We derive characteristic polynomials and other structural polynomials formed from the cofactors of the characteristic matrix for these graphs. We also obtain spectra and structural polynomials for ladders with flipped weights and/or Möbius topology. In all cases, the structural polynomials for the composite graphs are expressed in terms of products of polynomials for graphs of half order. This form of the expressions allows global deductions about the transmission spectra of molecular devices in the graph-theoretical theory of ballistic molecular conduction.

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A formula for Eigenpairs of certain symmetric tridiagonal matrices

Abstract: A closed form expression is given for the eigenvalues and eigenvectors of a symmetric tridiagonal matrix of odd order whose diagonal elements are all equal and whose superdiagonal elements alternate between the values c and d. An implicit formula is given for the even order case.

Cited by 23 publications

References 4 publications

Localized molecular orbitals for excited states of polyenals, polyendials, and polyenones

Localized molecular orbitals for excited states of polyenals, polyendials, and polyenones

The metal-insulator transition in dimerized Hückel chains

Spectra and structural polynomials of graphs of relevance to the theory of molecular conduction

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