Analytic Results for the Eigenvalues of Certain Tridiagonal Matrices

Willms, Allan R.

doi:10.1137/070695411

Cited by 59 publications

(45 citation statements)

References 5 publications

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“…Indeed, if the reorganization energy λ is reduced to zero, the effective H PP n +• Hamiltonian becomes independent of x and thus isomorphous to the Hückel theory Hamiltonian for a linear π-conjugated polymer. Diagonalization of this Hamiltonian with zero main diagonal yields the ground state energy that evolves proportionally to cos[π/( n + 1)], 46 that approximates as a 1/ n trend for small n . Thus, for small λ or, equivalently, large H ab /λ values, the H PP n +• ground state energies should initially evolve in approximate 1/ n fashion.…”

Section: Resultsmentioning

confidence: 99%

Key Role of End-Capping Groups in Optoelectronic Properties of Poly-p-phenylene Cation Radicals

et al. 2014

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Poly-p-phenylenes (PPs) are prototype systems for understanding the charge transport in π-conjugated polymers. In a combined computational and experimental study, we demonstrate that the smooth evolution of redox and optoelectronic properties of PP cation radicals toward the polymeric limit can be significantly altered by electron-donating iso-alkyl and iso-alkoxy end-capping groups. A multiparabolic model (MPM) developed and validated here rationalizes this unexpected effect by interplay of the two modes of hole stabilization: due to the framework of equivalent p-phenylene units and due to the electron-donating end-capping groups. A symmetric, bell-shaped hole in unsubstituted PPs becomes either slightly skewed and shifted toward an end of the molecule in iso-alkyl-capped PPs or highly deformed and concentrated on a terminal unit in PPs with strongly electron-donating iso-alkoxy capping groups. The MPM shows that the observed linear 1/n evolution of the PP cation radical properties toward the polymer limit originates from the hole stabilization due to the growing chain of p-phenylene units, while shifting of the hole toward electron-donating end-capping groups leads to early breakdown of these 1/n dependencies. These insights, along with the readily applicable and flexible multistate parabolic model, can guide studies of complex donor–spacer–acceptor systems and doped molecular wires to aid the design of the next generation materials for long-range charge transport and photovoltaic applications.

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Section: Resultsmentioning

confidence: 99%

Key Role of End-Capping Groups in Optoelectronic Properties of Poly-p-phenylene Cation Radicals

et al. 2014

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“…For a proof of the next result we refer to Yueh (2005); see Willms (2008) for further eigenvalue formulas for tridiagonal matrices. …”

Section: Theorem 845 1 the Eigenvalues Of A N Are The Distinct Reamentioning

confidence: 97%

The Mathematics of Networks of Linear Systems

Fuhrmann

Helmke

2015

Universitext

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“…where = π/(2L + 2) Second, the matrix T (q, q −1 ) (with a = q and b = q −1 ) has the eigenvalues [12,14,15]:…”

Section: B Analytically Tractable Tridiagonal Matricesmentioning

confidence: 99%

Simulating quantum circuits by adiabatic computation: Improved spectral gap bounds

et al. 2020

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Adiabatic quantum computing is a framework for quantum computing that is superficially very different to the standard circuit model. However, it can be shown that the two models are computationally equivalent. The key to the proof is a mapping of a quantum circuit to an an adiabatic evolution, and then showing that the minimum spectral gap of the adiabatic Hamiltonian is at least inverse polynomial in the number of computational steps L. In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for L 1 is mins ∆ π 2 /[8(L + 1) 2 ], an improvement over previous estimates. Our first method is a direct approach based on an eigenstate ansatz, while the the second uses Weyl's theorem to leverage known exact results into a bound for the gap. Our results suggest that it may be possible to use these methods to find bounds for spectral gaps of Hamiltonians in other scenarios.

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Analytic Results for the Eigenvalues of Certain Tridiagonal Matrices

Cited by 59 publications

References 5 publications

Key Role of End-Capping Groups in Optoelectronic Properties of Poly-p-phenylene Cation Radicals

Key Role of End-Capping Groups in Optoelectronic Properties of Poly-p-phenylene Cation Radicals

The Mathematics of Networks of Linear Systems

Simulating quantum circuits by adiabatic computation: Improved spectral gap bounds

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