Quantum Interference, Graphs, Walks, and Polynomials

Abstract

In this paper we explore quantum interference in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley-Hamilton theorem for characteristic polynomials and the Coulson-Rushbrooke pairing theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Green's function for electron transmission in terms of the odd powers of the vertex adjacency matrix orHückel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are cancelled by others, leading to another class of quantum interference. For non-alternant hydrocarbons, the finite Green's function expansion may include both even and odd powers. Nevertheless, QI can in some circumstances come about for non-alternants, from cancellation of odd and even-length walk terms. We report some progress, but not a complete resolution of the problem of understanding the coefficients in the expansion of the Green's function in a power series of the adjacency matrix, these coefficients being behind the cancellations that we have mentioned. And we introduce a perturbation theory for transmission as well as some potentially useful infinite power series expansions of the Green's function.

Quantum Interference in Real-Time Electron-Dynamics: Gaining Insights from Time-Dependent Configuration Interaction Simulations

“…Tsuji and Hoffmann [27] have shown and discussed with great clarity that in conduction calculations based on the HMO model Hamiltonian, QI can be predicted for a pair of atoms µ and ν using the zeroth-order Green's function…”

“…From a theoretical stand point, a number of studies have established qualitative relationships between observed/predicted QI trends with conjugation patterns in hydrocarbons-destructive QI in cross-conjugated molecules and constructive QI in linearly conjugated ones [26][27][28]. In particular, Markussen et al have presented a set of graphical rules to predict if a molecular structure can lead to QI or not [24].…”

Parity symmetry in a number of problems of quantum and structural chemistry

“…Простейший подход к GF основывается на схемах TB, которые использовались еще в начале зарождения SME [59]. Но поразительно, что модели TB и поныне продолжают довольно часто применяться в SME, и недавно вышел интересный обзор Хоффмана [11] и сотр. по чисто хюккелевской теории молекулярной проводимости.…”

“…В данном обзоре этот термин означает записанный выше специальный случай (1), когда элемент симметрии антикоммутирует с базисным оператором, отражающим, возможно, и приближенно, некие фундаментальные свойства молекулы. Стоит отметить, что хотя ряд симметрийных аспектов близких задач был истолкован еще в статье Макинтоша [7], квантовохический контекст требует иных акцентов и дальнейших обобщений, в част-https://doi.org/10.26565/2220-637X-2019-32-01 ности в свете современных проблем теории электрической проводимости органических молекул [8][9][10][11][12]. Наше изложение, объединяющее различные задачи, включает и новые результаты, но больше опирается на прежние и совсем недавние работы, изложенные теперь в едином ключе.…”

Quantum Interference in Real-Time Electron-Dynamics: Gaining Insights from Time-Dependent Configuration Interaction Simulations

“…Exploiting QI for practical purposes requires that decoherence effects are minimal; it is an experimentally established fact that with increase in temperature, destructive interference effects are lost resulting in enhanced conductivity [23]. ones [26][27][28]. In particular, Markussen et al have presented a set of graphical rules to predict if a molecular structure can lead to QI or not [24].…”