Unexpected effects of third-order cross-terms in heteronuclear spin systems under simultaneous radio-frequency irradiation and magic-angle spinning NMR

We recently noted [R. K. Harris, P. Hodgkinson, V. Zorin, J.-N. Dumez, B. Elena, L. Emsley, E. Salager, and R. Stein, Magn. Reson. Chem. 48, S103 (2010)10.1002/mrc.2636] anomalous shifts in apparent 1H chemical shifts in experiments using 1H homonuclear decoupling sequences to acquire high-resolution 1H NMR spectra for organic solids under magic-angle spinning (MAS). Analogous effects were also observed in numerical simulations of model 13C,1H spin systems under homonuclear decoupling and involving large 13C,1H dipolar couplings. While the heteronuclear coupling is generally assumed to be efficiently suppressed by sample spinning at the magic angle, we show that under conditions typically used in solid-state NMR, there is a significant third-order cross-term from this coupling under the conditions of simultaneous MAS and homonuclear decoupling for spins directly bonded to 1H. This term, which is of the order of 100 Hz under typical conditions, explains the anomalous behaviour observed on both 1H and 13C spins, including the fast dephasing observed in 13C{1H} heteronuclear spin-echo experiments under 1H homonuclear decoupling. Strategies for minimising the impact of this effect are also discussed

A new approach to the method of source-sink potentials for molecular conduction

We re-derive the tight-binding source-sink potential (SSP) equations for ballistic conduction through conjugated molecular structures in a form that avoids singularities. This enables derivation of new results for families of molecular devices in terms of eigenvectors and eigenvalues of the adjacency matrix of the molecular graph. In particular, we define the transmission of electrons through individual molecular orbitals (MO) and through MO shells. We make explicit the behaviour of the total current and individual MO and shell currents at molecular eigenvalues. A rich variety of behaviour is found. A SSP device has specific insulation or conduction at an eigenvalue of the molecular graph (a root of the characteristic polynomial) according to the multiplicities of that value in the spectra of four defined device polynomials. Conduction near eigenvalues is dominated by the transmission curves of nearby shells. A shell may be inert or active. An inert shell does not conduct at any energy, not even at its own eigenvalue. Conduction may occur at the eigenvalue of an inert shell, but is then carried entirely by other shells. If a shell is active, it carries all conduction at its own eigenvalue. For bipartite molecular graphs (alternant molecules), orbital conduction properties are governed by a pairing theorem. Inertness of shells for families such as chains and rings is predicted by selection rules based on node counting and degenerac

Hierarchical Representations with Signatures for Large Expression Management

We describe a method for managing large expressions in symbolic computations which combines a hierarchical representation with signature calculations. As a case study, the problem of factoring matrices with non-polynomial entries is studied. Gaussian Elimination is used

Linear Algebra Using Maple's LargeExpressions Package

The package LargeExpressions has been available in Maple for a number of years, but it is not well known. It provides tools for managing large expressions. In this paper, we describe a new application of this tool to the LU factoring of matrices. We describe a function that factors a matrix and expresses the results using a hierarchical representation. As part of the LU factoring, we introduce several strategies for pivoting, veiling an expression and zero-recognition in our function. All these strategies can be chosen based on the application

Hierarchical representations with signatures for large expression managemen

We describe a method for managing large expressions in symbolic computations which combines a hierarchical representation with signature calculations. As a case study, the problem of factoring matrices with exponential polynomial entries is studied. Gaussian Elimination is used. Results on the complexity of the approach together with benchmark calculations are given. We begin by giving our definitions of hierarchical representations and signatures for this poster. Definition 1. An exponential polynomial p over Z and over a set of independent variables {x1,..., xm} is a polynomial p ∈ Z[x1,..., xm, y1,..., ym], where yk = a xk, k = 1,..., m, and where a is either the base of natural logarithms or a (non-zero) integer. Definition 2. A hierarchical representation (HR) over Z and over a set of independent variables {x1,..., xm} is an ordered list [S1, S2,..., Sl] of symbols, together with an associated list [D1, D2,..., Dl] of definitions of the symbols. For each Si with i ≥ 1, there is a definition Di of the form Si = f(σ1, σ2,..., σk) where f ∈ Z[σ1,..., σk], and each σj is either a symbol in [S1, S2,..., Si−1] or an exponential polynomial in the independent variables. A given expression can have different HR, i.e. different lists of definitions [D1, D2,...]. The strategy used to assign the symbols during the generation of expressions will be something that can be varied by the implementation. It is important to test zero for the expressions in hierarchical representations. Obviously expanding them to test zero is not a good way. The idea of using signatures of the expressions in HR is to provide a probabilistic zero-test, at a lower cost