Intramolecular surfaces for vicinal proton–proton coupling constants 3JHH

Equations for the intramolecular surfaces of the 3JHH coupling constants in ethane, ethylene, and acetylene are formulated, and the corresponding coefficients are estimated from calculations at the DFT/B3LYP level. The chosen variables are changes in bond lengths, in the torsion angle φ between the coupled protons Ha and Hb, in bond angles, and in dihedral angles. The 3JHH surface of ethane is formulated as an extended Karplus equation with the coefficients of a truncated Fourier series on the torsion angle φ expanded as second-order Taylor series in the chosen variables taking into account the invariance of 3JHH under reflections and rotations of nuclear coordinates. Partial vibrational contributions from linear and square terms corresponding to changes in the geometry of the Ha − Ca − Cb − Hb fragment are important while those from cross terms are small with a few exceptions. The 3JHH surface of ethane is useful to predict contributions to 3JHH from changes in local geometry of derivatives but vibrational contributions are predicted less satisfactorily. The predicted values at the B3LYP/BS2 level of the 3JHH couplings (vibrational contributions at 300 K) from equilibrium geometries are 9.79 (−0.17) for acetylene, and 17.08 (1.93) and 10.73(0.93) for the trans and cis couplings of ethylene.

A new algorithm to accelerate harmonic analysis of time series

The Lomb periodogram has been traditionally a tool that allows us to elucidate if a frequency turns out to be important for explaining the behaviour of a given time series. Many linear and nonlinear reiterative harmonic processes that are used for studying the spectral content of a time series take into account this periodogram in order to avoid including spurious frequencies in their models due to the leakage problem of energy from one frequency to others. However, the estimation of the periodogram requires long computation time that makes the harmonic analysis slower when we deal with certain time series. Here we propose an algorithm that accelerates the extraction of the most remarkable frequencies from the periodogram, avoiding its whole estimation of the harmonic process at each iteration. This algorithm allows the user to perform a specific analysis of a given scalar time series. As a result, we obtain a functional model made of (1) a trend component, (2) a linear combination of Fourier terms, and (3) the so-called mixed secular terms by reducing the computation time of the estimation of the periodogram.