Real space electron delocalization, resonance, and aromaticity in chemistry

Chemists explaining a molecule’s stability and reactivity often refer to the concepts of delocalization, resonance, and aromaticity. Resonance is commonly discussed within valence bond theory as the stabilizing effect of mixing different Lewis structures. Yet, most computational chemists work with delocalized molecular orbitals, which are also usually employed to explain the concept of aromaticity, a ring delocalization in cyclic planar systems which abide certain number rules. However, all three concepts lack a real space definition, that is not reliant on orbitals or specific wave function expansions. Here, we outline a redefinition from first principles: delocalization means that likely electron arrangements are connected via paths of high probability density in the many-electron real space. In this picture, resonance is the consideration of additional electron arrangements, which offer alternative paths. Most notably, the famous 4n + 2 Hückel rule is generalized and derived from nothing but the antisymmetry of fermionic wave functions.

A Radial-Based Centralized Kriging Method for System Reliability Assessment

System reliability assessment is a challenging task when using computationally intensive models. In this work, a radial-based centralized Kriging method (RCKM) is proposed for achieving high efficiency and accuracy. The method contains two components: Kriging-based system most probable point (MPP) search and radial-based centralized sampling. The former searches for the system MPP by progressively updating Kriging models regardless of the nonlinearity of the performance functions. The latter refines the Kriging models with the training points (TPs) collected from pregenerated samples. It concentrates the sampling in the important high-probability density region. Both components utilize a composite criterion to identify the critical Kriging models for system failure. The final Kriging models are sufficiently accurate only at those sections of the limit states that bound the system failure region. Its efficiency and accuracy are demonstrated via application to three examples.

Improvement of hydrological model calibration by selecting multiple parameter ranges

The parameters of hydrological models are usually calibrated to achieve a good performance of the model, owing to the highly non-linear problem of hydrology process modelling. However, parameter calibration efficiency has a direct relation with parameter range. Furthermore, parameter range selection is affected by probability distribution of parameter values, parameter sensitivity and correlation. A newly proposed method is introduced to select and coordinate parameter ranges for improving the calibration of hydrological models with multiple parameters. At first, the probability distribution characteristics of single parameter value was analysed based on 100 samples obtained from independent calibration with initial parameter range and the distribution type (i.e. normal, exponential and uniform distributions) determined for single parameter. Then, the way to select the optimal range for single parameter was demonstrated by comparing different reduced and extended ranges corresponding to the distribution. Next, parameter correlation and sensibility were estimated to coordinate range selection of single parameter and the optimal combination of ranges for all parameters obtained. The results show that the probability of calibrated parameter values of Xinanjiang model takes on the normal or exponential distributions. For normal distribution, selecting the range of high probability density from the initial range is much more efficient for calibration. For exponential distribution, if the initial range can not be extended, selecting the range of high probability density contributes to high objective function. If the initial range can be extended, it is better to make the exponential distribution convert into normal distribution by doubling the range along X-axis direction and subsequently select the range according to normal distribution. Moreover, the coordination of range selection of single parameters makes the calibration of models with multiple parameters more efficient and effective.

Simplifying Neural Networks by Soft Weight-Sharing

One way of simplifying neural networks so they generalize better is to add an extra term to the error function that will penalize complexity. Simple versions of this approach include penalizing the sum of the squares of the weights or penalizing the number of nonzero weights. We propose a more complicated penalty term in which the distribution of weight values is modeled as a mixture of multiple gaussians. A set of weights is simple if the weights have high probability density under the mixture model. This can be achieved by clustering the weights into subsets with the weights in each cluster having very similar values. Since we do not know the appropriate means or variances of the clusters in advance, we allow the parameters of the mixture model to adapt at the same time as the network learns. Simulations on two different problems demonstrate that this complexity term is more effective than previous complexity terms.