Complex band structure with non-orthogonal basis set: analytical properties and implementation in the SIESTA code

The complex band structure, although not directly observable, determines many properties of a material where the periodicity is broken, such at surfaces, interfaces and defects. Furthermore, its knowledge helps in the interpretation of electronic transport calculations and in the study of topological materials. Here we extend the transfer matrix method, often used to compute the complex bands, to electronic structures constructed using an atomic non-orthogonal basis set. We demonstrate that when the overlap matrix is not the identity, the non-orthogonal case, spurious features appear in the analytic continuation of the band structure to the complex plane. The properties of these are studied both numerically and analytically and discussed in the context of existing literature. Finally, a numerical implementation to extract the complex band structure from periodic calculations carried out with the density functional theory code SIESTA is presented. This is constructed as a simple post-processing tool, and it is therefore amenable to high-throughput studies of insulators and semiconductors.

Optimizing weighted gene co-expression network analysis with a multi-threaded calculation of the topological overlap matrix

Biomolecular networks are often assumed to be scale-free hierarchical networks. The weighted gene co-expression network analysis (WGCNA) treats gene co-expression networks as undirected scale-free hierarchical weighted networks. The WGCNA R software package uses an Adjacency Matrix to store a network, next calculates the topological overlap matrix (TOM), and then identifies the modules (sub-networks), where each module is assumed to be associated with a certain biological function. The most time-consuming step of WGCNA is to calculate TOM from the Adjacency Matrix in a single thread. In this paper, the single-threaded algorithm of the TOM has been changed into a multi-threaded algorithm (the parameters are the default values of WGCNA). In the multi-threaded algorithm, Rcpp was used to make R call a C++ function, and then C++ used OpenMP to start multiple threads to calculate TOM from the Adjacency Matrix. On shared-memory MultiProcessor systems, the calculation time decreases as the number of CPU cores increases. The algorithm of this paper can promote the application of WGCNA on large data sets, and help other research fields to identify sub-networks in undirected scale-free hierarchical weighted networks. The source codes and usage are available at https://github.com/do-somethings-haha/multi-threaded_calculate_unsigned_TOM_from_unsigned_or_signed_Adjacency_Matrix_of_WGCNA.

Algorithm optimization for weighted gene co-expression network analysis: accelerating the calculation of Topology Overlap Matrices with OpenMP and SQLite

MotivationWeighted gene co-expression network analysis (WGCNA) is an R package that can search highly related gene modules. The most time-consuming step of the whole analysis is to calculate the Topological Overlap Matrix (TOM) from the Adjacency Matrix in a single thread. This study changes it to multithreading.ResultsThis paper uses SQLite for multi-threaded data transfer between R and C++, uses OpenMP to enable multi-threading and calculates the TOM via an adjacency matrix on a Shared-memory MultiProcessor (SMP) system, where the calculation time decreases as the number of physical CPU cores increases.Availability and implementationThe source code is available at https://github.com/do-somethings-haha/[email protected]

An Algorithm to Correct for the CASSCF Active Space in Multiscale QM/MM Calculations Based on Geometry Ensembles

The convolution of the excitation energies, computed by the complete active space self-consistent field (CASSCF) or other CAS-based methods, of an ensemble of geometries generated by molecular dynamic simulations is a usual recipe to obtain the absorption spectrum or the density of states of a chromophore. This approach requires that all the considered geometries have the same molecular orbitals within the active space. However, the different geometrical features and/or the different influence of the solvent or biological environments along the sample geometries makes the preservation of the active space a challenging task, which is usually ignored. In this work, we present an algorithm to correct for the active space of geometry ensembles in CASSCF calculations. The algorithm is based on the calculation of the molecular orbital overlap matrix between a previously selected reference geometry, with the desired active space, and each of the sampled geometries. Depending on the value of the overlap matrix elements, the algorithm determines whether one or more pairs of molecular orbitals of the sampled geometry have to be swapped for a subsequent CASSCF calculation. We have applied the developed algorithm to quantum mechanics/molecular mechanics CASSCF/MM and CASPT2/MM calculations for sets of geometries of the five canonical nucleobases in aqueous solution obtained from classical molecular dynamics simulations. The algorithm shows a very good efficacy since it recovered the correct active space for 76\% of the geometries which presented undesired molecular orbitals in the active space after the first CASSCF wavefunction optimization. In addition, the importance of having the same orbitals within the active space for all the geometries is discussed based on the computed density of states for the solvated nucleobases.

An Algorithm to Correct for the CASSCF Active Space in Multiscale QM/MM Calculations Based on Geometry Ensembles

The convolution of the excitation energies, computed by the complete active space self-consistent field (CASSCF) or other CAS-based methods, of an ensemble of geometries generated by molecular dynamic simulations is a usual recipe to obtain the absorption spectrum or the density of states of a chromophore. This approach requires that all the considered geometries have the same molecular orbitals within the active space. However, the different geometrical features and/or the different influence of the solvent or biological environments along the sample geometries makes the preservation of the active space a challenging task, which is usually ignored. In this work, we present an algorithm to correct for the active space of geometry ensembles in CASSCF calculations. The algorithm is based on the calculation of the molecular orbital overlap matrix between a previously selected reference geometry, with the desired active space, and each of the sampled geometries. Depending on the value of the overlap matrix elements, the algorithm determines whether one or more pairs of molecular orbitals of the sampled geometry have to be swapped for a subsequent CASSCF calculation. We have applied the developed algorithm to quantum mechanics/molecular mechanics CASSCF/MM and CASPT2/MM calculations for sets of geometries of the five canonical nucleobases in aqueous solution obtained from classical molecular dynamics simulations. The algorithm shows a very good efficacy since it recovered the correct active space for 76\% of the geometries which presented undesired molecular orbitals in the active space after the first CASSCF wavefunction optimization. In addition, the importance of having the same orbitals within the active space for all the geometries is discussed based on the computed density of states for the solvated nucleobases.

Overlap matrix concentration in optimal Bayesian inference

We consider models of Bayesian inference of signals with vectorial components of finite dimensionality. We show that under a proper perturbation, these models are replica symmetric in the sense that the overlap matrix concentrates. The overlap matrix is the order parameter in these models and is directly related to error metrics such as minimum mean-square errors. Our proof is valid in the optimal Bayesian inference setting. This means that it relies on the assumption that the model and all its hyper-parameters are known so that the posterior distribution can be written exactly. Examples of important problems in high-dimensional inference and learning to which our results apply are low-rank tensor factorization, the committee machine neural network with a finite number of hidden neurons in the teacher–student scenario or multi-layer versions of the generalized linear model.

A Democratic Mapping between the Shell Model and the IBM

A method is proposed to connect states of the shell model and the interacting boson model and derive the boson model hamiltonian from shell-model data. This novel mapping technique is based on the properties of the shell-model overlap matrix. An application to the /7/2 shell is presented and the results of the new mapping are compared with the standard OAI results.