Gaussian Basis Functions Research Articles

VeloxChem: A Python‐driven density‐functional theory program for spectroscopy simulations in high‐performance computing environments

AbstractAn open‐source program named VeloxChem has been developed for the calculation of electronic real and complex linear response functions at the levels of Hartree–Fock and Kohn–Sham density functional theories. With an object‐oriented program structure written in a Python/C++ layered fashion, VeloxChem enables time‐efficient prototyping of novel scientific approaches without sacrificing computational efficiency, so that molecular systems involving up to and beyond 500 second‐row atoms (or some 10,000 contracted and in part diffuse Gaussian basis functions) can be routinely addressed. In addition, VeloxChem is equipped with a polarizable embedding scheme for the treatment of the classical electrostatic interactions with an environment that in turn is modeled by atomic site charges and polarizabilities. The underlying hybrid message passing interface (MPI)/open multiprocessing (OpenMP) parallelization scheme makes VeloxChem suitable for execution in high‐performance computing cluster environments, showing even slightly beyond linear scaling for the Fock matrix construction with use of up to 16,384 central processing unit (CPU) cores. An efficient—with respect to convergence rate and overall computational cost—multifrequency/gradient complex linear response equation solver enables calculations not only of conventional spectra, such as visible/ultraviolet/X‐ray electronic absorption and circular dichroism spectra, but also time‐resolved linear response signals as due to ultra‐short weak laser pulses. VeloxChem distributed under the GNU Lesser General Public License version 2.1 (LGPLv2.1) license and made available for download from the homepage https://veloxchem.org.This article is categorized under: Software > Quantum Chemistry Electronic Structure Theory > Density Functional Theory Theoretical and Physical Chemistry > Spectroscopy

Intermolecular Interactions in Molecular Organic Crystals upon Relaxation of Lattice Parameters

Crystal structure prediction is based on the assumption that the most thermodynamically stable structure will crystallize first. The existence of other structures such as polymorphs or from counterenantiomers requires an accurate calculation of the electronic energy. Using atom-centered Gaussian basis functions in periodic Density Functional Theory (DFT) calculations in Turbomole, the performance of two dispersion-corrected functionals, PBE-D3 and B97-D, is assessed for molecular organic crystals of the X23 benchmark set. B97-D shows a MAE (mean absolute error) of 4 kJ/mol, compared to 9 kJ/mol for PBE-D3. A strategy for the convergence of lattice energies towards the basis set limit is outlined. A simultaneous minimization of molecular structures and lattice parameters shows that both methods are able to reproduce experimental unit cell parameters to within 4–5%. Calculated lattice energies, however, deviate slightly more from the experiment, i.e., by 0.4 kJ/mol after unit cell optimization for PBE-D3 and 0.5 kJ/mol for B97-D. The accuracy of the calculated lattice energies compared to the experimental values demonstrates the ability of current DFT methods to assist in the quest for possible polymorphs and enantioselective crystallization processes.

Matrix Elements of One Dimensional Explicitly Correlated Gaussian Basis Functions

Kinetic, potential and overlap matrix elements of one dimensional correlated Gaussians multiplied by polynomial factors are presented. These matrix elements can be used to calculate energies of one dimensional cold atom systems, or to construct a tensorial product to calculate energies in 2 or 3 dimensional systems with a nonspherical potential.

Efficient mixture control chart pattern recognition using adaptive RBF neural network

In this paper, a stochastic gradient method based adaptive version of the radial basis function neural network has proposed to map the pattern features of the control chart patterns in different categories to recognize their belonging class. Adaptiveness has given over the spreadness and centers of Gaussian basis function appeared in the hidden nodes of the radial basis function neural network. Along with normal abnormalities in patterns, the mixture of different abnormal patterns has also considered capturing the worst possible conditions of abnormalities in real time. The advantages of the proposed method have appeared as very high recognition accuracy, minimum error in learning and generalize performance with small training dataset in control chart pattern recognition. Achieved performance has compared with the state of art results available in the literature which has applied feature based recognition using Support vector machine and Genetic algorithm. The proposed method has enhanced the recognition generalization of control chart patterns with simplicity in design and high level of decision confidence. The performances have achieved through the simulation-based experiments over a huge number of patterns containing ten different types of pattern and on average, 99.99% accuracy has achieved.

Resolution-of-the-identity approximation for complex-scaled basis functions.

A resolution-of-the-identity (RI) approximation for two-electron integrals over Gaussian basis functions with a complex-scaled exponent is presented. Such functions are used in non-Hermitian quantum mechanics to represent electronic resonances by L2 integrable wave functions with complex energies. We have implemented this new RI approximation for second-order Møller-Plesset perturbation (MP2) theory as well as for the Coulomb and exchange contributions in Hartree-Fock (HF) theory. We discuss the differences to the standard RI approximation of Hermitian quantum mechanics and demonstrate the utility of the non-Hermitian RI-MP2 and RI-HF methods by computations of the orientation-dependent ionization rates of CO, C6H6, and C10H8 in static electric fields. Our results illustrate that RI-MP2 correctly describes correlation effects in molecular electronic resonances while the computational cost is low enough to allow for investigations of medium-sized molecules.

Multipole-based distance-dependent screening of Coulomb integrals.

We derive a new estimate for two-electron repulsion integrals (ERIs), when evaluated within a local atomic basis set. It is based on the multipole expansion and provides a rigorous upper bound of an ERI for well-separated charge distributions. The scheme is generally applicable in any formalism that uses ERIs. We employ it here to screen for potentially negligible contributions in the calculation of the Fock exchange matrix. Using Gaussian basis functions, we show that the estimate allows us to accelerate the construction of the exchange matrix by up to a factor of two without introducing further approximations.

Correlated Gaussians and Low-Discrepancy Sequences

Within the Correlated Gaussian Method the parameters of the Gaussian basis functions are often chosen stochastically using pseudo-random sequences. We show that alternative low-discrepancy sequences, also known as quasi-random sequences, provide bases of better quality.

Toward First-Principles-Level Polarization Energies in Force Fields: A Gaussian Basis for the Atom-Condensed Kohn-Sham Method.

The last 20 years of force field development have shown that even well parametrized classical models need to at least approximate the dielectric response of molecular systems-based, e.g., on atomic polarizabilities-in order to correctly render their structural and dynamic properties. Yet, despite great advances most approaches tend to be based on ad hoc assumptions and often insufficiently capture the dielectric response of the system to external perturbations, such as, e.g., charge carriers in semiconducting materials. A possible remedy was recently introduced with the atom-condensed Kohn-Sham density-functional theory approximated to second order (ACKS2), which is fully derived from first principles. Unfortunately, specifically its reliance on first-principles derived parameters so far precluded the widespread adoption of ACKS2. Opening up ACKS2 for general use, we here present a reformulation of the method in terms of Gaussian basis functions, which allows us to determine many of the ACKS2 parameters analytically. Two sets of parameters depending on exchange-correlation interactions are still calculated numerically, but we show that they could be straightforwardly parametrized owing to the smoothness of the new basis. Our approach exhibits three crucial benefits for future applications in force fields: i) efficiency, ii) accuracy, and iii) transferability. We numerically validate our Gaussian augmented ACKS2 model for a set of small hydrocarbons which shows a very good agreement with density-functional theory reference calculations. To further demonstrate the method's accuracy and transferability for realistic systems, we calculate polarization responses and energies of anthracene and tetracene, two major building blocks in organic semiconductors.

A 3-D Topology Optimization of Magnetic Cores for Wireless Power Transfer Device

This paper presents the topology optimization of the magnetic core for a wireless power transfer (WPT) device. In this optimization, the coil and magnetic-core shapes for WPT are represented by the Gaussian basis functions. They are sequentially optimized using the genetic algorithm and 3-D finite-element analysis so that the coupling coefficient of the WPT device is maximized. Moreover, a robust optimization method is proposed to keep the interlinkage flux as large as possible against misalignment in the coils. It is shown by the computational and experimental results that the optimized device outperforms the conventional devices.

Excited state dynamics initiated by an electromagnetic field within the Variational Multi-Configurational Gaussian (vMCG) method

The Variational Multi-Configurational Gaussian (vMCG) approach offers a framework to perform exact trajectory-based quantum dynamics. Herein we use two model vibronic coupling Hamiltonians of pyrazine to explore, for the first time, the influence of the coupling between the external field and the Gaussian basis functions (GBFs) in vMCG on the dynamics. We show that when the excitation pulse is short compared to the nuclear dynamics, vertical projection without a field and explicit description of the external field converge. For longer pulses, a sizeable change is observed. We demonstrate that comparatively few GBFs are sufficient to provide qualitative agreement to MCTDH dynamics and a quantitative agreement can be achieved using ∼100 GBFs. Longer pulses require more GBFs due to the prolonged coupling between the ground and excited states. Throughout this work the single set formalism offers the fastest convergence.

Robust adaptive consensus of nonstrict-feedback multi-agent systems with quantized input and unmodeled dynamics

This paper deals with the distributed robust adaptive neural network tracking control for a class of nonstrict-feedback multi-agent systems (MASs) with unmodeled dynamics and input quantized by the hysteresis quantizer. A decomposition technique is utilized to handle the problem that only finite and discrete control values are available to control system. Meanwhile, the radial basis function neural networks (RBFNNs) are employed for approximation of the packaged unknown nonlinearities induced by recursive design. By introducing the structural characteristic of Gaussian basis function (see Lemma 5), the effects from the nonstrict-feedback are effectively compensated and the backstepping-based method can be successfully implemented. Then, by combining the dynamic signal with adaptive backstepping control technique, a robust adaptive distributed quantized tracking control scheme is developed for each follower. It is proved that all closed-loop signals are semi-globally uniformly ultimately bounded and the asymptotic consensus tracking is guaranteed. Finally, the effectiveness of the control method is illustrated by two simulation examples.

Shaping and controlling stabilisation graphs for calculating stable complex resonance energies

ABSTRACTStabilisation technique is a traditional approach for studying resonances, such as the metastable autoionisation states. Stabilisation graphs are usually calculated by scaling the employed basis functions. Over the years, many methods for extracting the resonance complex energies from these graphs have been developed, nowadays, they are usually evaluated via analytical continuation schemes. However, many times it is difficult to obtain a well-behaved stabilisation graphs upon basis-set scaling. Herein, we illustrate that it is possible to shape stabilisation graphs to behave physically by adjusting the relevant Gaussian basis function exponents. We illustrate this for the excited Li–He and states [with He)], which serve as an excellent example since generating well-behaved graphs in this case is challenging. We show that by controlling and shaping the stabilisation graphs we can obtain converged complex energies, which are calculated by the resonances via Padé approach (a recently proposed analytical continuation scheme). Nevertheless, the concepts presented here are general to any stabilisation-graph based method.

Segment-level probabilistic sequence kernel and segment-level pyramid match kernel based extreme learning machine for classification of varying length patterns of speech

In this work, we address some issues in the classification of varying length patterns of speech represented as sets of continuous-valued feature vectors using kernel methods. Kernels designed for varying length patterns are called as dynamic kernels. We propose two dynamic kernels namely segment-level pyramid match kernel (SLPMK) and segment-level probabilistic sequence kernel (SLPSK) for classification of long duration speech, represented as varying length sets of feature vectors using extreme learning machine (ELM). SLPMK and SLPSK are designed by partitioning the speech signal into increasingly finer segments and matching the corresponding segments. SLPSK is built upon a set of Gaussian basis functions, where half of the basis functions contain class-specific information while the other half implicates the common characteristics of all the speech utterances of all classes. The computational complexity of SVM training algorithms is usually intensive, which is at least quadratic with respect to the number of training examples. It is difficult to deal with the immense amount of data using traditional SVMs. For reducing the training time of classifier we propose to use a simple algorithm namely ELM. ELM refers to a wider type of generalized single hidden layer feedforward networks (SLFNs) whose hidden layer need not be tuned. In our work, we proposed to explore kernel based ELM to exploit dynamic kernels. We study the performance of the ELM-based classifiers using the proposed SLPSK and SLPMK for speech emotion recognition and speaker identification tasks and compare with other kernels for varying length patterns. Experimental studies showed that proposed ELM-based approach offer a 10–12% of relative improvement over baseline approach, and a 3–9% relative improvement over ELMs/SVMs using other state-of-the-art dynamic kernels.

Method of Moments Simulation of Modulated Metasurface Antennas With a Set of Orthogonal Entire-Domain Basis Functions

A family of orthogonal and entire-domain basis functions (named Fourier-Bessel) is proposed for the analysis of circular modulated metasurface (MTS) antennas. In the structures at hand, the MTS is accounted for in the electric field integral equation (EFIE) as a sheet transition impedance boundary condition on the top of a grounded dielectric slab. The closed-form Hankel transform of the Fourier-Bessel basis functions (FBBFs) allows one to use a spectral domain formulation in the method-of-moments (MoM) solution of the EFIE. Moreover, these basis functions are fully orthogonal, which implies that they are able to represent the global evolution of the current distribution in a compact form. FBBFs also present a better filtering capability of their spectrum compared to other well-known orthogonal families such as the Zernike functions. The obtained MoM matrix is sparse and compact, and it is thus very well-conditioned and can be efficiently computed and inverted. The numerical results based on the proposed decomposition are presented and compared with those based on the use of the Gaussian ring basis functions and with the full-wave analysis of MTS antennas implemented with small printed elements. A very good agreement is observed.

Integral partition bounds for fast and effective screening of general one-, two-, and many-electron integrals.

We introduce tight upper bounds for a variety of integrals appearing in electronic structure theories. These include electronic interaction integrals involving any number of electrons and various integral kernels such as the ubiquitous electron repulsion integrals and the three- and four-electron integrals found in explicitly correlated methods. Our bounds are also applicable to the one-electron potential integrals that appear in great number in quantum mechanical (QM), mixed quantum and molecular mechanical (QM/MM), and semi-numerical methods. The bounds are based on a partitioning of the integration space into balls centered around electronic distributions and their complements. Such a partitioning leads directly to equations for rigorous extents, which we solve for shell pair distributions containing shells of Gaussian basis functions of arbitrary angular momentum. The extents are the first general rigorous formulation we are aware of, as previous definitions are based on the inverse distance operator 1/r12 and typically only rigorous for simple spherical Gaussians. We test our bounds for six different integral kernels found throughout quantum chemistry, including exponential, Gaussian, and complementary error function based forms. We compare to previously developed estimates on the basis of significant integral counts and their usage in both explicitly correlated second-order Møller-Plesset theory (MP2-F12) and density functional theory calculations employing screened Hartree-Fock exchange.

Optimization of the nonlinear parameters of the correlated Gaussian basis functions with imaginary-time propagation

Optimization of the nonlinear parameters of the correlated Gaussian basis functions with imaginary-time propagation

Topology Optimization of Metamaterial Using Gaussian-Basis Functions

Topology Optimization of Metamaterial Using Gaussian-Basis Functions

Kaonic deuterium from realistic antikaon-nucleon interaction

We present precise three-body calculations for the spectrum of kaonic deuterium with a realistic antikaon-nucleon interaction. Thanks to the precise measurement of kaonic hydrogen, it is now possible to construct realistic $\bar KN$ interactions which reproduce the whole set of experimental data in the threshold region. Employing such realistic interactions, the energy of the three-body system of kaonic deuterium is determined with the accuracy of eV, by expanding its wave function with a large number of correlated Gaussian basis functions. The level shift and width of the 1S state are found to be 670 eV and 1016 eV, respectively. The improved Deser formulas work reasonably well to estimate the shift and width of kaonic hydrogen, but their application to kaonic deuterium does not give an accurate estimation. It is shown that the result is sensitive to the I = 1 component of the $\bar KN$ interaction, which will be further constrained in future experiments.

Neutral atoms and ion energies, accurate ionization potential, and electron affinities by polynomial generator coordinate Hartree–Fock method

We have developed accurate Gaussian basis functions obtained with the polynomial generator coordinate Hartree-Fock (p-GCHF) method for H, Zn, and Ga-Kr atoms. These basis sets have been applied in the calculation of nonrelativistic energies for neutral atoms, monovalent cations, monovalent anions, ionization potential (IP), and electron affinity (EA), with the objective of proving the quality of the basis set generated by the p-GCHF method. The total energies calculated for neutral atoms and monovalent cations and respective IP were minimally affected by the addition of polarization functions and their precision was comparable to the values reported in the literature. The relative errors were lower than 6.0 \texttimes\ 10$^{-5}$% and 7.0 \texttimes\ 10$^{-5}$% for neutral atoms and monovalent cations, respectively. The IP results were strictly equal to numerical Hartree-Fock (NHF) calculations and comparable to some experimental values. For monovalent anions, the nonrelativistic total energies were better than the Slater-type functions results and the relative errors were lower than 0.05% when compared to NHF. The EA results were the same as those obtained with NHF calculations reported in the literature for heavier elements. For IP and EA, our results followed the same periodic tendency when compared with experimental data.

Linear Scaling Pseudo Fermi-Operator Expansion for Fractional Occupation.

Recursive Fermi-operator expansion methods for the calculation of the idempotent density matrix are valid only at zero electronic temperature with integer occupation numbers. We show how such methods can be modified to include fractional occupation numbers of an approximate or pseudo Fermi-Dirac distribution and how the corresponding entropy term of the free energy is calculated. The proposed methodology is demonstrated and evaluated for different electronic structure methods, including density functional tight-binding theory, Kohn-Sham density functional theory using numerical orbitals, and quantum chemistry Hartree-Fock theory using Gaussian basis functions.