Slater-type Functions Research Articles

Modified saddle-point method applied to high-order above-threshold ionization and high-order harmonic generation: Slater-type versus asymptotic ground-state wave function

Modified saddle-point method applied to high-order above-threshold ionization and high-order harmonic generation: Slater-type versus asymptotic ground-state wave function

Basis Set Requirements of σ-Functionals for Gaussian- and Slater-Type Basis Functions and Comparison with Range-Separated Hybrid and Double Hybrid Functionals.

σ-Functionals belong to the class of Kohn-Sham (KS) correlation functionals based on the adiabatic-connection fluctuation-dissipation theorem and are technically closely related to the random phase approximation (RPA). They have the same computational demand as the latter, with the computational effort of an energy evaluation for both methods being lower than that of a preceding hybrid DFT calculation for typical systems but yield much higher accuracy, reaching chemical accuracy of 1 kcal/mol for quantities such as reactions and transition energies in main group chemistry. In previous work on σ-functionals, rather large Gaussian basis sets have been used. Here, we investigate the actual basis set requirements of σ-functionals and present three setups that employ smaller Gaussian basis sets ranging from quadruple-ζ (QZ) to triple-ζ (TZ) quality and represent a good compromise between accuracy and computational efficiency. Furthermore, we introduce an implementation of σ-functionals based on Slater-type basis sets and present two setups of QZ and TZ quality for this implementation. We test the accuracy of these setups on a large database of various physical properties and types of reactions, as well as equilibrium geometries and vibrational frequencies. As expected, the accuracy of σ-functional calculations becomes somewhat lower with a decreasing basis set size. However, for all setups considered here, calculations with σ-functionals are clearly more accurate than those within the RPA and even more so than those of the conventional KS methods. For the smallest setup using Gaussian-type basis functions and Slater-type basis functions, we introduce a reparametrization that reduces the loss in accuracy due to the basis set error to some extent. A comparison with the range-separated hybrid ωB97X-V and the double hybrid DSD-BLYP-D3 shows that σ functionals outperform in accuracy both of these accurate and, for their class, representative functionals.

Importance profiles. Visualization of atomic basis set requirements

Recent developments in fully numerical methods promise interesting opportunities for new, compact atomic orbital (AO) basis sets that maximize the overlap to fully numerical reference wave functions, following the pioneering work of Richardson and coworkers from the early 1960s. Motivated by this technique, we suggest a way to visualize the importance of AO basis functions employing fully numerical wave functions computed at the complete basis set limit: the importance of a normalized AO basis function |α⟩ centered on some nucleus can be visualized by projecting |α⟩ on the set of numerically represented occupied orbitals |ψi⟩ as I0(α)=∑i⟨α|ψi⟩⟨ψi|α⟩ . Choosing α to be a continuous parameter describing the AO basis, such as the exponent of a Gaussian-type orbital or Slater-type orbital basis function, one is then able to visualize the importance of various functions. The proposed visualization I0(α) has the important property 0⩽I0(α)⩽1 which allows unambiguous interpretation. We also propose a straightforward generalization of the importance profile for polyatomic applications I(α) , in which the importance of a test function |α⟩ is measured as the increase in projection from the atomic minimal basis. We exemplify the methods with importance profiles computed for atoms from the first three rows, and for a set of chemically diverse diatomic molecules. We find that the importance profile offers a way to visualize the atomic basis set requirements for a given system in an a priori manner, provided that a fully numerical reference wave function is available.

An information-theoretic approach to basis-set fitting of electron densities and other non-negative functions.

The numerical ill-conditioning associated with approximating an electron density with a convex sum of Gaussian or Slater-type functions is overcome by using the (extended) Kullback-Leibler divergence to measure the deviation between the target and approximate density. The optimized densities are non-negative and normalized, and they are accurate enough to be used in applications related to molecular similarity, the topology of the electron density, and numerical molecular integration. This robust, efficient, and general approach can be used to fit any non-negative normalized functions (e.g., the kinetic energy density and molecular electron density) to a convex sum of non-negative basis functions. We present a fixed-point iteration method for optimizing the Kullback-Leibler divergence and compare it to conventional gradient-based optimization methods. These algorithms are released through the free and open-source BFit package, which also includes a L2-norm squared optimization routine applicable to any square-integrable scalar function.

Calculation of One- and Two-Center Overlap Like Quantum Similarity Integrals over ψ^((α)) Exponential Type Functions

One- and two-center overlap like quantum similarity integrals over ψ^((α)) functions are evaluated using the one-center expansion method. These integrals are expressed through expansion coefficients and usual two-center overlap integrals. The expressions derived in this work have no restrictions for the values of orbital parameters, quantum numbers and internuclear distances. Since these integrals of ψ^((α)) functions are calculated for the first time, the comparison is made with literature values obtained from Slater type functions. The computation results are in good agreement with literature values. The algorithm presented in this study could be useful when exponential type functions are employed in quantum similarity measures of atomic and molecular systems.

Exponential type orbitals with hyperbolic cosine function basis sets for isoelectronic series of the atoms Be to Ne

Abstract Exponential type orbital with hyperbolic cosine basis functions, proposed recently for Hartree–Fock–Roothaan calculations of neutral atoms, are studied in detail for the calculations of isoelectronic series of atoms from Be to Ne. Calculations are performed for the neutral and the first 20 cationic members of the isoelectronic series of each atom in its ground state. Three of the most popular exponential type orbitals (Slater type functions, B functions and ψ (α) functions with α = 2) are combined with modified hyperbolic cosine function cosh(βr + γ) to improve the basis function quality within the minimal basis sets framework. Performances of the basis functions are compared with each other by using the same number of variational parameters in them. The obtained results are also compared with numerical Hartree–Fock and extended Slater type basis set results. The presented accuracy of the minimal basis descriptions of atomic systems supports the usage of these unconventional basis functions in electronic structure and property calculations.

Energy calculations of the (2P2 1D); (3D2 1G) and (4F2 1I) doubly excited states of helium isoelectronic sequence (z ≤ 20) via the modified atomic orbital theory

We report in this paper the total energies of the 2p2 1D, 3d2 1G, 4f2 1I doubly excited states of Helium isoelectronic sequence with nuclear charge Z ≤ 20. Calculations are performed using the Modified Atomic Orbital Theory (MAOT) [1;2] in the framework of a variational procedure. The purpose of this study required a mathematical development of the Hamiltonian applied to Slater-type wave function [3] combining with Hylleraas-type wave function [4]. The study leads to analytical expressions which are carried out under special MAXIMA computational program. This proposed MAOT variational procedure, leads to accurate results in good agreement as well as with available other theoretical results than experimental data. In the present work, a new correlated wave function is presented to express analytically the total energies for each 2p2 1D, 3d2 1G, 4f 2 1I Doubly Excited States (DES) in the He-like systems. The present accurate data may be a useful guideline for future experimental and theoretical studies in the (nl2 ) systems.

Electron nuclear dynamics with plane wave basis sets: complete theory and formalism.

Electron nuclear dynamics (END) is an ab initio quantum dynamics method that adopts a time-dependent, variational, direct, and non-adiabatic approach. The simplest-level (SL) END (SLEND) version employs a classical mechanics description for nuclei and a Thouless single-determinantal wave function for electrons. A higher-level END version, END/Kohn-Sham density functional theory, improves the electron correlation description of SLEND. While both versions can simulate various types of chemical reactions, they have difficulties to simulate scattering/capture of electrons to/from the continuum due to their reliance on localized Slater-type basis functions. To properly describe those processes, we formulate END with plane waves (PWs, END/PW), basis functions able to represent both bound and unbound electrons. As extra benefits, PWs also afford fast algorithms to simulate periodic systems, parametric independence from nuclear positions and momenta, and elimination of basis set linear dependencies and orthogonalization procedures. We obtain the END/PW formalism by extending the Thouless wave function and associated electron density to periodic systems, expressing the energy terms as functionals of the latter entities, and deriving the energy gradients with respect to nuclear and electronic variables. END/ PW has a great potential to simulate electron processes in both periodic (crystal) and aperiodic (molecular) systems (the latter in a supercell approach). Following previous END studies, END/PW will be applied to electron scattering processes in proton cancer therapy reactions.

Modified Atomic Orbital Calculations of Energy of the (2<i>s</i><sup>2</sup> <sup>1</sup><i>S</i>) Ground-State, the (2<i>p</i><sup>2</sup> <sup>1</sup><i>D</i>); (3<i>d</i><sup>2</sup> <sup>1</sup><i>D</i>) and (4<i>f</i><sup>2</sup> <sup>1</sup><i>I</i>) Doubly Excited States of Helium Isoelectronic Sequence from H<sup>-</sup> to Ca<sup>18+</sup>

We report in this paper the ground-state energy 2s2 1S and total energies of doubly excited states 2p2 1D, 3d2 1D, 4f2 1I of the Helium isoelectronic sequence from H- to Ca18+. Calculations are performed using the Modified Atomic Orbital Theory (MAOT) in the framework of a variational procedure. The purpose of this study required a mathematical development of the Hamiltonian applied to Slater-type wave function [1] combining with Hylleraas-type wave function [2]. The study leads to analytical expressions which are carried out under special MAXIMA computational program. This first proposed MAOT variational procedure, leads to accurate results in good agreement as well as with available other theoretical results than experimental data. In the present work, a new correlated wave function is presented to express analytically the total energies for the 2s2 1S ground state and each doubly 2p2 1D, 3d2 1D, 4f2 1I excited states in the He-like systems.The present accurate data may be a useful guideline for future experimental and theoretical studies in the (nl2) systems.

Comment on “Fourier transform of hydrogen-type atomic orbitals”

Podolsky and Pauling (Phys. Rev. 34, 109 (1929) doi: 10.1103/PhysRev.34.109 ) were the first ones to derive an explicit expression for the Fourier transform of a bound-state hydrogen eigenfunction. Yükçü and Yükçü (Can. J. Phys. 96, 724 (2018) doi: 10.1139/cjp-2017-0728 ), who were apparently unaware of the work of Podolsky and Pauling or of the numerous other earlier references on this Fourier transform, proceeded differently. They expressed a generalized Laguerre polynomial as a finite sum of powers, or equivalently, they expressed a bound-state hydrogen eigenfunction as a finite sum of Slater-type functions. This approach looks very simple, but it leads to comparatively complicated expressions that cannot match the simplicity of the classic result obtained by Podolsky and Pauling. It is, however, possible to reproduce not only Podolsky and Pauling’s formula for the bound-state hydrogen eigenfunction, but to obtain results of similar quality also for the Fourier transforms of other, closely related, functions, such as Sturmians, Lambda functions, or Guseinov’s functions, by expanding generalized Laguerre polynomials in terms of so-called reduced Bessel functions.

Structure and spectrum of the hydrated electron. A combined quantum chemical statistical mechanical simulation

In the current work, in order to model a free electron solvated in water, we have implemented a method to represent the molecular charge distribution as a set of Slater-type functions within the framework of a QM/MM method called QMSTAT. Also, it has been introduced in this method a new approach to calculate the intermolecular induction energy in the presence of very strong electric fields. Unlike most previous potentials, our method uses the ab initio energies of the electron-water complex as reference to obtain the few fitted parameters that the model needs, instead of the ab initio potential of the electron.A Metropolis-Monte Carlo simulation of 15 million steps has been carried out in a system that consists of an electron solvated in 200 water molecules explicitly described. The obtained results predict a vertical electron detachment energy of 3.30 eV and a radius of gyration of 2.38 Å, which are in very good agreement with the experimental data of 3.27 ± 0.10 eV and 2.45 Å, respectively. We have also obtained the absorption spectrum of the hydrated electron from our simulation. The maximum of the spectrum agrees very well with the experimental result and the experimental width at half-maximum is only 0.11 eV wider than the calculated curve. The calculations predict the same blue tail as the experimental result but with a less smooth shape. Also, our results indicate that the transfer of intensity from the transition arriving to the highest of the three p states to transitions arriving to higher-lying states contributes to the intensity enhancement of the blue tail of the spectrum. Regarding the solvated structure of the free electron, we have found that if the basis center is used as the electron site to calculate the correlation functions, the results predict a structure where the electron places itself on top of a central water molecule surrounded by the remaining water molecules in a very organized and stable arrangement. However, if the maximum of the electron density is used as electron site the obtained picture shows an electron located in a very fluctuating cavity that collapses and reforms continuously.

Accurate computation of screened Coulomb potential integrals in numerical Hartree–Fock programs

The frequently-used Zk and Yk functions in the framework of numerical Hartree–Fock method are formulated for atomic systems embedded in weakly-coupled plasmas modeled by Debye–Hückel potential. The Gegenbauer expansion is used to decompose the screened interelectronic Coulomb potential. Two numerical methods, the integration and differentiation ones, are developed to accurately and efficiently calculate the Zk and Yk integrals with each one showing special advantages in the implementation of numerical Hartree–Fock programs. Comparing to the analytical results specialized for Slater-type radial functions, it is generally concluded that the integration method possesses more accuracy and robustness than the differentiation one. Calculations on the two-electron C4+ ion in three different screening models are performed to demonstrate the applicability and stability of present methods in a wider range of screening parameters than existing Hartree–Fock calculations. With benchmark Hartree–Fock energies obtained in this work and accurate non-relativistic energies calculated based on Hylleraas-CI basis functions, it is found that the correlation energies are smallest in the fully screening model.

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.

Numerical calculation of the Spherical Bessel Transform from Gaussian quadrature in the complex-plane

A numerical method for the calculation of spherical Bessel transforms via Gaussian quadrature in the complex plane is presented. The method is evaluated by transforming Slater and Gaussian-type spherical coordinate basis functions, used in electronic structure calculations, from position space to momentum space. The feasibility and efficiency of the method is explored for different regions of momenta and different orders of the spherical Bessel transform. The results illustrate that in general the method performs very well in the large p regions when applied to the Slater-type functions. On the other hand, a parameter has to be introduced for the application to Gaussian-type functions in order to avoid cancellation. In this case, the method performs well but accuracy is lost at larger p. Use of the parameter in application to Slater functions yields accurate results for all regions of p.

Efficient algorithm for expanding theoretical electron densities in canterakis-zernike functions.

An algorithm for the efficient computation of Canterakis-Zernike moments of theoretically computed molecular electron densities and rotationally invariant Fingerprint indices derived from them is reported. The algorithm is suitable for any density expressed in terms of Gaussian- or Slater-type functions within the Linear Combination of Atomic Orbitals framework at any level of computation. Electron density is expressed as a one-center expansion of real regular spherical harmonics times radial factors by means of translation techniques, which facilitates the efficient computation of the moments in terms of a single one-dimension numerical integration. The performance of the algorithm is analyzed showing that the computation of radial factors in the quadrature points is responsible for almost all computational time. The procedure is applicable to any density obtained with standard packages for molecular structure calculations. © 2018 Wiley Periodicals, Inc.

Tucker Tensor Analysis of Matérn Functions in Spatial Statistics

Abstract In this work, we describe advanced numerical tools for working with multivariate functions and for the analysis of large data sets. These tools will drastically reduce the required computing time and the storage cost, and, therefore, will allow us to consider much larger data sets or finer meshes. Covariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to compute and store, especially in three dimensions. Therefore, we approximate covariance functions by cheap surrogates in a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of Matérn- and Slater-type functions with varying parameters and demonstrate numerically that their approximations exhibit exponentially fast convergence. We prove the exponential convergence of the Tucker and canonical approximations in tensor rank parameters. Several statistical operations are performed in this low-rank tensor format, including evaluating the conditional covariance matrix, spatially averaged estimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse, and Cholesky decomposition of a large covariance matrix. Low-rank tensor approximations reduce the computing and storage costs essentially. For example, the storage cost is reduced from an exponential 𝒪 ⁢ ( n d ) {\mathcal{O}(n^{d})} to a linear scaling 𝒪 ⁢ ( d ⁢ r ⁢ n ) {\mathcal{O}(drn)} , where d is the spatial dimension, n is the number of mesh points in one direction, and r is the tensor rank. Prerequisites for applicability of the proposed techniques are the assumptions that the data, locations, and measurements lie on a tensor (axes-parallel) grid and that the covariance function depends on a distance, ∥ x - y ∥ {\|x-y\|} .

Jj-Coupling-based atomic self-consistent-field calculations with relativistic effective core potentials and two-component spinors

A self-consistent-field (SCF) program for the calculation of atomic energies and wave functions defined in jj-coupling using two-component atomic spinors and relativistic effective core potentials (RECPs) is described. The code is based on the linear combination of atomic orbitals SCF algorithm for atomic states defined in LS-coupling developed by Roothaan and Bagus. Hamiltonian matrix elements with respect to one- and two-electron operators, including RECPs, are calculated for two-component atomic spinor basis functions of either Gaussian-type orbitals (GTOs) or Slater-type orbitals (STOs). Electronic states are defined as eigenfunctions of the total angular momentum squared operator and tables of the required vector coupling coefficients that define such pure states are provided. In addition, one or more GTO expansions of large- and small-core RECPs and their corresponding GTO basis sets are supplied for all elements Z=3 through Z=118. Optimized two-component basis sets of STOs or GTOs can be calculated for use in molecular structure codes based on RECPs and relativistic electronic structure theory. Atomic asymptotic state energies at the SCF level of theory for analysis of molecular dissociation limits can be studied. Program summaryProgram title: jjatomProgram Files doi:http://dx.doi.org/10.17632/zs4twp8r67.1Licensing provisions: GPLv3Programming language: Fortran 90Nature of problem: Relativistic atomic energies and wave functions; optimization of two component spinor basis sets.Solution method: Atomic spinors defined in jj-coupling and expansions in two-component spinor basis functions of Gaussian- or Slater-type functions. Self-consistent field optimization of the total energy. Used for optimization of two-component atomic spinor basis sets and calculations of valence spectra of heavy elements.

English

The paper focuses on a low-rank tensor structured representation of Slatertype and Hydrogen-like orbital basis functions that can be used in electronic structure calculations. Standard packages use the Gaussian-type basis functions which allow us to analytically evaluate the necessary integrals. Slater-type and Hydrogen-like orbital functions are physically more appropriate, but they are not analytically integrable. A numerical integration is too expensive when using the standard discretization techniques due the dimensionality of the problem. However, it can be effectively performed using the tensor representation of basis functions. Furthermore, this approach can take advantage of parallel computing.

Tensor Approximation of Slater-Type Orbital Basis Functions

This paper deals with a tensor representation of the Slater-type orbital basis functions. Localized basis systems are primarily used in electronic structure calculations. A choice of the system is usually limited to Gaussian-type orbitals due to the impossibility of the analytical evaluation of necessary integrals using other basis types. Unfortunately, it is not possible to use direct discretization techniques due to the dimensionality of the problem so the numerical integration is problematic. Tensor Numerical Methods overcome this problem by using special data representations, which are discussed. Finally, it is demonstrated how to use these methods to construct a tensor approximation of Slater-type orbital basis functions including an error estimate and its numerical verification.

Angular distributions in the double ionization of DNA bases by electron impact

Ab initio calculations of the five-fold differential cross sections for electron-impact double ionization of thymine, cytosine, adenine and guanine are performed in the first Born approximation for an incident energy close to 5500 eV. The wavefunctions of the DNA bases are constructed using the multi-center wave functions from the Gaussian 03 program. These multi-center wave functions are converted into single-center expansions of Slater-type functions. For the final state, the two ejected electrons are described by two Coulomb wave functions. The electron–electron repulsion between the two ejected electrons is also taken into account. Mechanisms of the double ionization are discussed for each case and the best choices of the kinematical parameters are determined for next experiments.