Gaussian-type Orbital Basis Sets Research Articles

Implementation of time-dependent Hartree–Fock in real space

Abstract Time-dependent Hartree–Fock (TDHF) is one of the fundamental post-Hartree–Fock (HF) methods to describe excited states. In its Tamm-Dancoff form, equivalent to Configuration Interaction Singles, it is still widely used and particularly applicable to big molecules where more accurate methods may be unfeasibly expensive. However, it is rarely implemented in real space, mostly because of the expensive nature of the exact-exchange potential in real space. Compared to widely used Gaussian-type orbitals (GTO) basis sets, real space often offers easier implementation of equations and more systematic convergence of Rydberg states, as well as favorable scaling, effective domain parallelization, flexible boundary conditions, and ability to treat model systems. We implemented TDHF in the Octopus real-space code as a step toward linear-response hybrid time-dependent density-functional theory (TDDFT), other post-HF methods, and ensemble density-functional theory methods involving exact exchange. Calculation of HF’s non-local exact exchange is very expensive in real space. We overcome this limitation with Octopus’ implementation of Adaptively Compressed Exchange, and find the appropriate mixing scheme and starting point to complete the ground-state calculation in a practical amount of time, and thus enable TDHF. We compared our results to those from GTOs on a set of small molecules and confirmed close agreement of results, though with larger deviations than in the case of semi-local TDDFT. We find that convergence of TDHF demands a finer real-space grid than semi-local TDDFT. We also present the subtleties in benchmarking a real-space calculation against GTOs, relating to Rydberg and vacuum states.

On the geometry dependence of the nuclear magnetic resonance chemical shift of mercury in thiolate complexes: A relativistic density functional theory study.

Thiolate containing mercury(II) complexes of the general formula [Hg(SR) ] have been of great interest since the toxicity of mercury was recognized. 199Hg nuclear magnetic resonance spectroscopy (NMR) is a powerful tool for characterization of mercury complexes. In this work, the Hg shielding constants in a series of [Hg(SR) ] complexes are therefore investigated computationally with particular emphasis on their geometry dependence. Geometry optimizations and NMR chemical shift calculations are performed at the density functional theory (DFT) level with both the zeroth-order regular approximation (ZORA) and four-component relativistic methods. The four exchange-correlation (XC) functionals PBE0, PBE, B3LYP, and BLYP are used in combination with either Dyall's Gaussian-type (GTO) or Slater-type orbitals (STOs) basis sets. Comparing ZORA and four-component calculations, one observes that the calculated shielding constants for a given molecular geometry have a constant difference of 1070 ppm. This confirms that ZORA is an acceptable relativistic method to compute NMR chemical shifts. The combinations of four-component/PBE0/v3z and ZORA/PBE0/QZ4P are applied to explore the geometry dependence of the isotropic shielding. For a given coordination number, the distance between mercury and sulfur is the key factor affecting the shielding constant, while changes in bond and dihedral angles and even different side groups have relatively little impact.

Band structures and Z2 invariants of two-dimensional transition metal dichalcogenide monolayers from fully relativistic Dirac-Kohn-Sham theory using Gaussian-type orbitals

Two-dimensional (2D) materials exhibit a wide range of remarkable phenomena, many of which owe their existence to the relativistic spin-orbit coupling (SOC) effects. To understand and predict properties of materials containing heavy elements, such as the transition-metal dichalcogenides (TMDs), relativistic effects must be taken into account in first-principles calculations. We present an all-electron method based on the four-component Dirac Hamiltonian and Gaussian-type orbitals (GTOs) that overcomes complications associated with linear dependencies and ill-conditioned matrices that arise when diffuse functions are included in the basis. Until now, there has been no systematic study of the convergence of GTO basis sets for periodic solids either at the nonrelativistic or the relativistic level. Here we provide such a study of relativistic band structures of the 2D TMDs in the hexagonal (2H), tetragonal (1T), and distorted tetragonal (1T') structures, along with a discussion of their SOC-driven properties (Rashba splitting and $\mathbb{Z}_2$ topological invariants). We demonstrate the viability of our approach even when large basis sets with multiple basis functions involving various valence orbitals (denoted triple- and quadruple-$\zeta$) are used in the relativistic regime. Our method does not require the use of pseudopotentials and provides access to all electronic states within the same framework. Our study paves the way for direct studies of material properties, such as the parameters in spin Hamiltonians, that depend heavily on the electron density near atomic nuclei where relativistic and SOC effects are the strongest.

All-Electron BSE@GW Method for K-Edge Core Electron Excitation Energies.

We present an accurate computational approach to calculate absolute K-edge core electron excitation energies as measured by X-ray absorption spectroscopy. Our approach employs an all-electron Bethe-Salpeter equation (BSE) formalism based on GW quasiparticle energies (BSE@GW) using numeric atom-centered orbitals (NAOs). The BSE@GW method has become an increasingly popular method for the computation of neutral valence excitation energies of molecules. However, it was so far not applied to molecular K-edge excitation energies. We discuss the influence of different numerical approximations on the BSE@GW calculation and employ in our final setup (i) exact numeric algorithms for the frequency integration of the GW self-energy, (ii) G0W0 and BSE starting points with ∼50% of exact exchange, (iii) the Tamm-Dancoff approximation and (iv) relativistic corrections. We study the basis set dependence and convergence with common Gaussian-type orbital and NAO basis sets. We identify the importance of additional spatially confined basis functions as well as of diffuse augmenting basis functions. The accuracy of our BSE@GW method is assessed for a benchmark set of small organic molecules, previously used for benchmarking the equation-of-motion coupled cluster method [Peng et al., J. Chem. Theory Comput., 2015, 11, 4146], as well as the medium-sized dibenzothiophene (DBT) molecule. Our BSE@GW results for absolute excitation energies are in excellent agreement with the experiment, with a mean average error of only 0.63 eV for the benchmark set and with errors <1 eV for the DBT molecule.

Tuning the quasi-harmonic treatment of crystalline ionic liquids within the density functional theory.

Five ionic liquids are selected for benchmarking the performance of quasi-harmonic density functional theory (DFT) calculations of structural, phonon, and thermodynamic properties of their crystals. Data predicted by individual computational setups are sorted, establishing a distinct hierarchy among the first-principles approaches. PBE-D3 and B3LYP-D3 functionals are coupled with various plane wave and Gaussian-type orbital (GTO) basis sets. Propagation of the basis set superposition error and of the imperfections of both functionals into finite-temperature properties is discussed in detail. PBE-D3 together with a triple-zeta GTO basis set often yields the most accurate predictions of predicted molar volume and heat capacity with errors at 1% and 8%, respectively, representing the state-of-the-art for quasi-harmonic DFT calculations for crystalline ionic liquids. Fortuitous error cancellation between the basis-set superposition (overbinding) and PBE imperfection (overexpanding) strongly affects the overall accuracy, unlike the case of B3LYP/GTO calculations, impeding systematic convergence of the methodology towards higher accuracy.

Approaching the basis set limit in Gaussian-orbital-based periodic calculations with transferability: Performance of pure density functionals for simple semiconductors.

Simulating solids with quantum chemistry methods and Gaussian-type orbitals (GTOs) has been gaining popularity. Nonetheless, there are few systematic studies that assess the basis set incompleteness error (BSIE) in these GTO-based simulations over a variety of solids. In this work, we report a GTO-based implementation for solids and apply it to address the basis set convergence issue. We employ a simple strategy to generate large uncontracted (unc) GTO basis sets that we call the unc-def2-GTH sets. These basis sets exhibit systematic improvement toward the basis set limit as well as good transferability based on application to a total of 43 simple semiconductors. Most notably, we found the BSIE of unc-def2-QZVP-GTH to be smaller than 0.7 mEh per atom in total energies and 20 meV in bandgaps for all systems considered here. Using unc-def2-QZVP-GTH, we report bandgap benchmarks of a combinatorially designed meta-generalized gradient approximation (mGGA) functional, B97M-rV, and show that B97M-rV performs similarly (a root-mean-square-deviation of 1.18eV) to other modern mGGA functionals, M06-L (1.26eV), MN15-L (1.29eV), and Strongly Constrained and Appropriately Normed (SCAN) (1.20eV). This represents a clear improvement over older pure functionals such as local density approximation (1.71eV) and Perdew-Burke-Ernzerhof (PBE) (1.49eV), although all these mGGAs are still far from being quantitatively accurate. We also provide several cautionary notes on the use of our uncontracted bases and on future research on GTO basis set development for solids.

Simulated infrared and Raman spectroscopy, complex dielectric function and refractive index dataset of monoclinic C2/m stoichiometric clinochlore Mg6Si4O10(OH)8 as obtained from Density Functional Theory.

This article reports a simulated dataset of the vibrational (infrared and Raman) and optical properties (complex dielectric function and refractive index) of clinochlore, an important mineral belonging to the phyllosilicate family [1]. The data here reported were calculated from ab initio Density Functional Theory (DFT) simulations at B3LYP level, including a correction for the dispersive forces (B3LYP-D* approach) and all-electron Gaussian-type orbitals basis sets. This dataset was calculated between 0 cm–1 and 4000 cm–1 and comprises infrared, reflectance and Raman spectra, frequency-dependent complex dielectric function and complex refractive index of clinochlore. The data was validated against available experimental spectroscopic results reported in literature and can be of help in several application fields, for instance fundamental georesource exploration and exploitation, in applied mineralogy, geology, material science, and as a reference to assess the quality of other theoretical approaches.

Static Polarizabilities at the Basis Set Limit: A Benchmark of 124 Species.

Benchmarking molecular properties with Gaussian-type orbital (GTO) basis sets can be challenging, because one has to assume that the computed property is at the complete basis set (CBS) limit, without a robust measure of the error. Multiwavelet (MW) bases can be systematically improved with a controllable error, which eliminates the need for such assumptions. In this work, we have used MWs within Kohn–Sham density functional theory to compute static polarizabilities for a set of 92 closed-shell and 32 open-shell species. The results are compared to recent benchmark calculations employing the GTO-type aug-pc4 basis set. We observe discrepancies between GTO and MW results for several species, with open-shell systems showing the largest deviations. Based on linear response calculations, we show that these discrepancies originate from artifacts caused by the field strength and that several polarizabilies from a previous study were contaminated by higher order responses (hyperpolarizabilities). Based on our MW benchmark results, we can affirm that aug-pc4 is able to provide results close to the CBS limit, as long as finite difference effects can be controlled. However, we suggest that a better approach is to use MWs, which are able to yield precise finite difference polarizabilities even with small field strengths.

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.

Accurate and balanced anisotropic Gaussian type orbital basis sets for atoms in strong magnetic fields.

In high magnetic field calculations, anisotropic Gaussian type orbital (AGTO) basis functions are capable of reconciling the competing demands of the spherically symmetric Coulombic interaction and cylindrical magnetic (B field) confinement. However, the best available a priori procedure for composing highly accurate AGTO sets for atoms in a strong B field [W. Zhu et al., Phys. Rev. A 90, 022504 (2014)] yields very large basis sets. Their size is problematical for use in any calculation with unfavorable computational cost scaling. Here we provide an alternative constructive procedure. It is based upon analysis of the underlying physics of atoms in B fields that allow identification of several principles for the construction of AGTO basis sets. Aided by numerical optimization and parameter fitting, followed by fine tuning of fitting parameters, we devise formulae for generating accurate AGTO basis sets in an arbitrary B field. For the hydrogen iso-electronic sequence, a set depends on B field strength, nuclear charge, and orbital quantum numbers. For multi-electron systems, the basis set formulae also include adjustment to account for orbital occupations. Tests of the new basis sets for atoms H through C (1 ≤ Z ≤ 6) and ions Li+, Be+, and B+, in a wide B field range (0 ≤ B ≤ 2000 a.u.), show an accuracy better than a few μhartree for single-electron systems and a few hundredths to a few mHs for multi-electron atoms. The relative errors are similar for different atoms and ions in a large B field range, from a few to a couple of tens of millionths, thereby confirming rather uniform accuracy across the nuclear charge Z and B field strength values. Residual basis set errors are two to three orders of magnitude smaller than the electronic correlation energies in multi-electron atoms, a signal of the usefulness of the new AGTO basis sets in correlated wavefunction or density functional calculations for atomic and molecular systems in an external strong B field.

Calculation of Araki-Sucher correction for many-electron systems

In this paper we consider the evaluation of the Araki-Sucher correction for arbitrary many-electron atomic and molecular systems. This contribution appears in the leading order quantum electrodynamics corrections to the energy of a bound state. The conventional one-electron basis set of Gaussian-type orbitals (GTOs) is adopted; this leads to two-electron matrix elements which are evaluated with help of generalised the McMurchie-Davidson scheme. We also consider the convergence of the results towards the complete basis set. A rigorous analytic result for the convergence rate is obtained and verified by comparing with independent numerical values for the helium atom. Finally, we present a selection of numerical examples and compare our results with the available reference data for small systems. In contrast with other methods used for the evaluation of the Araki-Sucher correction, our method is not restricted to few-electron atoms or molecules. This is illustrated by calculations for several many-electron atoms and molecules.

Effects of fluorine content on the elastic behavior of topaz [Al2SiO4(F,OH)2

In this work, we modeled the structure, the compressional behavior and the physical properties of topaz over six different fluorine contents and a wide range of pressure, using a quantum mechanical approach based on periodic boundary conditions. We adopted the density functional theory using the B3LYP functional and all-electron Gaussian-type orbitals basis sets. An atomic level description of the athermal ( T = 0 K) pressure-induced structural modification of topaz is provided. From the compression results we obtained the athermal bulk modulus ( K T0 ), its first derivative ( K ′) and the athermal volume at zero pressure ( V 0 ) by a third-order Birch-Murnaghan equation fit. The results show that K T0 increases with fluorine content. The compressional pattern is anisotropic, as observed by the axial compressibility and second-order elastic constants calculations. We observed that the compression involves three different mechanism, polyhedral contraction, polyhedral tilting and hydrogen bonding, all of them influenced, with different extent, by the fluorine content in topaz. Recent experimental results obtained by single-crystal X-ray and neutron diffraction of specific topaz compositions are in very good agreement with our simulations, which further extend the knowledge of the structural and elastic properties of topaz over a wider range of fluorine content.

Angle-dependent strong-field molecular ionization rates with tuned range-separated time-dependent density functional theory.

Strong-field ionization and the resulting electronic dynamics are important for a range of processes such as high harmonic generation, photodamage, charge resonance enhanced ionization, and ionization-triggered charge migration. Modeling ionization dynamics in molecular systems from first-principles can be challenging due to the large spatial extent of the wavefunction which stresses the accuracy of basis sets, and the intense fields which require non-perturbative time-dependent electronic structure methods. In this paper, we develop a time-dependent density functional theory approach which uses a Gaussian-type orbital (GTO) basis set to capture strong-field ionization rates and dynamics in atoms and small molecules. This involves propagating the electronic density matrix in time with a time-dependent laser potential and a spatial non-Hermitian complex absorbing potential which is projected onto an atom-centered basis set to remove ionized charge from the simulation. For the density functional theory (DFT) functional we use a tuned range-separated functional LC-PBE*, which has the correct asymptotic 1/r form of the potential and a reduced delocalization error compared to traditional DFT functionals. Ionization rates are computed for hydrogen, molecular nitrogen, and iodoacetylene under various field frequencies, intensities, and polarizations (angle-dependent ionization), and the results are shown to quantitatively agree with time-dependent Schrödinger equation and strong-field approximation calculations. This tuned DFT with GTO method opens the door to predictive all-electron time-dependent density functional theory simulations of ionization and ionization-triggered dynamics in molecular systems using tuned range-separated hybrid functionals.

Semiconducting and optical properties of selected binary compounds by linear response DFT+U and hybrid functional methods

This contribution focuses on bulk band gap and dielectric response calculations for copper phosphide Cu3P, using hybrid functionals and linear response DFT+U methods, considering both plane wave and Gaussian-type orbitals basis sets. The semiconducting state of Cu3P is reproduced in good quantitative agreement with experiments when using these methods, at variance with the description provided by standard DFT calculations based on GGA functionals, predicting Cu3P to be a semimetal. Additional DFT+U calculations were performed on zinc blende cadmium selenide CdSe, to further investigate the effect of U on semiconducting binary compounds. For both Cu3P and CdSe, the materials’ band gap depends strongly on the on-site repulsion of the nonmetal unfilled p-orbitals, while hybrid functionals still perform well. This suggests that the use of the Hubbard term in DFT+U calculations should be carefully checked for all elements when dealing with semiconducting binary compounds.

Implementation of density functional embedding theory within the projector-augmented-wave method and applications to semiconductor defect states

We report a new implementation of the density functional embedding theory (DFET) in the VASP code, using the projector-augmented-wave (PAW) formalism. Newly developed algorithms allow us to efficiently perform optimized effective potential optimizations within PAW. The new algorithm generates robust and physically correct embedding potentials, as we verified using several test systems including a covalently bound molecule, a metal surface, and bulk semiconductors. We show that with the resulting embedding potential, embedded cluster models can reproduce the electronic structure of point defects in bulk semiconductors, thereby demonstrating the validity of DFET in semiconductors for the first time. Compared to our previous version, the new implementation of DFET within VASP affords use of all features of VASP (e.g., a systematic PAW library, a wide selection of functionals, a more flexible choice of U correction formalisms, and faster computational speed) with DFET. Furthermore, our results are fairly robust with respect to both plane-wave and Gaussian type orbital basis sets in the embedded cluster calculations. This suggests that the density functional embedding method is potentially an accurate and efficient way to study properties of isolated defects in semiconductors.

Unconventional bond functions for quantum chemical calculations

New types of bond function (BF) basis sets are proposed and tested for quantum chemical applications. First, BF basis sets constituted of conventional Gaussiantype orbitals (GTO) are considered. Both the exponents and the positions of the BFs are optimized, but, in contrast to previous studies, the position of each BF shell is varied separately. Second, new types of basis functions, the general ellipsoidal Gaussian-type orbitals (EGTOs), are proposed for quantum chemical applications. The EGTOs are distorted spherical GTOs and, as such, are expected to be well suited for describing the polarized charge densities in molecular environments. EGTOs can be used either as atom-centered (AC) basis functions or as BFs. In this study, the latter possibility is explored, and BF basis sets including EGTOs are optimized and compared to those containing only conventional GTO BFs. The performance of the developed GTO and EGTO BF basis sets is assessed for Hartree–Fock and density functional calculations against conventional AC GTO basis sets. Our results show that using GTO BF basis sets, the results are significantly improved, while the number of the basis functions can be decreased by about 10 %, which is not dramatic; however, the average angular momentum quantum number in the BF sets is significantly lower. The accuracy of the computed energies can be further increased by about 15 % if EGTO BFs are used.

In silico infrared and Raman spectroscopy under pressure: the case of CaSnO3 perovskite.

The CaSnO3 perovskite is investigated under geochemical pressure, up to 25 GPa, by means of periodic ab initio calculations performed at B3LYP level with local Gaussian-type orbital basis sets. Structural, elastic, and spectroscopic (phonon wave-numbers, infrared and Raman intensities) properties are fully characterized and discussed. The evolution of the Raman spectrum of CaSnO3 under pressure is reported to remarkably agree with a recent experimental determination [J. Kung, Y. J. Lin, and C. M. Lin, J. Chem. Phys. 135, 224507 (2011)] as regards both wave-number shifts and intensity changes. All phonon modes are symmetry-labeled and bands assigned. The single-crystal total spectrum is symmetry-decomposed into the six directional spectra related to the components of the polarizability tensor. The infrared spectrum at increasing pressure is reported for the first time and its main features discussed. All calculations are performed using the Crystal14 program, taking advantage of the new implementation of analytical infrared and Raman intensities for crystalline materials.

Ab initio modeling of wall structure and shape in perovskite-based nanotubes

A large-scale first-principles simulation of the structure and stability of SrZrO3 and BaZrO3 single- and double-wall nanotubes with different chiralities and diameters was performed using the periodic PBE0 method and the basis set of localized Gaussian-type atomic orbitals. The initial structures of nanotubes were obtained by the rolling up of slabs cut from perovskite bulk phases and consisting of two or four alternating (001) ZrO2, SrO or BaO layers. Significant structural reconstruction was found in 4-layer singe-walled and in double-walled nanotubes. If the distance between the single-wall components of double-walled nanotubes is less than approximately 5.0Å, they inclined to merge to stable polyhedron-shaped tubular objects consisting of blocks with distorted cubic perovskite structure. A comparison of the data obtained with the results of our previous works shows that the stability of perovskite nanotubes with merged walls increases in the following sequence of the parent phases: SrZrO3<BaZrO3≈SrTiO3<BaTiO3. Calculated stability correlates with a ratio RII/RIV of ionic radii of group II and IV metals.

Investigation of the bulk and surface properties of CdSe: insights from theory.

CdSe, in the form of quantum dots, is a semiconductor material extensively used for photovoltaic applications. We have performed a comprehensive density functional theory investigation of the geometrical and electronic properties of CdSe bulk crystals (for both the wurtzite and zinc blende phases) as well as their wurtzite (10-10) surface. Several combinations of Gaussian-type orbital basis sets and exchange-correlation functionals have been tested with a periodic formalism. The computational method for further studies was selected on the basis of the comparison of computed geometrical and electronic properties to available experimental data for the bulk crystals of CdSe, as well as to additional projector-augmented wave calculations. In a second step, the so-defined protocol was used to successfully simulate CdSe wurtzite nanocrystals exposing their nonpolar (10-10) surface. Most importantly, we have shown that the presented computational protocol is accurate to model not only bulk crystals but also surfaces, thus providing a powerful theoretical tool to simulate the light harvesting components of quantum dot sensitized solar cells.

Density functional investigation of the thermophysical and thermochemical properties of talc [Mg3Si4O10(OH)2

The knowledge of the P, T behavior of talc is very important in mineralogical–petrological and geophysical research fields because talc can be considered a hydrous phase that can recycle water into the Earth’s mantle and also an important mineral in both industrial and technological applications. However, very few works have been presented to fully characterize the thermodynamic properties of this mineral, especially at atomic scale. In a previous work, we modeled the structural and mechanical properties of talc using the B3LYP-D* hybrid density functional, which included a correction for the dispersive forces and all-electron Gaussian-type orbital basis sets. The results were in good agreement with single-crystal X-ray and neutron diffraction experimental data. Here, we extend the investigation to the thermochemical and thermophysical properties of talc using the same density functional approach and the quasi-harmonic approximation, providing the thermal equation of state, the heat capacity and the entropy of the mineral at different P, T conditions.