Gaussian-type Orbital Basis Research Articles

Efficient Hartree–Fock exchange algorithm with Coulomb range separation and long-range density fitting

Separating the Coulomb potential into short-range and long-range components enables the use of different electron repulsion integral algorithms for each component. The short-range part can be efficiently computed using the analytical algorithm due to the locality in both the Gaussian-type orbital basis and the short-range Coulomb potentials. The integrals for the long-range Coulomb potential can be approximated with the density fitting method. A very small auxiliary basis is sufficient for the density fitting method to accurately approximate the long-range integrals. This feature significantly reduces the computational efforts associated with the N4 scaling in density fitting algorithms. For large molecules, the range separation and long-range density fitting method outperforms the conventional analytical integral evaluation scheme employed in Hartree–Fock calculations and provides more than twice the overall performance. In addition, this method offers a higher accuracy compared to conventional density fitting methods. The error in the Hartree–Fock energy can be easily reduced to 0.1 μEh per atom or smaller.

Lah Number Mediated Analytical Two‐center Two‐electron Integrals of Coulomb Green Function of H‐like Orbitals for 1Σ States

AbstractEnergetics of two‐center two‐electron (2c–2e) systems carry challenges in theoretical understanding of Schrödinger equation (SE) for well‐known divergence of Coulomb interactions and nuclear separation (R) in modified H‐like AOs, Slater type orbitals (STOs), Gaussian type orbitals (GTOs), B‐spline, Sturmian function and etc. employed to VBT and MOT. Certain elegant computational and analytical techniques were developed for STO, GTO and other square integrable basis set within Born‐Oppenheimer (BO) approximation. STOs and GTOs have an essential limitation of absence of radial nodes. Thus, analytical treatment has become an urge for H‐like AOs. We have considered the diatomic molecules only for the sake of simplicity. Employing Sheffer identity in associated Laguerre polynomial/Whittaker‐M function forms of H‐like AOs and transforming integrals into elliptic coordinates with two nuclei on two foci furnishes exact, analytical and simple Coulomb integrals (Js) in terms of R. Lah number originated from for nuclear coordinates only due to Sheffer identity shows that energetics of diatomic molecules can be anticipated as extremum function of R. Therefore, the optimization of potential energy surface (PES) of electrons as gradient of R may lead to σ‐bond formation. In this paper, we have developed diagonal Js for bound states of H2 molecule.

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.

Modeling surface spin polarization on ceria-supported Pt nanoparticles

In this work, we employ density functional theory simulations to investigate possible spin polarization of CeO2-(111) surface and its impact on the interactions between a ceria support and Pt nanoparticles. With a Gaussian type orbital basis, our simulations suggest that the CeO2-(111) surface exhibits a robust surface spin polarization due to the internal charge transfer between atomic Ce and O layers. In turn, it can lower the surface oxygen vacancy formation energy and enhance the oxide reducibility. We show that the inclusion of spin polarization can significantly reduce the major activation barrier in the proposed reaction pathway of CO oxidation on ceria-supported Pt nanoparticles. For metal-support interactions, surface spin polarization enhances the bonding between Pt nanoparticles and ceria surface oxygen, while CO adsorption on Pt nanoparticles weakens the interfacial interaction regardless of spin polarization. However, the stable surface spin polarization can only be found in the simulations based on the Gaussian type orbital basis. Given the potential importance in the design of future high-performance catalysts, our present study suggests a pressing need to examine the surface ferromagnetism of transition metal oxides in both experiment and theory.

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.

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.

Density Functional Theory for Molecular and Periodic Systems Using Density Fitting and Continuous Fast Multipole Methods.

An implementation of Kohn-Sham density functional theory within the TURBOMOLE program package with Gaussian-type orbitals (GTO) as basis functions is reported that treats molecular and periodic systems of any dimensionality on an equal footing. Its key component is a combination of density fitting/resolution of identity (DF) approximation and continuous fast multipole method (CFMM) applied for the electronic Coulomb term. This DF-CFMM scheme operates entirely in the direct space and partitions Coulomb interactions into far-field part evaluated using multipole expansions and near-field contribution calculated employing density fitting. Computational efficiency and favorable scaling behavior of our implementation approaching O(N) for the formation of Kohn-Sham matrix is demonstrated for various molecular and periodic systems including three-dimensional models with unit cells containing up to 640 atoms and 19072 GTO basis functions.

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.

Construction of complex STO-NG basis sets by the method of least squares and their applications

Electronic resonance state energies and photoionization cross sections of atoms and molecules are calculated with the complex basis function method by using mixture of appropriate complex basis functions representing one-electron continuum orbitals and the usual real basis functions for the remaining bound state orbitals. The choice of complex basis functions has long been a central difficulty in such calculations. To address this challenge, we constructed complex Slater-type orbital represented by N-term Gaussian-type orbitals (cSTO-NG) basis sets using the method of least squares. Three expansion schemes are tested: (1) expansion in complex Gaussian-type orbitals, (2) expansion in real Gaussian-type orbitals, and (3) expansion in even-tempered real Gaussian-type orbitals. By extending the Shavitt–Karplus integral transform expression to cSTO functions, we have established a mathematical foundation for these expansions. To demonstrate the efficacy of this approach, we have applied these basis sets to the calculation of the lowest Feshbach resonance of H2 and the photoionization cross section of the He atom including autoionization features due to doubly excited states. These calculations produce acceptably accurate results compared with past calculations and experimental data in all cases examined here.

Towards an accurate representation of the continuum in calculations of electron, positron and laser field interactions with molecules

Gaussian-type orbitals (GTOs) are the most common choice of basis functions in calculations of electronic structure of molecules, i.e. for the description of bound electrons. The main advantage of this approach is the analytic form of the multicentre molecular integrals. For the same reason GTOs have been adopted as basis functions for the description of the unbound particle in many ab-initio calculations of electron, positron and laser fields interacting with molecules. However, the accurate description of the unbound particle using GTOs may become very difficult and in some cases numerically unstable. We describe an approach for the representation of the continuum in which the unbound particle is described using a mixed GTO and B-spline basis set in a manner which exploits the best features of these functions. Analytical expressions for the GTO-only molecular integrals allow us to accurately represent the part of the wavefunction close to the target, while the B-splines enable us to represent accurately the wavefunction's tail, corresponding to the unbound particle, over a wide range of kinetic energies. The main challenge posed by this approach is the accurate and rapid numerical evaluation of a large number of mixed BTO/GTO molecular integrals. We propose a scheme for this integral calculation in which the overlap integrals between GTOs and B-spline functions play a central role and demonstrate that they can be calculated rapidly and accurately.

Spin-State-Corrected Gaussian-Type Orbital Basis Sets

Recently, we reported that the basis set has a profound influence on the computed values for spin-state splittings [J. Phys. Chem. A 2008, 112, 6384]. In particular, small Gaussian-type orbital (GTO) basis sets were shown to be unreliable for the prediction of them. Here, we report simple modifications of the small Pople-type Gaussian-type orbital basis sets (3-21G, 3-21G*, 6-31G, 6-31G*), which correct their faulty behavior for the spin-state energies. We have investigated the basis sets for a set of 13 first-row transition-metal complexes for which reliable reference data have been obtained at the OPBE/TZ2P(STO) level. For several systems, we have used single and double spin-contamination corrections to avoid ambiguity of the results because of spin contamination, that is, the energies and geometries were obtained for the pure spin states. The spin ground states as predicted by the spin-state-corrected GTO basis sets (s6-31G, s6-31G*) are in complete agreement with the reference Slater-type orbital (STO) data, while those of the original basis sets and a recent modification by Baker and Pulay (m6-31G*) are not for all cases. The spin-state-corrected GTO basis sets also improve upon the original and modified basis sets for the accuracy of geometry optimization, while the accuracy of the vibrational frequencies is as good or better. At a limited additional cost, one therefore obtains very reliable results for these important spin-state energies.

Importance of the Basis Set for the Spin-State Energetics of Iron Complexes

We have performed a systematic investigation of the influence of the basis set on relative spin-state energies for a number of iron compounds. In principle, with an infinitely large basis set, both Slater-type orbital (STO) and Gaussian-type orbital (GTO) series should converge to the same final answer, which is indeed what we observe for both vertical and relaxed spin-state splittings. However, we see throughout the paper that the STO basis sets give consistent and rapidly converging results, while the convergence with respect to the basis set size is much slower for the GTO basis sets. For example, the large GTO basis sets that give good results for the vertical spin-state splittings of compounds 1-3 (6-311+G**, Ahlrichs VTZ2D2P) fail for the relaxed spin-state splittings of compound 4 (where 1 is Fe-(PyPepS)2 (PyPepSH 2 = N-(2-mercaptophenyl)-2-pyridinecarboxamide), 2 is Fe(tsalen)Cl (tsalen = N, N'-ethylenebis-(thiosalicylideneiminato)), 3 is Fe(N(CH2-o-C6H4S) 3)(1-Me-imidazole), and 4 is FeFHOH). Very demanding GTO basis sets like Dunning's correlation-consistent (cc-pVTZ, cc-pVQZ) basis sets are needed to achieve good results for these relaxed spin states. The use of popular (Pople-type) GTO, effective core potentials basis set (ECPB), or mixed ECPB(Fe):GTO(rest) basis sets is shown to lead to substantial deviations (2-10 kcal/mol, 14-24 kcal/mol for 3-21G), in particular for the high spin states that are typically placed at too low energy. Moreover, the use of an effective core potential in the ECPB basis sets results in spin-state splittings that are systematically different from the STO-GTO results.

A comparison between plane wave and Gaussian-type orbital basis sets for hydrogen bonded systems: Formic acid as a test case

The formic acid molecule, its dimers, and its molecular crystal are adopted as test systems to compare results obtained with plane wave (PW) basis sets and norm-conserving pseudopotentials to all-electron Gaussian-type orbital (GTO) calculations. The CPMD and CRYSTAL06 codes, respectively, are applied with the PBE, PW91, and BLYP density functionals. Hydrogen bonding is the leading interaction in the dimers and the crystal. In the latter, dispersive and weak C-H...O interactions are also relevant. Irrespective of the adopted functional, for all considered structures PW and GTO results converge smoothly as a function of the quality of the adopted basis sets to the same values for structures, energies of interaction, and harmonic vibrational features. To achieve a high level of mutual agreement the use of GTO basis sets of at least of triple-zeta quality including one set of polarization functions and PW basis sets with a kinetic energy cutoff higher than 110 Ry is recommended. Pros and cons of both approaches for studying molecular crystals are also discussed.

Density-functional generalized-gradient and hybrid calculations of electromagnetic properties using Slater basis sets.

In this paper we extend our density-functional theory calculations, with generalized gradient approximation and hybrid functionals, using Slater-type orbitals (STOs), to the determination of second-order molecular properties. The key to the entire methodology involves the fitting of all STO basis function products to an auxiliary STO basis, through the minimization of electron-repulsion integrals. The selected properties are (i) dipole polarizabilities, (ii) nuclear magnetic shielding constants, and (iii) nuclear spin-spin coupling constants. In all cases the one-electron integrals involving STOs were evaluated by quadrature. The implementation for (ii) involved some complexity because we used gauge-including atomic orbitals. The presence of two-electron integrals on the right-hand side of the coupled equations meant that the fitting procedure had to be implemented. For (iii) in the hybrid case, fitting procedures were again required for the exchange contributions. For each property we studied a number of small molecules. We first obtained an estimate of the basis set limit using Gaussian-type orbitals (GTOs). We then showed how it is possible to reproduce these values using a STO basis set. For (ii) a regular TZ2P quality STO basis was adequate; for (i) the addition of one set of diffuse functions (determined by Slater's rules) gave the required accuracy; for (iii) it was necessary to add a set of 1s functions, including one very tight function, to give the desired result. In summary, we show that it is possible to predict second-order molecular properties using STO basis sets with an accuracy comparable with large GTO basis sets. We did not encounter any major difficulties with either the selection of the bases or the implementation of the procedures. Although the energy code (especially in the hybrid case) may not be competitive with a regular GTO code, for properties we find that STOs are more attractive.

Charge transfer and “band lineup” in molecular electronic devices: A chemical and numerical interpretation

We present first-principles based calculation of charge transfer and “band lineup” in molecular electronic devices using as an example the device formed by a phenyldithiolate molecule bridging two gold electrodes and local-spin-density-functional theory with a Gaussian-type orbital basis. We show that significant charge transfer from the metal to the molecule occurs, reflecting the partially ionic character of the sulfur–gold bond and localized in the interfacial region. Such charge transfer increases the electrostatic potential in the molecule which changes the molecular energy level structures. The interaction between the molecular orbitals under the self-consistent potential and the surface metal states determines the lineup of molecular levels relative to the metal Fermi level. We also discuss the implications of our work on device engineering at the molecular scale.

Quantum dynamical manifolds 5. Hydrogen mass-spacetime

The newly developed methods of quantum dynamical manifold theory (QDMT) are used to determine a theory of the differential geometry and topology of the quantum dynamical manifold of the hydrogen nuclear-electron system. This provides an ab initio theory of the structure of nuclear-atomic system including the quarks, gluons, and (W+,Z0,W−) in the proton and the atomic orbital electron in a way that treats the strong, electromagnetic, weak, and gravitational forces from a unified QDMT viewpoint. The calculations proposed resemble self-consistent Green's function ones in Fourier space. The quasi-particles provide the basis for computing the globally defined connection needed for determining the self-consistent gauge potentials. This approach allows the simultaneous determination of coupling constants and masses along with the quasi-particle amplitudes. All integrals needed can be analytically computed using a Gaussian-type orbital basis. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 369–393, 2000

Use of STOs in Hartree-Fock calculations: Error analysis and variance-minimized pseudospectral method

In this study it is demonstrated that STO (Slater-type orbital) basis sets are particularly well suited to pseudospectral Hartree–Fock calculations. The reduction of two-electron integrals, to ones that are (at worst) equivalent to a one-electron integral over three centers, eliminates the need for slowly convergent one-center expansions. This allows all integrals to be calculated quickly and accurately in either spherical or ellipsoidal coordinates. A new variance-minimized variant of the pseudospectral method is derived and applied to a number of small closed-shell molecules. The performance of the algorithm is assessed relative to purely spectral calculations employing STO and GTO (Gaussian-type orbital) basis sets. The pseudospectral operator is used to assess the errors contained in solutions found by the purely spectral method. The suitability of a number of different de-aliasing set types is also examined. Orthogonal sets of hydrogen-like eigenfunctions were found to be optimal. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1537–1548, 1999

Pseudonatural Orbitals as a Basis for the Superposition of Configurations. I. He2+

The use of pseudonatural orbitals (PNO) is proposed to improve the rate of convergence in the superposition of configurations (SOC). Natural orbitals are determined for selected electron pairs in the Hartree—Fock field of the n−2 electron core and are then used as the basis for the total SOC calculation. Since these natural orbitals are not natural for the n-electron system they are considered false or pseudonatural orbitals when used in the n-electron problem. The PNO basis has been applied to He2+ and H3 to test the convergence. Complete results are reported here only for He2+. The PNO's are quite successful in speeding up the convergence of the SOC and rendering the calculation of correlation energy quite practical in general. Gaussian-type orbitals (GTO) are used throughout and were not a serious impediment to obtaining quantitative accuracy. In fact the large number of unoccupied Hartree—Fock orbitals consequent upon the use of a GTO basis permit a straightforward determination of the PNO orbitals.