Slater-type Orbital Basis Research Articles

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.

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.

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.

Optimization of the r2SCAN-3c Composite Electronic-Structure Method for Use with Slater-Type Orbital Basis Sets.

The “Swiss army knife” composite density functional electronic-structure method r2SCAN-3c (J. Chem. Phys.2021, 154, 064103) is extended and optimized for the use with Slater-type orbital basis sets. The meta generalized-gradient approximation (meta-GGA) functional r2SCAN by Furness et al. is combined with a tailor-made polarized triple-ζ Slater-type atomic orbital (STO) basis set (mTZ2P), the semiclassical London dispersion correction (D4), and a geometrical counterpoise (gCP) correction. Relativistic effects are treated explicitly with the scalar-relativistic zeroth-order regular approximation (SR-ZORA). The performance of the new implementation is assessed on eight geometry and 74 energy benchmark sets, including the extensive GMTKN55 database as well as recent sets such as ROST61 and IONPI19. In geometry optimizations, the STO-based r2SCAN-3c is either on par with or more accurate than the hybrid density functional approximation M06-2X-D3(0)/TZP. In energy calculations, the overall accuracy is similar to the original implementation of r2SCAN-3c with Gaussian-type atomic orbitals (GTO), but basic properties, intermolecular noncovalent interactions, and barrier heights are better described with the STO approach, resulting in a lower weighted mean absolute deviation (WTMAD-2(STO) = 7.15 vs 7.50 kcal mol–1 with the original method) for the GMTKN55 database. The STO-optimized r2SCAN-3c outperforms many conventional hybrid/QZ approaches in most common applications at a fraction of their cost. The reliable, robust, and accurate r2SCAN-3c implementation with STOs is a promising alternative to the original implementation with GTOs and can be generally used for a broad field of quantum chemical problems.

Accurate 57Fe Mössbauer Parameters from General Gaussian Basis Sets.

The prediction of isomer shifts in 57Fe Mossbauer spectra is typically achieved by building calibration lines using the values of the density at the nuclear position. Using Slater-type orbital basis or large and specific Gaussian-type orbital basis has been thus far mandatory to achieve accurate predictions with density functional theory methods. In this work, we show that replacing the value of the density at the nucleus by the density integrated in a sphere of radius 0.06 au centered on the Fe nuclei yields excellent calibration lines (r2 = 0.976) with a high predictive power (q2 = 0.975, MAE = 0.055 mm·s–1) while using the conventional def2-TZVP basis set and X-ray geometrical parameters. Our data set comprises 69 57Fe-containing compounds and 103 signals. We also find B3LYP performing significantly better than the PW91 functional.

Tsallis q‐exponentials as atomic orbitals in two‐electron systems

AbstractThe use of simple q‐exponential functions, derived from Tsallis entropy distribution, as basis set of atomic orbitals, is explored within the non‐relativistic limit. As compared to Gaussian or to STO (Slater type orbitals) basis set, considering q as variational parameter, q‐exponentials functions lower considerably the energy of two electron systems, always respecting the Ritz variation principle, where E ≥ E0, with E0 being the true energy. The q parameter attains the highest value of 1.209 for Z = 1 (H−) slowly going down toward 1 as Z grows (for Z = 36 q = 1.005225, Kr+34). Interestingly, the correlation energy as determined considering the exact solution by the calculations of Pekeris with the current ones, shows a regular variation as a function of the q parameter. Moreover, the study confirms the link between correlation energy and Shannon information entropy.

An efficient hybrid scheme for time dependent density functional theory.

A hybrid approach able to perform Time Dependent Density Functional Theory (TDDFT) simulations with the same accuracy as that of hybrid exchange-correlation (xc-) functionals but at a fraction of the computational cost is developed, implemented, and validated. The scheme, which we name Hybrid Diagonal Approximation (HDA), consists in employing in the response function a hybrid xc-functional (containing a fraction of the non-local Hartree-Fock exchange) only for the diagonal elements of the omega matrix, while the adiabatic local density approximation is employed for the off-diagonal terms. HDA is especially (but not exclusively) advantageous when using Slater type orbital basis sets and allows one to employ them in a uniquely efficient way, as we demonstrate here by implementing HDA in a local version of the Amsterdam Density Functional code. The new protocol is tested on NH3, C6H6, and the [Au25(SCH3)18]- cluster as prototypical cases ranging from small molecules to ligand-protected metal clusters, finding excellent agreement with respect to both full kernel TDDFT simulations and experimental data. Additionally, a specific comparison test between full kernel and HDA is considered at the Casida level on seven other molecular species, which further confirm the accuracy of the approach for all investigated systems. For the [Au25(SCH3)18]- cluster, a speedup by a factor of seven is obtained with respect to the full kernel. The HDA, therefore, promises to provide a quantitative description of the optical properties of medium-sized systems (nanoclusters) at an affordable cost, thanks to its computational efficiency, especially in combination with a complex polarization algorithm version of TDDFT.

A Quadratic Pair Atomic Resolution of the Identity Based SOS-AO-MP2 Algorithm Using Slater Type Orbitals

We report a production level implementation of pair atomic resolution of the identity (PARI) based second-order Møller–Plesset perturbation theory (MP2) in the Slater type orbital (STO) based Amsterdam Density Functional (ADF) code. As demonstrated by systematic benchmarks, dimerization and isomerization energies obtained with our code using STO basis sets of triple-ζ-quality show mean absolute deviations from Gaussian type orbital, canonical, basis set limit extrapolated, global density fitting (DF)-MP2 results of less than 1 kcal/mol. Furthermore, we introduce a quadratic scaling atomic orbital based spin-opposite-scaled (SOS)-MP2 approach with a very small prefactor. Due to a worst-case scaling of , our implementation is very fast already for small systems and shows an exceptionally early crossover to canonical SOS-PARI-MP2. We report computational wall time results for linear as well as for realistic three-dimensional molecules and show that triple-ζ quality calculations on molecules of several hundreds of atoms are only a matter of a few hours on a single compute node, the bottleneck of the computations being the SCF rather than the post-SCF energy correction.

Analytical evaluation of relativistic molecular integrals. II: Method of computation for molecular auxiliary functions involved

The Slater-type orbital basis with non-integer principal quantum numbers is a physically and mathematically motivated choice for molecular electronic structure calculations in both non-relativistic and relativistic theory. The non-analyticity of these orbitals at $r=0$, however, requires analytical relations for multi-center integrals to be derived. This is nearly insurmountable. Previous papers by present authors eliminated this difficulty. Highly accurate results can be achieved by the procedure described in these papers, which place no restrictions on quantum numbers in all ranges of orbital parameters. The purpose of this work is to investigate computational aspects of the formulae given in the previous paper. It is to present a method which helps to increase computational efficiency. In terms of the processing time, evaluation of integrals over Slater-type orbitals with non-integer principal quantum numbers are competitive with those over Slater-type orbitals with integer principal quantum numbers.

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.

Time-dependent density functional theory calculations for the excitation spectra of III-V ternary alloys

We adopted the time-dependent density functional theory (TDDFT) within the linear augmented Slater-type orbitals (LASTO) basis and the cluster averaging method to compute the {\color{red}excitation} spectra of III-V ternary alloys with arbitrary concentration $x$. The TDDFT was carried out with the use of adiabatic meta-generalized gradient approximation (mGGA), which contains the $1/q^2$ singularity in the dynamical exchange-correlation kernel ($f_{XC,00}(\mathbf{q})$) as $q\rightarrow 0$. We found that by using wave functions obtained in local density approximation (LDA) while using mGGA to compute self-energy correction to the band structures, we can get {\color{red} good overall} agreement between theoretical results and experimental data for the excitation spectra. Thus, our studies provide some insight into the theoretical calculation of optical spectra of semiconductor alloys.

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.

Combining Slater-type orbitals and effective core potentials

We present a general methodology to evaluate matrix elements of the effective core potentials (ECPs) within one-electron basis set of Slater-type orbitals (STOs). The scheme is based on translation of individual STO distributions in the framework of Barnett-Coulson method. We discuss different types of integrals which naturally appear and reduce them to few basic quantities which can be calculated recursively or purely numerically. Additionally, we consider evaluation of the STOs matrix elements involving the core polarisation potentials (CPP) and effective spin-orbit potentials. Construction of the STOs basis sets designed specifically for use with ECPs is discussed and differences in comparison with all-electron basis sets are briefly summarised. We verify the validity of the present approach by calculating excitation energies, static dipole polarisabilities and valence orbital energies for the alkaline earth metals (Ca, Sr, Ba). Finally, we evaluate interaction energies, permanent dipole moments and ionisation energies for barium and strontium hydrides, and compare them with the best available experimental and theoretical data.

Calibrating Reaction Enthalpies: Use of Density Functional Theory and the Correlation Consistent Composite Approach in the Design of Photochromic Materials.

Acquisition of highly accurate energetic data for chromium-containing molecules and various chromium carbonyl complexes is a major step toward calibrating bond energies and thermal isomerization energies from mechanisms for Cr-centered photochromic materials being developed in our laboratories. The performance of six density functionals in conjunction with seven basis sets, utilizing Gaussian-type orbitals, has been evaluated for the calculation of gas-phase enthalpies of formation and enthalpies of reaction at 298.15 K on various chromium-containing systems. Nineteen molecules were examined: Cr(CO)6, Cr(CO)5, Cr(CO)5(C2H4), Cr(CO)5(C2ClH3), Cr(CO)5(cis-(C2Cl2H2)), Cr(CO)5(gem-(C2Cl2H2)), Cr(CO)5(trans-(C2Cl2H2)), Cr(CO)5(C2Cl3H), Cr(CO)5(C2Cl4), CrO2, CrF2, CrCl2, CrCl4, CrBr2, CrBr4, CrOCl2, CrO2Cl2, CrOF2, and CrO2F2. The performance of 69 density functionals in conjunction with a single basis set utilizing Slater-type orbitals (STO) and a zeroth-order relativistic approximation was also evaluated for the same test set. Values derived from density functional theory were compared to experimental values where available, or values derived from the correlation consistent composite approach (ccCA). When all reactions were considered, the functionals that exhibited the smallest mean absolute deviations (MADs, in kcal mol-1) from ccCA-derived values were B97-1 (6.9), VS98 (9.0), and KCIS (9.4) in conjunction with quadruple-ζ STO basis sets and B97-1 (9.3) in conjunction with cc-pVTZ basis sets. When considering only the set of gas-phase reaction enthalpies (ΔrH°gas), the functional that exhibited the smallest MADs from ccCA-derived values were B97-1 in conjunction with cc-pVTZ basis sets (9.1) and PBEPBE in conjunction with polarized valence triple-ζ basis set/effective core potential combination for Cr and augmented and multiple polarized triple-ζ Pople style basis sets (9.5). Also of interest, certainly because of known cancellation of errors, PBEPBE with the least-computationally expensive basis set combination considered in the present study (valence double-ζ basis set/effective core potential combination for Cr and singly-polarized double-ζ Pople style basis sets) also provided reasonable accuracy (11.1). An increase in basis set size was found to have an improvement in accuracy for the best performing functional (B97-1).

NEW PHOSPHORUS COMPOUNDS K[PCL3(X)] (X= SCN, CN): PREPARATION AND DFT AND SPECTROSCOPIC STUDIES

Two new phosphorus complexes, potassium trichlorothiocyanophosphate (III) (PTCTCP; K[PCl 3 (SCN)]) and potassium trichlorocyanophosphate (III) (PTCCP; K[PCl 3 (CN)]) were synthesized from the reaction of KSCN and KCN, respectively, with PCl 3 . The chemical formulas and compositions of these compounds were determined by elemental analysis and spectroscopic methods, such as phosphorus-31 nuclear magnetic resonance (NMR) spectroscopy ( 31 P-NMR), Fourier transform infrared (FTIR) spectroscopy, ultraviolet-visible (UV-Vis) spectroscopy and mass spectrophotometry. All of the theoretical calculations and determinations of the properties of these compounds were performed as part of the Amsterdam Density Functional (ADF) program. Excitation energies were assessed using time-dependent perturbation density functional theory (TD-DFT). In addition, the molecular geometry was optimized and the frequencies and excitation energies were calculated using standard Slater-type orbital (STO) basis sets with triple-zeta quality double plus polarization functions (TZ2P) for all of the atoms. The assignment of the principal transitions and total densities of state (TDOS) for orbital analysis were performed using the GaussSum 2.2 program.

Simulating Third-Order Nonlinear Optical Properties Using Damped Cubic Response Theory within Time-Dependent Density Functional Theory.

A general implementation for damped cubic response properties is presented using time-dependent density functional theory (TDDFT) and Slater-type orbital basis sets. To directly calculate two-photon absorption (TPA) cross sections, we also present an implementation of a reduced damped cubic response approach. Validation of the implementations includes a detailed comparison between response theory and the sum-over-states approach for calculating the nonlinear optical properties of LiH, as well as a comparison between the simulated and experimental TPA and third-harmonic generation (THG) spectra for the dimethylamino-nitrostilbene (DANS) molecule. The study of LiH demonstrates the incorrect pole structure obtained in response theory due to the adiabatic approximation typically employed for the exchange-correlation kernel. For DANS, we find reasonable agreement between simulated and experimental TPA and THG spectra. Overall, this work shows that care must be taken when calculating higher-order response functions in the vicinity of one-photon poles due to the approximate kernels typically used in the simulations.

Application of Löwdin's canonical orthogonalization method to the Slater-type orbital configuration-interaction basis set

We apply L€ canonical orthogonalization method to investigate the linearly dependent problem arising from the variational calculation of atomic systems using Slater-type orbital configuration-interaction (STO-CI) basis functions. With a specific arithmetic precision used in numerical computations, the nonorthogonal STO-CI basis is easily linearly dependent when the number of basis functions is sufficiently large. We show that L€ canonical orthogonalization method can successfully overcome such problem and simultaneously reduce the dimension of basis set. This is illustrated first through an S-wave model He atom, and then the real two-electron atoms in both the ground and excited states. In all of these calculations, the variational bound state energies of the two-electron systems are obtained in reasonably high accuracy using over-redundant STO-CI bases, however, without using extended high-precision technique. V C 2015 Wiley Periodicals, Inc.

Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. III. Case study of the beryllium dimer

In this paper we present results of ab initio calculations for the beryllium dimer with a basis set of Slater-type orbitals (STOs). Nonrelativistic interaction energy of the system is determined using the frozen-core full configuration interaction calculations combined with high-level coupled-cluster correction for inner-shell effects. We have developed STO basis sets, ranging in quality from double to sextuple $\ensuremath{\zeta}$, which are used in these computations. Principles of their construction are discussed and several atomic benchmarks are presented. Relativistic effects of order ${\ensuremath{\alpha}}^{2}$ are calculated perturbatively by using the Breit-Pauli Hamiltonian and are found to be significant. We also estimate the leading-order QED effects. Influence of the adiabatic correction is found to be negligible. Finally, the electronic binding energy of the beryllium dimer is determined to be $929.0\phantom{\rule{0.16em}{0ex}}\ifmmode\pm\else\textpm\fi{}\phantom{\rule{0.16em}{0ex}}1.9\phantom{\rule{0.16em}{0ex}}{\text{cm}}^{\ensuremath{-}1}$, in a very good agreement with the recent experimental value.

Ultrafast Estimation of Electronic Couplings for Electron Transfer between π-Conjugated Organic Molecules.

Simulation of charge transport in organic semiconducting materials requires the development of strategies for very fast yet accurate estimation of electronic coupling matrix elements for electron transfer between organic molecules (transfer integrals, Hab). A well-known relation that is often exploited for this purpose is the approximately linear dependence of electronic coupling with respect to the overlap of the corresponding diabatic state wave functions for a given donor-acceptor pair. Here we show that a single such relation can be established for a large number of different π-conjugated organic molecules. In our computational scheme the overlap of the diabatic state wave function is simply estimated by the overlap of the highest singly occupied molecular orbital of donor and acceptor, projected on a minimum valence shell Slater-type orbital (STO) basis with optimized Slater decay coefficients. After calibration of the linear relation, the average error in Hab as obtained from the STO orbital overlap is a factor of 1.9 with respect to wave function-theory validated DFT calculations for a diverse set of π-conjugated organic dimers including small arenes, arenes with S, N, and O heteroatoms, acenes, porphins, and buckyballs. The crucial advantage of the scheme is that the STO orbital overlap calculation is analytic. This leads to speedups of 6 orders of magnitude with respect to reference DFT calculations, with little loss of accuracy in the regime relevant to charge transport in organics.

Computation of screened two‐electron matrix elements

Computational studies are presented for atoms in screening environments. Numerical solution of atomic Hartree–Fock equations in the Slater type orbitals basis are presented for screened Debye electron–electron as well as electron–nucleus interaction. Slater integrals are evaluated using a Gauss–Laguerre quadrature. Detailed numerical results are presented for various atoms and Debye lengths establishing the method as well as showing the opposing effects of electron–nucleus and electron–electron screening that may induce ambiguity in physical properties extracted from experimental plasma data. A detailed discussion of convergence properties is given. The relevant outcome of the present study revealed the fact that the present expansion method and the previously published Legendre expansion method (Winkler, Int. J. Quantum Chem. 2010, 110, 3129) have their best convergence properties in different regions of the Debye screening length. This is important for applications. In addition, the new expansion method can be implemented for other screening potentials, where other approaches fail. © 2013 Wiley Periodicals, Inc.