Basis Set Optimization Research Articles

A Visual Representation for Accurate Local Basis Set Construction and Optimization: A Case Study of SrTiO3 with Hybrid DFT Functionals

The linear combination of atomic orbitals (LCAO) method is advantageous for calculating important bulk and surface properties of crystals and defects in/on them. Compared to plane wave calculations and contrary to common assumptions, hybrid density functional theory (DFT) functionals are actually less costly and easier to implement in LCAO codes. However, choosing the proper basis set (BS) for the LCAO calculations representing Guassian-type functions is crucial, as the results depend heavily on its quality. In this study, we introduce a new basis set (BS) visual representation, which helps us (1) analyze the collective behavior of individual atoms’ shell exponents (s, p, and d), (2) better compare different BSs, (3) identify atom-type invariant relationships, and (4) suggest a robust method for building a local all-electron BS (denoted as BS1) from scratch for each atom type. To compare our BS1 with the others existing in the literature, we calculate the basic bulk properties of SrTiO3 (STO) in cubic and tetragonal phases using several hybrid DFT functionals (B3LYP, PBE0, and HSE06). After adjusting the exact Hartree–Fock (HF) exchange of PBEx, HSEx, and the state-of-the-art meta-GGA hybrid r2SCANx functionals, we find the r2SCAN15 and HSE27 for BS1, with the amount of exact HF exchange of 0.15 and 0.27, respectively, perform equally well for reproducing several most relevant STO properties. The proposed robust BS construction scheme has the advantage that all parameters of the obtained BS can be reoptimized for each new material, thus increasing the quality of DFT calculation predictions.

Basis Set Generation and Optimization in the NISQ Era with Quiqbox.jl.

In the noisy intermediate-scale quantum era, ab initio computation of electronic structure problems has become one of the major benchmarks for identifying the boundary between classical and quantum computational power. Basis sets play a key role in the electronic structure methods implemented on both classical and quantum devices. To investigate the consequences of single-particle basis sets, we propose a framework for more customizable basis set generation and optimization. This framework allows composite basis sets to go beyond typical basis set frameworks, such as atomic basis sets, by introducing the concept of mixed-contracted Gaussian-type orbitals. These basis set generations set the stage for more flexible variational optimization of basis set parameters. To realize this framework, we have developed an open-source software package named "Quiqbox" in the Julia programming language. We demonstrate various examples of using Quiqbox for basis set optimization and generation, ranging from optimizing atomic basis sets on the Hartree-Fock level, preparing the initial state for variational quantum eigensolver computation, and constructing basis sets with completely delocalized orbitals. We also include various benchmarks of Quiqbox for basis set optimization and ab initial electronic structure computation.

BasisOpt: A Python package for quantum chemistry basis set optimization.

The accuracy and efficiency of molecular quantum chemical calculations depend critically on the basis set used. However, the development of novel basis sets is hindered because much of the literature relies on the use of opaque processes and tools that are not publicly available. We present here BasisOpt, a tool for the automated optimization of basis sets with an easy-to-use framework. It features an open and accessible workflow for basis set optimization that can be easily adapted to almost any quantum chemistry program, a standardized approach to testing basis sets, and visualization of both the optimized basis sets and the optimization process. We provide examples of usage in realistic basis set optimization scenarios where: (i) a density fitting basis set is optimized for He, Ne, and Ar; (ii) the exponents of the def2-SVP basis are re-optimized for a set of molecules rather than atoms; and (iii) a large, almost saturated basis of sp primitives is automatically reduced to (10s5p) while achieving the lowest energy for such a basis set composition.

Material-Specific Optimization of Gaussian Basis Sets against Plane Wave Data.

Since in periodic systems, a given element may be present in different spatial arrangements displaying vastly different physical and chemical properties, an elemental basis set that is independent of physical properties of materials may lead to significant simulation inaccuracies. To avoid such a lack of material specificity within a given basis set, we present a material-specific Gaussian basis optimization scheme for solids, which simultaneously minimizes the total energy of the system and optimizes the band energies when compared to the reference plane wave calculation while taking care of the overlap matrix condition number. To assess this basis set optimization scheme, we compare the quality of the Gaussian basis sets generated for diamond, graphite, and silicon via our method against the existing basis sets. The optimization scheme of this work has also been tested on the existing Gaussian basis sets for periodic systems such as MoS2 and NiO, yielding improved results.

The completeness properties of Gaussian‐type orbitals in quantum chemistry

AbstractIn this work, I extend results on the convergence of Gaussian basis sets in quantum chemistry, previously shown for ground‐state hydrogenic wavefunctions, to orbitals of arbitrary angular momentum. I give rigorous proofs of their asymptotic behavior, and demonstrate for methods with regular potential operators—in particular, Hartree–Fock and Kohn–Sham density functional theory—that the assumption of completeness is correct under fairly lenient conditions. The final result under the correct norm is that the convergence in energy follows , where M is the number of Gaussians and k is a positive constant, generalizing previous results due to Kutzelnigg. This then yields prescriptions for accelerated convergence using even‐tempered Gaussians, which could be used as initial guesses in future basis set optimizations.

Gaussian Basis Sets for Crystalline Solids: All-Purpose Basis Set Libraries vs System-Specific Optimizations.

It is customary in molecular quantum chemistry to adopt basis set libraries in which the basis sets are classified according to either their size (triple-ζ, quadruple-ζ, ...) and the method/property they are optimal for (correlation-consistent, linear-response, ...) but not according to the chemistry of the system to be studied. In fact the vast majority of molecules is quite homogeneous in terms of density (i.e., atomic distances) and types of bond involved (covalent or dispersive). The situation is not the same for solids, in which the same chemical element can be found having metallic, ionic, covalent, or dispersively bound character in different crystalline forms or compounds, with different packings. This situation calls for a different approach to the choice of basis sets, namely a system-specific optimization of the basis set that requires a practical algorithm that could be used on a routine basis. In this work we develop a basis set optimization method based on an algorithm–similar to the direct inversion in the iterative subspace–that we name BDIIS. The total energy of the system is minimized together with the condition number of the overlap matrix as proposed by VandeVondele et al. [VandeVondele et al. J. Chem. Phys.2007, 227, 114105]. The details of the method are here presented, and its performance in optimizing valence orbitals is shown. As demonstrative systems we consider simple prototypical solids such as diamond, graphene sodium chloride, and LiH, and we show how basis set optimizations have certain advantages also toward the use of large (quadruple-ζ) basis sets in solids, both at the DFT and Hartree–Fock level.

Computational procedure to an accurate DFT simulation to solid state systems

The density functional theory has become increasingly common as a methodology to explain the properties of crystalline materials because of the improvement in computational infrastructure and software development to perform such computational simulations. Although several studies have shown that the characteristics of certain classes of materials can be represented with great precision, it is still necessary to improve the methods and strategies in order to achieve more realistic computational modeling. In the present work, strategies are reported in a systematic way for the accurate representation of crystalline systems. The crystalline compound chosen for the study as a case test was BaMoO4, both because of its potential technological application and because of the low accuracy of the simulations previously reported in the literature. The computational models were carried out with the B3LYP and WC1LYP functionals selected from an initial set containing eight hybrid functionals in conjunction with an all-electron basis set. Two different strategies were applied for improving the description of the initial models, both involving atomic basis set optimization and Hartree-Fock exchange percentage adjustment. The results obtained with the two strategies show a precision of structural parameters, band gap energy, and vibrational properties never before presented in theoretical studies of BaMoO4. Finally, a flowchart of good calculation practices is elaborated. This can be of great value for the organization and conduction of calculations in new research.

BSSE-correction scheme for consistent gaussian basis sets of double- and triple-zeta valence with polarization quality for solid-state calculations.

Revised versions of our published pob-TZVP [Peintinger, M. F.; Oliveira, D. V. and Bredow, T., J. Comput. Chem., 2013, 34 (6), 451-459.] and unpublished pob-DZVP basis sets, denoted as pob-TZVP-rev2 and pob-DZVP-rev2, have been derived for the elements HBr. It was observed that the pob basis sets suffer from the basis set superposition error (BSSE). In order to reduce this effect, we took into account the counterpoise energy of hydride dimers as an additional parameter in the basis set optimization. The overall performance, portability, and SCF stability of the resulting rev2 basis sets are significantly improved compared to the original pob basis sets. © 2019 Wiley Periodicals, Inc.

ОПТИМИЗАЦИЯ ПАРАМЕТРОВ БАЗИСНОГО НАБОРА ДЛЯ DFT МОДЕЛИРОВАНИЯ НАНОМАТЕРИАЛОВ

Atomic-like orbital basis sets allow efficient calculation of nanomaterial’s surface properties within the density functional theory. However, unlike plane wave basis sets, they require thorough optimization on a reference system before modelling systems of interest. We considered the basis set optimization procedure for several structures: bulk tantalum carbide, oxygen molecule, bulk lithium, and α -carbyne. We showed that during the optimization procedure not only the total energy of a reference system should be monitored but other physical characteristics (bond length and atomic charges) too. Moreover, optimal basis parameters could not correspond to the minimum total energy of a reference system to get the correct physical properties. We obtained optimal orbital parameters, which can be used for modelling of the following systems: oxygen adsorption on tantalum carbide surface, and Li-functionalized carbyne. Considering oxygen adsorption on TaC surface and Li-functionalization of carbyne, we also demonstrated that the basis set optimization influences binding energies and atomic charges of an adsorbent and a surface.

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.

Choice of hybrid functional and basis set optimization to calculate the structural, electronic, mechanical, and vibrational properties of BaSnO3

This work aimed to study the influence of seven hybrid functionals (B3LYP, B1WC, B3PW, PBE0, PBESOL0, SOGGAXC, and WC1LYP) and selected basis sets in the CRYSTAL software library on the calculated structural, electronic, mechanical, and vibrational properties of BaSnO3 under the framework of periodic density functional theory. Based on the results showing the best approximation of the experimental band gap energy and lattice parameter, the PBE0 functional and the Baranek basis set were used to optimize the outer valence shell functions, αsp and αd, in order to improve the quality of calculated properties. An analysis was done to determine the density of states, band structure, elastic constants, and infrared (IR) active modes. The results were compared with other reported quantum mechanical calculations and available experimental data.

Optimized Basis Sets for the Environment in the Domain-Specific Basis Set Approach of the Incremental Scheme.

Minimal basis sets, denoted DSBSenv, based on the segmented basis sets of Ahlrichs and co-workers have been developed for use as environmental basis sets for the domain-specific basis set (DSBS) incremental scheme with the aim of decreasing the CPU requirements of the incremental scheme. The use of these minimal basis sets within explicitly correlated (F12) methods has been enabled by the optimization of matching auxiliary basis sets for use in density fitting of two-electron integrals and resolution of the identity. The accuracy of these auxiliary sets has been validated by calculations on a test set containing small- to medium-sized molecules. The errors due to density fitting are about 2-4 orders of magnitude smaller than the basis set incompleteness error of the DSBSenv orbital basis sets. Additional reductions in computational cost have been tested with the reduced DSBSenv basis sets, in which the highest angular momentum functions of the DSBSenv auxiliary basis sets have been removed. The optimized and reduced basis sets are used in the framework of the domain-specific basis set of the incremental scheme to decrease the computation time without significant loss of accuracy. The computation times and accuracy of the previously used environmental basis and that optimized in this work have been validated with a test set of medium- to large-sized systems. The optimized and reduced DSBSenv basis sets decrease the CPU time by about 15.4% and 19.4% compared with the old environmental basis and retain the accuracy in the absolute energy with standard deviations of 0.99 and 1.06 kJ/mol, respectively.

First example of the correlated calculation of the one-bond tellurium-carbon spin-spin coupling constants: Relativistic effects, vibrational corrections, and solvent effects.

This work reports on the comprehensive calculation of the NMR one-bond spin-spin coupling constants (SSCCs) involving carbon and tellurium, (1) J((125) Te,(13) C), in four representative compounds: Te(CH3 )2 , Te(CF3 )2 , Te(CCH)2 , and tellurophene. A high-level computational treatment of (1) J((125) Te,(13) C) included calculations at the SOPPA level taking into account relativistic effects evaluated at the 4-component RPA and DFT levels of theory, vibrational corrections, and solvent effects. The consistency of different computational approaches including the level of theory of the geometry optimization of tellurium-containing compounds, basis sets, and methods used for obtainig spin-spin coupling values have also been discussed in view of reproducing the experimental values of the tellurium-carbon SSCCs. Relativistic corrections were found to play a major role in the calculation of (1) J((125) Te,(13) C) reaching as much as almost 50% of the total value of (1) J((125) Te,(13) C) while relativistic geometrical effects are of minor importance. The vibrational and solvent corrections account for accordingly about 3-6% and 0-4% of the total value. It is shown that taking into account relativistic corrections, vibrational corrections and solvent effects at the DFT level essentially improves the agreement of the non-relativistic theoretical SOPPA results with experiment. © 2016 Wiley Periodicals, Inc.

An eigenvalue problem with orthogonality constraints and its applications to excited state calculations

A simple and easily implemented technique for taking orthogonality constraints into account , which has been proposed earlier was used to investigate the effect of the finite dimensional approximation and basis set optimization on the accuracy of excited state energies of the H2 + and H3 ++ molecular ions.

Identifying Raman and Infrared Vibrational Motions of Erythritol Tetranitrate

The vibrational bands of erythritol tetranitrate (ETN) were measured experimentally with both Raman spectroscopy and attenuated total reflectance Fourier transform infrared (ATR FT-IR) spectroscopy. Seventy-two (3N-6) vibrational modes were predicted for ETN using density functional theory calculations performed using the B3LYP/6-31G* density functional basis set and geometry optimization. Raman spectroscopy and ATR FT-IR were used to measure observable Raman and IR signatures between 140 and 3100 wavenumbers (cm(-1)). Within this spectral range, 32 Raman bands and 21 IR bands were measured and identified by their predicted vibrational motion. The spectroscopic and theoretical analysis of ETN performed will advance the detection and identification capabilities of field measuring instruments for this explosive.

Auxiliary Basis Sets for Density Fitting in Explicitly Correlated Calculations: The Atoms H-Ar.

Auxiliary basis sets specifically matched to the correlation consistent cc-pVnZ-F12 and cc-pCVnZ-F12 orbital basis sets for the elements H-Ar have been optimized at the density-fitted second-order Møller-Plesset perturbation theory level of theory for use in explicitly correlated (F12) methods, which utilize density fitting for the evaluation of two-electron integrals. Calculations of the correlation energy for a test set of small to medium sized molecules indicate that the density fitting error when using these auxiliary sets is 2 to 3 orders of magnitude smaller than the F12 orbital basis set incompleteness error. The error introduced by the use of these fitting sets within the resolution-of-the-identity approximation of the many-electron integrals arising in F12 theory has also been assessed and is demonstrated to be negligible and well-controlled. General guidelines are proposed for the optimization of density fitting auxiliary basis sets for use with F12 methods for other elements.

Electronic, structural, and elastic properties of metal nitrides XN (X = Sc, Y): A first principle study

We utilized a simple, robust, first principle method, based on basis set optimization with the BZW-EF method, to study the electronic and related properties of transition metal mono-nitrides: ScN and YN. We solved the KS system of equations self-consistently within the linear combination of atomic orbitals (LCAO) formalism. It is shown that the band gap and low energy conduction bands, as well as elastic and structural properties, can be calculated with a reasonable accuracy when the LCAO formalism is used to obtain an optimal basis. Our calculated, indirect electronic band gap (\documentclass[12pt]{minimal}\begin{document}${\rm E}^\mathrm{\Gamma -X}_g$\end{document}EΓ−Xg) is 0.79 (LDA) and 0.88 eV (GGA) for ScN. In the case of YN, we predict an indirect band gap (\documentclass[12pt]{minimal}\begin{document}${\rm E}^\mathrm{\Gamma -X}_g$\end{document}EΓ−Xg) of 1.09 (LDA) and 1.15 eV (GGA). We also calculated the equilibrium lattice constants, the bulk moduli (Bo), effective masses, and elastic constants for both systems. Our calculated values are in excellent agreement with experimental ones where the latter are available.

Benchmark Full Configuration Interaction Calculations on the Lowest-Energy (2)P and (4)P States of the Three-Electron Harmonium Atom.

Full configuration interaction calculations carried out in conjunction with careful optimization of basis sets and judicious extrapolation schemes for 12 values of the confinement strength ω ranging from 0.1 to 1000.0 provide benchmark energies for the (2)P ground state and the (4)P first excited state of the three-electron harmonium atom, allowing for numerical verification of the recently obtained second-order energy coefficients and confirming the few available results of Monte Carlo studies. The final energy values, obtained by correcting the extrapolated data for residual errors in the low-order energy coefficients, possess accuracy of ca. 20 μHartree for the doublet state and ca. 10 μHartree for the quartet one, making them suitable for calibration and testing of approximate electron correlation methods of quantum chemistry. The energy limits for individual angular momenta ranging from 1 to 4 are also available, facilitating comparisons with results of calculations involving finite basis sets. An example of application involving the BLYP and B3LYP functionals is provided.

Spectroscopic, molecular structure characterizations and quantum chemical computational studies of ( E)-5-(diethylamino)-2-[(2-fluorophenylimino)methyl]phenol

The molecular structure and spectroscopic properties of ( E)-5-(diethylamino)-2-[(2-fluorophenylimino)methyl]phenol were characterized by X-ray diffraction, IR and UV/Vis spectroscopy. These properties of title compound were also investigated from calculative point of view. Geometry optimization in gas phase was performed using DFT method with B3LYP applying 6-311G(d,p) basis set and geometry optimizations in solvent media were performed with the same level of theory by the polarizable continuum model (PCM). TD-DFT calculations starting from optimized geometry were carried out in both gas and solution phase to calculate excitation energies of title compound. In addition, while the non-linear optical properties were computed, thermodynamic properties were obtained at the optimized geometry with the same level of theory. The intramolecular proton transfer process from enol form to keto form was investigated using DFT method with B3LYP applying 6-311G(d,p) basis set. Transition state structure in EtOH was performed with the same level of theory by the polarizable continuum model (PCM).

Bond functions in CEPA-PNO computations of electronic structure. Basis set optimization studies in N2H2. Application to ozone and the stability of the bound cyclic conformer of O3

Extensive optimization of s and p Gaussian bond functions used to augment (9s, 5p|5s) → [5s, 3p | 3s | atomic basis in N2H2 at the SCF and CEPA-PNO level leads to simple rules for the use of bond functions to efficiently replace nuclear-centered higher angular momentum function basis augmentation. (9s, 5p) → [5s2p] + s, p bond function basis computations featuring SCF bond function optimization and geometry optimization at the CEPA-PNO level yield significantly better wave functions for bent and cyclic ozone than previous studies and reveal the great stability of the bound cyclic conformer of O3.