Numerical Atomic Orbitals Research Articles

Assessment of the exchange-hole dipole moment dispersion correction for the energy ranking stage of the seventh crystal structure prediction blind test.

The seventh blind test of crystal structure prediction (CSP) methods substantially increased the level of complexity of the target compounds relative to the previous tests organized by the Cambridge Crystallographic Data Centre. In this work, the performance of density-functional methods is assessed using numerical atomic orbitals and the exchange-hole dipole moment dispersion correction (XDM) for the energy-ranking phase of the seventh blind test. Overall, excellent performance was seen for the two rigid molecules (XXVII, XXVIII) and for the organic salt (XXXIII). However, for the agrochemical (XXXI) and pharmaceutical (XXXII) targets, the experimental polymorphs were ranked fairly high in energy amongst the provided candidate structures and inclusion of thermal free-energy corrections from the lattice vibrations was found to be essential for compound XXXI. Based on these results, it is proposed that the importance of vibrational free-energy corrections increases with the number of rotatable bonds.

Charge self-consistent dynamical mean field theory calculations in combination with linear combination of numerical atomic orbitals framework based density functional theory

Charge self-consistent dynamical mean field theory calculations in combination with linear combination of numerical atomic orbitals framework based density functional theory

Influence of atomistic features in plasmon-exciton coupling and charge transfer driven by a single molecule in a metallic nanocavity.

When an organic molecule is placed inside a plasmonic cavity formed by two metallic nanoparticles (MNP) under illumination, the electronic excitations of the molecule couple to the plasmonic electromagnetic modes of the cavity, inducing new hybrid light-matter states called polaritons. Atomistic abinitio methods accurately describe the coupling between MNPs and molecules at the nanometer scale and allow us to analyze how atomistic features influence the interaction. In this work, we study the optical response of a porphine molecule coupled to a silver nanoparticle dimer from first principles, within the linear-response time-dependent density functional theory framework, using the recently developed Python Numeric Atomic Orbitals implementation to compute the optical excitations. The optical spectra show the splitting of the resonances of the plasmonic dimer and the molecule into two distinct polaritons, a characteristic feature of the strong light-matter coupling regime. Our results stress the importance of atomistic features, such as the gap configuration in determining the plasmon-exciton coupling strength and in the emergence of molecule-mediated charge-transfer plasmon (CTP) resonances at lower frequencies. Moreover, we show that the strength of the CTP resonance can be tuned by shifting the alignment of the molecular energy levels with respect to the Fermi level of the MNPs.

Machine-Learning-Based Interatomic Potentials for Group IIB to VIA Semiconductors: Toward a Universal Model.

Rapid advancements in machine-learning methods have led to the emergence of machine-learning-based interatomic potentials as a new cutting-edge tool for simulating large systems with ab initio accuracy. Still, the community awaits universal interatomic models that can be applied to a wide range of materials without tuning neural network parameters. We develop a unified deep-learning interatomic potential (the DPA-Semi model) for 19 semiconductors ranging from group IIB to VIA, including Si, Ge, SiC, BAs, BN, AlN, AlP, AlAs, InP, InAs, InSb, GaN, GaP, GaAs, CdTe, InTe, CdSe, ZnS, and CdS. In addition, independent deep potential models for each semiconductor are prepared for detailed comparison. The training data are obtained by performing density functional theory calculations with numerical atomic orbitals basis sets to reduce the computational costs. We systematically compare various properties of the solid and liquid phases of semiconductors between different machine-learning models. We conclude that the DPA-Semi model achieves GGA exchange-correlation functional quality accuracy and can be regarded as a pretrained model toward a universal model to study group IIB to VIA semiconductors.

GPAW: An open Python package for electronic structure calculations.

We review the GPAW open-source Python package for electronic structure calculations. GPAW is based on the projector-augmented wave method and can solve the self-consistent density functional theory (DFT) equations using three different wave-function representations, namely real-space grids, plane waves, and numerical atomic orbitals. The three representations are complementary and mutually independent and can be connected by transformations via the real-space grid. This multi-basis feature renders GPAW highly versatile and unique among similar codes. By virtue of its modular structure, the GPAW code constitutes an ideal platform for the implementation of new features and methodologies. Moreover, it is well integrated with the Atomic Simulation Environment (ASE), providing a flexible and dynamic user interface. In addition to ground-state DFT calculations, GPAW supports many-body GW band structures, optical excitations from the Bethe-Salpeter Equation, variational calculations of excited states in molecules and solids via direct optimization, and real-time propagation of the Kohn-Sham equations within time-dependent DFT. A range of more advanced methods to describe magnetic excitations and non-collinear magnetism in solids are also now available. In addition, GPAW can calculate non-linear optical tensors of solids, charged crystal point defects, and much more. Recently, support for graphics processing unit (GPU) acceleration has been achieved with minor modifications to the GPAW code thanks to the CuPy library. We end the review with an outlook, describing some future plans for GPAW.

Ab initio electronic structure calculations based on numerical atomic orbitals: Basic fomalisms and recent progresses

AbstractThe numerical atomic orbital (NAO) basis sets offer a computationally efficient option for electronic structure calculations, as they require fewer basis functions compared with other types of basis sets. Moreover, their strict localization allows for easy combination with current linear scaling methods, enabling efficient calculation of large physical systems. In recent years, NAO bases have become increasingly popular in modern electronic structure codes. This article provides a review of the ab initio electronic structure calculations using NAO bases. We begin by introducing basic formalisms of the NAO‐based electronic structure method, including NAO base set generation, self‐consistent calculations, force, and stress calculations. We will then discuss some recent advances in the methods based on the NAO bases, such as real‐time dependent density functional theory (rt‐TDDFT), efficient implementation of hybrid functionals, and other advanced electronic structure methods. Finally, we introduce the ab initio tight‐binding model, which can be generated directly after the self‐consistent calculations. The model allows for efficient calculation of electronic structures, and the associated topological, and optical properties of the systems.This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Electronic Structure Theory > Density Functional Theory Structure and Mechanism > Computational Materials Science

Automatic Generation of Accurate and Cost-Efficient Auxiliary Basis Sets.

We have recently discussed an algorithm to automatically generate auxiliary basis sets (ABSs) of the standard form for density fitting (DF) or resolution-of-the-identity (RI) calculations in a given atomic orbital basis set (OBS) of any form, such as Gaussian-type orbitals, Slater-type orbitals, or numerical atomic orbitals [J. Chem. Theory Comput. 2021, 17, 6886]. In this work, we study two ways to reduce the cost of such automatically generated ABSs without sacrificing their accuracy. We contract the ABS with a singular value decomposition proposed by Kállay [J. Chem. Phys. 2014, 141, 244113], used here in a somewhat different setting. We also drop the high-angular momentum functions from the ABS, as they are unnecessary for global fitting methods. Studying the effect of these two types of truncations on Hartree-Fock (HF) and second-order Møller-Plesset perturbation theory (MP2) calculations on a chemically diverse set of first- and second-row molecules within the RI/DF approach, we show that accurate total and atomization energies can be achieved by a combination of the two approaches with significant reductions in the size of the ABS. While the original approach yields ABSs whose number of functions NbfABS scales with the number of functions in the OBS, NOBSbf, as NABSbf = γNOBSbf with the prefactor , the reduction schemes of this work afford results of essentially the same quality as the original unpruned and uncontracted ABS with γ ≈ 5-6, while an accuracy that may suffice for routine applications is achievable with a further reduced ABS with γ ≈ 3-4. The observed errors are similar at HF and MP2 levels of theory, suggesting that the generated ABSs are highly transferable and can also be applied to model challenging properties with high-level methods.

Implementation of the meta-GGA exchange-correlation functional in numerical atomic orbital basis: With systematic testing on SCAN, rSCAN, and r2SCAN functionals.

Kohn-Sham density functional theory (DFT) is nowadays widely used for electronic structure theory simulations, and the accuracy and efficiency of DFT rely on approximations of the exchange-correlation functional. By including the kinetic energy density τ, the meta-generalized-gradient approximation (meta-GGA) family of functionals achieves better accuracy and flexibility while retaining the efficiency of semi-local functionals. For example, the strongly constrained and appropriately normed (SCAN) meta-GGA functional has been proven to yield accurate results for solid and molecular systems. We implement meta-GGA functionals with both numerical atomic orbitals and plane wave bases in the ABACUS package. Apart from the exchange-correlation potential, we also discuss the evaluation of force and stress. To validate our implementation, we perform finite-difference tests and convergence tests with the SCAN, rSCAN, and r2SCAN meta-GGA functionals. We further test water hexamers, weakly interacting molecules from the S22 dataset, as well as 13 semiconductors using the three functionals. The results show satisfactory agreement with previous calculations and available experimental values.

Efficient implementation of analytical gradients for periodic hybrid functional calculations within fitted numerical atomic orbitals from NAO2GTO.

The NAO2GTO scheme provides an efficient way to evaluate the electron repulsion integrals (ERIs) over numerical atomic orbitals (NAOs) with auxiliary Gaussian-type orbitals (GTOs). However, the NAO2GTO fitting will significantly impact the accuracy and convergence of hybrid functional calculations. To address this issue, here we propose to use the fitted orbitals as a new numerical basis to properly handle the mismatch between NAOs and fitted GTOs. We present an efficient and linear-scaling implementation of analytical gradients of Hartree-Fock exchange (HFX) energy for periodic HSE06 calculations with fitted NAOs in the HONPAS package. In our implementation, the ERIs and their derivatives for HFX matrix and forces are evaluated analytically with the auxiliary GTOs, while other terms are calculated using numerically discretized GTOs. Several integral screening techniques are employed to reduce the number of required ERI derivatives. We benchmark the accuracy and efficiency of our implementation and demonstrate that our results of lattice constants, bulk moduli, and band gaps of several typical semiconductors are in good agreement with the experimental values. We also show that the calculation of HFX forces based on a master-worker dynamic parallel scheme has a very high efficiency and scales linearly with respect to system size. Finally, we study the geometry optimization and polaron formation due to an excess electron in rutile TiO2 by means of HSE06 calculations to further validate the applicability of our implementation.

Compatibility of DFT+U with non-collinear magnetism and spin-orbit coupling within a framework of numerical atomic orbitals

We report the extension of the density-functional theory plus Hubbard U (DFT+U) method to the case of non-collinear magnetism and spin-orbit coupling in a framework of numerical atomic orbitals. Both the Hubbard repulsion term U, and the exchange J parameters are explicitly included and treated separately. The occupation numbers of the localized orbitals belonging to the correlated shell are computed from the projections of the Kohn-Sham eigenfunctions onto a set of non-overlapping, orthogonal, localized projectors. We provide the detailed expressions for the total energy, forces and stresses including the Pulay corrections. Our implementation on the version 5.0 of the siesta package has been validated with simulations carried out in isolated atoms and bulk solids including atoms with a strong spin-orbit coupling.

XDM-corrected hybrid DFT with numerical atomic orbitals predicts molecular crystal lattice energies with unprecedented accuracy.

Molecular crystals are important for many applications, including energetic materials, organic semiconductors, and the development and commercialization of pharmaceuticals. The exchange-hole dipole moment (XDM) dispersion model has shown good performance in the calculation of relative and absolute lattice energies of molecular crystals, although it has traditionally been applied in combination with plane-wave/pseudopotential approaches. This has limited XDM to use with semilocal functional approximations, which suffer from delocalization error and poor quality conformational energies, and to systems with a few hundreds of atoms at most due to unfavorable scaling. In this work, we combine XDM with numerical atomic orbitals, which enable the efficient use of XDM-corrected hybrid functionals for molecular crystals. We test the new XDM-corrected functionals for their ability to predict the lattice energies of molecular crystals for the X23 set and 13 ice phases, the latter being a particularly stringent test. A composite approach using a XDM-corrected, 25% hybrid functional based on B86bPBE achieves a mean absolute error of 0.48 kcal mol-1 per molecule for the X23 set and 0.19 kcal mol-1 for the total lattice energies of the ice phases, compared to recent diffusion Monte-Carlo data. These results make the new XDM-corrected hybrids not only far more computationally efficient than previous XDM implementations, but also the most accurate density-functional methods for molecular crystal lattice energies to date.

Density Functional Theory Plus Dynamical Mean Field Theory within the Framework of Linear Combination of Numerical Atomic Orbitals: Formulation and Benchmarks.

The combination of density functional theory with dynamical mean-field theory (DFT+DMFT) has become a powerful first-principles approach to tackle strongly correlated materials in condensed matter physics. The wide use of this approach relies on robust and easy-to-use implementations, and its implementation in various numerical frameworks will increase its applicability on the one hand and help crosscheck the validity of the obtained results on the other. In this work, we develop a formalism within the linear combination of numerical atomic orbital (NAO) basis set framework, which allows for merging of NAO-based DFT codes with DMFT quantum impurity solvers. The formalism is implemented by interfacing two NAO-based DFT codes with three DMFT impurity solvers, and its validity is testified by benchmark calculations for a wide range of strongly correlated materials, including 3d transition metal compounds, lanthanides, and actinides. Our work not only enables DFT+DMFT calculations using popular and rapidly developing NAO-based DFT code packages but also facilitates the combination of more advanced beyond-DFT methodologies available in these codes with the DMFT machinery.

Localized Resolution of Identity Approach to the Analytical Gradients of Random-Phase Approximation Ground-State Energy: Algorithm and Benchmarks.

We develop and implement a formalism which enables calculating the analytical gradients of particle-hole random-phase approximation (RPA) ground-state energy with respect to the atomic positions within the atomic orbital basis set framework. Our approach is based on a localized resolution of identity (LRI) approximation for evaluating the two-electron Coulomb integrals and their derivatives, and the density functional perturbation theory for computing the first-order derivatives of the Kohn-Sham (KS) orbitals and orbital energies. Our implementation allows one to relax molecular structures at the RPA level using both Gaussian-type orbitals (GTOs) and numerical atomic orbitals (NAOs). Benchmark calculations against previous implementations show that our approach delivers adequate numerical precision, highlighting the usefulness of LRI in the context of RPA gradient evaluations. A careful assessment of the quality of RPA geometries for small molecules reveals that post-KS RPA systematically overestimates the bond lengths. We furthermore optimized the geometries of the four low-lying water hexamers-cage, prism, cyclic, and book isomers, and determined the energy hierarchy of these four isomers using RPA. The obtained RPA energy ordering is in good agreement with that yielded by the coupled cluster method with single, double and perturbative triple excitations, despite that the dissociation energies themselves are appreciably underestimated. The underestimation of the dissociation energies by RPA is well corrected by the renormalized single excitation correction.

First-principles calculations of the surface states of doped and alloyed topological materials via band unfolding method

One of the most remarkable characteristics of topological materials is that they have special surface states, which are determined by the topological properties of their bulk materials. Angle resolved photoemission spectroscopy (ARPES) is a powerful tool to explore surface states, which allows to further investigate the topological phase transitions. However, it is very difficult to compare the first-principles calculated band structures to the ARPES results, when the systems are doped or alloyed, because the band structures are heavily folded. We develop an efficient band unfolding method based on numerical atomic orbitals (NAOs). We apply this method to study the surface states of the non-magnetically and magnetically doped topological insulators Bi2Se3 and the topological crystalline insulators Pb1−xSnxTe.

DFT+U within the framework of linear combination of numerical atomic orbitals.

We present a formulation and implementation of the density functional theory (DFT)+U method within the framework of linear combination of numerical atomic orbitals (NAO). Our implementation not only enables single-point total energy and electronic-structure calculations but also provides access to atomic forces and cell stresses, hence allowing for full structure relaxations of periodic systems. Furthermore, our implementation allows one to deal with non-collinear spin texture, with the spin-orbit coupling (SOC) effect treated self-consistently. The key aspect behind our implementation is a suitable definition of the correlated subspace when multiple atomic orbitals with the same angular momentum are used, and this is addressed via the "Mulliken charge projector" constructed in terms of the first (most localized) atomic orbital within the d/f angular momentum channel. The important Hubbard U and Hund J parameters can be estimated from a screened Coulomb potential of the Yukawa type, with the screening parameter either chosen semi-empirically or determined from the Thomas-Fermi screening model. Benchmark calculations are performed for four late transition metal monoxide bulk systems, i.e., MnO, FeO, CoO, and NiO, and for the 5d-electron compounds IrO2. For the former type of systems, we check the performance of our DFT+U implementation for calculating bandgaps, magnetic moments, electronic band structures, as well as forces and stresses; for the latter, the efficacy of our DFT+U+SOC implementation is assessed. Systematic comparisons with available experimental results, especially with the results from other implementation schemes, are carried out, which demonstrate the validity of our NAO-based DFT+U formalism and implementation.

Straightforward and Accurate Automatic Auxiliary Basis Set Generation for Molecular Calculations with Atomic Orbital Basis Sets.

Density fitting (DF), also known as the resolution of the identity (RI), is a widely used technique in quantum chemical calculations with various types of atomic basis sets─Gaussian-type orbitals, Slater-type orbitals, as well as numerical atomic orbitals─to speed up density functional, Hartree-Fock (HF), and post-HF calculations. Traditionally, custom auxiliary basis sets are hand-optimized for each orbital basis set; however, some automatic schemes have also been suggested. In this work, we propose a simple yet numerically stable automated scheme for forming auxiliary basis sets with the help of a pivoted Cholesky decomposition, which is applicable to any type of atomic basis function. We exemplify the scheme with proof-of-concept calculations with Gaussian basis sets and show that the proposed approach leads to negligible DF/RI errors in HF and second-order Møller-Plesset (MP2) total energies of the non-multireference part of the W4-17 test set when used with orbital basis sets of at least polarized triple-ζ quality. The results are promising for future applications employing Slater-type orbitals or numerical atomic orbitals, as well as schemes based on the use of local fitting approaches. Global fitting approaches can also be used, in which case the high-angular-momentum functions produced by the present scheme can be truncated to minimize the computational cost.

Strategy for constructing compact numerical atomic orbital basis sets by incorporating the gradients of reference wavefunctions

We develop an algorithm to construct high-quality numerical atomic orbital (NAO) basis sets suitable for large-scale, efficient density-functional calculations. The key idea behind this algorithm is that, in addition to fitting the reference wavefunctions themselves generated by plane-wave based calculations of chosen target systems, the first derivatives of the reference wavefunctions are also taken into account as the fitting target. By doing so, the quality of the generated NAO basis sets is significantly improved in the sense that the same level of numerical precision can be achieved with smaller basis set sizes or with reduced cutoff radii of the NAOs.

Accurate stress calculations based on numerical atomic orbital bases: Implementation and benchmarks

We present the detailed formalism for stress calculations within the framework of numerical atomic orbital (NAO) bases, as implemented in the first-principles electronic code package ABACUS. The validity and numerical precision of our implementation are benchmarked against reference results obtained using the finite difference method as well as those calculated using plane wave (PW) bases. The equation of state for α-quartz calculated using our NAO-based implementation is in excellent agreement with the experimental data. We further show that the stresses calculated in terms of NAO bases converge much faster with respect to the energy cutoff than those calculated using PW bases. This enables accurate stress calculations at relatively low cost. An analysis is provided to elucidate the origin of this behavior.

All-electron periodic G0W0 implementation with numerical atomic orbital basis functions: Algorithm and benchmarks

We present an all-electron, periodic ${G}_{\text{0}}{W}_{\text{0}}$ implementation within the numerical atomic orbital (NAO) basis framework. A localized variant of the resolution-of-the-identity (RI) approximation is employed to significantly reduce the computational cost of evaluating and storing the two-electron Coulomb repulsion integrals. We demonstrate that the error arising from localized RI approximation can be reduced to an insignificant level by enhancing the set of auxiliary basis functions, used to expand the products of two single-particle NAOs. An efficient algorithm is introduced to deal with the Coulomb singularity in the Brillouin zone sampling that is suitable for the NAO framework. We perform systematic convergence tests and identify a set of computational parameters, which can serve as the default choice for most practical purposes. Benchmark calculations are carried out for a set of prototypical semiconductors and insulators, and compared to independent reference values obtained from an independent ${G}_{\text{0}}{W}_{\text{0}}$ implementation based on linearized augmented plane waves (LAPWs) plus high-energy localized orbitals (HLOs) basis set, as well as experimental results. With a moderate (FHI-aims tier 2) NAO basis set, our ${G}_{\text{0}}{W}_{\text{0}}$ calculations produce band gaps that typically lie in between the standard LAPW and the $\mathrm{LAPW}+\mathrm{HLO}$ results. Complementing tier 2 with highly localized Slater-type orbitals (STOs), we find that the obtained band gaps show an overall convergence towards the $\mathrm{LAPW}+\mathrm{HLO}$ results. The algorithms and techniques developed in this work pave the way for efficient implementations of correlated methods within the NAO framework.

Efficient Hybrid Density Functional Calculations for Large Periodic Systems Using Numerical Atomic Orbitals

We present an efficient, linear-scaling implementation for building the (screened) Hartree-Fock exchange (HFX) matrix for periodic systems within the framework of numerical atomic orbital (NAO) basis functions. Our implementation is based on the localized resolution of the identity approximation by which two-electron Coulomb repulsion integrals can be obtained by only computing two-center quantities-a feature that is highly beneficial to NAOs. By exploiting the locality of basis functions and efficient prescreening of the intermediate three- and two-index tensors, one can achieve a linear scaling of the computational cost for building the HFX matrix with respect to the system size. Our implementation is massively parallel, thanks to a MPI/OpenMP hybrid parallelization strategy for distributing the computational load and memory storage. All these factors add together to enable highly efficient hybrid functional calculations for large-scale periodic systems. In this work, we describe the key algorithms and implementation details for the HFX build as implemented in the ABACUS code package. The performance and scalability of our implementation with respect to the system size and the number of CPU cores are demonstrated for selected benchmark systems up to 4096 atoms.