Atomic Orbital Basis Sets Research Articles

First-principles calculations of LiNbO3 optical properties: From far-infrared to ultraviolet

We perform first-principles calculations of optical properties for ferroelectric phase of LiNbO3 crystal using density functional theory (DFT) for wide range of wavelengths, from far-infrared (IR) to ultraviolet. We study frequency dependence of complex dielectric tensor and related quantities, such as refractive and reflection indices, absorption coefficients, etc. Our calculation incorporates advantages of numerical approaches based on atomic-orbital all-electron Gaussian-type basis sets, as it is realized in CRYSTAL14 program. We compared predictions obtained in general-gradient approach with PBESOL exchange-correlation functional and in hybrid approach with PBESOL0 functional, and we have found that hybrid PBESOL0 functional is more applicable to describe the wide set of the experimental data.

Off‐center Gaussian functions: Applications toward larger basis sets, post‐second‐order correlation treatment, and truncated virtual orbital space in investigations of noncovalent interactions

AbstractThis work is a continuation of our pilot study on the use of off‐center s‐type Gaussian functions in noncovalent interaction calculations. A grid of s‐type Gaussians surrounding the molecule is intended to substitute for the presence of diffuse basis functions, which are important for accurate description of noncovalent interactions. The advantage over the use of diffuse functions is reduction or elimination of linear dependency issues in the atomic orbital basis set, which often causes convergence problems. Placement of a grid of s‐functions on the surface of the molecule allows more favorable scaling of the total number of basis functions with the molecular size to be achieved. In this article, we present several innovations on the concept, namely the parametrization and assessment of grids with triple‐ζ quality basis set; use in post‐second‐order correlation treatment (methods such as Møller–Plesset third‐order or coupled‐cluster); and combination with modified virtual orbital approaches, namely the frozen natural orbitals and optimized virtual orbital space methods.

Velocity-gauge real-time TDDFT within a numerical atomic orbital basis set

The interaction of laser fields with solid-state systems can be modeled efficiently within the velocity-gauge formalism of real-time time dependent density functional theory (RT-TDDFT). In this article, we discuss the implementation of the velocity-gauge RT-TDDFT equations for electron dynamics within a linear combination of atomic orbitals (LCAO) basis set framework. Numerical results obtained from our LCAO implementation, for the electronic response of periodic systems to both weak and intense laser fields, are compared to those obtained from established real-space grid and Full-Potential Linearized Augmented Planewave approaches. Potential applications of the LCAO based scheme in the context of extreme ultra-violet and soft X-ray spectroscopies involving core-electronic excitations are discussed.

Kohn-Sham potentials from electron densities using a matrix representation within finite atomic orbital basis sets.

We revisit the static response function-based Kohn-Sham (KS) inversion procedure for determining the KS effective potential that corresponds to a given target electron density within finite atomic orbital basis sets. Instead of expanding the potential in an auxiliary basis set, we directly update the potential in its matrix representation. Through numerical examples, we show that the reconstructed density rapidly converges to the target density. Preliminary results are presented to illustrate the possibility of obtaining a local potential in real space from the optimized potential in its matrix representation. We have further applied this matrix-based KS inversion approach to density functional embedding theory. A proof-of-concept study of a solvated proton transfer reaction demonstrates the method's promise.

Real-time time dependent density functional theory with numerical atomic orbital basis set: methodology and applications

Real-time time dependent density functional theory (rt-TDDFT) approach directly provides the time domain evolution of electronic wave functions together with ionic movements, presenting a versatile way of real time tracking ultrafast dynamics and phenomena either in perturbative regime or in non-perturbative regime. Thus, rt-TDDFT is a unique ab initio quantum method applicable for the exploration of strong field physics that is beyond the linear response theory. Numerical implementations of the rt-TDDFT based on planewaves and real-space grids have been demonstrated in recent years. However, the above two methods are suitable for the efficient treatment of low energy excitation on the scale of a few electron volts in a small size system. In this paper, we present a state-of-the-art real-time TDDFT approach as implemented in the time dependent ab initio package (TDAP). By employing atomic orbital basis sets, which are small in size and fast in performance, we are able to simulate a large-size system for long electronic propagation time with less computational cost while maintaining relatively high accuracy. The length and velocity-gauge of electromagnetic field are both implemented, showing the flexibility and credibility in applying our methods to various laser induced phenomena in diverse systems including solids, interfaces and two-dimensional materials. Furthermore, recently developed k-resolved algorithm ensures the possibility of handling the problems with a unit cell approach, which significantly reduces the formidable computational costs of traditional rt-TDDFT simulations. Detailed flow and implementation of this method are discussed in this paper, and several quintessential examples for applications are introduced. First, we use the present method to calculate the photoabsorption properties of armchair graphene nanoribbons and monitor the excitation details with momentum resolution. Then, we simulate laser melting of silicon, which captures the most important features of nonthermal melting observed in experiment, and further reveals that it can be attributed to drastic laser-induced change in bonding electron density and subsequent decrease in the melting barrier. After that, a model MoS2/WS2 bilayer system is used as an example to show how our method can be used to monitor the electronic dynamics in such a van der Waals heterostructure. Finally, we show the possibility of controlling the electron dynamic process to enhance high harmonic generation intensity and generate isolated attosecond pulse in monolayer MoS2 via two-color field. Most of the above examples present new ideas in their respective areas and demonstrate that our method has a great potential application in studying interesting ultrafast dynamics phenomena in a wide range of quantum systems.

Reliable and computationally affordable prediction of the energy gap of (TiO2)n (10 ≤ n ≤ 563) nanoparticles from density functional theory.

The optical gap (Ogap) of a set of (TiO2)n nanoclusters and nanoparticles with n = 10-563 and different morphologies such as spherical, octahedral, lamellar, or tubular finite structures is investigated based on a relativistic all-electron description along with a numerical atomic centered orbital basis set. Two different functionals are used, PBE and PBEx, the former corresponds to a standard implementation of the generalized gradient approximation (GGA) and the latter to a hybrid functional with 12.5% of Fock exchange which reproduces the band gap of bulk TiO2 anatase and rutile. It is shown that the inclusion of exchange Fock in the PBE functional promotes a systematic energy gap opening of 1.25 eV relative to the PBE values. Remarkably, a linear correlation is found between PBEx and PBE Ogap calculated values with concomitant similar correlations for the HOMO and LUMO orbital energies. However, it appears that PBEx induces a larger downshift on the HOMO orbital than the upshift observed on the LUMO one. The fact that the PBEx hybrid functional was shown to reproduce the experimental energy gaps of stoichiometric and reduced TiO2 bulk phases leads to a suitable and practical way to successfully estimate Ogap of TiO2 nanoparticles containing up to thousands of atoms with PBEx precision from computationally affordable PBE calculations.

Structural, electronic, and mechanical properties of cubic TiO2: A first-principles study

We present an analysis of structural, electronic, and mechanical properties of cubic titanium dioxide (TiO2) using an all electron orthogonalzed linear combinations of atomic orbitals (OLCAO) basis set under the framework of density functional theory (DFT). The structural property, especially the lattice constant a, and the electronic properties such as the band diagram and density of states (DOS) are studied and analyzed. The mechanical properties such as bulk moduli, shear moduli, Young’s Moduli, and Poison’s ratio are also investigated thoroughly. The calculations are carried out on shear moduli and anisotropy factor for cubic TiO2. The Vickers hardness is also tested for fluorite and pyrite cubic-structured TiO2. Furthermore, the results are compared with the previous theoretical and experimental results. It is found that DFTbased simulation produces results which are approximation to experimental results, whereas the calculated elastic constants are better than the previous theoretical and experimental values.

Detailed Wave Function Analysis for Multireference Methods: Implementation in the Molcas Program Package and Applications to Tetracene.

High-level multireference computations on electronically excited and charged states of tetracene are performed, and the results are analyzed using an extensive wave function analysis toolbox that has been newly implemented in the Molcas program package. Aside from verifying the strong effect of dynamic correlation, this study reveals an unexpected critical influence of the atomic orbital basis set. It is shown that different polarized double-ζ basis sets produce significantly different results for energies, densities, and overall wave functions, with the best performance obtained for the atomic natural orbital (ANO) basis set by Pierloot et al. Strikingly, the ANO basis set not only reproduces the energies but also performs exceptionally well in terms of describing the diffuseness of the different states and of their attachment/detachment densities. This study, thus, not only underlines the fact that diffuse basis functions are needed for an accurate description of the electronic wave functions but also shows that, at least for the present example, it is enough to include them implicitly in the contraction scheme.

Ab initio study of the structure and electronic properties of magnesium and calcium nitrates and their crystal hydrates

The crystal structures and electronic properties of magnesium and calcium nitrates, magnesium nitrate hexahydrate, and calcium nitrate tetrahydrate are studied at the density functional theory level by a hybrid functional in the basis set of localized atomic orbitals using the CRYSTAL14 program code. Atomic structural parameters, atomic charges, bond populations, energy and electron spatial distributions are calculated. The mainly electrostatic nature of interactions between nitro groups and water molecules is shown. The spectrum of the density of states of crystal hydrates, in comparison with nitrates, contains additional bands due to the presence of water. In the spectra of unoccupied states a gap is observed: the anionic gap is ~6.5 eV and the cationic gap is ~8.8 eV.

Density-based errors in mixed-basis mean-field electronic structure, with implications for embedding and QM/MM methods

We consider mean-field electronic structure calculations with subsystems that employ different atomic-orbital basis sets. A major source of error arises in charge-manifestation reactions (including ionization, electron attachment, or deprotonation) due to electronic density artifacts at the subsystem interface. The underlying errors in the electronic density can be largely eliminated with Fock-matrix corrections or by avoiding the use of a minimal basis set in the low-level region. These corrections succeed by balancing the electronegativity of atoms at the subsystem interface, much as link-atoms in QM/MM calculations rely upon balancing the electronegativity of atoms in the truncated QM region.

Fast iterative solution of the Bethe–Salpeter eigenvalue problem using low-rank and QTT tensor approximation

In this paper, we propose and study two approaches to approximate the solution of the Bethe–Salpeter equation (BSE) by using structured iterative eigenvalue solvers. Both approaches are based on the reduced basis method and low-rank factorizations of the generating matrices. We also propose to represent the static screen interaction part in the BSE matrix by a small active sub-block, with a size balancing the storage for rank-structured representations of other matrix blocks. We demonstrate by various numerical tests that the combination of the diagonal plus low-rank plus reduced-block approximation exhibits higher precision with low numerical cost, providing as well a distinct two-sided error estimate for the smallest eigenvalues of the Bethe–Salpeter operator. The complexity is reduced to O(Nb2) in the size of the atomic orbitals basis set, Nb, instead of the practically intractable O(Nb6) scaling for the direct diagonalization. In the second approach, we apply the quantized-TT (QTT) tensor representation to both, the long eigenvectors and the column vectors in the rank-structured BSE matrix blocks, and combine this with the ALS-type iteration in block QTT format. The QTT-rank of the matrix entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, No<Nb, hence the overall asymptotic complexity for solving the BSE problem by the QTT approximation is estimated by O(log⁡(No)No2). We confirm numerically a considerable decrease in computational time for the presented iterative approaches applied to various compact and chain-type molecules, while supporting sufficient accuracy.

Ab initiotensorial electronic friction for molecules on metal surfaces: Nonadiabatic vibrational relaxation

Molecular adsorbates on metal surfaces exchange energy with substrate phonons and low-lying electron-hole pair excitations. In the limit of weak coupling, electron-hole pair excitations can be seen as exerting frictional forces on adsorbates that enhance energy transfer and facilitate vibrational relaxation or hot-electron-mediated chemistry. We have recently reported on the relevance of tensorial properties of electronic friction [M. Askerka et al., Phys. Rev. Lett. 116, 217601 (2016)] in dynamics at surfaces. Here we present the underlying implementation of tensorial electronic friction based on Kohn-Sham density functional theory for condensed phase and cluster systems. Using local atomic-orbital basis sets, we calculate nonadiabatic coupling matrix elements and evaluate the full electronic friction tensor in the Markov limit. Our approach is numerically stable and robust, as shown by a detailed convergence analysis. We furthermore benchmark the accuracy of our approach by calculation of vibrational relaxation rates and lifetimes for a number of diatomic molecules at metal surfaces. We find friction-induced mode-coupling between neighboring CO adsorbates on Cu(100) in a $c(2\ifmmode\times\else\texttimes\fi{}2)$ overlayer to be important for understanding experimental findings.

Liquid Water through Density-Functional Molecular Dynamics: Plane-Wave vs Atomic-Orbital Basis Sets

We determine and compare structural, dynamical, and electronic properties of liquid water at near ambient conditions through density-functional molecular dynamics simulations, when using either plane-wave or atomic-orbital basis sets. In both frameworks, the electronic structure and the atomic forces are self-consistently determined within the same theoretical scheme based on a nonlocal density functional accounting for van der Waals interactions. The overall properties of liquid water achieved within the two frameworks are in excellent agreement with each other. Thus, our study supports that implementations with plane-wave or atomic-orbital basis sets yield equivalent results and can be used indiscriminately in study of liquid water or aqueous solutions.

Approaching the basis set limit for DFT calculations using an environment-adapted minimal basis with perturbation theory: Formulation, proof of concept, and a pilot implementation.

Recently developed density functionals have good accuracy for both thermochemistry (TC) and non-covalent interactions (NC) if very large atomic orbital basis sets are used. To approach the basis set limit with potentially lower computational cost, a new self-consistent field (SCF) scheme is presented that employs minimal adaptive basis (MAB) functions. The MAB functions are optimized on each atomic site by minimizing a surrogate function. High accuracy is obtained by applying a perturbative correction (PC) to the MAB calculation, similar to dual basis approaches. Compared to exact SCF results, using this MAB-SCF (PC) approach with the same large target basis set produces <0.15 kcal/mol root-mean-square deviations for most of the tested TC datasets, and <0.1 kcal/mol for most of the NC datasets. The performance of density functionals near the basis set limit can be even better reproduced. With further improvement to its implementation, MAB-SCF (PC) is a promising lower-cost substitute for conventional large-basis calculations as a method to approach the basis set limit of modern density functionals.

Ambient-Potential Composite Ewald Method for ab Initio Quantum Mechanical/Molecular Mechanical Molecular Dynamics Simulation.

A new approach for performing Particle Mesh Ewald in ab initio quantum mechanical/molecular mechanical (QM/MM) simulations with extended atomic orbital basis sets is presented. The new approach, the Ambient-Potential Composite Ewald (CEw) method, does not perform the QM/MM interaction with Mulliken charges nor electrostatically fit charges. Instead the nuclei and electron density interact directly with the MM environment, but in a manner that avoids the use of dense Fourier transform grids. By performing the electrostatics with the underlying QM density, the CEw method avoids self-consistent field instabilities that have been encountered with simple charge mapping procedures. Potential of mean force (PMF) profiles of the p-nitrophenyl phosphate dissociation reaction in explicit solvent are computed from PBE0/6-31G* QM/MM molecular dynamics simulations with various electrostatic protocols. The CEw profiles are shown to be stable with respect to real-space Ewald cutoff, whereas the PMFs computed from truncated and switched electrostatics produce artifacts. PBE0/6-311G**, AM1/d-PhoT, and DFTB2 QM/MM simulations are performed to generate two-dimensional PMF profiles of the phosphoryl transesterification reactions with ethoxide and phenoxide leaving groups. The semiempirical models incorrectly produce a concerted ethoxide mechanism, whereas PBE0 correctly produces a stepwise mechanism. The ab initio reaction barriers agree more closely to experiment than the semiempirical models. The failure of Mulliken-charge QM/MM-Ewald is analyzed.

Optical response of silver clusters and their hollow shells from linear-response TDDFT

We present a study of the optical response of compact and hollow icosahedral clusters containing up to 868 silver atoms by means of time-dependent density functional theory. We have studied the dependence on size and morphology of both the sharp plasmonic resonance at 3–4 eV (originated mainly from sp-electrons), and the less studied broader feature appearing in the 6–7 eV range (interband transitions). An analysis of the effect of structural relaxations, as well as the choice of exchange correlation functional (local density versus generalised gradient approximations) both in the ground state and optical response calculations is also presented. We have further analysed the role of the different atom layers (surface versus inner layers) and the different orbital symmetries on the absorption cross-section for energies up to 8 eV. We have also studied the dependence on the number of atom layers in hollow structures. Shells formed by a single layer of atoms show a pronounced red shift of the main plasmon resonances that, however, rapidly converge to those of the compact structures as the number of layers is increased. The methods used to obtain these results are also carefully discussed. Our methodology is based on the use of localised basis (atomic orbitals, and atom-centered and dominant-product functions), which bring several computational advantages related to their relatively small size and the sparsity of the resulting matrices. Furthermore, the use of basis sets of atomic orbitals also allows the possibility of extending some of the standard population analysis tools (e.g. Mulliken population analysis) to the realm of optical excitations. Some examples of these analyses are described in the present work.

ChemInform Abstract: Small Atomic Orbital Basis Set First‐Principles Quantum Chemical Methods for Large Molecular and Periodic Systems: A Critical Analysis of Error Sources

AbstractReview: 123 refs.

Accelerating Ab Initio Path Integral Simulations via Imaginary Multiple-Timestepping.

This work investigates the use of multiple-timestep schemes in imaginary time for computationally efficient ab initio equilibrium path integral simulations of quantum molecular motion. In the simplest formulation, only every n(th) path integral replica is computed at the target level of electronic structure theory, whereas the remaining low-level replicas still account for nuclear motion quantum effects with a more computationally economical theory. Motivated by recent developments for multiple-timestep techniques in real-time classical molecular dynamics, both 1-electron (atomic-orbital basis set) and 2-electron (electron correlation) truncations are shown to be effective. Structural distributions and thermodynamic averages are tested for representative analytic potentials and ab initio molecular examples. Target quantum chemistry methods include density functional theory and second-order Møller-Plesset perturbation theory, although any level of theory is formally amenable to this framework. For a standard two-level splitting, computational speedups of 1.6-4.0x are observed when using a 4-fold reduction in time slices; an 8-fold reduction is feasible in some cases. Multitiered options further reduce computational requirements and suggest that quantum mechanical motion could potentially be obtained at a cost not significantly different from the cost of classical simulations.

Accurate tight-binding Hamiltonians for two-dimensional and layered materials

We present a scheme to controllably improve the accuracy of tight-binding Hamiltonian matrices derived by projecting the solutions of plane-wave ab initio calculations on atomic orbital basis sets. By systematically increasing the completeness of the basis set of atomic orbitals, we are able to optimize the quality of the band structure interpolation over wide energy ranges including unoccupied states. This methodology is applied to the case of interlayer and image states, which appear several eV above the Fermi level in materials with large interstitial regions or surfaces such as graphite and graphene. Due to their spatial localization in the empty regions inside or outside of the system, these states have been inaccessible to traditional tight-binding models and even to ab initio calculations with atom-centered basis functions.

A reduced basis approach for calculation of the Bethe–Salpeter excitation energies by using low-rank tensor factorisations

ABSTRACTThe Bethe–Salpeter equation (BSE) is a reliable model for estimating the absorption spectra in molecules and solids on the basis of accurate calculation of the excited states from first principles. Direct diagonalisation of the BSE matrix is practically intractable due to O(N6) complexity scaling in the size of the atomic orbital basis set, N. In this paper, we introduce and analyse a reduced basis approach to computation of the Bethe–Salpeter excitation energies which can lead to a relaxation of the numerical costs down to O(N3). The BSE operator is specified in terms of the two-electron integrals in the Hartree–Fock molecular orbital basis and the respective energies, calculated by the tensor-based solver described in previous works. The reduced basis method includes two steps. First, the diagonal plus low-rank approximation to fully populated blocks in the BSE matrix is calculated, enabling an easier partial eigenvalue solver for a large auxiliary system relying only on matrix–vector multiplications with rank-structured matrices. Second, a small subset of eigenvectors from the auxiliary eigenvalue problem is selected to build a projection of the exact BSE system onto this reduced basis set. Numerical tests demonstrate the ϵ-rank bounds for the blocks of the BSE matrix on examples of some compact molecules. The accuracy of the reduced basis approach vs. the effective matrix rank is illustrated.