Atom-centered Functions Research Articles

Ionic forces and stress tensor in all-electron density functional theory calculations using an enriched finite-element basis

The enriched finite-element basis---wherein the finite-element basis is enriched with atom-centered numerical functions---has recently been shown to be a computationally efficient basis for systematically convergent all-electron density functional theory (DFT) ground-state calculations. In this work, we present the expressions to compute variationally consistent ionic forces and stress tensor for all-electron DFT calculations in the enriched finite-element basis. In particular, we extend the formulation of configurational forces [P. Motamarri and V. Gavini, Phys. Rev. B 97, 165132 (2018)] to the enriched finite-element basis and elucidate the additional contributions arising from the enrichment functions. We demonstrate the accuracy of the formulation by comparing the computed forces and stresses for various benchmark systems with those obtained from finite differencing the ground-state energy. Further, we also benchmark our calculations against the Gaussian basis for molecular systems and against the linearized augmented plane wave with local orbitals basis for periodic systems.

H2O2 adsorption and dissociation on various CeO2 (111) surface models: a first-principles study

Periodic density functional theory (DFT) calculations using the hybrid PBE0 functional and atom-centered Gaussian functions as basis sets were carried out to investigate the absorption and the first steps involved in the decomposition of hydrogen peroxide (H2O2) on three different models of the ceria (111) surface. One of the models is a clean surface, and the others are defective and partially hydroxylated ceria surfaces. On the clean surface, we found that the minimum energy path of hydrogen peroxide decomposition involves a three-step process, i.e., adsorption, deprotonation, and formation of the peroxide anion, stabilized through its interaction with the surface at a Ce (IV) site, with activation barriers of less than about 0.5 eV. The subsequent formation of superoxide anions and molecular oxygen species is attributed to electron transfer from the reactants to the Ce (IV) ions underneath. On the defective surface, H2O2 dissociation is an energetically downhill reaction thermodynamically driven by the healing of the O vacancies, after the reduction and decomposition of H2O2 into oxygen and water. On the hydroxylated surface, H2O2 is first adsorbed by forming a favorable H-bond and then undergoes heterolytic dissociation, forming two hydroxyl groups at two vicinal Ce sites.

Mapping valence electron distributions with multipole density formalism using 4D-STEM

Recent advancement in aberration correction and detector technology opened a door to various applications using 4D-STEM, which yields a diffraction pattern for each scanning position within a crystal unit-cell in scanning transmission electron microscopy (STEM) and generates incredible amounts of data in momentum space. Currently 4D-STEM analysis relies on the center-of-mass of the diffraction patterns in electric field and charge density mapping. It only derives the total projected charge density and is limited to phase objects, e.g. extremely thin samples. Here, we propose a new analytical method to accurately map aspherical valence electron distributions with atom-centered multipolar functions formalism using the whole 4D-STEM dataset. We demonstrate that, with the full dynamical calculations for various sample thicknesses, the method is sensitive not only to the miniscule charge transfer, but also to the atomic site symmetry and aspherical electron orbitals. The process of the refinement is much more robust and reliable than quantitative convergent beam electron diffraction.

Coupled-cluster treatment of molecular strong-field ionization.

Ionization rates and Stark shifts of H2, CO, O2, H2O, and CH4 in static electric fields have been computed with coupled-cluster methods in a basis set of atom-centered Gaussian functions with a complex-scaled exponent. Consideration of electron correlation is found to be of great importance even for a qualitatively correct description of the dependence of ionization rates and Stark shifts on the strength and orientation of the external field. The analysis of the second moments of the molecular charge distribution suggests a simple criterion for distinguishing tunnel and barrier suppression ionization in polyatomic molecules.

Reflections on the MOLECULE integral program (and its developer)

ABSTRACTIn this essay, I review some of the advances made by Jan Almlöf in the calculation of molecular integrals over Gaussian-type basis functions, and in the use of molecular symmetry to both produce integrals over symmetry-adapted combinations of basis functions and to ensure that these latter were constructed from the minimum possible number of distinct integrals over atom-centred functions. The survey is, as already noted, an essay, historical in nature, and I crave the reader's indulgence that it is structured as an essay and not as a conventional scientific manuscript.

Efficient floating diffuse functions for accurate characterization of the surface-bound excess electrons in water cluster anions.

In this work, the effect of diffuse function types (atom-centered diffuse functions versus floating functions and s-type versus p-type diffuse functions) on the structures and properties of three representative water cluster anions featuring a surface-bound excess electron is studied and we find that an effective combination of such two kinds of diffuse functions can not only reduce the computational cost but also, most importantly, considerably improve the accuracy of results and even avoid incorrect predictions of spectra and the EE shape. Our results indicate that (a) simple augmentation of atom-centered diffuse functions is beneficial for the vertical detachment energy convergence, but it leads to very poor descriptions for the singly occupied molecular orbital (SOMO) and lowest unoccupied molecular orbital (LUMO) distributions of the water cluster anions featuring a surface-bound excess electron and thus a significant ultraviolet spectrum redshift; (b) the ghost-atom-based floating diffuse functions can not only contribute to accurate electronic calculations of the ground state but also avoid poor and even incorrect descriptions of the SOMO and the LUMO induced by excessive augmentation of atom-centered diffuse functions; (c) the floating functions can be realized by ghost atoms and their positions could be determined through an optimization routine along the dipole moment vector direction. In addition, both the s- and p-type floating functions are necessary to supplement in the basis set which are responsible for the ground (s-type character) and excited (p-type character) states of the surface-bound excess electron, respectively. The exponents of the diffuse functions should also be determined to make the diffuse functions cover the main region of the excess electron distribution. Note that excessive augmentation of such diffuse functions is redundant and even can lead to unreasonable LUMO characteristics.

Permutation invariant potential energy surfaces for polyatomic reactions using atomistic neural networks.

The applicability and accuracy of the Behler-Parrinello atomistic neural network method for fitting reactive potential energy surfaces is critically examined in three systems, H + H2 → H2 + H, H + H2O → H2 + OH, and H + CH4 → H2 + CH3. A pragmatic Monte Carlo method is proposed to make efficient choice of the atom-centered mapping functions. The accuracy of the potential energy surfaces is not only tested by fitting errors but also validated by direct comparison in dynamically important regions and by quantum scattering calculations. Our results suggest this method is both accurate and efficient in representing multidimensional potential energy surfaces even when dissociation continua are involved.

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.

Variable van der Waals Radii Derived From a Hybrid Gaussian Charge Distribution Model for Continuum-Solvent Electrostatic Calculations

Abstract We introduce a hybrid Gaussian charge distribution model (HGM) that partitions the molecular electron density into overlapping spherical atomic domains. The semi-empirical HGM consists of atom-centered spherical Gaussian functions and discrete point charges, which are optimized to reproduce the electrostatic potential on the molecular surface as well as the number of electrons in atom-centered and certain off-atom-centered spherical regions as closely as possible. In contrast, our previous Gaussian charge distribution model [J. Chem. Phys. 129, 014509 (2008)] contained only spherical Gaussian functions and was not required to reproduce the number of electrons in off-atom-centered regions. Variable van der Waals (vdW) radii fluctuating around the Bondi radii are derived from the HGM based on the isodensity contour concept and further employed to define the molecular cavity in our quantum mechanical/Poisson–Boltzmann/surface area model as well as the polarizable continuum model. The variable vdW radii produce more accurate solvation free energies for 31 neutral molecules than the Bondi radii for both continuum solvent models (CSM) consistently. Moreover, for H atoms, the linear dependence of the atomic radii on the atomic partial charges is identified.

Benchmark calculations of excess electrons in water cluster cavities: balancing the addition of atom-centered diffuse functions versus floating diffuse functions.

Diffuse functions have been proved to be especially crucial for the accurate characterization of excess electrons which are usually bound weakly in intermolecular zones far away from the nuclei. To examine the effects of diffuse functions on the nature of the cavity-shaped excess electrons in water cluster surroundings, both the HOMO and LUMO distributions, vertical detachment energies (VDEs) and visible absorption spectra of two selected (H2O)24(-) isomers are investigated in the present work. Two main types of diffuse functions are considered in calculations including the Pople-style atom-centered diffuse functions and the ghost-atom-based floating diffuse functions. It is found that augmentation of atom-centered diffuse functions contributes to a better description of the HOMO (corresponding to the VDE convergence), in agreement with previous studies, but also leads to unreasonable diffuse characters of the LUMO with significant red-shifts in the visible spectra, which is against the conventional point of view that the more the diffuse functions, the better the results. The issue of designing extra floating functions for excess electrons has also been systematically discussed, which indicates that the floating diffuse functions are necessary not only for reducing the computational cost but also for improving both the HOMO and LUMO accuracy. Thus, the basis sets with a combination of partial atom-centered diffuse functions and floating diffuse functions are recommended for a reliable description of the weakly bound electrons. This work presents an efficient way for characterizing the electronic properties of weakly bound electrons accurately by balancing the addition of atom-centered diffuse functions and floating diffuse functions and also by balancing the computational cost and accuracy of the calculated results, and thus is very useful in the relevant calculations of various solvated electron systems and weakly bound anionic systems.

A novel Gaussian-Sinc mixed basis set for electronic structure calculations.

A Gaussian-Sinc basis set methodology is presented for the calculation of the electronic structure of atoms and molecules at the Hartree-Fock level of theory. This methodology has several advantages over previous methods. The all-electron electronic structure in a Gaussian-Sinc mixed basis spans both the "localized" and "delocalized" regions. A basis set for each region is combined to make a new basis methodology-a lattice of orthonormal sinc functions is used to represent the "delocalized" regions and the atom-centered Gaussian functions are used to represent the "localized" regions to any desired accuracy. For this mixed basis, all the Coulomb integrals are definable and can be computed in a dimensional separated methodology. Additionally, the Sinc basis is translationally invariant, which allows for the Coulomb singularity to be placed anywhere including on lattice sites. Finally, boundary conditions are always satisfied with this basis. To demonstrate the utility of this method, we calculated the ground state Hartree-Fock energies for atoms up to neon, the diatomic systems H2, O2, and N2, and the multi-atom system benzene. Together, it is shown that the Gaussian-Sinc mixed basis set is a flexible and accurate method for solving the electronic structure of atomic and molecular species.

Additional diffuse functions in basis sets for dipole-bound excited states of anions

The existence of dipole-bound excited states has largely been shown by previous computation through the use of the standard LCAO-MO creation with numerous and diffuse atom-centered functions. Earlier work has suggested that the standard aug-cc-pVDZ basis set augmented with a small set of diffuse Rydberg-like orbitals is sufficient for the computation of dipole-bound ground states in anions. In this work, we explore the addition of four, even-tempered s-type functions augmenting the aug-cc-pVDZ set for the computation of dipole-bound excited states. It is shown herein that the inclusion of these orbitals is more cost-effective and internally accurate than atom-centered functions. Both vertical and adiabatic excitation energies are negligibly affected by the centering location of these additional diffuse functions. However, the excited state harmonic vibrational frequencies can be influenced by the placement of these additional diffuse functions, especially for the lowest energy modes.

Dispersion-correcting potentials can significantly improve the bond dissociation enthalpies and noncovalent binding energies predicted by density-functional theory.

Dispersion-correcting potentials (DCPs) are atom-centered Gaussian functions that are applied in a manner that is similar to effective core potentials. Previous work on DCPs has focussed on their use as a simple means of improving the ability of conventional density-functional theory methods to predict the binding energies of noncovalently bonded molecular dimers. We show in this work that DCPs developed for use with the LC-ωPBE functional along with 6-31+G(2d,2p) basis sets are capable of simultaneously improving predicted noncovalent binding energies of van der Waals dimer complexes and covalent bond dissociation enthalpies in molecules. Specifically, the DCPs developed herein for the C, H, N, and O atoms provide binding energies for a set of 66 noncovalently bonded molecular dimers (the "S66" set) with a mean absolute error (MAE) of 0.21 kcal/mol, which represents an improvement of more than a factor of 10 over unadorned LC-ωPBE/6-31+G(2d,2p) and almost a factor of two improvement over LC-ωPBE/6-31+G(2d,2p) used in conjunction with the "D3" pairwise dispersion energy corrections. In addition, the DCPs reduce the MAE of calculated X-H and X-Y (X,Y = C, H, N, O) bond dissociation enthalpies for a set of 40 species from 3.2 kcal/mol obtained with unadorned LC-ωPBE/6-31+G(2d,2p) to 1.6 kcal/mol. Our findings demonstrate that broad improvements to the performance of DFT methods may be achievable through the use of DCPs.

Localized overlap algorithm for unexpanded dispersion energies

First-principles-based, linearly scaling algorithm has been developed for calculations of dispersion energies from frequency-dependent density susceptibility (FDDS) functions with account of charge-overlap effects. The transition densities in FDDSs are fitted by a set of auxiliary atom-centered functions. The terms in the dispersion energy expression involving products of such functions are computed using either the unexpanded (exact) formula or from inexpensive asymptotic expansions, depending on the location of these functions relative to the dimer configuration. This approach leads to significant savings of computational resources. In particular, for a dimer consisting of two elongated monomers with 81 atoms each in a head-to-head configuration, the most favorable case for our algorithm, a 43-fold speedup has been achieved while the approximate dispersion energy differs by less than 1% from that computed using the standard unexpanded approach. In contrast, the dispersion energy computed from the distributed asymptotic expansion differs by dozens of percent in the van der Waals minimum region. A further increase of the size of each monomer would result in only small increased costs since all the additional terms would be computed from the asymptotic expansion.

Extension of the B3LYP–dispersion-correcting potential approach to the accurate treatment of both inter- and intra-molecular interactions

We recently demonstrated that dispersion-correcting potentials (DCPs), which are atom-centered Gaussian-type functions that were developed for use with B3LYP (Torres and DiLabio in J Phys Chem Lett 3:1738–1744, 2012), greatly improved the ability of the underlying functional to predict non-covalent interactions. However, the recent application of the B3LYP–DCP approach to study the β-scission of the cumyloxyl radical led to a calculated barrier height that was over-estimated by ca. 8 kcal/mol. We demonstrate in the present work that the source of this error arises from the previously developed carbon atom DCPs, which erroneously alters the electron density in the C–C covalent-bonding region. In this work, we developed a new C-DCP with a form that was expected to less strongly influence the electron density in the covalent bonding region. Tests of the new C-DCP, in conjunction with previously published H-, N-, and O-DCPs, with B3LYP–DCP/6-31+G(2d,2p) on the S66, S22B, HSG-A, and HC12 databases of non-covalently interacting dimers showed that it is one of the most accurate methods available for treating intermolecular interactions, giving mean absolute errors (MAEs) of 0.19, 0.27, 0.16, and 0.18 kcal/mol, respectively. Additional testing on the S12L database of very large complexation systems gave an MAE of 2.6 kcal/mol, demonstrating that the B3LYP–DCP/6-31+G(2d,2p) approach to be one of the best-performing and most feasible methods for treating large systems containing significant non-covalent interactions. Finally, we showed that the modeling of C–C-making/C–C-breaking chemistry is well predicted using the newly developed DCPs. In addition to predicting a barrier height for the β-scission of the cumyloxyl radical, that is, within 1.7 kcal/mol of the high-level value, application of B3LYP–DCP/6-31+G(2d,2p) to 10 databases that include reaction barrier heights and energies, isomerization energies, and relative conformation energies gives performance that is among the best of all available dispersion-corrected density-functional theory approaches.

Intensity oscillations in the carbon 1s ionization cross sections of 2-butyne

Carbon 1s photoelectron spectra for 2-butyne (CH3C≡CCH3) measured in the photon energy range from threshold to 150 eV above threshold show oscillations in the intensity ratio C2,3/C1,4. Similar oscillations have been seen in chloroethanes, where the effect has been attributed to EXAFS-type scattering from the substituent chlorine atoms. In 2-butyne, however, there is no high-Z atom to provide a scattering center and, hence, oscillations of the magnitude observed are surprising. The results have been analyzed in terms of two different theoretical models: a density-functional model with B-spline atom-centered functions to represent the continuum electrons and a multiple-scattering model using muffin-tin potentials to represent the scattering centers. Both methods give a reasonable description of the energy dependence of the intensity ratios.

Symmetry-adapted Wannier functions in the maximal localization procedure

A procedure to construct symmetry-adapted Wannier functions in the framework of the maximally-localized Wannier function approach[Marzari and Vanderbilt, Phys. Rev. B \textbf{56}, 12847 (1997); Souza, Marzari, and Vanderbilt, \textit{ibid.} \textbf{65}, 035109 (2001)] is presented. In this scheme the minimization of the spread functional of the Wannier functions is performed with constraints that are derived from symmetry properties of the specified set of the Wannier functions and the Bloch functions used to construct them, therefore one can obtain a solution that does not necessarily yield the global minimum of the spread functional. As a test of this approach, results of atom-centered Wannier functions for GaAs and Cu are presented.

Extracting chemical information from plane wave calculations by a 3D ‘fuzzy atoms’ analysis

Bond order and valence indices have been calculated by the method of the three-dimensional ‘fuzzy atoms’ analysis, using the numerical molecular orbitals obtained from plane wave DFT calculations, i.e., without introducing any external atom-centered functions. Weight functions of both Hirshfeld and Becke types have been applied. The results are rather close to the similar ‘fuzzy atoms’ ones obtained by using atom-centered basis sets and agree well with the chemical expectations, stressing the power of the genuine chemical concepts.

A divide and conquer real-space approach for all-electron molecular electrostatic potentials and interaction energies

A computational scheme to perform accurate numerical calculations of electrostatic potentials and interaction energies for molecular systems has been developed and implemented. Molecular electron and energy densities are divided into overlapping atom-centered atomic contributions and a three-dimensional molecular remainder. The steep nuclear cusps are included in the atom-centered functions making the three-dimensional remainder smooth enough to be accurately represented with a tractable amount of grid points. The one-dimensional radial functions of the atom-centered contributions as well as the three-dimensional remainder are expanded using finite element functions. The electrostatic potential is calculated by integrating the Coulomb potential for each separate density contribution, using our tensorial finite element method for the three-dimensional remainder. We also provide algorithms to compute accurate electron-electron and electron-nuclear interactions numerically using the proposed partitioning. The methods have been tested on all-electron densities of 18 reasonable large molecules containing elements up to Zn. The accuracy of the calculated Coulomb interaction energies is in the range of 10(-3) to 10(-6) E(h) when using an equidistant grid with a step length of 0.05 a(0).

Are There Atomic Orbitals in a Molecule?

Effective atomic orbitals (AOs) have been calculated by the method of the "fuzzy atoms" analysis by using the numerical molecular orbitals (MOs) obtained from plane-wave DFT calculation, i.e., without introducing any atom-centered functions. The results show that in the case of nonhypervalent atoms there are as many effective AOs with non-negligible occupation numbers, as many orbitals are in the classical minimal basis set of the given atom. This means that, for nonhypervalent systems, it is possible to present the MOs as sums of effective atomic orbitals that resemble very much the atomic minimal basis orbitals of the individual atoms (or their hybrids). For hypervalent atoms some additional orbitals basically of d-type are also of some importance; they are necessary to describe the back-donation to these positive atoms. It appears that the d-type orbitals play a similar role also for strongly positive carbon atoms. The method employed here is also useful to decide whether the use of polarization functions of a given type is a matter of conceptual importance or has only a numerical effect.