All-electron Results Research Articles

Adaptively compressed exchange in the linearized augmented plane wave formalism

We present an implementation of the adaptively compressed exchange (ACE) operator in the linearized-augmented plane waves formalism. ACE is a low-rank representation of the Fock exchange that avoids any loss of precision for the total energy. Our study shows that this property remains in the all-electron case, as we apply this method in nonrelativistic total-energy calculations with a hybrid exchange-correlation functional PBE0. The obtained data for light atoms and molecules are within a few $\ensuremath{\mu}\mathrm{Ha}$ of the precise multiresolution-analysis calculations. Aside from ACE, another key ingredient to achieve such a high precision with Fock exchange was the use of high-energy local orbitals. Finally, we use this implementation to calculate PBE0 gaps in solids and compare the results to other all-electron results.

Correlation Consistent Basis Sets and Core Polarization Potentials for Al-Ar with ccECP Pseudopotentials.

New correlation consistent basis sets for the second-row atoms (Al–Ar) to be used with the neon-core correlation consistent effective core potentials (ccECPs) have been developed. The basis sets, denoted cc-pV(n+d)Z-ccECP (n = D, T, Q), include the “tight”-d functions that are known to be important for second-row elements. Sets augmented with additional diffuse functions are also reported. Effective core polarization potentials (CPPs) to account for the effect of core–valence correlation have been adjusted for the same elements, and two different forms of the CPP cutoff function have been analyzed. The accuracy of both the basis sets and the CPPs is assessed through benchmark calculations at the coupled-cluster level of theory for atomic and molecular properties. Agreement with all-electron results is much improved relative to the basis sets that originally accompanied the ccECPs; moreover, the combination of cc-pV(n+d)Z-ccECP and CPPs is found to be a computationally efficient and accurate alternative to including core electrons in the correlation treatment.

High Accuracy Transition Metal Effective Cores for the Many-Body Diffusion Monte Carlo Method.

Practical applications of the real-space diffusion Monte Carlo (DMC) method require the removal of core electrons, where currently localization approximations of semilocal potentials are generally used in the projector. Accurate calculations of complex solids and large molecules demand minimizing the impact of approximated atomic cores. Prior works have shown that the errors from such approximations can be sizable in both finite and periodic systems. In this work, we show that a class of differential pseudopotentials, known as pseudo-Hamiltonians, can be constructed for the 3d transition metal atoms, entirely removing the need for any localization scheme in the DMC projector. As a proof of principle, we demonstrate the approach for the case of Co. In order to minimize errors in the pseudo-Hamiltonian at the many-body level, we generalize the recently proposed correlation-consistent pseudopotential generation scheme to successively close semilocal representations of the differential potentials. Our generation scheme successfully produces potentials tailored specifically for real space projector quantum Monte Carlo methods with low error at the many-body level, i.e., with many-body scattering properties very close to relativistic all-electron results. In particular, we show that the agreement with respect to atomic and molecular quantities reach chemical accuracy in many cases─on par with the most accurate semilocal pseudopotentials available. Further, our pseudo-Hamiltonian generation scheme utilizes standard quantum chemistry codes designed only to work with semilocal pseudopotentials, enabling straightforward generation of pseudo-Hamiltonians for additional elements in future works.

Norm-Conserving Pseudopotentials and Basis Sets to Explore Actinide Chemistry in Complex Environments.

We have developed a new set of norm-conserving pseudopotentials and companion Gaussian basis sets for the actinide (An) series (Ac-Lr) using the Goedecker, Teter, and Hutter (GTH) formalism with the Perdew, Burke, and Ernzerhof (PBE) exchange-correlation functional of generalized gradient approximation. To test the accuracy and reliability of the newly parameterized An-GTH pseudopotentials and basis sets, a variety of benchmarks on actinide-containing molecules were carried out and compared to all-electron and available experimental results. The new pseudopotentials include both medium- ([Xe]4f14) and large-core ([Xe]4f145d10) options that successfully reproduce the structures and energetics, particularly redox processes. The medium-core size set, in particular, reproduces all-electron calculations over multiple oxidation states from 0 to VII, whereas the large-core set is suitable only for the early series elements and low oxidation states. The underlying reason for these transferability issues is discussed in detail. This work fills a critical void in the literature for studying the chemistry of 5f-block elements in the condensed phase.

Robust All-Electron Optimization in Orbital-Free Density-Functional Theory Using the Trust-Region Image Method.

We present a Gaussian-basis implementation of orbital-free density-functional theory (OF-DFT) in which the trust-region image method (TRIM) is used for optimization. This second-order optimization scheme has been constructed to provide benchmark all-electron results with very tight convergence of the particle-number constraint, associated chemical potential, and electron density. It is demonstrated that, by preserving the saddle-point nature of the optimization and simultaneously optimizing the density and chemical potential, an order of magnitude reduction in the number of iterations required for convergence is obtained. The approach is compared and contrasted with a new implementation of the nested optimization scheme put forward by Chan, Cohen, and Handy. Our implementation allows for semilocal kinetic-energy (and exchange-correlation) functionals to be handled self-consistently in all-electron calculations. The all-electron Gaussian-basis setting for these calculations will enable direct comparison with a wide range of standard high-accuracy quantum-chemical methods as well as with Kohn-Sham density-functional theory. We expect that the present implementation will provide a useful tool for analyzing the performance of approximate kinetic-energy functionals in finite systems.

Density functional theory plane-wave/pseudopotential calculations of the equation of state of rhenium in the terapascal regime

The equation of state (EOS) of hexagonal close-packed rhenium has been previously determined up to 1.5 TPa (V/V0 = 0.46) using all-electron (AE) density functional theory (DFT) including core relativistic effects [Phys. Rev. B 100, 174107 (2019)]. This work focuses on assessing the validity of pseudopotentials at such high-pressures, above 1 terapascal (TPa), by comparing the EOS obtained using plane-wave/pseudopotentials (PW/PPs) with previous AE calculations. Both ONCV pseudopotentials tested (PBE and PBEsol) yield results in good agreement with all-electron calculations. The fitted parameters of the Rose-Vinet EOS, the bulk modulus, B0 (380(2) GPa and 379(3) GPa for the PBE and PBEsol PPs), and its pressure derivative, B0’(4.58(2) and 4.59(3) for the PBE and PBEsol PPs), are in good agreement with previous all-electron results (B0=367(5) GPa and B0’=4.64(3)). The agreement is worse for the equilibrium volume at zero pressure, V0 (199.9(2) Bohr3 and 199.2(2) Bohr3 for PBE and PBEsol PPs, versus 196.7(3) Bohr3 for AE). Even so, for a given V/V0 ratio, the pressures estimated using the EOS derived from the PPs and AE calculations are in very good agreement, thus suggesting that the PPs used in this study are valid even at such high compression regime.

Diffusion quantum Monte Carlo calculations with a recent generation of effective core potentials for ionization potentials and electron affinities

Pseudopotentials are an essential ingredient in diffusion quantum Monte Carlo (DMC) calculations to increase efficiency substantially. A new generation of effective core potentials (ccECP) has been recently developed for DMC calculations. In this paper, performance of DMC using ccECP potentials on total energies, ionization potentials (IPs) and electron affinities (EAs) of some second- and third-row atoms and molecules is investigated systematically with different types of trial wave functions. DMC results are compared with those of high-level coupled-cluster methods extrapolated to complete basis set limit (CC-CBS). Error of ccECP potentials on IPs and EAs is also evaluated through a comparison with those from all-electron calculations. Our results show that mean errors in DMC energies with the ccECP potentials are smaller than those with the pseudopotentials developed by Burkatzki, Filippi, and Dolg (BFD), when the same type of trial wave functions is adopted. Mean absolute deviations (MADs) on IPs of DMC compared with those of CC-CBS are about 1.6 kcal/mol with single-determinant-Jastrow trial wave functions, and 1 kcal/mol with multideterminant-Jastrow trial wave functions using either ccECP or BFD potentials. MADs on EAs with DMC using the ccECP potentials are about 1 kcal/mol and slightly larger than those with the BFD potentials. Our results show that ccECP potentials are able to provide reliable IPs and EAs in DMC calculations. Accuracy of IPs and EAs from DMC calculations using ccECP potentials is similar to that with the BFD potentials, although mean error in total DMC energies with ccECP potentials is smaller. Furthermore, error of ccECP potentials in DMC calculations on IPs and EAs is smaller than that of BFD potentials compared with all-electron results.

Calculation of spin-spin zero-field splitting within periodic boundary conditions: Towards all-electron accuracy

For high-spin centers, one of the key spectroscopic fingerprints is the zero-field splitting (ZFS) addressable by electron paramagnetic resonance. In this paper, an implementation of the spin-spin contribution to the ZFS tensor within the projector augmented-wave (PAW) formalism is reported. We use a single-determinant approach proposed by M. J. Rayson and P. R. Briddon [Phys. Rev. B 77, 035119 (2008)], and complete it by adding a PAW reconstruction term which has not been taken into account before. We benchmark the PAW approach against a well-established all-electron method for a series of diatomic radicals and defects in diamond and cubic silicon carbide. While for some of the defect centers the PAW reconstruction is found to be almost negligible, in agreement with the common assumption, we show that in general it significantly improves the calculated ZFS towards the all-electron results.

Virtual Orbital Many-Body Expansions: A Possible Route towards the Full Configuration Interaction Limit.

It is demonstrated how full configuration interaction (FCI) results in extended basis sets may be obtained to within sub-kJ/mol accuracy by decomposing the energy in terms of many-body expansions in the virtual orbitals of the molecular system at hand. This extension of the FCI application range lends itself to two unique features of the current approach, namely, that the total energy calculation can be performed entirely within considerably reduced orbital subspaces and may be so by means of embarrassingly parallel programming. Facilitated by a rigorous and methodical screening protocol and further aided by expansion points different from the Hartree-Fock solution, all-electron numerical results are reported for H2O in polarized core-valence basis sets ranging from double-ζ (10 e, 28 o) to quadruple-ζ (10 e, 144 o) quality.

Diffusion Monte Carlo for Accurate Dissociation Energies of 3d Transition Metal Containing Molecules.

Transition metals and transition metal compounds are important to catalysis, photochemistry, and many superconducting systems. We study the performance of diffusion Monte Carlo (DMC) applied to transition metal containing dimers (TMCDs) using single-determinant Slater-Jastrow trial wavefunctions and investigate the possible influence of the locality and pseudopotential errors. We find that the locality approximation can introduce nonsystematic errors of up to several tens of kilocalories per mole in the absolute energy of Cu and CuH if Ar or Mg core pseudopotentials (PPs) are used for the 3d transition metal atoms. Even for energy differences such as binding energies, errors due to the locality approximation can be problematic if chemical accuracy is sought. The use of the Ne core PPs developed by Burkatzki et al. (J. Chem. Phys. 2008, 129, 164115), the use of linear energy minimization rather than unreweighted variance minimization for the optimization of the Jastrow function, and the use of large Jastrow parametrizations reduce the locality errors. In the second section of this article, we study the general performance of DMC for 3d TMCDs using a database of binding energies of 20 TMCDs, for which comparatively accurate experimental data is available. Comparing our DMC results to these data for our results that compare best with experiment, we find a mean unsigned error (MUE) of 4.5 kcal/mol. This compares well with the achievable accuracy in CCSDT(2)Q (MUE = 4.6 kcal/mol) and the best all-electron DFT results (MUE = 4.5 kcal/mol) for the same set of systems (Truhlar et al. J. Chem. Theory Comput. 2015, 11, 2036-2052). The mean errors in DMC depend less on the exchange-correlation functionals used to generate the trial wavefunction than the corresponding mean errors in the underlying DFT calculations. Furthermore, the QMC results obtained for each molecule individually vary less with the functionals used. These observations are relevant for systems such as molecules interacting with transition metal surfaces where the DFT functionals performing best for molecules (hybrids) do not yield improvements in DFT. Overall, the results presented in this article yield important guidelines for both the assessment of the achievable accuracy with DMC and the design of DMC calculations for systems including transition metal atoms.

Optimization algorithm for the generation of ONCV pseudopotentials

We present an optimization algorithm to construct pseudopotentials and use it to generate a set of Optimized Norm-Conserving Vanderbilt (ONCV) pseudopotentials for elements up to Z=83 (Bi) (excluding Lanthanides). We introduce a quality function that assesses the agreement of a pseudopotential calculation with all-electron FLAPW results, and the necessary plane-wave energy cutoff. This quality function allows us to use a Nelder–Mead optimization algorithm on a training set of materials to optimize the input parameters of the pseudopotential construction for most of the periodic table. We control the accuracy of the resulting pseudopotentials on a test set of materials independent of the training set. We find that the automatically constructed pseudopotentials (http://www.quantum-simulation.org) provide a good agreement with the all-electron results obtained using the FLEUR code with a plane-wave energy cutoff of approximately 60 Ry.

Systematic pseudopotentials from reference eigenvalue sets for DFT calculations

Pseudopotential-based Density-Functional Theory (DFT) permits the calculation of material properties with a modest computational effort, besides an acknowledged tradeoff of generating and testing pseudopotentials that reproduce established benchmark structural and electronic properties. To facilitate the needed benchmarking process, here we present a pragmatic method to optimize pseudopotentials for arbitrary materials directly from eigenvalue sets consistent with all-electron results. This method thus represents a much needed pragmatic route for the creation and assessment of sensitive pseudopotentials for DFT calculations that has been exemplified within the context of the SIESTA code. Comprehensive optimized pseudopotentials, basis sets, and lattice parameters are provided for twenty chemical elements in the bulk, and for both LDA and GGA exchange–correlation potentials. This method helps addressing the following issues: (i) the electronic dispersion and structural properties for Ge, Pd, Pt, Au, Ag, and Ta better agree with respect to all-electron results now, (ii) we provide the expected metallic behavior of Sn in the bulk – which comes out semiconducting when using available pseudopotentials, (iii) we create a validated pseudopotential for LDA-tungsten, and (iv) we create the first Bi pseudopotential for SIESTA that reproduces well-known electron and hole pockets at the L and T points. We investigated the transferability of these pseudopotentials and basis sets, and predict a new phase for two-dimensional tin as well.

Relativistic Small-Core Pseudopotentials for Actinium, Thorium, and Protactinium

Small-core pseudopotentials for actinium, thorium, and protactinium have been energy-adjusted to multiconfiguration Dirac-Hartree-Fock reference data based on the Dirac-Coulomb-Breit Hamiltonian and the Fermi nucleus model. Corresponding optimized valence basis sets of polarized valence quadruple-ζ quality are presented. Atomic test calculations for the first four ionization potentials show satisfactory results at both the Hartree-Fock and the multireference averaged coupled-pair functional level. Highly correlated Fock-space coupled cluster calculations demonstrate that the new pseudopotentials yield ionization potentials, which are in excellent agreement with corresponding all-electron results and experimental data. The pseudopotentials and basis sets supplement a similar set previously published for uranium.

Consistent treatment of charged systems within periodic boundary conditions: The projector augmented-wave and pseudopotential methods revisited

The ab initio calculation of charged defect properties in solids is not straightforward because of the delicate interplay between the long-range Coulomb interaction and the periodic boundary conditions. We derive the projector augmented-wave (PAW) energy and Hamiltonian with special care taken on the potentials from the Coulomb interaction. By explicitly treating the background compensation charge, we find additional terms in the total energy of the charged cells and in the potential. We show that these background terms are needed to accurately reproduce all-electron calculations of the formation energy of a charged defect. In particular, the previous PAW expressions were spuriously sensitive to the pseudization conditions and this artifact is removed by the background term. This PAW derivation also provides insights into the norm-conserving pseudopotential framework.We propose then an alternative definition for the total energy of charged cells and for the Kohn-Sham potential within this framework that better approximates the all-electron results.

Scalable properties of metal clusters: A comparative DFT study of ionic-core treatments

To assess various ionic-core approximations we carried out density functional calculations on a series of octahedral palladium and gold model clusters Mn with n=13–147. We compared results for average bond lengths, cohesive energies, vertical ionization potentials, and electron affinities to the corresponding all-electron scalar relativistic results. We used extrapolated bulk values to compare with experiments. The results of the projector-augmented wave method agree best with the all-electron results. When modeling palladium–gold nanoalloy particles, one should keep in mind that current ECP or PAW modeling of palladium moieties appears to be more accurate than modeling of gold species.

Generalized relativistic effective core potential calculations of the adiabatic potential curve and spectroscopic constants for the ground electronic state of the Ca2molecule

The potential curve, dissociation energy, equilibrium internuclear distance, and spectroscopic constants for the ground state of the Ca2 molecule are calculated with the help of the generalized relativistic effective core potential method, which allows one to exclude the inner core electrons from the calculations and to take the relativistic effects into account effectively. Extensive generalized correlation basis sets were constructed and used. The scalar relativistic coupled cluster method with corrections for high-order cluster amplitudes is used for the correlation treatment. The results are analyzed and compared with the experimental data and corresponding all-electron results. © 2013 Wiley Periodicals, Inc.

Molecular electrostatic potential at the atomic sites in the effective core potential approximation

Considering calculations of the molecular electrostatic potential at the atomic sites (MEP@AS) in the presence of effective core potentials (ECP), we found that the consequent use of the definition of MEP@AS based on the energy derivative with respect to nuclear charge leads to a formula that differs by one term from the result of simple application of Coulomb's law. We have developed a general method to analytically treat derivatives of ECP with respect to nuclear charge. Benchmarking calculations performed on a set of simple molecules show that our formula leads to a systematic decrease in the error connected with the introduction of ECP when compared to all-electron results. Because of a straightforward implementation and relatively low costs of the developed procedure we suggest to use it by default.

Comparison of density functional approximations and the finite-temperature Hartree-Fock approximation in warm dense lithium

We compare the behavior of the finite-temperature Hartree-Fock model with that of thermal density functional theory using both ground-state and temperature-dependent approximate exchange functionals. The test system is bcc Li in the temperature-density regime of warm dense matter (WDM). In this exchange-only case, there are significant qualitative differences in results from the three approaches. Those differences may be important for Born-Oppenheimer molecular dynamics studies of WDM with ground-state approximate density functionals and thermal occupancies. Such calculations require reliable regularized potentials over a demanding range of temperatures and densities. By comparison of pseudopotential and all-electron results at T=0 K for small Li clusters of local bcc symmetry and bond lengths equivalent to high density bulk Li, we determine the density ranges for which standard projector augmented wave (PAW) and norm-conserving pseudopotentials are reliable. Then, we construct and use all-electron PAW data sets with a small cutoff radius that are valid for lithium densities up to at least 80 g/cm{3}.

On the use of effective core potentials in the calculation of magnetic properties, such as magnetizabilites and magnetic shieldings

State-of-the art effective core potentials (ECPs) that replace electrons of inner atomic cores involve non-local potentials. If such an effective core potential is added to the Hamiltonian of a system in a magnetic field, the resulting Hamiltonian is not gauge invariant. This means, magnetic properties such as magnetisabilities and magnetic shieldings (or magnetic susceptibilities and nuclear magnetic resonance chemical shifts) calculated with different gauge origins are different even for exact solutions of the Schrödinger equation. It is possible to restore gauge invariance of the Hamiltonian by adding magnetic field dependent terms arising from the effective core potential. Numerical calculations on atomic and diatomic model systems (potassium mono-cation and potassium dimer) clearly demonstrate that the standard effective core potential Hamiltonian violates gauge invariance, and this affects the calculation of magnetisabilities more strongly than the calculation of magnetic shieldings. The modified magnetic field dependent effective core potential Hamiltonian is gauge invariant, and therefore it is the correct starting point for distributed gauge origin methods. The formalism for gauge including atomic orbitals (GIAO) and individual gauge for localized orbitals methods is worked out. ECP GIAO results for the potassium dimer are presented. The new method performs much better than a previous ECP GIAO implementation that did not account for the non-locality of the potential. For magnetic shieldings, deviations are clearly seen, but they amount to few ppm only. For magnetisabilities, our new ECP GIAO implementation is a major improvement, as demonstrated by the comparison of all-electron and ECP results.

ChemInform Abstract: The Pseudopotential Approximation in Electronic Structure Theory

AbstractReview: 204 refs.