arXiv subjects

Susi Lehtola

Publications and source records attributed to Susi Lehtola.

54 records · Page 3Linked to original sources

arXiv2020

Efficient implementation of the superposition of atomic potentials initial guess for electronic structure calculations in Gaussian basis sets

The superposition of atomic potentials (SAP) approach has recently been shown to be a simple and efficient way to initialize electronic structure calculations [S. Lehtola, J. Chem. Theory Comput. 15, 1593 (2019)]. Here, we study the differences between effective potentials from fully numerical density functional and optimized effective potential calculations for fixed configurations. We find that the differences are small, overall, and choose exchange-only potentials at the local density approximation level of theory computed on top of Hartree-Fock densities as a good compromise. The differences between potentials arising from different atomic configurations are also found to be small at this level of theory. Furthermore, we discuss the efficient Gaussian-basis implementation of SAP via error function fits to fully numerical atomic radial potentials. The guess obtained from the fitted potentials can be easily implemented in any Gaussian-basis quantum chemistry code in terms of two-electron integrals. Fits covering the whole periodic table from H to Og are reported for non-relativistic as well as fully relativistic four-component calculations that have been carried out with fully numerical approaches.

physics.comp-ph↗

arXiv2020

Polarized Gaussian basis sets from one-electron ions

We demonstrate that basis sets suitable for electronic structure calculations can be obtained from simple accuracy considerations for the hydrogenic one-electron ions $Y^{(Y-1)+}$ for $Y\in[1,Z]$, necessitating no self-consistent field calculations at all. It is shown that even-tempered basis sets with parameters from the commonly-used universal Gaussian basis set (UGBS) [E. V. R. de Castro and F. E. Jorge, J. Chem. Phys. 108, 5225 (1998)] reproduce non-relativistic spin-restricted spherical Hartree-Fock total energies from fully numerical calculations to better accuracy than UGBS, which is shown to exhibit huge errors for some elements, e.g. 0.19 $E_{h}$ for Th$^+$ and 0.13 $E_{h}$ for Lu as it has been parametrized for a single atomic configuration. Having shown the feasibility of the one-electron approach, partially energy-optimized basis sets are formed for all atoms in the periodic table, $1\leq Z\leq118$, by optimizing the even-tempered parameters for $Z^{(Z-1)+}$. As the hydrogenic Gaussian basis sets suggested in the present work are built strictly from first principles, also polarization shells can be obtained in the same fashion in contrast to previous approaches. The accuracy of the polarized basis sets is demonstrated by calculations on a small set of molecules by comparison to fully numerical reference values, which show that chemical accuracy can be reached even for challenging cases like SF$_6$. The present approach is straightforward to extend to relativistic calculations, and could facilitate studies beyond the established periodic table.

physics.comp-ph↗

arXiv2020

An overview of self-consistent field calculations within finite basis sets

A uniform derivation is presented of the self-consistent field equations in a finite basis set. Both restricted and unrestricted Hartree-Fock (HF) theory as well as various density functional (DF) approximations are considered. The unitary invariance of the HF and DF models is discussed, paving the way for the use of localized molecular orbitals. The self-consistent field equations are derived in a non-orthogonal basis set, and their solution is discussed in the presence of linear dependencies in the basis set. It is argued why iterative diagonalization of the Kohn-Sham-Fock matrix leads to the minimization of the total energy. Alternative methods for the solution of the self-consistent field equations via direct minimization as well as stability analysis are also briefly discussed. Explicit expressions are given for the contributions to the Kohn-Sham-Fock matrix up to meta-GGA functionals. Range-separated hybrids and non-local correlation functionals are also briefly discussed.

physics.comp-ph↗

arXiv2020

CASSCF with Extremely Large Active Spaces using the Adaptive Sampling Configuration Interaction Method

The complete active space self-consistent field (CASSCF) method is the principal approach employed for studying strongly correlated systems. However, exact CASSCF can only be performed on small active spaces of ~20 electrons in ~20 orbitals due to exponential growth in the computational cost. We show that employing the Adaptive Sampling Configuration Interaction (ASCI) method as an approximate Full CI solver in the active space allows CASSCF-like calculations within chemical accuracy (<1 kcal/mol for relative energies) in active spaces with more than ~50 active electrons in ~50 active orbitals, significantly increasing the sizes of systems amenable to accurate multiconfigurational treatment. The main challenge with using any selected CI-based approximate CASSCF is the orbital optimization problem; they tend to exhibit large numbers of local minima in orbital space due to their lack of invariance to active-active rotations (in addition to the local minima that exist in exact CASSCF). We highlight methods that can avoid spurious local extrema as a practical solution to the orbital optimization problem. We employ ASCI-SCF to demonstrate lack of polyradical character in moderately sized periacenes with up to 52 correlated electrons and compare against heat-bath CI on an iron porphyrin system with more than 40 correlated electrons.

physics.comp-ph↗

arXiv2020

On the accurate reproduction of strongly repulsive interatomic potentials

Knowledge of the repulsive behavior of potential energy curves $V(R)$ at $R\to0$ is necessary for understanding and modeling irradiation processes of practical interest. $V(R)$ is in principle straightforward to obtain from electronic structure calculations; however, commonly-used numerical approaches for electronic structure calculations break down in the strongly repulsive region due to the closeness of the nuclei. In the present work, we show by comparison to fully numerical reference values that a recently developed procedure [S. Lehtola, J. Chem. Phys. 151, 241102 (2019)] can be employed to enable accurate linear combination of atomic orbitals calculations of $V(R)$ even at small $R$ by a study of the seven nuclear reactions He2 <=> Be, HeNe <=> Mg, Ne2 <=> Ca, HeAr <=> Ca, MgAr <=> Zn, Ar2 <=> Kr, and NeCa <=> Zn.

physics.comp-ph↗

arXiv2019

Communication: Curing basis set overcompleteness with pivoted Cholesky decompositions

The description of weakly bound electronic states is especially difficult with atomic orbital basis sets. The diffuse atomic basis functions that are necessary to describe the extended electronic state generate significant linear dependencies in the molecular basis set, which may make the electronic structure calculations ill-convergent. We propose a method where the over-complete molecular basis set is pruned by a pivoted Cholesky decomposition of the overlap matrix, yielding an optimal low-rank approximation that is numerically stable; the pivot indices determining a reduced basis set that is complete enough to describe all the basis functions in the original over-complete basis. The method can be implemented either by a simple modification to the usual canonical orthogonalization procedure, which hides the excess functions and yields fewer efficiency benefits, or by generating custom basis sets for all the atoms in the system, yielding significant cost reductions in electronic structure calculations. The pruned basis sets from the latter choice allow accurate calculations to be performed at a lower cost even at the self-consistent field level, as illustrated on a solvated (H2O)24- anion. Our results indicate that the Cholesky procedure allows one to perform calculations with accuracies close to standard augmented basis sets with cost savings which increase with the size of the basis set, ranging from 9% fewer functions in single-ζ basis sets to 28% fewer functions in triple-ζ basis sets.

physics.chem-ph↗

arXiv2019

Fully numerical calculations on atoms with fractional occupations. Range-separated exchange functionals

A recently developed finite element approach for fully numerical atomic structure calculations [S. Lehtola, Int. J. Quantum Chem. 119, e25945 (2019)] is extended to the description of atoms with spherically symmetric densities via fractionally occupied orbitals. Specialized versions of Hartree-Fock as well as local density and generalized gradient approximation density functionals are developed, allowing extremely rapid calculations at the basis set limit on the ground and low-lying excited states even for heavy atoms. The implementation of range-separation based on the Yukawa or complementary error function (erfc) kernels is also described, allowing complete basis set benchmarks of modern range-separated hybrid functionals with either integer or fractional occupation numbers. Finally, computation of atomic effective potentials at the local density or generalized gradient approximation levels for the superposition of atomic potentials (SAP) approach [S. Lehtola, J. Chem. Theory Comput. 15, 1593 (2019)] that has been shown to be a simple and efficient way to initialize electronic structure calculations is described. The present numerical approach is shown to afford beyond microhartree accuracy with a small number of numerical basis functions, and to reproduce literature results for the ground states of atoms and their cations for $1 \leq Z \leq 86 $. Our results indicate that the literature values deviate by up to 10 μEh from the complete basis set limit. The numerical scheme for the erfc kernel is shown to work by comparison to results from large Gaussian basis set calculations from the literature. Spin-restricted ground states are reported for Hartree-Fock and Hartree-Fock-Slater calculations with fractional occupations for $1 \leq Z \leq 118$.

physics.comp-ph↗

arXiv2019

A review on non-relativistic fully numerical electronic structure calculations on atoms and diatomic molecules

The need for accurate calculations on atoms and diatomic molecules is motivated by the opportunities and challenges of such studies. The most commonly-used approach for all-electron electronic structure calculations in general - the linear combination of atomic orbitals (LCAO) method - is discussed in combination with Gaussian, Slater a.k.a. exponential, and numerical radial functions. Even though LCAO calculations have major benefits, their shortcomings motivate the need for fully numerical approaches based on, e.g. finite differences, finite elements, or the discrete variable representation, which are also briefly introduced. Applications of fully numerical approaches for general molecules are briefly reviewed, and their challenges are discussed. It is pointed out that the high level of symmetry present in atoms and diatomic molecules can be exploited to fashion more efficient fully numerical approaches for these special cases, after which it is possible to routinely perform all-electron Hartree-Fock and density functional calculations directly at the basis set limit on such systems. Applications of fully numerical approaches to calculations on atoms as well as diatomic molecules are reviewed. Finally, a summary and outlook is given.

physics.chem-ph↗

arXiv2019

Stretched or noded orbital densities and self-interaction correction in density functional theory

Semi-local approximations to the density functional for the exchange-correlation energy of a many-electron system necessarily fail for lobed one-electron densities, including not only the familiar stretched densities but also the less familiar but closely-related noded ones. The Perdew-Zunger (PZ) self-interaction correction (SIC) to a semi-local approximation makes that approximation exact for all one-electron ground- or excited-state densities and accurate for stretched bonds. When the minimization of the PZ total energy is made over real localized orbitals, the orbital densities can be noded, leading to energy errors in many-electron systems. Minimization over complex localized orbitals yields nodeless orbital densities, which reduce but typically do not eliminate the SIC errors of atomization energies. Other errors of PZ SIC remain, attributable to the loss of the exact constraints and appropriate norms that the semi-local approximations satisfy, and suggesting the need for a generalized SIC. These conclusions are supported by calculations for one-electron densities, and for many-electron molecules. While PZ SIC raises and improves the energy barriers of standard generalized gradient approximations (GGA's) and meta-GGA's, it reduces and often worsens the atomization energies of molecules. Thus PZ SIC raises the energy more as the nodality of the valence localized orbitals increases from atoms to molecules to transition states. PZ SIC is applied here in particular to the SCAN meta-GGA, for which the correlation part is already self-interaction-free. That property makes SCAN a natural first candidate for a generalized SIC.

physics.chem-ph↗

arXiv2019

Fully numerical Hartree-Fock and density functional calculations. II. Diatomic molecules

We present the implementation of a variational finite element solver in the HelFEM program for benchmark calculations on diatomic systems. A basis set of the form $χ_{nlm}(μ,ν,ϕ)=B_{n}(μ)Y_{l}^{m}(ν,ϕ)$ is used, where $(μ,ν,ϕ)$ are transformed prolate spheroidal coordinates, $B_{n}(μ)$ are finite element shape functions, and $Y_{l}^{m}$ are spherical harmonics. The basis set allows for an arbitrary level of accuracy in calculations on diatomic molecules, which can be performed at present with either nonrelativistic Hartree--Fock (HF) or density functional (DF) theory. Hundreds of DFs at the local spin-density approximation (LDA), generalized gradient approximation (GGA) and the meta-GGA level can be used through an interface with the Libxc library; meta-GGA and hybrid DFs aren't available in other fully numerical diatomic program packages. Finite electric fields are also supported in HelFEM, enabling access to electric properties. We introduce a powerful tool for adaptively choosing the basis set by using the core Hamiltonian as a proxy for its completeness. HelFEM and the novel basis set procedure are demonstrated by reproducing the restricted open-shell HF limit energies of 68 diatomic molecules from the $1^{\text{st}}$ to the $4^{\text{th}}$ period with excellent agreement with literature values, despite requiring \emph{orders of magnitude} fewer parameters for the wave function. Then, the electric properties of the BH and N2 molecules under finite field are studied, again yielding excellent agreement with previous HF limit values for energies, dipole moments, and dipole polarizabilities, again with much more compact wave functions than what were needed in the literature references. Finally, HF, LDA, GGA, and meta-GGA calculations of the atomization energy of N2 are performed, demonstrating the superb accuracy of the present approach.

physics.chem-ph↗

arXiv2019

Fully numerical Hartree-Fock and density functional calculations. I. Atoms

Although many programs have been published for fully numerical Hartree--Fock (HF) or density functional (DF) calculations on atoms, we are not aware of any that support hybrid DFs, which are popular within the quantum chemistry community due to their better accuracy for many applications, or that can be used to calculate electric properties. Here, we present a variational atomic finite element solver in the HelFEM program suite that overcomes these limitations. A basis set of the type $χ_{nlm}(r,θ,ϕ)=r^{-1}B_{n}(r)Y_{l}^{m}(\hat{\boldsymbol{r}})$ is used, where $B_{n}(r)$ are finite element shape functions and $Y_{l}^{m}$ are spherical harmonics, which allows for an arbitrary level of accuracy. HelFEM supports nonrelativistic HF and DF including hybrid functionals, which are not available in other commonly available program packages. Hundreds of functionals at the local spin density approximation (LDA), generalized gradient approximation (GGA), as well as the meta-GGA levels of theory are included through an interface with the Libxc library. Electric response properties are achievable via finite field calculations. We introduce an alternative grid that yields faster convergence to the complete basis set than commonly used alternatives. We also show that high-order Lagrange interpolating polynomials yield the best convergence, and that excellent agreement with literature HF limit values for electric properties, such as static dipole polarizabilities, can be achieved with the present approach. Dipole moments and dipole polarizabilities at finite field are reported with the PBE, PBEh, TPSS, and TPSSh functionals. Finally, we show that a recently published Gaussian basis set is able to reproduce absolute HF and DF energies of neutral atoms, cations, as well as anions within a few dozen microhartrees.

physics.chem-ph↗

arXiv2019

Fully numerical electronic structure calculations on diatomic molecules in weak to strong magnetic fields

We present fully numerical electronic structure calculations on diatomic molecules exposed to an external magnetic field at the unrestricted Hartree-Fock limit, using a modified version of a recently developed finite element program, HelFEM. We have performed benchmark calculations on a few low-lying states of H2, HeH+, LiH, BeH+, BH, and CH+ as a function of the strength of an external magnetic field parallel to the molecular axis. The employed magnetic fields are in the range of $B=[0,10]~B_0$ atomic units, where $B_0 \approx 2.35 \times 10^5$ T. We have compared the results of the fully numerical calculations to ones obtained with the LONDON code using a large uncontracted gauge-including Cartesian Gaussian (GICG) basis set with exponents adopted from the Dunning aug-cc-pVTZ basis set. By comparison to the fully numerical results, we find that the basis set truncation error in the gauge-including Gaussian basis set is of the order of 1 kcal/mol at zero field, that the truncation error grows rapidly when the strength of the magnetic field increases, and that the largest basis set truncation error at $B=10~B_0$ exceeds 1000 kcal/mol. Studies in larger Gaussian basis sets suggest that reliable results can be obtained in GICG basis sets at fields stronger than $B=B_0$, provided that a sufficient coverage of higher-angular-momentum functions is included in the basis set.

physics.chem-ph↗

arXiv2019

An assessment of initial guesses for self-consistent field calculations. Superposition of Atomic Potentials: simple yet efficient

Electronic structure calculations, such as in the Hartree-Fock or Kohn-Sham density functional approach, require an initial guess for the molecular orbitals. The quality of the initial guess has a significant impact on the speed of convergence of the self-consistent field (SCF) procedure. Popular choices for the initial guess include the one-electron guess from the core Hamiltonian, the extended Hückel method, and the superposition of atomic densities (SAD). Here, we discuss alternative guesses obtained from the superposition of atomic potentials (SAP), which is easily implementable even in real-space calculations. We also discuss a variant of SAD which produces guess orbitals by purification of the density matrix that could also be used in real-space calculations, as well as a parameter-free variant of the extended Hückel method, which resembles the SAP method and is easy to implement on top of existing SAD infrastructure. The performance of the core Hamiltonian, the SAD and the SAP guesses as well as the extended Hückel variant is assessed in non-relativistic calculations on a dataset of 259 molecules ranging from the first to the fourth periods by projecting the guess orbitals onto precomputed, converged SCF solutions in single- to triple-ζ basis sets. It is shown that the proposed SAP guess is the best guess on average. The extended Hückel guess offers a good alternative, with less scatter in accuracy.

physics.chem-ph↗

arXiv2018

Orbital optimization in the perfect pairing hierarchy. Applications to full-valence calculations on linear polyacenes

We describe the implementation of orbital optimization for the models in the perfect pairing hierarchy [Lehtola et al, J. Chem. Phys. 145, 134110 (2016)]. Orbital optimization, which is generally necessary to obtain reliable results, is pursued at perfect pairing (PP) and perfect quadruples (PQ) levels of theory for applications on linear polyacenes, which are believed to exhibit strong correlation in the π space. While local minima and σ-π symmetry breaking solutions were found for PP orbitals, no such problems were encountered for PQ orbitals. The PQ orbitals are used for single-point calculations at PP, PQ and perfect hextuples (PH) levels of theory, both only in the π subspace, as well as in the full σπ valence space. It is numerically demonstrated that the inclusion of single excitations is necessary also when optimized orbitals are used. PH is found to yield good agreement with previously published density matrix renormalization group (DMRG) data in the π space, capturing over 95% of the correlation energy. Full-valence calculations made possible by our novel, efficient code reveal that strong correlations are weaker when larger bases or active spaces are employed than in previous calculations. The largest full-valence PH calculations presented correspond to a (192e,192o) problem.

cond-mat.str-el↗

arXiv2017

Cluster decomposition of full configuration interaction wave functions: a tool for chemical interpretation of systems with strong correlation

Approximate full configuration interaction (FCI) calculations have recently become tractable for systems of unforeseen size thanks to stochastic and adaptive approximations to the exponentially scaling FCI problem. The result of an FCI calculation is a weighted set of electronic configurations, which can also be expressed in terms of excitations from a reference configuration. The excitation amplitudes contain information on the complexity of the electronic wave function, but this information is contaminated by contributions from disconnected excitations, i.e. those excitations that are just products of independent lower-level excitations. The unwanted contributions can be removed via a cluster decomposition procedure, making it possible to examine the importance of connected excitations in complicated multireference molecules which are outside the reach of conventional algorithms. We present an implementation of the cluster decomposition analysis and apply it to both true FCI wave functions, as well as wave functions generated from the adaptive sampling CI (ASCI) algorithm. The cluster decomposition is useful for interpreting calculations in chemical studies, as a diagnostic for the convergence of various excitation manifolds, as well as as a guidepost for polynomially scaling electronic structure models. Applications are presented for (i) the double dissociation of water, (ii) the carbon dimer, (iii) the π space of polyacenes, as well as (iv) the chromium dimer. While the cluster amplitudes exhibit rapid decay with increasing rank for the first three systems, even connected octuple excitations still appear important in Cr$_2$, suggesting that spin-restricted single-reference coupled-cluster approaches may not be tractable for some problems in transition metal chemistry.

cond-mat.str-el↗

arXiv2017

Theory and applications of generalized Pipek--Mezey Wannier functions

The theory for the generation of Wannier functions within the generalized Pipek--Mezey approach [Lehtola, S.; Jónsson, H. J. Chem. Theory Comput. 2014, 10, 642] is presented and an implementation thereof is described. Results are presented for systems with periodicity in one, two and three dimensions as well as isolated molecules. The generalized Pipek--Mezey Wannier functions (PMWF) are highly localized orbitals consistent with chemical intuition where a distinction is maintained between σ- and π-orbitals. The PMWF method is compared with the so-called maximally localized Wannier functions (MLWF) that are frequently used for the analysis of condensed matter calculations. Whereas PMWFs maximize the localization criterion of Pipek and Mezey, MLWFs maximize that of Foster and Boys and have the disadvantage of mixing σ- and π-orbitals in many cases. The PMWF orbitals turn out to be as localized as the MLWF orbitals as evidenced by cross-comparison of the values of the PMWF and MLWF objective functions for the two types of orbitals. Our implementation in the atomic simulation environment (ASE) is compatible with various representations of the wave function, including real-space grids, plane waves and linear combinations of atomic orbitals. The projector augmented wave formalism for the representation of atomic core electrons is also supported. Results of calculations with the GPAW software are described here, but our implementation can also use output from other electronic structure software such as ABINIT, NWChem and VASP.

cond-mat.mtrl-sci↗

arXiv2016

Cost-effective description of strong correlation: efficient implementations of the perfect quadruples and perfect hextuples models

Novel implementations based on dense tensor storage are presented for the singlet-reference perfect quadruples (PQ) [Parkhill, Lawler, and Head-Gordon, J. Chem. Phys. 130, 084101 (2009)] and perfect hextuples (PH) [Parkhill and Head-Gordon, J. Chem. Phys. 133, 024103 (2010)] models. The methods are obtained as block decompositions of conventional coupled-cluster theory that are exact for four electrons in four orbitals (PQ) and six electrons in six orbitals (PH), but that can also be applied to much larger systems. PQ and PH have storage requirements that scale as the square, and as the cube of the number of active electrons, respectively, and exhibit quartic scaling of the computational effort for large systems. Applications of the new implementations are presented for full-valence calculations on linear polyenes (C n H n+2 ), which highlight the excellent computational scaling of the present implementations that can routinely handle active spaces of hundreds of electrons. The accuracy of the models is studied in the π space of the polyenes, in hydrogen chains (H 50 ), and in the π space of polyacene molecules. In all cases, the results compare favorably to density matrix renormalization group values. With the novel implementation of PQ, active spaces of 140 electrons in 140 orbitals can be solved in a matter of minutes on a single core workstation, and the relatively low polynomial scaling means that very large systems are also accessible using parallel computing.

cond-mat.str-el↗

arXiv2015

Nanoplasmonics simulations at the basis set limit through completeness-optimized, local numerical basis sets

We present an approach for generating local numerical basis sets of improving accuracy for first-principles nanoplasmonics simulations within time-dependent density functional theory. The method is demonstrated for copper, silver, and gold nanoparticles that are of experimental interest but computationally demanding due to the semi-core d-electrons that affect their plasmonic response. The basis sets are constructed by augmenting numerical atomic orbital basis sets by truncated Gaussian-type orbitals generated by the completeness-optimization scheme, which is applied to the photoabsorption spectra of homoatomic metal atom dimers. We obtain basis sets of improving accuracy up to the complete basis set limit and demonstrate that the performance of the basis sets transfers to simulations of larger nanoparticles and nanoalloys as well as to calculations with various exchange-correlation functionals. This work promotes the use of the local basis set approach of controllable accuracy in first-principles nanoplasmonics simulations and beyond.

cond-mat.mtrl-sci↗