Computational modeling of the electron momentum density

The properties and the functionality of materials are determined to a large extent by their electronic structure. The electronic structure can be examined through the electron momentum density, which is classically equivalent to the velocity distribution of the electrons. Changes in the structure of materials induce changes on their electronic structure, which in turn are reflected as changes in the electron momentum densities that can be routinely measured using, e.g., x-ray Compton scattering. The changes in the momentum density can be linked back to the structural changes the system has experienced through the extensive use of computational modeling. This procedure naturally requires using a model matching the accuracy of the experiment, which is constantly improving as the result of the ongoing development of synchrotron radiation sources and beam line instrumentation. However, the accuracies of the current computational methods have not been hitherto established. This thesis focuses on developing the methods used to compute the electron momentum density in order to achieve an accuracy comparable to that of the experiment. The accuracies of current quantum chemical methods that can be used to model the electron momentum density are established. The completeness-optimization scheme is used to develop computationally efficient basis sets for modeling the electron momentum density at the complete basis set limit. A novel, freely available software program that can be used to perform all of the necessary electronic structure calculations is also introduced.Materiaalien ominaisuudet sekä toiminnallisuus ovat pitkälti niiden elektronisen rakenteen määräämiä. Elektronirakennetta voidaan tutkia elektroniliikemäärätiheyden avulla, joka vastaa klassisesti elektronien nopeusjakaumaa. Materiaalien rakenteessa tapahtuvat muutokset muuttavat niiden elektronirakennetta, joka puolestaan heijastuu niiden liikemäärätiheyksiin joka voidaan rutiininomaisesti mitata käyttämällä esimerkiksi röntgen-Compton-sirontaa. Liikemäärätiheydessä tapahtuvat muutokset voidaan yhdistää systeemissä tapahtuneisiin rakennemuutoksiin käyttämällä laskennallista mallinnusta. Tämä luonnollisesti vaatii sellaisen mallin käyttämistä, jonka tarkkuus on verrattavissa mittaustuloksen tarkkuuteen, joka taas paranee jatkuvasti synkrotronisäteilylähteiden ja mittauslaitteistojen kehityksen vuoksi. Nykyisten mallinnusmenetelmien tarkkuutta ei ole kuitenkaan vielä määritetty. Tässä väitöskirjassa kehitetään elektroniliikemäärätiheyden mallinnusmenetelmiä, tarkoituksena saavuttaa kokeisiin verrattavissa oleva tarkkuus. Nykyaikaisten kvanttikemiallisten menetelmien tarkkuudet määritetään. Täydellisyysoptimointimenetelmää käytetään laskennallisesti tehokkaiden, elektroniliikemäärätiheyden mallintamiseen suunnattujen kantajoukkojen muodostamiseen, joiden tulokset ovat kantajoukkorajalla. Esittelemme myös uuden, vapaasti saatavilla oleva ohjelman, jolla voidaan suorittaa kaikki mallintamisessa tarvittavat elektronirakennelaskut

Automatic Generation of Accurate and Cost-efficient Auxiliary Basis Sets

We have recently discussed an algorithm to automatically generate auxiliary basis sets (ABSs) of the standard form for density fitting (DF) or resolution-of-the-identity (RI) calculations in a given atomic orbital basis set (OBS) of any form [J. Chem. Theory Comput. 2021, 17, 6886]. In this work, we study two ways to reduce the cost of such automatically generated ABSs without sacrificing their accuracy. We contract the ABS with a singular value decomposition proposed by K\'allay [J. Chem. Phys. 2014, 141, 244113], used here in a somewhat different setting. We also drop the high-angular momentum functions from the ABS, as they are unnecessary for global fitting methods. Studying the effect of these two types of truncations on Hartree--Fock (HF) and second-order M{\o}ller--Plesset perturbation theory (MP2) calculations on a chemically diverse set of first- and second-row molecules within the RI/DF approach, we show that accurate total and atomization energies can be achieved by a combination of the two approaches with significant reductions in the size of the ABS. While the original approach yields ABSs whose number of functions $N_{\text{bf}}^{\text{ABS}}$ scales with the number of functions in the OBS, $N_{\text{OBS}}^{\text{bf}}$, as $N_{\text{ABS}}^{\text{bf}}=\gamma N_{\text{OBS}}^{\text{bf}}$ with the prefactor $\gamma\approx\mathcal{O}(10)$, the reduction schemes of this work afford results of essentially the same quality as the original unpruned and uncontracted ABS with $\gamma\approx5\text{-}6$, while an accuracy that may suffice for routine applications is achievable with a further reduced ABS with $\gamma\approx3\text{-}4$. The observed errors are similar at HF and MP2 levels of theory, suggesting that the generated ABSs are highly transferable, and can also be applied to model challenging properties with high-level methods.Comment: 20 pages, 6 figure

A call to arms: making the case for more reusable libraries

The traditional foundation of science lies on the cornerstones of theory and experiment. Theory is used to explain experiment, which in turn guides the development of theory. Since the advent of computers and the development of computational algorithms, computation has risen as the third cornerstone of science, joining theory and experiment on an equal footing. Computation has become an essential part of modern science, amending experiment by enabling accurate comparison of complicated theories to sophisticated experiments, as well as guiding by triage both the design and targets of experiments and the development of novel theories and computational methods. Like experiment, computation relies on continued investment in infrastructure: it requires both hardware (the physical computer on which the calculation is run) as well as software (the source code of the programs that performs the wanted simulations). In this Perspective, I discuss present-day challenges on the software side in computational chemistry, which arise from the fast-paced development of algorithms, programming models, as well as hardware. I argue that many of these challenges could be solved with reusable open source libraries, which are a public good, enhance the reproducibility of science, and accelerate the development and availability of state-of-the-art methods and improved software.Comment: 13 pages, 1 figur

Meta-GGA density functional calculations on atoms with spherically symmetric densities in the finite element formalism

Density functional calculations on atoms are often used for determining accurate initial guesses as well as generating various types of pseudopotential approximations and efficient atomic-orbital basis sets for polyatomic calculations. To reach the best accuracy for these purposes, the atomic calculations should employ the same density functional as the polyatomic calculation. Atomic density functional calculations are typically carried out employing spherically symmetric densities, corresponding to the use of fractional orbital occupations. We have described their implementation for density functional approximations (DFAs) belonging to the local density approximation (LDA) and generalized gradient approximation (GGA) levels of theory as well as Hartree-Fock (HF) and range-separated exact exchange [S. Lehtola, Phys. Rev. A 2020, 101, 012516]. In this work, we describe the extension to meta-GGA functionals using the generalized Kohn-Sham scheme, in which the energy is minimized with respect to the orbitals, which in turn are expanded in the finite element formalism with high-order numerical basis functions. Furnished with the new implementation, we continue our recent work on the numerical well-behavedness of recent meta-GGA functionals [S. Lehtola and M. A. L. Marques, J. Chem. Phys. 2022, 157, 174114]. We pursue complete basis set (CBS) limit energies for recent density functionals, and find many to be ill-behaved for the Li and Na atoms. We report basis set truncation errors (BSTEs) of some commonly used Gaussian basis sets for these density functionals and find them to be strongly functional dependent. We also discuss the importance of density thresholding in DFAs and find that all of the functionals studied in this work yield total energies converged to $0.1\ \mu E_{h}$ when densities smaller than $10^{-11}a_{0}^{-3}$ are screened out.Comment: 30 pages, 2 figure

Accurate reproduction of strongly repulsive interatomic potentials

Knowledge of the repulsive behavior of potential energy curves V (R) at R -> 0 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 this 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 He-2 (sic) Be, HeNe (sic) Mg, Ne-2 (sic) Ca, HeAr (sic) Ca, MgAr (sic) Zn, Ar-2 (sic) Kr, and NeCa (sic) Zn.Peer reviewe

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 Huckel 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 Huckel 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 Huckel variant is assessed in nonrelativistic calculations on a data set 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-zeta basis sets. It is shown that the proposed SAP guess is the best guess on average. The extended Huckel guess offers a good alternative, with less scatter in accuracy.Peer reviewe

Importance profiles. Visualization of basis set superposition errors

Recent developments in fully numerical methods promise interesting opportunities for new, compact atomic orbital (AO) basis sets that maximize the overlap to fully numerical reference wave functions, following the pioneering work of Richardson and coworkers from the early 1960s. Motivated by this technique, we suggest a way to visualize the importance of AO basis functions in polyatomic calculations, employing fully numerical calculations at the complete basis set (CBS) limit: the importance of a normalized AO basis function $|\alpha\rangle$ centered on some nucleus can be visualized by projecting $|\alpha\rangle$ on the set of numerically represented occupied orbitals $|\psi_{i}\rangle$ as $I(\alpha)=\sum_{i}\langle\alpha|\psi_{i}\rangle\langle\psi_{i}|\alpha\rangle$. Choosing $\alpha$ to be a continuous parameter describing the orbital basis, such as the exponent of a Gaussian-type orbital (GTO) or Slater-type orbital (STO) basis function, one is then able to visualize the importance of various functions on various centers in various molecules. The proposed visualization $I(\alpha)$ has the important property $0\leq I(\alpha)\leq1$ which allows unambiguous interpretation. We exemplify the method with importance profiles computed for atoms from the first three rows in a set of chemically diverse diatomic molecules. We find that the method offers a good way to visualize basis set superposition errors: the non-orthonormality of AO basis functions on different atomic centers is unambiguously revealed by the importance profiles computed for the ghost atom in an atomic calculation performed in the numerical basis set for a diatomic molecule.Comment: 10 pages, 3 figure

Fully numerical calculations on atoms with fractional occupations and range-separated exchange functionals

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, the 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 micro-Hartree accuracy with a small number of numerical basis functions, and to reproduce the literature results for the ground states of atoms and their cations for 1Peer reviewe