L'objectif de cette thèse est de fournir des bornes d'erreur pour des problèmes aux valeurs propres linéaires et non linéaires issus du calcul de structure électronique, en particulier celui de l'état fondamental avec la théorie de la fonctionnelle de la densité. Ces bornes d'erreur reposent principalement sur des estimations a posteriori. D'abord, nous étudions un phénomène de compensation d'erreur de discrétisation pour un problème linéaire aux valeurs propres, grâce à une analyse a priori de l'erreur sur l'énergie. Ensuite, nous présentons une analyse a posteriori pour le problème du laplacien aux valeurs propres discrétisé par une large classe d'éléments finis. Les bornes d'erreur proposées pour les valeurs propres simples et leurs vecteurs propres associés sont garanties, calculables et efficaces. Nous nous concentrons alors sur des problèmes aux valeurs propres non linéaires. Nous proposons des bornes d'erreur pour l'équation de Gross-Pitaevskii, valables sous des hypothèses vérifiables numériquement, et pouvant être séparées en deux composantes venant respectivement de la discrétisation et de l'algorithme itératif utilisé pour résoudre le problème non linéaire aux valeurs propres. L'équilibrage de ces composantes d'erreur permet d'optimiser les ressources numériques. Enfin, nous présentons une méthode de post-traitement pour le problème de Kohn-Sham discrétisé en ondes planes, améliorant la précision des résultats à un faible coût de calcul. Les solutions post-traitées peuvent être utilisées soit comme solutions plus précises du problème, soit pour calculer une estimation de l'erreur de discrétisation, qui n'est plus garantie, mais néanmoins proche de l'erreur. / The objective of this thesis is to provide error bounds for linear and nonlinear eigenvalue problems arising from electronic structure calculation. We focus on ground-state calculations based on Density Functional Theory, including Kohn-Sham models. Our bounds mostly rely on a posteriori error analysis. More precisely, we start by studying a phenomenon of discretization error cancellation for a simple linear eigenvalue problem, for which analytical solutions are available. The mathematical study is based on an a priori analysis for the energy error. Then, we present an a posteriori analysis for the Laplace eigenvalue problem discretized with finite elements. For simple eigenvalues of the Laplace operator and their corresponding eigenvectors , we provide guaranteed, fully computable and efficient error bounds. Thereafter, we focus on nonlinear eigenvalue problems. First, we provide an a posteriori analysis for the Gross-Pitaevskii equation. The error bounds are valid under assumptions that can be numerically checked, and can be separated in two components coming respectively from the discretization and the iterative algorithm used to solve the nonlinear eigenvalue problem. Balancing these error components allows to optimize the computational resources. Second, we present a post-processing method for the Kohn-Sham problem, which improves the accuracy of planewave computations of ground state orbitals at a low computational cost. The post-processed solutions can be used either as a more precise solution of the problem, or used for computing an estimation of the discretization error. This estimation is not guaranteed, but in practice close to the real error.

The Kohn-Sham density functional theory (KS-DFT) finds an approximate solution for the many-electron problem for the ground state energy and density by solving the self-consistent one-electron Schr\"{o}dinger equations. KS-DFT would be an exact theory if we could find the precise form of exchange-correlation energy $(E_{xc})$. However, this would not be computationally feasible. The density functional approximations (DFAs) are designed to be exact in the limit of uniform densities. They require a parametrization of the correlation energy per electron $(\varepsilon_c)$ of the uniform electron gas (UEG). These DFAs take the parametrizations of correlation energy as their input since the exact analytical form of $\varepsilon_c$ is still unknown. Almost all the DFAs of higher rungs of Jacob's ladder employ an additional function on top of $\varepsilon_c$ for approximating their correlation energy. Exchange energies in these DFAs are also approximated by applying an enhancement factor to the exchange energy per electron of the UEG. Exchange-correlation energy is the glue that holds the atoms and molecules together. The correlation energy is an important part of ``nature's glue" that binds one atom to another, and it changes significantly when the bonding of the molecule changes. It is a measure of the effect of Coulomb repulsion due to electronic mutual avoidance and is necessarily negative. We compared three parametrizations of the correlation energy per electron of the uniform electron gas to the original and the corrected density parameter interpolation (DPI), which is almost independent of QMC input, and with the recent QMC of Spink \textit{et al.}, which extends the Ceperley-Alder results to fractional spin polarization and higher densities or smaller Seitz radius $r_s$. These three parametrizations are Perdew-Zunger or PZ 1981, Vosko-Wilk-Nusair or VWN 1980, and Perdew-Wang or PW 1992. The three parametrizations (especially the sophisticated PW92) are closer to the constraint satisfying DPI and are very close to the high-density limit rather than the QMC results of Spink \textit{et al.}. These DFAs suffer from self-interaction error (SIE) which arises due to an imperfect cancellation of self-Hartree energy by self-exchange-correlation energy of a single fully occupied orbital. The self-interaction correction (SIC) method introduced by Perdew and Zunger (PZ) in 1981 to remove the SIE encounters a size-extensivity problem when applied to the Kohn-Sham (KS) orbitals. Hence, we make use of Fermi L\"owdin orbitals (FLO) for applying the PZ-SIC to the density functional approximations (DFAs). FLOs are the unitary transformation of the KS orbitals localized at the Fermi orbital descriptor (FOD) positions and then orthonormalized using L\"owdin's symmetric method. The PZ-SIC makes any approximation exact only in the region of one-electron density and no correction if applied to the exact functional. But it spoils the slowly varying (in space) limits of the uncorrected approximate functionals, where those functionals are right by construction. Hence, scaling of PZ-SIC is required such that it remains intact in the region of one-electron density and scales down in the region of many-electron densities. The PZ-SIC improves the performance of DFAs for the properties that involve significant SIE, as in stretched bond situations, but overcorrects for equilibrium properties where SIE is insignificant. This overcorrection is often reduced by LSIC, local scaling of the PZ-SIC to the local spin density approximation (LSDA). We propose a new scaling factor to use in an LSIC-like approach that satisfies an additional important constraint: the correct coefficient of Z in the asymptotic expansion of the $E_{xc}$ for atoms of atomic number Z, which is neglected by LSIC. LSIC and LSIC+ are scaled by functions of the iso-orbital indicator $z_{\sigma}$ that distinguishes one-electron regions from many-electron regions. LSIC+ applied to LSDA works better than LSDA-LSIC and the Perdew, Burke, and Ernzerhof (PBE) generalized gradient approximation (GGA) and gives comparable results to the strongly constrained and appropriately normed (SCAN) meta-GGA in predicting the total energies of atoms, atomization energies, barrier heights, ionization potentials, electron affinities, and bond-length of molecules. LSDA-LSIC and LSDA-LSIC+ both fail to predict interaction energies involving weaker bonds, in sharp contrast to their earlier successes. It is found that more than one set of localized SIC orbitals can yield a nearly degenerate energetic description of the same multiple covalent bonds, suggesting that a consistent chemical interpretation of the localized orbitals requires a new way to choose their Fermi orbital descriptors. A spurious correction to the exact functional would be found unless the self-Hartree and exact self-exchange-correlation terms of the PZ-SIC energy density were expressed in the same gauge. Therefore, LSIC and LSIC+ are applied only to LSDA since only LSDA has the exchange-correlation (xc) energy density in the gauge of the Hartree energy density. The transformation of energy density that achieves the Hartree gauge for the exact xc functional can be applied to approximate functionals. The use of this compliance function guarantees that scaled-down self-interaction correction (sdSIC) will make no spurious non-zero correction to the exact functional and transforms the xc energy density into the Hartree gauge. We start from the interior scaling of PZ-SIC and end at exterior scaling after the gauge transformation. SCAN-sdSIC evaluated on SCAN-SIC total and localized orbital densities is applied to the highly accurate SCAN functional, which is already much better than LSDA. Hence, the predictive power of SCAN-sdSIC is much better, even though it is scaled by $z_\sigma$ too. It provides good results for several ground state properties discussed here, including the interaction energy of weakly bonded systems. SCAN-sdSIC leads to an acceptable description of many equilibrium properties, including the dissociation energies of weak bonds. However, sdSIC fails to produce the correct asymptotic behavior $-\frac{1}{r}$ of xc potential. The xc potential as seen by the outermost electron will be $\frac{-X_{HO}^{sd}}{r}$ where HO labels the highest occupied orbital and hence doesn't guarantee a good description of charge transfer. The optimal SIC that remains to be developed might be PZ-SIC evaluated on complex Fermi-L\"owdin orbitals (with nodeless orbital densities) and Fermi orbital descriptors chosen to minimize a measure of the inhomogeneity of the orbital densities. / Physics

Le présent travail concerne l’avancement des approximations de l’énergie d’échange- corrélation (XC) de la théorie fonctionnelle de la densité (DFT) de Kohn-Sham (KS) basée sur l’approche du facteur de corrélation (CF). Le travail est organisé en trois parties où chaque partie est construite sur des modèles et méthodes précédents. La première partie du travail introduit une nouvelle condition physique à travers la déri- vation du développement en série du quatrième ordre du trou d’échange exact. La dérivation détaillée des formules requises est suivie d’une analyse approfondie qui montre que le terme de quatrième ordre peut ajouter des informations supplémentaires importantes qui sont par- ticulièrement pertinentes pour les molécules par rapport aux atomes. Sur la base de ces résultats, nous explorons les fonctionnelles d’échange qui dépendent du terme de quatrième ordre de l’expansion du trou d’échange. Nous constatons également que les développements d’ensembles de base gaussiens, fréquemment utilisés dans les codes de structure électronique, donnent des représentations insatisfaisantes du terme de quatrième ordre. La deuxième partie de ce travail porte sur la mise en œuvre de nouvelles versions du modèle CF initial [J. P. Precechtelova, H. Bahmann, M. Kaupp et M. Ernzerhof, J. Chem. Phys. 143, 144102 (2015)] dans lequel le trou XC est approximé. Étant donné que diverses contraintes satisfaites par le trou XC sont connues, des approximations peuvent être conçues pour éviter en grande partie des ajustements empiriques. Dans l’approche CF, le trou XC est écrit comme le produit d’un trou d’échange multiplié par un facteur de corrélation. Une contrainte importante satisfaite par le modèle CF est qu’il reproduit correctement l’éner- gie d’échange exacte dans la limite de haute densité. Ceci est réalisé en utilisant l’énergie d’échange exacte par particule comme variable d’entrée, c’est-à-dire que le modèle CF s’ap- puie sur l’échange exact. Des variations du modèle CF initial sont proposées qui assurent que la réponse exacte est obtenue dans la limite homogène. De plus, nous appliquons une correction à la profondeur du trou XC qui est conçue pour capturer une forte corrélation. Les fonctions d’échange-corrélation qui s’appuient sur un échange exact, comme les hybrides, échouent souvent pour les systèmes qui présentent une corrélation électronique importante. Malgré ce fait et malgré la réduction de l’empirisme à un seul paramètre dans CF, des énergies d’atomisation précises sont obtenues pour des composés de métaux de transition fortement corrélés. Le modèle CF montre des résultats significativement supérieurs aux fonctionnelles populaires comme Perdew-Burke-Ernzerhof (PBE), PBE hybride et Tao-Perdew-Staroverov- Scuseria (TPSS). La troisième partie du travail s’appuie sur les modèles CF précédents développés dans notre groupe et aborde l’erreur d’auto-interaction à un électron et introduit un modèle de facteur de corrélation modifié où f C (r, u) est construit tel qu’il se réduit à un dans les régions à un électron d’un système à plusieurs électrons. Ce trou XC avec une correction d’auto- interaction est ensuite utilisé pour générer la fonctionnelle énergie XC correspondante. La nouvelle fonctionnelle est évaluée en l’implémentant dans un programme KS et en calculant diverses propriétés moléculaires. Nous constatons que, dans l’ensemble, une amélioration significative est obtenue par rapport aux versions précédentes du modèle de facteur de cor- rélation. / The present work is concerned with the advancement of approximations to the exchangecorrelation (XC) energy of Kohn-Sham (KS) density functional theory (DFT) based on the correlation factor (CF) approach. The work is organized in three parts where each part is build upon previous models and methods. The first part of the work introduces a new physical condition through the derivation of the fourth-order series expansion of the exact exchange hole. The detailed derivation of the required formulas is followed by a thorough analysis that shows that the fourth-order term can add important additional information that is particularly relevant for molecules compared to atoms. Drawing on these findings, we explore exchange functionals that depend on the fourth-order term of the expansion of the exchange hole. We also find that Gaussian basis set expansions, frequently used in electronic structure codes, result in unsatisfactory representations of the fourth-order term. The second part of this work addresses the implementation of new versions of the initial CF model [J. P. Precechtelova, H. Bahmann, M. Kaupp, and M. Ernzerhof, J. Chem. Phys. 143, 144102 (2015)] in which the XC hole is approximated. Since various constraints satisfied by the XC hole are known, approximations to it can be designed which largely avoid empirical adjustments. In the CF approach, the XC-hole is written as a product of an exchange hole times a correlation factor. An important constraint satisfied by the CF model is that it correctly reproduces the exact exchange energy in the high density limit. This is achieved by employing the exact exchange-energy per particle as an input variable, i.e., the CF model builds on exact exchange. Variations of the initial CF model are proposed which ensure that the exact answer is obtained in the homogeneous limit. Furthermore, we apply a correction to the depth of the XC-hole that is designed to capture strong correlation. Exchangecorrelation functionals that build on exact exchange, such as hybrids, often fail for systems that exhibit sizeable electron correlation. Despite this fact and despite the reduction of empiricism to a single parameter within CF, accurate atomization energies are obtained for strongly-correlated transition metal compounds. The CF model significantly improves upon widely used functionals such as Perdew-Burke-Ernzerhof (PBE), PBE hybrid, and Tao-Perdew-Staroverov-Scuseria (TPSS) density functionals. The third part of the work builds on the previous CF models developed in our group and addresses the one-electron, self-interaction error and introduces a modified correlation factor model where fC(r, u) is constructed such that it reduces identically to one in oneelectron regions of a many-electron system. This self-interaction corrected XC-hole is then used to generate the corresponding XC-energy functional. The new functional is assessed by implementing it into a KS program and by calculating various molecular properties. We find that, overall, a significant improvement is obtained compared to previous versions of the correlation factor model.

This thesis concerns quadratic response properties and their application to properties in Jablonski diagrams such as resonant two-photon absorption and excited state absorption. Our main interest lies in optical power limiting applications, and in this context, molecules containing heavy metal atoms prove superior. Therefore, we are interested in how relativity affects these properties, and in order to assess this, a four-component relativistic framework is adopted. To properly address the molecular properties of interest, both relativistic effects and electron correlation need to be accounted for. These two properties are not additive, and, therefore, correlation needs to be incorporated into the four-component framework. We present the implementation of quadratic response properties at the four-component density functional level of theory. For second-harmonic generation, we have, with numerical examples, demonstrated that correlation and relativity are indeed not additive and that the inclusion of noncollinear magnetization is of little importance. We report that both electron correlation as well as relativity strongly affect results for second-harmonic generation. For example, relativity alone reduces the µβ-response signal by 62% and 75% for meta- and ortho-bromobenzene, respectively, and enhances the same response by 17% and 21% for meta- and ortho-iodobenzene, respectively. In the four-component framework, we present the implementations of single and double residues of the quadratic response function, which allows for the evaluation of resonant two-photon absorption cross sections and excited state properties. Using these tools, we discuss different levels of approximation to the relativistic Hamiltonian and we demonstrate that for two-photon absorption, a proper treatment of relativistic effects qualitatively alters the spectrum. For example, already for an element as light as neon, significant differences are seen between the relativistic and nonrelativistic spectra as triplet transitions acquire substantial absorption cross sections in the former case. Finally, quantum mechanics in conjunction with electrodynamics is applied to determine clamping levels in macroscopic samples. The microscopic properties of the optically active chromophores are determined by response theory, and then, electrodynamics is used to describe the interactions between the chromophores and incident laser pulses. Using this approach a series of molecules have been investigated and their performances have been compared and ranked in order to find novel materials for optical power limiting applications.

The structures of AgSin (n = 1 ¡V 16) clusters are investigated using first-principles calculations. Our studies suggest that AgSin clusters with n = 7, 10, and 15 are relatively stable isomers and that these clusters prefer to be exohedral rather than endohedral. Moreover, doping leaves the inner core structure of the clusters largely intact. Additionally, the plot of fragmentation energies as a function of silicon atoms shows that the AgSin are favored to dissociate into one Ag atom and Sin clusters. Alternative pathways exist for n > 7 (except n = 11 and 16) in which the AgSin cluster dissociates into a stable Si7 and a smaller fragment AgSin􀀀7. The AgSi11 and AgSi16 cluster dissociates into a stable Si10 and a small fragment AgSi. Lastly, our analysis indicates that doping of Ag atom significantly decreases the gaps between the highest occupied molecular orbital and the lowest unoccupied molecular orbital for n > 7.

The structures of AuSin (n = 1 - 16) clusters are investigated systematically using first-principles calculations. The lowest energy isomers exhibit preference toward exohedral rather than endohedral structure. Our studies suggest that AuSin clusters with n = 5 and 10 are relatively stable isomers. We found no significant alteration in the cluster¡¦s inner core structure for sizes n= 6, 7, 10, 11, 12, 14, and 15 even in the presence of doping. Moreover, analysis of fragmentation energies is presented in detail. Our studies further indicate that doping of Au atom significantly decreases the gaps between the highest occupied molecular orbital and the lowest unoccupied molecular orbital for n > 7. Additionally, we report on similar results obtained for CuSin (n = 1 - 16) and AgSin (n = 14, 15, and 16) and compared them with those on AuSin clusters. Next, the low energy isomers for certain sizes of CuSin (n = 10 -16 ) clusters are selected for further optimizations using Gaussian 03 package. We found that for CuSin (n = 12 - 16 ), the endohedral isomers have lower energies than their exohedral counterparts, consistent with a recent study by Janssens et al. [15] in which a similar trend was observed.