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Electron Correlation Energies in AtomsMcCarthy, Shane 09 February 2012 (has links)
No description available.
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Análise e aplicação do limite de Lieb-Oxford na teoria do funcional da densidade / Analysis and application of the Lieb-Oxford bound in density-functional theoryOdashima, Mariana Mieko 08 June 2010 (has links)
Simulações de propriedades de estrutura eletrônica possuem fundamental importância para a física do estado sólido e química quântica. A teoria do funcional da densidade (DFT) é atualmente o método de estrutura eletrônica mais empregado, desde escalas atômicas e nanoscópicas até aglomerados biomoleculares. A acurácia da DFT depende essencialmente de aproximações para os efeitos de troca e correlação, para as quais existem vínculos a serem satisfeitos como forma de controlar sua construção. Esse é um tópico de grande importância, pois a construção de melhores funcionais é necessária para uma descrição cada vez mais precisa dos efeitos de muitos corpos na DFT. No presente trabalho, investigamos o comportamento da energia de troca e correlação e o desenvolvimento de funcionais aproximados sob a ótica de um vínculo universal de sistemas de interação Coulombiana, o limite inferior de Lieb-Oxford. Primeiramente apresentamos evidências de que em diversas classes de sistemas a energia de troca e correlação é distante do limite de Lieb-Oxford. A redução do limite foi implementada nos funcionais Perdew-Burke-Erzenhof (PBE), porém a forma com que o vínculo é implementado apenas aumentou a energia de troca. Propusemos em seguida que o limite de Lieb-Oxford não fosse utilizado apenas para determinar o valor de um parâmetro, como em PBE, mas que fosse ponto-de-partida de uma nova forma família de funcionais, do tipo hiper-GGA. Exploramos uma construção não-empírica, com implementação pós-autoconsistente. A particular forma proposta se beneficiou da redução do limite Lieb-Oxford, obtendo resultados satisfatórios para as energias de correlação. / Electronic-structure calculations play a fundamental role in solid-state physics and quantum chemistry. Density-functional theory (DFT) is today the most-widely used electronic-structure method, from atomic and nanoscopic scales to biomolecular aggregates. The accuracy of DFT depends essentially on approximations to the exchange and correlation energy, which are controlled by exact constraints. This is a very important issue, since the improvement of functionals is the key to a better description of many-body effects. In the present work, we investigate the exchange-correlation energy and approximate functionals from the viewpoint of an universal constraint on interacting Coulomb systems: the Lieb-Oxford lower bound. Initially we present evidence that for several classes of systems (atoms, ions, molecules and solids), the actual exchange-correlation energies are far from the Lieb-Oxford lower bound. A tighter form of this bound was conjectured; implemented in the Perdew-Burke-Erzenhof (PBE) functionals, and tested for atoms, molecules and solids. Finally, we propose to use the Lieb-Oxford bound not just to fix the value of a parameter as in PBE, but as a starting point for a new family of hyper-GGA functionals. For these, we explored a non-empirical construction, investigating its performance for atoms and small molecules post-selfconsistently. The particular HGGA proposed benefited from the tightening of the Lieb-Oxford bound and exhibited satisfactory correlation energies.
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Análise e aplicação do limite de Lieb-Oxford na teoria do funcional da densidade / Analysis and application of the Lieb-Oxford bound in density-functional theoryMariana Mieko Odashima 08 June 2010 (has links)
Simulações de propriedades de estrutura eletrônica possuem fundamental importância para a física do estado sólido e química quântica. A teoria do funcional da densidade (DFT) é atualmente o método de estrutura eletrônica mais empregado, desde escalas atômicas e nanoscópicas até aglomerados biomoleculares. A acurácia da DFT depende essencialmente de aproximações para os efeitos de troca e correlação, para as quais existem vínculos a serem satisfeitos como forma de controlar sua construção. Esse é um tópico de grande importância, pois a construção de melhores funcionais é necessária para uma descrição cada vez mais precisa dos efeitos de muitos corpos na DFT. No presente trabalho, investigamos o comportamento da energia de troca e correlação e o desenvolvimento de funcionais aproximados sob a ótica de um vínculo universal de sistemas de interação Coulombiana, o limite inferior de Lieb-Oxford. Primeiramente apresentamos evidências de que em diversas classes de sistemas a energia de troca e correlação é distante do limite de Lieb-Oxford. A redução do limite foi implementada nos funcionais Perdew-Burke-Erzenhof (PBE), porém a forma com que o vínculo é implementado apenas aumentou a energia de troca. Propusemos em seguida que o limite de Lieb-Oxford não fosse utilizado apenas para determinar o valor de um parâmetro, como em PBE, mas que fosse ponto-de-partida de uma nova forma família de funcionais, do tipo hiper-GGA. Exploramos uma construção não-empírica, com implementação pós-autoconsistente. A particular forma proposta se beneficiou da redução do limite Lieb-Oxford, obtendo resultados satisfatórios para as energias de correlação. / Electronic-structure calculations play a fundamental role in solid-state physics and quantum chemistry. Density-functional theory (DFT) is today the most-widely used electronic-structure method, from atomic and nanoscopic scales to biomolecular aggregates. The accuracy of DFT depends essentially on approximations to the exchange and correlation energy, which are controlled by exact constraints. This is a very important issue, since the improvement of functionals is the key to a better description of many-body effects. In the present work, we investigate the exchange-correlation energy and approximate functionals from the viewpoint of an universal constraint on interacting Coulomb systems: the Lieb-Oxford lower bound. Initially we present evidence that for several classes of systems (atoms, ions, molecules and solids), the actual exchange-correlation energies are far from the Lieb-Oxford lower bound. A tighter form of this bound was conjectured; implemented in the Perdew-Burke-Erzenhof (PBE) functionals, and tested for atoms, molecules and solids. Finally, we propose to use the Lieb-Oxford bound not just to fix the value of a parameter as in PBE, but as a starting point for a new family of hyper-GGA functionals. For these, we explored a non-empirical construction, investigating its performance for atoms and small molecules post-selfconsistently. The particular HGGA proposed benefited from the tightening of the Lieb-Oxford bound and exhibited satisfactory correlation energies.
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Assessment of the scaled Perdew-Zunger self-interaction correction applied to three levels of density functional approximationsBhattarai, Puskar, 0000-0002-5613-7028 January 2021 (has links)
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
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On the physisorption of water on graphene: a CCSD(T) studyVoloshina, Elena, Usvyat, Denis, Schütz, Martin, Dedkov, Yuriy, Paulus, Beate 02 April 2014 (has links) (PDF)
The electronic structure of the zero-gap two-dimensional graphene has a charge neutrality point exactly at the Fermi level that limits the practical application of this material. There are several ways to modify the Fermi-level-region of graphene, e.g. adsorption of graphene on different substrates or different molecules on its surface. In all cases the so-called dispersion or van der Waals interactions can play a crucial role in the mechanism, which describes the modification of electronic structure of graphene. The adsorption of water on graphene is not very accurately reproduced in the standard density functional theory (DFT) calculations and highly-accurate quantum-chemical treatments are required. A possibility to apply wavefunction-based methods to extended systems is the use of local correlation schemes. The adsorption energies obtained in the present work by means of CCSD(T) are much higher in magnitude than the values calculated with standard DFT functional although they agree that physisorption is observed. The obtained results are compared with the values available in the literature for binding of water on the graphene-like substrates. / Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.
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Propriedades eletrônicas e estruturais de clusters metálicos via métodos ab initio / Eletronic and strustural properties of metal clusters by ab initio methodsDamasceno Junior, Jose Higino 25 September 2015 (has links)
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Previous issue date: 2015-09-25 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / Clusters systems are very different from molecules or their bulk materials, since
they exhibit many specific properties. As example, the bond in metallic clusters of metallic
atoms is intermediate between metallic and covalent bonding. In general, the structural and
electronic properties of these systems are very difficult to measure experimentally, and
therefore theoretical modeling is very important in characterizing them. In this thesis, we
employed ab initio methods to study metallic clusters such as the aluminum hydride
clusters as well as a few aromatic metal clusters. The optimized geometries of the studied
clusters have been determined using DFT. The electronic structures of these systems were
investigated using the QMC methods. The calculations were carried out within the
Variational (VMC) and fixed-node diffusion (DMC) quantum Monte Carlo methods. The
calculations are also performed in the Hartree-Fock (HF) approximation in order to analyze
the impact of electron correlation. With regards the aluminum hydride clusters, the total
atomic binding energy impact varies from ~20% up to about ~50%, whereas for the
electron binding energy it ranges from ~1% up to ~73%. The decomposition of the electron
binding energies clearly shows that both charge redistribution and electron correlation are
important in determining the detachment energies, whereas electrostatic and exchange
interactions are responsible for the ionization potential. For the aromatic metal clusters, the
presence of a dopant plays important role in their electronic properties enhancing their
binding energy, electron affinity, hardness and resonance energy. / Clusters são sistemas bastante diferentes de moléculas e sólidos, pois exibem
propriedades bastante peculiares. Por exemplo, a ligação em um cluster metálico tem
intensidade intermediária entre as ligações covalentes e metálicas. Em geral, as
propriedades eletrônicas e estruturais desses sistemas são bastante difíceis de serem
medidas experimentalmente e, portanto, uma modelagem teórica é muito importante na
caracterização desses. Nesta Tese, utilizamos métodos ab initio para estudar clusters
metálicos, tal como clusters de hidretos de alumínio assim como também alguns clusters
metálicos aromáticos. As estruturas geométricas dos clusters estudados foram otimizadas
via DFT. A estrutura eletrônica desses clusters foi investigada usando o método de Monte
Carlo Quântico Variacional (MCQD) e de difusão (MCQD) com aproximação de nós fixos.
Os cálculos também foram realizados a partir da aproximação de Hartree-Fock, afim de se
analisar o impacto da energia de correlação eletrônica. Para os hidretos de alumínio, a
energia de correlação eletrônica tem impacto na energia total de ligação variando de 20% a
50%. Da mesma maneira, a energia de ligação de um elétron ao cluster tem grande
contribuição da energia de correlação eletrônica, variando de 1% a 73%. A decomposição
da energia de ligação mostra claramente que a relaxação e a correlação eletrônica são
importantes na determinação da afinidade eletrônica, enquanto que a interação de troca
eletrostática é responsável pelo potencial de ionização. Para os clusters aromáticos, a
presença do dopante desempenha um importante papel nas propriedades desses clusters,
uma vez que otimiza a energia de ligação, a afinidade eletrônica, a dureza e a energia de
ressonância.
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On the physisorption of water on graphene: a CCSD(T) studyVoloshina, Elena, Usvyat, Denis, Schütz, Martin, Dedkov, Yuriy, Paulus, Beate January 2011 (has links)
The electronic structure of the zero-gap two-dimensional graphene has a charge neutrality point exactly at the Fermi level that limits the practical application of this material. There are several ways to modify the Fermi-level-region of graphene, e.g. adsorption of graphene on different substrates or different molecules on its surface. In all cases the so-called dispersion or van der Waals interactions can play a crucial role in the mechanism, which describes the modification of electronic structure of graphene. The adsorption of water on graphene is not very accurately reproduced in the standard density functional theory (DFT) calculations and highly-accurate quantum-chemical treatments are required. A possibility to apply wavefunction-based methods to extended systems is the use of local correlation schemes. The adsorption energies obtained in the present work by means of CCSD(T) are much higher in magnitude than the values calculated with standard DFT functional although they agree that physisorption is observed. The obtained results are compared with the values available in the literature for binding of water on the graphene-like substrates. / Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.
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Utilisation de l’apprentissage automatique pour approximer l’énergie d’échange-corrélationCuierrier, Étienne 01 1900 (has links)
Le sujet de cette thèse est le développement de nouvelles approximations à l’énergie
d’échange-corrélation (XC) en théorie de la fonctionnelle de la densité (DFT).
La DFT calcule l’énergie électronique d’une molécule à partir de la densité électronique,
une quantité qui est considérablement plus simple que la fonction d’onde. Cette théorie a
été développée durant les années 1960 et elle est devenue la méthode de choix en chimie
quantique depuis 1990, grâce à un ratio coût/précision très favorable. En pratique, la DFT
est utilisée par les chercheurs et l’industrie pour prédire des spectres infrarouges, la longueur
des liens chimiques, les barrières d’activation, etc. Selon l’approche Kohn-Sham, seulement
le terme de l’énergie XC est inconnu et doit être approximé. Les chapitres de ce texte sont
des articles consacrés au développement d’approches non locales et à l’utilisation de
l’apprentissage automatique pour améliorer la précision et/ou la rapidité des calculs de l’énergie
XC.
Le premier article de cette thèse concerne le développement d’approximations non locales
au trou XC [Cuierrier, Roy, et Ernzerhof, JCP (2021)]. Notre groupe de recherche a précédemment
développé la méthode du facteur de corrélation (CFX) [Pavlíková Přecechtělová,
Bahmann, Kaupp, et Ernzerhof, JCP (2015)] et malgré les résultats supérieurs de CFX
comparativement aux fonctionnelles courantes en DFT pour le calcul de l’énergie, cette approche
n’est pas exacte pour les systèmes uniélectroniques. Les méthodes non locales telles que le
facteur X [Antaya, Zhou, et Ernzerhof, PRA (2014)] corrigent ce problème. Ainsi, le but
du projet du premier article est de combiner CFX avec le facteur X, afin de former un facteur XC
exact pour l’atome d’hydrogène, tout en conservant les bonnes prédictions de CFX
pour les molécules. Nos résultats montrent que notre modèle non local est exact pour les
systèmes uniélectroniques, cependant, la densité électronique non locale a un comportement
fortement oscillatoire qui rend difficile la construction du facteur XC et la qualité de ses
prédictions pour les propriétés moléculaires est inférieure aux fonctionnelles hybrides. Notre
étude permet de fournir une explication concernant l’échec des méthodes non locales en
chimie, comme l’approximation de la densité pondérée [Gunnarsson, Jonson, et Lundqvist,
PLA (1976)]. Les nombreuses oscillations de la densité non locale limitent la performance
des facteurs XC qui sont trop simples et qui ne peuvent pas atténuer ces oscillations.
vLe sujet du deuxième article de cette thèse [Cuierrier, Roy et Ernzerhof, JCP (2021)]
est relié aux difficultés rencontrées durant le premier projet. L’apprentissage automatique
(ML) est devenu une méthode populaire dans tous les domaines de la science. Les réseaux de
neurones artificiels (NN) sont particulièrement puissants, puisqu’ils permettent un contrôle
et une flexibilité considérables lors de la construction de fonctions approximatives. Ainsi,
nous utilisons un NN pour modéliser le trou X à partir de contraintes physiques. Durant le
premier projet de cette thèse, nous avons observé qu’une fonction mathématique simple n’est
pas adaptée pour être combinée avec la densité non locale, les NN pourraient donc être un
outil utile pour approximer un trou X. Néanmoins, ce chapitre s’intéresse à la densité locale,
avant de s’attaquer à la non-localité. Les résultats que nous avons obtenus lors du calcul
des énergies X des atomes montrent le potentiel des NN pour construire automatiquement
des modèles du trou X. Une deuxième partie de l’article suggère qu’un NN permet d’ajouter
d’autres contraintes à des approximations du trou X déjà existantes, ce qui serait utile pour
améliorer CFX. Sans les NN, il est difficile de trouver une équation analytique pour accomplir
cette tâche. L’utilisation du ML est encore récente en DFT, mais ce projet a contribué à
montrer que les NN ont beaucoup d’avenir dans le domaine de la construction de trou XC.
Finalement, le dernier chapitre concerne un projet [Cuierrier, Roy, Wang, et Ernzerhof,
JCP (2022)] qui utilise aussi des NN en DFT. Des travaux précédents du groupe ont montré
que le terme de quatrième ordre du développement en série de puissances de la distance
interélectronique du trou X (Tσ (r)) [Wang, Zhou, et Ernzerhof, PRA (2017)] est un ingrédient
utile pour améliorer les approximations du calcul de l’énergie X pour les molécules. Cependant,
il n’a pas été possible de construire un modèle qui satisfait le deuxième et le quatrième
terme du développement en série de puissances simultanément. Ainsi, avec l’expertise
développée en apprentissage automatique lors du deuxième projet, le but de l’étude du troisième
article est d’utiliser Tσ (r) comme une variable d’entrée à un NN qui approxime l’énergie
X. Nous avons montré qu’en utilisant comme ingrédients la fonctionnelle de PBE, Tσ (r) et
un NN, il est possible de s’approcher de la qualité des résultats d’une fonctionnelle hybride
(PBEh) pour le calcul d’énergies d’atomisation, de barrières d’activation et de prédiction de
la densité électronique. Cette étude démontre que Tσ (r) contient de l’information utile pour
le développement de nouvelles fonctionnelles en DFT. Tσ (r) est en principe plus rapide à
calculer que l’échange exact, donc nos fonctionnelles pourraient s’approcher de l’exactitude
d’une fonctionnelle hybride, tout en étant plus rapides à calculer. / The subject of this thesis is the development of new approximations for the exchange-
correlation (XC) energy in Density Functional Theory (DFT).
DFT calculates the electronic energy from the electronic density, which is a considerably
simpler quantity than the wave function. DFT was developed during the 1960s and became
the most popular method in quantum chemistry during the 1990s, thanks to its favourable
cost/precision ratio. In practice, DFT is used by scientists and the industry to predict
infrared spectra, bond lengths, activation energies, etc. The Kohn-Sham approach in DFT
is by far the most popular, since it is exact in theory and only the XC functional has to be
be approximated. The exact form of the XC functional is unknown, thus the development
of new approximations for the XC functional is an important field of theoretical chemistry.
In this thesis, we will describe the development of new non-local methods and the use of
machine learning to improve the prediction and the efficiency of the calculation of XC energy.
The first article in this thesis [Cuierrier, Roy, and Ernzerhof, JCP (2021)] is about the
development of non-local approximations of the XC hole. Our research group previously
developed the correlation factor approach (CFX) [Pavlíková Přecechtělová, Bahmann, Kaupp, and
Ernzerhof, JCP (2015)]. The prediction of CFX for molecular properties compares favourably
to other common functionals. However, CFX suffers from one-electron self-interaction error
(SIE). Non-local models such as the X factor [Antaya, Zhou, and Ernzerhof, PRA (2014)]
can fix the SIE, thus the goal of this project is to combine CFX with the X factor to build a
non-local XC factor. We show that our method is exact for one-electron systems, however,
our simple XC factor is not appropriate for the oscillatory behaviour of the non-local density
and the results for molecules are inferior when compared to hybrid functionals. Our study
provides an explanation of why non-local models, such as the weighted density
approximation [Gunnarsson, Jonson, and Lundqvist, PLA (1976)], are not as successful as the common
DFT functionals (PBE, B3LYP, etc.) in chemistry. The non-local electronic density is an
elaborate function and often has a large number of local minima and maxima. The
development of functionals using simple XC factors does not lead to satisfying results for the
prediction of molecular energies. Therefore, a sophisticated XC factor that could attenuate
the oscillatory shape of the non-local density is required.
viiThe second article [Cuierrier, Roy, and Ernzerhof, JCP(2021)] addresses the difficulties
observed for the development of non-local functionals during the first project. Machine
learning (ML) is a useful technique that is gaining popularity in many fields of science, including
DFT. Neural networks (NN) are particularly powerful, since their structure allows
considerable flexibility to approximate functions. Thus, in this chapter, we use a NN to approximate
the X hole by considering many of its known physical and mathematical constraints during
the training of the NN. The results we obtain, for the calculation of energies of atoms using
the NN, reveal the potential of this method for the automation of the construction of X
holes. The second part of the paper shows that an NN can be used to add more constraints
to an existing X hole approximation, which would be quite useful to improve CFX. The X
hole obtained for a stretched H2 molecule is promising when compared to the exact values.
ML is still a new tool in DFT and our work shows that it has considerable potential for the
construction of XC hole approximations.
Finally, the last chapter [Cuierrier, Roy, Wang, and Ernzerhof, JCP (2022)] describes a
project that also uses NN. In a previous work by our group, it is shown that the fourth-order
term of the expansion of the X hole (Tσ (r)) could improve the calculation of the X energy for
molecules [Wang, Zhou, and Ernzerhof, PRA (2017)]. However, developing an equation that
satisfies both the second and fourth-order terms simultaneously proved difficult. Thus, using
the expertise in ML we developed during the second project, we build a new NN that uses
the fourth-order term of the expansion of the X hole as a new ingredient to approximate
the XC energy. Starting from the PBE functional, we trained a NN to reproduce the X
energy of the hybrid functional PBEh. Our results show that this approach is a considerable
improvement compared to PBE for the calculation of atomization energies, barrier heights
and the prediction of electronic density. This study confirms that the fourth-order term of
the expansion of the X hole does include useful information to build functionals in DFT.
Since the calculation of the fourth-order term has a more favourable computational scaling
compared to the exact exchange energy, our new functionals could lead to faster calculations
in DFT.
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Développements et applications de méthodes pour la description de l’énergie de corrélation dans les molécules et les solides / Developments and applications of methods for the description of correlation energy in molecules and solidsClaudot, Julien 05 July 2018 (has links)
Les fonctionnelles de la densité couramment utilisées ont rencontrées un succès spectaculaire dans la modélisation des systèmes physiques, chimiques, et biologiques. Toutefois, elles se sont avérées inadaptées pour décrire certaines situations, comme par exemple les forces de dispersion de London ou les phénomènes de corrélation forte. Dans le cadre de cette thèse, nous nous sommes intéressés à des développements récents de la formulation de l’énergie de corrélation exprimée à partir du théorème de fluctuation-dissipation et connexion adiabatique, visant à pallier ces problèmes. En particulier, différentes implémentations des méthodes au-delà de l’approximation de la phase aléatoire, qui permettent la prise en compte de la contribution d’échange dans le calcul de l’énergie de corrélation, ont été comparées. Ensuite, afin de réduire drastiquement la complexité numérique, une procédure d’orthogonalisation des vecteurs utilisées pour représenter la matrice diélectrique a été développée. Ces méthodes ont ensuite été appliquées au calcul de l’énergie de liaison de petits complexes moléculaires. La formulation de l’énergie de corrélation de la théorie de perturbation de Møller-Plesset dans le contexte matrice diélectrique est aussi présentée et testée. En parallèle, des calculs utilisant les méthodes semi-empiriques numériquement efficaces ont été conduits sur trois ensembles de molécules afin d’en tester les performances concernant les énergies de liaisons en les comparant aux valeurs de références disponibles dans la littérature / Commonly used density functionals have encountered a spectacular success in the modelling of physical, chemical or biological systems. However, they have proven to be unsuitable to describe some situations, such as London’s dispersion forces or strong correlation behaviour. In this thesis, we have been interested in recent developments in the formulation of the correlation energy from the adiabatic connection fluctuation dissipation theorem, to overcome these problems. In particular, different implementations of methods beyond the random phase approximation, which allow to take into account the exchange contribution in the computation of the correlation energy, have been compared. Then, in order to drastically decrease the numerical complexity, an orthogonalization procedure of the vectors used to represent the dielectric matrix has been developed. Then these approaches were applied to the calculation of the binding energy of small molecular complexes. The formulation of the correlation energy of the Møller-Plesset perturbation theory within the dielectric matrix context is also presented and tested. In parallel, calculations using numerically efficient semi-empirical methods were conducted over three molecular sets in order to test their performances regarding the binding energies by comparing them to reference values available in the literature
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Construction of exchange and exchange-correlation functionalsWang, Rodrigo 04 1900 (has links)
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.
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