The choice of basic variables in current-density functional theory

The selection of basic variables in current-density functional theory and formal properties of the resulting formulations are critically examined. Focus is placed on the extent to which the Hohenberg--Kohn theorem, constrained-search approach and Lieb's formulation (in terms of convex and concave conjugation) of standard density-functional theory can be generalized to provide foundations for current-density functional theory. For the well-known case with the gauge-dependent paramagnetic current density as a basic variable, we find that the resulting total energy functional is not concave. It is shown that a simple redefinition of the scalar potential restores concavity and enables the application of convex analysis and convex/concave conjugation. As a result, the solution sets arising in potential-optimization problems can be given a simple characterization. We also review attempts to establish theories with the physical current density as a basic variable. Despite the appealing physical motivation behind this choice of basic variables, we find that the mathematical foundations of the theories proposed to date are unsatisfactory. Moreover, the analogy to standard density-functional theory is substantially weaker as neither the constrained-search approach nor the convex analysis framework carry over to a theory making use of the physical current density

Computing optical properties of large systems

In recent years, time-dependent density-functional theory (TDDFT) has been the method of choice for calculating optical excitations in medium sized to large systems, due to its good balance between computational cost and achievable accuracy. In this thesis, TDDFT is reformulated to fit the framework of the linear-scaling density-functional theory (DFT) code ONETEP. The implementation relies on representing the optical response of the system using two sets of localised, atom centered, in situ optimised orbitals in order to ideally describe both the electron and the hole wavefunctions of the excitation. This dual representation approach requires only a minimal number of localised functions, leading to a very efficient algorithm. It is demonstrated that the method has the capability of computing low energy excitations of systems containing thousands of atoms in a computational effort that scales linearly with system size. The localised representation of the response to a perturbation allows for the selective convergence of excitations localised in certain regions of a larger system. The excitations of the whole system can then be obtained by treating the coupling between different subsystems perturbatively. It is shown that in the limit of weakly coupled excitons, the results obtained with the coupled subsystem approach agree with a full treatment of the entire system, with a large reduction in computational cost. The strengths of the methodology developed in this work are demonstrated on a number of realistic test systems, such as doped p-terphenyl molecular crystals and the exciton coupling in the Fenna-Matthews-Olson complex of bacteriochlorophyll. It is shown that the coupled subsystem TDDFT approach allows for the treatment of system sizes inaccessible by previous methods.Open Acces

Repulsively diverging gradient of the density functional in the Reduced Density Matrix Functional Theory

The Reduced Density Matrix Functional Theory (RDMFT) is a remarkable tool for studying properties of ground states of strongly interacting quantum many body systems. As it gives access to the one-particle reduced density matrix of the ground state, it provides a perfectly tailored approach to studying the Bose-Einstein condensation or systems of strongly correlated electrons. In particular, for homogeneous Bose-Einstein condensates as well as for the Bose-Hubbard dimer it has been recently shown that the relevant density functional exhibits a repulsive gradient (called the Bose-Einstein condensation force) which diverges when the fraction of non-condensed bosons tends to zero. In this paper, we show that the existence of the Bose-Einstein condensation force is completely universal for any type of pair-interaction and also in the non-homogeneous gases. To this end, we construct a universal family of variational trial states which allows us to suitably approximate the relevant density functional in a finite region around the set of the completely condensed states. We also show the existence of an analogous repulsive gradient in the fermionic RDMFT for the $N$-fermion singlet sector in the vicinity of the set of the Hartree-Fock states. Finally, we show that our approximate functional may perform well in electron transfer calculations involving low numbers of electrons. This is demonstrated numerically in the Fermi-Hubbard model in the strongly correlated limit where some other approximate functionals are known to fail.Comment: v6: the published version. New J. Phys 202

Pascual Jordan, his contributions to quantum mechanics and his legacy in contemporary local quantum physics

After recalling episodes from Pascual Jordan's biography including his pivotal role in the shaping of quantum field theory and his much criticized conduct during the NS regime, I draw attention to his presentation of the first phase of development of quantum field theory in a talk presented at the 1929 Kharkov conference. He starts by giving a comprehensive account of the beginnings of quantum theory, emphasising that particle-like properties arise as a consequence of treating wave-motions quantum-mechanically. He then goes on to his recent discovery of quantization of ``wave fields'' and problems of gauge invariance. The most surprising aspect of Jordan's presentation is however his strong belief that his field quantization is a transitory not yet optimal formulation of the principles underlying causal, local quantum physics. The expectation of a future more radical change coming from the main architect of field quantization already shortly after his discovery is certainly quite startling. I try to answer the question to what extent Jordan's 1929 expectations have been vindicated. The larger part of the present essay consists in arguing that Jordan's plea for a formulation without ``classical correspondence crutches'', i.e. for an intrinsic approach (which avoids classical fields altogether), is successfully addressed in past and recent publications on local quantum physics.Comment: More biographical detail, expansion of the part referring to Jordan's legacy in quantum field theory, 37 pages late

Correcting deficiencies in approximate density functionals

In the last fifty years, approximate density functional theory (DFT) has become firmly established as the de facto standard for electronic structure calculations in chemistry. Although the theory itself is formally exact, approximations must be made for the unknown exchange–correlation (XC) functional, and whilst many successful approximate functionals exist, a number of deficiencies still persist, leading to many cases where the approximation breaks down completely. This thesis addresses two prevalent deficiencies, and examines some novel approaches to reducing and eliminating them. Chapter 1 provides a background to electronic structure theory, with particular reference to the approximate solution of the electronic Schrödinger equation through ab initio wavefunction methods. Chapter 2 then provides the formal justification for DFT as an alternative to wavefunction-based methods, and outlines common approximations to the XC functional. Two prominent deficiencies of approximate DFT are discussed: delocalisation error due to non-linearity in the energy variation with number of electrons, and incorrect long-range behaviour of the XC potential. Chapter 3 examines a system-dependent tuning technique for the range-separated hybrid class of XC functionals, whereby the range-separation parameter is non-empirically tuned to self-consistent energy-linearity conditions, which has been successfully used to improve the calculation of quantities affected by the delocalisation error. A full, systematic assessment of this tuning technique is provided, and it is demonstrated that the success of the technique is aided by a convenient cancellation of errors. In Chapter 4, the tuned functionals are applied to quantities relevant to conceptual DFT. It is shown that functionals tuned to the energy conditions of Chapter 3 remain appropriate for calculation of the electronegativity from orbital energies, however the density variation with number of electrons — described by the Fukui function — is better modelled by conventional non-tuned functionals. Finally, an entirely new approach to functional development is provided in Chapter 5. The behaviour of a functional under density scaling is used to impose homogeneity constraints on a simple functional form, culminating in an electron-deficient functional that satisfies the appropriate energy-linearity condition and exhibits the correct asymptotic XC potential

Ein einheitliches Approximationsschema für Dichtefunktionaltheorien: Ein auf Kraftbilanz basierender Ansatz

First-principle methods as a way of understanding various fundamental phenomena that occur in nature is an active field of research in condensed matter physics and other related fields. There is great interest in the study of how a system or a property changes when an external perturbation is applied to it, say, by switching on a magnetic field or probing the system with a laser. Many successful theoretical developments have been made over the years to specifically treat these different situations. However using one of these theories out of its assigned setting, by construction, does not always guarantee a suitable outcome and some interesting features may not be captured. This is partly due to the approximations that are used in these methods which are geared to only specific external perturbations or properties. It is therefore of importance to have a theory that can, in a consistent way, treat these various settings and allow for the qualitative study of the changes that occur when different external stimuli (magnetic fields, lasers,...) are applied to a system. We propose here such an approach that contains all the ingredients necessary to perform such a qualitative study. In this thesis we present a unifying scheme to determine exchange correlation potentials in density and current density functional theories including vector potentials. The standard energy-based approach to determine functionals is not used here. Instead this approach relies on the equations of motion of particular current densities and is viable both for the ground state and the time-dependent setting. We aim at directly approximating the density-potential mapping thereby avoiding subtleties that arise from functional differentiability and also the costly optimized effective potential procedure of orbital-dependent energy functionals. We then show that the different density functional theories are connected through these equations of motion and demonstrate this for a local-exchange approximation. We show how these exchange-type approximations reduce to the usual local density approximation in the case of a homogeneous system. We highlight what is not captured when approximations for simple settings are used in more complex ones. In addition, these equations of motion provide a way to numerically construct density-potential mappings for different density functional theories and we show this particularly for a ground state lattice setting including the Peierl’s phase. All these show that this equation-of-motion-based approach bears many interesting advantages and provides a new path for approximations in density functional theories. Moreover, it sets a path for a more complete understanding of the properties of molecules or solids subject to different external stimuli.Ab-inition Methoden zum Verständnis grundlegender Phänomene, die in der Natur auftreten, sind ein aktives Forschungsfeld in der Festkörperphysik und anderen verwandten Bereichen. Eine der Hauptfragestellungen hierbei ist, wie sich ein System und seine Eigenschaften ändern, wenn sich eine externe Größe ändert, beispielsweise indem ein Magnetfeld eingeschaltet oder das System mit einem Laser getrieben wird. Im Laufe der Jahre wurden viele erfolgreiche Theorien entwickelt, um spezifische solche Situationen zu behandeln. Die Verwendung dieser Theorien in anderen Situationen führt jedoch häufig nicht zu einem zufriedenstellenden Ergebnis, da wichtige Merkmale nicht erfasst werden. Dies ist teilweise auf die Näherungen zurückzuführen, die bei diesen Methoden verwendet werden, und welche nur für spezielle externe Störungen oder spezifische Eigenschaften zulässig sind. Es ist daher wichtig, eine Theorie zu haben, die alle Spezialfälle auf konsistente Art und Weise behandelt. Wir stellen einer Herangehensweise vor, die alle Bestandteile enthält, die zur Durchführung einer solchen qualitativen Studie erforderlich sind. Wir stellen in dieser Arbeit ein einheitliches Schema von Näherungen zur Bestimmung von Austauschkorrelationspotentialen (einschließlich Vektorpotentialen)in Dichte- und Stromdichtefunktionaltheorien vor. Der standardmäßige energiebasierte Ansatz zur Bestimmung von Funktionalen wird hier nicht verwendet. Stattdessen beruht dieser Ansatz auf den Bewegungsgleichungen bestimmter Stromdichten und ist sowohl für den Grundzustand als auch für die zeitabhängige Situation gültig. In dem Ansatz wollen wir die Dichte-Potential-Abbildung direkt approximieren, wodurch Problema vermieden werden, die sich aus der funktionalen Differenzierbarkeit ergeben, sowie das kostspielige optimierte effektive Potentialverfahren von orbitalabhängigen Energiefunktionalen. Wir zeigen dann, dass die verschiedenen Dichtefunktionaltheorien durch diese Bewegungsgleichungen verbunden sind, und demonstrieren dies für eine lokale Austauschnäherung. Wir zeigen, wie sich diese Näherungen vom Austauschtyp bei einem homogenen System auf die übliche lokale-Dichte-Näherung reduzieren. Durch das Anwenden dieser komplexeren Näherungen auf einfachere Systeme ergeben sich Effekte, die in den anderen Theorien nicht sichtbar werden. Darüber hinaus bieten diese Bewegungsgleichungen eine Möglichkeit, Dichte-Potential-Abbildungen für verschiedene Dichtefunktionaltheorien numerisch zu untersuchen, und wir zeigen dies insbesondere für ein simplest System realisiert auf einem Gitter im Grundzustand welches eine Peierl-Phase enthält. All dies zeigt, dass dieser auf Bewegungsgleichungen basierende Ansatz viele interessante Vorteile bietet und einen neuen Weg für Approximationen in Dichtefunktionaltheorien bietet. Darüber hinaus zeigt die neue Näherungstheorie wie man die Untersuchung von Eigenschaften von Molekülen und Festkörpern unter dem Einfluß von verschiedenen externen Störungen vereinheitlichen kann

Ein einheitliches Approximationsschema für Dichtefunktionaltheorien: Ein auf Kraftbilanz basierender Ansatz

First-principle methods as a way of understanding various fundamental phenomena that occur in nature is an active field of research in condensed matter physics and other related fields. There is great interest in the study of how a system or a property changes when an external perturbation is applied to it, say, by switching on a magnetic field or probing the system with a laser. Many successful theoretical developments have been made over the years to specifically treat these different situations. However using one of these theories out of its assigned setting, by construction, does not always guarantee a suitable outcome and some interesting features may not be captured. This is partly due to the approximations that are used in these methods which are geared to only specific external perturbations or properties. It is therefore of importance to have a theory that can, in a consistent way, treat these various settings and allow for the qualitative study of the changes that occur when different external stimuli (magnetic fields, lasers,...) are applied to a system. We propose here such an approach that contains all the ingredients necessary to perform such a qualitative study. In this thesis we present a unifying scheme to determine exchange correlation potentials in density and current density functional theories including vector potentials. The standard energy-based approach to determine functionals is not used here. Instead this approach relies on the equations of motion of particular current densities and is viable both for the ground state and the time-dependent setting. We aim at directly approximating the density-potential mapping thereby avoiding subtleties that arise from functional differentiability and also the costly optimized effective potential procedure of orbital-dependent energy functionals. We then show that the different density functional theories are connected through these equations of motion and demonstrate this for a local-exchange approximation. We show how these exchange-type approximations reduce to the usual local density approximation in the case of a homogeneous system. We highlight what is not captured when approximations for simple settings are used in more complex ones. In addition, these equations of motion provide a way to numerically construct density-potential mappings for different density functional theories and we show this particularly for a ground state lattice setting including the Peierl’s phase. All these show that this equation-of-motion-based approach bears many interesting advantages and provides a new path for approximations in density functional theories. Moreover, it sets a path for a more complete understanding of the properties of molecules or solids subject to different external stimuli.Ab-inition Methoden zum Verständnis grundlegender Phänomene, die in der Natur auftreten, sind ein aktives Forschungsfeld in der Festkörperphysik und anderen verwandten Bereichen. Eine der Hauptfragestellungen hierbei ist, wie sich ein System und seine Eigenschaften ändern, wenn sich eine externe Größe ändert, beispielsweise indem ein Magnetfeld eingeschaltet oder das System mit einem Laser getrieben wird. Im Laufe der Jahre wurden viele erfolgreiche Theorien entwickelt, um spezifische solche Situationen zu behandeln. Die Verwendung dieser Theorien in anderen Situationen führt jedoch häufig nicht zu einem zufriedenstellenden Ergebnis, da wichtige Merkmale nicht erfasst werden. Dies ist teilweise auf die Näherungen zurückzuführen, die bei diesen Methoden verwendet werden, und welche nur für spezielle externe Störungen oder spezifische Eigenschaften zulässig sind. Es ist daher wichtig, eine Theorie zu haben, die alle Spezialfälle auf konsistente Art und Weise behandelt. Wir stellen einer Herangehensweise vor, die alle Bestandteile enthält, die zur Durchführung einer solchen qualitativen Studie erforderlich sind. Wir stellen in dieser Arbeit ein einheitliches Schema von Näherungen zur Bestimmung von Austauschkorrelationspotentialen (einschließlich Vektorpotentialen)in Dichte- und Stromdichtefunktionaltheorien vor. Der standardmäßige energiebasierte Ansatz zur Bestimmung von Funktionalen wird hier nicht verwendet. Stattdessen beruht dieser Ansatz auf den Bewegungsgleichungen bestimmter Stromdichten und ist sowohl für den Grundzustand als auch für die zeitabhängige Situation gültig. In dem Ansatz wollen wir die Dichte-Potential-Abbildung direkt approximieren, wodurch Problema vermieden werden, die sich aus der funktionalen Differenzierbarkeit ergeben, sowie das kostspielige optimierte effektive Potentialverfahren von orbitalabhängigen Energiefunktionalen. Wir zeigen dann, dass die verschiedenen Dichtefunktionaltheorien durch diese Bewegungsgleichungen verbunden sind, und demonstrieren dies für eine lokale Austauschnäherung. Wir zeigen, wie sich diese Näherungen vom Austauschtyp bei einem homogenen System auf die übliche lokale-Dichte-Näherung reduzieren. Durch das Anwenden dieser komplexeren Näherungen auf einfachere Systeme ergeben sich Effekte, die in den anderen Theorien nicht sichtbar werden. Darüber hinaus bieten diese Bewegungsgleichungen eine Möglichkeit, Dichte-Potential-Abbildungen für verschiedene Dichtefunktionaltheorien numerisch zu untersuchen, und wir zeigen dies insbesondere für ein simplest System realisiert auf einem Gitter im Grundzustand welches eine Peierl-Phase enthält. All dies zeigt, dass dieser auf Bewegungsgleichungen basierende Ansatz viele interessante Vorteile bietet und einen neuen Weg für Approximationen in Dichtefunktionaltheorien bietet. Darüber hinaus zeigt die neue Näherungstheorie wie man die Untersuchung von Eigenschaften von Molekülen und Festkörpern unter dem Einfluß von verschiedenen externen Störungen vereinheitlichen kann

Properties of exact density functionals for electronic quantum transport

Density functional theory and its extension in the nonequilibrium regime, time-dependent density functional theory, are powerful tools for predicting the structures, energies and dynamics of electronic systems. Their usefulness derives from the Kohn-Sham scheme whereby a system of real, interacting particles is replaced by a fictitious system of non-interacting particles subject to an effective external potential instead of a pairwise particle-particle interaction. The Kohn-Sham universe yields the same observable phenomena as that predicted by standard quantum mechanics so long as the effective external potential is known. However, for the vast majority of systems it is not known, and the usually local (in time and space) functional approximations employed do not capture the physics of true nonlocal interactions. In this thesis, the exact charge and current densities of model quantum transport devices described by nonlocal potentials are studied and methods for reverse-engineering the corresponding exact Kohn-Sham effective external potential for time-dependent and steady-state density functional theory approaches to the same systems are presented, as well as the resulting exact potentials themselves. Features of improved functionals for calculating approximate Kohn-Sham systems are demonstrated. These functionals are suggested to be very different from existing functionals employed, describing not potentials but electric and magnetic fields, and have a strong dependence on the local and semilocal charge and current density

Time Dependent Density Functional Theory of Dynamical Response in 3d and 4d Metals

Two different but complementary quantum mechanical many-body problems are investigated. These problems include both static and dynamic aspects of the electronelectron interaction in real materials. In Chapter One, we take up the cases of Ag and Ni with a microscopic evaluation of the dielectric function and loss function using the formalism of time dependent density functional theory and all-electron techniques. We address the striking line shapes that have been recently observed via inelastic scattering experiments. The present work reveals three relevant energy scales for excitations in the selected systems. These scales are argued to be generic to a large number of 3d and 4d metals, and include the threshold for excitation of d electrons, final state energies, and the plasmon energy. Our results for Ag corroborate the experimental interpretation of the anomalous dispersion of the nominal plasmon loss, and shed new light on the striking line shape as well as predicting an anomalous dispersion of the nominal plasmon lifetime. In agreement with experiment, the theoretical loss spectrum of Ni is found to be equally complex with two prominent loss features at ~22eV and ~28 eV. The ab initio results demonstrate that both phase space and a strong modulation of d → p transitions lead to the predicted behavior. Moreover, in contrast to the canonical description that has been used to describe these features, we find them to be quite different from plasma oscillations. In Chapter Two, we address static properties of the electron-electron interaction as it pertains to ground state properties. In the exchange-only method one approximates the exchange-correlation energy functional of density functional theory by its Hartree-Fock form, ensuring that the method adheres to several scaling laws and identities which are violated by the local density and generalized gradient approximations. Although there is no formal correspondence to eigenvalue gaps determined by photoemission or inverse photoemission, we find that exchange-only results partially remove the discrepancy between these energy gaps and those obtained based on the local density approximation. However, we also find marked discrepancies with other recent theoretical treatments. Suggestions for future research are made