As experimental techniques begin to probe electronic motions in increasing detail, the need is arising for compact and informative visualisations of simulations of such processes. The inherent challenge is that a full many-electron wavefunction is a high-dimensional object, representing the complicated correlations of strongly repulsive bodies in a small molecular volume. A general procedure is needed to distill this to a smaller amount of information that does not rely on any specific level of approximation. The result should allow for easy and intuitive interpretation while drawing out nontrivial aspects of the underlying many-body dynamics, such as the complex phase information inherent of a nonstationary state. Current visualisation schemes based on physical observables or the qualitative information contained in simple wavefunctions, such as time-dependent configuration-interaction-singles (TD-CIS) and time-dependent self-consistent-field (TD-SCF), are discussed. This information is compared to an analysis based on the one-body reduced density operator (1-RDO), which is well-defined for general wavefunctions. It is seen that the distinction between two paradigms of many-body dynamics, electron transport and energy transport, is reflected in the coherences of a difference-1-RDO, or lack thereof.
As experimental techniques begin to probe electronic motions in increasing detail, the need is arising for compact and informative visualisations of simulations of such processes. The inherent challenge is that a full many-electron wavefunction is a high-dimensional object, representing the complicated correlations of strongly repulsive bodies in a small molecular volume. A general procedure is needed to distill this to a smaller amount of information that does not rely on any specific level of approximation. The result should allow for easy and intuitive interpretation while drawing out nontrivial aspects of the underlying many-body dynamics, such as the complex phase information inherent of a nonstationary state. Current visualisation schemes based on physical observables or the qualitative information contained in simple wavefunctions, such as time-dependent configuration-interaction-singles (TD-CIS) and time-dependent self-consistent-field (TD-SCF), are discussed. This information is compared to an analysis based on the one-body reduced density operator (1-RDO), which is well-defined for general wavefunctions. It is seen that the distinction between two paradigms of many-body dynamics, electron transport and energy transport, is reflected in the coherences of a difference-1-RDO, or lack thereof.
The problem of quantum dynamics in open systems has gained attention in recent decades and not the least due to the advances made in quantum transport in molecular systems. The main motivation behind quantum transport and molecular electronics is the futuristic goal to be able at some point to replace, or to complement, the silicon-based technology and to make the electronic devices faster. On a fundamental level, one has to deal with time-dependent processes where electron-electron or electron- phonon interactions are of great importance, and they can cause profound quantitative and qualitative changes on the physical and dynamical properties of electronic systems compared to the non-interacting case. Most of the studies of quantum transport have been focused on the steady-state description while neglecting the short-time dynamics. However, the dynamical effects are of great importance since fast- switching processes play a pivotal role in the operation of future devices. We studied the problem of time-dependent electron transport through the Anderson impurity model by using many-body perturbation theory (MBPT) together with Keldysh Green’s functions as well as with time-dependent density functional theory (TDDFT). These methods were compared with numerically exact time-dependent density-matrix renormalization group (tDMRG) method. We found that the many-body perturbation theory results beyond Hartree-Fock approximation were in close agreement with tDMRG results. In addition we studied the possibility of multistablity in the density and current of an interacting nanoscale junction as well as how to reversibly switch between the multiple solutions in time domain. An accurate theoretical treatment of electron correlation even in as simple model as an interacting electron gas at metallic densities still continues to be a challenge; especially description of features in the photoemission spectra due to electron correlations provides a theoretical challenge. The many-body perturbation theory yields a systematic way to study electron-electron (electron-phonon) correlations in various systems. One of the widely used approximations in MBPT is the GW approximation in which the bare interaction line is replaced with screened interaction line in the first order exchange diagram. The GW approximation gives good estimates for the band gap values close to experimental ones but especially the self-consistent GW approximation has a number of deficiencies like washing out of plasmon features and overestimation of bandwidths compared to experiment. One way to improve GW calculations is to include vertex corrections. Unfortunately, the straightforward inclusion of vertex corrections yields negative spectra in some frequency regions. We developed a diagrammatic approach to construct self-energy approximations with positive spectral properties. Our approach consists of expressing a self-energy of response diagram as a product of half-diagrams after which a minimal set of additional diagrams is identified to construct a perfect square. We applied this method to study vertex corrections in a homogeneous electron gas. In addition we analyzed the diagrammatic content of photocurrent with density functional theory. The expression for the photocurrent was obtained as an integral over the Kohn-Sham spectral function renormalized by effective potentials that depend on the exchange correlation kernel of current density functional theory. The expression for the photocurrent gives us the angular dependence of the photocurrent but it does not provide a direct access to the kinetic energy distribution of the photoelectrons.
In our analysis of the dynamic one-electron difference-density matrices arising from time-dependent many-body states, we have observed two very distinct modes for the migration of an excitation through a system. In one case, coherences in the density matrix give rise to single particle and hole states that appear to migrate as independent units. Though interacting, they are independent in the sense that the single-electron state that describes an individual particle or hole is everywhere entangled with the same state of the electrons that have been integrated/traced out. These pseudo-wavefunctions carry nontrivial phase information (nonzero momentum), which is consistent with their observed direction of travel. The other case is dominated by a transition that can be thought of as the closing of a particle--hole pair in one location, entangled with the opening of such a pair in another location. In this case, the spatially-dependent phase of a given particle or hole state lies very near the real axis, reminiscent of time-independent molecular orbitals.
In no wave function of the type (1) [product of single determinants for each spin] is there a statistical correlation between the positions of the electrons with antiparallel spin. The purpose of the aforementioned generalization of (1) is to allow for such correlations. This will lead to an improvement of the wave function and, therefore, to a lowering of the energy value. E. Wigner, Phys. Rev. 46, 1002 (1934) Summary Although the exact many-body wavefunction cannot be written down in general, we do know of some of its properties. For example, there are differences between the wavefunctions of insulators and metals and the cusp condition gives the behavior as any two charges approach each other. In this chapter we also discuss approximate wavefunctions, ways to judge their accuracy and how to include electronic correlation. Examples of many-body wavefunctions are the Slater–Jastrow (pair product) wavefunction and its generalization to pairing and backflow wavefunctions. In other places in this book, we argue that it is not necessary to look at the many-body wavefunctions explicitly because they are unwieldy; the one- and two-body correlation functions discussed in Ch. 5 are sufficient to determine the energy and give information on the excitation spectra. However, these correlation functions do not always contain all information of interest. In principle, the ground-state energy of a many-electron system is a functional of the density, but the very derivation of the theorem invokes the many-body wavefunction, as expressed explicitly in Eq. (4.16). The effects of antisymmetry are manifest in the correlation functions, but antisymmetry is most simply viewed as a property of the wavefunction; electronic correlation is fundamentally a result of properties of the many-body wavefunction. Studying many-body wavefunctions provides a very useful, different point of view of many-body physics from the approaches based on correlation functions. Many of the most important discoveries in physics have come about by understanding the nature of wavefunctions, such as the Laughlin wavefunction for the fractional quantum Hall effect, the BCS wavefunction for superconductors, p-wave pairing in superfluid 3 He, and the Heitler– London approach for molecular binding. The role of Berry's phases has brought out the importance of the phase of the wavefunction in determining properties of quantum systems. This has led to new understanding of the classification of insulators, metals, superconductors, vortices, and other states of condensed matter.
In this thesis the electron transport through interacting quantum wires is studied.The time-dependent properties of the system are investigated by using the non-equilibrium Green function technique.This technique provides a way to include ultra-fast timedependent processes, arbitrary strong external fields and also brings forward an effective treatment of electron-electron and electron-phonon interactions.The electronelectron interactions are included via various self-energy approximations at Hartree-Fock, second-order Born and GW levels.These approximations are called conserving since they fulfill the physical conservation laws like the charge conservation law.The interacting central system is coupled to one-dimensional tight-binding leads, which are considered effectively as noninteracting.The system is perturbed by a bias voltage after which the time-evolution of the system is followed by Kadanoff-Baym equations.The non-equilibrium steady-state value of the current depends on the used self-energy approximation, which indicates the importance of the history-dependence in the calculation of current.This is an important observation in the aspect of time-dependent density functional theory, since it is one indication for the need of functionals with memory.The steady-state current is independent of the initial state, which is in accordance with the memory-loss theorem.The transient current shows small oscillations, which are consequence of the density oscillations especially in the first and last site.The density oscillations in the sites are effect of the electron transition from the lead to the central system and vice versa.
In this paper, we propose a generic and systematic approach for study of the electronic structure for atoms or molecules. In particular, we address the issue of single particle states, or orbitals, which should be one of the most important aspects of a quantum many-body theory. We argue that the single-particle Green function provides a most general scheme for generating these single particle states or orbitals. We call them the correlated atomic or molecular orbitals to make a distinction from those determined from Hartree−Fock equation. We present the calculation of the single particle properties (i.e., the electron affinities (EA′s) and ionization potentials (IP′s)) for the H2O molecule using the correlated molecular orbitals in the context of quantum chemistry with a second-order self energy. We also calculate the total ground state energy with a single Slater wavefunction determined only from the hole states. Comparisons are made with available experimental data as well as with those from the Hartree−Fock or density functional theory (DFT) calculations. We conclude that the correlated atomic or molecular orbital approach provides a strictest and most powerful method for studying the single-particle properties of atoms or molecules. It also gives a better total energy than do the Hartree−Fock and DFT even at the single Slater determinant level. It promises that a correlation theory based on the correlated atomic or molecular orbitals will become an approach which possesses the advantages and also overcomes their shortcomings of current quantum chemistry methods based on either the conventional quantum many-body theory or the DFT.
Correlated electronic systems involve strong shortrange repulsion. At integer filling, the primary effect of correlation is the emergence of an insulating state where band theory predicts a metal. Understanding the transition from a metal to this ‘Mott insulator’, as interaction strength is increased, and the effect of doping the Mott state, are classic problems in quantum many-body physics. They have been explored for decades, starting with the approximate solutions of Hubbard. The real breakthrough came in the 1990s with the advent of dynamical mean field theory, which mapped the correlated system to an embedded single site, and described the detailed spectral evolution across the Mott transition. The only deficiency in this approach is the neglect of spatial correlations, and, possibly, the lack of some ‘visual intuition’ about the transition. We discuss a complementary approach, which provides a reasonable description of Mott physics in a real-space setting and allows a certain degree of visualization. This article provides a pedagogical review of our approach and presents some illustrative results.