The Green function formalism introduced by Keldysh for irreversible processes is applied to the calculation of the time-dependent charge exchange probabilities in scattering problems. Considering the initial state of two non-interacting subsystems described by an independent particle model, their interaction is taken in a first stage, as evolving in time according to a time-dependent Hartree - Fock (TDHF) scheme. The equations of motion for the two-times Green functions are subsequently obtained in Dyson-like form by including the residual electronic correlations expanded perturbatively up to second order. It is shown that, by solving these equations of motion, this in turn allows one to calculate the two-particle correlation functions at equal times, as required to describe the fractions of charge in each subsystem. This procedure contrasts with that of considering expectation values of the correlation functions taken between the corrected TDHF state up to an equivalent order in the perturbation. In order to test the advantages of this method we have applied this scheme to the scattering of an atom from a three-substrate atom chain described by an Anderson Hamiltonian, where comparisons with exact solutions can be easily established. We found that, in this case, the fractions of charges carried by the scattered particle obtained with our proposal compare fairly well with exact results, within an ample range of parameter selection.
The Green function formalism introduced by Keldysh for irreversible processes is applied to the calculation of the time-dependent charge exchange probabilities in scattering problems. Considering the initial state of two non-interacting subsystems described by an independent particle model, their interaction is taken in a first stage, as evolving in time according to a time-dependent Hartree - Fock (TDHF) scheme. The equations of motion for the two-times Green functions are subsequently obtained in Dyson-like form by including the residual electronic correlations expanded perturbatively up to second order. It is shown that, by solving these equations of motion, this in turn allows one to calculate the two-particle correlation functions at equal times, as required to describe the fractions of charge in each subsystem. This procedure contrasts with that of considering expectation values of the correlation functions taken between the corrected TDHF state up to an equivalent order in the perturbation. In order to test the advantages of this method we have applied this scheme to the scattering of an atom from a three-substrate atom chain described by an Anderson Hamiltonian, where comparisons with exact solutions can be easily established. We found that, in this case, the fractions of charges carried by the scattered particle obtained with our proposal compare fairly well with exact results, within an ample range of parameter selection.
In this paper we derive a formula for the time-dependent Hartree-Fock dielectric function by using Green's-function theory. We develop a set of self-consistent equations for the self energy, polarization, and Green's function which can be iterated to obtain successively more accurate approximations for the polarization. In this procedure the Hartree and Hartree-Fock approximations occur naturally as the first and second steps in the iterations. As the iterations are carried out, we obtain not only the expressions for the polarization, but also the one-electron Hartree-Fock equations, so that there is no question as to which polarization formula corresponds to which equation. In this way we obtain an expression for the dielectric function which clearly and directly corresponds to the Hartree-Fock equation and which is therefore called the time-dependent Hartree-Fock (TDHF) dielectric function. In this TDHF formula for the dielectric function, the first term is the time-dependent Hartree or random-phase approximation dielectric function. We show that in order to be consistent with the theory, one should use Hartree wave functions and energies to compute random-phase approximation dielectric functions and Hartree-Fock wave functions and energies to compute TDHF dielectric functions. Numerical results are presented for the free-electron gas.
Within the Landauer approach to conduction discussed in the previous chapter, electron interactions have been included only at the mean-field level. This is quite a strong approximation, especially in nanojunctions, where large current densities are common. I therefore want to go beyond this level of description. I can follow two different routes. (1) I can develop a functional theory of quantum correlations via time-dependent effective single-particle equations, as I will do in Chapter 7 where I will use dynamical density-functional theories within the micro-canonical approach to conduction. (2) I can try to solve the time-dependent Schrödinger equation 1.16 – or its mixed-state version, the Liouville-von Neumann equation 1.60 – directly. Written this way, this last proposition seems hopeless. In reality, we can employ a many-body technique, known as the non-equilibrium Green's function formalism (NEGF), also referred to as the Keldysh formalism (Keldysh, 1964; Kadanoff and Baym, 1962), which allows us, at least in principle, to do just that: solve the time-dependent Schrödinger equation for an interacting many-body system exactly, from which one can, in principle, calculate the time-dependent current. This is done by solving equations of motion for specific time-dependent single-particle Green's functions, from which the physical properties of interest, such as the charge and current densities, can be obtained. I have stressed the term “time-dependent” several times, because, as I will show in a moment, the NEGF is “exact” only when one solves the time-dependent Schrödinger equation for a closed quantum system, subject to deterministic perturbations: the system is closed but not necessarily isolated. These perturbations may drive the system far away from its initial state of thermodynamic equilibrium (Keldysh, 1964).
The equation of motion for nonequilibrium Green functions is derived within the framework of the Schwinger and Keldysh formalism of perturbation expansion. For nonequilibrium distribution Green functions, the equation of motion derived from quantum mechanics contains undefined singularities, whose explicit form depends on the specific initial or boundary condition. In the present work, the exact expression of singular terms is found in the equation of motion from the time-looped perturbation theory in which the adiabatic initial condition is implied. Unlike the usual Dyson perturbation formalism or the well known Kadanoff-Baym equation of motion, our resulting equation can be adopted directly for calculations without the graphical analysis, which depends on the specific form of the Hamiltonian. On the basis of this equation of motion, the procedure of a nonperturbative solution is outlined and potential applications are briefly discussed.
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.