Time-dependent quantities are calculated in the linear response limit for a correlated one dimensional model atom driven by an external quadrupolar time-dependent field. Besides the analysis of the time-evolving energy change in the correlated two-particle system, and orthogonality of initial and final states, Mehler's formula is applied in order to derive a point-wise decomposition of the time-dependent one-matrix in terms of time-dependent occupation numbers and time-dependent orthonormal, natural orbitals. Based on such exact spectral decomposition on the time domain, Rényi's entropy is also investigated. Considering the structure of the exact time-dependent one-matrix, an independent-particle model is defined from it which contains exact information on the single-particle probability density and probability current. The resulting noninteracting auxiliary state is used to construct an effective potential and discuss its applicability.
We determine the exact time-dependent non-idempotent one-particle reduced density matrix and its spectral decomposition for a harmonically confined two-particle correlated one-dimensional system when the interaction terms in the Schrödinger Hamiltonian are changed abruptly. Based on this matrix in coordinate space we derivea precise condition for the equivalence of the purity and the overlap-square of the correlated and non-correlated wave functions as the system evolves in time. This equivalence holds only if the interparticle interactions are affected, while the confinement terms are unaffected within the stability range of the system. Under this condition we also analyze various time-dependent measures of entanglement and demonstrate that, depending on the magnitude of the changes made in the Schrödinger Hamiltonian, periodic, logarithmically incresing or constant value behavior of the von Neumann entropy can occur.
Time-dependent density functional theory (TDDFT) provides a successful approach to calculate excitation energies of atomic and molecular systems. In part I of this work we present a double-pole approximation (DPA) to the response equations of TDDFT. The double-pole approximation provides an exact description of systems with two strongly coupled excitations which are isolated from the rest of the spectrum. In contrast to the traditional single- pole approximation of TDDFT the DPA also yields corrections to the Kohn-Sham oscillator strengths. Several critical pole separations can be identified, e.g. we find that the pole coupling can cause transitions to vanish entirely from the optical spectrum. We also demonstrate how to invert the double-pole solution which allows us to predict matrix elements of the exchange- correlation kernel fxc from experimental input. This can serve as benchmark for the construction of future approximations for the kernel fxc. Reduced density matrix functional theory (RDMFT) has emerged recently as promising candidate to treat strongly correlated electronic many-body systems beyond traditional density functional theory (DFT). The research within RDMFT was so far focussed on the static theory. In this work we attempt some first steps towards a time-dependent generalization of RDMFT. In part II we derive equations of motion for natural orbitals and occupation numbers. Using the equation of motion for the occupation numbers we show that an adiabatic extension of presently known ground-state functionals of static RDMFT always leads to occupation numbers which are constant in time. From the stationary conditions of the equations of motion for the N-body correlations (correlated parts of the N-body matrices) we derive a new class of ground-state functionals which can be used in static RDMFT. Applications are presented for a one-dimensional model system where the time-dependent many-body Schrödinger equation can be propagated numerically. We use optimal control theory to find optimized laser pulses for transitions in a model for atomic Helium. From the numerically exact correlated wavefunction we extract the exact time evolution of natural orbitals and occupation numbers for (i) laser-driven Helium and (ii) electron-ion scattering. Part III of this work considers time-dependent quantum transport within TDDFT. We present an algorithm for the calculation of extended eigenstates of single- particle Hamiltonians which is especially tailored to a finite-difference discretization of the Schrödinger equation. We consider the propagation of finite mesoscopic systems and demonstrate the limitations of such an approach. To overcome the shortcomings of a description of quantum transport in terms of finite systems we develop a time-propagation scheme for extended states which utilizes a mixed basis representation. Our discretization scheme allows to treat central device and lead regions on the same footing thus preventing artificial reflections at grid boundaries.
We use the natural orbitals to define an independent particle system, from which the exact one-particle density matrix can be obtained with an ensemble of degenerate determinantal ground states. Also defining explicit phases for the orbitals, and admitting functionals that are dependent on those phases, time-dependent equations for the orbitals and occupation numbers are obtained from an action principle. The wrong polarizability and lack of double excitations of straightforward adiabatic time-dependent density matrix functional theory are then corrected, and the important symmetry χ(ω)=χ{*}(-ω), lost in previous ad hoc improvements, is restored. The extension of the response calculations beyond the occupied-virtual pairs, which are the only ones admitted in time-dependent density functional theory, leads to greatly improved response properties.
Based on the correlated ground-state wave function of an exactly solvable interacting one-dimensional two-electron model Hamiltonian we address the switch-off of the external confining interaction in order to calculate the exact time-evolving wave function from a correlated initial state. The Hartree-Fock approximation for the initial state of the model system is considered as well. By taking the differences of kinetic and potential energy terms obtained with these treatments, the time-dependent components of Wigner's correlation energy are determined.
The optimized effective potential (OEP) method is a formally exact approach to treat orbital-dependent exchange-correlation functionals. The first part of this chapter reviews the static OEP method. Orbital-dependent exchange-correlation functionals are implicit density functionals. The associated exchange-correlation potential is constructed via an integral equation. In particular, the exact exchange potential is defined in this way, starting from the Fock exchange energy functional. The resulting exchange potential has the correct asymptotic behaviour, and has a discontinuity as a function of particle number. Several properties and applications of the static OEP are reviewed. The second part of the chapter discusses the time-dependent OEP method. Again, the time-dependent exchange-correlation potential is constructed with an integral equation, which is derived from a variational principle. Two case studies are discussed, which illustrate the memory dependence and the discontinuity of the exchange correlation potential. The linear-response limit of the time-dependent OEP is introduced.
Based on the correlated ground-state wave function of an exactly solvable interacting one-dimensional two-electron model Hamiltonian we address the switch-off of confining and interparticle interactions to calculate the exact time-evolving wave function from a prescribed correlated initial state. Using this evolving wave function, the time-dependent pair probability function $\mathcal{R}({x}_{1},{x}_{2},t)\ensuremath{\equiv}{n}_{2}({x}_{1},{x}_{2},t)/[n({x}_{1},t)\phantom{\rule{0.16em}{0ex}}n({x}_{2},t)]$ is determined via the pair density
n2(x1,x2,t) and single-particle density
n(x,t). It is found that $\mathcal{R}(0,0,t=\ensuremath{\infty})=\mathcal{R}(0,0,t=0)>1$, and
R(x1,x2,t∗)=1 at a finite
t∗ for $\ensuremath{\Lambda}\ensuremath{\ne}0$ interparticle interaction strength in the initial two-electron model. By expanding
n(x,t) in an infinite sum of closed-shell products of time-dependent normalized single-particle states and time-dependent occupation numbers ${P}_{k}(\ensuremath{\Lambda},t)$, the von Neumann entropy $S(\ensuremath{\Lambda},t)=\ensuremath{-}{\ensuremath{\sum}}_{k=0}^{\ensuremath{\infty}}{P}_{k}(t)\mathrm{ln}{P}_{k}(t)$ is calculated as well. The such-defined information entropy is zero at ${t}^{*}(\ensuremath{\Lambda})$ and its maximum in time is $S(\ensuremath{\Lambda},t=\ensuremath{\infty})=S(\ensuremath{\Lambda},t=0)$.