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Physical Review A

Time-dependent density-matrix functional in Liouville space and the optical response of many-electron systems

Vladimir Chernyak, Shaul Mukamel

Published Nov 1, 1995• 13 citations
Publisher

Abstract

A time-dependent functional approach is developed for calculating the dynamical optical response and ground-state properties of nonrelativistic fermion systems. The functional (effective action) uses the particle-hole components of the reduced single-electron density matrix as variables and is constructed using the natural orbitals as a reference. It is defined on pairs of trajectories that constitute a Liouville space path in the phase space of the classical analog of the quantum system. A perturbative ``semiclassical'' calculation of the effective Liouville-space action is used to incorporate correlation effects in the linear optical response.

DOI
10.1103/physreva.52.3601

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A time-dependent functional approach is developed for calculating the dynamical optical response and ground-state properties of nonrelativistic fermion systems. The functional (effective action) uses the particle-hole components of the reduced single-electron density matrix as variables and is constructed using the natural orbitals as a reference. It is defined on pairs of trajectories that constitute a Liouville space path in the phase space of the classical analog of the quantum system. A perturbative ``semiclassical'' calculation of the effective Liouville-space action is used to incorporate correlation effects in the linear optical response.
The time-dependent density-functional (TDDFT) equations may be written either for the Kohn-Sham orbitals in Hilbert space or for the single-electron density matrix in Liouville space. A collective-oscillator, quasiparticle, representation of the density response of many-electron systems which explicitly reveals the relevant electronic coherence sizes is developed using the Liouville-space representation of adiabatic TDDFT. Closed expressions for the nonlinear density-density response are derived, eliminating the need to solve nonlinear integral equations, as required in the Hilbert-space formulation of the response.
A new approach for the calculation of the linear response of both the ground and thermal-equilibrium states of many-electron systems is proposed via an integral formulation of the density-functional theory. Based on the path-integral representation of the one-particle Green's function, the frequency-dependent linear-response function for the Kohn-Sham noninteracting system is expressed as an explicit functional of the Kohn-Sham local potential, instead of the summation over the single-electron orbitals. Thus the dynamic linear density response can be determined as the solution of self-consistent equations which only need as input the total electron density of the ground or the equilibrium state. Exchange and correlation effects are incorporated. Because orbitals are not employed, the formulation provides the possibility for calculations of linear-response properties, such as dynamic polarizability and photoabsorption cross sections of systems with very many electrons. The present formulation can also be applied to general fermions.
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.
A linear-scaling first-principles quantum mechanical method is developed to evaluate the optical responses of large molecular systems. Instead of a many-body wave function, the equation of motion is solved for the reduced single-electron density matrix in the time domain. The locality of the reduced single-electron density matrix is utilized to ensure that computational time scales linearly with system size. The two-electron Coulomb integrals are evaluated with the fast multipole method, and the calculation of exchange-correlation quadratures utilizes the locality of an exchange-correlation functional and the integral prescreening technique. As an illustration, the resulting time-dependent density-functional theory is used to calculate the absorption spectra of polyacetylene oligomers and linear alkanes. The linear-scaling of computational time versus the system size is clearly demonstrated.
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.
The adiabatic approximation is problematic in time-dependent density matrix functional theory. With pure density matrix functionals (invariant under phase change of the natural orbitals) it leads to lack of response in the occupation numbers, hence wrong frequency dependent responses, in particular α(ω→0)≠α(0) (the static polarizability). We propose to relinquish the requirement that the functional must be a pure one-body reduced density matrix (1RDM) functional, and to introduce additional variables which can be interpreted as phases of the one-particle states of the independent particle reference system formed with the natural orbitals, thus obtaining so-called phase-including natural orbital (PINO) functionals. We also stress the importance of the correct choice of the complex conjugation in the two-electron integrals in the commonly used functionals (they should not be of exchange type). We demonstrate with the Löwdin-Shull energy expression for two-electron systems, which is an example of a PINO functional, that for two-electron systems exact responses (polarizabilities, excitation energies) are obtained, while writing this energy expression in the usual way as a 1RDM functional yields erroneous responses.