Reduced Single-electron Density Matrix Research Articles

Imaging of transition charge densities involving carbon core excitations by all X-ray sum-frequency generation.

X-ray diffraction signals from the time-evolving molecular charge density induced by selective core excitation of chemically inequivalent carbon atoms are calculated. A narrowband X-ray pulse selectively excites the carbon K-edge of the -CH3 or -CH2F groups in fluoroethane (CH3-CH2F). Each excitation creates a distinct core coherence which depends on the character of the electronic transition. Direct propagation of the reduced single-electron density matrix, using real-time time-dependent density functional theory, provides the time-evolving charge density following interactions with external fields. The interplay between partially filled valence molecular orbitals upon core excitation induces characteristic femtosecond charge migration which depends on the core-valence coherence, and is monitored by the sum-frequency generation diffraction signal. This article is part of the theme issue 'Measurement of ultrafast electronic and structural dynamics with X-rays'.

Time-Dependent Density Functional Theory for Open Systems and Its Applications.

Photovoltaic devices, electrochemical cells, catalysis processes, light emitting diodes, scanning tunneling microscopes, molecular electronics, and related devices have one thing in common: open quantum systems where energy and matter are not conserved. Traditionally quantum chemistry is confined to isolated and closed systems, while quantum dissipation theory studies open quantum systems. The key quantity in quantum dissipation theory is the reduced system density matrix. As the reduced system density matrix is an O(M! × M!) matrix, where M is the number of the particles of the system of interest, quantum dissipation theory can only be employed to simulate systems of a few particles or degrees of freedom. It is thus important to combine quantum chemistry and quantum dissipation theory so that realistic open quantum systems can be simulated from first-principles. We have developed a first-principles method to simulate the dynamics of open electronic systems, the time-dependent density functional theory for open systems (TDDFT-OS). Instead of the reduced system density matrix, the key quantity is the reduced single-electron density matrix, which is an N × N matrix where N is the number of the atomic bases of the system of interest. As the dimension of the key quantity is drastically reduced, the TDDFT-OS can thus be used to simulate the dynamics of realistic open electronic systems and efficient numerical algorithms have been developed. As an application, we apply the method to study how quantum interference develops in a molecular transistor in time domain. We include electron-phonon interaction in our simulation and show that quantum interference in the given system is robust against nuclear vibration not only in the steady state but also in the transient dynamics. As another application, by combining TDDFT-OS with Ehrenfest dynamics, we study current-induced dissociation of water molecules under scanning tunneling microscopy and follow its time dependent dynamics. Given the rapid development in ultrafast experiments with atomic resolution in recent years, time dependent simulation of open electronic systems will be useful to gain insight and understanding of such experiments. This Account will mainly focus on the practical aspects of the TDDFT-OS method, describing the numerical implementation and demonstrating the method with applications.

Shape-Dependent Electronic Excitations in Metallic Chains

The electronic excitations of silver chains with different geometries (linear, circle, arc and zigzag chains) have been investigated at the time-dependent density functional theory level, by solving the equation of motion of the reduced single-electron density matrix in the real-time domain. A scaling parameter 0 ≤ λ ≤ 1 has been introduced to adjust the two-electron contributions during propagation in the time domain in a way that allows us to distinguish different electronic excitations — plasmon and single-particle excitations. The dipole responses, in particular the plasmon resonances of these metallic chains to an external δ-pulse, show a strong dependence on their geometric structures. In most cases, the dipole responses of these chains possess great tunability when altering their geometric parameters — the radius of the circle and arc chains, and the bond angle of the zigzag chains. Some excitations in these chains also show a wide tunable excitation energy range, more than 1 eV, making it possible to fine-tune the excitations of the metallic chains at an atomic scale.

Plasmon resonances in linear noble-metal chains

The electronic excitations of three noble-metall chains--copper, silver, and gold--have been investigated at the time-dependent density functional theory level. The reduced single-electron density matrix is propagated according to the Liouville-von Neumann equation in the real-time domain after an impulse excitation. The propagation in the real-time domain enables us to investigate the formation and size evolution of electronic excitations in these metallic chains with different number of atoms, up to a total of 26 atoms. The longitudinal oscillations at lower excitation energies are dominated by s → p transitions in these chains and have collective or central resonances, while the first peak involving d → p transitions in the longitudinal mode appears at a higher excitation energy and shows collective resonances. In the transverse oscillations, there are in most cases d → p transitions in each resonance, which can be attributed to either central or end resonances. Convergence of the oscillations, in particular those involving the collective and central resonances in the three noble-metal chains can only be observed for chains with 18 atoms or more. Different spectroscopic characteristics among these three metallic chains can be attributed to their different electronic structures, in particular the relativistic effects in the gold chains have a dramatic effect on their electronic structures and excitations.

Time-dependent quantum transport: An efficient method based on Liouville-von-Neumann equation for single-electron density matrix

Basing on our hierarchical equations of motion for time-dependent quantum transport [X. Zheng, G. H. Chen, Y. Mo, S. K. Koo, H. Tian, C. Y. Yam, and Y. J. Yan, J. Chem. Phys. 133, 114101 (2010)], we develop an efficient and accurate numerical algorithm to solve the Liouville-von-Neumann equation. We solve the real-time evolution of the reduced single-electron density matrix at the tight-binding level. Calculations are carried out to simulate the transient current through a linear chain of atoms, with each represented by a single orbital. The self-energy matrix is expanded in terms of multiple Lorentzian functions, and the Fermi distribution function is evaluated via the Padè spectrum decomposition. This Lorentzian-Padè decomposition scheme is employed to simulate the transient current. With sufficient Lorentzian functions used to fit the self-energy matrices, we show that the lead spectral function and the dynamics response can be treated accurately. Compared to the conventional master equation approaches, our method is much more efficient as the computational time scales cubically with the system size and linearly with the simulation time. As a result, the simulations of the transient currents through systems containing up to one hundred of atoms have been carried out. As density functional theory is also an effective one-particle theory, the Lorentzian-Padè decomposition scheme developed here can be generalized for first-principles simulation of realistic systems.

Transient electronic dynamics of noninteracting open systems beyond linear response

We present analytical results on the transient current response of noninteracting openelectronic systems under time-dependent external voltages in both linear- andnonlinear-response regimes. The derivations are based on an equation of motion formalismfor the system reduced single-electron density matrix (Zheng et al 2007 Phys.Rev. B 75 195127). Dissipative interactions between the system and leads aretreated by the nonequilibrium Green’s function approach. The linear-responsedynamics is characterized by the analytical admittance spectrum of the opensystem, through which the quantum coherent transport properties are mapped toequivalent classical circuits. The nonlinear-response current spectrum not only resolvesthe intrinsic energetic configuration of the system, but also reflects the uniquedynamical features due to the transient characteristics of the applied voltages.

Probing valence electronic wave-packet dynamics by all x-ray stimulated Raman spectroscopy: A simulation study

The femtosecond impulsive stimulated x-ray Raman signal from quinolinol is simulated and analyzed using the doorway-window representation of pump-probe spectroscopy. A valence electronic wave packet prepared by the pump, tuned on resonance with a given core hole, is localized in the vicinity of a selected atom and probed by a window localized on a different atom, selected by the probe pulse. All valence electronic states within the pulse bandwidths can be observed with high spatial and temporal resolution by monitoring the variation of the signal with the delay between the pulses. Natural orbital representation of the reduced single-electron density matrix is used to visualize the dynamics of the valence wave packet, described as a linear combination of determinants made of occupied and unoccupied Kohn-Sham orbitals.

Time-dependent density-functional theory for open systems

By introducing the self-energy density functionals for the dissipative interactions between the reduced system and its environment, we develop a time-dependent density-functional theory formalism based on an equation of motion for the Kohn-Sham reduced single-electron density matrix of the reduced system. Two approximate schemes are proposed for the self-energy density functionals, the complete second order approximation and the wide-band limit approximation. A numerical method based on the wide-band limit approximation is subsequently developed and implemented to simulate the steady and transient current through various realistic molecular devices. Simulation results are presented and discussed.

Density matrix based time-dependent density functional theory and the solution of its linear response in real time domain

A density matrix based time-dependent density functional theory is extended in the present work. Chebyshev expansion is introduced to propagate the linear response of the reduced single-electron density matrix upon the application of a time-domain delta-type external potential. The Chebyshev expansion method is more efficient and accurate than the previous fourth-order Runge-Kutta method and removes a numerical divergence problem. The discrete Fourier transformation and filter diagonalization of the first-order dipole moment are implemented to determine the excited state energies. It is found that the filter diagonalization leads to highly accurate values for the excited state energies. Finally, the density matrix based time-dependent density functional is generalized to calculate the energies of singlet-triplet excitations.

Linear-scaling time-dependent density-functional theory

A linear-scaling time-dependent density-functional theory is developed to evaluate the optical response of large molecular systems. The two-electron Coulomb integrals are evaluated with the fast multipole method, and the calculation of exchange-correlation quadratures utilizes the locality of exchange-correlation functional within the adiabatic local density approximation and the integral prescreening technique. Instead of many-body wave function, the equation of motion is solved for the reduced single-electron density matrix in the time domain. Based on its ``nearsightedness'', the reduced density matrix cutoffs are employed to ensure that the computational time scales linearly with the system size. As an illustration, the resulting time-dependent density-functional theory is used to calculate the absorption spectra of linear alkanes, and the linear scaling of computational time versus the system size is clearly demonstrated.

Localized-density-matrix implementation of time-dependent density-functional theory

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.

A dynamic mean-field theory for dissipative interacting many-electron systems

A non-Markovian theory of dissipative many-electron systems in the presence of arbitrary coherent fields is presented. The new formulation is constructed via the second-quantization of a complete second-order quantum dissipation theory followed by the random phase or time-dependent Hartree–Fock approximation for electron–electron correlation. The theory is shown to be of the Fermi–Dirac thermal equilibrium statistics limit if electron–electron correlation is neglected. The key quantity is the reduced single-electron density matrix instead of the reduced density matrix of the system, which leads to the drastic reduction of numerical simulation time. The validity of the dynamic mean-field approximation is tested in a model two-electron spin-conversed system with Ohmic dissipation in the presence of excitation field.

Excitation and dissipation of interacting many-electron system

A new formalism based on the equation of motion for the reduced single-electron density matrix has been developed to simulate the dissipative interacting many-electron systems. The electron correlation is treated within the random phase approximation. The Markovian bath is adopted. The resulting nonradiative dissipative term is of Lindblad-like form. The formalism is applied to a butadiene molecule embedded in a liquid or solid environment, where the valence electrons are taken as the system and the nuclear vibrational modes as the bath. Various excitations are found decaying differently, e.g., different absorption peak widths are observed.

Intraband terahertz emission from coupled semiconductor quantum wells: A model study using the exciton representation

Intraband emission resulting from the second-order polarization induced by a femtosecond pump field in semiconductor nanostructures is calculated. The two-band model is transformed to the exciton representation, and exciton-phonon coupling represented by arbitrary spectral densities is incorporated into the equations of motion for the relevant exciton variables through Redfield superoperators. Time-resolved coherent and incoherent terahertz radiation as well as the incoherent optical fluorescence are calculated for a one-dimensional model of coupled quantum wells in an external electric field with on-site Coulomb interaction between electrons and holes and an overdamped Brownian oscillator spectral density representing the electron-phonon coupling. A doorway-window picture based on the reduced single-electron density matrix is developed for the analysis of both signals. Disorder and phonon-induced dephasing effects on the various signals are discussed and compared.

Generalized linear-scaling localized-density-matrix method

A generalized linear scaling localized-density-matrix (LDM) method is developed to adopt the nonorthonormal basis set and retain full Coulomb differential overlap integrals. To examine its validity, the method is employed to evaluate the absorption spectra of polyacetylene oligomers containing up to 500 carbon atoms. The semiempirical Hamiltonian for the π electrons includes explicitly the overlap integrals among the π orbitals, and is determined from the ab initio Hartree–Fock reduced single-electron density matrix. Implementation of the generalized LDM method at the ab initio molecular orbital calculation level is discussed.

Density-matrix–electronic-oscillator representation of optical spectroscopy of semiconductor nanocrystals

The optical response of CdSe semiconductor nanocrystals is investigated using the reduced single-electron density matrix in real space, calculated by means of the time-dependent Hartree–Fock technique. The spectroscopic signatures of exciton confinement are analyzed using the frequency-dependent electronic coherence matrix (off-diagonal density-matrix elements). The effects of Hartree and the Fock (exchange) type Coulomb interactions on the exciton binding energy are discussed. The latter result in almost dark excitons situated energetically below the main transition. Off-diagonal Coulomb matrix elements lead to larger exciton binding energies compared with previous calculations, and result in a better agreement of the size dependence of the lowest optical transition with experiment.

Nonlinear Polarizabilities of Donor−Acceptor Substituted Conjugated Polyenes

The polarizabilities of conjugated polyenes with 8−36 carbon atoms are calculated using the time dependent Hartree−Fock (TDHF) technique. By examining the reduced single-electron density matrix, we explore the relations among the ground state chemical bonding structure and off-resonant and resonant susceptibilities. The effects of a donor−acceptor substitution on oligomers of different sizes are investigated.

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

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.

Real-space coupled-oscillator approach to the radiative decay of conjugated polymers.

The optical response of conjugated polymers is calculated using a one-dimensional tight-binding model with Coulomb interactions, and adopting the multipolar form for the electromagnetic field Hamiltonian. By formulating the problem using the reduced single-electron density matrix and its evolution in Liouville space, we obtain a real-space description for the elementary electronic excitations. A unified expression for the linear response including radiative decay, which is valid for arbitrary chain size compared with the optical wavelength, is described. For small sizes it shows exciton confinement effects on the radiative decay and for larger sizes it fully incorporates polariton effects.

Master equation for nonstationary states of an electron interacting with phonons

We consider the dissipative effects of electron-phonon interactions on electron states which are not eigenstates of the pure electron hamiltonian. A master-equation formalism is developed to calculate the time evolution of the occupation probabilities of these states as well as other elements of the reduced single-electron density matrix. Conditions are found which ensure that the master equations are markoffian. It is suggested that markofficity rather than approximate stationarity is the important criterion for the usefulness of a basis of states for the description of the evolution of a physical system.