Multiconfiguration Time-dependent Hartree-Fock Research Articles

Efficient simulation of multielectron dynamics in molecules under intense laser pulses: implementation of the multiconfiguration time-dependent Hartree–Fock method based on the adaptive finite element method

We present an implementation of the multiconfiguration time-dependent Hartree–Fock method based on the adaptive finite element method for molecules under intense laser pulses. For efficient simulations, orbital functions are propagated by a stable propagator using the short iterative Arnoldi scheme and our implementation is parallelized for distributed memory computing. This is demonstrated by simulating high-harmonic generation from a water molecule and achieves a simulation of multielectron dynamics with overwhelmingly less computational time, compared to our previous work [Sawada, R. et al. Phys. Rev. A, 2016, 93, 023434].

Investigation of the multielectron dynamics of CO in intense laser fields by the effective potential analysis of natural orbitals

We simulated the multielectron dynamics of CO in multicolor pulses of a near-IR fundamental ω and harmonics up to 4ω by using the multiconfiguration time-dependent Hartree-Fock method. The dynamics were analyzed in terms of effective potentials that govern the time evolution of molecular orbitals. The effective potential for the highest occupied orbital has a hump barrier responsible for anisotropic ionization of CO and the shape of a hump significantly changes at different internuclear distances. The electronic response became nonadiabatic when a 4ω-field was included, resulting in the enhancement in ionization and the extension of the high-order harmonic generation cutoff.

Efficient adaptive exponential time integrators for nonlinear Schrödinger equations with nonlocal potential

The performance of exponential-based numerical integrators for the time propagation of the equations associated with the multiconfiguration time-dependent Hartree–Fock (MCTDHF) method for the approximation of the multi-particle Schrödinger equation in one space dimension is assessed. Among the most popular integrators such as Runge–Kutta methods, time-splitting, exponential integrators and Lawson methods, exponential Lawson multistep methods with one predictor–corrector step provide the best stability and accuracy at the least effort. This assessment is based on the observation that the evaluation of the nonlocal terms associated with the potential is the computationally most demanding part of such a calculation in our setting. In addition, the predictor step provides an estimator for the local time-stepping error, thus allowing for adaptive time-stepping which reflects the smoothness of the solution and enables to reliably control the accuracy of a computation in a robust way, without the need to guess an optimal stepsize a priori. One-dimensional model examples are studied to compare different time integrators and demonstrate the successful application of our adaptive methods.

Manipulation of Multielectron Dynamics of Molecules by Fourier-Synthesized Intense Laser Pulses: Effective Potential Analysis of CO

We present a theoretical investigation as to how multielectron dynamics of CO are manipulated by Fourier-synthesized intense laser pulses. The pulses used are assumed to be comprised of harmonics up to the fourth order. The multiconfiguration time-dependent (TD) Hartree-Fock (MCTDHF) method, where the multielectron wavefunction Ψ(t) is expressed as a linear combination of various electron configurations, is employed to simulate the dynamics of CO interacting with Fourier-synthesized pulses. The multielectron nature such as electron correlation is quantified by using our effective potential approach. To begin with, the time-dependent natural orbitals {ϕj(r,t)} which diagonalize the first order reduced density matrix are obtained from Ψ(t), where r is the one-electron coordinate. The effective potentials υjeff(r,t) that determine the dynamics of ϕj(r,t) are then derived from the equations of motion for {ϕj(r,t)}. υjeff(r,t) consists of the one-body part υ1(t) including the interaction with the laser electric field ε(t) and the two-body part υ2,j(t) originating from electron-electron interaction. In this way, the role of electron correlation can be quantified by comparing υjeff(r,t) with those obtained by the TDHF method, where Ψ(t) is approximated by a single Slater determinant. We found a very similar profile in υ5σeff(r,t) of the 5σ highest occupied molecular orbital for both near-infrared one-color (ω) and directionally asymmetric ω+2ω two-color pulses; when ε(t) points from the nucleus C to O, a hump appears in υ5σeff(r,t) only 2 bohrs outward from C. The hump formation, which originates from the field-induced change in υ2,5σ(t) (especially, due to electron correlation), is responsible for preferential electron ejection from the C atom side (experimentally observed anisotropic ionization). A coherent superposition of ω and 2ω fields with an appropriate relative phase thus works as a one-color pulse of which either positive or negative peaks are filtered out. More sophisticated manipulation is possible by adding higher harmonics to a synthesized field. We show that the 5σ orbital can be squeezed toward the inside of the potential valley in υ5σeff(r,t), which encloses the molecule at a radius of ∼7 bohrs (semicircle in the region of z <0), by adjusting the phases of a ω+2ω+3ω+4ω field. The hump and valley formation in υ5σeff(r,t) are closely correlated with domains of increasing and decreasing electron density, respectively.

Effects of intrachain disorder on photoexcitation in conjugated polymer chains

The luminescence efficiency of conjugated polymers has been a central topic in the study of light emitting. The effect of disorder plays an important role in generating excitons after the conjugated polymers have been excited by photons. In this paper, by using the Su-Schriffer-Heeger model, which has been modified to include intrachain disorder and electron correlation, we investigate the effects of disorder on the photoexcitation, especially on the yield of excitons in a conjugated polymer chain. We adopt the multi-configurational time-dependent Hartree–Fock method to solve the multi-electron time-dependent Schrödinger equation and the Newtonian equation of motion for the lattice vibration. The results show that after the photoexcitation relaxation process, the products of the disordered polymer chain are qualitatively distinct from those of the pure polymer chain. While the pairs of polarons can be generated directly after the photoexcitation in a pure polymer chain, the disorder favors excitons as the products of the photoexcitation, and the yield of excitons depends crucially on the kind and strength of the disorder. Furthermore, the influences of the electron correlation and the conjugation length on the yield of excitons are also discussed. Specifically, we find that in the case of diagonal disorder, when the conjugation length is short and the diagonal disorder is weak, the excitons are mainly generated by the recombination of two lattice defects with a high yield of excitons which will be reduced as the conjugation length increases. The excitons tend to be generated directly with a low yield of excitons which is enlarged as the disorder gets stronger when the conjugation length is long or the diagonal disorder is strong. The on-site Coulomb repulsion favors the generation of excitons as well. The case of off-diagonal disorder is similar to that of diagonal disorder except that the on-site Coulomb potential favors the generation of excitons in the weak disorder regime but depresses the generation of excitons in the strong disorder regime. When both diagonal and off-diagonal disorders are considered, the yield of excitons is dominated by the off-diagonal disorder. We hope that our investigations can provide useful guidance and help for designing organic photoelectric materials and devices.

Molecular dynamic simulation of photogenerated soliton pairs in trans-polyacetylene

Abstract The formation and evolution of photogenerated soliton pairs in trans-polyacetylene is simulated by using a nonadiabatic dynamical method. Electron–lattice interaction as well as electron correlation effects are both included by employing the Su–Schrieffer–Heeger+Pariser–Parr–Pople model. We adopt the multiconfigurational time-dependent Hartree–Fock formulas which allows for the distinguishing between charged and neutral soliton pairs. It is found that charged soliton pairs are created in the initial stage of the evolution after photoexcitation. Subsequently, the formed charged soliton pairs transform into neutral ones during the dynamical evolutions. The conversion time from charged to neutral state depends on the strength of Coulomb interactions. However, neutral soliton pairs are suppressed in the non-interacting limit, which is in good agreement with the previous estimates based on the Su–Schrieffer–Heeger model. We hope these results can provide further insight into some of the basic issues of interest in photogenerated processes in conducting polymers.

Static-field ionization model of He-like ions for diagnostics of light-field intensity

We study static-field ionization of He-like ions with nuclear charge number in the range of $2\le Z \le 36$. Both the tunneling and over-the-barrier regimes are considered. We calculate the ionization rates by three approximate methods: a fitting formula based on the Perelemov-Popov-Terent'ev (PPT) formula, a perturbative expansion in powers of $1/Z$, and a single-active electron approximation, and compare them with reference ionization rates computed by the multiconfiguration time-dependent Hartree-Fock (MCTDHF) approach. The relative deviation of the rates computed by the PPT-based fitting formula and the third-order perturbation theory from the rates computed by the MCTDHF approach is found to be around 10\%. We discuss quantitatively the contribution from the exchange interaction during the ionization by comparing the single-active electron rates and the reference rates in which multielectron effects are included. We find that a single-active electron approximation, where the exchange interaction is neglected, results in the overestimation of the ionization rates by 30\% for He and 2\% for He-like Kr, showing that the magnitude of the effect of the exchange interaction scales approximately as $1/Z$.

Excited-state populations in the multiconfiguration time-dependent Hartree–Fock method

We study time-dependent populations in the excited states of many-electron atoms and molecules with the multiconfiguration time-dependent Hartree–Fock method and reveal the non-stationary behavior of the excited-state populations during field-free propagation originating from the nonlinear character of the equations of motion. We calculate the time-dependent populations of the three lowest singlet S states in a helium atom for two different cases of intense laser-atom interaction. In the first case, He is excited by a VUV pulse with the central wavelength of 108 nm, which is two-photon resonant with the transition. In the second case, He is excited by an off-resonant UV laser pulse with the central wavelength of 200 nm. We obtain the stationary wave functions of the excited states by imaginary time propagation and show that the populations oscillate as a function of time even after the laser pulse has vanished. The time-dependent variations in the populations in the respective eigenstates can be sufficiently suppressed when the number of orbitals adopted in the expansion of the wave function exceeds eight.

Time-dependent effective potential for the ultrafast correlated electron dynamics of molecules in intense laser fields: Application to anisotropic ionization of CO

By using the multiconfiguration time-dependent (TD) Hartree-Fock (MCTDHF) method, we simulated the multielectron dynamics of a CO molecule irradiated by near-IR two-cycle pulses with different carrier envelope phases. The ionization rate calculated is higher when the laser electric field ε (t) points from the nucleus C to O than the opposite case, in agreement with the results of two-color ionization experiments. The mechanism of anisotropic tunnel ionization in CO was examined by converting the obtained multielectron dynamics to the representation in terms of TD natural orbitals {ϕj (t)}. Within the framework of MCTDHF, we derived the unitary equations of motion for {ϕj (t)}. From the derived equations, we defined the TD effective potentials that govern the dynamics of {ϕj (t)}. In for the 5σ HOMO, a narrow hump that originates from two-body electron-electron repulsion is formed on the top of the field-induced distorted barrier near the nucleus C when ε (t) points from C to O, which is responsible for the directional anisotropy of tunnel ionization. For 4σ HOMO-2, a high barrier to suppress ionization is formed in when ε (t) points from C to O, in correlation with the electron-electron interaction with a 5σ electron on route to ionization. For the opposite phase, becomes barrierless, which enhances high-harmonic generation through ϕ 4σ(t).

Convergence of exponential Lawson-multistep methods for the MCTDHF equations

We consider exponential Lawson multistep methods for the time integration of the equations of motion associated with the multi-configuration time-dependent Hartree–Fock (MCTDHF) approximation for high-dimensional quantum dynamics. These provide high-order approximations at a minimum of evaluations of the computationally expensive nonlocal potential terms, and have been found to enable stable long-time integration. In this work, we prove convergence of the numerical approximation on finite time intervals under minimal regularity assumptions on the exact solution. A numerical illustration shows adaptive time propagation based on our methods.

Inter-Coulombic Decay in Laterally Arranged Quantum Dots Controlled by Polarized Lasers

The inter-Coulombic decay (ICD) process, where one electronically excited species relaxes while the neighboring one is concomitantly ionized, has been recently discovered likewise in atomic, molecular, biological, and nanostructured systems. Any theoretical prediction of it relies strongly on an accurate treatment of the involved resonance and continuum states. Here, we describe laser-induced ICD in quantum dots with electron dynamics at a multiconfiguration time-dependent Hartree–Fock level for the first time for a two-dimensional continuum, such that ICD control with laser polarization is within reach. This explicit electron dynamics is possible through the efficient Multigrid POTFIT tensor decomposition of the Coulomb interaction on a grid. Conclusively, ICD turns out to be much faster in laterally arranged self-assembled or lithographic quantum dots connected to a two-dimensional wetting-layer continuum than in previously investigated dots in nanowires.

Time-dependent multiconfiguration method applied to laser-driven H2+

We apply the extended multiconfiguration time-dependent Hartree-Fock method to the simulation of a hydrogen molecular ion exposed to an intense laser pulse. By comparing the results obtained by this method with the results obtained by a method in which the time-dependent Schr\"odinger equation is solved directly on a three-dimensional grid, we find that the results obtained by these two methods are in good agreement with each other when the number of time-dependent expansion terms exceeds 8. We further compare the results with those obtained by the conventional two-state Born-Oppenheimer approximation. In order to interpret the resultant time-dependent wave functions, we decompose the total wave function into the natural electronic and protonic orbitals that diagonalize the single-particle density matrices and find that the pair of electronic and protonic natural orbitals carrying the largest population describes the vibrational excitation, while the pair carrying the second largest population describes the dissociation into $\text{H}+{\mathrm{H}}^{+}$. We also examine the time-dependent motion of the protons in ${\mathrm{H}}_{2}^{+}$ in terms of the time-dependent adiabatic potential-energy curves, which are defined as the instantaneous eigenvalues of the Hamiltonian matrix governing the time-dependent motion of the protonic orbitals. We show that two potential minima are formed on the lowest-energy adiabatic potential-energy curve and that the nuclear wave packet localized in the inner minimum corresponds to the bound vibrational motion, while the nuclear wave packet localized in the outer minimum corresponds to the dissociation.

Time-dependent geminal method applied to laser-driven beryllium

We introduce the time-dependent geminal method, in which the total wave function is written as an antisymmetrized product of time-dependent geminals. A geminal is a two-electron orbital depending on the coordinates of two electrons, and each geminal is expanded as a sum of products of time-dependent one-electron orbitals. The equation of motion for the geminal coefficients similar to the time-dependent Hartree-Fock equation is derived. The evaluation of the largest eigenvalues of the second-order reduced density matrix is proposed as a way to measure the extent of the intergeminal correlation in a time-dependent wave function. Using the time-dependent geminal method, we simulate the evolution of the time-dependent wave function of a beryllium atom exposed to an intense laser pulse at two different wavelengths, 400 and 10 nm. The results are compared to those obtained by the time-dependent Hartree-Fock method and by the multiconfiguration time-dependent Hartree-Fock method.

A single-electron picture based on the multiconfiguration time-dependent Hartree–Fock method: application to the anisotropic ionization and subsequent high-harmonic generation of the CO molecule

The mechanisms of anisotropic near-IR tunnel ionization and high-order harmonic generation (HHG) in a CO molecule are theoretically investigated by using the multiconfiguration time-dependent Hartree–Fock (MCTDHF) method developed for the simulation of multielectron dynamics of molecules. The multielectron dynamics obtained by numerically solving the equations of motion (EOMs) in the MCTDHF method is converted to a single orbital picture in the natural orbital representation where the first-order reduced density matrix is diagonalized. The ionization through each natural orbital is examined and the process of HHG is classified into different optical paths designated by a combinations of initial, intermediate and final natural orbitals. The EOMs for natural spin-orbitals are also derived within the framework of the MCTDHF, which maintains the first-order reduced density matrix to be a diagonal one throughout the time propagation of a many-electron wave function. The orbital dependent, time-dependent effective potentials that govern the dynamics of respective time-dependent natural orbitals are deduced from the derived EOMs, of which the temporal variation can be used to interpret the motion of the electron density associated with each natural spin-orbital. The roles of the orbital shape, multiorbital ionization, linear Stark effect and multielectron interaction in the ionization and HHG of a CO molecule are revealed by the effective potentials obtained. When the laser electric field points to the nucleus O from C, tunnel ionization from the C atom side is enhanced; a hump structure originating from multielectron interaction is then formed on the top of the field-induced distorted barrier of the HOMO effective potential. This hump formation, responsible for the directional anisotropy of tunnel ionization, restrains the influence of the linear Stark effect on the energy shifts of bound states.

Multiconfiguration time-dependent Hartree-Fock treatment of electron correlation in strong-field ionization of H2 molecules

Electron correlation plays an important role in the multielectron interactions of many physical and chemical processes.The investigation of correlation effects in the non-perturbative electronic dynamics (e.g.non-sequential double ionization) when atoms and molecules are subjected to strong laser fields requires non-perturbative theoretical treatments. The direct numerical integration of the time-dependent Schrödinger equation successfully explains many experimental results,but it is computationally prohibitive for systems with more than two electrons.There is clearly a need for a theory which can treat correlation dynamics self-consistently in strong time-dependent electric fields.In this paper we develop a three-dimensional multiconfiguration time-dependent Hartree-Fock method,which can be applied to the non-perturbative electronic dynamics for diatomic molecules,and it can also investigate the effect of electron correlation in strong-field ionization of H2 molecules.This method adopts the prolate spheroidal coordinates (which can treat the two-center Coulomb potential accurately) and the finite-element method together with discrete-variable representation (which lowers the calculation burden from two-electron integrations).For the temporal propagation,we use the efficient short iterative Lanczos algorithm for the equation which governs the configuration expansion coefficients,while an eight-order Runge-Kutta (RK) method and an Bulirsch-Stoer (BS) extrapolation method,both with adaptive precision controls,are implemented to solve the nonlinear orbital equation.While both methods yield correct results,the BS method displays a better stability in the realtime propagation,while the RK method demands less computation.The alignment-dependent ionization probabilities of H2 molecules in intense extreme ultraviolet pulses are calculated.Comparisons between multi-configuration and single-configuration results show that electron correlation has little effect on the single ionization process,but it plays an important role in double ionization,leading to the decrease in the ionization probability.The double ionization probability from the single-configuration space 1σ is about three times larger that from 4σ1π.The ionization probability increases monotonically when the alignment angle increases from 0° to 90°, yielding a ratio of 2.6(1.5) between 90° and 0° for the double (single) ionization process.This method presents the basis for the future study of electron correlation in strong-field processes.

Time-Dependent Two-Particle Reduced Density Matrix Theory: Application to High-Harmonic Generation

We simulate the electronic response of atoms to strong laser pulses by propagating the time-dependent two-particle reduced density matrix (TD-2RDM). The propagation of the 2RDM requires carefully designed closure approximations to decouple the equation of motion for the 2RDM from the higher members of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Recent improvements in the closure approximation allow the accurate calculation of high-harmonic spectra from beryllium and neon targets where mean field theories such as time-dependent Hartree-Fock and time-dependent density functional theory show sizable discrepancies to multiconfigurational time-dependent Hartree-Fock (MCTDHF) reference calculations. We trace these differences back to the ionization dynamics. We find that the accurate description of the ionization dynamics is the key to achieve agreement with high-harmonic spectra from MCTDHF.

Effect of disorder on exciton dissociation in conjugated polymers**Project supported by the National Natural Science Foundation of China (Grant Nos. 11474218 and 11575116).

By using a multi-configurational time-dependent Hartree–Fock (MCTDHF) method for the time-dependent Schrödinger equation and a Newtonian equation of motion for lattice, we investigate the disorder effects on the dissociation process of excitons in conjugated polymer chains. The simulations are performed within the framework of an extended version of the Su–Schrieffer–Heeger model modified to include on-site disorder, off-diagonal, electron–electron interaction, and an external electric field. Our results show that Coulomb correlation effects play an important role in determining the exciton dissociation process. The electric field required to dissociate an exciton can practically impossibly occur in a pure polymer chain, especially in the case of triplet exciton. However, when the on-site disorder effects are taken into account, this leads to a reduction in mean dissociation electric fields. As the disorder strength increases, the dissociation field decreases effectively. On the contrary, the effects of off-diagonal disorder are negative in most cases. Moreover, the dependence of exciton dissociation on the conjugated length is also discussed.

Optimized pulses for Raman excitation through the continuum: Verification using the multiconfigurational time-dependent Hartree-Fock method

We have verified a mechanism for Raman excitation of atoms through continuum levels previously obtained by quantum optimal control using the multiconfigurational time-dependent Hartree-Fock (MCTDHF) method. For the optimal control, which requires running multiple propagations to determine the optimal pulse sequence, we used the computationally inexpensive time-dependent configuration interaction singles (TDCIS) method. TDCIS captures all of the necessary correlation of the desired processes but assumes that ionization pathways reached via double excitations are not present. MCTDHF includes these pathways and all multiparticle correlations in a set of time-dependent orbitals. The mechanism that was determined to be optimal in the Raman excitation of the Ne $1{s}^{2}2{s}^{2}2{p}^{5}3{p}^{1}$ valence state via the metastable $1{s}^{2}2{s}^{1}2{p}^{6}3{p}^{1}$ resonance state involves a sequential resonance-valence excitation. First, a long pump pulse excites the core-hole state, and then a shorter Stokes pulse transfers the population to the valence state. This process represents the first step in a multidimensional x-ray spectroscopy scheme that will provide a local probe of valence electronic correlations. Although at the optimal pulse intensities at the TDCIS level of theory the MCTDHF method predicts multiple ionization or excitation ionization of the atom, at slightly lower intensities (reduced by a factor of about 4) the TDCIS mechanism is shown to hold qualitatively. Quantitatively, the MCTDHF populations are reduced from the TDCIS calculations by a factor of 4.

Single-photon double ionization: renormalized-natural-orbital theory versus multi-configurational Hartree–Fock

The N-particle wavefunction has too many dimensions for a direct time propagation of a many-body system according to the time-dependent Schrödinger equation (TDSE). On the other hand, time-dependent density functional theory (TDDFT) tells us that the single-particle density is, in principle, sufficient. However, a practicable equation of motion for the accurate time evolution of the single-particle density is unknown. It is thus an obvious idea to propagate a quantity which is not as reduced as the single-particle density but less dimensional than the N-body wavefunction. Recently, we have introduced time-dependent renormalized-natural-orbital theory (TDRNOT). TDRNOT is based on the propagation of the eigenfunctions of the one-body reduced density matrix, the so-called natural orbitals. In this paper we demonstrate how TDRNOT is related to the multi-configurational time-dependent Hartree–Fock (MCTDHF) approach. We also compare the performance of MCTDHF and TDRNOT versus the TDSE for single-photon double ionization (SPDI) of a 1D helium model atom. SPDI is one of the effects where TDDFT does not work in practice, especially if one is interested in correlated photoelectron spectra, for which no explicit density functional is known.

Electron correlation in beryllium: Effects in the ground state, short-pulse photoionization, and time-delay studies

We apply a three-dimensional (3D) implementation of the time-dependent restricted-active-space self-consistent-field (TD-RASSCF) method to investigate effects of electron correlation in the ground state of Be as well as in its photoionization dynamics by short XUV pulses, including time-delay in photoionization. First, we obtain the ground state by propagation in imaginary time. We show that the flexibility of the TD-RASSCF on the choice of the active orbital space makes it possible to consider only relevant active space orbitals, facilitating the convergence to the ground state compared to the multiconfigurational time-dependent Hartree-Fock method, used as a benchmark to show the accuracy and efficiency of TD-RASSCF. Second, we solve the equations of motion to compute photoelectron spectra of Be after interacting with a short linearly polarized XUV laser pulse. We compare the spectra for different RAS schemes, and in this way we identify the orbital spaces that are relevant for an accurate description of the photoelectron spectra. Finally, we investigate the effects of electron correlation on the magnitude of the relative time-delay in the photoionization process into two different ionic channels. One channel, the ground state channel in the ion, is accessible without electron correlation. The other channel is only accessible when including electron correlation. The time-delay is highly sensitivity to the choice of the active space, and hence to the account of electron-electron correlation.