Multiconfiguration Time-dependent Hartree-Fock Research Articles

Probing autoionizing states of molecular oxygen with XUV transient absorption: Electronic-symmetry-dependent line shapes and laser-induced modifications

The dynamics of autoionizing Rydberg states of oxygen are studied using attosecond transient absorption technique, where extreme ultraviolet (XUV) initiates molecular polarization and near infrared (NIR) pulse perturbs its evolution. Transient absorption spectra show positive optical density (OD) change in the case of $ns\sigma_g$ and $nd\pi_g$ autoionizing states of oxygen and negative OD change for $nd\sigma_g$ states. Multiconfiguration time-dependent Hartree-Fock (MCTDHF) calculation are used to simulate the transient absorption spectra and their results agree with experimental observations. The time evolution of superexcited states is probed in electronically and vibrationally resolved fashion and we observe the dependence of decay lifetimes on effective quantum number of the Rydberg series. We model the effect of near-infrared (NIR) perturbation on molecular polarization and find that the laser induced phase shift model agrees with the experimental and MCTDHF results, while the laser induced attenuation model does not. We relate the electron state symmetry dependent sign of the OD change to the Fano parameters of the static absorption lineshapes.

High-harmonic spectra from time-dependent two-particle reduced-density-matrix theory

The accurate description of the non-linear response of many-electron systems to strong-laser fields remains a major challenge. Methods that bypass the unfavorable exponential scaling with particle number are required to address larger systems. In this paper we present a fully three-dimensional implementation of the time-dependent two-particle reduced density matrix (TD-2RDM) method for many-electron atoms. We benchmark this approach by a comparison with multi-configurational time-dependent Hartree-Fock (MCTDHF) results for the harmonic spectra of beryllium and neon. We show that the TD-2RDM is very well-suited to describe the non-linear atomic response and to reveal the influence of electron-correlation effects.

Time-dependent restricted-active-space self-consistent-field theory with space partition

Aiming at efficient numerical analysis of time-dependent (TD) many-electron dynamics of atoms involving multielectron continua, the TD restricted-active-space self-consistent-field theory with space partition (TD-RASSCF-SP) is presented. The TD-RASSCF-SP wave function is expanded in terms of TD configuration-interaction coefficients with Slater determinants composed of two kinds of TD orbitals: $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ orbitals are defined to be nonvanishing in the inner region ($\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}$), a small volume around the atomic nucleus, and $\stackrel{\ifmmode \check{}\else \v{}\fi{}}{M}$ orbitals are nonvanishing in the large outer region ($\stackrel{\ifmmode \check{}\else \v{}\fi{}}{V}$). For detailed discussion of the SP strategy, the equations of motion are derived by two different formalisms for comparison. To ensure continuous differentiability of the wave function across the two regions, one of the formalisms makes use of the property of the finite-element discrete-variable-representation (FEDVR) functions and introduces additional time-independent orbitals. The other formalism is more general and is based on the Bloch operator as in the $R$-matrix theory, but turns out to be less practical for numerical applications. Hence, using the FEDVR-based formalism, the numerical performance is tested by computing double-ionization dynamics of atomic beryllium in intense light fields. To achieve high accuracy, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ should be set large to take into account the strong many-electron correlation around the nucleus. On the other hand, $\stackrel{\ifmmode \check{}\else \v{}\fi{}}{M}$ can be set much smaller than $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ for capturing the weaker correlation between the two outgoing photoelectrons. As a result, compared with more accurate multiconfigurational TD Hartree-Fock (MCTDHF) method, the TD-RASSCF-SP method may achieve comparable accuracy in the description of the double-ionization dynamics. There are, however, difficulties related to the stiffness of the equations of motion of the TD-RASSCF-SP method, which makes the required time step for this method smaller than the one needed for the MCTDHF approach.

Decomposition of the configuration-interaction coefficients in the multiconfiguration time-dependent Hartree-Fock method.

An approximate implementation of the multiconfiguration time-dependent Hartree-Fock method is proposed, in which the matrix of configuration-interaction coefficients is decomposed into a product of matrices of smaller dimension. The applicability of this method in which all the configurations are kept in the expansion of the wave function, while the configuration-interaction coefficients are approximately calculated, is discussed by showing the results on three model systems: a one-dimensional model of a beryllium atom, a one-dimensional model of a carbon atom, and a one-dimensional model of a chain of four hydrogen atoms. The time-dependent electronic dynamics induced by a few-cycle, long-wavelength laser pulse is found to be well described at a lower computational cost compared to the standard multiconfiguration time-dependent Hartree-Fock treatment. Drawbacks of the method are also discussed.

Instabilities and inaccuracies of multi-configuration time-dependent Hartree-Fock

We demonstrate that the widely used multi-configuration time-dependent Hartree- Fock method is restricted to a certain class of applications and fails for scenarios where periods of low entanglement occur during the propagation. By using illustrative and physically relevant examples, based on the Hubbard model of solid state physics, we show the existence of serious instabilities in the method itself and demonstrate that the method does not converge with respect to electron correlations. Possible cures of the approach are discussed.

Implementation of the multiconfiguration time-dependent Hatree-Fock method for general molecules on a multiresolution Cartesian grid

We report a three-dimensional numerical implementation of multiconfiguration time-dependent Hartree-Fock (MCTDHF) based on a multi-resolution Cartesian grid, with no need to assume any symmetry of molecular structure. We successfully compute high-harmonic generation (HHG) of H2 and H2O. The present implementation will open a way to the first-principle theoretical study of intense-field and attosecond-pulse induced ultrafast phenomena in general molecules.

Direct extraction of intense-field-induced polarization in the continuum on the attosecond time scale from transient absorption

Author(s): Li, X; Haxton, DJ; Gaarde, MB; Schafer, KJ; McCurdy, CW | Abstract: A procedure is suggested for using transient absorption spectroscopy above the ionization threshold to measure the polarization of the continuum induced by an intense optical pulse. In this way transient absorption measurement can be used to probe subfemtosecond intense field dynamics in atoms and molecules. The method is based on an approximation to the dependence of these spectra on time delay between an attosecond XUV probe pulse and an intense pump pulse that is tested over a wide range of intensities and time delays by all-electrons-active calculations using the multiconfiguration time-dependent Hartree-Fock method in the case of neon.

Electron correlation effects on polarons recombination in coupled polymer chains

The recombination dynamics of singlet and triplet oppositely charged polarons under the influence of electron–electron (e–e) interactions in coupled polymer chains are investigated using a multi-configurational time-dependent Hartree–Fock (MCTDHF) method. During recombination processes, singlet and triplet intrachain excitons are important products. By calculating the yields of the singlet and triplet intrachain excitons as a function of the on-site and long-range e–e interactions, it is found that the yields of the singlet and triplet intrachain excitons both decrease with increasing on-site e–e interactions. On the other hand, as the long-range e–e interactions increase, the yields of singlet intrachain excitons initially increase and then maintain a constant value, while the yields of the triplet intrachain excitons decrease. Our results show that the long-range e–e interaction is of fundamental importance and improves the luminescence efficiency in coupled polymer chains. Finally, the influence of the polymer chain length on the yields of singlet and triplet intrachain excitons is discussed.

Implementation of fixed-nuclei polyatomic MCTDHF capability and the future with nuclear motion

SynopsisWe discuss the implementation (https://commons.lbl.gov/display/csd/LBNL-AMO-MCTDHF) of Multiconfiguration Time-Dependent Hartree-Fock for polyatomic molecules using a Cartesian product grid of sinc basis functions, and present absorption cross sections and other results calculated with it.

Long-range Coulomb interaction effects on polarons in conjugated polymers

Combining the Su–Schrieffer–Heeger and Pariser–Parr–Pople model (SSH+PPP), we have studied the influence of electron–electron interactions on the motion and dissociation of a polaron in the presence of an electric field, with emphasis on the long-range Coulomb interactions. The multiconfigurational time-dependent Hartree–Fock (MCTDHF) formalism was used to compute the electron–electron interactions. How the saturated velocity and the critical dissociation electric field of the polaron are related to the on-site Coulomb repulsion and long-range Coulomb interactions has been investigated. It was found that the on-site Coulomb interaction does not favor the motion of a polaron. There is a critical strength of the long-range Coulomb interaction for which the polaron takes on the most localized configuration. Comparing with the results obtained using the extended Hubbard model, we found that competition between the long-range Coulomb interactions and the on-site Coulomb interaction leads to a non-monotonic dependence of both the saturated velocity and the critical dissociation electric field on the long-range Coulomb interactions.

Qualitative failure of a multiconfiguration method in prolate spheroidal coordinates in calculating dissociative photoionization ofH2+

A formulation of a multiconfiguration time-dependent Hartree-Fock (MCTDHF) method with nuclear motion is tested by application to a three-body breakup problem, the dissociative photoionization cross section of the ${\mathrm{H}}_{2}^{+}$ ion. The representation of the wave function in terms of a set of Slater determinants used for all nuclear geometries, with a prescribed parametric dependence upon the nuclear geometry such that the cusps follow the nuclei, times a complete basis expansion in the nuclear degrees of freedom shows promise as a method for treating nonadiabatic electronic and nuclear motion in molecules. However, the method used here for diatomics, in which the parametric dependence is prescribed through the choice of prolate spheroidal coordinates, produces qualitatively incorrect steplike behavior in the calculated cross section near onset. Modifications to the prolate spheroidal coordinate system that would improve this nonadiabatic diatomic MCTDHF representation are proposed.

Electron Correlation Effects in Polaron-Pair Recombination in Conjugated Polymers**Supported by the National Natural Science Foundation of China under Grant No 11474218.

Within a Su–Schriffer–Heeger model modified to include electron-electron interaction and an external electric field, we investigate the dynamics of oppositely charged polarons in a polymer chain in the presence of both electron-phonon and electron-electron interactions under the influence of an external electric field. We adopt a multi-configurational time-dependent Hartree–Fock method for the time-dependent Schrödinger equation and the Newtonian equation of motion for a lattice. Our results show that the on-site Coulomb interaction is of fundamental importance and favors the recombination between the pairs of polarons, and the yield of excitons depends crucially on the strength of the on-site Coulomb interaction U. Furthermore, the influence of the nearest neighbor interaction V is also discussed.

Propagating two-particle reduced density matrices without wave functions

Describing time-dependent many-body systems where correlation effects play an important role remains a major theoretical challenge. In this paper we develop a time-dependent many-body theory that is based on the two-particle reduced density matrix (2-RDM). We develop a closed equation of motion for the 2-RDM employing a novel reconstruction functional for the three-particle reduced density matrix (3-RDM) that preserves norm, energy, and spin symmetries during time propagation. We show that approximately enforcing $N$-representability during time evolution is essential for achieving stable solutions. As a prototypical test case which features long-range Coulomb interactions we employ the one-dimensional model for lithium hydride (LiH) in strong infrared laser fields. We probe both one-particle observables such as the time-dependent dipole moment and two-particle observables such as the pair density and mean electron-electron interaction energy. Our results are in very good agreement with numerically exact solutions for the $N$-electron wavefunction obtained from the multiconfigurational time-dependent Hartree-Fock method.

Two methods for restricted configuration spaces within the multiconfiguration time-dependent Hartree-Fock method

The multiconfiguration time-dependent Hartree-Fock (MCTDHF) method has shown promise in calculating electronic dynamics in molecules driven by strong and high-energy lasers. It must incorporate restricted configuration spaces (meaning that a particular combination of Slater determinants is used, instead of full configuration interaction) to be applied to big systems. Two different Ans\atze are used to determine the essential term in the equations. The first Ansatz is the Lagrangian variational principle. The explicit, complete MCTDHF equations of motion, satisfying that principle, for arbitrary configuration spaces, are given. The property that a restricted configuration list must satisfy in order for the Lagrangian and McLachlan variational principles to give different results is identified. The second Ansatz keeps the density matrix block diagonal among equivalent orbitals, in a generalization of the method of Worth [J. Chem. Phys. 112, 8322 (2000)]. The methods perform well in calculating the dynamics of Be and ${\mathrm{BC}}^{2+}$ subject to ultrafast, ultrastrong lasers in severely truncated Hilbert spaces, although they exhibit differing degrees of numerical stability as implemented.

Ultrafast population transfer to excited valence levels of a molecule driven by x-ray pulses

First-principles quantum-mechanical calculations of an intense-field ultrafast two-color core-hole stimulated Raman process in nitric oxide are presented. They employ the multiconfiguration time-dependent Hartree-Fock (MCTDHF) method with all 15 electrons active. These calculations demonstrate a robust excitation localized on an atom through a core-electron stimulated Raman transition, the first step in proposed stimulated x-ray Raman spectroscopy experiments. A total population transfer of approximately 41% into valence excited states and 30% ionization is obtained via two concurrent 1.31-fs pulses with maximum intensities of 0.5 and $3\ifmmode\times\else\texttimes\fi{}{10}^{17}\text{W}{\mathrm{cm}}^{\ensuremath{-}2}$. It is found that both resonant and nonresonant (via the continuum) Raman transitions contribute. All aspects of these calculations except for ac Stark shifts are converged with a modest basis of 11 orbitals, demonstrating the efficiency of MCTDHF for the treatment of nonperturbative electronic dynamics in molecules.

The structure of approximate two electron wavefunctions in intense laser driven ionization dynamics

The structure of approximate two-electron wavefunctions in strong-field-driven ionization dynamics is investigated in depth, both theoretically and numerically. Theoretical analyses clarify that for two-electron singlet systems, the previously proposed time-dependent extended Hartree–Fock (TD-EHF) method (1995 Phys. Rev. A 51 3999) is equivalent to the multiconfiguration time-dependent Hartree–Fock method with two occupied orbitals. The latter wavefunction is further transformed into the natural expansion form, enabling the direct propagation of the natural orbitals (NOs). These methods, as well as the conventional time-dependent Hartree–Fock (TDHF) method, are numerically assessed as regards providing a description of the ionization dynamics of a one-dimensional helium atom model. This numerical analysis (i) explains the reason behind the well-known failure of the TDHF method to describe tunneling ionization, (ii) demonstrates the interpretive power of the TD-EHF wavefunction in both the original nonorthogonal formulation and the NO-based formulation, and (iii) highlights different manifestations of the electron correlation (an effect beyond the single-determinant description), in tunneling ionization, high harmonic generation, and nonsequential double ionization. Possible extensions of the NO basis approach to multielectron systems are briefly discussed.

Spin-Unrestricted Multi-Configuration Time-Dependent Hartree Fock Theory

Based on spin-unrestricted hartree fock theory, we present the spin unrestricted multi-configuration time dependent hartree fock theory (UMCTDHF) to describe the electron correlation dynamics of systems interacting with laser field. The positive spin orbitals and the negative spin orbitals are propagated in their own subspace respectively. The spin orbital in the spin-down subspace acts with that in the spin-up subspace by the reduced density matrix and mean field operator. The ground energy is acquired by propagating the trial wave function in the imaginary time by using spin-restricted MCTDHF (RMCTDHF) and UMCTDHF respectively. Then the ionization probabilities and the electrons energies are calculated by using RMCTDHF and UMCTDHF when the laser field is present. The ionization probability calculated with UMCTDHF agrees with the previous theoretical reports very well. The UMCTDHF method is accurate and applicable for open shell system beyond the capability of the RMCTDHF method.

The complex scaled multiconfigurational time-dependent Hartree-Fock method for studying resonant states: Application to the 2 s2 He Feshbach resonance

The complex scaled multiconfigurational time-dependent Hartree-Fock method (CMCTDHF also called the complex scaled multiconfigurational linear response method) is used to study the doubly excited He 2s(2) Feshbach resonance energy and lifetime. Various even-tempered Gaussian basis set/complete active space (CAS) combinations are examined. The results are compared with the complex scaled full configuration interaction (FCI) [P. R. Zdanska and N. Moiseyev, J. Chem. Phys. 123, 194105 (2005)], complex-scaled equation-of-motion coupled-cluster method with single and double substitutions [K. B. Bravaya, D. Z. Zuev, E. Epifanovsky, and A. I. Krylov, J. Chem. Phys. 138, 124106 (2013)], and exact numerical calculations [Y. K. Ho, Phys. Rev. A 23, 2137 (1981); E. Lindroth, Phys. Rev. A 49, 4473 (1994); A. Bürgers, D. Wintgren, and J.-M. Ross, J. Phys. B 28, 3163 (1995); and N. Elander, S. Levin, and E. Yarevsky, Phys. Rev. A 67, 062508 (2003)]. It is shown that CMCTDHF is able to provide highly accurate resonance parameters with a fairly small size of basis set and CAS, where the computational resources required are much less than the complex scaled FCI method and exact numerical results.

Non-Born–Oppenheimer molecular wave functions of H2 by extended multi-configuration time-dependent Hartree–Fock method

A ground state wave function of a one-dimensional H2 molecule is derived numerically by the extended multi-configuration time-dependent Hartree–Fock (MCTDHF) method. The electronic orbitals and the amplitudes for the nuclear motion constituting the ground-state wave function are derived by solving the coupled equations of motion by the imaginary time propagation. Comparisons with the results obtained by the Born–Huang (BH) expansion method as well as with the exact wave function reveal that the memory size required in the extended MCTDHF method is about two orders of magnitude smaller than in the BH expansion method.

Optimal multiconfiguration approximation of anN-fermion wave function

We propose a simple iterative algorithm to construct the optimal multi-configuration approximation of an $N$-fermion wave function. That is, $M\geq N $ single-particle orbitals are sought iteratively so that the projection of the given wave function in the $C_M^N$-dimensional configuration subspace is maximized. The algorithm has a monotonic convergence property and can be easily parallelized. The significance of the algorithm on the study of entanglement in a multi-fermion system and its implication on the multi-configuration time-dependent Hartree-Fock (MCTDHF) are discussed. The ground state and real-time dynamics of spinless fermions with nearest-neighbor interactions are studied using this algorithm, discussing several subtleties.