Time-dependent Hartree-Fock Equations Research Articles

Effect of ionization asymmetry on high harmonic generation from oriented CO in orthogonal two-color fields

We investigate the high harmonic generation from oriented CO molecules in orthogonal two-color (OTC) fields by solving the three-dimensional time-dependent Hartree–Fock equation. The infrared fundamental laser pulse with frequency ω polarized along the molecular axis drives the harmonic generation process, while a weak second harmonic field with frequency ω 1 = 2ω perturbs the electron dynamics along the direction perpendicular to molecular axis. We observe non-integer order harmonics of ω 1 generated from CO with its polarization perpendicular to the molecular axis, which differs from aligned H2 under the same laser fields. Combining the Lewenstein model with the classical electronic trajectory analysis, we find the ionization asymmetry along molecular axis in polar molecules under the cross modulation of the OTC fields results in the difference of high harmonics between CO and H2.

Entrainment effects in neutron-proton mixtures within the nuclear energy-density functional theory: Low-temperature limit

Mutual entrainment effects in cold neutron-proton mixtures are studied in the framework of the self-consistent nuclear energy-density functional theory. Exact expressions for the mass currents, valid for both homogeneous and inhomogeneous systems, are directly derived from the time-dependent Hartree-Fock equations with no further approximation. The equivalence with the Fermi-liquid expression is also demonstrated. Focusing on neutron-star cores, a convenient and simple analytical formulation of the entrainment matrix in terms of the isovector effective mass is found, thus allowing to relate entrainment phenomena in neutron stars to isovector giant dipole resonances in finite nuclei. Results obtained with different functionals are presented. These include the Brussels-Montreal functionals, for which unified equations of state of neutron stars have been recently calculated.

Non-Markovian pure dephasing in a dielectric excited by a few-cycle laser pulse

We develop the theory of pure dephasing in a solid exposed to an ultrashort laser pulse beyond the commonly used Markov approximation. This approach takes into account the finite cutoff energy of the bath and can be applied to both many-particle and phonon environments. With numerical simulations performed with the time-dependent Hartree-Fock equations, we investigate how the excitation probability and high-harmonic generation are described by different models of decoherence. It is shown that the time-dependent rates allow for temporally high dephasing to successfully reproduce the main features of high-harmonics spectrum and avoid an overestimation of the charge carrier population after the pulse, which is a common problem of the Markov approximation.

Effect of long-range Coulomb interactions on electron transport in a nanoscale one-dimensional ring

The effect of long-range Coulomb interactions on electron transport in a one-dimensional ring is investigated by a numerical method with the time-dependent Hartree–Fock equation. Our results show that long-range Coulomb interactions induce a multi-electron wave packet state. Moreover, the number of electrons in the multi-electron wave packet varies depending on the strength of the long-range Coulomb interactions.

Photoabsorption spectra of small Na clusters: TDHF and BSE versus CI and experiment

Time-dependent Hartree-Fock (TDHF) and Bethe-Salpeter equation (BSE) methods are benchmarked against configuration interaction (CI) calculations of excitation energies and photoabsorption cross-sections for small closed shell sodium clusters with up to six atoms in several low-energy configurations. The mean absolute deviation of lowest excitation energies of each symmetry in these clusters for TDHF or BSE calculations from CI results is about 0.1 eV. The Thomas-Reiche-Kuhn (TRK) sum rule is satisfied to within numerical accuracy for both TDHF and BSE calculations, however, the Tamm-Dancoff approximations applied to either method for these systems yield optical cross-sections which grossly violate the TRK sum rule. TDHF and BSE calculations for photoabsorption cross-sections of ${\mathrm{Na}}_{8}$ and ${\mathrm{Na}}_{20}$ clusters are compared to experiment and found to be in good agreement. A feature in the cross-section of ${\mathrm{Na}}_{20}$ above 4 eV is found to be caused by a large density of optical transitions rather than an incipient volume plasmon with a short lifetime.

A Dirac field interacting with point nuclear dynamics

The system describing a single Dirac electron field coupled with classically moving point nuclei is presented and studied. The model is a semi-relativistic extension of corresponding time-dependent one-body Hartree-Fock equation coupled with classical nuclear dynamics, already known and studied both in quantum chemistry and in rigorous mathematical literature. We prove local existence of solutions for data in $$H^\sigma $$ with $$\sigma \in [1,\frac{3}{2}[$$. In the course of the analysis a second new result of independent interest is discussed and proved, namely the construction of the propagator for the Dirac operator with several moving Coulomb singularities.

Quantal diffusion description of multinucleon transfers in heavy-ion collisions

Employing the stochastic mean-field (SMF) approach, we develop a quantal diffusion description of the multi-nucleon transfer in heavy-ion collisions at finite impact parameters. The quantal transport coefficients are determined by the occupied single-particle wave functions of the time-dependent Hartree-Fock equations. As a result, the primary fragment mass and charge distribution functions are determined entirely in terms of the mean-field properties. This powerful description does not involve any adjustable parameter, includes the effects of shell structure and is consistent with the fluctuation-dissipation theorem of the non-equilibrium statistical mechanics. As a first application of the approach, we analyze the fragment mass distribution in $^{48}\mathrm{Ca}+{}^{238}\mathrm{U}$ collisions at the bombarding energy $E_{\text{c.m.}}=193$ MeV and compare the calculations with the experimental data.

The TDHF code Sky3D version 1.1

The nuclear mean-field model based on Skyrme forces or related density functionals has found widespread application to the description of nuclear ground states, collective vibrational excitations, and heavy-ion collisions. The code Sky3D solves the static or dynamic equations on a three-dimensional Cartesian mesh with isolated or periodic boundary conditions and no further symmetry assumptions. Pairing can be included in the BCS approximation for the static case. The code is implemented with a view to allow easy modifications for including additional physics or special analysis of the results.

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.

Beyond integrability: Baryon-baryon backward scattering in the massive Gross-Neveu model

Due to integrability, baryon-baryon scattering in the massless Gross-Neveu model at large N features only forward elastic scattering. A bare mass term breaks integrability and is therefore expected to induce backward elastic scattering as well as inelastic reactions. We confirm these expectations by a study of baryon-baryon scattering in the massive Gross-Neveu model near the non-relativistic limit. This restriction enables us to solve the time-dependent Hartree-Fock equations with controlled approximations, using a combination of analytical methods from an effective field theory and the numerical solution of partial differential equations.

Natural excitation orbitals from linear response theories: Time-dependent density functional theory, time-dependent Hartree-Fock, and time-dependent natural orbital functional theory.

Straightforward interpretation of excitations is possible if they can be described as simple single orbital-to-orbital (or double, etc.) transitions. In linear response time-dependent density functional theory (LR-TDDFT), the (ground state) Kohn-Sham orbitals prove to be such an orbital basis. In contrast, in a basis of natural orbitals (NOs) or Hartree-Fock orbitals, excitations often employ many orbitals and are accordingly hard to characterize. We demonstrate that it is possible in these cases to transform to natural excitation orbitals (NEOs) which resemble very closely the KS orbitals and afford the same simple description of excitations. The desired transformation has been obtained by diagonalization of a submatrix in the equations of linear response time-dependent 1-particle reduced density matrix functional theory (LR-TDDMFT) for the NO transformation, and that of a submatrix in the linear response time-dependent Hartree-Fock (LR-TDHF) equations for the transformation of HF orbitals. The corresponding submatrix is already diagonal in the KS basis in the LR-TDDFT equations. While the orbital shapes of the NEOs afford the characterization of the excitations as (mostly) simple orbital-to-orbital transitions, the orbital energies provide a fair estimate of excitation energies.

Stochastic time-dependent Hartree-Fock methods for fermions: Quasiprobability distributions, master equations, and convergence towards exact quantum dynamics

The time-dependent fermionic Hartree-Fock equations can be stochastically extended in such a way as to become the exact representation of quantum dynamics. This fact was first observed in the work of Juillet and Chomaz [Phys. Rev. Lett. 88, 142503 (2002)]. During the past decade, this observation has led to the emergence of a whole family of stochastic wave-function methods for fermions. The common feature of all these methods is that they are based on the expansion of the density operator over the dyadic product of the two fermionic Slater determinant states. In this work, we develop a unified and rigorous foundation for this family of methods. We find a general form of stochastic equations and describe the sufficient conditions under which these methods converge towards exact quantum dynamics. To achieve these goals, we employ the representation of quantum dynamics in generalized phase space. In particular, we consider the quasiprobability distributions which emerge in these stochastic methods and their master equations. It is shown that the convergence towards exact quantum dynamics is controlled by the problem of boundary terms. We provide an example of stochastic Hartree-Fock method which is well-defined and free from this problem.

A New Method and a New Scaling for Deriving Fermionic Mean-Field Dynamics

We introduce a new method for deriving the time-dependent Hartree or Hartree-Fock equations as an effective mean-field dynamics from the microscopic Schroedinger equation for fermionic many-particle systems in quantum mechanics. The method is an adaption of the method used in [Pickl, Lett. Math. Phys., 97(2):151-164, 2011] for bosonic systems to fermionic systems. It is based on a Gronwall type estimate for a suitable measure of distance between the microscopic solution and an antisymmetrized product state. We use this method to treat a new mean-field limit for fermions with long-range interactions in a large volume. Some of our results hold for singular attractive or repulsive interactions. We can also treat Coulomb interaction assuming either a mild singularity cutoff or certain regularity conditions on the solutions to the Hartree(-Fock) equations. In the considered limit, the kinetic and interaction energy are of the same order, while the average force is subleading. For some interactions, we prove that the Hartree(-Fock) dynamics is a more accurate approximation than a simpler dynamics that one would expect from the subleading force. With our method we also treat the mean-field limit coupled to a semiclassical limit, which was discussed in the literature before, and we recover some of the previous results. All results hold for initial data close (but not necessarily equal) to antisymmetrized product states and we always provide explicit rates of convergence.

Mean-field evolution of fermionic systems

We study the dynamics of interacting fermionic systems, in the mean-field regime. We consider initial states which are close to quasi-free states and prove that, under suitable assumptions on the inital data and on the many-body interaction, the quantum evolution of the system is approximated by a time-dependent quasi-free state. In particular we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent Hartree-Fock equation. Our theorems allow to describe the dynamics of both pure states (zero temperature states) and mixed states (positive temperature states). Our results hold for all times, and give effective estimates on the rate of convergence towards the Hartree-Fock evolution. The results on pure states are based on joint works with N. Benedikter and B. Schlein, [5, 6]; while those on mixed states are based on a joint work with N. Benedikter, V. Jaksic, C. Saffirio and B. Schlein, [7].

Ab initio non-relativistic spin dynamics.

Many magnetic materials do not conform to the (anti-)ferromagnetic paradigm where all electronic spins are aligned to a global magnetization axis. Unfortunately, most electronic structure methods cannot describe such materials with noncollinear electron spin on account of formally requiring spin alignment. To overcome this limitation, it is necessary to generalize electronic structure methods and allow each electron spin to rotate freely. Here, we report the development of an ab initio time-dependent non-relativistic two-component spinor (TDN2C), which is a generalization of the time-dependent Hartree-Fock equations. Propagating the TDN2C equations in the time domain allows for the first-principles description of spin dynamics. A numerical tool based on the Hirshfeld partitioning scheme is developed to analyze the time-dependent spin magnetization. In this work, we also introduce the coupling between electron spin and a homogenous magnetic field into the TDN2C framework to simulate the response of the electronic spin degrees of freedom to an external magnetic field. This is illustrated for several model systems, including the spin-frustrated Li3 molecule. Exact agreement is found between numerical and analytic results for Larmor precession of hydrogen and lithium atoms. The TDN2C method paves the way for the ab initio description of molecular spin transport and spintronics in the time domain.

Nuclear spin scissors – new type of collective motion

The coupled dynamics of the orbital and spin scissors modes is studied with the help of the Wigner Function Moments method on the basis of Time Dependent Hartree-Fock equations in the harmonic oscillator model including spin orbit potential plus quadrupole- quadrupole and spin-spin residual interactions. The relation between our results and the recent experimental data is discussed.

The TDHF code Sky3D

The nuclear mean-field model based on Skyrme forces or related density functionals has found widespread application to the description of nuclear ground states, collective vibrational excitations, and heavy-ion collisions. The code Sky3D solves the static or dynamic equations on a three-dimensional Cartesian mesh with isolated or periodic boundary conditions and no further symmetry assumptions. Pairing can be included in the BCS approximation for the static case. The code is implemented with a view to allow easy modifications for including additional physics or special analysis of the results. Program summaryProgram title: Sky3DCatalogue identifier: AESW_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AESW_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 43187No. of bytes in distributed program, including test data, etc.: 1423973Distribution format: tar.gzProgramming language: Fortran 90. The OpenMP version requires a relatively recent compiler; it was found to work using gfortran 4.6.2 or later and the Intel compiler version 12 or later.Computer: All computers with a Fortran compiler supporting at least Fortran 90.Operating system: All operating systems with such a compiler. Some of the Makefiles and scripts depend on a Unix-like system and need modification under Windows.Has the code been vectorized or parallelized?: Yes, Runs under OpenMP and MPI, unlimited number of processors can be used.RAM: 1 GBClassification: 17.16, 17.22, 17.23.External routines: LAPACK, FFTW3Nature of problem:The time-dependent Hartree–Fock equations can be used to simulate nuclear vibrations and collisions between nuclei for low energies. This code implements the equations based on a Skyrme energy functional and also allows the determination of the ground-state structure of nuclei through the static version of the equations. For the case of vibrations the principal aim is to calculate the excitation spectra by Fourier-analyzing the time dependence of suitable observables. In collisions, the formation of a neck between nuclei, the dissipation of energy from collective motion, processes like charge transfer and the approach to fusion are of principal interest.Solution method:The nucleonic wave function spinors are represented on a three-dimensional Cartesian mesh with no further symmetry restrictions. The boundary conditions are always periodic for the wave functions, while the Coulomb potential can also be calculated for an isolated charge distribution. All spatial derivatives are evaluated using the finite Fourier transform method. The code solves the static Hartree–Fock equations with a damped gradient iteration method and the time-dependent Hartree–Fock equations with an expansion of the time-development operator. Any number of initial nuclei can be placed into the mesh in with arbitrary positions and initial velocities.Restrictions:The reliability of the mean-field approximation is limited by the absence of hard nucleon–nucleon collisions. This limits the scope of applications to collision energies about a few MeV per nucleon above the Coulomb barrier and to relatively short interaction times. Similarly, some of the missing time-odd terms in the implementation of the Skyrme interaction may restrict the applications to even–even nuclei.Unusual features:The possibility of periodic boundary conditions and the highly flexible initialization make the code also suitable for astrophysical nuclear-matter applications.Running time:The running time depends strongly on the size of the grid, the number of nucleons, and the duration of the collision. For a single-processor PC-type computer it can vary between a few minutes and weeks.

Mean-field dynamics of fermions with relativistic dispersion

We extend the derivation of the time-dependent Hartree-Fock equation recently obtained by Benedikter et al. [“Mean-field evolution of fermionic systems,” Commun. Math. Phys. (to be published)] to fermions with a relativistic dispersion law. The main new ingredient is the propagation of semiclassical commutator bounds along the pseudo-relativistic Hartree-Fock evolution.

MULTICONFIGURATION HARTREE–FOCK THEORY FOR PSEUDORELATIVISTIC SYSTEMS: THE TIME-DEPENDENT CASE

In [Setting and analysis of the multi-configuration time-dependent Hartree–Fock equations, Arch. Ration. Mech. Anal.198 (2010) 273–330] the third author has studied in collaboration with Bardos, Catto and Mauser the nonrelativistic multiconfiguration time-dependent Hartree–Fock system of equations arising in the modeling of molecular dynamics. In this paper, we extend the previous work to the case of pseudorelativistic atoms. We show the existence and the uniqueness of global-in-time solution to the underlying system under technical assumptions on the energy of the initial data and the charge of the nucleus. Moreover, we prove that the result can be extended to the case of neutron stars when the number of electrons is less than a critical number N cr .

The semiclassical limit of the time dependent Hartree–Fock equation: The Weyl symbol of the solution

We study the time evolution of the Weyl symbol of a solution of the time dependent Hartree Fock equation, assuming that for t=0, it has a Weyl symbol which is integrable in the phase space, such as all its derivatives. We prove that the solution has the same property for all t, and we give an asymptotic expansion, in L1 sense, of this Weyl symbol.