Time-dependent Hartree-Fock Method Research Articles

Excited states of molecular and crystalline acetylene: application of TDHF and BSE via density fitting methods

Low-lying states in small, multiply bonded hydrocarbon molecules may be divided into valence and Rydberg excitations. Excited states of the cubic phase of the acetylene molecular crystal are calculated using the time-dependent Hartree–Fock (TDHF) method. TDHF and Bethe–Salpeter equation (BSE) calculations are performed on molecular CH in order to interpret these excited states. Computationally inexpensive TDHF and BSE methods are compared to previous CASPT2, MRCI and EOM-CCSD calculations and experiment. Localised, lowest energy excited states in the CH molecular crystal mirror those in the gas phase. In the next lowest excitations the electron and hole separate over one or two shells of molecular neighbours. A density fitting method for calculating the Coulomb matrix elements which arise in time-dependent Hartree Fock theory (TDHF) in periodic systems is implemented in the Exciton code and described briefly.

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.

Giant dipole resonance and shape evolution in Nd isotopes within TDHF method

The isovector giant dipole resonance (IVGDR) in even–even Nd isotopes from A = 124 to A = 160 is studied in the framework of time-dependent Hartree–Fock (TDHF) with Skyrme forces SkI3, SVbas, SLy5 and SLy6. The dipole strength is calculated and compared with the experimental data on photon absorption cross section σγ. An overall agreement between them is obtained. The dipole strengths in 124–140Nd and 152–160Nd are predicted. In addition, the correlation between the quadrupole deformation parameter β2 and the splitting of the giant dipole resonance (GDR) spectra is studied. The results confirm that is proportional to β2. Shape phase transition in Nd isotopes is also investigated in the light of IVGDR.

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).

Influence of the tensor interaction on heavy-ion fusion cross sections

[Background] While the tensor interaction has been shown to significantly affect the nuclear structure of exotic nuclei, its influence on nuclear reactions has only recently been investigated. The primary reason for this neglect is the fact that most studies of nuclear dynamics do not include the tensor force at all in their models. Indeed, only a few Skyrme parametrizations consider the tensor interaction in parameter determination. With modern research facilities extending our ability to probe exotic nuclei, a correct description of nuclear dynamics and heavy-ion fusion is vital to both supporting and leading experimental efforts. [Method] Fusion cross sections are calculated using ion-ion potentials generated by the fully microscopic density-constrained time-dependent Hartree-Fock (DC-TDHF) method with the complete Skyrme tensor interaction. [Results] For light nuclei, the tensor force only slightly changes the sub-barrier fusion cross sections at very low energies. Heavier nuclei, however, begin to exhibit a substantial hindrance effect in the sub-barrier region. This effect is strongest in spin-unsaturated systems, though can manifest in other configurations as well. Static, ground state deformation effects of the tensor force can also affect cross sections by shifting the fusion barrier. [Conclusions] The tensor interaction has a measurable effect on the fusion cross sections of nuclei spanning the nuclear chart. The effect comes from both static effects present in the ground state and dynamic processes arising from the time evolution of the system. This motivates the development of a modern Skyrme parameter set that includes all time-odd and tensor terms and that studies moving forward should include the tensor force to ensure a more robust and complete description of nuclei.

Density-constrained time-dependent Hartree-Fock-Bogoliubov method

Background: The Density-constraint Time-dependent Hartree-Fock method is currently the tool of choice to predict fusion cross-sections. However, it does not include pairing correlations, which have been found recently to play an important role. Purpose: To describe the fusion cross-section with a method that includes the superfluidity and to understand the impact of pairing on both the fusion barrier and cross-section. Method: The density-constraint method is tested first on the following reactions without pairing, $^{16}$O+$^{16}$O and $^{40}$Ca+$^{40}$Ca. A new method is developed, the Density-constraint Time-dependent Hartree-Fock-Bogoliubov method. Using the Gogny-TDHFB code, it is applied to the reactions $^{20}$O+$^{20}$O and $^{44}$Ca+$^{44}$Ca. Results: The Gogny approach for systems without pairing reproduces the experimental data well. The DC-TDHFB method is coherent with the TDHFB fusion threshold. The effect of the phase-lock mechanism is shown for those reactions. Conclusions: The DC-TDHFB method is a useful new tool to determine the fusion potential between superfluid systems and to deduce their fusion cross-sections.

Time-dependent Hartree-Fock plus Langevin approach for hot fusion reactions to synthesize the Z=120 superheavy element

We develop a novel approach to fusion reactions for syntheses of superheavy elements, which combines the time-dependent Hartree-Fock (TDHF) method with a dynamical diffusion model based on the Langevin equation. In this approach, the distance of the closest approach for the capture process is estimated within the TDHF approach, which is then plugged into the dynamical diffusion model as an initial condition. We apply this approach to hot fusion reactions leading to formation of the element $Z=120$, that is, the $^{48}$Ca+$^{254,257}$Fm, $^{51}$V+$^{249}$Bk, and $^{54}$Cr+$^{248}$Cm reactions. Our calculations indicate that the distances of the closest approach for these systems are similar to each other and thus the difference in the probabilities of evaporation residue formation among those reaction systems originates mainly from the evaporation process, which is sensitive to the fission barrier height and the excitation energy of a compound nucleus.

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.

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.

Photoinduced collective mode, inhomogeneity, and melting in a charge-order system

We theoretically investigate photoresponses of a correlated electron system upon stimuli of a pulsed laser light. Real-time dynamics of an interacting spinless fermion model on a one-dimensional chain, as a model of charge order (CO), are numerically simulated using the time-dependent Hartree-Fock method. In particular, we discuss the differences between two situations as the initial state:the homogeneous order and the presence of a domain wall, i.e., a kink structure embedded in the CO bulk. Coherent dynamics are seen in the former case: When the frequency of the pump light $\omega_{p}$ is varied, along with single particle excitations across the CO gap ($\Delta_\textrm{CO}$), the resonantly-excited collective phase mode near $\omega_{p} \simeq \Delta_\textrm{CO}/2$ efficiently destabilizes CO. In clear contrast, in the latter case, when $\omega_{p}$ is tuned at such in-gap frequencies and the intensity of light is sufficiently large,inhomogeneity spreads out from the kink to the bulk region through kink creations. Moreover, even stronger intensity induces the inhomogeneous melting of CO where the CO gap is destroyed.

Isotopic trends of quasifission and fusion-fission in the reactions Ca48+Pu239,244

Background: Quasifission and fusion-fission are primary mechanisms to prevent the production of superheavy elements. The recent experimental measurements reveal that the fusion-evaporation cross section in the $3n$ reaction channel of $^{48}$Ca+$^{239}$Pu is 50 times lower than using $^{244}$Pu as target nucleus. However, the precise mechanisms of this remarkable isotopic dependence are not well understood. Purpose: To understand the experimental observation of the rapid decrease of stability of superheavy nuclei as the neutron number decreases, the theoretical studies of quasifission and fusion-fission in connection with experimental production for $Z$=114 flerovium isotopes are required to investigate the possible differences in reaction mechanisms induced by these two targets. Methods: We propose an approach called TDHF+HIVAP to take into account both the evolution of dinuclear system and the deexcitation of compound nucleus, which combines the microscopic time-dependent Hartree-Fock (TDHF) method for the fusion and quasifission dynamics with the statistical evaporation model HIVAP for fusion-fission dynamics. Results: ......The quantum shell effect displays a crucial role in both the quasifission and the fusion-fission processes. The quasifission is considerably reduced and the survival probability is enhanced around one order of magnitude in the reaction using $^{244}$Pu target as compared to the $^{239}$Pu case. Conclusions: The studies by using TDHF+HIVAP method well account for the experimental observations and the present method clearly shows its applicability in the reaction mechanisms of quasifission and fusion-fission dynamics. The experimental and theoretical results encourage the use of neutron-rich targets for the production of new superheavy elements.

Diagnostic and control of linear and nonlinear optical effects in selected self-assembled metallophthalocyanine chlorides nanostructures

In this paper we described a new self-assembly phenomenon of the metallophthalocyanine chlorides nanostructures and its influence on the linear optical properties as well as the Second Harmonic Generation process. The self-assembly phenomenon were achieved through an annealing process carried out immediately after the deposition process. The studied nanostructures were subjected to the annealing process for 24 h and the temperature of the process was equal to 525 K. We discussed experimental results and theoretical calculations of structural, linear and nonolinear optical properties for aluminum and gallium phthalocyanine chlorides. The linear and second-order nonlinear optical properties for these compounds were investigated at microscopic and macroscopic levels. The electric dipole moments and dispersion-free first hyperpolarizabilities were determined by quantum chemical calculations based on Density Functional Theory. Ab-initio quantum mechanical calculations (time-dependent Hartree-Fock method) for the studied metallophthalocyanine chlorides were carried out to compute the frequency-dependent first hyperpolarizabilities and second-order susceptibilities at the wavelengths used in SHG measurements. Our results shed light on the linear and nonlinear optical properties of the nanostructures. The results showed that second harmonic signal is strong and polarized, and this polarizing effect was achieved by controlling the arrangement of the molecules inside the formed nanostructures. Our results also reveal potential application of the nanostructures not only for nonlinear optics but also for thermal sensor devices.

Multi-Electron Effects in Attosecond Transient Absorption of CO Molecules**Supported by the National Basic Research Program of China under Grant No 2013CB922203, the National Natural Science Foundation of China under Grant No 11374366, the Innovation Foundation of National University of Defense Technology under Grant No B110204, and the Hunan Provincial Innovation Foundation for Postgraduate under Grant No CX2011B010.

Using the fully propagated time-dependent Hartree–Fock method, we identify that both the dynamic core polarization and multiorbital contributions are important in the attosecond transient absorption of CO molecules. The dynamics of core electrons effectively modifies the behaviors of electrons in the highest occupied molecular orbital, resulting in the modulation of intensity and position of the absorption peaks. Depending on the alignment angles, different inner orbitals are identified to contribute, and even dominate the total absorption spectra. As a result, multi-electron fingerprints are encoded in the absorption spectra, which shed light on future applications of attosecond transient absorption in complex systems.

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.

Attosecond photoionization dynamics in neon

We study the role of electron-electron correlation in the ground-state of Ne, as well as in photoionization dynamics induced by an attosecond XUV pulse. For a selection of central photon energies around 100 eV, we find that while the mean-field time-dependent Hartree-Fock method provides qualitatively correct results for the total ionization yield, the photoionization cross section, the photoelectron momentum distribution as well as for the time-delay in photoionization, electron-electron correlation is important for a quantitative description of these quantities.

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.

Transport properties of isospin asymmetric nuclear matter using the time-dependent Hartree-Fock method

Background: The study of deep-inelastic reactions of nuclei provide a vehicle to investigate nuclear transport phenomena for a full range of equilibration dynamics. These inquires provide us the ingredients to model such phenomena and help answer important questions about the nuclear Equation of State (EOS) and its evolution as a function of neutron-to-proton $(N/Z)$ ratio. Purpose: The motivation is to examine the real-time dynamics of nuclear transport phenomena and its dependence on $(N/Z)$ asymmetry from a microscopic point of view to avoid any pre-conceived assumptions about the involved processes. Method: Time-dependent Hartree-Fock (TDHF) method in full 3D is employed to calculate deep-inelastic reactions of $^{78}$Kr+$^{208}$Pb and $^{92}$Kr+$^{208}$Pb systems at 8.5~MeV$/A$. The impact parameter and energy-loss dependence of relevant observables are calculated. In addition, density constrained TDHF method is used to compute excitation energies of the primary fragments. The statistical deexcitation code GEMINI is utilized to examine the final reaction products. Results: The kinetic energy loss and sticking times as a function of impact parameter are calculated. Final properties of the fragments (charge, mass, scattering angle, kinetic energy) are computed. Conclusions: We find a smooth dependence of the energy loss, $E_\mathrm{loss}$, on the impact parameter for both systems. On the other hand the transfer properties for low $E_\mathrm{loss}$ values are very different for the two systems but become similar in the higher $E_\mathrm{loss}$ regime. The mean life time of the charge equilibration process, obtained from the final $(N-Z)/A$ value of the fragments, is shown to be $\sim 0.5$~zs. This value is slightly larger (but of the same order) than the value obtained from reactions at Fermi energies.

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.

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.