Finite-element Discrete Variable Representation Research Articles

Implementation of the Time-Dependent Complete-Active-Space Self-Consistent-Field Method for Diatomic Molecules.

We present a computational approach that implements the time-dependent complete-active-space self-consistent-field method, as introduced in [Phys. Rev. A 88, 023402 (2013)]. Our implementation addresses the challenge of diatomic molecules subjected to an intense laser pulse by considering the full dimensionality of the problem using prolate spheroidal coordinates. The method incorporates the gauge-invariant frozen-core approximation, boosts the evaluation of the electron-electron interaction term using finite-element discrete-variable representation with Neumann expansion, and utilizes an exponential time differencing scheme tailored for the stable propagation of the stiff nonlinear orbital functions. We have successfully applied this methodology to study high-harmonic generation in diatomic molecules such as H2, LiH, and N2, shedding light on the impact of electron correlations in these systems.

Multiorbital impact on strong-field ionization of HCl molecules

We have put forward the time-dependent Hartree-Fock approach to investigate ionization dynamics of HCl molecules subjected to strong few-cycle laser pulses, which is based on the finite-element discrete-variable representation in prolate spheroidal coordinates and the Crank-Nicolson method. It is demonstrated that the electron is favored to be freed by the peak electric field pointing towards the H atom for the HCl molecule possessing the bonding length $R=2.5$ a.u., while the ionization prefers to occur for the maximum electric field pointing from the H nucleus to Cl nucleus with increasing internuclear separations. Analysis indicates that the ionization yield of $5\ensuremath{\sigma}$ electrons becomes significant with respect to that of $2\ensuremath{\pi}$ electrons with stretching bonding lengths, and the relative contributions of the ionization from $5\ensuremath{\sigma}$ and $2\ensuremath{\pi}$ electrons change as a function of internuclear separations, which gives rise to the above distinct oriented ionization of HCl molecules.

Laser-assisted XUV double ionization of helium atoms: Intensity dependence of joint angular distributions**Project supported by the National Natural Science Foundation of China (Grant Nos. 11774131 and 91850114).

We investigate the intensity effect of ultrashort assisting infrared laser pulse on the single-XUV-photon double ionization of helium atoms by solving full six-dimensional time-dependent Schrödinger equation with implement of finite element discrete variable representation. The studies of joint energy distributions and joint angular distributions of the two photoelectrons reveal the competition for ionized probabilities between the photoelectrons with odd parity and photoelectrons with even parity in single-XUV-photon double ionization process in the presence of weak infrared laser field, and such a competition can be modulated by changing the intensity of the weak assisting-IR laser pulses. The emission angles of the two photoelectrons can be adjusted by changing the laser parameters as well. We depict how the assisting-IR laser field enhances and/or enables the back-to-back and side-by-side emission of photoelectrons created in double ionization process.

Accurate computation of above threshold ionization spectra for stretched in strong laser fields

Investigations on the simplest benchmark system can reveal most underlying mechanisms for the intricate dynamics of molecular systems induced by strong laser pulses. However, due to the two-center Coulomb potential and the highly nonlinear nature of the electron dynamics, the accurate computation of the above threshold ionization spectra remains challenging, especially at large internuclear distances and high laser intensities. In the present work, we implement a new Gauss-quadrature approximation (GA) in the framework of finite element discrete variable representation to solve the time-dependent Schrödinger equation of in strong laser fields. By using this GA, one can arrive at a matrix representation of the first derivative operator that keeps its anti-hermiticity. This crucial feature allows a very stable propagation of the wavefunction under the usual Lanczos scheme. Combining with a wavefunction splitting in the asymptotic region, we show that our present numerical method can reliably deal with the electronic dynamics of stretched molecules at large internuclear distances for high laser intensities and long pulse durations. Accurate photoelectron momentum distributions under these conditions are presented and the distinct features due to the two-center potential are discussed.

Correlation dynamics in double photoionization of excited helium atom by a single ultrashort XUV pulse

SynopsisWe study single- and two-photon double ionization of helium by short XUV pulses by numerically solving the time-dependent Schrödinger equation in full dimensionality within a finite element discrete variable representation scheme. We discuss the joint energy and angular distributions, identify sequential and non-sequential contributions in the double ionization by ultrashort pulse, and track the dynamics of the ionization process in distinct double ionization regimes.

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.

Fully differential single-photon double photoionization of atomic magnesium

The valence-shell double ionization of atomic magnesium is calculated using a grid-based representation of the $3{s}^{2}$ electron configuration in the presence of a fully occupied frozen-core configuration of the remaining ten electrons. Atomic orbitals are constructed from an underlying finite-element discrete variable representation that facilitates accurate representation of the interaction between the inner-shell electrons with those entering the continuum. Length and velocity gauge results are compared with recent theoretical calculations and experimental measurements for the total double-, single-, and triple-differential cross sections, particularly at the photon energy of 55.49 eV for the last one. Comparison between the similar processes of double ionization of the $n{s}^{2}$ atoms helium, beryllium, and magnesium further illuminates the role of valence-shell electron correlation in atomic targets with heliumlike electronic configurations and symmetry.

Ionization dynamics ofC2H2in intense laser fields: Time-dependent Hartree-Fock approach

We investigate the multielectron ionization dynamics of C2H2 subjected to intense few-cycle laser fields with the time-dependent Hartree-Fock approach, in which one-center method is employed with the finite-element discrete-variable representation and B-spline functions. It is found that, when the photon energy is close to the energy gap between the two inner orbitals, an inner-orbital single-photon resonant transition phenomenon occurs and plays an important role in the ionization process of the molecule. Furthermore, the ionization of inner electrons surpasses that of the electrons in the highest occupied molecular orbital (HOMO) at high laser intensities, which may be attributed to the effect of the spatial distributions of molecular orbitals. Moreover, this phenomenon will be enhanced by a reduction of the screen for electrons in the HOMO due to the ionization of inner electrons.

Adiabatic versus diabatic approach to multichannel Coulomb scattering for mutual neutralisation reaction [formula omitted

In this paper it is demonstrated that the split-potential driven Schrödinger approach to two-body Coulomb multichannel quantum scattering in a diabatic framework presented by us in a previous paper (Volkov et al., 2011) also can be formulated within an adiabatic framework. The new formulation of the theory is numerically realised using finite element discrete variable representation. The method is applied to a realistic model of the fundamental mutual neutralisation reaction H++H-→H2∗→H(1)+H(n) described in terms of the seven lowest 1Σg+ electronic states of the H2 molecule. The obtained cross sections are in good agreement with other methods applied to the same model.

Mapped Finite Element Discrete Variable Representation

Efficient numerical solver for the Schrödinger equation is very important in physics and chemistry. The finite element discrete variable representation (FE-DVR) was first proposed by Rescigno and Mc-Curdy [Phys. Rev. A 62, 032706 (2000)] for solving quantum-mechanical scattering problems. In this work, an FE-DVR method in a mapped coordinate was proposed to improve the efficiency of the original FE-DVR method. For numerical demonstration, the proposed approach is applied for solving the electronic eigenfunctions and eigenvalues of the hydrogen atom and vibrational states of the electronic state 3Σg+ of the Cs2 molecule which has long-range interaction potential. The numerical results indicate that the numerical efficiency of the original FE-DVR has been improved much using our proposed mapped coordinate scheme.

Photoionization of helium by attosecond pulses: Extraction of spectra from correlated wave functions

We investigate the photoionization spectrum of helium by attosecond XUV pulses both in the spectral region of doubly excited resonances as well as above the double ionization threshold. In order to probe for convergence, we compare three techniques to extract photoelectron spectra from the wavepacket resulting from the integration of the time-dependent Schroedinger equation in a finite-element discrete variable representation basis. These techniques are: projection on products of hydrogenic bound and continuum states, projection onto multi-channel scattering states computed in a B-spline close-coupling basis, and a technique based on exterior complex scaling (ECS) implemented in the same basis used for the time propagation. These methods allow to monitor the population of continuum states in wavepackets created with ultrashort pulses in different regimes. Applications include photo cross sections and anisotropy parameters in the spectral region of doubly excited resonances, time-resolved photoexcitation of autoionizing resonances in an attosecond pump-probe setting, and the energy and angular distribution of correlated wavepackets for two-photon double ionization.

Application of discrete variable representation to planar ${\rm H}_2^+$H2+ in strong xuv laser fields

We present an efficient and accurate grid method to study the strong field dynamics of planar H(2)(+) under Born-Oppenheimer approximation. After introducing the elliptical coordinates to the planar H(2)(+), we show that the Coulomb singularities at the nuclei can be successfully overcome so that both bound and continuum states can be accurately calculated by the method of separation of variables. The time-dependent Schrödinger equation (TDSE) can be accurately solved by a two-dimensional discrete variable representation (DVR) method, where the radial coordinate is discretized with the finite-element discrete variable representation for easy parallel computation and the angular coordinate with the trigonometric DVR which can describe the periodicity in this direction. The bound states energies can be accurately calculated by the imaginary time propagation of TDSE, which agree very well with those computed by the separation of variables. We apply the TDSE to study the ionization dynamics of the planar H(2)(+) by short extreme ultra-violet (xuv) pulses, in which case the differential momentum distributions from both the length and the velocity gauge agree very well with those calculated by the lowest order perturbation theory.

Photoionization of atoms encapsulated by cages using the power-exponential potential

The systems of confined atoms in cages have received considerable attention for decades due to interesting phenomena arising from the effect of cage environment on the atom. For early theoretical work based on empirical model potentials, the Dirac δ-potential, i.e. the so-called bubble potential, and the attractive short-range spherical shell potential are conventionally used for the description of interaction between the valence electron of confined atom and the cage. In this work, the power-exponential potential with a flexible confining shape is proposed to model the cages. The methods of complex scaling in the finite-element discrete variable representation are implemented to investigate the hydrogen, hydrogen-like ions and alkali metals encapsulated by the cages. The energy spectrum varying with the confining well depth exhibits avoided crossings. The influence of cage on atomic photoionization leading to the oscillation behaviour or the so-called confinement resonances in cross sections is demonstrated in a variety of confined atomic systems. In comparisons with existing predictions using the Dirac δ-potential and the attractive short-range spherical shell potentials, our results show the significant influence of cage thickness and smooth shell boundary on the photoionization. The drastic changes of cross sections due to the character of cage are presented and discussed for the encaged lithium and sodium atoms. The present model is useful for clarifying the boundary effect of confining shell on the endohedral atoms.

Photoionization of Endohedral Atoms in Fullerene Cages

The finite-element discrete variable representation combined with the method of complex coordinate rotation is implemented to investigate the hydrogenic atoms and alkali metals encapsulated by the fullerene cages. The energy levels varying with the confining potential of the fullerene cage exhibit avoided crossings caused by the so-called mirror collapse from the switch of near degenerate states. The effects of fullerene cages on photoionization of confined atoms leading to the oscillation behavior and confinement resonances in photoionization cross sections are demonstrated. The results of cross sections for hydrogen-like lithium ion as a function of the cage radius and shell thickness are presented. The emergence of the Cooper minima due to the influence of the fullerene cages is observed for endohedral lithium and sodium atoms. Comparisons are made to the existing predictions.

Attosecond streaking of molecules in the low-energy region studied by a wavefunction splitting scheme

We present a theoretical study of the low-energy photoelectron spectra of hydrogen molecular ion generated by a single attosecond pulse in the presence of an infrared (IR) laser field. In order to investigate this type of attosecond streaking of molecules, we developed a very efficient grid-based numerical method to solve the two-centre time-dependent Schrödinger equation (TDSE) in the prolate spheroidal coordinates. Specifically, the radial coordinate is discretized with the finite-element discrete variable representation (FE-DVR) for easy parallel computation and the angular coordinate with the usual DVR. A wavefunction splitting scheme is utilized to reduce the demanding requirement of the computational resource to solve the corresponding TDSE when an IR field is present. After verification of the accuracy and efficiency of our method, we then apply it to investigate the attosecond streaking spectra of H+2 in the low-energy region. In contrast to the usual attosecond streaking in the high-energy region, part of the low-energy electrons may be driven back to rescatter with the residual two-centre core. Very interesting interference structures are present in the low-energy region. When the internuclear distance is small, they are very similar to what we have recently observed in the atomic case.

Photoionization cross sections of hydrogen impurities in spherical quantum dots using the finite-element discrete-variable representation

The method of complex-coordinate rotation in a finite-element discrete variable representation is implemented to study the photoionization of hydrogen impurity in spherical quantum dots which are modelled by finite oscillator and Gaussian potentials. The level splitting and crossing of hydrogen impurities due to the variation of dot radii and confining potentials are presented, as well as the shifting down of the energy of hydrogenic impurity toward the combined energy of the free hydrogen and the amplitude of the confining potential. We illustrate the staircaselike behavior in ground and excited energy curves varying with dot radii. The variation of photoionization cross sections with photoelectron energies are given for several dot radii. The cross sections as functions of quantum-dot radii and confining potentials for specific photon energies showing resonancelike profiles are observed and discussed.

Breakup of the alignedH2molecule by xuv laser pulses: A time-dependent treatment in prolate spheroidal coordinates

We have carried out calculations of the triple-differential cross section for one-photon double ionization of molecular hydrogen for a central photon energy of $75$~eV, using a fully {\it ab initio}, nonperturbative approach to solve the time-dependent \Schro equation in prolate spheroidal coordinates. The spatial coordinates $\xi$ and $\eta$ are discretized in a finite-element discrete-variable representation. The wave packet of the laser-driven two-electron system is propagated in time through an effective short iterative Lanczos method to simulate the double ionization of the hydrogen molecule. For both symmetric and asymmetric energy sharing, the present results agree to a satisfactory level with most earlier predictions for the absolute magnitude and the shape of the angular distributions. A notable exception, however, concerns the predictions of the recent time-independent calculations based on the exterior complex scaling method in prolate spheroidal coordinates [Phys.~Rev.~A~{\bf 82}, 023423 (2010)]. Extensive tests of the numerical implementation were performed, including the effect of truncating the Neumann expansion for the dielectronic interaction on the description of the initial bound state and the predicted cross sections. We observe that the dominant escape mode of the two photoelectrons dramatically depends upon the energy sharing. In the parallel geometry, when the ejected electrons are collected along the direction of the laser polarization axis, back-to-back escape is the dominant channel for strongly asymmetric energy sharing, while it is completely forbidden if the two electrons share the excess energy equally.

Time-dependent second-order Born calculations for model atoms and molecules in strong laser fields

Using the finite-element discrete variable representation of the nonequilibrium Green's function (NEGF) we extend previous work [K. Balzer et al., Phys. Rev. A 81, 022510 (2010)] to nonequilibrium situations and compute---from the two-time Schwinger-Keldysh-Kadanoff-Baym equations---the response of the helium atom and the heteronuclear molecule lithium hydride to laser fields in the uv and xuv regimes. In particular, by comparing the one-electron density and the dipole moment to time-dependent Hartree-Fock results on one hand and the full solution of the time-dependent Schr\"odinger equation on the other hand, we demonstrate that the time-dependent second Born approximation carries valuable information about electron-electron correlation effects. Also, we outline an efficient distributed memory concept which enables a parallel and well-scalable algorithm for computing the NEGF in the two-time domain.

Double photoionization of excited lithium and beryllium

We present total, energy-sharing, and triple differential cross sections for one-photon, double ionization of lithium and beryllium starting from aligned, excited $P$ states. We employ a recently developed hybrid atomic orbital with numerical grid method based on the finite-element discrete-variable representation and exterior complex scaling. Comparisons with calculated results for the ground-state atoms, as well as analogous results for ground-state and excited helium, serve to highlight important selection rules and show some interesting effects that relate to differences between inter- and intrashell electron correlation.

Optimizing the FEDVR-TDCC code for exploring the quantum dynamics of two-electron systems in intense laser pulses

To efficiently solve the three-dimensional (3D) time-dependent linear and nonlinear Schrödinger equation, we have developed a large-scale parallel code RSP-FEDVR [B. I. Schneider, L. A. Collins, and S. X. Hu, Phys. Rev. E 73, 036708 (2006)], which combines the finite-element discrete variable representation (FEDVR) with the real-space product algorithm. Using the similar algorithm, we have derived an accurate approach to solve the time-dependent close-coupling (TDCC) equation for exploring two-electron dynamics in linearly polarized intense laser pulses. However, when the number (N) of partial waves used for the TDCC expansion increases, the FEDVR-TDCC code unfortunately slows down, because the potential-matrix operation scales as ∼O(N2) . In this paper, we show that the full potential-matrix operation can be decomposed into a series of small-matrix operations utilizing the sparse property of the [N×N] potential matrix. Such optimization speeds up the FEDVR-TDCC code by an order of magnitude for N=256 . This may facilitate the ultimate solution to the 3D two-electron quantum dynamics in ultrashort intense optical laser pulses, where a large number of partial waves are required.