The properties of interest include electronic configurations of the ground states, ionization potentials, electron affinities, and excitation energies, which are associated with the spectroscopic and chemical behavior of these elements, and are therefore of considerable interest.
The quality of the calculations is assessed by applying the same methods to lighter homologs, where the experimental information is available. This comparison shows very good agreement, within a few hundredths of an electronvolt, and similar accuracy is expected for the SHEs. Many of the properties predicted for the SHEs differ significantly from what may be expected by straightforward extrapolation of lighter homologies, demonstrating that the structure and chemistry of SHEs are strongly affected by relativity, making determination of their place in the Periodic Table a challenge.
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Purchase Instant Access. View Preview. Learn more Check out. Abstract Electronic structure and atomic properties of the transactinide or superheavy elements SHEs are reviewed. We use two-electron full dimensional calculations full-2e as benchmark for our haCC computations. We solve the two-electron TDSE using an independent particle basis of the form:. We use the same type of single center expansion for all the benchmark computations. A complete description of this method will be presented elsewhere . The purpose of the two-electron calculations is to demonstrate to which extent these fully correlated calculations are reproduced by the haCC approach.
For that we use the same tSURFF propagation times and identical box sizes when comparing the two types of calculations. Full convergence of the two-electron calculation in propagation time and box size is not discussed in the present paper. In this section, we present photoelectron spectra from helium and beryllium atoms and from the hydrogen molecule with linearly polarized laser fields computed with the above described coupled channels formalism.
We also present the single photon ionization cross-sections for the beryllium atom and the wavelength dependence of the ionization yield for the hydrogen molecule to compare with other existing calculations. We use cos2 envelope pulses for all the calculations and the exact pulse shape is given as. We compare our results for helium and the hydrogen molecule with full-2e calculations  and for beryllium with effective two electron model calculations.
The convergence of the full-2e benchmark calculations and the haCC calculations were done systematically and independently. The radial finite element basis consisted ofhigh order polynomials with typical orders and the total number of radial basis functions was such that there were functions per atomic unit.
The angular momenta requirement strongly depends on the wavelength. The longer wavelengths needed larger number of angular momenta for convergence. All the calculations presented are converged with respect to the single electron basis parameters like the order and the box size, well below the differences caused by inclusion of ionic states. Hence, we only present various observables as a function of the number of ionic states. The storage requirements with the algorithms that we use are dictated by the two particle reduced density matrices.
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This is not a large requirement in the context of the currently available computational resources. In order to avoid replication, these objects were stored in shared memory. The computation times vary widely depending on the wavelength and the number of ionic states in the basis. They scale with the square of the number of ionic states. For the cases presented here, the required times range from 0. These times also have a strong dependence on the exact time propagators used and a discussion on the suitable time propagators is out of the scope of this work.
Helium is the largest atom that can be numerically treated in full dimensionality. With linearly polarized laser fields, the symmetry of the system can be used to reduce the problem to five-dimensions.
The energies of the helium ionic states are n2 for principal quantum number n. The first two ionic states are separated by 1. This has been a motivation to treat helium as an effective single electron system with XUV and longer wavelengths in some. Figure 1. Photoelectron spectra from helium with three-cycle, 21 nm laser pulse with a peak intensity of W cm Left figure: ground state channel 1s , Right figure: a first excited state channel 2pz.
The upper panels show spectra obtained with a full-2e and haCC calculations with different number of ionic states included as indicated in the legend. Here, n is the principal quantum number. The lower panels show relative errors of haCC calculations with respect to full-2e calculations. The inset shows the 2s2p resonance see main text. We examine below the validity of treating helium as an effective single electron system, by comparing haCC calculations with full dimensional calculations at different wavelengths. The one and two photon ionization peaks of 1s and 2pz channel spectra are shown.
The relative errors of haCC calculations are computed with respect to the full dimensional calculation. The single photon peak of the 1s channel is computed to a few percent accuracy, except for a feature around 1. The resonant feature can be identified with the 2s2p doubly excited state , which is reproduced to few percent accuracy with the addition of 2nd shell ionic states. While the position of the resonance is reproduced accurately in the calculations presented here, the propagation time was well below the life-time of this resonance which is reflected in the width of the feature that is well above the natural line width.
A broadband few cycle XUV pulse tends to excite the initial state into a band of final states which may include many correlated intermediate states. Here, the intrinsic limitations of any coupled channels approach that is based on ionic bound states only are exposed. Firstly, a correlated intermediate state with a bound character needs large number ofionic states to be correctlyrepresented. Secondly, the ionic bound states based on Gaussian basis sets do not have the exact asymptotic behavior.
This can lead to an inaccuracy in length gauge dipole matrix elements. Finally, the absence of ionic continuum states in our approach is another possible source of inaccuracy. For obtaining long-lived resonance structures by a time-dependent method one must, as a general feature of such methods, propagate for times at least on the scale of the life time of the resonance. The only alternative is to independently solve the stationary resonant scattering problem and decompose the time-dependent solution after the end ofthe pulse into the corresponding scattering continuum.
Solving the scattering problem, however, is a computationally very demanding task by itself. For obtaining the resonances with tSURFF, one can simply propagate until the resonance has decayed completely and all flux has passed through the surface where the flux is collected. At this point it should be remarked that the relevant information about resonances maybe generated more efficiently by stationary methods like time-independent complex scaling [27,28].
With haCC, due to its very compact representation, we can easily propagated much longer to obtain the resonances to any desired accuracy. The ratio of 1.
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The positions of the resonances are accurate on the level of a. Here we had chosen a box size of 45 a. Hence, only these states are. T is the time propagation in the units of laser cycles. Left panel: 1s ionic channel, right panel: 2pz ionic channel. Figure 3. Total photoelectron spectra from helium with Left figure: three-cycle, nm laser pulse with a peak intensity of W cm-2, Right figure: three-cycle, nm laser pulse with a peak intensity of 3 X W cm The lower panels show relative errors of haCC calculations with respect to full-2e calculation.
One can obtain the higher excited states by increasing the box size, at the penalty of a larger discretization and somewhat longer propagation times, see discussion in section 2. Similar as long lived resonances, threshold behavior of the spectrum near energy zero only emerges with propagation time, figure 2. Further distortions near threshold are due to the effective truncation ofthe Coulomb tail in the absorbing region.
Accuracies at the lowest energies can be pushed by increasing both, simulation box size and propagation times. Figure 3 shows total photoelectron spectra from helium at and nm wavelengths. The exact laser parameters are indicated in the figure captions. Addition of second and third shell ionic states improves the accuracy of the spectra to few percent level in the important regions of the spectrum.
Addition of more ionic states, does not improve the accuracy further. This is possibly due to the missing continuum of the second electron that is needed to fully describe the polarization of the ionic core. This is consistent with the knowledge that at longer wavelengths, it is the ionization. Table 1. Energies of the used single ionic states of beryllium relative to the ground state ion. Figure 4. Photoelectron spectra from the beryllium atom. Left figure: ground state channel spectra with three-cycle, 21 nm laser pulse with a peak intensity of W cm Right figure: total spectra with three-cycle, nm laser pulse with a peak intensity of W cm The upper panels show spectra obtained with effective-2e and haCC calculations with different number of ionic states included as indicated in the legend.
The lower panels show relative errors of haCC calculations with respect to the effective-2e calculations. Our findings show that helium at long wavelengths can be approximated as a single channel system. Beryllium is a four electron system that is often treated as a two electron system due to the strong binding of its inner two electrons.
The third ionization potential of beryllium is With photon energies below this third ionization potential, it can be safely treated as an effective two electron system. This allows us to have a benchmark for our spectra by adapting the simple Coulomb potential to an effective potential in our two electron code. We use the effective potential given in  for our benchmark calculations. We refer to these as 'effective-2e' calculations. Table 1 lists the energies of the first 8 ionic states of beryllium relative to the ground ionic state.
As the ionic excitation energies are much smaller than in Helium one would expect inter-channel couplings to play a greater role. Figure 4 shows photoelectron spectra from beryllium with 21 and nm wavelength laser pulses. The exact parameters are indicated in the figure caption. The relative errors ofspectra from the haCC calculations are computed with respect to the effective-2e calculations. At 21 nm, the one and two photon ionization peaks of ground state channel spectra are shown. Here, the single photon ionization process itselfneeds more than the ground ionic state to produce accurate photoelectron spectra.
Adding more ionic states improves the accuracy to a few percent level. We find that the close energetic spacing of beryllium ionic states leads to a greater possibility of inter-channel coupling, which requires more than the ground ionic state to be well represented. Also, at nm we need more than the ground ionic state to compute realistic spectra.
With the addition of 1s23s and 1s23p states, a structure similar to the one predicted by the benchmark calculation develops around 10 eV. This structure maybe identified with the lowest resonance 1s22p3s at The coupled channels calculations with the number of ionic states considered here, however do not reproduce the structure on the second peak exactly. This points to a feature of a coupled channels basis that the correct representation of a strongly correlated state that has bound character requires a large number of ionic states.
Figure 5. Single photoionization cross-sections for beryllium in the photon range of eV. Presented are results from haCC calculations with 4,5,8 ionic states. It has been shown through examples in section 4. Lithium, the smallest alkali metal, also has been successfully modeled as a single electron system in an effective potential, for example in . We find that beryllium needs at least two ionic states, 1s22s and 1s22p, for a realistic modeling. It serves as a first simple example where single electron models break down and multiple channels need to be considered.
In figure 5, we present the single photon ionization cross-sections as a function of photon energy from our haCC method and compare them with the cross-sections calculated with TD-RASCI method  and R-matrix method  and with experimental results from . As we are computing the rate, the exact size of the inner region does not play a role. The norm drop reaches a steady state irrespective of the inner region size.
We used for our computations presented here a cycle continuous wave laser pulse with a three cycle cos2 ramp up and ramp down and with an intensity of W cm We checked convergence with respect to the pulse duration and the inner region size, and the computations are converged well below the differences seen by addition of ionic states in the basis. All the theoretical results lie in this range except at low energies. In the haCC calculations, including more than 4 ionic states does not change the cross-sections.
In this energy range, the cross-sections from haCC calculations show a dependence on the number ofionic states included. This modulation maybe attributed to the presence ofauto-ionizing states in this region. Table 3 in  presents a list of resonances that appear in beryllium electronic structure. The first ionization potential is 9. With photon energies around 20 eV, the resulting photoelectron reaches continuum region where a number ofresonances are present. As correlated resonances need many ionic states to be well represented in a coupled channels basis, this may explain the dependence of the cross-section on the number of ionic states in eV photon range.
6.3 Development of Quantum Theory
The hydrogen molecule in linearlypolarized laser fields parallel to the molecular axis, with fixed nuclei has the same symmetry as helium in linearly polarized laser fields. The off-centered nuclear potential, however,. Figure 6. Photoelectron spectra from H2 with a three-cycle 21 nm laser pulse with a peak intensity of W cm 2. The upper panels show spectra obtained with full-2e and haCC calculations with different number of ionic states I included as indicated in the legend. The lower panels show relative errors of haCC calculations with respect to the full-2e calculation.
While the number of basis functions can be reduced through a choice of a more natural coordinate system like prolate spheroidal coordinates for diatomics , the challenge of computing two electron integrals remains. As a benchmark for spectra, we use results from a full dimensional calculation, that expands the wavefunction in a single center basis. Figure 6 shows photoelectron spectra from H2 at 21 nm wavelength.
The exact laser parameters are given in the figure caption. The ground state log and first excited state 1oU channel spectra are shown. We find that, at this wavelength, a single ionic state is not sufficient to produce accurate photoelectron spectra. With the addition of more ionic states, there is a systematic improvement in the accuracy of the calculations. The single photon ionization to the shake-up channel 1ou is also computed to a few percent accuracy with 11 ionic states. We find that the single ionization continuum of H2 is more complex than helium and it needs more than a single ionic state.
With 4 and 6 ionic states, we find artefacts on the two photon peaks. Then parts of the correlation contained in Q cannot be presented in that basis such that a non-zero correlated state. This spurious correlated state moves to higher energy with addition of ionic states. A straight forward solution to this problem is to compute this state and project it out from the basis. But this would require locating the spurious state in the eigenvalue spectrum, which is very demanding for large Hamiltonians. Fortunately, by their dependence on the number of ionic states, artefacts of this kind are easily detected and can be moved out of the region of interest by using sufficiently many ionic states.
Such artefacts are a natural consequence ofanyansatzofthekind 5 and need to be monitored. Figure 7 shows total photoelectron spectra at and nm wavelengths. Addition of more ionic states helps reproduce additional resonant features in the spectrum. Inclusion of up to 6 ionic states reproduces the feature around 0. We find that with H2 at longer wavelengths, ground ionic state is sufficient to compute realistic spectra and only for resonant features a large number ionic states is required. Figure 8 shows total ionization yield as a function of photon energy in the range 0.
Figure 7. Total photoelectron spectra from H2 with—left figure: three-cycle nm laser pulse with a peak intensity of W cm 2. Right figure: three-cycle nm laser pulse with a peak intensity of W cm The dashed vertical lines separate different multi-photon ionization regimes. The haCC calculations shown were performed using two ionic states, convergence was verified by performing four-state calculations at selected points.
The vertical lines in the figure separate different multi-photon ionization regimes. The most conspicuous discrepancies between haCC and CI appear in the range 0. The discrepancies maybe a result of the intrinsic limitations or the convergence of either calculation. Also note that the results in  were shifted by 0. Although these differences are miniscule for energies they may indicate for somewhat larger deviations in the wave functions and the values of ionization potentials give a measure for the accuracy of the computations.
Our full-2e computations that, in principle, could help to resolve the discrepancy are expensive and have not been pushed to an accuracy level which would allow to decide between the two results. However, we believe that the present level of agreement between haCC and CI is quite satisfactory and supports the validity of both approaches. The hybrid anti-symmetrized coupled channels method introduced here opens the route to the reliable ab initio calculation of fully differential single photo-emission spectra from atoms and small molecules for a broad range of photon energies.
It unites advanced techniques for the solution of the TDSE for one- and two-electron systems in strong fields with state of the art quantum chemistry methods for the accurate description of electronic structure and field-induced bound state dynamics. Key ingredients for the successful implementation are good performance of tSURFF for the computation of spectra from comparatively small spatial domains on the one hand and access to the well established technology of quantum chemistry on the other hand.
In future implementations, it maybe sufficient to output from a given package the generalized one, two, and three-electron density matrices together with generalized Dyson orbitals, both defined in the present paper. It turned out to be instrumental for accurate results that haCC allows for the inclusion of neutral states in a natural fashion and at very low computational cost.
Several new techniques were introduced and implemented for the establishment of the method. Most notably, the mixed gauge approach  turned out to be crucial for being able to take advantage of the field-free electronic structure in presence of a strong field without abandoning the superior numerical properties of a velocity-gauge like calculation.
The finite element method used for single-electron strong-field dynamics is convenient, but certainly not the only possible choice. Low-rank updates are used in several places for the efficient computation of the inverses of the large overlap matrix and to control the linear dependency problems arising from anti-symmetrizing the essentially complete finite elements basis against the Hartree-Fock orbitals. We have made an effort to explore the potential range of applicability of the method by performing computations in a wide range of parameters on a few representative systems, where results can be checked against essentially complete methods.
An Alternative Approach to the Problem of CNT Electron Energy Band Structure
Spectra for the He atom were independently obtained from fully correlated two-electron calculations. An interesting observation is that in the long wavelength regime indeed a single ionization channel produces correct results, justifying ex post wide spread model approaches of the strong field community. As a note of caution, we recall that this is only possible as the fully correlated initial state is routinely included in the haCC scheme. At short wavelength, the ionic excited state dynamics plays a larger role and reliable results require inclusion of up to 9 ionic channels.
With this we could correctly resolve also the peak due to He's doubly excited state. The second atomic system, Be, was chosen to expose the role ofelectronic dynamics in the ionic states. While the 1s core electrons are energetically well-separated and no effect of their dynamics was discernable in a comparison with a frozen core model, the narrow spaced ionic states preclude single channel models. Depending on the observable and on desired accuracies, a minimum of two ionic channels had to be used.
For the comparison of H2 photoionization and photoelectron spectra, we could refer to literature and supplemented the data with full two-electron calculations. At nm, H2 can be treated as a single channel system. At intermediate wavelengths, we find the need for at least two ionic channels, and we could obtain a fair agreement with comparison data.
Here one has to take into consideration that all alternative methods operate near the limits oftheir applicability. With this set of results we demonstrated the correctness of the method and its essential features. In our calculations, also the fundamental limitations of the approach were exposed. Clearly, the field-induced dynamics of the ionic part must be describable by a few states with bound character. Note that the problem is partly mitigated by the possibility to include fully correlated ground as well as singly- and doubly-excited states with bound-state character that are known to appear in the dynamics.
The method in its present implementation can be applied to many electron atoms  and small molecules such as N2 and CO2, which will be reported in a forthcoming publication. The ionic states in these molecules are closely spaced as in beryllium and hence they would also need several ionic states in the basis for convergence.
We have no reliable heuristics to a priori estimate the number ofstates needed for a given accuracy. From the present experience, we expect that at long wavelengths, for example at nm, inclusion of ionic states in the range of eV below the first ionization threshold maybe sufficient for convergence. This translates to about ionic states in the basis for these systems which is a feasible problem. At the moment, the computation of the two-electron integrals poses a mild technical limitation for such calculations, and an improvement of the presently rather straight-forward algorithm is needed for going to.
Treating molecular systems with lower symmetry leads to a further fill in of the Hamiltonian matrix due to the large number of allowed transitions. Such calculations appear quite feasible as well, however at comparatively higher resource consumption than the few hours on a eight-core machine needed for the majority of the results presented here. Another limitation arises when the molecule becomes too large for computing even strong field single-electron dynamics over its complete extension.
Also, for the single electron part, we use at present single-center expansions, which perform notoriously poorly if scattering centers are distributed over more than a few atomic units. This limitation may well be overcome by a more versatile single-electron discretization, though at significant implementation effort. Other potential extensions are to double-emission. Combining such already sizable calculations with a dication described by quantum chemistry in the same spirit as here maybe feasible.
The formula presented can be readily extended to include that case.