Interfacing the B-Spline R-Matrix and R-Matrix with Time Dependence Computer Codes: An Update

Abstract

As a continuation of Schneider et al., Atoms 2022 10, 26, we report recent progress in the development and deployment of the interface between the computational codes B-Spline R-matrix (BSR) and R-Matrix with Time dependence (RMT). These advances have been achieved within the context of the $LS$-coupling scheme. In its current state, the interface handles atomic target states described by single configurations and supports the Fano–Racah phase convention, as required by RMT. As first example of an application, we use the interface to investigate multiphoton single ionization of helium exposed to a linearly polarized laser field with wavelengths between 280 and 316 nm and a peak intensity of $3\times {10}^{14}$ W/cm². As a second example, we consider high-order harmonic generation (HHG) in carbon, driven by an intense 30-cycle laser field at 800 nm and a peak intensity of $1\times {10}^{12}$ W/cm².

1. Introduction

The B-Spline R-matrix (BSR) approach [1] is an alternative formulation of the celebrated ab initio R-matrix method [2], designed for applications to atomic and molecular systems. The BSR method, along with its computational implementation released in 2006, was developed by the late Oleg Zatsarinny. Compared to the previous computational implementations of the R-matrix method [3,4], the BSR code stands out for employing nonorthogonal orbitals to represent both the bound and continuum one-electron orbitals. Furthermore, BSR introduced B-splines as basis functions into the R-matrix paradigm.

On its own, the BSR code can be used to obtain atomic structure information for bound states, such as energy levels and oscillator strengths. Its applications [5] also cover the study of atomic continuum processes, including electron–atom and electron–ion scattering, as well as radiative processes such as bound–bound transitions, weak-field photoionization, and polarizabilities. Other observables, such as scattering cross sections, can be obtained by interfacing BSR with asymptotic codes, e.g., (P)STGF [6,7] and FARM [8], through the so-called H file [3].

Note that the study of the observables, processes, and physical quantities mentioned above is carried out within the framework of time-independent quantum-mechanical equations. However, time-dependent dynamics can also be investigated through the time-dependent extension of the R-matrix method. Such an extension was first proposed by Burke and Burke [9], who outlined the general idea already in 1997. Further important steps in the timeline were the explicit application to the hydrogen atom in 2008 [10] and the subsequent expansion to multielectron atoms and ions in a nonrelativistic framework in 2011 [11]. We refer the interested reader to [12], where a detailed historical review is presented.

In parallel to the development of the time-dependent extension of R-matrix theory, several advancements were made in the computational aspects of the method1, with the release of the RMT code [13] in 2020 representing a major milestone in the field. Essentially, RMT is a computational program that propagates the initial wave function in time according to the Time-Dependent Schrödinger Equation (TDSE) for a given multielectron atom, ion, or molecule interacting with short-pulse intense laser light, while accounting for correlation effects.

The RMT suite has recently benefited from the creation of interfaces with the Belfast R-matrix code [3,13,14], as well as UKRmol+ [13,15]. This has enabled the study of time-dependent spin-orbit dynamics in krypton [16], semi-relativistic RABBITT (Reconstruction of Attosecond Beating by Interferences of Two-photon Transitions) calculations [17], and the study of fully nonperturbative multielectron processes in molecules driven by ultrashort arbitrarily polarized strong laser fields [18]. However, as outlined in [12], it was clear that the atomic input data required for RMT could alternatively be generated via the BSR code. In this way, instead of using the Belfast codes [3,4], one could exploit the use of nonorthogonal orbitals in the target description that BSR offers. As a result, correlation effects could be incorporated into the atomic target expansion through individually optimized, term-dependent, nonorthogonal sets of one-electron orbitals. This would lead to compact target configuration-interaction expansions (see, e.g., [19]) with a smaller number of correlation orbitals, thereby promising reduced CPU time and high accuracy in TDSE simulations.

The interoperability between the BSR and RMT codes was not built in by default; therefore, integrating both programs requires the development of an appropriate interface. In fact, it was Barry Schneider who played a key role in promoting and supporting the development of this interface, as announced in [12], with the present paper representing its first continuation. A detailed background and motivation for the interface are provided in [12]. In this context, the goal of the present work is twofold: (i) We report an update on the progress made on the interface between BSR and RMT that, at present, operates within a nonrelativistic framework for atomic systems, where the $LS$-coupling scheme is appropriate. (ii) This paper serves as a brief user guide for the interface in its current state.

The present manuscript is organized as follows. In Section 2, we begin by describing the major modification incorporated into the BSR code: the implementation of the Fano-Racah phase convention. We then describe the utilities that constitute the interface in Section 3. To illustrate the new code, we apply the interface to treat multiphoton ionization of helium, with results shown in Section 4.1. Additionally, we calculate high-order harmonic generation (HHG) in carbon, a less studied system in this context, with results displayed in Section 4.2. Finally, a summary and outlook are provided in Section 5.

2. Modifications to BSR

Several changes were introduced to the BSR code at different levels (including the main programs, libraries2, and utilities), while no modifications were made to the RMT code. In this Section, we provide a brief description of the most significant one: the modification in the phase convention for the spherical harmonics.

2.1. Phase Convention

The BSR code was originally developed under the Condon-Shortley phase convention for spherical harmonics, whereas the RMT package (as well as the Belfast codes) adopts the Fano-Racah convention. The relationship between the two conventions [2,21] is simply given by

$${\mathcal{Y}}_{\mathcal{l}m}(\theta ,\phi )={\mathrm{i}}^{\mathcal{l}}{\mathrm{Y}}_{\mathcal{l}m}(\theta ,\phi ),$$

(1)

where ${\mathcal{Y}}_{\mathcal{l}m}(\theta ,\phi )$ denotes the spherical harmonic in the Fano-Racah convention, and ${\mathrm{Y}}_{\mathcal{l}m}(\theta ,\phi )$ represents the standard spherical harmonic in the Condon-Shortley convention, both of degree ℓ and order m. While the change in phase convention may seem like a minor issue, it alters the sign of the wave function amplitudes at the boundary, the long-range coefficients, the dipole matrix elements, and other quantities. Thus, it was necessary to develop an alternative version of the BSR code that operates using the Fano-Racah phase convention. Such a version will soon be made publicly available through a repository.

The change in the phase convention primarily affects the bsr_breit and mult programs, as they perform the angular integrations required to express the matrix elements of the Hamiltonian and electric dipole operators. The major change occurs in the subroutines zno_breit and ang_zclkl, which are ultimately responsible for computing the relevant angular integrals associated with the one- and two-electron operators. How to account for the effect of the change in phase convention in these subroutines can be inferred from [22] and [23].

2.2. Target States

The modification of the phase convention also modifies the input BSR data coming from the Multi-Configuration Hartree-Fock (MCHF) code [24,25], which also uses the Condon–Shortley convention. This means that the c-files, which contain the multiconfiguration expansions, must be amended accordingly. Depending on the parity of the configuration, the change in the phase convention may alter the sign of the coefficients in these expansions; see Section 5 in [26]. Currently, the interface only handles single target configurations, for which no modification is necessary. Under this restriction, the c-files from MCHF can be used without issues.

3. Interface: New Utilities

Four new BSR utilities have been developed to generate the input files required by RMT: Splinedata, Splinewaves, H, d00, and d. These tools constitute the interface between BSR and RMT and can be found in [27]. A brief description of each utility is provided below.

3.1. bsr_Splinedata

Based on the BSR input file knot.dat, bsr_Splinedata generates the Splinedata file, which contains all relevant information about the B-splines, including the knot sequence, order, number of intervals, and other auxiliary parameters.

3.2. bsr_Splinewaves

Employing the output of bsr_hd3, this utility generates the radial orbitals for each channel in the B-spline representation, providing the coefficients of the corresponding linear combinations (see Eq. (12.5) in [1]). For each channel, these coefficients are stored in the output file Splinewaves. The implementation of the utility is based on Zatsarinny’s tool collect_rsol.

3.3. bsr_H

This is an updated version of Zatsarinny’s program sum_hh, see [1]. The new utility produces the H file with several inconsistencies from its predecessor corrected, such as the ordering of amplitudes, the handling of vanishing Buttle corrections, the correct implementation of the MORE parameter, among others.

3.4. bsr_d

This utility generates both the d and d00 files. To construct d, it invokes mult3 to compute dipole matrix elements in the length form between the R-matrix states stored in Splinewaves. The d00 file, in turn, contains information on allowed dipole transitions in the $LS$-coupling scheme. This version extends the utility bsr_dd by including dipole matrix elements between target states. Currently, only the length form is supported; the velocity form is under development but not yet available.

3.5. Usage

Figure 1 shows an example script illustrating how the interface is deployed in practice. The utilities follow the same coding and usage conventions established by Zatsarinny for the BSR code. For instance, with the exception of bsr_H, each utility produces a log file, providing the user with relevant information about the corresponding calculation.

4. Example Illustrations: Multiphoton Ionization of Helium and High-Order Harmonic Generation in Carbon Atoms

In this section, we illustrate the new code with two examples, namely multiphoton ionization (MPI) of helium and high-order harmonic generation (HHG) in carbon. While helium has been studied extensively, the six-electron carbon case is more representative due to the initial configuration $\left(1s^{2}2s^{2}2p^2\right)$.

4.1. MPI in Helium

As an application, we consider a helium atom subjected to an intense linearly polarized laser pulse that induces multiphoton single ionization. For concreteness, we consider a 3-state model for the ionic ${\mathrm{He}}^{+}$ target, which includes the $1s$, $2s$, and $2p$ orbitals. Since the target description for this simple system involves only orthogonal orbitals, we are not yet exploiting one of the main features of BSR. Nevertheless, this 3-state model provides the simplest testing ground for simultaneously verifying the correct functioning of the interface and the proper implementation of the Fano-Racah phase convention in BSR.

To generate reference RMT results, we also used the RMATRX-II (RMII) code [28] to produce the input data for the 3-state model described above. In this approach, analytical expressions for the hydrogenic target states were employed. These reference results were then compared with those obtained using the BSR-RMT interface, where, in contrast, numerical target orbitals derived from MCHF [24] were used. To minimize potential numerical discrepancies between the two sets of results, we modified the RMII code to adopt the same B-spline basis as BSR.

We varied the laser wavelength between 280 and 316 nm while keeping the peak intensity fixed at $3.0\times {10}^{14}$ W/cm². The pulse used in the simulations had a sine-squared temporal envelope and a duration of 30 optical cycles. We employed an R-matrix radius of 40 a.u. and expanded the continuum orbitals inside the R-matrix box using 60 B-splines of order 8. For all wavelengths considered, the size of the outer region was at least 1000 a.u., thereby ensuring that no absorbing boundary condition was required.

Once the time-evolved wave function was obtained via RMT, we focused on the probability ($P_{\mathrm{ngs}}$) of the helium atom not remaining in its ground state, defined as

$$P_{\mathrm{ngs}}=1-\underset{t\to \infty }{lim}{\left|\langle {\psi }_0|\psi \left(t\right)\rangle \right|}^2,$$

(2)

where ${\psi }_0$ denotes the ground state wave function.

In our simulations, we took the maximum value of the total angular momentum involved in the close-coupling expansion up to 30 (61 partial waves), the value for which we observed convergence in (2). Figure 2 shows $P_{\mathrm{ngs}}$ as a function of the wavelength. One can immediately notice the perfect, although entirely expected, overlap between the BSR-RMT and RMII-RMT duos.3 Additionally, to corroborate the qualitative physical aspects of both RMT results, we included benchmark data from the single active electron approximation used in [29] and its extension described in [30], which considers ℓ-dependent angular momentum potentials. Overall, there is qualitative agreement between the (ℓ-)SAE and the RMT results, including the pronounced peak in $P_{\mathrm{ngs}}$ around 295 nm. At this wavelength, six photons are needed to ionize the atom. However, the summed energy of five of them matches the energy gap between the ${\left(1s^2\right)}^{1}S$ and ${\left(1s2p\right)}^{1}P$ states of $\approx 20.964$ eV, thereby providing a resonant stepping stone.

For a better comparison between the RMT results, we considered their relative deviation

$$RD\equiv \left|{\displaystyle \frac{P_{\mathrm{ngs}}^{\left(\mathrm{RMII}\right)}-P_{\mathrm{ngs}}^{\left(\mathrm{BSR}\right)}}{P_{\mathrm{ngs}}^{(\mathrm{RMII})}}}\right|$$

(3)

where the superscripts indicate the input data used in the calculation of $P_{\mathrm{ngs}}$, i.e., either from RMII or BSR. For all wavelengths considered, this deviation does not exceed $3\times {10}^{-8}$, as shown in Figure 2. This is a clear indication that the interface is functioning as intended.

As an additional check, we ran RMT with deliberately wrong input obtained from BSR with the Condon-Shortley phase convention. Figure 3 depicts the outcome of this exploration in terms of $P_{\mathrm{ngs}}$. While noticeable differences are present, there is qualitative, almost even quantitative, agreement. Note, however, that only channels involving the $2p$ target orbital in our 3-state model are affected by the change of the phase convention. In contrast, the channels involving the $1s$ and $2s$ orbitals remain unaffected. Furthermore, for this set of laser parameters, a 1-state model (with just $1s$) is sufficient to capture the main physical features of the multiphoton process [31].

4.2. HHG in Carbon

As a second example, we consider a carbon atom initially in its ${\left(1s^{2}2s^{2}2p^2\right)}^1{S}^e$ state subjected to a linearly polarized laser field, thereby producing high-order harmonic generation. Specifically, we use a near-infrared, 30-cycle laser field with a central wavelength of 800 nm and a peak intensity of $1\times {10}^{12}$ W/cm². The target, ${\mathrm{C}}^{+}$, is described by the single configuration ${\left(1s^{2}2s^{2}2p\right)}^{2}P$. As performed for He, we used the RMII-RMT pair to produce reference results for comparison with those obtained from BSR-RMT. To ensure consistent target descriptions in both BSR and RMII, we interpolated the MCHF numerical one-electron orbitals using Slater-type functions, as required by RMII. The same B-spline set as for He was used for C. Total angular momenta up to 10 were included in the close-coupling expansion, resulting in 21 partial waves. No absorbing boundary condition was employed, and the following parameter values were used in RMT: lplusp=0, and adjust_GS_energy=.false., GS_finast=1. This setting ensures that the carbon atom is initially in the lowest energy ${}^{1}S^e$ state.

As soon as the pulse is over, we calculate the spectral density (HHG spectrum) for each pair, which represents the dipole-radiated energy per frequency $\omega$. It is given by [32]

$$S\left(\omega \right)={\displaystyle \frac{2}{3\pi c^3}}{\left|\tilde{a}\left(\omega \right)\right|}^2,\tilde{a}\left(\omega \right)={\int }_{-\infty }^{\infty }a\left(t\right)e^{i\omega t}dt.$$

(4)

Here, $a\left(t\right)$ is the second derivative with respect to time of the induced dipole moment, $d\left(t\right)=-\langle \psi \left(t\right)\left|z\right|\psi \left(t\right)\rangle$. The HHG spectrum is shown in Figure 4. One can immediately notice the excellent agreement between the results. This is a clear indication that the interface is functioning as intended for multielectron systems.

5. Conclusions and Outlook

In this contribution, we described the current status of the interface between the BSR and RMT codes. This is the first update since the announcement by Barry Schneider and coauthors in 2022 [12]. At the moment, the first version of the interface supports atoms described within the $LS$-coupling scheme, with the additional constraint that the ionic target states are represented by a single configuration. Currently, we are working on the implementation of multiconfiguration expansions for the target states in BSR, described within the Fano-Racah phase convention. In addition, we are exploring the incorporation of relativistic corrections into the interface via the Breit-Pauli Hamiltonian in the intermediate $J\mathcal{l}K$-coupling scheme, similar to the RMATRX-I interface [16]. In the near future, we envision adding the BSR-RMT interface to the suite of state-of-the-art codes available on the Atomic, Molecular, and Optical Sciences Gateway (AMOSGateway) [33], where BSR is already installed [34].