Breit–Pauli R-Matrix approach for the time-dependent investigation of ultrafast processes

Abstract

We have refactored the Breit–Pauli R-Matrix integral package within the RMatrxI package to employ a B-Spline basis to allow for level-resolved time-dependent R-Matrix calculations involving a laser pulse. The B-Spline approach independently verifies the accuracy of the current integral package pstg1r.f, but requires greater flexibility at the R-Matrix boundary when describing the continuum wavefunctions. This adaptation can be integrated with either the subsequent serial or parallel Breit–Pauli suite of codes.

Program summary

Program Title: BSplineStg1

Program Files doi: http://dx.doi.org/10.17632/prk6fsn56y.1

Licensing provisions: GPLv2

Programming language: Fortran 2003

Nature of problem: Previously, the time-dependent R-Matrix with Time-dependent (RMT) codes could only be used with atomic data from atomic physics codes which did not include spin–orbit effects. When we desire to include spin–orbit effects in a level resolved Time-dependent R-Matrix calculation, we must use a different atomic structure and integral package, such as the RMatrixI package. However, while the RMatrixI package is able to include spin–orbit effects, it was not previously compatible for use with RMT.

Solution method: To enable compatibility between RMatrixI and RMT, we write an alternative to the Stg1 code within the RMatrixI package (titled BSplineStg1) that uses B-Spline techniques to create and describe continuum orbitals. The necessary one- and two-electron integrals for the remainder of the RMatrixI package are calculated using Gauss–Legendre integration. The remainder of the Breit–Pauli R-Matrix codes require no further modification.

Introduction

In recent years, a number of advances in laser technology have enabled the real-time observation of ultrafast electron dynamics [1]. Advances in Free Electron Laser (FEL) technology (bringing pulses of higher intensity at higher frequency) and deeper understanding of High Harmonic Generation (HHG) processes (enabling pulses over shorter timescales — the current record is $43$ attoseconds [2]) mean that the field of attosecond-physics has an ever-expanding array of methods to observe and control processes mediated through electron-dynamics.

Examples of these methods include a variety of spectroscopy techniques (most notably Attosecond Transient Absorption Spectroscopy (ATAS) [3], [4]), pump-probe techniques [5], and interferometry techniques [6], [7]. These recent experimental, and their corresponding theoretical, advances have brought a greater degree of understanding to the study of non-relativistic electron dynamics. However there are still many relatively unexplored aspects of the field, a notable example being the impact of spin–orbit effects on ultrafast processes. While the introduction of spin–orbit effects to ultrafast systems brings increased complexity, the study of such systems might bring the promise of a potential exploitation of such semi-relativistic effects in the observation and control of electrons.

For example, it is not well understood how spin–orbit effects will affect HHG processes. It can be speculated that, for example, it might be possible to exploit spin–orbit interaction effects to increase HHG yield. Similarly, it is not known in detail how spin–orbit interactions affect the dynamics of core-hole states in atoms heavier than Ar over ultrafast timescales. This is especially important as the spin–orbit induced transfer between differing spin multiplicities has been shown to be important in a number of biological processes (e.g. sight, therapy of skin diseases), and other non-biological processes such as fluorescence [8].

There is a need for the development of ab-initio theoretical techniques to model ultrafast spin–orbit dynamics, ideally incorporating a full description of electron correlation. There exist a number of tools that are capable of studying non-relativistic dynamics, examples include the Helium Finite-Difference (FD) method [9], time-dependent close coupling (TDCC) methods [10], [11], and Time-Dependent Density Functional Theory (TDDFT) [12], among others. However, few are capable of studying systems where the spin–orbit interaction plays a role in the dynamics. Of note is the time-dependent configuration-interaction singles (TDCIS) method [13], which has previously studied spin–orbit effects in HHG. However the TDCIS method aims to describe only single excitations from closed shell systems, limiting the range of core dynamics that can be included in the model.

Another promising method is the R-Matrix with Time-dependence (RMT) method [14]. The RMT method is built upon time-independent R-Matrix codes whose effectiveness has been demonstrated over the previous decades. These codes are then combined with Finite-Difference (FD) techniques to enable time-propagation of the system. As with most R-Matrix methods, RMT considers the interaction of all valence and core electrons, enabling the ab-initio study of all desired core and outer electron dynamics. Examples of processes investigated using RMT include ultrafast photoionisation [15], rescattering [16], HHG [17], attosecond transient absorption spectroscopy [3], and double ionisation processes [18], [19], [20]. Where available, excellent agreement with experimental data has been achieved in each of these cases. Furthermore, RMT has been able to study processes in a variety of system sizes, from hydrogen [14] up to xenon (while omitting relativistic effects) [17], and has recently been extended to study small molecules [21].

For effective modelling of the dynamics within the atom, RMT requires atomic structure data, which to date has been obtained using the RMatrixII package [22], [23]. However, to carry out RMT calculations with spin–orbit interactions included, it is necessary to obtain atomic data containing the corresponding spin–orbit interaction, which will be described using a jK coupling scheme. While the RMatrixII package is not yet able to include spin–orbit interactions, and only produces data in the LS coupling scheme, the RMatrixI package [24] does have the capability to calculate the required semi-relativistic atomic data.

The first development of RMatrixI occurred in the late 1970s [24], [25], and in recent years, the capability to exploit parallel computing facilities has been incorporated into the package. This enables the investigation of more challenging systems requiring a large number of target states — a capability that the RMatrixII package does not yet have. An example of recent work that required a level-resolved description including the spin–orbit interaction was electron-impact excitation of neutral molybdenum (a system with 42 electrons), for use in the study of fusion energy sources [26]. The purpose of this work is to enable the calculation of the time-independent jK-coupled atomic data calculated with the RMatrixI codes which can be used by the RMT method to obtain time-dependent semi-relativistic data on ultrafast atomic physics processes.

A complete description of the RMatrixI package can be found in [24]. The following paragraphs briefly describe the basic RMatrixI functionality to enable a discussion of the changes necessary to adapt the RMatrixI package for RMT calculations. We begin by describing the most basic functionality of the RMatrixI package. To produce atomic data, RMatrixI must perform the following tasks:

• “Stg1” — The purpose of Stg1 is to read in the bound orbitals and calculate continuum orbitals that are orthogonal to the bound orbitals. Required one- and two-electron integrals involving bound and continuum orbitals are then calculated. This stage is the focus of the work here, and is discussed in detail in the next section.

• “Stg2” — If a non-relativistic calculation is required, the $N$ electron target states are calculated through diagonalisation. These targets are coupled to spherical harmonics to obtain channels and symmetries. Dipole interactions and long range interactions between states are then calculated. If a Breit–Pauli calculation is required, only the construction of channels and dipole matrices in an LS coupled description is performed.

• “StgjK” — Reads in Hamiltonian symmetries and dipole matrices between LS symmetries (which might already include mass–velocity and Darwin corrections) and transforms them into jK coupled Hamiltonian matrices, and reduced dipole matrix files. If a non-relativistic calculation is desired, Stg3 is run directly after Stg2, and StgjK is omitted.

• “Stg3” — Stg3 finds a set of diagonalised $N+1$ electron states for each symmetry, and then calculates the dipole interaction between the $N+1$ electron eigenstates. After Stg3 all atomic data necessary to run time-dependent calculations is present.

In order to adapt the RMatrixI package for use in time-dependent calculations, there are two main changes that must be implemented. Firstly, substantial changes must be made to the method for calculating the continuum orbital to meet the differing requirements of time-independent and time-dependent calculations. For example, in time-independent R-Matrix calculations, it is useful for the continuum orbital basis to cover as wide a range of energies as possible. For this, a numerical basis (calculated using an algorithm based on the one in [27]) is used, with states differing by a constant energy spacing with a constant derivative at the boundary between the inner and outer regions. The regularity of energy in this setup allows a “Buttle correction” to the energy to be calculated to account for this truncation, where the effect of the higher energy states truncated from the basis is estimated with a quadratic fit to the basis functions used.

Time-dependent calculations, however, require the connection between the inner and outer regions to be established for every time-step. Hence, a highly accurate representation of the wavefunction is needed, which the Buttle correction is not able to provide. Therefore, instead of the numerical grid based basis, we here choose to implement a B-Spline basis [28] for the generation of continuum orbitals. B-Splines have a number of useful properties that would seem to make them a natural choice for the description of RMatrixI continuum orbitals intended for time-dependent calculation. Firstly, they are polynomials of order $k$, meaning that they can form a good approximation for the Coulomb-function-like wavefunction. Secondly, they are local functions. This leads to a convenient numerical behaviour, as inaccuracy introduced through cancellations over the entire basis function are less likely to occur. Thirdly, they do not need a Buttle correction, meaning that exact wavefunction matching at the boundary can occur.

There are a number of examples of previous time-dependent R-Matrix work where B-Spline techniques are used to describe continuum orbitals [16], [17], [23], [29], [30]. There are also furtherR-Matrix examples of use of B-Splines in time-independent calculations, both in other atomic time-independent methods [31], [32], [33] and molecular methods [34], [35].

We note that it is possible to make the necessary changes with the modification of only the Stg1 code. As such, Stg2 and Stg3 can be used in their original form for time-dependent calculations (the angular momentum algebra and coupling between the inner and outer regions are not affected by the requirements of time-dependent calculations). As a result, in this work to extend time-dependent capability to RMatrixI we are only concerned with writing a new version of the Stg1 codes. It is not necessary to modify any other part of the RMatrixI inner region package.