Abstract
We formulate a computationally efficient timeindependent method based on the multielectron molecular Rmatrix formalism. This method is used to calculate transition matrix elements for the multiphoton ionization of atoms and molecules under the influence of a perturbative field. The method relies on the partitioning of space which allows us to calculate the infiniterange freefree dipole integrals analytically in the outer region, beyond the range of the initial bound wave function. This approach is valid for an arbitrary order, that is, any number of photons absorbed both in the bound and the continuum part of the spectrum (below and abovethreshold ionization). We calculate generalized multiphoton cross sections and angular distributions of different systems (H, He, \(\hbox {H}_{{2}}\), \(\hbox {CO}_{{2}}\)) and validate our approach by comparison with data from the literature.
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Introduction
Multiphoton ionization (MPI) and its variant resonanceenhanced multiphoton ionization (REMPI) in atoms and molecules have a range of important applications ranging from laserinduced plasma generation^{1,2}, chemical diagnostics^{3,4,5}, chiral recognition^{6,7,8,9,10} to laserfilamentation^{11,12,13,14,15}, harmonic^{16,17,18} and highharmonic^{19} generation and photoelectron spectroscopy^{20}. Given its practical importance accurate data on MPI are surprisingly scarce and striking discrepancies in the MPI cross sections remain in the literature despite recent advances in the experimental technology^{1}. This highlights the role of theory in supplying the missing data.
In the last two decades many experiments probing fundamental aspects of MPI have been carried out. Simulations of the multiphoton perturbative processes can provide valuable insight, whether due to the perturbative field used, or thanks to specific selection rules in effect that restrict the process to a specific Nphoton transition. An example of the latter is the RABITT mechanism for measuring the photoionization time delays^{21,22,23,24}, including its multisideband (multiphoton) variant^{25}. Similarly, absorption of counterrotating circularly polarised photons via several indistinguishable interfering pathways was shown to lead to characteristic electron vortices in the momentum angular distribution^{26,27,28,29}. Recently, a twophoton ATI of helium atoms by finite, fewfemtosecond pulses was investigated using a timedependent method^{30}, building on earlier monochromaticpulse results from perturbation theory^{31,32,33}. For a general introduction to the field of manyphoton abovethreshold ionisation (ATI) see^{34} and references therein.
While MPI of atoms, particularly of the hydrogenlike type, has been thoroughly studied in the past^{35}, the literature on MPI of molecules is much more scarce due to the lack of the spherical symmetry and associated difficulties with the description of the intermediate and final wave functions of the system in the continuum. An exception to this rule is the multielectron molecular Rmatrix Floquet (RMF) approach^{36} based on an earlier atomic RMF theory^{37} and applied to MPI of \(\hbox {H}_{{2}}\)^{38}. With the exception of twophoton crosssections for molecular hydrogen^{39,40,41,42} and nitrogen^{43} and general (\(N + 0\)) MPI of molecules in DFT approach^{44}, multiphoton processes in molecules are studied using timedependent approaches for dynamics in arbitrary external fields, see e.g.^{23,24} and references in^{45}; this is computationally very demanding and doesn’t provide direct access to the multiphoton transition matrix elements. The RMF approach for multielectron systems^{46} is conceptually closest to the method developed here. Nevertheless, a general method specialized to the evaluation of accurate multiphoton matrix elements for molecules is not presently available.
In this work we bridge the gap in theoretical methodology by formulating, implementing and testing a timeindependent ab initio Rmatrix approach for the multiphoton ionization of multielectron molecules. The stationary character of our method is crucial since it allows us to use highlevel descriptions for the electronic structure of all bound and continuum wave functions involved while avoiding the computationally expensive time evolution of the wave function.
The onephoton molecular ionization problem has been formulated within the stationary Rmatrix theory^{47} and implemented within the molecular package UKRmol+^{48}. It requires calculation of the final stationary photoionization state, which is the solution of the full Schrödinger equation with the incomingwave boundary conditions. Twophoton (\(2+0\)) cross sections for photon energies below the singlephoton ionization threshold have been calculated for molecular hydrogen^{45,49} by a similar approach.
In this work we generalize the molecular Rmatrix photoionization methodology to all photon orders, see Fig. 1. The principal problem of the calculation is the evaluation of the freefree matrix elements. In the Rmatrix formulation of photoionization this aspect is simplified by splitting of the coordinate space into the inner and outer region. In the inner region, exact exchange and multielectron correlation effects are included using configuration interaction. In the outer region, where a single electron moves in the multipole potential of the residual molecule, the problem is treated using analytic techniques.
After a brief review of the computational method in the next two sections, we present results for multiphoton ionization of helium, molecular hydrogen and carbon dioxide across a continuous range of photon energies probing REMPI and nonresonant MPI. The full exposition of the theory is given in Appendix A. It is split into two parts. For clarity, in Appendices A.1 and A.2, we first discuss the \((N + 1)\)photon ionization, whose description requires fewer conceptual extensions of the onephoton ionization problem than the general \(N+M\) process. In Appendix A.3 we describe the general \((N + M)\)photon process.
Overview of the molecular Rmatrix method
A detailed description of the Rmatrix theory and its molecular implementation has been given elsewhere^{45,47,48,50}. Here we limit ourselves to the definition of the key quantities needed for the development of the theory presented in this work.
The basic idea of this method is to divide the space by a sphere of radius \(r=a\) (often called “Rmatrix radius”) into an inner region, where multielectron interactions including exchange are important, and an outer region, where exchange and correlation between the continuum electron and the residual target are negligible. The Rmatrix, constructed in the inner region, is the Green’s function for the oneelectron outerregion problem evaluated on the boundary and provides the link between both regions. In the following we denote all spinspace coordinates of the (\(N+1\))/N electrons by \({\mathbf {X}}_{N+1}/{\mathbf {X}}_{N}\).
In the outer region the \((N+1)\)electron wave function can be written as a sum of direct products of the bound wave functions of the \(N\)electron residual molecule and the oneelectron wave functions of the continuum electron:
where \({\overline{\phi }}_{p}^{\Gamma _{p}}\) are the channel wave functions^{45} of space symmetry \(\Gamma _{p}\) defined as the residual Nelectron state coupled to the real spherical harmonic \(X_{l_{p},m_{p}}(\hat{{\mathbf {r}}}_{N+1})\) and spin \(\sigma _{N+1}\) of the continuum electron in the outer region; \(F_{p}(r)\) is the radial channel wave function of the continuum electron.
In the inner region any solution \(\Psi _{E}\) of the timeindependent Schrödinger equation for energy E can be expressed as a linear combination of the Rmatrix eigenstates \(\psi _k\)
where the form of the \(A_{k}(E)\) coefficients depends on the choice of the asymptotic boundary conditions for the outerregion solution^{47}. The \((N+1)\)electron eigenstates \(\psi _k\) are expressed on the CloseCoupling level in terms of “continuum configurations” \({\mathscr {A}}\Phi _i^N\eta _{ij}\) and “\(L^2\) configurations” \(\chi _m^{N+1}\) as
where \({\mathscr {A}}\) indicates the antisymmetrization operation, \(\eta _{ij}({\mathbf {r}}_{N+1}, \sigma _{N+1})\) are continuum spinorbitals dependent on the position vector \({\mathbf {r}}_{N+1}\) and spin \(\sigma _{N+1}\) with a nonzero amplitude on the Rmatrix sphere, \(\chi _m^{N+1}\) are configurations not containing continuum orbitals. The summation over \(i\) runs over the subset of all residual ion eigenstates included in the model; \(j\) runs over those continuum orbitals that are coupled by symmetry to the respective residual ion states and \(m\) runs over configurations generated from the molecular orbitals fully contained inside the inner region. The coefficients \(c_{ijk}\) and \(b_{mk}\) are obtained by diagonalizing the Hamiltonian using equation
where \({\hat{L}}\) is the Bloch operator^{48,50}
Multiphoton ionization
In this section atomic units are used exclusively. In the leadingorder perturbation theory (LOPT) the total generalized \(K\)photon ionization cross section for a fixed orientation of the molecule^{51}
is proportional to the squared magnitude of the \(K\)photon transition matrix element^{51}
and has dimension length\(^{2K}\) \(\times \) time\(^{K  1}\). Here \({\hat{D}}_{c_k}\) is the projection of the dipole operator along the polarization \(c_k\) of the \(k\)th absorbed photon, \(\Psi _i\) is the bound initial state of the molecule with energy \(E_i\), that is usually sufficiently well described by one of the innerregion Rmatrix eigenstates, \(\Psi _{f\varvec{k}_f}^{()}\) is the final stationary photoionization state as defined in^{47} and \({\hat{G}}_k^{(+)}\) is the Green’s operator of the full Hamiltonian \({\hat{H}}\) of the system, evaluated at energy \(E_{i}+k\omega \),
For simplicity, we denote the aggregated energy of the first \(k\) absorbed photons as \(k\omega \), implying that all have the same energy, but in principle the energies of the photons can be different and the product \(k\omega \) replaced by a sum of their energies.
The formula (7) for the transition matrix element is equivalent to solution of \(K  1\) inhomogeneous Schrödinger equations for stationary intermediate states \(\Psi _{i + j\omega }^{(+)}\),
also discussed in^{44}, followed by evaluation of the final dipole transition,
The boundary conditions for the intermediate states are chosen to correspond to the physical outgoingwave solution. As long as the combined photon energies are insufficient to ionize the target, the boundary condition is asymptotically zero. In the Rmatrix formulation the bound intermediate states (excited states of the molecule) are accurately represented by the Rmatrix eigenstates as long as the Rmatrix radius is large enough to contain them fully: see^{45} where the \(N+0\) cross sections for \(\hbox {H}_2\) were calculated this way.
Once the combined energy of \(j\) photons exceeds the ionization threshold, the situation changes. The righthand side that drives the equation for the first unbound intermediate state \(\Psi _{i+j\omega }^{(+)}\) is proportional to a bound state limited to the inner region, allowing us to use a purely outgoing solution in the outer region,
where \(Z\) is the residual ion charge, \(H_l^+(\eta ,\rho )\) is the CoulombHankel function, \(X_{lm}(\hat{\varvec{r}})\) is the real spherical harmonic, \(\Phi _p\) is a state of the residual ion and the index \(p\) labels the asymptotic oneelectron photoionization channels accessible after absorption of \(j\) photons. In writing (11) we have neglected higher multipoles of the electronmolecule interaction. However, the theory can be formulated to take them into account too, by means of an asymptotic expansion^{50}. Nevertheless, in photoionization processes of neutral molecules the Coulomb interaction dominates and the higher multipoles can be safely neglected. It is very practical to consider closed channels too when the channel thresholds are approached from below, so that not only the continuum states but also highly excited bound states can “leak” from the inner region. For charged residual ions, the radial function of a closed channel is the exponentially decreasing real Whittaker function^{52},
where \(\kappa _p = \sqrt{2 E_p}\) is the magnitude of the imaginary momentum associated with the closed channel \(p\).
The full exposition of the theory is given in Appendix A. Here we briefly note that the Rmatrix method makes it possible to write equations for the intermediate states in the inner region that automatically contain the desired boundary condition in the outer region. The inner region solutions can be then unambiguously extended into the outer region by means of the special functions \(H_p^+\) and \(W_p\). The transition elements (9) needed to calculate the cross sections are then calculated as a sum of contributions from the inner and outer region, employing the expansions (2) and (1). In the outer region, this requires integration of highly oscillatory integrals from the boundary between the regions all the way to infinity. This integration is managed using asymptotic forms of the Coulomb (or Whittaker) functions and repeated integration by parts; see Appendix C for details. When, for a given photon energy, the Rmatrix radius chosen is insufficient for use of the asymptotic theory, numerical integration can be used for some radial interval as well.
Results and discussion
As the first application of the newly developed method we choose the hydrogen atom, where semianalytic calculations have been performed by other authors long ago, see Fig. 2a. We note that in this case all “asymptotic” forms discussed above and in Appendix A are actually exact solutions valid throughout all space which allows us to test our implementation (a custom oneelectron code) and validate the presented Rmatrix theory. After separation of the angular degrees of freedom, this problem is onedimensional. The radial basis in the inner region consisted of 1000 equally spaced radial Bsplines and extended up to radius \(a = 500\) a.u. There are no surprises, the results obtained perfectly match the old calculations by Klarsfeld^{53} and Karule^{54}. In the calculations for H, He and \(\hbox {H}_2\) (see below), we have not used the continuation of the bound intermediate states by means of Eq. (12). Instead, a large enough Rmatrix radius was used to contain these states. Resulting datasets for this and other discussed calculations are available in tabular form in the Supplementary Information to this article.
Twoelectron targets: helium atom and \(\hbox {H}_2\)
Still in the atomic domain, we performed a less trivial calculation of twophoton ionization of helium using UKRmol+^{48}. This required implementing the multichannel version of our multiphoton approach into the UKRmol+ codes. The molecular Rmatrix package was used to construct and diagonalize the innerregion Hamiltonian (4) for all required irreducible representations of the target’s point group symmetry (here restricted to \(\hbox {D}_{{2h}}\) as the largest available Abelian point group). Then, transition dipole elements between the Rmatrix eigenstates are calculated for use in \(d_{\text {inn}}\), Eq. (28), and in the evaluation of the righthand sides of the equations for the intermediate states, Eq. (22). Apart from this, the package also provides the transition dipole elements between the ionic states in (30), the boundary amplitudes \(w_{kp}\), the real Gaunt’s coefficients for Eq. (32) and the necessary components of the final stationary photoionization state: the photoionization coefficients \(A_{pj}^{()}\) for Eq. (24) and the \(K\)matrices needed for construction of the \(S\)matrix in Eq. (26).
The size of the inner region was set to \(a = 100\) a.u. As basic building blocks of the wave function of the residual ion we used HartreeFock orbitals of \(\hbox {He}^+\) calculated in Psi4^{55} with the Gaussian basis set daugccpVDZ. For the centreofmasscentred continuum basis we used the partial wave expansion up to \(\ell = 4\) and a radial basis set consisting of 200 evenly spaced Bsplines. A full CI model was used for the \(L^2\) expansion in Eq. (3). To evaluate the outer region integrals we used 5 terms of the expansion (48) and \(P = 3\) in Eq. (50). The results are shown in Fig. 2b and compared with the available calculated results^{31,32,33}. We can see a perfect agreement with earlier calculations below the first core excited resonance, i.e. resonant transition He\(^+\)(1s)–He\(^+\)(2s) in the residual ion, at around \(41\) eV. The deviation occurring there and for higher photon energies is a direct consequence of the chosen Gaussian basis set, which is spatially limited and cannot represent diffuse excited states of the ion. Additionally, our calculations don’t account for the double ionization channels opening at 39.5 eV. These are expected limitations of the molecular package, which however do not contradict the validity of the presented approach.
The narrow intervals of energies around the oneelectron ionization thresholds (in every channel) are the only problematic regions for this method. Below the threshold, one should see a series of Feshbach resonances converging to the threshold. This is difficult to represent accurately because the highly doublyexcited states responsible for these resonances are increasingly spatially extended requiring a very large Rmatrix sphere to represent them sufficiently accurately. Additionally, the asymptotic expansion of the CoulombHankel functions, written in terms of the argument \(\rho = kr\), becomes inapplicable close to the threshold (from above or from below) unless an extremely large Rmatrix radius is used. Nevertheless, sufficiently far away from the threshold the theory works flawlessly and the results are very satisfying.
In Fig. 3 we compare one, two, three and fourphoton cross sections for ionisation of \(\hbox {H}_2\) by a field polarised parallel to the molecular axis, with up to 3 photon absorptions in the continuum (highenergy section of the last curve). The repeated Feshbach resonance patterns below the thresholds of three, two and onephoton ionisation corresponding to excitation of a metastable state by several photons are well observable. There resonances correspond to intermediate excitations of the neutral molecule into one of the higher lying neutral bound states, ionized by the remaining photons. As before, in the vicinity of the thresholds the present method does not provide good results and the data were omitted from the plots. For this calculation we used static exchange model with the HartreeFock orbitals of the ion \(\hbox {H}_2^+\) built from the atomic Gaussian basis set ccpVDZ, Rmatrix radius \(a = 150\) a.u. and a continuum basis formed of 225 uniformly spaced radial Bsplines and partial wave expansion up to \(\ell = 4\). The positions of the nuclei of both the initial neutral molecule and the residual ion were fixed at the equilibrium internuclear distance of \(\hbox {H}_2\), which is 1.4 atomic units.
Further, in Fig. 4 we present calculated cross sections for twophoton ionization of \(\hbox {H}_2\) employing a full CI model with the augccpVTZ basis set that was used also in^{45} (called “ATZ” there). The polarization of the field was chosen parallel to the molecular axis (Fig. 4a) or perpendicular to it (Fig. 4b). The radius of the inner region was set to \(a = 100\) a.u. in order to converge the results around the coreexcited resonances well. To represent the continuum we included 150 evenly spaced Bsplines of order 6 for partial waves up to \(\ell = 6\). The curve is interrupted at several energies around the one and twophoton ionization thresholds (the latter are not marked), where the asymptotic expansion of Coulomb functions does not converge properly. The transition responsible for a coreexcited resonance occurs within the residual ion rather than in the photoelectron, and so the momenta of the continuum functions in Eq. (29) are then very similar. This leads to problems with the asymptotic integrations, see in particular Eq. (58), where the convergence of higher terms depends also on the assumption that \(k  k'r \gg 1\) in the denominator. However, extending the Rmatrix radius and possibly reducing the number of terms in the asymptotic expansion in Eq. (50) (we used \(P=3\) in this calculation) mitigates this deficiency close to the resonances. Below the onephoton ionization threshold, the cross sections are compared to calculations of Apalategui and Saenz^{41} and Morales et al.^{40} with almost perfect agreement, disregarding small energy (horizontal) shifts arising from different relative energies of the ground state and the residual ion as obtained from the different methods. The only apparent disagreement is visible before the second Feshbach resonance in the first plot (parallel polarization), where the results of Apalategui and Saenz lack of the sharp “shoulder” below 15 eV visible in results of Morales et al. as well as in ours. This region was shown in^{45} to be sensitive to the molecular structure model used in the calculation. As in the case of helium, the description and energies of the higher excitations of \(\hbox {H}_2^+\) manifesting as coreexcited resonances are also most likely limited by the atomic basis set and the resonances might shift to lower energies when a more diffuse basis set is used. As the cross section is dominated by the contributions of the resonances (both coreexcited and Feshbach ones), shifts in their positions translate to significant changes in the overall magnitude of the cross section.
Figure 5a–c shows the molecularorientationaveraged data: the total isotropic cross section \(\sigma _0\) and the asymmetry parameters \(\beta _2\) and \(\beta _4\), respectively, calculated by transforming the matrix elements (7) to the spherical basis and performing averaging over molecular orientations parametrized by Euler angles as in^{42,44}; see Appendix D. Alternatively, the isotropic cross section \(\sigma _0\) can be obtained directly in the real spherical harmonic basis and molecular frame by averaging only over polarizations \(\varvec{\epsilon }\) of the photons and disregarding the dependence on the direction of the photoelectron momentum, by use of the formula^{56,57}
The results below the onephoton ionization thresholds are compared to other published calculations^{39,41,42}. We see a good agreement with the calculation of Apalategui and Saenz^{41}; the small systematic difference in magnitude of the cross section before the first resonance is discussed in Appendix D, where we show that it probably comes from a typo in the codes of Apalategui and Saenz evaluating their formula for the orientational average. We also observe a qualitative agreement with the calculations of Ritchie and McGuire^{39} and Demekhin et al.^{42} Here, the quantitative differences also follow from very different ionization thresholds used; whereas Demekhin et al. employ the experimental value of the adiabatic first ionization threshold between ground vibronic states (15.43 eV), in the present calculation we use our calculated value for the vertical threshold, which—for the employed model—comes out as 16.43 eV. The large 1 eV difference is responsible for most of the relative shift of the resonance features between the datasets. Furthermore, the quantitatively different behaviour is related to the choice of different quantum chemical models: The models used in^{39} and^{42} appear to provide results much closer to the static exchange model discussed earlier, see Fig. 5d–f, where also our calculated onephoton vertical first ionization threshold (15.55 eV) accidentally agrees better with the adiabatic experimental value. However, the more complete full CI model, albeit yielding a different vertical ionization potential, includes additional effects not captured by the previous simpler models.
Multielectron target: \(\hbox {CO}_2\)
Finally, we calculate photoelectron angular distributions for the twophoton ionization of a comparatively larger molecule: \(\hbox {CO}_2\), see Fig. 6. Here we used a highquality electronic model of the molecule introduced in^{47,58}, which was already demonstrated to yield very accurate onephoton cross section. The target model is based on the use of molecular orbitals of the neutral molecule in its equilibrium geometry (C–O distance 1.1621 Å) optimized using the complete active space selfconsistent field method (CASSCF) with ccpVTZ basis. Here we focus on the lowenergy region with photon energies up to 25 eV (i.e. total absorbed energy up to 50 eV) in which the continuum partial wave expansion converged for \(\ell = 6\). Rmatrix radius of only \(a = 10\) a.u. was sufficient and allowed to use Gaussian functions to represent the continuum orbitals. All molecular integrals were computed in quadruple precision using GBTOlib^{48} which allowed to avoid numerical linear dependencies and retain all continuum orbitals in the basis. Convergent singlephoton cross sections were obtained including 300 states of the residual ion in the CloseCoupling expansion (3), see^{58}. The radius of \(a = 10\) a.u. is mostly too small for the asymptotic integrals to be applicable. Therefore the necessary onedimensional outerregion integrations (34) have been performed numerically using the trapezoidal Romberg integration up to the radius \(b = 100\) a.u. and analytically in the asymptotic part. Calculation with \(b = 200\) a.u. was done for a subset of energies to check convergence away from the onephoton thresholds. To allow the intermediate bound states to extend beyond the inner region boundary we included closed channels (12) in the outer region wave function. As in the earlier onephoton calculation^{58}, the energy of the ground state has been manually shifted by \(1.1\) eV to recover the experimental first ionization threshold 13.78 eV^{59}.
In general, the partial isotropic cross sections for ionization into all four presented final states of \(\hbox {CO}_2^+\) exhibit a rapid rise with the photon energy, reaching the maximum one or two eV before the abovethreshold ionization threshold, from which they decrease. Qualitatively, this rise and decrease in the belowthreshold twophoton ionization cross section is somewhat similar to the behaviour of the onephoton photoabsorption cross section measurement^{59} below the onephoton ionization threshold, see Fig. 7. The forest of resonances in the energy range 7–10 eV is associated with the twophoton thresholds of the residual ion states A, B, and C. The prominent pair of narrow isolated resonances that follows, located at approximately 11 eV, corresponds to intermediate excitation of the neutral molecule by the first photon to the lowest dipoleallowed states \({}^1\Sigma _u^+\) and \({}^1\Pi _u\) for parallel and perpendicular orientation of the molecule with respect to the polarization of the ionizing field, respectively. Right after these two resonances, further twophoton ionization channels open and the picture gets more complicated. The cross sections peak around the resonance associated with the excitation to the second \({}^1\Sigma _u^+\) state of neutral \(\hbox {CO}_2\) at 12.4 eV. Beyond the first onephoton threshold the cross sections decrease and exhibit further resonance structures corresponding to higher autoionizing excited states, as well as resonances associated with further onephoton thresholds. In this calculation, 300 ionic states were included in the model, Eq. (3), to correctly describe the polarization effects in the investigated range of energies. This is an order of magnitude more than the handful of states used for full CI calculations with He and \(\hbox {H}_2\), with the first excited final state not even appearing before the ATI region. In the case of \(\hbox {CO}_2\) these twophoton thresholds are much more densely spaced in energy and scattered all over the energy range of interest.
From the computational perspective, the most demanding part of the molecular multiphoton calculations is typically the evaluation of the molecular integrals needed in UKRmol+ to construct the molecular Hamiltonian. Among the discussed calculations, the molecular integrals were the most resourcedemanding in the case of the fullCI model of \(\hbox {H}_2\) with Rmatrix radius of 100 a.u., where their calculation took a little over 50 h on a 40core machine, producing 120 GiB of data. The remaining preparatory steps in UKRmol+ are typically faster; the diagonalization of the Hamiltonian and evaluation of transition dipole elements between its eigenstates for fullCI \(\hbox {H}_2\) took 6 h on the same machine. The evaluation of the twophoton ionization cross sections themselves for all 8000 distinct photon energies plotted in Fig. 5a–c and all \(\sim \)1700 final photoionization channels by the method proposed here amounted only to a little over 1 h on a common 10core workstation.
However, the complexity of the calculation of multiphoton ionization amplitudes discussed in this article is strongly affected by the number of channels, as well as by the abovethreshold photon absorption order. For instance, disregarding the preparatory calculations in UKRmol+, the evaluation of the one/two/three /fourphoton ionization cross sections of \(\hbox {H}_2\) in the static exchange model (with 12 outer region channels) for Fig. 3 took 1 s/1 s/16 s/1.5 h on a 10core workstation, while the calculation of the twophoton ionization cross sections of \(\hbox {CO}_2\) in Fig. 6 (up to \(\sim \) 11,000 channels, 680 distinct photon energies) took around 40 h on a 36core server machine, mostly because of the need for the numerical integration in the outer region due to the use of a very small Rmatrix radius. For \(\hbox {CO}_2\), the UKRmol+ stage was comparatively fast, consisting of 40 minutes for integral calculation in quadruple precision on 36 cores, less than 1 h for Hamiltonian diagonalization and around 2 h for calculation of the Kmatrices needed in the outer region. Generally, it is advisable to extend the Rmatrix radius as much as possible, so that the numerical integration in the outer region is avoided (or at least minimized) and a high resolution in energies becomes computationally cheap. This is feasible in UKRmol+ thanks to the availability of Bspline orbitals for construction of the continuum basis.
Conclusion
In this article we have presented a method for the calculation of multiphoton ionization amplitudes and cross sections for arbitrary photon orders. The method is based on the Rmatrix approach, which accurately solves the timeindependent Schrödinger equation in the close vicinity of the target system (atom, molecule, molecular cluster, ...) and uses an analytic asymptotic expansion of the wave function elsewhere. The last point enables analytic treatment of the multidimensional oscillatory integrals that contribute to photon absorptions in abovethreshold ionization. At the same time, the flexibility of the inner region allows us to treat complex targets using variational quantumchemistry methods of the configurationinteraction type to describe the multielectron dynamics.
Neglecting the channel coupling in the outer region, i.e. the longrange multipole potentials, is possible when the Coulomb interaction of the ejected electron with the residual ion is the dominant interaction. This narrows down the applicability of the method as presented here to situations where the ionized target has a nonzero charge or a negligible dipole moment; however, this is not a serious limitation, because most of the studied targets are neutral and the residual ions are singly charged. In any case the method can be straightforwardly extended to take into account channel coupling in the outer region by means of a multichannel asymptotic expansion of the outgoing solution^{50}. The dipole integrals implied by this ansatz would have the same form as those worked out here.
The three cornerstones of the method are the following:

Embedding of the purely outgoing boundary condition in the Schrödinger equation via Bloch operator (Appendix A).

Reduction of the effective rank of the inhomogeneous system of equations by means of the ShermanMorrisonWoodbury formula (Appendix B).

General algorithm for evaluation of the multidimensional asymptotic CoulombGreen’s integrals (Appendix C).
For accurate results, this method presently requires the inner region to have a large enough size, so that the asymptotic formulas for Coulomb functions and their integrals can be applied. This can pose a problem close to thresholds, where linear momenta in some channels are small, the corresponding channel wave functions oscillate slowly and the asymptotic formulas require large distances to achieve the requested accuracy (or even validity). This can be tackled by further partitioning of the outer region into the “transition” part (say, for radii \(a\) to \(b\)) and the “asymptotic” part (from \(b\) to \(+\infty \)), where in the former one the integrals are evaluated numerically, while in the latter the analytic approach is used. In the presented results we have only used such splitting in the case of the twophoton ionization of \(\hbox {CO}_2\) and aimed at large inner region radii otherwise.
We validated the theory by comparison of the generalized multiphoton cross sections to available published data for atomic hydrogen, helium and molecular hydrogen. We also provide original results for twophoton ionization of \(\hbox {CO}_2\), below and above the ATI threshold. However, photoionization cross sections are not the only domain of applicability of this theory. The access to accurate transition amplitudes for polyatomic molecules can be used to study a range of interference phenomena (e.g. two and multiphoton RABITT, electron vortices) using a timeindependent approach, even though these have been traditionally the domain of timedependent calculations^{23,24,25}, or of approximative calculations that extend asymptotic wave functions all the way to the origin^{21}. Our work opens the way to accurate calculations of multiphoton processes in multielectron molecules. This is the direction of our further research.
Data availability
All data generated during this study are included in this published article and its Supplementary Information files.
References
Sharma, A. et al. Counting the electrons in a multiphoton ionization by elastic scattering of microwaves. Sci. Rep. 8, 2874. https://doi.org/10.1038/s4159801821234y (2018).
Vagin, K. Y. & Uryupin, S. A. Collective modes of plasma formed by multiphoton ionization of rarefied gas. Plasma Sources Sci. Technol. 29, 035005. https://doi.org/10.1088/13616595/ab5e28 (2020).
Sharma, A. et al. Diagnostics of CO concentration in gaseous mixtures at elevated pressures by resonance enhanced multiphoton ionization and microwave scattering. J. Appl. Phys. 128, 141301. https://doi.org/10.1063/5.0024194 (2020).
Boesl, U. et al. Resonanceenhanced multiphoton ionization: A speciesselective ion source for analytical timeofflight mass spectroscopy. Chemosphere 29, 1429–1440. https://doi.org/10.1016/00456535(94)902763 (1994).
Ryszka, M. et al. Dissociative multiphoton ionization of isolated uracil and uraciladenine complexes. Int. J. Mass Spectrom. 396, 48–54. https://doi.org/10.1016/j.ijms.2015.12.006 (2016).
Kastner, A. et al. Intermediate state dependence of the photoelectron circular dichroism of fenchone observed via femtosecond resonanceenhanced multiphoton ionization. J. Chem. Phys. 147, 013926. https://doi.org/10.1063/1.4982614 (2017).
Beaulieu, S. et al. Multiphoton photoelectron circular dichroism of limonene with independent polarization state control of the boundbound and boundcontinuum transitions. J. Chem. Phys. 149, 134301. https://doi.org/10.1063/1.5042533 (2018).
Comby, A. et al. Realtime determination of enantiomeric and isomeric content using photoelectron elliptical dichroism. Nat. Commun. 9, 5212. https://doi.org/10.1038/s41467018076099 (2018).
Goetz, R. E., Koch, C. P. & Greenman, L. Perfect control of photoelectron anisotropy for randomly oriented ensembles of molecules by XUV REMPI and polarization shaping. J. Chem. Phys. 151, 074106. https://doi.org/10.1063/1.5111362 (2019).
Lux, C. et al. Circular dichroism in the photoelectron angular distributions of camphor and fenchone from multiphoton ionization with femtosecond laser pulses. Angew. Chem. Int. Ed. 51, 5001–5005. https://doi.org/10.1002/anie.201109035 (2012).
Zvorykin, V. D., Shutov, A. V. & Ustinovskii, N. N. Review of nonlinear effects under TWpower PS pulses amplification in GARPUNMTW Ti:sapphireKrF laser facility. Matter Radiat. Extremes 5, 045401. https://doi.org/10.1063/5.0004130 (2020).
Shutov, A. V. et al. Major pathway for multiphoton air ionization at 248nm laser wavelength. Appl. Phys. Lett. 111, 224104. https://doi.org/10.1063/1.5006939 (2017).
Dharmadhikari, A. K., Dharmadhikari, J. A. & Mathur, D. Visualization of focusing–refocusing cycles during filamentation in BaF2. Appl. Phys. B 94, 259. https://doi.org/10.1007/s0034000833177 (2008).
Smetanin, I. V., Levchenko, A. O., Shutov, A. V., Ustinovskii, N. N. & Zvorykin, V. D. Role of coherent resonant nonlinear processes in the ultrashort KrF laser pulse propagation and filamentation in air. Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. Atoms 369, 87–91. https://doi.org/10.1016/j.nimb.2015.10.032 (2016).
Koga, J. K. et al. Simulation and experiments of the laser induced breakdown of air for femtosecond to nanosecond order pulses. J. Phys. D Appl. Phys. 43, 025204. https://doi.org/10.1088/00223727/43/2/025204 (2009).
Plummer, M. & Noble, C. J. Resonant enhancement of harmonic generation in argon at 248 nm. J. Phys. B At. Mol. Opt. Phys. 35, L51–L58. https://doi.org/10.1088/09534075/35/2/101 (2002).
Ackermann, P., Münch, H. & Halfmann, T. Resonantlyenhanced harmonic generation in argon. Opt. Express 20, 13824–13832. https://doi.org/10.1364/OE.20.013824 (2012).
Ivanov, I. A. & Kheifets, A. S. Resonant enhancement of generation of harmonics. Phys. Rev. A 78, 053406. https://doi.org/10.1103/PhysRevA.78.053406 (2008).
Brown, A. C., Hutchinson, S., Lysaght, M. A. & van der Hart, H. W. Interference between competing pathways in atomic harmonic generation. Phys. Rev. Lett. 108, 063006. https://doi.org/10.1103/PhysRevLett.108.063006 (2012).
Champenois, E. G. et al. Involvement of a lowlying Rydberg state in the ultrafast relaxation dynamics of ethylene. J. Chem. Phys. 144, 014303. https://doi.org/10.1063/1.4939220 (2016).
Dahlström, J. M. et al. Theory of attosecond delays in laserassisted photoionization. Chem. Phys. 414, 53–64. https://doi.org/10.1016/j.chemphys.2012.01.017 (2013).
Dahlström, J. M., L’Huillier, A. & Maquet, A. Introduction to attosecond delays in photoionization. J. Phys. B At. Mol. Opt. Phys. 45, 183001. https://doi.org/10.1088/09534075/45/18/183001 (2012).
Vos, J. et al. Orientationdependent stereo Wigner time delay and electron localization in a small molecule. Science 360, 1326–1330. https://doi.org/10.1126/science.aao4731 (2018).
Serov, V. V. & Kheifets, A. S. Time delay in XUV/IR photoionization of \(\text{ H}_2\)O. J. Chem. Phys. 147, 204303. https://doi.org/10.1063/1.4993493 (2017).
Bharti, D. et al. Decomposition of the transition phase in multisideband schemes for reconstruction of attosecond beating by interference of twophoton transitions. Phys. Rev. A 103, 022834. https://doi.org/10.1103/PhysRevA.103.022834 (2021).
Ngoko Djiokap, J. M. et al. Electron vortices in photoionization by circularly polarized attosecond pulses. Phys. Rev. Lett. 115, 113004. https://doi.org/10.1103/PhysRevLett.115.113004 (2015).
Pengel, D., Kerbstadt, S., Englert, L., Bayer, T. & Wollenhaupt, M. Control of threedimensional electron vortices from femtosecond multiphoton ionization. Phys. Rev. A 96, 043426. https://doi.org/10.1103/PhysRevA.96.043426 (2017).
Armstrong, G. S. J. et al. Modeling tomographic measurements of photoelectron vortices in counterrotating circularly polarized laser pulses. Phys. Rev. A 100, 063416. https://doi.org/10.1103/PhysRevA.100.063416 (2019).
Kerbstadt, S., Eickhoff, K., Bayer, T. & Wollenhaupt, M. Odd electron wave packets from cycloidal ultrashort laser fields. Nat. Commun. 10, 658. https://doi.org/10.1038/s41467019086017 (2019).
Boll, D. I. R., Fojón, A. O., McCurdy, C. W. & Palacios, A. Angularly resolved twophoton abovethreshold ionization of helium. Phys. Rev. A 99, 023416. https://doi.org/10.1103/PhysRevA.99.023416 (2019).
Shakeshaft, R. Twophoton single and double ionization of helium. Phys. Rev. A 76, 063405. https://doi.org/10.1103/PhysRevA.76.063405 (2007).
Feng, L. & van der Hart, H. W. Twophoton double ionization of He. J. Phys. B At. Mol. Opt. Phys. 36, L1–L7. https://doi.org/10.1088/09534075/36/1/101 (2003).
Sánchez, I., Bachau, H. & Cormier, E. Theory of twophoton spectroscopy of autoionizing states in helium and beryllium. J. Phys. B At. Mol. Opt. Phys. 28, 2367–2384. https://doi.org/10.1088/09534075/28/12/007 (1995).
Eberly, J. H., Javanainen, J. & Rzążevski, K. Abovethreshold ionisation. Phys. Rep. 204, 331–383. https://doi.org/10.1088/01430807/9/4/004 (1991).
Manakov, N. L., Marmo, S. I. & Sviridov, S. A. Method for calculating multiphoton abovethreshold processes in atoms: Twophoton abovethreshold ionization. J. Exp. Theor. Phys. 108, 557–570. https://doi.org/10.1134/S1063776109040025 (2009).
Burke, P. G., Colgan, J., Glass, D. H. & Higgins, K. RmatrixFloquet theory of molecular multiphoton processes. J. Phys. B At. Mol. Opt. Phys. 33, 143. https://doi.org/10.1088/09534075/33/2/302 (2000).
Burke, P. G., Francken, P. & Joachain, C. J. RmatrixFloquet theory of multiphoton processes. Europhys. Lett. 13, 617–622. https://doi.org/10.1209/02955075/13/7/008 (1990).
Colgan, J., Glass, D. H., Higgins, K. & Burke, P. G. Rmatrix Floquet theory of molecular multiphoton processes: II. Multiphoton ionization of H2. J. Phys. B At. Mol. Opt. Phys. 34, 2089–2106. https://doi.org/10.1088/09534075/34/11/303 (2001).
Ritchie, B. & McGuire, E. J. Twophoton ionization of \(\text{ H}_2\). Phys. Rev. A 24, 2532–2540. https://doi.org/10.1103/PhysRevA.24.2532 (1981).
Morales, F., Martín, F., Horner, D. A., Rescigno, T. N. & McCurdy, C. W. Twophoton double ionization of \(\text{ H}_2\) at 30eV using exterior complex scaling. J. Phys. B At. Mol. Opt. Phys. 42, 134013. https://doi.org/10.1088/09534075/42/13/134013 (2009).
Apalategui, A. & Saenz, A. Multiphoton ionization of the hydrogen molecule \(\text{ H}_2\). J. Phys. B At. Mol. Opt. Phys. 35, 1909–1928. https://doi.org/10.1088/09534075/35/8/309 (2002).
Demekhin, P. V., Lagutin, B. M. & Petrov, I. D. Theoretical study of angularresolved twophoton ionization of H\({}_{2}\). Phys. Rev. A 85, 023416. https://doi.org/10.1103/PhysRevA.85.023416 (2012).
Larsen, K. A. et al. Role of dipoleforbidden autoionizing resonances in nonresonant onecolor twophoton single ionization of \(\text{ N}_2\). Phys. Rev. A 102, 063118. https://doi.org/10.1103/PhysRevA.102.063118 (2020).
Toffoli, D. & Decleva, P. Density functional theory for molecular multiphoton ionization in the perturbative regime. J. Chem. Phys. 137, 134103. https://doi.org/10.1063/1.4754820 (2012).
Benda, J. et al. Perturbative and nonperturbative photoionization of \(\text{ H}_2\) and \(\text{ H}_2\)O using the molecular “Rmatrix with time’’ method. Phys. Rev. A 102, 052826. https://doi.org/10.1103/PhysRevA.102.052826 (2020).
Joachain, C. J. RmatrixFloquet theory of multiphoton processes: Concepts, results and perspectives. J. Mod. Opt. 54, 1859–1882. https://doi.org/10.1080/09500340701308724 (2007).
Harvey, A., Brambila, D. S., Morales, F. & Smirnova, O. An Rmatrix approach to electronphotonmolecule collisions: Photoelectron angular distributions from aligned molecules. J. Phys. B At. Mol. Opt. Phys. 47, 215005. https://doi.org/10.1088/09534075/47/21/215005 (2014).
Mašín, Z., Benda, J., Gorfinkiel, J. D., Harvey, A. G. & Tennyson, J. UKRmol+: A suite for modelling electronic processes in molecules interacting with electrons, positrons and photons using the Rmatrix method. Comput. Phys. Commun. 249, 107092. https://doi.org/10.1016/j.cpc.2019.107092 (2020).
Brown, A. C. et al. RMT: Rmatrix with timedependence. Solving the semirelativistic, timedependent Schrödinger equation for general, multielectron atoms and molecules in intense, ultrashort, arbitrarily polarized laser pulses. Comput. Phys. Commun. 250, 107062. https://doi.org/10.1016/j.cpc.2019.107062 (2020).
Burke, P. G. RMatrix Theory of Atomic Collisions (Springer, 2011).
Drake, G. W. (ed.) Springer Handbook of Atomic, Molecular, and Optical Physics (Springer, 2006).
Olver, F. W. J. et al. NIST Digital Library of Mathematical Functions (2020). Release 1.0.26 of 15th March 2020.
Klarsfeld, S. Twophoton ionization of atomic hydrogen in the ground state. Lettere al Nuovo Cimento 3, 395–398. https://doi.org/10.1007/BF02819081 (1970).
Karule, E. Twophoton ionisation of atomic hydrogen simultaneously with onephoton ionisation. J. Phys. B At. Mol. Phys. 11, 441–447. https://doi.org/10.1088/00223700/11/3/015 (1978).
Smith, D. G. A. et al. PSI4 1.4: Opensource software for highthroughput quantum chemistry. J. Chem. Phys. 152, 184108. https://doi.org/10.1063/5.0006002 (2020).
Zamastil, J. & Benda, J. Quantum Mechanics and Electrodynamics (Springer, 2017).
Andrews, D. L. & Thirunamachandran, T. On threedimensional rotational averages. J. Chem. Phys. 67, 5026–5033. https://doi.org/10.1063/1.434725 (1977).
Mašín, Z. et al. Electron correlations and precollision in the recollision picture of high harmonic generation. J. Phys. B At. Mol. Opt. Phys. 51, 134006. https://doi.org/10.1088/13616455/aac598 (2018).
Chan, W. F., Cooper, G. & Brion, C. E. The electronic spectrum of carbon dioxide. Discrete and continuum photoabsorption oscillator strengths (6–203 eV). Chem. Phys. 178, 401–413. https://doi.org/10.1016/03010104(93)85079N (1993).
Homeier, H. H. H. & Steinborn, E. O. Some properties of the coupling coefficients of real spherical harmonics and their relation to Gaunt coefficients. J. Mol. Struct. THEOCHEM 368, 31–37. https://doi.org/10.1016/S01661280(96)90531X (1996).
Aymar, M. & Crance, M. Twophoton ionisation of atomic hydrogen in the presence of onephoton ionisation. J. Phys. B At. Mol. Opt. Phys. 13, L287–L292. https://doi.org/10.1088/00223700/13/9/002 (1980).
Cormier, E., Bachau, H. & Zhang, J. Discretization techniques applied to the study of multiphoton excitation of resonances in helium. J. Phys. B At. Mol. Opt. Phys. 26, 4449–4463. https://doi.org/10.1088/09534075/26/23/016 (1993).
Maquet, A., Véniard, V. & Marian, T. A. The Coulomb Green’s function and multiphoton calculations. J. Phys. B At. Mol. Opt. Phys. 31, 3743–3764. https://doi.org/10.1088/09534075/31/17/004 (1998).
Higham, N. J. Accuracy and Stability of Numerical Algorithms (Society for Industrial and Applied Mathematics, 2002).
Gao, B. & Starace, A. Numerical methods for freefree radiative transition matrix elements. Comput. Phys. 1, 70. https://doi.org/10.1063/1.4903436 (1987).
Acknowledgements
The authors thank Dr. Thomas Meltzer for careful reading of the manuscript and for helpful suggestions. Z.M. and J.B. acknowledge support of the Czech Science Foundation as the project GA CR 2015548Y and support of the PRIMUS programme of the Charles University as the project PRIMUS/20/SCI/003. Some computational resources were supplied by the project “eInfrastruktura CZ” (eINFRA LM2018140) provided within the program Projects of Large Research, Development and Innovations Infrastructures. This work was also supported by The Ministry of Education, Youth and Sports from the Large Infrastructures for Research, Experimental Development and Innovations project “IT4Innovations National Supercomputing Center–LM2015070”.
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J.B. independently developed the theory, implemented it into the UKRmol+ code, ran all calculations and wrote the first draft of the manuscript. Z.M. supervised the work and wrote parts of the manuscript.
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Supplementary Information
Appendices
Appendices
A Multiphoton ionization in Rmatrix approach
A.1 Incorporation of the boundary condition
In the following, the channel angular momentum numbers \(l_{p}m_{p}\) will be collectively denoted by the index p so the CoulombHankel function and the real spherical harmonic become \(H_p^+(kr)\) and \(X_p\) respectively. We will not discuss the closed channels explicitly. The decreasing Whittaker function \(W_p\) is simply the CoulombHankel function \(H_p^+\) with positiveimaginary momentum and has the same asymptotic expansion as \(H_p^+\). In the following text we do not distinguish these two functions and “open channels” can be understood to also include all weakly closed channels, whose penetration into the outer region needs to be considered to compensate for the finite inner region.
To embed the boundary condition (11) into the Schrödinger equation we proceed in the following way. First, we introduce the standard Bloch operator (5) into the equation for the intermediate state by adding it to and subtracting from the Hamiltonian,
We expand the wave function of the intermediate state in the Rmatrix eigenstates \(\psi _m\) of the hermitian operator \({\hat{H}} + {\hat{L}}\) in the inner region,
and project the Eq. (14) on \(\psi _m\), resulting in
where the matrix elements of the Bloch operator between the inner region eigenstates \(\psi _m\) and \(\psi _n\) can be expressed as
by means of the boundary amplitudes \(w_{pm}\) and radial derivatives \(w_{pn}'\) of those eigenstates when projected on the oneelectron outer channels,
Now, we can take advantage of the partially known asymptotic form (10) of the soughtafter wave function. The boundary amplitude and the derivative of the radial channel wave function computed from the innerregion solution (15) are
Comparing these expressions with (11) and eliminating the unknown expansion coefficient \(a_p\) then yields
where the boundary logarithmic derivative \(\Lambda _p(a)\) stands for
Substituting (20) into the Schrödinger equation (16) leads to the final form
This equation provides the correct inner region expansion coefficients of (15) compatible with the boundary condition (10). The outer region expansion coefficients \(a_p\) can be evaluated from (19) and (11) as
The solution of Eq. (22) can be optimized using the ShermanMorrisonWoodbury formula. For details see Appendix B.
A.2 Freefree matrix elements
The final (detector) wave function can be expressed in a similar way as the intermediate states. In the inner region we can write the partial wave expansion of (2),
where the partialwave photoionization coefficients \(A_{pk}^{()}\) follow from the onephoton ionization theory^{47}, \(\sigma _p = \arg \Gamma (\ell _p + 1  {\mathrm i}Z/k_f)\) is the Coulomb phase shift and \(\varvec{k}_f\) the final momentum of the photoelectron after ionization into the residual state \(\Phi _f\). In the outer region it is
where the radial function has the form^{50}
and \(S_{qp}\) is the element of the Smatrix coupling the energyaccessible outer channels \(q\) and \(p\). In the case of absorption of only one photon in the continuum (e.g. absorption of the second, IR photon in the RABITT scheme) the evaluation of the dipole matrix element (9) can be split into inner region and outer region contributions,
The symbol \(D_{cqk}\) is the matrix element of the component \(c\) of the dipole operator between the states \(\Phi _q\) and \(\Phi _k\) of the residual ion and partial waves \(X_q\) and \(X_k\) of the ejected electron,
The symbol \(G_{l_1 m_1 l_2 m_2}^{l_3 m_3}\) denotes the real Gaunt’s coefficient (integral over three real spherical harmonics)^{60}. In Eq. (30) we have used the definition of the dipole matrix operator \({\hat{D}}_{c}\) and its separation into the ionic and oneelectron parts:
To evaluate the contribution to the dipole matrix element from the outer region we need the integrals
where \(s_{1}\) and \(s_{2}\) stand for any combination of \(+\) and −. These integrals can be evaluated for large values of \(a\) by replacing the CoulombHankel functions by their asymptotic series, using the method of repeated integration by parts^{61,62}. This method is explained in detail in Appendix C.
A.3 Higherorder abovethreshold ionization
When we go to higher orders (or energies) and it becomes possible that one or more photons leading to a given intermediate state are absorbed in the continuum, the righthand side of Eq. (14) is nonzero in the outer region or even polynomially diverges with radius. This behaviour is transferred also to the solution itself, see Fig. 8. The method then needs a generalization. Now, there is a source term both in the inner region and in the outer region. The value of the wave function at the Rmatrix region boundary is a combination of these two sources. The inner region source, as before, generates an outgoing CoulombHankel wave in each accessible channel. In addition, given that the outer region channels are not coupled, the contribution from the outer region to channel \(p\) can be directly obtained employing the CoulombGreen’s function
where \(F_{p}(k_{p}r)\) is the regular Coulomb function^{52}. The Eq. (35) is only valid for open channels. For energyforbidden channels \(p\) one should use the negativeenergy Green’s function involving the real Whittaker functions^{63}. In this work, though, we disregard contribution of closed channels to the 3photon (and higher) ionization and limit ourselves to large Rmatrix radii instead in such cases. For this reason we do not use the negativeenergy Green’s function, even though the extension is straightforward.
Combining the two contributions to the outer wave function gives
At the inner region boundary we then have
where
Evaluation of this integral is discussed in Appendix C. At the same time, the channel function \(f_p\) and its derivative need to satisfy Eq. (19), leading to a generalization of (20),
The additional term that reflects the effect of the outer region source in the inner region does not depend on the sought expansion coefficients \(c_n\) and hence can be transferred to the righthand side, yielding the final, fundamental equation of this method
The radial integration in \(\langle \psi _m  {\hat{D}}_j  \Psi _{i + (j1)\omega }^{(+)} \rangle \) is limited to the inner region since the state \(\psi _m\) is restricted to it. The additional term on the righthand side compensates for this truncation. Given the updated boundary formula (37), the formula for the outer region expansion coefficients has to be generalized to
Together with (36), we now have the complete prescription for the intermediate state (given by \(c_n\) and \(a_p\)) in the whole configuration space that can be used in the evaluation of the final transition dipole elements, or as a seed for intermediate states of even higher order. Of course, when the righthand side of (14) is limited to the inner region, then \(\beta _p = 0\) and the equations in A.3 simplify to their restricted forms introduced earlier in A.1.
Evaluation of \(d_{\text {out},p}\) (as well as of \(\beta _p\), which has the same form) gets progressively more and more complicated with the increasing order, requiring integrals of dimension increasing with the number of photons absorbed in the continuum, e.g. for 3 and 4 photons absorbed above the onephoton ionization threshold:
etc. For simplicity, the radial and momentumdependent arguments have been suppressed, but it should be clear that the Green’s function separates parts of the expression that depend on adjacent \(r_i\) and \(r_{i+1}\). These nested integrals arise because the highorder intermediate states entering the formulas (9) and (39) no longer have the simple asymptotic proportional to \(a_p H_p^+\) and contain, as the abovethreshold absorption order increases, an increasing number of integrals over the Green’s function, see Eq. (36). Evaluation of the required multidimensional integrals is discussed in Appendix C.
The overall asymptotic complexity of the method is exponential, \(O(\exp N)\), in the number of absorptions below the ionisation threshold due to the exponentially rising number of possible combinations of absorbed photon polarizations, and even factorial, \(O(M!)\), in the number of absorptions in the continuum, above the ionisation threshold, as discussed in Appendix C. However, as the practical applications of perturbative theory are generally limited to low orders anyway, the rapidly rising complexity for highorder abovethreshold ionisation is not particularly worrying.
B Efficient solution of “diagonal plus low rank” equation
In the standard Rmatrix theory of collisions the inversion of the innerregion Hamiltonian, Eq. (4), needs to be performed only once. However, since equation (22) needs to be solved for each photon energy independently, it pays off to implement this solution efficiently. As written, the matrix of the set of equations has rank equal to the number of inner region eigenstates, which may be a very large number \(N\). Solution of such equation has the asymptotic complexity \(O(N^3)\). Fortunately, it has a very specific form, sometimes called “diagonal plus low rank” matrix, where the low rank refers to the second term of the matrix in Eq. (22) being constructed from boundary amplitude matrices, which have one of the dimensions equal to the number of photoionization channels \(n\), while typically \(n \ll N\). In this case, one can make use of the ShermanMorrisonWoodbury formula^{64},
which allows expressing the solution of the \(N\)by\(N\) system using the solution of an \(n\)by\(n\) one, where \(N\) is the number of inner region eigenstates and \(n\) is the number of the outer region channels. In this case the solution is
where \({\mathbf {P}} = (E_{\text {tot}}  {\mathbf {E}})^{1}\) is a diagonal \(N\)by\(N\) matrix. Note that the rank of the inner inversion in (46) needs to be limited to the number of open channels \(n_o \le n\). Now the most timeconsuming operation becomes the construction of the matrix \({\mathbf {w}} {\mathbf {P}} {\mathbf {w}}^\top \), consisting of \(O(n_o^2 N)\) multiplications, which for \(n_{o} < N\) outperforms the \(O(N^3)\) operations required to solve the original system, particularly for a small number of open channels.
C Evaluation of multidimensional CoulombGreen dipole integrals
Multidimensional integrals of products of coordinate powers, CoulombHankel functions and CoulombGreen’s functions from a large radius \(a\) to infinity in each coordinate can be expressed as a sum of “triangular” integrals over \(a< r_1< r_2< \dots< r_M < +\infty \), in order to fix the assignment of the coordinates in the Green’s function (35). Then, the regular Coulomb functions \(F_l\) are rewritten using CoulombHankel functions \(H_l^\pm \):
All CoulombHankel functions are in turn expanded using the asymptotic series^{52}
where \(a = 1 + l + {\mathrm i} s\eta \), \(b = l + {\mathrm i} s\eta \), \(\eta = Z / k\), \(\rho = kr\), \(\phi _l(\eta ,\rho ) = \rho  \eta \log (2\rho )  \pi l / 2 + \sigma \), \(\sigma =\arg \Gamma (l+1+{\mathrm i}\eta )\) and \(s\) is either \(+1\) or \(1\). As a result, the final integrand in (39) will become a product of oscillatory exponentials and (possibly negative) coordinate powers. In general, such integral does not have a well defined value. We circumvent this by introduction of an auxiliary damping factor \(\exp (cr_i)\) with positive constant \(c\) into the integrand for each coordinate \(r_i\) as suggested in^{65}. Following the integration we perform the limit \(c \rightarrow 0+\); the value of the integral stays finite and welldefined. All formulas in this section are to be understood in this limit. The prototypical onedimensional integral
with \(\theta (r) = s_1\theta _1(r) + s_2\theta _2(r)\), where \(\theta _i(r) = k_i r + Z\log (2k_ir)/k_i  \pi l_i / 2 + \sigma _i\), can be evaluated by repeated integration by parts (limited to \(P\) iterations) as done in^{61,62}, leading to
where \(\gamma (r) = {\mathrm i}/\theta '(r)\) and the terms \(T_t(r)\) satisfy the recursive definition
The primes indicate derivatives with respect to \(r\). For evaluation of the terms \(T_t\), the derivatives of \(f(r)\) at \(r = a\) need to be known, which is easy for the onedimensional case (49), where \(f(r)\) is simply a (small, or even negative) power of \(r\). The sum in (50) is asymptotic series similar to (48); higher terms are proportional to high derivatives of negative powers of the coordinate and to high derivatives of the phase \(\theta (r)\) at large distances, which rapidly decrease in magnitude with each new term, provided that the evaluation radius \(r = a\) is sufficiently large. For modest radii, though, considering fewer terms leads to better results.
Nested “triangular” integrals of the type
arising in absorption of at least two photons in the continuum can be calculated similarly by recursion, based on the observation that
and so for a twodimensional integral
That is, the integrand at any level has the same form “function times exponential” and the same approach can be used to evaluate the integral as for lower orders. For the secondorder integral, the only additional information needed to evaluate it is the knowledge of the value and derivatives of \({\tilde{f}}_2 = f_2 g_1\) at \(r = a\), which can be precalculated before evaluating the second integral. The need of \(P\) derivatives of \({\tilde{f}}_2\) translates, via the chain rule, to the need to know \(P\) derivatives of \(g_1\), which in turn just raises the number of derivatives of \(f_1\) that need to be precalculated beforehand, because derivatives of the terms \(T_t\) are then needed, too,
The algorithm for evaluation of the nested “triangular” exponential integral of arbitrary dimension \(N\) then begins by calculation of derivatives of \(f_i(r)\) up to some high order dependent on \(M\) and \(P\), followed by recurrent evaluation of terms \(T_t\) and the integrals \(I\) (and their needed derivatives) for subsequently higher dimensions.
Note that, as would be expected from the multidimensional nature of the problem, while the asymptotic computational complexity for calculation of a single “triangular” integral using this approach is manageable \(O(M^3 P^2)\), the computational complexity of the whole algorithm is asymptotically proportional to \(M!\). This scaling is a result of the \(r_1< r_2< ... < r_M\) unique orderings (permutations) are required. The corresponding number of independent triangular integrals to be calculated is proportional to \(2^M\) for all splittings of the regular Coulomb function \(F_l\) into the CoulombHankel functions \(H_l^\pm \), and also up to \(n^M\) (\(n\) being the typical number of channels in an irreducible representation) for evaluation of a multidimensional integral for each possible (dipole coupled) sequence of intermediate state outer region channels. Recall that \(M\) refers to the number of photons absorbed in the continuum. Consequently, the problem is computationally tractable only for modest abovethreshold ionisation orders.
When the Rmatrix radius \(r = a\) is too small for the asymptotic procedures to be valid, it is possible to employ numerical integration between \(r = a\) and some sufficiently large \(r = b\) that is suitable for application of the asymptotic methods. The numerical integration typically needs to be employed for slowly oscillating functions only. Outside of the \((a,b)^M\) hypercube, e.g. in \((a,b)^{M1} \times (b,+\infty )\), it can be combined with the analytic formulas.
Even though they are not necessary—as we have just outlined a general algorithm—for convenience and illustration we include below the explicit forms of the one and twodimensional integrals. The onedimensional case is
where \(u = s_1 k_1 + s_2 k_2\) and \(v = Z(s_1/k_1 + s_2/k_2)\). The coefficient \(a_{mpq}\) is defined recursively as
and \(a_{mpq} = 0\) for invalid indices (\(p < 0\), \(q < 0\) or \(q > p\)). For two dimensions, where additionally \(\phi (r) = s_3\theta _3(r) + s_4\theta _4(r)\), \({\tilde{\gamma }}(r) = {\mathrm i}/(\phi '(r) + \theta '(r))\), \({\tilde{u}} = s_1 k_1 + s_2 k_2 + s_3 k_3 + s_4 k_4\), \({\tilde{v}} = Z(s_1/k_1 + s_2/k_2 + s_3/k_3 + s_4/k_4)\) and \({\tilde{m}} = m_1 + m_2\), we have
with the recursive definition of the constant \(b_{{\tilde{m}}pqstt'}\)
with the initial condition \(b_{{\tilde{m}}pq000} = 1\), and \(b_{{\tilde{m}}pqstt'} = 0\) whenever \(t < 0\), \(t > s\), \(t' < 0\) or \(t' > s\).
D Orientation averaging
The laboratoryframe differential cross section of ionization of a sample of randomly oriented molecules by absorption of \(n\) photons with given laboratoryframe polarizations \(p_1, \dots , p_n\) (\(p_{i}\) is equal to 0 for linear polarization or \(\pm 1\) for a circular polarization) is
and has the form of a Legendre expansion
where \(\sigma _0 = b_0\) is the isotropic generalized integral cross section, \(\beta _L = b_L/\sigma _0\) are the dimensionless asymmetry parameters, and the quantities \(b_L\) are
Here, \(M_{fi, l_f m_f, m_n \dots m_1}\) are terms of the partial wave expansion of (7); that is, \(l_f\) and \(m_f\) are the molecularframe angular quantum numbers of the partial wave, while \(m_1, \dots , m_n\) label the components of the dipole operator in spherical basis in the molecular frame. The expression for \(b_L\) was obtained by: expressing the molecularframe expansion of \(M_{fi}\) in terms of the laboratoryframe quantum numbers \(m_f'\) and \(p_1, \dots , p_n\),
expressing the product \(Y_{l_f m_f'} Y_{\lambda _f \mu _f'}^*\) that arises in Eq. (64) as a series in \(Y_{LM}\); recursively combining the rotational matrices associated with a partial wave with those associated with the dipole operator; and eventually using the orthogonality of the rotational matrices.
In the main text we mention a discrepancy in Fig. 5a between the orientationaveraged isotropic cross section for twophoton ionization of \(\hbox {H}_2\) calculated in this work and the results of Apalategui and Saenz^{41}. The authors of the latter article break down the total cross section into components \(\sigma _a, \sigma _b, \dots , \sigma _g\), some of which they explicitly plot. In Fig. 9 we reproduce those components and demonstrate that if we flip the sign of the interference term \(\sigma _f\) with respect to the definition (7f) in the reference, we obtain their results (as far as our calculated transition matrix elements permit); cf. thin broken blue curve to thick solid blue curve in Fig. 9. With the exact definition of \(\sigma _f\) in the article, however, we obtain our results; the two red curves in Fig. 9 lie on top of each other. Based on this observation, together with the fact that we obtained our results also in Cartesian basis via independent formula (13), which does not treat \(\sigma _f\) separately, we conclude that the authors of the earlier calculation used an incorrect sign of \(\sigma _f\) when evaluating the final orientation averaged cross section and that the agreement between the calculations is then actually much better than Fig. 5a suggests. Nevertheless, the formula (7) of their paper appears to be correct.
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Benda, J., Mašín, Z. Multiphoton above threshold ionization of multielectron atoms and molecules using the Rmatrix approach. Sci Rep 11, 11686 (2021). https://doi.org/10.1038/s4159802189733z
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DOI: https://doi.org/10.1038/s4159802189733z
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