Topical Review: Extracting Molecular Frame Photoionization Dynamics from Experimental Data

herein are categorized by reduced χ̄ 2 values ( χ̄ 2 = χ 2 /( N − N v a r s ), where N is the number of data points, and N v a r s the number of parameters) unless otherwise noted. • Further details of the codebase and numerical implementation can be found in Sect. [sec:numerical-notes] . This methodology amounts to a statistical sampling of the solution hyperspace, which may be expected to contain some local minima in general high-dimensional cases. Running in parallel on 20 (logical) cores of an AMD Threadripper 2950X based workstation, this required 5GB of RAM and took on the order of 2 hours (note that the time per fit cycle had large variance, since convergence time depends on the start parameters); further benchmarks for the current codebase can be found in the PEMtk documentation . Bootstrapping basics: fit results

are summarised broadly in Fig. [509194], which shows the best fit results over the fits. A few (tentative) conclusions can be quickly drawn from this: 1. The results appear to indicate a well-defined, single global minima was found, with χ̄ 2 = 2.291 × 10 −3, and 203 fits (20%) obtained this value. 2. There are additional close-lying bands with χ̄ 2 < 2.3 × 10 −3 . In the test case, 349 fits (35%) obtained this region of the fitting space. These results may be due to local minima, or simply fits which didn’t quite converge for numerical reasons (e.g. reached the upper limit of iterations in fitting protocol); further exploration to identify whether these sub-sets of parameters are significantly different is therefore required. 3. There are multiple bands of fits which converged to higher χ̄ 2 ; since there is a large step to the first of these ( χ̄ 2 = 2.312 × 10 −3 ), and they appear with much lower frequency, these are most likely indicative of local minima. Bootstrapping basics: retrieved parameter sets & fidelity To drill down into the results, the values and clustering of the retrieved parameter sets can be investigated, as well as the overall quality of the fitting. Fig. [494229] illustrates the spread in retrieved parameters over the best fit region, and Fig. [743962] shows the overall quality of the fits in this region. For the purposes of discussion, each band observed in Fig. [509194] will be described as a χ̄ 2 cluster or group, assumed to equate to a viable set of retrieved parameters, and the best such set should equate to the "true" values, i.e. the photoionization matrix elements, if the methodology is successful and there is a well-defined global minima. Again, some tentative conclusions can be drawn: 1. The results for the magnitudes show very little difference within each χ̄ 2 cluster, and closer inspection reveals that values between clusters are typically within 5%, with the exception of the |3, ±1⟩ case which shows a larger spread. This indicates a good agreement in all results. 2. The phase results broadly fall into two classes, (a) cases which show good agreement within a χ̄ 2 cluster, and (b) cases which have two values per χ̄ 2 cluster. Additionally there is a larger spread in the values between clusters in phase space, and this is again most apparent for the |3, ±1⟩ case, which spans the full − π ≤ ϕ ≤ π phase range. Note that there is no spread in the ϕ 1, −1 (phase) parameter, this was fixed as a reference phase in the current case, with ϕ = −0.861. 3. The well-defined grouping here indicates that the retrieved parameter sets are unique for each χ̄ 2 cluster, and each case can be regarded as a distinct, viable, parameter set to be futher investigated; a priori it is expected that the lowest χ̄ 2 should be the true result, but in practice this may not always hold depending on the data. 4. The parameters with split/pairs of values are indicative of cases where there is an insensitivity in the fit. In this case, the |3, ±1⟩ case shows an insensitivity to the sign of the phase, which appears as a splitting due to the mod ( π ) nature of the result. There is a slight bias in the results towards one of the pair in each case, though this may be a statistical artiefact. (This type of effect will be discussed further below.) To further investigate the parameter sets, a correlation pair/matrix representation can be used. For the phases, an example is shown in Fig. [888108] . In this plot each parameter is shown as a function of all other parameters for each set (fit), and coloured by χ̄ 2 . This results in a rather complex, but informative, representation. In this example the phases were “corrected", with the reference value set to zero in order to illustrate general trends (i.e. as would typically be the case when an absolute reference value is not known a priori), which results in a zero spread for this parameter (top left panel in Fig. [888108] ). Additionally, the ranges are wrapped to mod ( π ), keeping positive values only, thus collapsing the ± pairs observed in Fig. [494229] for ϕ 3, ±1 . The bottom row and final column on the right of Fig. [888108] show patterns of the parameters with χ̄ 2 value. Of particular note here are the curves visible in most cases: these indicate the curvature of the χ̄ 2 hypersurface with respect to the given coordinate (parameter), where a steeper curve indicates how well defined a given parameter is. The remaining correlation plots indicate the spread of individual parameters (diagonal panels), and parameter-paramter correlations (off-diagonal panels). In general good clustering is observed for the lowest χ̄ 2 values, consistent with the expectations from Fig. [494229], with more spread apparent in higher-valued results. For the phases with ± pair splittings - for example the 2nd column in Fig. [888108] - characteristic V shapes are observed, fanning out from the best-fit cluster as a function of χ̄ 2 ; these correspond to cases where the ± pairs are not wrapped (corrected) to the same value, hence indicate larger uncertainties in these parameter sets. Some outlier parameter sets can also be seen in this plot, for instance the single points visible to the right of the panels in the 2nd column of Fig. [888108], which are correlated with larger χ̄ 2 values (bottom panel of the column). To investigate the fidelity of the results in this test case the best parameter set(s) can be compared with the input matrix elements, and results are presented in Table [tab:matE] . In this comparison, uncertainties on the retrieved parameters were estimated statistically, based on the standard deviation in the best fit results for the lowest χ̄ 2 group; in this case it is of note that no “experimental" uncertainties were included in the fitting, so the estimates herein indicate purely the error in the retrieval procedure with noisy data (without noise, essentially perfect results are found in this case). • The final results are generally quite close to the inputs. In this case, the 10% random noise essentially translates to a similar retrieval uncertainty in the magnitudes, although there is also a notable decrease in accuracy with the |3, ±1⟩ case, and significant increase in accuracy for the σ u cases. • The absolute phase values appear to be quite far off in some cases, but the phase corrected values typically look quite reasonable; this is consistent with a lack of sensitivity in the test dataset to the sign of the phases (as discussed elsewhere), but the remapped phases (0 : π ) are generally precise. Additionally, the “absolute" phases given are averaged over all ± pairs, hence end up quite close to zero, whilst the “corrected" phases are remapped before averaging. In general the former is, of course, not desirable, but serves to illustrated the type of issue that may arise in a retrieval protocol. • With the exception of |3, 0⟩ and |1 − 1⟩, all remapped parameters are within 10% of the reference values. • For the ϕ 3, 0 it appears that the value is not well-defined in the data, and is different from the reference value by ∼ π /4. • For ϕ 1 − 1 the absolute error is, indeed, small, but since this is close to zero, and should be defined as zero in the phase corrected case, the percentage error appears large, although the absolute value is close to the true value. • The doubly-degenerate | π u , l, ±1⟩ cases are found to have approximately the same magnitudes and phases in the free fit. In this case this is expected from the input values, hence indicates a good fit result. In general, if known a priori, such constraints can also be included in the fitting protocol, and should result in faster and more precise results where there are significant symmetry constraints on the results. • The differences between the data and reference values are much larger than the standard deviation of the fits. This is indicative of a good (close/singular) batch of fits, with a unique global minimum, but reveals that the results obtained are not perfect - as expected for noisey data. Adding more data-points to the fit, and/or using higher fidelity data would help in this case. • Alternative uncertainty estimations can be obtained from the individual parameter set results, based on the curvature of the χ 2 hyperspace w.r.t. each parameter, or w.r.t. to all other parameters . In testing, only the first approach was considered, making use of values returned by the lmfit library routines from inversion of the Hessian matrix (see Sect. [sec:numerical-notes] ); however, this value was not found to be useful in many cases here, with relative errors into the thousands of %, likely due to the strongly-correlated nature of the fit. The 2nd approach has been used previously, but is significantly more time-consuming (each test value necessitates a refitting of all other parameters), hence statistical uncertainties may represent the best approach for fast and robust estimations. • As noted above, although noise was added to the simulated data prior to fitting, experimental/data uncertainties were not included in the fitting. In general these may be expected to be relatively small for β parameters obtained via imaging-type experiments with good S:N (for instance, the results of Refs. ), but may remain significant for absolute count-rates (i.e. β 0, 0 ), and consequently are expected to map to larger uncertainties in the absolute magnitudes. However, fitting to count-rate normalised data (angular distributions only) can mitigate this effect, as was explored in the original demonstration of the technique, which returned accurate relative magnitudes and MFPADs for the X -state with experimental uncertainities typically < 10%. | comp | m | p | labels | |||| | Cont | l | m | μ | |||| | PU | 1 | -1 | 1 | 1.163-1.354j | 1.785 | -0.861 | 1,-1 | | 1 | -1 | 1.163-1.354j | 1.785 | -0.861 | 1,1 | || | 3 | -1 | 1 | -0.803-0.017j | 0.803 | -3.120 | 3,-1 | | | 1 | -1 | -0.803-0.017j | 0.803 | -3.120 | 3,1 | || | SU | 1 | 0 | 0 | -2.317+1.359j | 2.686 | 2.611 | 1,0 | | 3 | 0 | 0 | 1.106-0.087j | 1.109 | -0.079 | 3,0 | | Type | m | p | pc | labels | || | Cont | l | m | |||| | PU | 1 | -1 | 1.614 | -0.859 | 0.000 | 1,-1 | | 1 | 1.614 | -0.861 | 0.005 | 1,1 | || | 3 | -1 | 1.143 | 0.144 | 2.059 | 3,-1 | | | 1 | 1.142 | 0.143 | 2.059 | 3,1 | || | SU | 1 | 0 | 2.653 | 2.357 | 2.987 | 1,0 | | 3 | 0 | 1.144 | -0.785 | 0.154 | 3,0 | | l,m | 1,-1 | 1,1 | 3,-1 | 3,1 | 1,0 | 3,0 | || | Type | Source | dType | |||||| | m | mean | num | 1.614 | 1.614 | 1.143 | 1.142 | 2.653 | 1.144 | | ref | num | 1.785 | 1.785 | 0.803 | 0.803 | 2.686 | 1.109 | | | diff | % | 10.581 | 10.600 | 29.733 | 29.715 | 1.264 | 3.030 | | | num | -0.171 | -0.171 | 0.340 | 0.339 | -0.034 | 0.035 | || | std | % | 0.129 | 0.136 | 0.267 | 0.265 | 0.002 | 0.007 | | | num | 0.002 | 0.002 | 0.003 | 0.003 | 0.000 | 0.000 | || | diff/std | % | 8190.232 | 7766.406 | 11153.884 | 11193.713 | 79089.050 | 44706.225 | | | p | mean | num | -0.859 | -0.861 | 0.144 | 0.143 | 2.357 | -0.785 | | ref | num | -0.861 | -0.861 | -3.120 | -3.120 | 2.611 | -0.079 | | | diff | % | 0.193 | 0.000 | 2264.893 | 2274.558 | 10.797 | 89.979 | | | num | 0.002 | 0.000 | 3.265 | 3.264 | -0.254 | -0.706 | || | std | % | 1.636 | 0.000 | 1249.292 | 1255.151 | 5.709 | 17.101 | | | num | 0.014 | 0.000 | 1.801 | 1.801 | 0.135 | 0.134 | || | diff/std | % | 11.803 | inf | 181.294 | 181.218 | 189.126 | 526.167 | | | pc | mean | num | 0.000 | 0.005 | 2.059 | 2.059 | 2.987 | 0.154 | | ref | num | 0.000 | 0.000 | 2.259 | 2.259 | 2.811 | 0.782 | | | diff | % | nan | 100.000 | 9.740 | 9.737 | 5.893 | 407.124 | | | num | 0.000 | 0.005 | -0.201 | -0.200 | 0.176 | -0.628 | || | std | % | nan | 243.707 | 0.257 | 0.461 | 0.227 | 4.433 | | | num | 0.000 | 0.013 | 0.005 | 0.010 | 0.007 | 0.007 | || | diff/std | % | nan | 41.033 | 3791.665 | 2109.989 | 2599.980 | 9183.815 | Bootstrapping basics: density matrix representation An alternative test of fidelity can be investigated via a density matrix representation of the results, this is shown in Fig. [998904] . These plots show the real and imaginary components of the density matrices for the ionization continuum (as defined in Sect. [sec:density-mat-basic], Eqn. [eqn:radial-density-mat] ), normalised to the maximum (complex) value. Overall the agreement is good for the real values (right column of Fig. [998904] ), as expected from the values in Table [tab:matE], and the differences (bottom row) are typically < 10 %. Similar results are observed for the imaginary values (left column), which are shown for the unsigned phase case only (corresponding to phases set ‘pc’ as per Table [tab:matE] ). For the signed phase case (not shown, but corresponding to phases set ‘p’ as per Table [tab:matE] ) the loss of signs leads to larger differences in the off-diagonal density terms, and the possibility of phase flips, which can result in inverted patterns visible in the imaginary part of the density matrix. However, as discussed elsewhere (e.g. Sect. [sec:bootstrap-fidelity] ), symmetry constraints may render the continuum insensitive to the sign of the phases, and the retrieved matrix elements may therefore still present a high-fidelity reconstruction of the continuum. Perhaps more interesting/useful in the density matrix representation is the visualisation of the phase relations between the photoionization matrix elements (the off-diagonal density matrix elements), and the ability to quickly check the overall pattern of the elements, hence confirm that no phase-relations are missing and orthogonality relations are fulfilled, or that very different patterns/sets of matrix elements were retrieved. Furthermore, the density matrix elements also provide a complete description of the photoionization event, and makes clear the equivalence of the “complete" photoionization experiments (and associated continuum reconstruction methods) with quantum tomography schemes . It can be used as the starting point for further analysis based on density matrix techniques - this is discussed, for instance, in Ref., and can also be viewed as a bridge between traditional methods in spectroscopy and AMO physics, and more recent concepts in the quantum information sciences (e.g. Refs. ). Bootstrapping basics: obtaining MFPADs With the matrix elements to hand, the MFPADs can also be computed, for any arbitrary molecular alignment and polarization. Some examples are shown in Fig. [454268], which shows results for the two sets of retrieved matrix elements discussed above (raw, and with phase-correction), along with the reference computational results (as used to originally generate the simulated data) and the differences between the phase-corrected case and the reference results. A number of points are illustrated by the figure: • The original/input matrix elements produce the MFPADs shown in the top row of the figure, which provide a reference for the fidelity of the reconstructed MFPADs. • Polarization geometries are illustrated for four cases (see Fig. [781808] for the MF reference coords), labelled z, x, y, d, where z is parallel to the bond axis, x, y perpendicular and d “diagonal" relative to the z -axis ( θ z = π /4). In this case symmetry dictates that z ( σ u continuum) is cylindrically symmetric, the x, y cases are a conjugate π u pair, and that the diagonal case mixes these components, hence breaks the symmetry and is sensitive to the relative phase between the continua. • The 2nd row illustrates the results from the phase-corrected parameters. These show generally good agreement with the reference case, apart from slightly exaggerated off-diagonal lobes for the x, y results, consistent with the larger magnitudes for |3, ±1⟩ in the reconstructed matrix elements. • The 3rd row shows difference plots between the reference and phase-corrected results. This emphasizes the differences in the |3, ±1⟩ component magnitudes. The z case shows very good agreement with the reference case (note the smaller magnitudes), whilst the d case indicates some differences, again attributable to the differences in |3, ±1⟩ component magnitudes. • The bottom row illustrates the case of the phase-averaged (but not “corrected") parameters. This serves to highlight the sensitivity of the PADs to the relative phases of the matrix elements - here the x, y results show poor agreement with the reference case due to averaging over the ± ϕ pairs, and this also affects the d case. Overall, with careful treatment of the phases, it is clear that high-fidelity MFPADs can be recovered in the current case. Although the difference plots indicate some differences between the reconstructed and reference case, the absolute changes in the form of the MFPADs are fairly minor. The lack of an absolute phase is not an issue in general for MFPAD recovery, although this does constitute a loss of relative phase in the continuum wavefunctions as a function of energy. Further bootstrapping: information content & sensitivity As well as considering the results from full fits of the data, the inherent sensitivity of various aspects of the problem can also be investigated. In general, this will depend on the details of the problem at hand (symmetry, ADMs etc.), but can in essence be considered independently of the matrix elements themselves via “channel functions" or equivalent (see Sect. [sec:channel-funcs] and Sects. [appendix:formalism], [sec:channel-funs-full] for full definitions). In the PEMtk routines, the various component tensors are computed and packaged as a basis set prior to fitting, and can be further examined independently. (For full details see the PEMtk docs .) Fig. [776753] illustrates the geometric coupling term B L, M (Eqn. [eq:BLM-func-defn] ), hence the sensitivity of different (L,M) terms to the matrix element products. This term incorporates the coupling of the partial wave pairs, | l, m ⟩ and | l ′, m ′⟩, into the term B L, M , where { L, M } are observable total angular momenta, hence indicates which terms are allowed for a given set of partial waves - in the current test case, l = 1, 3 only (as defined by the known matrix elements). This is essentially a way to visualize the general selection rules into the observable: for instance, only terms l = l ′ and m = − m ′ contribute to the overall photoinoization cross-section term ( L = 0, M = 0). However, since these terms are fairly simply followed algebraically in this case, via the rules inherent in the 3j product, this is not particularly insightful. These visualizations will become more useful when dealing with real sets of matrix elements, and specific polarization geometries, which will further modulate the B L, M terms. For example, in the AF M is further restricted by M = S − R p (Eqn. [eq:delta-func-defn] ); in the current case with S = 0, R p = 0 (this is typically the case for a cylindrically-symmetric experimental configuration), only M = 0 terms will contribute. Fig. [652406] is more complicated, and illustrates the tensor product Λ R ⊗ E P R ( ê ) ⊗ Δ L, M ( K, Q, S ) ⊗ A Q, S K ( t ), expanded over all quantum numbers. This term, therefore, incorporates all of the dependence (or response) of the AF- β L M s on the polarisation state, and the axis distribution. In this case, it’s clear that there’s a significant response to the alignment in the L = 0, 2 terms, some response in L = 4 and - for the most part - no significant contribution from higher-order terms (threshold of 0.01), for the selected set of ADMs. This visualisation is potentially useful for planning measurements sensitive to certain properties, for example, in this case the L = 6 term is significant only over a small t -range, so this region could be targeted experimentally to obtain data more sensitive to higher-order l -wave term couplings (per Fig. [776753] ). Conversely, the L = 0, 2 response terms are quite symmetric over the half-revival, so making experimental measurements at t -points symmetrically over this feature will provide redundant, but not additional, information content to the dataset for matrix-element retrieval. Also of note is the opposite response of the μ = μ ′ = 0 terms and μ ≠ μ ′ (with μ = ±1, μ ′ = ∓1) to the alignment, indicating different response and sensitivities in the polarization projections. This is particularly apparent in the L = 6 term, where the μ ≠ μ ′ terms drop below threshold. Finally, Fig. [676540] illustrates the full AF channel (response) function 𝛶 L, M u, ζ ζ ′ (Eqns. [eqn:channel-fns], [eq:channel-func-defn-AF] ), which is essentially the complete geometric basis set, hence equivalent to the AF- β L M (Eqn. [eq:BLM-tensor-AF] ) if the ionization matrix elements were set to unity. This illustrates not only the coupling of the geometric terms into the observable L, M, but also how the partial wave | l, m ⟩ terms map to the observables. A few observations in this case: • The largest reponse is in the total cross-section ( β 0, 0 ), which is 2 times larger than any other term; • this response is similar for both the l = 1 and l = 3 contributions, and all m, hence we expect similar sensitivity to both partial cross-sections (continua) in this case. • For L > 0, l ≠ l ′ are present, indicative of the sensitivity of the PADs to cross-terms (interferences). • The response is, as might be expected, distinct for terms with l = l ′ and l ≠ l ′, and as a function of L, l, l ′. • The parameters indicate enhanced sensitivity to higher-order terms ( L = 6) just after the main revival feature. This was already apparent in Fig. [652406], and is a consequence of the larger K = 4, 6 terms in the ADMs in this region (see Fig. [720080] ). In the current case this is not particularly important for the matrix element retrieval, but in general may be significant in more complex cases with larger l present. These results demonstrate the high level of detail that can be obtained from a set of channel functions, and how such a basis set can aid in both planning and interpretation of experimental measurements in terms of the sensitivity to given channels. Many outstanding questions remain, in particular general metrics of total information content are not yet defined, nor are general methods for determination of a sufficient information content for retrieval in a given case. Such exploration is currently a topic of ongoing research, although it is also of note that some authors have previously explored related issues (see, for example, Refs. ). Further bootstrapping: sub-sample size & fidelity In cases with noisy data, and/or where uncertainties in fitted parameters remain large, fitting to larger and/or alternative sub-sets of the data may be pursued. An obvious route to improvement of the retrieved matrix elements is via the inclusion of additional data points. In the original experimental demonstration, after an initial 10-point fit, further data was included, up to 89 data points, to minimise uncertainties in the retrieved matrix elements. 2 Another, similar, route is via traditional statistical bootstrapping, in which fitting is tested for various randomly-selected sub-sets of the data (but the size of the fitted data set is not varied). Additionally, if the channel functions (fitting basis set) are investigated (see Sect. [sec:bootstrapping-info-sensitivity] ), additional data points may be selected to enhance the sensitivity to certain partial wave components. In the current case, work is ongoing, but some effort has been made to investigate sampling strategies. In particular, preliminary work using different sample sizes plus additional statistical weightings (statistical bootstrapping) has yielded very promising results. For example, a dataset of 30 t-points equally spaced over the range 3.5 - 4.5 ps (as compared to 4 - 5 ps in the previous demonstration above), with random (Poissionian) statistical weightings additionally applied, yielded: • 30 % convergence (best χ 2 ) in 100 test fits. • Standard deviations on the retrieved parameters of < 10 −4 (cf. Table [tab:matE] ). • Fideleities on retrieved parameters typically < 10%. However, whilst this appears promising, it is of note that the trends in the fidelity of the retrieved matrix element remained similar to those shown in Table [tab:matE], with the l = 3 cases less well-defined. This indicates that a more fundamental information-content limitation remained. Additionally, other tests with the same initial dataset but different statistical weightings did not yield the same high percentage of converged fits with low standard deviation (hence retrieval quality), indicating that more careful and methodical work is required here, perhaps with more sophisticated statistical techniques applied. Bootstrapping technique outlook & future directions In the current case, it is clear that the bootstrapping methodology for obtaining full photoionization matrix elements from time-domain, AF datasets, works well. The current numerical routines in the PEMtk package (and back-end libraries) are relatively stable and fast, and amenable to detailed inspection. However, a range of outstanding questions and routes of investigation remain, for instance: • Scaling to larger problems (larger molecules, more matrix elements). This is currently under investigation, but - based on previous work - it is anticipated that small polyatomic molecules should be tractable to the basic approach. • More sophisticated approaches, for instance careful sub-selection of data based on the channel functions, data obtained for additional polarization geometries or with shaped laser pulses . Such approaches may be expected to yield higher fidelity reconstructions for small polyatomic systems, and may be required for more complex cases. • Faster numerical routines, in particular via GPU-based numerics. • Implementation of additional fitting routines, both from standard numerical methods, and also from related specialised domain problems, for example phase-retrieval methods developed for optical interferograms and spectrograms (e.g. general FROG-type retrieval methods, ptychography/holographic techniques ) and homotopy approaches may be applicable. • Correlations and overlaps with other related/emerging photoionization techniques may also prove fruitful. For instance, photoionization matrix elements are of interest in high-harmonic spectroscopy schemes, angle-resolved RABBITT and general attosecond “clocking" and time-delay measurement techniques. In many cases these methods are also directly sensitive to the energy-dependence of the matrix elements, thus providing additional information relative to a basic 1-photon ionization study (albeit with additional complexity), and have recently been of great theoretical interest in the ab initio photoionization community . Machine learning (ML), and particularly recent “deep learning" techniques (often also generically referred to as AI), are currently in vogue and evolving rapidly. Such approaches may also present interesting opportunities for MF retrieval problems. Perhaps the main strength of these methods is that, in favourable cases, very complex problems may be treated without complete computation and/or understanding of the underlying physics (e.g. the recent success of DeepMind/AlphaFold in protein folding ); the use of ML/AI may, therefore, be particularly interesting for cases which are otherwise intractable to a full ab initio analysis, e.g. complicated molecular dynamics or strongly-coupled light-matter systems, where the step-wise and separable approaches detailed herein will currently fail, and closed-form equations/solutions do not exist. A secondary use may be as fast fitting algorithms for cases of the type discussed herein, where the physics is understood, albeit complicated. This is much less interesting scientifically since it does not present a “new" capability, although may still prove fruitful if the increase in speed (or robustness) is significant compared to current methods, e.g. to allow for real-time analysis during experimental runs. Furthermore, use in this vein, but with the aim of solving a fully coupled problem without the necessity of a bootstrapping or multi-step type of analysis (i.e. combining the various stages illustrated in Figs. [807606], [731792] into a single ML-based routine) is certainly worth pursuing, with the potential to create a method that simultaneously retrieves both the alignment and photoelectron properties from experimental data without additional researcher intervention. The difficulty in all cases will likely be the generation of a suitable data set for the ML/AI training procedure, which is required before the routines can be deployed on new problems. Significant introductory and general discussion to the (rapidly evolving) topic can be found online and, for example, in Refs. . Some recent use of ML/AI in the AMO context has proved successful, e.g. Refs. tackle specific data-analysis and reconstruction type problems; Refs. examine ML for ab initio computation of potential energy surfaces; Ref. provides a broad review of ML in the physical sciences, including quantum state reconstruction, as well as particle physics, cosmology and materials science. MF reconstruction via matrix inversion The key aspects of this method involve first fitting out the ADMs A Q S K ( t ) from measured β̄ L M u ( t ) in Eq. [eqn:beta-convolution-C] to retrieve the C̄ K Q S L M (the vector C l a b ), and construction of the matrix G L ′ M ′ K S L M P Δ q which facilitates the extraction of the molecular frame coefficients C P Δ q L M (the vector C m o l ). The steps in the process are depicted in Fig. [731792], which label’s primary inputs as the ‘AF/LF measurements’, referring to the β̄ L M u ( t ), and the ADMs A Q S K ( t ) calculated for a grid of fluence and temperature values by solving the TDSE using the appropriate Hamiltonian for impulsive alignment . The ‘Alignment Retrieval’ process requires solving the linear equations Eq. [eqn:beta-convolution-C] for the C̄ K Q S L M using these primary inputs over the entire gird of ADMs, and comparing to the measured data to find the bets fit solution. This provides the vector C l a b needed for the next step - solving the linear equations Eq. [eq:basic] . This also requires that the matrix G L ′ M ′ K S L M P Δ q be invertable. This matrix therefore encodes the uniquely accessible MF information in the experiment. Specifically, rows in the matrix link the LF parameter β̄ L M u ( t ) to an MF beta parameter β L M ( θ, χ ) that can be used to construct the MFPADs, θ and χ being polar and azimuthal angles of the laser polarization in the MF. Specifically, a particular row, or set of rows, being entirely zero may indicate that certain MF β L M are inaccessible by the set of LF measurements. For instance, in the G L ′ M ′ K S L M P Δ q matrix for the N 2 ( X 1 Σ g + ) → N 2 + ( X 2 Σ g + ) ionization of N 2 with a single photon, from a RWP excited by a linearly polarized pulse was computed. The results are reproduced herein for reference, see Fig. [931809], and this can be considered as an indicator of sensitivity in a somewhat analogous fashion to the channel functions shown in Figs. [776753] - [676540] . Rows with M = 1 were all determined to be identically zero, therefore only allowing the retrieval of MFPADs with the ionizing field parallel and perpendicular to the molecular axis. In this case, the partial wave analysis involved in the previously described method, along with the non-linear fitting necessarry to retireve the radial dipole matrix elements is avoided. However, only two of the MFPADs shown in Fig. [731792] extracted by bootstrapping are retrieved by this method. The utility of this method is better demonstrated by application to a polyatomic molecule of D n h symmetry. Gregory et al. also compute the G L ′ M ′ K S L M P Δ q matrix for C 2 H 4 ( 1 A g ) → C 2 H 4 + ( 2 B 3 u ) ionization of C 2 H 4, with D 2 h symmetry, from a RWP excited by linearly polarized light. In contrast with N 2, all rows of the G L ′ M ′ K S L M P Δ q are non-zero for C 2 H 4, but the system of linear equations is inconsistent since there are more MF parameters (in the vector C m o l ) than LF measurements (in the vector C l a b ). Nonetheless, the solution selected by Gregory et al. minimizes the retrieved vector C m o l and correctly reproduce the MFPADs for any orientation. The basic ( x, y, z ) cases are reproduced herein as Fig. [584598], indicating the high fidelity of the reconstructions (for arbitrary polarization geometries see Fig. 5 of Ref. ). This ‘extra’ information available in the LF measurement for C 2 H 4 compared to N 2 can be traced back to the fact that the RWP is 2D for an asymmetric top excited by linearly polaized light(cf. Eq. [eq:mfrealsig] ). While this method does not directly provide radial dipole matrix elements, the MFPADs produced can potentially be used as a constraint, along with additional experimental data, to determine these from a non-linear fit (see Sect. [sec:recon-from-MFPADs] ). Matrix element retrieval from MFPADs To complete the circle, one can also consider whether MFPADs - either directly measured or reconstructed via matrix inversion - contain sufficient information to retrieve the underlying matrix elements. The former case has already been the subject of several studies, for instance refs., and shown to work well for at least homo and hetero nuclear diatomics. In a sense the latter case can be considered as (yet another) bootstrapping-type scheme - MFPADs are obtained via a minimum effort/information route, then leveraged to obtain underlying properties. Fitting for MFPADs is currently implemented in the PEMtk suite in a similar manner to the AF fitting detailed in Sect. [sec:bootstrapping], based on the tensor formalism detailed in Sect. [sec:tensor-formulation] . An extensive numerical example is not presented herein, but in testing for the N 2 example case, e.g. using MFPADs as shown in Fig. [454268], matrix element retrieval was found to be possible in general. However, in line with previous observations for other methods (and the underlying physics), the fidelity and retrievable relations again depend on the exact nature and quality - generally the information content - of the MFPAD dataset (or, equivalently, the set of measured β L, M ). In testing, a number of different datasets were trialed, and some notes and observations are given below. MFPADs were tested at single ϵ only, as a function of polarization geometry, cf. the MFPADs as shown in Fig. [454268] . Data for fitting therefore consisted of parameters β L, M ( ϵ, R n̂ ), with L m a x = 6. For fitting intensity-normalised MFPADs (i.e. β 0, 0 = 1), for single or multiple polarization geometries from the ( x, y, z ) set: • This was found to be sufficient to determine the corresponding continuum matrix elements in this case, i.e. z polarization corresponds to the σ u continuum, and x, y to the π u continuum. (This separation may not be so clean in other cases, depending on the symmetry of the system.) • Matrix elements were constrained as expected, with only relative magnitudes and phases obtained (per continuum) in this manner. • Similarly, due to a lack of cross-terms between continua in this case, the phase relations between the σ u and π u continua were also undefined in this case. • In line with the AF reconstruction procedure, the sign of the phases was also undefined in this case. • Similarly, the fidelity on the retrieved matrix elements (for defined relations), was on the order of the uncertainties in the fitted dataset. For determination of additional matrix element relations, additional data can be incorporated in the fitting. • Incorporating absolute magnitudes ( β 0, 0 = σ ) allows for the relative magnitudes of the continua to be determined. • Incorporating additional suitable interferences, e.g. diagonal polarization (cf. Fig. [454268] ), in the dataset allows for additional relative phase relationships to be determined, e.g. between the σ u and π u continua. Retrieval of matrix elements vs. ϵ is also possible; however, it is again constrained by the presence (or otherwise) of interferences. In the basic case, each energy is treated independently, and relative phases as a function of energy are undefined. However, these may be approximately defined by imposing an energy constraint, e.g. defining that the phases are smooth vs. energy or follow a certain functional form (see, for an example, Ref. ). In general, an MF fitting procedure, whether from directly measured MFPADs, or reconstructed MFPADs, may be expected to work generally in principle, with similar caveats to the full AF-bootstrapping methodology of Sect. [sec:bootstrapping] (and the existing literature, see Sect. [sec:CompleteLit] ). Both methodologies are constrained by the symmetry of the system, and general information content. Notable differences are the implicit presence of cross-terms between continua in the AF case (Eqn. [eq:BLM-tensor-AF] ), which may be missing in the MF case (Eqn. [eq:BLM-tensor-MF] ), depending on the choice of polarization geometries. Outstanding questions remain similar for both cases, namely the fidelity of reconstruction, and the ability to scale-up to more complex cases (larger molecules, dynamic systems) and concomitant information content requirements, and are a subject of ongoing research. Summary & Outlook From this topical review, it is hoped that the reader has gained a solid grounding in photoionization physics, and general reconstruction methods, for both MF observable reconstruction and full matrix element retrieval (“complete" photoionization or quantum tomography treatments), and that a suitable toolkit and platform for interested researchers has been developed. Over the last decade or two, such experiments, as well as theoretical treatments and unified data-analysis techniques have developed significantly, and become more accessible to non-specialists. It is our hope that the new platform and tools discussed herein (Sect. [sec:resources] ) will prove useful, and help photoionization (and related) studies involving MF and full matrix element retrieval to (continue to) become more routine and successful. Whilst significant new work has been presented herein, development work is ongoing, with aims to make the code-base more robust and user-friendly, test new cases - particularly for more complex molecular systems, and dynamical cases - and implement new features, such as metrics for information content for a given basis set that will aid experimental planning, and further density-matrix based analysis methods. As well as publication in the literature, new developments and results will also be made available via the open-source platform, and it is hoped that other researchers will also contribute to grow these efforts over time. Meanwhile, experimental methodologies continue to improve and grow in sophistication, allowing more routes, and often more direct routes, to high information-content observables, whether in the lab or molecular frame, in a range of interactions and scenarios, including for larger molecules and probing dynamical effects. Recent work in this vein from the current authors and coworkers includes (atomic) matrix element retrieval from photoelectron imaging measurements with polarization multiplexing (via shaped laser pulses), (hyperfine) quantum beat spectroscopy from time-resolved photoelectron imaging experiments, and quantum tomographic determination of (LF) density matrices including electronic dynamics - theory and application to N H 3 (manuscript in preparation). These types of investigations indicate the potential for further general developments, and the utility of the MF retrieval and reconstruction techniques discussed herein as a foundation for a broader range of problems in molecular photoionization and dynamics, ultimately building to a class of molecular quantum state retrieval methods from photoelectron measurements .

The authors would like to thank the JPB editors - particularly A. Stolow - for inviting this submission, their encouragement, and their extreme patience during its (rather lengthy) preparation. Special thanks also to R. Boll, D. Rolles and A. Rudenko for comments & discussion on an early version of the manuscript, and technical notes on the state-of-the-art in coincidence measurements; M. Schuurman for general discussion; the referees for close-reading of the manuscript, and helpful suggestions for improvements and additional references. Resources The following online resources are available, and it is hoped that interested readers will find these useful, and contribute. Manuscipt text, live on Authorea (includes interactive figures) and Github . Post-publication reader comments & suggestions can be added directly via the web. Full bibliography, via a Zotero group “Molecular Frame PADs Measurements and Reconstruction" . Additional references and suggestions can be added to the group, and it is hoped that this will grow to provide a useful community resource. Code for AF computation and matrix element retrieval, as illustrated in Sect. [sec:Recon] . • Full Jupyter notebooks and source data are available on Figshare, DOI: 10.6084/m9.figshare.20293782 . This also includes a complete archive of the python software releases and environment used during this work .

forum at AMO Open Science . We hope interested readers will use this forum for general discussion on the topic, if only to suggest other venues for discussion. A Docker-based distribution of various codes for tackling photoionization problems is also available from the Open Photoionization Docker Stacks project, which aims to make these tools more accessible to interested researchers . Appendix A - Further reading The following sections provide further reading on specific topics for interested readers. In line with the aims of this manuscript, these sections are (very) far from comprehensive, but aim to provide a jumping-off point for the topics; it is hoped that a more comprehensive, crowd-sourced and ongoing listing will be generated via the “Molecular Frame PADs Measurements and Reconstruction" online bibliography on Zotero . MF experimental literature sample Reviews covering this class of experiment can be found in ; some (representative) examples from the literature include: • MFPADs for N2 core ionization • MFPADs from various diatomics with coincidence velocity-map imaging • MFPADs from CO K-shell photoionization and matrix element retrieval . • MFPADs and matrix element retrieval for N2 • MFPADs and matrix element retrieval for NO inner-valence ionization (this makes use of a general F L N ( θ e ) parametrized treatment of the MFPADs (see also ref. ), which provides a suitable reduction method for experimental data analysis for RF and MFPADs, and the potential for matrix element retrieval, in a somewhat similar manner to the tensor approach discussed herein). • NO2 RFPADs . • NO MFPADs (1s channels) . • Interference and entanglement in H2 double ionization in the MF . • General discussion of multi-particle imaging with COLTRIMS . • Imaging methane MFPADs with electron-ion-ion coincidences . • MFPADs from naphthalene ( C 10 H 8 ) via alignment and tomographic imaging . • MFPADs from H2 and D2 at 30eV photon energies (HHG source) . • MFPADs via alignment for Auger-Meitner electrons . • MFPADs from core ionization of carbon tetrafluoride ( C F 4 ), ethane ( C 2 H 6 ) and 1,1-difluoroethylene ( C 2 H 2 F 2 ). Theory literature sample A number of authors have developed relevant theory for photoionization problems. Whilst the fundamentals are similar, different formalisms have been derived for various specific cases, or to emphasize particular aspects of the problem. A few notable examples are listed here. • Early derivations for atomic PADs • Early derivations for molecular (LF)PADs, rotationally-resolved cases, and optically-active cases • Angular momentum transfer in LF and MFPADs • General MFPAD (“fixed-molecule") derivation • Atomic PADs and spin polarization • Resonant multiphoton PADs • Molecular orbital continuum decomposition • A simple-model description for MFPADs, including F L N function decomposition • Matrix element retrieval from time-resolved photoelectron imaging experiments . • Time-resolved PADs and dynamics (including molecular alignment) • LF, RF and MFPADs theory and determination (review article) • Photoionization with a focus on alignment/RWP effects, including numerical studies, and MFPAD case-study . • LF and MFPADs theory review and tutorials (with numerical examples) Complete experiments literature sample The topic of complete photoionization experiments has been recently reviewed in, see also older review articles . Some representative and noteworthy examples are also listed below (see also Sect. [appendix:MF-expt] above for cases involving MF measurements). • First experimental demonstration for photoionization of an atomic target by Berry and coworkers, who studied N a ( 2 P 1/2, 2 P 3/2 ). • First molecular demonstration for N O, from the Zare group, with state-resolved LF measurements including linear and circularly polarized fields (see also prior work ). • General theoretical discussion on complete experiments in atoms and molecules . • Time-resolved rotational-wavepacket methods applied to N O (narrow wavepacket), see also Refs. (review) and (theory & analysis). • Non-linear polyatomic demonstration with state-resolved LF measurements ( N H 3 ) . • Theoretical investigation of matrix element retrieval for photoelectron imaging experiments (see also for rotational wavepacket reconstruction from complementary methods including high-harmonic generation). • Complete experiments with polarization shaping . • Time-resolved rotational-wavepacket method applied to N 2 (broad wavepacket) and bootstrap technique development (as per Sect. [sec:bootstrapping] herein.) Appendix B - Further formalism A complete accounting of the formalism used herein is given in the following sections, expanding on the brief introduction of Sect. [sec:tensor-formulation], along with some additional notes for interested readers. Full Photoionization Formalism The equations for the β L M parameters, in the molecular and lab frames (MF & AF respectively), written in terms of geometric tensor parameters, are given in Sect. [sec:full-tensor-expansion] . The necessary tensor components are further detailed below. For further discussion and derivations see Refs., and Ref. for a summary including various different formalisms; further examples can also be found in the ePSproc documentation . For general discussion on tensor methods in atomic and molecular spectroscopy and related problems, see Refs. . Note, also, that several equivalent formalisms have been presented in the literature (Sect. [sec:theory-lit] ), often specialised to a particular problem or choice of phase conventions. In the following the equations are general, although the phase conventions are chosen to match those used in ePolyScat (Sect. [sec:mat-ele-conventions] ), hence the numerics for the analysis presented herein. In practice the ePSproc codes allow the user to set these conventions as desired. Electric field term $$E_{P,R}(\hat{e})=[e\otimes e^{*}]_{R}^{P}=[P]^{\frac{1}{2}}\sum_{p}(-1)^{R}\left(\begin{array}{ccc} 1 & 1 & P\\ p & R-p & -R \end{array}\right)e_{p}e_{R-p}^{*} \label{eq:EPR-defn-1}$$ Where e p and e R − p define the field strengths for the polarizations p and R − p, which are coupled into the spherical tensor E P R . For the simplest case of a linearly-polarized field, p = 0, and only terms P = 0, 2 with R = 0 (i.e. E 0, 0 and E 2, 0 ) are non-zero. Note this notation implicitly describes only the time-independent photon angular momentum coupling, but time-dependent/shaped laser fields can be readily incorporated by allowing for time-dependent fields e p ( t ) (see, for instance, Ref. ). term The coupling of the partial wave pairs, | l, m ⟩ and | l ′, m ′⟩, into the observable set of { L, M } is defined by a tensor contraction with two 3j terms . $$B_{L,M}=(-1)^{m}\left(\frac{(2l+1)(2l'+1)(2L+1)}{4\pi}\right)^{1/2}\left(\begin{array}{ccc} l & l' & L\\ 0 & 0 & 0 \end{array}\right)\left(\begin{array}{ccc} l & l' & L\\ -m & m' & M \end{array}\right) \label{eq:BLM-func-defn}$$ Note for the AF case the terms may be reindexed by M = S − R ′ (see below). This allows for all MF projections to contribute, rather than a single specified polarization geometry. Term A general geometric projection term can be defined in the MF and AF. For the MF projection term, Λ R ′, R ( R n̂ ): $$\Lambda_{R',R}(R_{\hat{n}})=(-1)^{(R')}\left(\begin{array}{ccc} 1 & 1 & P\\ \mu & -\mu' & R' \end{array}\right)D_{-R',-R}^{P}(R_{\hat{n}}) \label{eq:lambda-func-defn-MF}$$ This is similar to the E P, R term, and essentially rotates the field defined in the LF into the MF by a set of rotations (Euler angles) defined by R n̂ = { χ, Θ, Φ }. For the AF case, a simplified form can be used, since there is no single orientation/rotation defined in relation to the MF, and the relations are defined by the molecular axis distribution (see below). $$\bar{\Lambda}_{R'}=(-1)^{(R')}\left(\begin{array}{ccc} 1 & 1 & P\\ \mu & -\mu' & R' \end{array}\right)\equiv\Lambda_{R',R'}(R_{\hat{n}}=0) \label{eq:lambda-func-defn-AF}$$ The notation here implies that the electric field and axis distributions are expanded about the same axis ( R n̂ = 0), which corresponds to the simplest parallel align-probe field geometry. Additional frame rotation(s) may be applied in some cases (e.g. for crossed-polarization of the alignment and probe fields). Alignment term The axis distribution moments (ADMs) define the LF in this case, and are given above as a set of parameters A Q, S K ( t ). These give rise to couplings which can be written as: $$\Delta_{L,M}(K,Q,S)=(2K+1)^{1/2}(-1)^{K+Q}\left(\begin{array}{ccc} P & K & L\\ R & -Q & -M \end{array}\right)\left(\begin{array}{ccc} P & K & L\\ R' & -S & S-R' \end{array}\right) \label{eq:delta-func-defn}$$ Hence the coupling between the LF and MF is, effectively, defined by the final term in the AF observable (Eqn. [eq:BLM-tensor-AF] ): ∑ K, Q, S Δ L, M ( K, Q, S ) A Q, S K ( t ) And the observable is restricted to L m a x = | P + K |, and M = 0 only if Q = S = R = 0 (the usual cylindrically symmetric case). Dipole matrix elements and conventions Herein the numerical form of the dipole matrix elements is chosen to match the definitions used by ePolyScat . In the notation of Ref. : $$I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(\epsilon)=\langle\Psi_{i}^{p_{i},\mu_{i}}|\hat{d_{\mu}}|\Psi_{f}^{p_{f},\mu_{f}}\varphi_{klm}^{(-)}\rangle\label{eq:eps-I}$$ T μ 0 p i μ i , p f μ f ( θ k̂ , ϕ k̂ , θ n̂ , ϕ n̂ ) = ∑ l, m, μ I l, m, μ p i μ i , p f μ f ( ϵ ) Y l m * ( θ k̂ , ϕ k̂ ) D μ, μ 0 1 ( R n̂ ) $$I_{\mu_{0}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})=\frac{4\pi^{2}\epsilon}{cg_{p_{i}}}\sum_{\mu_{i},\mu_{f}}|T_{\mu_{0}}^{p_{i}\mu_{i},p_{f}\mu_{f}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})|^{2}\label{eq:eps-MFPAD}$$ In this formalism: • I l, m, μ p i μ i , p f μ f ( ϵ ) is the radial part of the dipole matrix element (as per Eqns. [eq:r-kllam], [eqn:I-zeta] herein), determined from the initial and final state electronic wavefunctions Ψ i p i , μ i and Ψ f p f , μ f , and the radial part of the photoelectron wavefunction φ k l m (−) (cf. ϕ l m ( k ) in Eqn. [eq:r-kllam], and where (−) denotes outgoing wave normalisation) and dipole operator $\hat{d_{\mu}}$. Here the wavefunctions are indexed by irreducible representation (i.e. symmetry) by the labels p i and p f , with components μ i and μ f respectively; l, m are angular momentum components, μ is the projection of the polarization into the MF. Each energy and irreducible representation corresponds to a calculation in ePolyScat. • T μ 0 p i μ i , p f μ f ( θ k̂ , ϕ k̂ , θ n̂ , ϕ n̂ ) is the full matrix element (expanded in polar coordinates) in the MF, where k̂ denotes the direction of the photoelectron k -vector, and n̂ the direction of the polarization vector n of the ionizing light. This is equivalent to the full photoelectron wavefunction denoted Ψ e ( k ) in Sect. [sec:dynamics-intro] . • Y l m * ( θ k̂ , ϕ k̂ ) is a spherical harmonic. Note the conjugate form here. • D μ, − μ 0 1 ( R n̂ ) is a Wigner rotation matrix element, with a set of Euler angles R n̂ = ( ϕ n̂ , θ n̂ , χ n̂ ), which rotates/projects the polarization into the MF . • I μ 0 ( θ k̂ , ϕ k̂ , θ n̂ , ϕ n̂ ) is the final (observable) MFPAD, for a polarization μ 0 and summed over all symmetry components of the initial and final states, μ i and μ f . Note that this sum can be expressed as an incoherent summation, since these components are (by definition) orthogonal. • g p i is the degeneracy of the state p i . As noted previously, in general there are multiple equivalent definitions for these terms used by different authors; the ePolyScat definitions above apply to the test matrix elements used herein (e.g. Table [tab:inputMatE] ), which lists I l, m, μ p i μ i , p f μ f ( ϵ ) for continuum symmetry components p f = σ u , π u . These matrix elements are normalised to the total cross-section (in Mb), and include symmetrization. For cases where symmetrization is not included in the matrix elements directly, it can be addressed via the use of symmetrized (or generalised) harmonics, which essentially provide correctly symmetrized expansions of spherical harmonics for a given irreducible representation, Γ . These can be defined by linear combinations of spherical harmonics (see Refs. for more): X h l Γ μ * ( θ, ϕ ) = ∑ λ b h l λ Γ μ Y l, λ ( θ, ϕ ) where: • Γ is an irreducible representation, • ( l, λ ) define the usual spherical harmonic indicies (rank, order) • b h l λ Γ μ are symmetrization coefficients, • index μ allows for indexing of degenerate components, • h indexs cases where multiple components are required with all other quantum numbers identical. The exact form of these coefficients will depend on the point-group of the system, see, e.g. Refs. ; for numerical implementation notes in PEMtk see Sect. [sec:numerical-notes] . Channel functions expansion Following the tensor components detailed above, the full form of the channel functions can be written as: $$\varUpsilon_{L,M}^{u,\zeta\zeta'}=(-1)^{M}(2P+1)^{\frac{1}{2}}E_{P-R}(\hat{e};\mu_{0})(-1)^{(\mu'-\mu_{0})}\Lambda_{R',R}(R_{\hat{n}};\mu,P,R,R')B_{L,-M}(l,l',m,m') \label{eq:channel-func-defn-MF}$$ $$\bar{\varUpsilon}_{L,M}^{u,\zeta\zeta'}=(-1)^{M}[P]^{\frac{1}{2}}E_{P-R}(\hat{e};\mu_{0})(-1)^{(\mu'-\mu_{0})}\bar{\Lambda}_{R'}(\mu,P,R')B_{L,S-R'}(l,l',m,m')\Delta_{L,M}(K,Q,S)A_{Q,S}^{K}(t) \label{eq:channel-func-defn-AF}$$ Final state density matrix As introduced in Sect. [sec:density-mat-basic], the (radial) density matrix can be expressed as the outer-product of the (radial) matrix elements. Following the channel function notation, it is also trivial to write the radial matrix elements in density matrix form in the ζ ζ ′ representation: ρ ζ ζ ′ = | ζ ⟩⟨ ζ ′| ≡ 𝕀 ζ, ζ ′ And the full final continuum state as a density matrix in the ζ ζ ′ representation (with the observable dimensions L, M explicitly included in the density matrix), which will also be dependent on the choice of channel functions ( u ): ρ L, M u, ζ ζ ′ = 𝛶 L, M u, ζ ζ ′ 𝕀 ζ, ζ ′ Here the density matrix can be interpreted as the final, LF/AF or MF density matrix (depending on the channel functions used), incorporating both the intrinsic and extrinsic effects (i.e. all channel couplings and radial matrix elements for the given measurement), with dimensions dependent on the unique sets of quantum numbers required - in the simplest case, this will just be a set of partial waves ζ = ( l, m ). Note that this final state is distinct from the “radial" density matrix (Eqn. [eqn:radial-density-mat] ), which encodes purely intrinsic (molecular scattering) photoionization dynamics (thus characterises the scattering event). The L, M notation indicates here that these dimensions should not be summed over, hence the tensor coupling into the β L, M u parameters can also be written in terms of the density matrix: β L, M u = ∑ ζ, ζ ′ ρ L, M u, ζ ζ ′ In fact, this form arises naturally since the β L, M u terms are the state multipoles (geometric tensors) defining the system, which can be thought of as a coupled basis equivalent of the density matrix representations (see, e.g., ref., Chpt. 4.). In a more traditional notation (cf. Eqn. [eq:cstate], see also Refs. ), the density operator can be expressed as: ρ ( t ) = ∑ L M ∑ K Q S A Q S K ( t )∑ ζ ζ ′ 𝛶 L, M u, ζ ζ ′ | ζ, Ψ + ⟩⟨ ζ, Ψ + | μ q ρ i μ q ′ * | ζ ′, Ψ + ⟩⟨ ζ ′, Ψ + | with ρ i = | Ψ i ⟩⟨ Ψ i |. This is, effectively, equivalent to an expansion in the various tensor operators defined above, in a state-vector notation. Matrix inversion formalism As discussed in Sect. [sec:matrix-inv-intro], and following Gregory et. al., the general formalism can also be rewritten with the ADMs separate (see also Reid & Underwood convolution form, ), and the LF/AF given as per Eqn. [eqn:beta-convolution-C] (cf. Eqn. [eq:BLM-tensor-AF] ). Herein, the notation from Gregory et. al. is rewritten slightly, as per Eqns. [eq:basic] - [eq:MPinversion] . In the current notation, the full expressions are written as per Eqn. [eqn:beta-convolution-C] for the LF/AF (cf. Eqn. [eq:BLM-tensor-AF] ): β̄ L, M ( E, t ) = ∑ K, Q, S C K, Q, S L, M ( E ) A Q, S K ( t ) And for the MF (cf. Eqn. [eq:BLM-tensor-MF] ): β L, M ( E, Ω ) = ∑ P, R, Δ q C P, R L, M ( E, Δ q ) D R, Δ q P ( Ω ) And the required coefficients defined as: C̄ K Q S L M ( ϵ ) = ∑ ζ ζ ′ 𝕀 ζ ζ ′ Γ, Γ ′ ( ϵ ) Γ K Q S ζ ζ ′ L M $$\begin{aligned} \Gamma_{KQS}^{\zeta\zeta'LM} & = & (-1)^{M}[P]^{\frac{1}{2}}E_{P-R}(\hat{e};\mu_{0})(-1)^{(\mu'-\mu_{0})}\bar{\Lambda}_{R'}(\mu,P,R')B_{L,S-R'}(l,l',m,m')\Delta_{L,M}(K,Q,S)\\ & = & \bar{\varUpsilon}_{L,M}^{u,\zeta\zeta'}/A_{Q,S}^{K}(t) \end{aligned}$$ C P R L M ( ϵ, Δ q ) = ∑ ζ ζ ′ 𝕀 ζ ζ ′ Γ, Γ ′ ( ϵ ) Γ P R Δ q ζ ζ ′ L M $$\Gamma_{PR\Delta q}^{\zeta\zeta'LM}=(-1)^{M}(2P+1)^{\frac{1}{2}}E_{P-R}(\hat{e};\mu_{0})(-1)^{(\mu'-\mu_{0})}\bar{\Lambda}_{\Delta q}(\mu,P,\Delta q)B_{L,-M}(l,l',m,m')$$ Where it is assumed that R ′ = Δ q, and that the rotational matrix element D R, Δ q P ( Ω ) is computed independently (note that the current numerical function Λ R ′, R ( R n̂ ; μ, P, R, R ′) could possibly be used directly here, but this version keeps the angle-dependence separate as per the matrix inversion formalism). Note also that Q = 0 only for the derivations in Gregory et. al., and the the inverse matrix as given in Eqn. [eq:MPinversion], although the results should generalise - see discussion in Ref. for details, particularly Sect. 3.2 and the appendices. Numerical implementation in ePSproc and PEMtk Photoionizaion calculations with ePSproc The ePSproc codebase aims to provide methods for post-processing with ePolyScat matrix elements (or equivalent matrix elements from other sources), including computation of AF and MF observables. The numerical implementation thus follows the conventions of ePolyScat, as given in Sect. [sec:mat-ele-conventions] . Additionally, various switches can also be set to define alternative choices, e.g. conjugate forms, use of real harmonics etc., for computation of observables. Since the code is open-source Python, users may also swap libraries/conventions to their preference. By default the following conventions/libraries are used: • Angular momentum functions (Wigner D and 3js) are currently implemented directly, or via the Spherical Functions library, and have been tested for consistency with the definitions in Zare (for details see the ePSproc docs ).).). • Spherical harmonics are defined with the usual physics conventions: orthonormalised, and including the Condon-Shortley phase. Numerically they are implemented directly or via SciPy’s sph_harmfunction (see the SciPy docs for details . Further manipulation and conversion between different normalisations can be readily implemented with the SHtools library . • General tensor handling and manipulation makes use of the Xarray library . Data handling and fitting with PEMtk The Photoelectron Metrology Toolkit (PEMtk) codebase aims to provide various general data handling routines for photoionization problems. At the time of writing, only fitting routines are implemented, along with some basic utility functions, and backend functionality from ePSproc. Further details can be found in the PEMtk documentation . The results presented in Sect. [sec:bootstrapping] make use of PEMtk routines, including functions provided to wrap matrix elements and ePSproc observable calculations for fitting, and analysis routines for identifying candidate matrix elements. The full analysis notebooks are available in the Figshare repository for this article . Non-linear optimization (fitting) is handled via the lmfit library, which implements and/or wraps a range of non-linear fitting routines in Python ; for the Levenberg-Marquardt least-squares minimization method used herein this wraps Scipy’s least_squares functionality , which therefore constituted the core minimization routine for the demonstration case. Although not demonstrated herein, computation of X h l Γ μ * ( θ, ϕ ) (Eqn. [eq:symm-harmonics] ) is also currently implemented in the PEMtk codebase, making use of libmsym (symmetry coefficients) and SHtools (general spherical harmonic handling and conversion). For worked examples, see the PEMtk docs . It is hoped that this will be a useful tool for tackling photoionization problems more generally, without a priori knowledge of the matrix elements for a given system. Footnotes 1 In the technical sense, alignment retains inversion symmetry in the LF, while orientation typically implies reduction of the LF symmetry to match the molecular point group symmetry. The term "orientation" is used herein as synonomous with the MF for an arbitrary molecular system, but in some cases - e.g. homonuclear diatomics - alignment may be sufficient for observation of MF observables. ↩︎ 2 This procedure was only tested for the X-state in the experimental case, not for the A or B-state datasets. and especially the supplementary materials for more details. ↩︎ Information & Authors Information Version history Copyright This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License

Authors Metrics & Citations Metrics Article Usage 580views 51downloads Citations Download citation Paul Hockett, Varun Makhija. Topical Review: Extracting Molecular Frame Photoionization Dynamics from Experimental Data. Authorea. 24 July 2024. DOI: https://doi.org/10.22541/au.172184927.77322709/v1 DOI: https://doi.org/10.22541/au.172184927.77322709/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu.