Electronic Supplementary Material (ESI) for Physical Chemistry Chemical Physics. This journal is © the Owner Societies 2018

Supporting Information Femtosecond infrared spectroscopy reveals the primary events of the ferrioxalate actinometer Steffen Straub, Paul Brünker, Jörg Lindner and Peter Vöhringer†*.

†Institut für Physikalische und Theoretische Chemie, Rheinische Friedrich-Wilhelms-Universität, Wegelerstraße 12, 53115 Bonn, Germany.

1. Various photochemical mechanisms for ferrioxalate

The chemical processes that are initiated upon ultraviolet excitation of an aqueous solution of potassium trisoxalatoferrate(III) provide the molecular basis of the “gold standard” of photochemical actinometry;1 namely, the ferrioxalate actinometer. In the field of photochemistry, actinometers are widespread analytical tools that allow for a precise measurement of the absolute internal radiative flux of a photochemical reactor. The ferrioxalate actinometer rests on the net light-induced photocoversion of ferric into ferrous iron III 3‒ II 2‒ 2‒ 2 [Fe (ox)3] + hν  2 [Fe (ox)2] + 2 CO2 + ox , (1) 2 where ox = C2O4. Hatchard and Parker originally proposed the photochemical processes following an optical excitation (eq. (2) below) to proceed via a sequence of inner-sphere (3a) and outer-sphere electron transfer reactions, (3b) and (3c), each of which are coupled to the loss of an oxalate ligand from the Fe(III) center: III 3‒ III 3‒ [Fe (ox)3] + hν  *[Fe (ox)3] (2)

III 3‒ II 2‒ ●‒ *[Fe (ox)3]  [Fe (ox)2] + ox (3a)

●‒ ●‒ ox ⇋ CO2 + CO2 (4) III 3‒ ●‒ II 2‒ 2‒ [Fe (ox)3] + ox  [Fe (ox)2] + 2 CO2 + ox (3b) III 3‒ ●‒ II 2‒ 2‒ [Fe (ox)3] + CO2  [Fe (ox)2] + CO2 + ox (3c) ●‒ In bulk water, the intermediate oxalate radical anions, C2O4 , are known to be in equilibrium ●‒ 3 with neutral CO2 and carbon dioxide radical anions, CO2 (eq. (4)). Both open-shell carbonaceous species can then transfer their unpaired electron to another ferric center (3b and c) II 2‒ 2‒ to yield a ferrooxalate, [Fe (C2O4)2] , the closed-shell free oxalate dianion, C2O4 , and neutral carbon dioxide. Figure S1: Correlation of the electronic absorption spectrum of aqueous ferrioxalate and the photochemical quantum yield of the ferrioxalate actinometer. The quantum yield data (symbols) are compiled from the literature. Circles from Ref. 2, squares from Ref. 4, diamonds from Ref. 5.

This Hatchard-Parker mechanism fully accounts for a photochemical quantum yield for Fe(II) production that exceeds a value of one as was observed in many independent experiments reported in the literature so far. 1 Thus, a single photon is capable of reducing more than a single iron(III) complex (cf. Figure S1) as is expressed in the net actinometer reaction (1).5 In addition, the actinometer’s photochemical quantum yield correlates in a unique fashion with the electronic spectrum of the ferrioxalate complex in aqueous solution. Formally, the UV/Vis spectrum can be divided into two regions (see Figure S1): ligand-field (d-d) transitions are observed for wavelengths longer than ~450 nm whereas only ligand-to-metal charge-transfer (LMCT) transitions can occur for shorter wavelength. A steep increase of the photochemical quantum yield of ferrioxalate at photon energies of 20000 cm‒1 was observed,1 which can clearly be correlated with the onset of the intense charge-transfer excitations. The first flash photolysis experiments6 were carried out with a time resolution in the microsecond regime and relied exclusively on a near-UV-to-visible (UV/Vis) detection. These studies were targeted at exploring the UV-induced photochemical kinetics of ferrioxalate in H2O solution and were able to detect a long-lived intermediate, which absorbed around 400 nm and which was found to decay according to first-order kinetics. Such a result provided some evidence that the rate controlling step cannot be the expected bimolecular reactions (3b and 3c) of the ferrioxalate ●‒ ●‒ with either of the two carbon-containing open-shell fragments, C2O4 or CO2 . Instead, Hatchard and Parker suggested that this intermediate is generated directly from the optically III 3‒ prepared excited state, *[Fe (C2O4)3] , and that is features some ligand-centered radical character, e.g. a ferrous oxalate to which an oxalate radical anion is still attached (cf. Figure S2, structure b): III 3‒ II ● 3‒ II 2‒ ●‒ *[Fe (ox)3]  [(ox)2Fe (OCOCO2 )]  [Fe (ox)2] + ox , (5) The sequence (5) is then followed by the equilibration (4). In such a mechanism, the formation of ●‒ ●‒ the free radicals, C2O4 and CO2 , occurs after a metal-oxygen bond cleavage and requires a prior electron transfer from the oxalate ligand to the iron center. Figure S2. Lewis structures. Ferric parent complex (a); a penta-coordinated ferrous intermediate bearing the oxalate radical anion as a monodentate ligand (b); a hexa-coordinated ferric intermediate bearing two carbon dioxide radicals as monodentate ligands (c); a ferric dioxalate (d); a ferric trioxalate with one of the three oxalate ligands bound in a monodentate fashion (e), and the ferrous dioxalate with a O- coordinated and bent carbon dioxide radical anion as presented in this paper (f).

Perone and coworkers7-8 conducted a spectro-electrochemical study on flash-photolyzed ferrioxalate solutions and proposed that the same intermediate corresponds to a species, which arises from a homolytic C‒C bond rupture and which features a ligand-centered biradical character as schematically depicted by structure c in Figure S2:

III 3‒ III ● 3‒ II 2‒ ●‒ *[Fe (ox)3]  [(ox)2Fe (OC O)2]  [Fe (ox)2] + ox . (6) In this alternative mechanism, the carbonaceous free radicals emerge in parallel to the dissociation of the metal-oxygen bond and their formation does not require a prior full ligand-to- metal one-electron transfer, i.e. quite in contrast to the original Hatchard-Parker mechanism. Later, a re-examination of the flash-induced kinetics in the near-UV/Vis was carried out by Cooper and DeGraff.9-10 The authors concluded that the first process of sequence (5) is practically indistinguishable from the three-body ligand dissociation

III 3‒ III ‒ ●‒ II 2‒ ●‒ *[Fe (ox)3]  [Fe (ox)2] + 2 CO2  [Fe (ox)2] + CO2 + CO2 (7) during which the oxidation state +III at the metal is fully preserved. In contrast to reactions (5) and (6), the reduction of the metal takes place here in a secondary bimolecular electron-transfer ●‒ from a CO2 primary fragment to the ferric center dressed with only two oxalate ligands (cf. structure d, Figure S2). Thus, the photochemical mechanisms of aqueous ferrioxalate can broadly be classified into “prompt” (sequence 5) and “delayed” (sequences 6 and 7) electron-transfer mechanisms. Near-UV/Vis flash-photolysis experiments with a much better time resolution of a few tens of nanoseconds have appeared in the literature only recently11-13 and their results seem to favor the prompt sequence (5). Yet, a direct pathway, which avoids the formation of an intermediate ●¯ through the direct coupling of the photoreduction with the C2O4 dissociation, also needs to be invoked to quantitatively reproduce the experimentally determined optical transients. At this stage, it becomes quite obvious that experiments with an even better time resolution, preferably on the ultimate femtosecond scale, are required to differentiate unmistakably between “prompt” and “delayed” electron transfer mechanisms. It is quite remarkable however, that published reports on such femtosecond spectroscopic experiments on ferrioxalate are very scarce. To date, the only existing literature, which investigates the ultrafast optical spectroscopy of ferrioxalate, is limited to probing in the UV/Vis.p14-17 (see also Ref. 18 for a summary of ultrafast UV/Vis pump-probe studies of ferric dicarboxylates containing citrate, tartarate, and lactate ligands). Interestingly, Rentzepis et al. used such femtosecond optical data only to complement ultrafast extended x-ray absorption fine structure (EXAFS) spectroscopy and to facilitate a data analysis of their x-ray data. Iron-K post-edge EXAFS between 7 keV and 8 keV is known to deliver very valuable structural information such as bond lengths and coordination numbers pertaining to the coordination sphere surrounding the metal. In its electronic ground state, the ferrioxalate parent complex features a high-spin electronic configuration (FeIII, d5) with S = 5/2 (sextet) and a molecular structure, in which all Fe‒O bond lengths are equal to 2.00 Å. During the first 2 ps after UV absorption, the EXAFS data are indicative of an increase of the Fe‒O bond distances to 2.16 Å. This structural III 3‒ expansion was attributed to the excited state, *[Fe (C2O4)3] ; in which the metal’s oxidation state is preserved. Most importantly, this period of bond length increase is purely non-reactive, i.e. it is not accompanied by an electron transfer or a dissociation. Subsequently, the Fe‒O distances decrease to 1.93 Å at 4 ps and 1.87 Å at 9 ps. Aided by electronic structure calculations, the authors assigned this phase of bond-length contraction to a heterolytic cleavage of a Fe‒O bond and not (!) to a homolytic cleavage as in expressed sequence 5) followed by the full ligand fragmentation. Instead, the ferric nature of the parent is conserved throughout the entire sequence

III 3‒ III 3‒ III ‒ ●‒ *[Fe (ox)3]  [(ox)2Fe (OCOCO2)]  [Fe (ox)2] + 2CO2 . (8) The evolution from five-fold to four-fold coordination of the iron(III) center (i.e. from structure e to structure d in Figure S2) takes place within less than 10 ps, whereas the necessary reduction of the metal is purely bimolecular in nature and as such occurs only on the microsecond scale. These results initiated additional time-resolved studies at the x-ray free electron laser facility, SACLA (XFEL), at RIKEN using advanced core-level spectroscopy.19 Specifically, utilizing the total fluorescence yield method, the precise spectral position of the iron-K edge was tracked as a function of time after UV-excitation. The so-called “edge-shift” is known to be a highly sensitive reporter of the oxidation state at the iron (1 eV per one electron oxidation/reduction). Indeed, a pronounced red-shift of the Fe-K-edge by more than 4 eV within 140 fs after UV-excitation was recorded, which was subsequently seen to decay to a value of ~3 eV on a time scale of about 3 ps. Guided once again by electronic structure calculations, these results served to sketch yet another primary reaction mechanism, in which the optically accessed excited state rapidly loses ●‒ CO2 and CO2 in a sequential manner to generate the final ferrous dioxalate product: III 3‒ II 3‒ [Fe (ox)3] + hν  *[Fe (ox)3] (9a)

II 3‒ II ● 3‒ *[Fe (ox)3]  [(ox)2Fe (OC O)] + CO2 (9b) II ● 3‒ II 2‒ ●‒ [(ox)2Fe (OC O)]  [Fe (ox)2] + CO2 + CO2 (9c) Note that in this interpretation the initial photoexcitation is interpreted as a full one-electron LMCT thus giving rise to an optically prepared excited state that exhibits the metal already at the final +II oxidation state. Finally, another ultrafast x-ray absorption study appeared in 2017, which was carried out with a laser-driven plasma source at NIST in Boulder and which essentially supported the conclusion drawn from the SACLA data that electron transfer precedes the ligand dissociation.20 The above review of the existing literature on the photochemical mechanism of ferrioxalate clearly demonstrates that despite all efforts from electronic spectroscopies in the UV/Vis and the x-ray regions, covering time scales from milliseconds to femtoseconds, sequence of primary events and associated time scales remains largely a mystery. It is apparent that still more input from ultrafast spectroscopy is needed to finally clarify the primary photochemical events leading to the photoconversion of the ferrioxalate actinometer. Strikingly, ultrafast vibrational spectroscopy such as UV-pump/mid-infrared probe spectroscopy, which is indeed capable of providing highly valuable information regarding the molecular and electronic structure of a molecular system following its optical excitation, has not been carried out yet on aqueous ferrioxalate. Such valuable data are lacking because the most characteristic vibrations of the complex, which are the oxalate stretching vibrations, absorb around 1700 cm‒1. Unfortunately, even in highly concentrated aqueous solutions, this spectral region is totally obscured by the overwhelming solvent absorption arising from the water bending mode at 1643 cm‒1. It is this very strong solvent background, which precluded in the past any reliable absorption measurements (stationary or time-resolved) in the mid-IR spectral region of our interest. In this paper, we have therefore explored the linear and time-dependent vibrational spectroscopy of ferrioxalate in heavy water (D2O) instead. Liquid D2O provides a usable free spectral range between 1250 cm‒1 and 2200 cm‒1 and its optical transmission at 1700 cm‒1 is as high as 50% when a 50 μm-thick layer is used. The transmission in this spectral region is dictated by the weak solvent bending-librational combination band.

2. Experimental and theoretical methods

Spectroscopy. Stationary UV/Vis spectra were recorded on a Shimadzu UV-160 spectrophotometer and stationary IR-spectra were obtained from a Nicolet 5700 (Thermo Fisher) FTIR spectrometer. Femtosecond UV-pump/mid-IR-probe spectroscopy was carried out with an ultrafast Ti:Sapphire front-end laser system (Newport Spectra Physics, Solstice Ace) delivering 50 fs-duration pulses at a center wavelength of 800 nm and a repetition rate of 1 kHz. A fraction of the output of the front-end was used to pump a home-built optical parametric amplifier (OPA), whose signal and idler pulses were frequency down-converted by difference frequency generation in a type-I AgGaS2-crystal thereby providing probe pulses tunable between 10 μm and 2.8 μm with an energy as high as 2 μJ and a bandwidth of about 200 cm–1. After removing residual signal and idler light with a coated Ge filter, the DFG beam was split into probe and reference pulses, the first of which are directed over a motorized delay stage. Both MIR beams were focused into the sample with a gold‐coated 90° off‐axis parabola (OAP) with an effective focal length of +100 mm. The pulses were collimated with an identical OAP and directed to a polychromator equipped with a liquid nitrogen cooled HgCdTe array detector (Infrared Associates MCT‐6400) for a detection of their spectra. Pump pulses at a wavelength of 266 nm were obtained by frequency- tripling a fraction of the front-end’s fundamental pulses in two consecutive type-I β-Ba(BO2)2- crystals, the first of which was cut for frequency doubling at 800 nm and the second was cut for sum-frequency mixing of 800 nm and 400 nm. The pump pulses were focused into the sample by a +400 mm fused silica lens at an angle of only 5° between pump and probe beam. To ensure homogeneous illumination conditions, the focus of the pump beam was located behind the sample such that its diameter was slightly larger than that of the probe beam (400 μm). The sample solution was circulated with a gear pump at a flow rate of ca. 100 mL min–1 through a home‐built flow cell that was equipped with two CaF2 windows held at a spacing of 100 μm. The whole pump-probe setup was purged with nitrogen to avoid absorption of the probe pulses by atmospheric carbon dioxide. Theory. Quantum chemical calculations were performed within the framework of density functional theory using the B3LYP21-22 hybrid functional together Ahlrichs’ triple-ζ basis set, def2-TZVP23 as implemented in Gaussian 09.24 Solvent effects were taken into account by the conductor-like polarizable continuum model (CPCM). Geometry optimizations were performed at all feasible spin multiplicities with various initial input structures and were considered to be converged if the built-in convergence criteria of “Opt=Tight” were fulfilled. The optimized structures were confirmed to be true minima of the potential energy surface by performing a harmonic normal mode analysis and verifying that all eigenvalues of the Hessian are positive. Vibrational frequencies were calculated analytically and not scaled. The agreement between theoretical and experimental spectra proved sufficient. Finally, theoretical IR spectra were obtained by convoluting the stick-spectrum with a Lorentzian profile having a full spectral width at half maximum (FWHM) of 12 cm–1. To get further insight into the electronic structures of the various putative intermediates a natural bond orbital analysis was carried out using the NBO 6.0 program package.25

3. Global analysis

A global analysis of the spectro-temporal evolution was attempted using the Glotaran application software developed by van Stokkum and coworkers26 for fitting superposition models to multi- dimensional data as originally introduced by the group of van Grondelle27. The results of a singular value decomposition of the Δ푂퐷(휈̃ ,푡)-data matrix when restricted to the wavenumber interval 1340 cm‾1 ≤ 휈̃ ≤ 1730 cm‾1 and to delays 0.5 ps ≤ t ≤ 1000 ps is displayed in Figure S3. Figure S3. Singular value decomposition of the delay and probe-wavenumber-dependent differential optical density for positive delays and for in the inner and peripheral CO regions of the oxalate ligands only.

There appear to be five dominant components whose left and right singular vectors are shown explicitly in the middle and right panel. Performing a global analysis by setting up a consecutive series of first-order reactions gives rise to time-dependent populations as well as species- associated spectra that are shown in Figure S4 together with the singular-value decomposition of the resultant residual data matrix. The species-associated difference spectra are clearly indicative of a continuous frequency-upshift of an induced absorption in the inner C–O region of the oxalate ligands around 1435 cm‾1, which is quantified more accurately using the peak-position analysis as described in Section 2.4 and in Figure 6b of the main paper. Likewise, the global analysis identifies an induced absorption in the peripheral C=O region near 1640 cm‾1, which experiences an initial frequency downshift followed by an upshift, i.e. highly suggestive of a complex spectro-temporal evolution, which is characterized in more detail in Figure 6a of the main paper.

Figure S4. Global analysis of the delay and probe-wavenumber-dependent differential optical density for positive delays and for in the inner and peripheral CO regions of the oxalate ligands only using a consecutive model with five first-order reactions. Left: populations, middle: species-associated spectra, right: SVD of the residual data matrix. The lifetimes of the consecutive reactions are 1.39 ps, 1.43 ps, 25.2 ps, 592 ps, and > 10 ns.

A complementary global analysis of the purely absorptive product spectrum, which is entirely equivalent to subtracting the properly weighted stationary FTIR spectrum of the sample from the species-associated spectra, emphasizes more clearly the spectral shifting dynamics originating from the vibrational relaxation of the replenished hot ground state (cf. Figure S5). Figure S5. Global analysis of the delay and probe-wavenumber-dependent purely absorptive product spectra for positive delays and for in the inner and peripheral CO regions of the oxalate ligands only using a consecutive model with five first-order reactions.

Because of the continuous spectral shifts occurring in both, the inner C–O and peripheral C=O stretching regions, we have decided to refrain from pursuing a more in-depth global analysis because in such cases, the results are often ambiguous and difficult to interpret as was discussed in detail by Marciniak and Lochbrunner.28 Instead, we have decided in favor of a spectral peak- position analysis whose technical details are outlined in the main paper. In Figure S6, we plot the time-dependent purely absorptive product spectrum in a color-shaded contour representation together with the time-dependent spectral shifts as obtained from this method.

Figure S6. Color-shaded contour representation of the spectro-temporal evolution of the purely absorptive product spectrum. The curves a) through c) represent the same multi-exponential fits to the time- dependent peak positions that are shown in Figure 6 a-c) of the main paper. The white dashed curve d) corresponds to curve a) but scaled and horizontally shifted along the probe wavenumber axis. The two absorption bands whose peak position is represented by curves a) and d) originate from the same species; namely, the hot electronic ground state undergoing vibrational relaxation. We finally note that the pronounced spectral dynamics revealed by Figure 5 and 6 of the main paper as well as Figure S6 may also be examined by means of a lifetime density map analysis.29

References

1 H. J. Kuhn, S. E. Braslavsky and R. Schmidt, Pure & Appl. Chem. 2004, 76, 2105-2146. 2 C. G. Hatchard and C. A. Parker, Proc. R. Soc. Lond. A 1956, 235, 518-536. 3 Q. G. Mulazzani, M. Dangelantonio, M. Venturi, M. Z. Hoffman and M. A. J. Rodgers, J. Phys. Chem. 1986, 90, 5347-5352. 4 C. H. Langford and C. A. Holubov, Inorg. Chim. Acta Lett. 1981, 53, L59-L60. 5 S. Goldstein and J. Rabani, J. Photochem. Photobiol. A 2008, 193, 50-55. 6 C. A. Parker and C. G. Hatchard, J. Phys. Chem. 1959, 63, 22-26. 7 R. A. Jamieson and S. P. Perone, J. Phys. Chem. 1972, 76, 830-839. 8 J. I. Patterson and S. P. Perone, J. Phys. Chem. 1973, 77, 2437-2440. 9 G. D. Cooper and B. A. DeGraff, J. Phys. Chem. 1971, 75, 2897-2902. 10 G. D. Cooper and B. A. DeGraff, J. Phys. Chem. 1972, 76, 2618-2625. 11 V. Nadtochenko and J. Kiwi, J. Photochem. Photobiol. A 1996, 99, 145-153. 12 I. P. Pozdnyakov, O. V. Kel, V. F. Plyusnin, V. P. Grivin and N. M. Bazhin, J. Phys. Chem. A 2008, 112, 8316-8322. 13 I. P. Pozdnyakov, O. V. Kel, V. F. Plyusnin, V. P. Grivin and N. M. Bazhin, J. Phys. Chem. A 2009, 113, 8820-8822. 14 J. Chen, H. Zhang, I. V. Tomov, X. L. Ding and P. M. Rentzepis, Chem. Phys. Lett. 2007, 437, 50-55. 15 J. Chen, H. Zhang, I. V. Tomov, M. Wolfsherg, X. L. Ding and P. M. Rentzepis, J. Phys. Chem. A 2007, 111, 9326-9335. 16 J. Chen, H. Zhang, I. V. Tomov and P. M. Rentzepis, Inorg. Chem. 2008, 47, 2024-2032. 17 J. Chen, A. S. Dvornikov and P. M. Rentzepis, J. Phys. Chem. A 2009, 113, 8818-8819. 18 I. P. Pozdnyakov, A. A. Melnikov, N. Tkachenko, S. V. Chekalin, H. Lemmetyinen and V. F. Plyusnin, Dalton Trans. 2014, 43, 17590-17595. 19 Y. Ogi, Y. Obara, T. Katayama, Y. I. Suzuki, S. Y. Liu, N. C.-M. Bartlett, N. Kurahashi, S. Karashima, T. Togashi, Y. Inubushi, K. Ogawa, S. Owada, M. Rubešova, M. Yabashi, K. Misawa, P. Slavíček and T. Suzuki, Struct. Dynam. 2015, 2, 034901 20 G. C. O'Neil, L. Miaja-Avila, Y. Il Joe, B. K. Alpert, M. Balasubramanian, D. M. Sagar, W. Doriese, J. W. Fowler, W. K. Fullagar, N. Chen, G. C. Hilton, R. Jimenez, B. Ravel, C. D. Reintsema, D. R. Schmidt, K. L. Silverman, D. S. Swetz, J. Uhlig and J. N. Ullom, J. Phys. Chem. Lett. 2017, 8, 1099-1104. 21 A. D. Becke, J. Chem. Phys. 1993, 98, 5648-5652. 22 P. J. Stephens, F. J. Devlin, C. F. Chabalowski and M. J. Frisch, J. Phys. Chem. 1994, 98, 11623-11627. 23 F. Weigend and R. Ahlrichs, Phys. Chem. Chem. Phys. 2005, 7, 3297-3305. 24 M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, N. J. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski and D. J. Fox, Gaussian Inc., Wallingford CT, 2009. 25 E. D. Glendening, J. K. Badenhoop, A. E. Reed, J. E. Carpenter, A. Bohmann, C. M. Morales, C. R. Landis and F. Weinhold, Theoretical Chemistry Institute, University of Wisconsin, 2013. 26 J. J. Snellenburg, S. P. Laptenok, R. Seger, K. M. Mullen and I. H. M. van Stokkum, J. Stat. Soft. 2012, 49, 1-22. 27 I. H. M. van Stokkum, D. S. Larsen and R. van Grondelle, Bba-Bioenergetics 2004, 1657, 82-104. 28 H. Marciniak and S. Lochbrunner, Chem. Phys. Lett. 2014, 609, 184-188. 29 G. F. Dorlhiac, C. Fare and J. J. van Thor, Plos Comput. Biol. 2017, 13, e1005528.