research papers
Alternative difference analysis scheme combining Rspace fit with global optimization XANES fit for Xray transient absorption spectroscopy
^{a}Beijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, People's Republic of China, and ^{b}University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China
^{*}Correspondence email: zhaohf@ihep.ac.cn
Timeresolved Xray absorption spectroscopy (TRXAS), based on the laserpump/Xrayprobe method, is powerful in capturing the change of the geometrical and electronic structure of the absorbing atom upon excitation. TRXAS data analysis is generally performed on the laseron minus laseroff difference spectrum. Here, a new analysis scheme is presented for the TRXAS difference fitting in both the extended Xray absorption finestructure (EXAFS) and the Xray absorption nearedge structure (XANES) regions. Rspace difference fitting could quickly provide the main quantitative structure change of the first shell. The XANES fitting part introduces a global nonderivative optimization algorithm and optimizes the local structure change in a flexible way where both the core calculation package and the search method in the fitting shell are changeable. The scheme was applied to the TRXAS difference analysis of Fe(phen)_{3} spin crossover complex and yielded reliable distance change and excitation population.
Keywords: difference XAFS fit; global optimization; Rspace; core/shell substitution; TRXAS.
1. Introduction
Timeresolved Xray absorption spectroscopy (TRXAS), based on the laserpump/Xrayprobe method, is a powerful probe for addressing the geometric and electronic structure of lightgenerated transient species (Chen et al., 2014; Chergui, 2016). Its time resolution capability is dependent on the Xray Generally, it can reach 100 ps temporal resolution using synchrotron radiation sources and less than 100 fs using Xray freeelectron lasers. TRXAS has been widely applied in and photophysics fields. Bressler et al. studied the lightinduced spin crossover of an iron complex (Bressler et al., 2009). Zhang et al. obtained a 0.01 Å highresolution structure change of the of an osmium complex (Zhang et al., 2014). Moonshiram et al. revealed the intermediate molecular structure of Co(II) and Co(I) photocatalyst in real H_{2} production (Moonshiram et al., 2016). Wen et al. studied BiFeO_{3} nanofilm and found an anisotropic change of the inplane contraction and outofplane extension (Wen et al., 2015).
To benefit from the well developed Xray absorption finestructure (XAFS) qspace was applied by Borfecchia et al. to study the photoactive metal complexes cis[Ru(bpy)_{2}(py)_{2}] (Borfecchia et al., 2014; Garino et al., 2014). Zhang et al. adopted a fit of the excitedstate reconstructed according to different excitation fractions (Zhang et al., 2015). FEFF, a popular XANES computation package based on the selfconsistent multiplescattering theory, is used in the study of the PtPOP system by van der Veen et al., who took Kas's Bayes XANES fit method to search for structure change information when the system is excited (van der Veen et al., 2010). The interpolation approach is effective in structure Smolentsev et al. used it to invoke FEFF and FDMNES in TRXAS fitting research on platinum dimer (Lockard et al., 2010). They also used the DFTMO method to simulate the preedge feature of different cobalt species (Moonshiram et al., 2016). MXAN (Benfatto et al., 2003), one of the widely used XANES fit algorithms, uses CONTINUUM to perform the calculation and adopts the MINUIT optimization package. MXAN has been applied to the characterization of the molecular excitation state of [Ru^{II}(bpy)_{3}]^{2+} by Benfatto et al. (2006).
methods developed over the last 40 years, different TRXAS treatments or packages have been developed in both the and XANES regions along with extensive TRXAS applications. The differential fitting method inHere we present an alternative data analysis scheme for TRXAS. It combines the Rspace fit with the global optimization XANES fit for TRXAS analysis. The fit of the difference spectrum is performed directly in both the and XANES regions. We adopt the Rspace fit to avoid one more reverse Fourier transform. The XANES difference spectrum fit is similar to that used in MXAN but it is more flexible, as we can choose different calculation cores, like FEFF (Rehr et al., 2010) or FDMNES (Joly et al., 2009), and different adaptive optimization algorithms as well to control the fit process.
2. Method
2.1. Workflow
We present here a TRXAS fit scheme in both the Rspace difference fit in the gives the average bond length changes of the first shell around the absorber, while the detailed structural change is derived in the XANES difference fit. Our XANES fit scheme is flexible, where both the search algorithm (shell) for variables and the spectrum calculation package (core) can be substituted by any suitable package.
and XANES regions where theThe workflow of our TRXAS fit scheme is composed of two blocks, as shown in Fig. 1, the difference fit (block 1) and the XANES difference fit (block 2): each one starts from finding the ground state structure and is followed by searching for the structural change of the intermediate state. The spectrum of the ground state or intermediate state is first calculated using the general formula (Newville, 2013) in block 1 or the selected calculation package in block 2. The theoretical difference spectrum is obtained according to the equation
where α is the fraction of the intermediate state, and and are the theoretical spectra of the ground state and the intermediate state, respectively. The theoretical difference spectrum is compared with the experimental difference spectrum to check their consistency in block 2. They are, however, firstly Fourier transformed into Rspace in block 1 before comparison. The structure of the intermediate state and α may be modified by a parameter search shell to perform the difference spectrum calculation again, following a comparison with the experimental spectrum until the consistency is acceptable.
We give here the definition of the goodness of fit, the minimum of which in the parameter space is what the difference fit searches for,
where x_{j} is the parameter to be fit, n is the number of independent parameters, m is the number of data points, ∊_{i} is the individual error in the experimental data set, and w_{i} is the When w_{i} = 1, the function reduces to the statistical χ^{2} function.
The conclusion of the difference fit in the
region is helpful for the XANES fit, as one could limit the varying range of some of the parameters in order to save the fit time.TRXAS data analysis is generally performed on the laseron minus laseroff difference spectrum. The difference highlights the structural change, since the difference spectrum eliminates the experimental system error. Our framework performs the fit directly to the difference spectrum instead of to the ground state and the excitation state spectrum separately. Moreover, preprocessing of the that the fraction of the intermediate state α in the system after pump is critical in producing the XANES fit. There are several ways to estimate α in TRXAS analysis. Moonshiram et al. estimated the Co(II) intermediate fraction using the Co(II) reference sample's preedge fingerprint (Moonshiram et al., 2016). This would fail when the analogue sample is difficult to synthesize or when it is unstable in the experiment, which occurs in most cases. We can estimate the fraction from an optical transient absorption (OTA) experiment carried out under the same conditions as for the TRXAS (Benfatto et al., 2006; Borfecchia et al., 2013), but some systems are OTA silent, the intermediate in the system cannot be detected by OTA spectroscopy. If we are unable to apply these two methods, an alternative choice for obtaining the intermediate fraction is by treating it as a variable like other structural parameters in the TRXAS fit (Lockard et al., 2010; Zhang et al., 2015). In our scheme, the intermediate fraction could be treated as a variable or constant.
spectrum, such as background subtraction, is not needed for the difference fit as the backgrounds are the same in the ground state and intermediate state. Obviously we can see from equation (1)We use Rspace instead of kspace for the fit. It is difficult to achieve a highquality TRXAS signal across the full range. Moreover, it is clear and straightforward to find the trivial difference between the simulation χ(R) and experiment in Rspace. Generally the major structure change in the system comes from the nearest atoms to the absorber, e.g. those in the first shell: we can limit the fitting domain to the first shell in Rspace during the fit, and the calculation of χ will be much faster as we do not need to consider scattering from other shells.
In the XANES fit, as described above, the core simulates the spectrum and the shell controls the parameter search. FEFF (Rehr et al., 2010) is used to carry out the XANES calculation by default in the scheme, and can be replaced by FDMNES (Joly et al., 2009) at present. The search method adopted by the shell is also substitutable, such as by the mesh adaptive direct search (MADS), the dividing rectangles method (Gablonsky & Kelley, 2001) or the improved stochastic ranking evolution strategy (Runarsson & Yao, 2000). We take here the MADS method as the default for the difference fit. It evolved from the generalized pattern search which is one of the modern grid searches. MADS has proven to be efficient (Audet et al., 2008, 2011) and is now widely used (Miiller et al., 2012; Eisenhower et al., 2012; BerrocalPlaza et al., 2014). Moreover, NOMAD, which is one of the packages that uses the MADS method, offers a parallel version (Audet et al., 2008).
2.2. Choice of the optimization algorithm
The optimization algorithm is important in the XANES fit. Traditional structure optimization in computation chemistry adopts and develops a series of local optimization algorithms with the initial Hessian guess method and the Hessian update method. However, in CALYPSO code (Wang et al., 2012) and the evolutionary optimization in the USPEX code (Glass et al., 2006), has achieved great success. The choice of optimization algorithm should also be deliberated in the XANES fit. First, we prefer to obtain a global minimum rather than a local minimum. Second, since the objective function of the XANES fit is given as a timeconsuming flow without analytic derivative information, instead of a simple formula versus the parameters, we do not know whether it is a convex problem before the fit, so generally we adopt a nonlinear optimization algorithm. Moreover, we decide to use nonderivative algorithms, as the XANES fit also cannot provide an analytical derivative, and the cost of a numerical derivative computation of the XANES fit is huge; also, the reliability of the numerical derivative is difficult to guarantee. Fortunately many global optimization algorithms do not need information about the derivative, so a global nonderivative algorithm is our choice. The deterministic algorithm and the stochastic and heuristics algorithm are the two main branches of global optimization. Normally the deterministic algorithm can perform a theoretical convergence analysis under given conditions, while some stochastic and heuristics algorithms can only provide a `probabilistic convergence guarantee' or `remain heuristics' (Pardalos et al., 2000). For the deterministic algorithm, taking MADS as an example, Audet et al. used MADS's hierarchy of convergence analysis (Audet & Dennis Jr, 2006) and proved that the lower triangular instance of MADS can produce a dense set of poll directions. Another example is the DIRECT algorithm. This can prove its ability of convergence to the Karush–Kuhn–Tucker point under mild conditions (Gablonsky & Kelley, 2001). Though some optimization algorithms can perform convergence analysis under given conditions, the theoretical condition is usually difficult to test. Therefore the benchmark of the algorithm is important in practice. From NLOPT library's benchmark (Kumar et al., 2016) and the derivativefree algorithms benchmark (Rios & Sahinidis, 2013), we can find that usually one optimization algorithm cannot solve all test problems. Also there is an optimization solution distribution of multiple optimization solver runs due to the algorithm itself or initial solution. Obviously, there is no single ultimate algorithm for all optimization problems, so the substitutable algorithm in the shell of the XANES fit is a good choice.
prediction, the application of heuristic global optimization algorithms, for example the particle swarm optimization in theWe notice that the correlation of structure parameters and intermediate fraction has not been studied thoroughly in TRXAS fitting. FITIT uses Kas's scheme of the conditioning number (Rehr et al., 2005) to deal with the correlation of structure parameters. In the study of pyrazolatebridged platinum dimer which employed FITIT to fit the fraction (Lockard et al., 2010), the correlation between fraction and structure parameters was not reported. MXAN does not give the correlation between structure parameters. Vorobeva et al. adopted MXAN in the study of excited Ti^{3+} in Al_{2}O (Vorobeva et al., 2009). They fit the fraction, but they could not distinguish between elongation and compression of the octahedron without a detailed study of the correlation between the structure change and fraction. Nowadays, methods have been developed for studying the correlation between parameters such as the maximal information coefficient (Reshef et al., 2011), distance correlation (Delicado & Smrekar, 2009) and mutual information estimator (Kraskov et al., 2004). It is expected that these methods can be applied in helping to deal with this problem.
3. Application
3.1. Sample and experimental description
We applied our scheme to a model spin crossover iron complex, 1,10phenanthroline iron(II) sulfate, Fe(II)(phen)_{3}. Fe(II)(phen)_{3} is a lowspin (LS) complex in the ground state, and becomes a highspin (HS) state upon photoexcitation due to spin crossover. It is important in dynamic magnetic research and has been well studied by TRXAS. The Nozawa group reconstructed the of the of Fe(II)(phen)_{3} and found a 0.17 Å bond elongation for the first shell compared with the LS ground state (Nozawa et al., 2010). Fe(II)(phen)_{3} was purchased from Alfa Aesar and was used without any further purification. The HS analogue complex, Fe(II)(2CH_{3}phen)_{3}, was synthesized. Its HS state results from of the methyl group. Fe(II)(2CH_{3}phen)_{3} not only provides a HS analogue for structure analysis but also helps to determine the excitation fraction.
TRXAS measurements were performed at beamline 11IDD of the Advanced Photon Source. An experimental description can be found elsewhere (Chen & Zhang, 2013). The Fe(II)(phen)_{3} laseron and laseroff spectra are shown in Fig. 2 along with of Fe(II)(2CH_{3}phen)_{3}.
In the _{3} and its HS analogue Fe(II)(2CH_{3}phen)_{3}. The data were processed by our scheme with the equation referring to that in Larch (Newville, 2013). The amplitude reduction factor was fixed to 0.9 and the was fixed to 6. The variables are the disorder factor σ^{2}, the energy shift E_{0} and the bond length r. The theoretical amplitude and phase are extracted from the 1.95 Å Fe—N single scattering path calculated by FEFF9. The Rspace fit is displayed in Fig. 3 and the fit results are listed in Table 1. The disorder factor and energy shift will be used in the following difference fit.
fit, we fit the groundstate Fe(II)(phen)

As stated above, the difference between laseron and laseroff highlights the excited signal. The difference spectrum (black) is displayed in Fig. 4, compared with the difference obtained from the HS analogue minus the ground state. The discrepancy reflects the excitation fraction; the excitation fraction is 37%. The reconstructed difference spectrum and the difference spectrum obtained from the excitation analogue are in good agreement. In the following we will use our new scheme to fit the difference spectrum in both the XANES and regions to obtain the structure change information and excitation fraction.
3.2. difference spectrum fitting
The reverse Fourier transform of the experimental difference spectrum is shown in Fig. 5. We find that the 1–1.9 Å Rspace window reveals the main features of the experimental spectrum. The kweight of the fit is 2. We used kspace in the 3–10 Å^{−1} region and Rspace in the 1–1.9 Å region for the fit.
We fit the intermediate's fraction and relative bond length change. The energy shift and the disorder factor, given in Table 1, are fixed during the fitting. The Rspace fit is shown in Fig. 6(a). We also performed the reverse Fourier transform of the Rspace fit result, as seen in Fig. 6(b). The Fe—N bond length is elongated by 0.17 Å under the excitation state. The excitation fraction is 37%. The results are in agreement with those from the HS analogue in Table 2. The difference fit can only provide bond length information, so next we perform the difference XANES fitting.
3.3. XANES difference fit
First we fit the groundstate structure. We used the FEFF9/FDMNES packages for the XANES calculation and took the NOMAD package to search for the optimized variables. Multiplescattering theory (MST) was selected in the FDMNES package. The initial structure of the LS ground state was taken from Yan et al. (2000). The radius of the cluster for selfconsistent multiple scattering is 6.5 Å, containing all of the atoms in the molecule. The real Hedin–Lundqvist exchangecorrelation potential was used in the calculation. The phenanthroline ring is taken as a rigid unit during the fit, and we chose the midpoint of the two ligand N atoms to be the representative of this unit. In total, nine structural variables representing three rigid bodies were fitted, three parameters (r, θ, φ) for each ligand. The best fits of FEFF and FDMNES are shown in Fig. 7(a). All of the features were reproduced by FEFF and they were much better than those reproduced by FDMNES, so we used FEFF's data here. The average Fe—N bond length from FEFF was 1.97 Å. The broadening given in FEFF by default is acceptable, as shown in Fig. 7(a), so no additional broadening was considered here. In the FDMNES fit, we used the arctangent style function representing the energydependent Lorentzian broadening width,
where E_{F} is the Γ_{Hole} is the core level broadening width, Γ_{m} is the maximum of the broadening width of the final state (high energy range) and E_{cent} is the energy whose width is half of Γ_{m}. Γ_{m}/E_{larg} represents the depth of the center of the arctangent function.
We fix Γ_{Hole} to be the default value 1.33 eV of iron in FDMNES. The fit result is E_{F} = 3.5 eV, Γ_{m} = 19.6 eV, E_{cent} = 16.4 eV, E_{larg} = 47.9 eV. When we use the finite difference full potential method (FDM) in FDMNES to calculate the spectrum based on the fitted structure via MST, the spectrum reproduces the feature better, as shown in Fig. 7. The fitted broadening parameters in FDM are: E_{F} = −3.6 eV, Γ_{m} = 23.5 eV, E_{cent} = 36.7 eV, E_{larg} = 50.0 eV, and the fitted fraction in FDM is 60%.
In the difference XANES fit, we used the FEFF9/FDMNES (MST) packages for the XANES calculation and took MADS optimization algorithms to search for the best optimized variables. Fit results using FEFF and other optimization algorithms can be found in the supporting information. We fixed the normalization factor and the energy shift of the intermediate state to be the same as those derived from the previous groundstate fit (in the FDMNES case, broadening parameters are also fixed). The fit results are listed in Table 2. Although FDMNES (MST)/NOMAD produces the lower Rfactor, the spectrum of the ground state derived by FDMNES is not good enough. Moreover, its intermediate state fraction is much larger than that from the intermediate analogue and that from the difference fit. The fit given by FEFF/NOMAD is shown in Fig. 7(b). The main features in the difference spectrum are reproduced. Coordinates of the intermediate structures are listed in the supporting information. The average bond length of the first shell extends to 0.11 Å, and the intermediate fraction is 39%, as shown in Table 3. The fitting results are close to those obtained in the Rspace fit and reported data (Nozawa et al., 2010).

The difference supporting information (Table S0). In `Direct_L Bound', we used the [−0.05 Å, 0.3 Å] boundary limitation of the bond length change. The limitation comes from the difference result. The lower bound is −0.05 Å because the gives the average bond length extension in which some bonds may contract. For comparison, in `Direct_L', we used the [−0.3 Å, 0.3 Å] boundary limitation of the bond length change. The boundary limitation from can improve by about 30% the efficiency of the fit.
result could be utilized in the XANES fitting loop, as listed in the4. Conclusion and outlook
We have developed a twostep scheme for data analysis of the TRXAS. First we fit the difference Rspace and obtained the bond length change of the first shell coordination; then we fit the difference XANES to obtain the threedimensional structure. The scheme is flexible in the XANES fit where both the spectrum calculation package and the global optimization algorithm for the variables are changeable. This scheme was applied in the analysis of a photoinduced spin crossover iron complex. The fit results are in agreement with the structure of the HS state analogue and the reported results. Next, some other calculation packages, such as XSPECTRA, based on the projector augmented wavefunction (Gougoussis et al., 2009), and OCEAN, based on the Bethe–Salpeter equation (Gilmore et al., 2015), will be included in our scheme. Also the correlation between the structure parameters and the intermediate fraction is a desirable research direction.
inSupporting information
Tables S0 to S10. Figures S1 to S8. DOI: https://doi.org/10.1107/S1600577517005719/rv5067sup1.pdf
Acknowledgements
We thank Qingyuan Meng and Guowei Huang in Professor Lizu Wu's group at Technical Institute of Physics and Chemistry for the synthesis of the Fe(II)(2CH_{3}phen)_{3} and the advice on the Fe(II)(phen)_{3} order. We are grateful to Xiaoyi Zhang at the Advanced Photon Source of the Argonne National Laboratory for the TRXAS data collection.
Funding information
Funding for this research was provided by: National Natural Science Foundation of China (award No. U1332205); Knowledge Innovation program of the Chinese Academy of Sciences (award No. KJCX2WN42).
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