Application of Singular Value Decomposition to photokinetic data of

Basic principle20

A series of photokinetic spectra can be written as a matrix M with the wavelength and the time as the two dimensions. The matrix M can be represented by a product of matrices Ufull and VT

full, possessing orthonormal columns and rows, respectively, and a rectangular diagonal

matrix Sfull by means of Singular Value Decomposition (SVD) (eq. (23), Figure 15).[149]

Thereby, columns in Ufull can be regarded as "basic spectra" that are linearly combined using

time-dependent factors in VT

full, i.e. the "basic kinetics", to yield the series of spectra M. The

singular values in Sfull give the "weights" by which the components of Ufull and VTfull are

considered. Application of SVD to photokinetic spectra of the AB(2Φ)-type gives only two large singular values, due to the fact that only two independent components are needed to fully

20 _{The basic idea and style of representation are adapted from a tutorial by Prof. N. P. Ernsting:} Decomposition of spectra of a thermodynamic equilibrium system, Humboldt-Universität zu Berlin,

represent the data. The other singular values are significantly smaller, representing random noise of the measurement, and can be dropped. Thus, truncated matrices U and VT _{are obtained}

each possessing two columns/rows containing two basic spectra and corresponding basic kinetics, respectively (eq. (24)). Note that at this point U and VT _{have no physical meaning.}

= ∙ ∙ (23)

≈ ∙ ∙ = ∙ ∙ ∙ = ∙ (24)

Figure 15. Principle of Singular Value Decomposition of photokinetic spectra and transformation to a

product of species associated spectra (SAS) and time-dependent concentration profiles KT_.

The photophysical kinetics of the AB(2Φ) system can be calculated by numerical integation of equations (18a-b), as described in the preceding section, assuming some reasonable values for the rate constants k1 and k2. This kinetic model, represented in a matrix

KT, can be transformed by a factor matrix F to the basic kinetics VT obtained from the SVD procedure (eq. (25)). F is calculated via the pseudoinverse (KT₎+ _{(eq. (26)), i.e. the obtained}

matrix Fopt contains the parameters that give the "best" projection of KT on VT by means of a

least squares procedure. Note thar for arbitrary values of k1 and k2 the product Fopt  KT is not

identical with VT_{. Thus, by varying the values for k}

1 and k2, calculation of KT, Fopt , and VTopt

and finally minimizing the sum of the quadratic errors E = (VT

opt-VT)2 of each element of the

matrices (eq. (28)) values for k1 and k2 are found that fit the kinetics VT obtained from SVD and thus describe the full photokinetic data in M. If the found kinetics KT_(k

1, k2) with optimized values for k1 and k2 represent the "true" photokinetics of the system, then the "true" spectra of both species ("species associated spectra", SAS) are represented by the product of U, S, and Fopt

= ∙ (25)

= ∙ = ∙ ∙ ∙ (26)

( , ) = ∙ ( , ) (27)

( , ) = ∑, ( , ) − _, (28)

It is important to note here that in principle this SVD based and other related procedures are able to decompose a series of spectra recorded during a physicochemical process into the (unknown) pure spectra of the species involved by fitting with a physical model. For example it is successfully applied for complicated protolytic equilibria or analysis of time-resolved absorption spectra.[149-150] _{However, in case of an reversible photoreaction of the AB(2Φ)-type}

the pure photokinetic data do not contain enough information to fully determine parameters k1 and k2 and thus to obtain the unknown spectrum of the isomer B.[143a] Instead, only the "pseudo rate" R = k1 + k2 is characteristic for a given series of photokinetic spectra.[141]

Consequently, application of the outlined procedure to a series of spectra of DAE 3d recorded upon irradiation with 313 nm light (Figure 16a) gives no defined minimum for parameters k1 and k2 during the optimization step. A global analysis by systematic variation of the parameters reveals that all pairs of k1/k2 fulfilling the relation k1 = R – k2 fit the experimental data (Figure 16b). Using these pairs of k1/k2 the corresponding species associated spectra SAS can be calculated. Thus, under the premise that none of the parameters can be negative and the obtained spectra have to be positive or equal zero at any wavelength, for the unknown isomer B only a range of spectra can be defined which contains the true one (Figure 16c).

Figure 16. SVD based evaluation of one set of photokinetic data of 3d (λirr = 313 nm). a) UV/Vis absorbance spectra of a solution of 3d in acetonitrile (c = 3.010-5 _{M) recorded under irradiation with}

313 nm light (I0 = 5.210-10 E s-1 cm-3, ferrioxalate actinometry) at 25 °C. Spectra were recorded every 12 s. b) Global analysis of the total quadratic error Log(Etotal) according to equation (28) by systematic variation of k1 and k2 in steps of 100 M-1. The red cross marks the "true" values for k1/k2 as obtained from fitting with two photokinetic datasets (vide infra). c) Range of plausible spectra for the ring-closed isomer as calculated from the SAS matrix.

SVD based analysis of photokinetic data recorded with two different irradiation wavelengths

To determine the spectra of the unknown isomers and the quantum yields from pure photokinetic data irradiation with a second wavelength is needed.[143a] _{Thereby, the assumption}

is made that quantum yields are wavelength independent.

This assumption is originally used in "Fischer's method" that calculates the spectrum of the unknown isomer from the difference of absorbance in the PSSs that are reached under irradiation with two different wavelengths.[142] _{Thereby, the ratio of the molar absorptivities of}

the two isomers has to be different at the two irradiation wavelengths. However, relying solely on the absorbance in the PSS can be erroneous, in particular when the observed difference in absorbance is small or no stable PSS is reached due to by-product formation. Both is typically the case for DAEs. Furthermore, if a thermal back reaction has to be considered Fisher's method can only be applied using high light intensities or low temperatures.[151]

Micheau and coworkers have shown that full evaluation of photochromic equilibria can be performed by simultaneous multi-variable curve fitting of several absorbance-time profiles recorded using two or more irradiation wavelength.[143b,152] _{For an AB(2Φ) system under}

irradiation with two different wavelengths and , five variables ( , , ε , , ) are unknown. They can be determined by simultaneously fitting at least 4 different absorbance- time profiles (with the irradiation/observation wavelengths: / , / , / and / ). However, following this approach on experimental data of DAE 3d it was found that although one set of parameters could be identified to give the best fit, multiple local minima were found depending on the initial guess, which makes the fitting procedure unreliable. This is an intrinsic problem of multi-variable curve fitting with five free variables.

Here it is shown that the application of SVD on photokinetic data recorded with two different irradiation wavelengths leads to the situation that only two unknowns (k1 and k2) have to be fitted simultaneously while the information on the molar absorptivities of the species is automatically obtained from the species associated spectra SAS. This greatly improves the reliability of the fitting procedure as only one global minimum is found, independent of the initial guess. Note that, in analogy to Fischer's method, wavelength independence of the quantum yields is assumed and the ratio of the molar absorptivities of the two isomers has to be different at the two irradiation wavelengths.

From measurements using two different irradiation wavelengths and two series of photokinetic data Mλ1 and Mλ2 are obtained. Both are subjected to SVD, as described above, to

yield truncated matrices U, S and VT _{for each set of data (equations (29) and (30)). At this point}

matrices U and S are combined to US. As both sets of photokinetic data originate from the same molecular species, the basic spectra collected in the matrices USλ1 and USλ2 have to be identical.

Thus USλ2 is replaced by the product of USλ1 and a transformation matrix X. Combining X and

and Mλ2 by a matrix product of identical basic spectra and two different basic kinetics

and _, (right side of eq. (30)). The transformation matrix X is calculated using the pseudoinverse ( ) (eq. (31)).

≈ ∙ (29)

≈ ∙ = ∙ ∙ = ∙ , (30)

= ( ) ∙ = ∙ ∙ ∙ (31)

As described above, the basic kinetics and _, can be transformed to a product

F  KT _{with K}T _{containing the concentration-time profiles obtained by numerical integration of}

rate equations and guessing starting values for the rate constants (equations (32) and (33)). This is done for the first set of photokinetic data assuming some values for k1 and k2, Fopt is

determined according to eq. (26), and the matrix SAS is calculated. Note that for the second set of photokinetic data Fopt has to be identical, as in the end the same species associated spectra SAS shall be obtained. However, the kinetic model for the second dataset contains rate constants k3 and k4 which are different to k1 and k2 due to different absorbance of the species at the irradiation wavelength. Nevertheless, k3 and k4 are related to k1 and k2, respectively, by their molar absorptivities at λ1 and λ2 (equations (33) and (34)). Importantly, the information on molar absorptivites can be found in the species associated spectra SAS, already calculated from

USλ1 and Fopt.

≈ ∙ ∙ ( , ) = ∙ ( , ) (32)

≈ ∙ ∙ ( , ) = ∙ ( , ) (33)

= ∙ (34)

= ∙ (35)

Thus, the overall optimization procedure is the following:

1) Starting values for k1 and k2 are guessed and is determined by numerical integration of rate equations (18a-b).

2) Fopt is calculated according to eq. (26) using .

4) From the SAS matrix information on the molar absorptivities of the two species at the

two irradiation wavelengths is taken to calculate k3 and k4 (eq. (34) and (35)), and hence to determine by numerical integration of rate equations (18a-b).

5) The matrices _, and _, are calculated (eq. (36) and (37)) and the overall sum of quadratic errors Etotal is determined according to eq. (38).

6) Etotal is minimized by variation of k1 and k2 and repetition of steps 1) - 5).

, ( , ) = ∙ ( , ) (36) , ( , ) = ∙ ( , ) (37) ( , ) = ∑ _, ( , ) − , , + ∑ _, ( , ) − _, , , (38)

The procedure was applied to two series of photokinetic data, the first under 313 nm (Figure 16a) and the second under 297 nm irradiation (Figure 17a). As can be seen from the irradiation spectra, molar absorptivities of the two isomers are fairly different at these two wavelengths. Importantly, SVD of both datasets and subsequent fitting of the kinetics provides a single global minimum for the parameters k1 and k2 (Figure 16b). The spectra of both isomers are obtained from the SAS matrix after the parameter optimization. Comparison of the calculated spectra with the spectra of the isolated isomers of 3d shows excellent agreement (Figure 16c). Finally the quantum yields are easily obtained by dividing k1 and k2 with molar absorptivities of the isomers at the irradiation wavelength, as given in the SAS matrix.

Figure 17. SVD based evaluation of two sets of photokinetic data of 3d (λ1 = 313 nm, λ2 = 297 nm). a) UV/Vis absorbance spectra of a solution of 3d in acetonitrile (c = 3.010-5 _{M) recorded under}

irradiation with 297 nm light (I0 = 6.610-10 E s-1 cm-3, ferrioxalate actinometry) at 25 °C. Spectra were recorded every 12 s. b) Global analysis of the total quadratic error Log(Etotal) according to equation (38) by systematic variation of k1 and k2 in steps of 100 M-1(k1) and 20 M-1 (k2). The red cross marks the minimum found during the optimization procedure. c) Comparison of spectra obtained from the SAS matrix after optimization and spectra of the isolated isomers of 3d.