Published: August 19th, 2021
This protocol introduces Franck-Condon Lineshape Analyses (FCLSA) of emission spectra and serves as a tutorial for the use of ARL Spectral Fitting software. The open-source software provides an easy and intuitive way to perform advanced analysis of emission spectra including excited state energy calculations, CIE color coordinate determination, and FCLSA.
The ARL Spectral Fitting application provides a free, publicly accessible, and fully transparent method for performing Franck-Condon Lineshape Analysis (FCLSA) on spectral data, in addition to CIE color coordinate determination and basic spectral processing. While some of the features may be found in commercial software or in programs made by academic research groups, we believe that ARL Spectral Fitting is the only application that possesses all three of the aforementioned attributes.
This program is intended as a standalone, GUI-based application for use by an average laboratory researcher without requiring any coding knowledge or proprietary software. In addition to the standalone executable hosted on ARL GitHub, the associated MATLAB files are available for use and further development.
FCLSA augments the information found in luminescence spectra, providing meaningful insight into the relationship between the ground and excited states of a photoluminescent species. This insight is achieved by modeling spectra with two versions (modes) of an equation that are characterized by either four or six parameters, depending on which mode is used. Once optimized, the value of each of these parameters can be used to gain insight into the molecule, as well as to perform further analysis (for example, the free energy content of the excited-state molecule). This application provides tools for easy by-hand fitting of imported data, as well as two methods for optimizing this fit-damped least-squares fitting, powered by the Levenberg-Marquardt algorithm, and derivative-free fitting utilizing the Nelder-Mead simplex algorithm. Furthermore, estimations of sample color can be performed and reported in CIE and RGB coordinates.
Photoluminescence measurements, comprising both fluorescence and phosphorescence spectra, are widely used throughout various academic fields and industrial applications1. Photocatalysts are increasingly used in organic synthesis for the production of complex and valuable target molecules2,3,4. In order to determine the energetics of photocatalysts, the excited state energy is routinely estimated using emission spectra. The development of novel lighting materials, such as organic light emitting diode (OLED) luminophores, necessitates that the observed color output be characterized and reported5,6. Commission international de l'éclairage (CIE) color coordinates are routinely used for this purpose7.
The purpose of the ARL Spectral Fitting application is to provide a quick and easy method to augment spectral data through meaningful analysis that is widely accessible both in terms of ease-of-use and availability (https://github.com/USArmyResearchLab/ARL_Spectral_Fitting). This software performs several routine spectral processing functions automatically for the user, including data normalization and conversion between wavelength, λ, and wavenumber, , units with appropriate intensity scaling as shown in the equation below1. The software is capable of handling a variety of input and output file formats. Several advanced analyses are easily performed using the software such as the calculation of CIE and chromaticity coordinates, color prediction, determination of the excited state free energy (ΔGES) in various units, and FCLSA for the determination of the FCLSA parameters8.
A graphical user interface (GUI)-based application was pursued because it allows any researcher to perform this analysis and requires no background knowledge of computer science. This application was written in MATLAB, using its App Designer tool. Outside of ARL Spectral Fitting, finding a publicly-accessible implementation of an application designed to perform Franck-Condon Lineshape Analysis is practically impossible. This is because research groups do not publicly release their implementations, preferring instead to keep them proprietary.
Franck-Condon Lineshape Analysis (FCLSA) is often used in the photophysical characterization of novel compounds because of the rich information it conveys about the molecule9,10,11,12,13,14. Each of the four parameters (six if in double mode) gives information about the excited state of the molecule. The energy quantity, or 0-0 energy gap, (E0) is the difference in zeroth energy levels of the ground and excited states of the molecule. The full width at half maximum (Δv½) informs about the widths of individual vibronic lines. The electron-vibrational coupling constant, or Huang-Rhys factor, (S) is a dimensionless calculation based on the equilibrium displacement between ground and excited states of the molecule15. Finally, the quantum spacing parameter (ħω) is the distance between vibrational modes that govern the nonradiative decay of a molecule.
The equations for single and double mode FCLSA are as follows:
where the parameters are as previously defined. In the double mode equation, S and ħω are separated into medium (M) and low (L) energy terms. is the intensity at the wavenumber v10,16,17,18. In both equations, the summation is performed over N quantum levels with a default value of N = 5, as is commonly used in the literature11, but any integer can be specified in the ARL Spectral Fitting Software under Settings | Fit.
1. Data import
2. Data processing
NOTE: The user may wish to perform data processing prior to the fitting process. Available processes include:
3. Manual fitting
NOTE: Based on the amount of structure visible in the spectrum, it may be highly advantageous to initialize the fitting parameters with appropriate estimates prior to optimization. This initialization can decrease the time required for optimization and helps to ensure that the values returned by optimization are realistic for the spectrum.
5. Chromaticity and free energy calculations
6. Data export
Using the fitting routine described above, Franck-Condon Lineshape Analysis was performed on two spectra that come pre-packaged with the application: the room temperature (292 K) and low temperature (77 K) emission spectra for 9,10-diphenylanthracene dissolved in toluene. Measurements were obtained using a spectrofluorometer with fluid solutions in 1 cm cuvettes and a standard cuvette holder for room temperature measurements. The low temperature measurements were obtained by immersing NMR tubes into liquid nitrogen in a dewar to generate frozen glass samples. All spectra were corrected for the detector response. A single mode fit was sufficient for the room temperature spectrum, while double mode was used to model the low temperature spectrum. Color analysis was performed on both spectra and found to yield similar estimates.
To fit the room temperature spectrum, by-hand adjustment was used after least-squares optimization with default customizations. The final parameter values obtained were as follows: E0 = 24380 cm-1, Δv½ = 1200 cm-1, S = 1.25, ħω = 1280 cm-1. The resulting coefficient of determination calculated was 0.99947 as shown in Figure 1. Calculation of free energy of the excited state using these parameter values yielded a value of 25,000 cm-1.
Simplex optimization was used to fit the low temperature spectrum. By-hand adjustment was not necessary after optimization. The final parameter values obtained were as follows: E0 = 24764 cm-1, Δv½ = 746 cm-1, S1 = 1.13, ħω1 = 1382 cm-1, S2 = 0.31, ħω2 = 651 cm-1. The resulting coefficient of determination calculated was 0.9991 as shown in Figure 2. Calculation of free energy of the excited state using these parameter values yielded a value of 25,700 cm-1.
Color analysis of the low temperature spectrum yielded the following results: chromaticity coordinate = [0.15819, 0.03349], CIE coordinate = [0.19571, 0.041432, 1], and predicted RGB value = [67, 0, 233]. The values obtained for the room temperature spectrum were very similar to that of the low temperature spectrum with unperceivable color differences.
Figure 1: Single mode fit of 9,10-diphenylanthracene (292 K): This figure shows the room temperature emission spectrum of 9,10-diphenylanthracene and its FCLSA fit function, achieved through least-squares optimization followed by by-hand adjustment of parameter values. This is an example of a loosely structured spectrum. Please click here to view a larger version of this figure.
Figure 2: Double mode fit of 9,10-diphenylanthracene (77 K): This figure shows the low temperature emission spectrum of 9,10-diphenylanthracene and its FCLSA fit function, achieved through a simplex optimization. This is an example of a highly structured spectrum. Please click here to view a larger version of this figure.
This application provides an easy and rapid analysis of emission spectra through two main methods commonly used in the photophysical community. The first is Franck-Condon Lineshape Analysis (FCLSA), which gives insight into the energetics and vibronic coupling associated with decay of excited state molecules back to their ground states. This is achieved by optimizing parameter values to maximize the goodness of fit of a spectrum using one of two possible FCLSA modelling equations. The second method of analysis provides insight into the observed color of the light emitted from the molecule. By combining tristimulus color curves with provided intensity data, the CIE coordinate can be calculated. This determination allows for the highly accurate color prediction of both absorption and emission spectra.
Experimental photoluminescence spectra are commonly measured using a photomultiplier tube (PMT) or charge coupled device (CCD) as a detector and plotted as emission intensity versus wavelength (nm). Many photophysical characterizations, including FCLSA and calculation of the free energy of the excited state, are performed in wavenumber space, as demonstrated by the use of (cm-1) in the corresponding equations above. In addition to the x-axis conversion, the emission intensity as measured versus wavelength, denoted as I(λ) must be converted to . This application automatically identifies the original x-axis units of imported spectral data as either wavelength (nm) or wavenumber (cm-1). By default, the application then converts the spectral data, normalizes the spectrum to unity at the maximum intensity peak, and plots the spectrum as "Normalized vs. wavenumber (cm-1)" to indicate that the correct intensity conversion was applied. Although it is recommended that all fitting be performed using wavenumber units, the application can also plot the spectrum as "Normalized I(λ) vs. wavelength (nm)" by following the instructions in section 2 above.
There are two optimization algorithms available for use in the application. The default option is damped least-squares, which utilizes the Levenberg-Marquardt algorithm21. Combining a version of gradient descent and the Gauss-Newton algorithm, this algorithm finds local, not necessarily global, minima. While this is a significant limitation, the algorithm offers advantages in its customizability-this method can take into account preferential weighting of data points, perform robust fitting, and display advanced goodness-of-fit statistics22. The alternative method of optimization is derivative-free, powered by the Nelder-Mead simplex algorithm23. This algorithm uses a heuristic method to return a global minimum of the given cost function (in this case, a sum of squared differences between predicted and observed intensities). The simplex method has been used for FCLSA before, though the code implementing it was never published24.
Both the least-squares and simplex optimization methods work best for structured spectra that exhibit narrow, well defined, and symmetric peaks. As spectra become less structured, meaning that they lose symmetry and the peaks broaden, these methods lead to less robust fits where parameters can become highly correlated. Typically, spectra recorded at low temperatures or in rigid media are more structured compared to those obtained near room temperature or in fluid solution12,25,26. The robust fit options included with the least-squares method can help to alleviate this issue. This problem can be significantly diminished if one or more of the parameters are fixed to a constant value during optimization. For instance, IR spectroscopy experiments can be used to determine relevant quantum spacing (ħω) values. Alternatively, relevant literature values can be used to set custom bounds for the parameters.
In some instances, the FCLSA fit, and parameters obtained from the optimization routines do not adequately represent the data even when robust fit options or fixed parameters are employed. This is a failure of the fitting algorithms and may be associated with the multiple FCLSA fitting parameters (potential overparameterization) or spectral shape of the data (featureless spectra). In these cases, further improvement of the fits may be obtained using a "by-hand fit" of the data with manipulation of the FCLSA parameters. The adequacy of such fits can be assessed visually and quantified by comparing goodness-of-fit statistics that are automatically included in the plot.
A general routine to follow for an accurate by-hand fit consists of the following five steps: First, determine an initial estimate for E0 manually or automatically using one of the three methods provided. By default, the parameter's value is assigned to the wavenumber associated with the highest intensity peak detected upon data import. Alternatively, the user can define E0 as the wavenumber at which the emission spectrum intersects its corresponding excitation spectrum. The final method to determine E0 uses the so-called X% Rule, where X = 1 or 10. In this method, E0 is assigned to a wavenumber X% of the full width at half-maximum (FWHM) intensity of the most prominent data peak assuming a Gaussian band shape. The second step in the by-hand fitting protocol is to calculate ħω based on quantum spacing observed in the structure of the emission spectrum. If possible, refer to the IR spectrum of the molecule and try to correlate the photoluminescence-based value to a strong band in the IR spectrum. Third, determine S based on the relative intensities of spectral peaks. Fourth, determine a rough Δv½ based on bandwidth. Fifth, iteratively readjust S and Δv½ as necessary.
The difficulty with performing FCLSA using broad, relatively featureless spectra was demonstrated through the fitting procedure for 9,10-diphenylanthracene in fluid solution at 292 K compared to that performed for the more structured spectrum obtained in frozen glass at 77 K. When fitting the room temperature spectrum, optimization returned an initial coefficient of determination of 0.9971 that was improved to 0.9994 through by-hand tuning of the parameters and visual inspection of the results. In contrast, by-hand fitting of the low temperature variant was unnecessary due to the fine structure of the spectrum that resulted in a coefficient of determination equal to 0.9991 after simplex optimization.
In many instances, both optimization routines (least-squares and simplex) return very similar results. This is indicative of them finding a global minimum for the FCLSA parameters. In general, the least-squares method tends to be better suited to data that is noisy, is not well structured, or contains many near-zero data points at the spectrum’s tails. Conversely, the simplex method tends to return better fits than the least-squares method for data that is well structured and possesses few outlier points. In these cases, the simplex method typically requires little by-hand pre-optimization of parameter values and no adjustment after optimization. For those cases in which the data’s noise or overall lack of structure prevents a high-quality fit using either of the provided optimization methods, it is recommended that the by-hand fitting method (see above) be employed with no subsequent optimization.
This application offers several advantages over previous implementations of Franck-Condon Lineshape Analysis. The first and most important advantage is that it is free, publicly accessible, and fully transparent. This is accomplished by posting the code to GitHub, providing access to anyone with a computer and internet connection (https://github.com/USArmyResearchLab/ARL_Spectral_Fitting). Not only can anyone access this application, but they can also view the underlying code. This provides an opportunity for community-sourced feedback and development. An additional advantage lies in the ease-of-use of this application. No background knowledge of computer science or command line interaction is required. Rather, this software employs a simple graphical user interface (GUI) that enables researchers of all backgrounds to perform the spectral analyses described above. Furthermore, this application provides the user multiple options for control over the optimization methods and can be used to determine the free energy of the excited state. Finally, the software calculates and reports several useful color values including chromaticity coordinates, CIE coordinates, RGB, and hexadecimal color codes. All of these analyses can be accomplished in seconds, requiring only that the user press a button.
The authors have nothing to disclose.
Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-20-2-0154. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
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