Module 3 – Calculation of Spectroscopic Properties

Figure 1 shows the module for calculating spectroscopic properties, such as experimental and theoretical intensity parameters, energy transfer rates, and emission quantum yield.

Figure 1. Module for spectroscopic properties calculations.

The primary objective of the theoretical protocol implemented in LUMPAC is focused on calculating the theoretical emission quantum yield. Therefore, the module for calculating the spectroscopic properties is structured into four submodules:

i) The first submodule determines the experimental intensity parameters ($\Omega_{\lambda}$) using the experimental emission or excitation spectrum.

ii) The second submodule calculates the theoretical intensity parameters by fitting the charge factors and polarizabilities to reproduce the experimental intensity parameters. By default, LUMPAC uses the QDC model [1] to adjust the intensity parameters, estimating the charge factors and polarizabilities from the adjustable Q, D, and C parameters, along with the atomic charge and electrophilic superdelocalizability of the coordination polyhedron atoms.

iii) The third submodule calculates the ligand-lanthanide ion energy transfer rates using the intensity parameters and the energies of the singlet and triplet excited states. If the lifetime of the ⁵D₀ emitter level of the Eu³⁺ complex is supplied, the theoretical emission quantum yield is quantified.

iv) Finally, the fourth submodule generates the theoretical absorption spectrum using the ORCA or GAUSSIAN output files.

The following sections will detail the calculation of the cited properties using LUMPAC.

Calculation of Experimental Intensity Parameters

Warning: Users must provide the experimental emission or excitation spectrum file to calculate the experimental intensity parameters.

The experimental intensity parameters for the Eu³⁺ ion are calculated using the following equation:

where the factor $\chi =n{{(n+2)}^{2}}/9$ is known as the Lorentz local field correction term. The refractive index ($n$) varies depending on the medium; for the solid state, $n$ = 1.5. ${\left\langle^7F_2\|U^{(\lambda)}\|^5D_0\right\rangle}^2$ = 0.0032 and ${\left\langle^7F_4\|U^{(\lambda)}\|^5D_0\right\rangle}^2$ = 0.0023 correspond to the squared reduced matrix elements of the unit operator. The A₀₁ term, representing the radiative emission rate of the ⁵D₀$\rightarrow$⁷F₁ transition, is calculated by Eq. $\eqref{eq::a01}$. The A₀₂ ( ⁵D₀$\rightarrow$⁷F₂) and A₀₄ ( ⁵D₀$\rightarrow$⁷F₄) quantities are given by Eq. $\eqref{eq::a0l}$.

The $S_{01}$ and $S_{0\lambda}$ parameters are the areas under the peaks of the ⁵D₀$\rightarrow$⁷F₁ and ⁵D₀$\rightarrow$⁷F$_\lambda$ transitions, respectively. The $\nu_{01}$ and $\nu_{0\lambda}$ quantities are the barycenter energies of ⁵D₀$\rightarrow$⁷F₁ and ⁵D₀$\rightarrow$⁷F$_\lambda$, respectively. For the Eu³⁺ ion, ⁵D₀$\rightarrow$⁷F₁ is assumed as the reference transition due to its magnetic dipole nature. As a result, this transition is practically independent of the chemical environment and exhibits minimal variation across different Eu³⁺ complexes.

As mentioned earlier, LUMPAC calculates the experimental intensity parameters for Eu³⁺ complexes using the experimental emission or excitation spectrum. From the emission spectrum, the areas corresponding to the ⁵D₀$\rightarrow$⁷F₁, ⁵D₀$\rightarrow$⁷F₂, and ⁵D₀$\rightarrow$⁷F₄ transitions must be defined. For the excitation spectrum, the areas of the ⁷F₀$\rightarrow$(⁵D₄, ⁵L₆, ⁵D₂, and ⁵D₁) transitions are required. The ⁷F₀$\rightarrow$⁵D₁ area is constant and serves as a calibration to determine $\Omega_2$ (⁷F₀$\rightarrow$⁵D₂), $\Omega_4$ (⁷F₀$\rightarrow$⁵D₄), and $\Omega_6$ (⁷F₀$\rightarrow$⁵L₆). LUMPAC provides a user-friendly interface for selecting these areas.[2]

Procedure for Calculating Intensity Parameters and Radiative Emission Rate with LUMPAC

1. Click on button (Figure 2) to open the .txt file of the emission or excitation spectrum, where wavelengths and intensities must be separated by a comma (“,”) or a space (“ ”).

Figure 2. LUMPAC interface for inserting the emission or excitation spectrum and generating the chromaticity diagram.

2. A chromaticity diagram can be generated from the selected emission spectrum by clicking on button (Figure 2). The chromaticity diagram of the [Eu(btfa)₃(bpy)] complex is shown in Figure 3, illustrating its emission color in the visible spectrum.

Figure 3. Chromaticity diagram of the [Eu(btfa)₃(bpy)] complex in dichloromethane.

3. For an emission spectrum, the areas of the ⁵D₀$\rightarrow$⁷F₂, ⁵D₀$\rightarrow$⁷F₁, and ⁵D₀$\rightarrow$⁷F₄ transitions, and optionally ⁵D₀$\rightarrow$⁷F₀, ⁵D₀$\rightarrow$⁷F₃, and ⁵D₀$\rightarrow$⁷F₅, must be appropriately selected using the LUMPAC interface (Figure 4).

It is important to specify the refractive index of the medium in which the spectrum was obtained to correct for light deviations caused by the medium. The emission spectrum of the [Eu(btfa)₃(bpy)] complex was obtained in dichloromethane (DCM), which has a refractive index of 1.424. The observed decay time ($\tau$) is another essential parameter. Combining $\tau$ and the radiative emission rate (A_rad), determined by selecting bands in the spectrum, the non-radiative emission rate (A_nrad) of the complex can then be calculated.

Figure 4. Procedure for selecting the areas under the main transitions for compounds based on the europium ion.

4. The emission spectrum areas can be selected by numerically entering the initial and final wavelengths of the band or by using the sliders . Figure 5 shows all possible selections.

Figure 5. Emission spectrum viewer with all possible areas selected and spectrum visualization options.

5. To calculate the spectroscopic properties using an excitation spectrum, select the “Excitation (JOEX)” option (Figure 6). The interface operates similarly to that of the emission spectrum, but refractive indices must be added individually for each band.

Figure 6. Interface for selecting the bands of the excitation spectrum.

6. After selecting the bands from the emission or excitation spectrum and adding the required data, click on button (Figure 2 and Figure 5) to execute the calculation of the experimental intensity parameters and radiative emission rate.

Warning: The data is saved in a file with the .lumpacexp extension, which adopts the same filename as the corresponding spectrum .txt file.

Theoretical Calculation of Intensity Parameters

Figure 7 shows the LUMPAC submodule, which calculates the theoretical intensity parameters. A nonlinear algorithm adjusts the charge factors ($g$) and polarizabilities ($\alpha$), used in the calculation of the $\gamma_p^t$ and $\Gamma_p^t$ parameters, respectively, to reproduce the experimental values of $\Omega_2$ and $\Omega_4$.

Figure 7. LUMPAC submodule for calculating the theoretical intensity parameters.

Procedure for Calculation of the Theoretical Intensity Parameters using LUMPAC

1. Click on button (Figure 8) to open the MOPAC output file (.out) containing the optimized geometry or the MOPAC input file (.mop) with the initial structure.

Figure 8 lists all quantities that can be included in the output file of the submodule for calculating the intensity parameters. By default, theoretical intensity parameters ($\Omega_{\lambda}$), forced electric dipole parameters ($\Omega_{\lambda}^{ED}$), charge factors ($g$), and polarizabilities ($\alpha$) are always calculated. Other data, such as the spherical coordinates of the coordination polyhedron, theoretical radiative emission rate (A_rad), and the effect of each ligand on A_rad, are selected by default but can be omitted, and additional options can be selected.

Figure 8. Input files supported by LUMPAC in calculation of the theoretical intensity parameters and optional quantities for the output file.

2. In LUMPAC 2.0, there are four different procedures for calculating the theoretical intensity parameters (Figure 9).

LUMPAC allows the simultaneous calculation of the theoretical intensity parameters for multiple files in the same directory by selecting the “Multiple” option (Figure 9). For LUMPAC to recognize the experimental intensity parameters for each complex, the .lumpacexp and .out files must have identical prefixes.

Figure 9. LUMPAC 2.0 interface for selecting the method of fitting theoretical intensity parameters.

2.1. By default, LUMPAC uses the QDC fitting (Figure 10). The g and α quantities, within physical limits, are calculated using three adjustable parameters (Q, D, and C).

The intensity parameters and the coordination number can be manually entered if they are not automatically loaded (Figure 10). The QDC fitting, initially implemented in the first version of LUMPAC and continued in version 2.0, assigns a different $g$ and $\alpha$ to each ligand atom, because these quantities are estimated as a function of the charge and electrophilic superdelocalizability of each atom, which vary among atoms. When applying the QDC fitting, the chemical partitioning can also be performed, accounting for the contribution of each ligand to A_rad, as illustrated in Figure 11).

Figure 10. LUMPAC interface dedicated to the QDC fitting.

2.2. Another option is to manually add the Q, D, and C values from a previous calculation, as shown in Figure 11.

Figure 11. Interface for manual input of the QDC parameters and ligand labels assignment to evaluate the chemical partitioning of the ligands on A_rad.

2.3. An alternative procedure for calculating intensity parameters applies to the previous adjustment, used by default before the implementation of the QDC model. In the previous adjustment, $g$ and $\alpha$ for the ligand atoms are arbitrarily grouped based on their chemical environment. LUMPAC provides a simple way for grouping atoms that share the same chemical environment (Figure 12).

The following atoms are grouped together (Figure 12) for the [Eu(btfa)₃(bpy)] complex: N8 and N9 (bipyridine nitrogens); O2, O3, O4, O5, O6, and O7 (β-diketone oxygens). Users can drag the ligand atoms from the “Ligand Atoms” box to their respective groups.

Figure 12. LUMPAC interface for defining the groups of ligand atoms based on the charge factors and polarizabilities.

2.4. Finally, $g$ and $\alpha$ determined previously can be manually entered for each ligand atom, as shown in Figure 13.

Figure 13. LUMPAC interface for manual insertion of $g$ and $\alpha$.

Because the radiative emission rate depends on $\Omega_6$, which is not measured experimentally, the calculations of the theoretical intensity parameters are essential. These calculations determine the contributions of the dynamic coupling ($\Omega_{\lambda}^{DC}$) and the electric dipole ($\Omega_{\lambda}^{ED}$) mechanisms, and $\Omega_{\lambda}^{ED}$ is useful for calculating the ligand-metal ion energy transfer rates via the multipolar mechanism.

Calculation of Energy Transfer Rate and Emission Quantum Yield

Figure 14 shows the module for calculating the energy transfer rates and, if the lifetime is provided, the emission quantum yield. These quantities are calculated using the intensity parameters estimated in the previous section, and the excited state energies contained in the ORCA output file.

Figure 14. Module for calculating energy transfer rates and emission quantum yield.

Procedure for Calculating Energy Transfer Rates and Emission Quantum Yield using LUMPAC

1. Click on button (Figure 15) to open the ORCA output file (.orcout), generated by the procedure in Module 2.

In LUMPAC 2.0, in addition to using the semiempirical excited states calculated in Module 2, data from DFT calculations performed with the ORCA and GAUSSIAN programs can be used. Another new feature of LUMPAC 2.0 is the possibility of calculating the energy transfer rates for Tb³⁺ complexes. The choice of the ion in question can be made as shown in Figure 15. However, the theoretical quantum yield cannot yet be determined Tb³⁺.

2. When the .orcout file is opened, the .omega file containing the previously calculated intensity parameters (Figure 15) will load automatically. To enable this automatic loading, the .orcout and .omega files must be in the same directory and have identical base names.

Attention: If the .omega file is not loaded automatically, manually provide the intensity parameters calculated through the LUMPAC interface or specify the corresponding .omega file.

Figure 15. Calculation of the energy transfer rates and emission quantum yield from the ORCA output file and theoretical intensity parameters.

3. To modify the energy transfer channels, intraligand decay rates, ligand states, and typical values of quantities important for calculating the ligand-lanthanide ion energy transfer rates, users must click on button (Figure 16).

It is possible to select energy transfer channels involving various excited acceptor levels of the Eu³⁺ ion: ⁵D₀, ⁵D₁, ⁵D₂, ⁵D₃, ⁵D₄, ⁵L₆, ⁵L₇, ⁵G₂, ⁵G₃, ⁵G₅, and ⁵G₆, excited from the ⁷F₀ or ⁷F₁ ground states. For Tb³⁺, the available acceptor levels are ⁵D₃, ⁵D₄, ⁵G₄, ⁵G₅, ⁵G₆, ⁵L₆, ⁵L₁₀, ⁵H₅, ⁵H₆, ⁵H₇, and ⁵F₅, excited from the ⁷F₅ or ⁷F₆ ground states. In LUMPAC 2.0, several different ligand levels can be considered in the rate equations. The S₀, S₁, and T₁ states are automatically selected by LUMPAC with the following typical rate values: 1$\times$10⁴ s^-1 for the S₀$\rightarrow$S₁ channel and 1$\times$10⁸ s^-1, 1$\times$10⁵ s^-1, and 1$\times$10⁶ s^-1 for the S₁$\rightarrow$T₁, T₁$\rightarrow$S₀, and S₁→S₀ channels, respectively (Figure 16). However, both the levels and rate values can be manually modified. To consider a new interaction between levels, simply provide the initial state, the final state, and the respective rate. For example, to include the T₅ state with decay rate of 5$\times$10⁹ s^-1 for S₁$\rightarrow$T₅ and 1×10¹⁰ s^-1 for T₅$\rightarrow$T₁, two lines must be added in the “Rate Equations” window (Figure 16): “S1 T1 5e9” and “T5 T1 1e10”.

Figure 16. Interface for modifying parameters for calculating the energy transfer rates.

When the “Effect of the ligand’s Decay Rates on the Quantum Yield” option is selected (Figure 16), LUMPAC varies the decay rates involving the ligand states. This process generates all possible combinations of intraligand decay rates from 1$\times$10⁰ s^-1 to 1$\times$10¹² s^-1, calculating the theoretical quantum yield for each combination. The results are stored in a .simul file, saved in the directory of the .orcout file.

4. Click on button (Figure 15 and Figure 16) to execute the calculation of energy transfer and back-transfer rates using the selected parameters. If the lifetime is provided, the emission quantum yield will also be calculated.

The .simul file with the results of the emission quantum yield evaluation as a function of the intraligand decay rates for the [Eu(btfa)₃(bpy)] complex is shown in Figure 17. The columns labeled “S1_T1”, “T1_S0”, and “S1_S0” refer to the S₁$\rightarrow$T₁, T₁$\rightarrow$S₀, and S₁$\rightarrow$S₀ decay rates, respectively. If an additional ligand level were included in the Rate Equations edit field (Figure 17), it would also be incorporated into the analysis. The “s_calc” column is the theoretical sensitization efficiency, calculated as the ratio between theoretical quantum yield and efficiency, while the “s_exp” column represents the experimental sensitization efficiency. The “yield” column displays the emission quantum yield for each rate combination, limited by the theoretical quantum efficiency. The “erro_s” column shows the error between the theoretical and experimental sensitization efficiency. The final three lines of Figure 17 present the correlation coefficient between the rates for each transfer channel and the quantum yield of the complex. Essentially, the correlation coefficient reveals how much each transfer channel influences the quantum yield. A positive value means that increasing the rate of that channel increases the quantum yield, while a negative value indicates that it decreases it. The absolute value of the coefficient indicates the strength of the effect.

Figure 17. Eu(btfa)₃(bpy).simul file displaying the emission quantum yield as function of the S₁$\rightarrow$T₁, T₁$\rightarrow$S₀, and S₁$\rightarrow$S₀ decay rates.

After the calculation finishes, an energy diagram is generated, as illustrated in Figure 18. This diagram displays the selected energy levels of the ligands and the Eu³⁺ ion on a common scale and also highlights the specified energy transfer and back-transfer channels.

Figure 18. Energy diagram of the [Eu(btfa)₃(bpy)] complex generated by LUMPAC after calculating the energy transfer rates.

Figure 19 shows the Eu(btfa)₃(bpy).lumpac output file containing all the calculated properties, with all options enabled in the “LUMPAC Printing Options” window (Figure 15).

Figure 19. LUMPAC output file for [Eu(btfa)₃(bpy)] showing all calculated spectroscopic properties.

Theoretical Calculation of the Absorption Spectrum

Figure 20 shows the module responsible for obtaining the theoretical absorption spectrum from the output file created by the ORCA and GAUSSIAN programs.

Figure 20. Module for generating the theoretical absorption spectrum from the ORCA and GAUSSIAN output file.

Procedure for Calculation of the Theoretical Absorption Spectrum using LUMPAC

1. Click on button (Figure 21) to open the ORCA output file (.orcout) and select the files listed in the List of Files for generating the spectrum.

Like the previous module, LUMPAC 2.0 also applies DFT data calculated by ORCA and GAUSSIAN for generating the theoretical absorption spectra.

Figure 21. LUMPAC interface for generating the absorption spectrum from the ORCA and GAUSSIAN output file.

Electronic transitions from the singlet ground state to the singlet excited states are allowed, with their probability proportional to the oscillator strength of the transition (fosc), as shown in Figure 22. The theoretical absorption spectrum is generated using these oscillator strengths and excitation energies, applying an arbitrary full width at half maximum (FWHM). Both the FWHM and the wavelength range can be modified to match the experimental spectrum, if needed.

Figure 22. Singlet energies and oscillator strengths (singlet$\rightarrow$singlet transitions) from the .orcout file, used to generate the theoretical absorption spectrum.

2. Click on button to calculate the theoretical absorption spectrum, as shown in Figure 23.

Figure 23 demonstrates the ability to overlay two spectra by selecting two different .orcout files. These files contain the excited states calculated for [Eu(btfa)₃(bpy)] and a generic Eu³⁺ complex, both located in the calculation directory. A click on the bar representing the band intensity provides the molecular orbital composition of the most significant transitions.

Figure 23. Absorption spectrum created by LUMPAC and options for visualizing excitation properties.

References

[1] J. D. L. Dutra, N. B. D. Lima, R. O. Freire, A. M. Simas, Sci. Rep., 2015, 5, 1–12.

[2] A. Ćirić, Ł. Marciniak, M. D. Dramićanin, Sci. Rep., 2022, 12, 1–10.