Article pubs.acs.org/JPCA
Theoretical Methodologies for Calculation of Judd−Ofelt Intensity Parameters of Polyeuropium Systems José Diogo L. Dutra,† José Wesley Ferreira,† Marcelo O. Rodrigues,‡ and Ricardo O. Freire*,† †
Pople Computational Chemistry Laboratory, Departamento de Química, Universidade Federal de Sergipe, 49100-000 São Cristóvão, Sergipe, Brazil ‡ Instituto de Química, Campus Universitário Darcy Ribeiro, P.O. Box 4478, CEP 70904-970 Brasília-DF, Brazil S Supporting Information *
ABSTRACT: When Judd−Ofelt intensity parameters of polynuclear compounds with asymmetric centers are calculated using the current procedure, the results are inconsistent. The problem arises from the fact that the experimental intensity parameters cannot be determined for each asymmetric polyhedron, and this precludes the individual theoretical adjustment. In this study, we then propose three different methods for calculation of these parameters of polyeuropium systems. The first, named the centroid method, proposes the calculation considering the center of the dimeric system as the half distances between the two europium centers. The second method, called the overlapped polyhedra method, proposes the calculation considering the overlapping of both europium polyhedra, and the last one, the individual polyhedron method, proposes the use of theoretical mean values of charge factors and polarizabilities associated with each europium−ligand atom bond to calculate the intensity parameters. One symmetric polyeuropium system and one asymmetric system were assessed by using the three methods. Among the methods assessed, the one based on the overlapped polyhedra produced more consistent results for the study of both kinds of systems.
1. INTRODUCTION It is indubitable that europium materials have attracted great attention due to the technological applications, such as electroluminescent devices,1 luminescent sensors for chemical species,2,3 optical fibers,4 lasers, UV dosimeters,5 and biologic markers.6 This worldwide interest may be associated with its singular optical properties, which justify the growing number of research studies and developments reached in recent years. The number of publications involving lanthanide luminescent systems has increased from 3500 in 2002 to nearly 6000 in 2012. The theoretical comprehension of optical properties of europium compounds has been an important platform for design7,8 of more efficient luminescent systems, because the calculation protocol for this ion is well-established.9 In a general manner, the ground state geometry calculation of a system is the first step of this protocol and can be done by using ab initio methodologies that represent the europium by an effective core potential,10,11 or by using the Sparkle models.12−14 Due its semiempirical nature, the Sparkle models are more efficient computationally; moreover, all Sparkle models present similar accuracy to what can be obtained via ab initio/ECP full geometry optimization calculations,15 so the are highly recommended for the study of systems containing a large number of atoms. © 2013 American Chemical Society
The spherical coordinates of the coordination polyhedron obtained in the first step are used for the calculation of the Judd−Ofelt intensity parameters, Ωλ (λ = 2, 4 and 6), and are still calculated by adjustment of charge factors (g) and polarizabilities (α) of each europium ligand bond to reproduce the experimental ones.9 In parallel, the optimized ground state geometry is also used in the singlet and triplet energy levels as input. The excited states calculations can be accomplished by using the time-dependent density functional theory methodology (TD-DFT)16 or the semiempirical INDO/S-CIS method.17,18The three intensity parameters that are required as input in the calculation of radiative decay rates (Arad), energy transfer, and back-transfer rates. These quantities are fundamental for the determination of the theoretical quantum yield.9 In other words, the Judd−Ofelt intensity parameters are important quantities in this procedure. For mononuclear Eu3+ materials, the theoretical protocol for determination of Ωλ values has shown satisfactory results.19 However, for polynuclear compounds with asymmetric centers, it was possible to observe that the classical procedure has inconsistent results. The problem is that the experimental intensity parameters cannot be determined for each asymmetric Received: October 4, 2013 Revised: December 5, 2013 Published: December 5, 2013 14095
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artifices to estimate the individual contribution of each polyhedron to the experimental luminescent properties. Because of that, the results obtained from each method are not necessarily related to each other directly. The focus of this work is only to evaluate which method allows the best explanation of the experimental data. 2.1. Centroid Method. The first approach, named the centroid method, was proposed for Eu3+ dimeric compounds that have an optical behavior like a single entity. In this case, it is experimentally impossible to obtain the values of intensity parameters for the individual centers. In this method, the europium centers are replaced by a pseudocenter placed at half the distance of the two europium ions, what we call the centroid. This way, all ligand atoms coordinated to distinct centers were considered coordinated to the centroid (Figure 1A). The adjustment is done by considering the spherical
polyhedron and this fact precludes the individual theoretical adjustment. In this work, we have proposed and evaluated three different theoretical procedures for the calculation of Judd−Ofelt intensity parameters of symmetric and asymmetric polynuclear europium systems. The new methodologies here proposed enable the complete luminescent study of each europium center in polynuclear europium systems.
2. METHODS The intensity parameters Ωλ (λ = 2, 4, and 6) are calculated using the eqs 1 and 2 derived from Judd−Ofelt theory.20,21 The theoretical parameters have been calculated by adjusting the charge factors (g) and polarizabilities (α), appearing in eqs 3 and 4, respectively, to reproduce the phenomenological (experimental) values of Ω2 and Ω4. λ − 1, λ + 1(odd) t (all)
Ωλ = (2λ + 1)
Bλtp =
∑
∑
t
p=0
|Bλtp|2 (2t + 1)
⎡ (λ + 1)(2λ + 3) ⎤1/2 2 t+1 ⟨r ⟩θ(t ,λ)γpt − ⎢ ⎥ ⎣ ⎦ ΔE 2λ + 1 × ⟨r λ⟩(1 − σλ)⟨f || C(λ) || f ⟩Γ tpδt , λ + 1
γpt =
Γ tp =
(1)
gj ⎛ 4π ⎞1/2 2 ⎜ ⎟ e ∑ ρj (2βj)t + 1 t + 1 Y pt*(θj ,φj) ⎝ 2t + 1 ⎠ Rj j
αj ⎛ 4π ⎞1/2 ⎜ ⎟ ∑ t + 1 Y pt*(θj ,φj) ⎝ 2t + 1 ⎠ Rj j
(2)
(3)
(4)
In this kind of calculation, it is necessary to provide the spherical coordinates of the coordination polyhedron of the system as input. In this study, we have used the crystallographic coordinates for the two evaluated compounds in all calculations. The spherical coordinates are presented in Tables S1 and S2 of Supporting Information. The radiative emission rate (Arad), taking into account the magnetic dipole (Smd) and forced electric dipole (Sed) mechanisms, is given by eq 5: A(5D0−7F J ) =
⎤ 64π 4v 3 ⎡ n(n2 + 2)2 Sed + n3Smd ⎥ ⎢ 3h(2J + 1) ⎣ 9 ⎦ (5)
The parameter Sed depends on the calculated intensity parameters, as can be seen in eq 6. Sed = e 2
∑ λ = 2,4,6
Ωλ |⟨5D0 || U (λ) || 7F J ⟩|2 (6)
The parameter Smd is quantified theoretically as being 9.6 × 10−42 esu2 cm2.22 As previously mentioned, the theoretical Judd−Ofelt parameters are determined by the fitting of the experimental quantities Ω2 and Ω4 by adjusting the charge factors (g) and polarizabilities (α). However, this procedure is incoherent when the system contains two different polyhedra. To solve this problem, three different methods are proposed here: (i) the centroid method (CM), (ii) the overlapped polyhedra method (OPM), and (iii) the individual polyhedron method (IPM). The three methods proposed do not have a physical background. Such methods are just purely mathematical
Figure 1. Representation of a hypothetical system illustrating the procedure adopted (A) by the centroid method, (B) overlapped polyhedra method, and (C) individual polyhedron method.
coordinates of all ligand atoms related to the centroid, which is located at half the distance between the two europium ions. The spherical coordinates of the system, comprised by all ligand atoms, are used as input in the adjustment of the charge factors (g) and polarizabilities (α) to reproduce the Ω2 and Ω4 experimental intensity parameters. The next step is to apply the 14096
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values of g and α for each Eu3+ center to calculate the individual theoretical intensity parameters. 2.2. Overlapped Polyhedra Method (OPM). In this method, initially, we overlap of two individual Eu3+ polyhedron, as shown in Figure 1B. This procedure allows us to obtain the spherical coordinates of all coordinating atoms associated with both polyhedra regarding a same europium center. From this new set of spherical coordinates, the values of g and α are then adjusted to reproduce the experimental intensity parameters. The next step is to apply the obtained values of g and α for each individual center to calculate their corresponding theoretical intensity parameters. 2.3. Individual Polyhedron Method (IPM). In the IPM method, the parameters g and α are adjusted from theoretical intensity parameters with experimental values. This procedure is separately carried out for each polyhedral (polyhedron 1 and polyhedron 2) by using the same experimental Ω2 and Ω4 intensity parameters as reference. For similar bonds in both centers, as illustrated by both oxigen atoms that connect the europium centers presented in Figure 1C, the final values of g and α are the average of the adjusted values. For bonds that were only in one center, the final value of charge factors and polarizability are exactly the same than those obtained by the adjustment performed individually. The same way as in the CM and OPM methods, the next step is to apply the set of g and α values to calculate the individual theoretical Ω2, Ω4 and Ω6 intensity parameters associated with each individual europium center.
Table 1. Intensity Parameters Calculated for Each Individual Polyhedron Obtained by the Centroid Method intensity parameters (10−20 cm2) individual adjustment exptl Ω2 Ω4 Ω6 Arad (s−1)
centroid method
polyhedron 1
polyhedron 2
4.33 1.55 13102.1 8946.0
3.11 0.13 0.13 145.36
3.32 0.14 0.15 151.79
4.32 1.51 211.84
Table 2. Intensity Parameters Calculated for Each Individual Polyhedron Obtained by the Overlapped Polyhedra Method intensity parameters (10−20 cm2) individual adjustment exptl Ω2 Ω4 Ω6 Arad (s−1)
overlapped method
polyhedron 1
polyhedron 2
4.32 1.51 0.31 203.13
1.63 1.05 0.20 114.73
1.32 0.99 0.20 104.64
4.32 1.51 211.84
Table 3. Intensity Parameters Calculated for Each Individual Polyhedron Obtained by the Individual Polyhedra Method intensity parameters (10−20 cm2) individual adjustment exptl Ω2 Ω4 Ω6 Arad (s−1)
3. APPLICATIONS 3.1. Symmetric Polyeuropium System. To evaluate the three methods, we initially chose a europium coordination polymer, [Eu(DPA)(HDPA)(H2O)2].4H2O (Figure 2),19 which has metal centers with similar coordination bonds. Figure 2A shows a polymeric 1D structure, whereas Figure 2B presents the dimeric unit used in our calculations.
4.32 1.51 211.84
polyhedron 1
polyhedron 2
1.76 1.60 0.33 127.22
2.79 1.23 0.33 152.64
Ω6 parameter are always overrated. This explains the high values of Ω6, and Arad obtained by the CM. Table 1 also shows that the values of Ω4 for each individual polyhedron were smaller than Ω6 parameter, and this is not in agreement with the expected result for this kind of compound. Nevertheless, it is important to highlight that the three intensity parameters calculated by the CM are very similar for both individual polyhedron, in accordance with the expected result because the two polyhedra are symmetric. Table 2 collects the values of intensity parameters and Arad obtained by the OPM. It can be observed that the OPM shows the values of Ω6 and Arad in good agreement with the experimental ones. The analysis of the individual parameters calculated for two individual polyhedron shows that, as expected, the Ω2 parameter is higher than that of Ω4, which, in turn, is greater than the Ω6 parameter. We can note that the sum of the individual Arad is 219.37 s−1,which is very close to the experimental value. The intensity parameters calculated by the IPM (Table 3) has also many of these positive features; however, it is possible to note that the adjusted values of Ω2 for each metal center are discrepant. In fact, it does not make physical sense, because both polyhedra are symmetric. In addition, the sum of the individual Arad contribution is approximately 32% higher than the experimental value. After this detailed analysis of the results obtained with these three methods, we can suggest that the OPM proved to be a more plausible methodology to investigate the compounds herein presented.
Figure 2. (A) Perspective view of the 1D [Eu(DPA)(HDPA)(H2O)2] coordination polymer. (B) Dimer fragment used for the intensity parameters calculations.
The intensity parameters calculated by using the three methods are presented in Tables 1−3. The values of Ω2 and Ω4 obtained by the three methods are in good agreement with experimental data. It is important to note that the Ω6 parameter obtained from CM was very high (1.31 × 10−16 cm2). This discrepancy is associated with the metal−ligand bond distance (Rt+1 j ) that appears in eqs 3 and 4. In cases of mononuclear europium compounds, the metal− ligand distance was on the order of 3 Å, whereas in the CM these distances can reach 6 Å. As a consequence, the values of 14097
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3.2. Asymmetric Polyeuropium System. The three methodologies evaluated for asymmetry were also performed for a tetramer europium compound.23 As we can observe in Figure 3A, this system possesses three symmetrical centers and one asymmetrical center.
observed in the study of the symmetrical system, the Arad calculated for the centroid system (696.05 s −1 ) was considerably higher than the experimental value. Table 5 shows a good agreement between the theoretical and experimental results. Although the intensity parameters values are low, the asymmetric system comprises one polyhedron of the type “polyhedron 1” presented in Figure 3B and three polyhedra of type “polyhedron 2” presented in Figure 3C. The Arad calculated for the overlapped polyhedron (483.35 s−1) is only 1.6% higher than the experimental value. As we can observe in Figure 3A, this system presents all four polyhedra connected. For this reason, it is expected that the Ω4 parameter for the system (6.4 × 10−20 cm2) is relatively high, because it reflects the rigidity of the system. Furthermore, it is expected that the values of Ω4 for the individual center are similar, because they are all interconnected. Lastly, as expected, Ω2 > Ω4 > Ω6. The results obtained by using the IPM seem to be consistent. However, it is possible to note that the averages of the individual values are very close to those of the experimental values. For example, the average value of Ω2 is 10.95 × 10−20 cm2, whereas the average of the experimental value is 11.14 × 10−20 cm2. The average values of Ω4, and Arad are 6.44× 10−20 cm2 and 478.29 s−1, respectively. This fact was not observed in the symmetric system.
Figure 3. (A) Perspective view of the [Eu4(ETA)9(OH)3(H2O)3)] compound. (B) and (C) are the two asymmetric polyhedra used in the intensity parameters calculations.
A first analysis of the results presented in Tables 4−6 suggests that the OPM had the most satisfactory results. The
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Table 4. Intensity Parameters Calculated for Each Individual Polyhedron Obtained by the Centroid Method
FINAL REMARKS Based on the results presented for these two systems, it is possible to assert that the three methods proposed in this study were able to predict the individual contribution from distinct metal centers for luminescent properties of the systems. The centroid method showed complications calculating the Ω6 parameter; nevertheless, this problem did not happen in the calculation of the Ω6 parameter associated with each individual polyhedron. The individual polyhedron method showed interesting and consistent results, but the analysis of the results obtained for the study of symmetric and asymmetric systems needs to be differentiated. Finally, it is clear that the approach based on the overlapped polyhedra method has produced more consistent results for the study of symmetric and asymmetric systems.
intensity parameters (10−20 cm2) individual adjustment Ω2 Ω4 Ω6 Arad (s−1)
exptl
centroid method
polyhedron 1
polyhedron 2
11.14 6.40
11.14 6.40 255.69 696.05
1.37 0.20 0.50 94.10
1.53 0.47 0.44 103.28
475.84
Table 5. Intensity Parameters Calculated for Each Individual Polyhedron Obtained by the Overlapped Polyhedra Method intensity parameters (10−20 cm2) individual adjustment Ω2 Ω4 Ω6 Arad (s−1)
exptl
centroid method
polyhedron 1
polyhedron 2
11.14 6.40
11.14 6.40 0.07 483.35
3.14 2.06 0.02 175.84
3.35 3.15 0.04 198.55
475.84
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S Supporting Information *
Tables containing the spherical coordinates of the coordination polyhedral of symmetric and asymmetric systems studied and Cif files (crystallographic structures) of [Eu(DPA)(HDPA)(H2O)2].4H2O and [Eu4(ETA)9(OH)3(H2O)3)] compounds. This material is available free of charge via the Internet at http://pubs.acs.org.
Table 6. Intensity Parameters Calculated for Each Individual Polyhedron Obtained by the Individual Polyhedron Method intensity parameters (10−20 cm2)
■
individual adjustment Ω2 Ω4 Ω6 Arad (s−1)
exptl
polyhedron 1
polyhedron 2
11.14 6.40
8.50 5.70 0.22 393.23
13.40 7.17 0.23 563.34
475.84
ASSOCIATED CONTENT
AUTHOR INFORMATION
Corresponding Author
*R. O. Freire: e-mail,
[email protected]; phone, (+55) 79 21056655. Notes
The authors declare no competing financial interest.
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results of the CM for tetramer structure (Table 4) show low Ω4 values. For the polyhedron 1 (Figure 3B), this method showed an unexpected behavior, because the value of the Ω4 parameter was smaller than the value of the Ω6 parameter. Moreover, as
ACKNOWLEDGMENTS The authors gratefully acknowledge CNPq (INCT/INAMI), FAPITEC/SE, and CAPES for its financial support. 14098
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