Chemical Partition of the Radiative Decay Rate of Luminescence of Europium Complexes

The spontaneous emission coefficient, Arad, a global molecular property, is one of the most important quantities related to the luminescence of complexes of lanthanide ions. In this work, by suitable algebraic transformations of the matrices involved, we introduce a partition that allows us to compute, for the first time, the individual effects of each ligand on Arad, a property of the molecule as a whole. Such a chemical partition thus opens possibilities for the comprehension of the role of each of the ligands and their interactions on the luminescence of europium coordination compounds. As an example, we applied the chemical partition to the case of repeating non-ionic ligand ternary complexes of europium(III) with DBM, TTA, and BTFA, showing that it allowed us to correctly order, in an a priori manner, the non-obvious pair combinations of non-ionic ligands that led to mixed-ligand compounds with larger values of Arad.

exp 0 5 7 The transition 5 D 0 → 7 F 1 is governed by a magnetic dipole mechanism and is therefore insensitive to electric dipole contributions. This transition is thus regarded as insensitive to the essentially static electric fields produced by the ligands and can be determined through the expression 28  is the barycenter, the weighted mean of the frequencies in cm −1 , corresponding the 5 D 0 → 7 F 1 transition. From this value, we can now compute the other radiative decay rates, with J from 0-6: are the energies of the barycenters of the respective transitions; and S[ 5 D 0 → 7 F J ] are the areas under the spectra corresponding to the respective transitions. Finally, the experimental intensity parameters can be calculated from 28 : 32 is the frequency of the transition in wavenumbers, χ is the Lorentz local-field correction term given by χ = n(n 2 + 2) 2 /9, and are the square reduced matrix elements whose values are 0.0032, 0.0023, and 0.0002 for λ = 2, 4, and 6, and J = λ in the case of europium 29 .
Another way of obtaining the Ω λ exp is from absorption spectra 30 . In such a procedure, the Ω λ exp are obtained by parameterization so that the calculated oscillator strengths match the experimental ones from the absorption spectra, and the parameterization error propagates to the intensity parameters. Besides, further errors may derive from uncertainties in the choice of theoretical transitions to relate to the experimental ones. Indeed, when Judd-Ofelt parameters are estimated from absorption spectra in this manner, errors are of the order of 10-20% 30 and that is why, in this article, we chose to evaluate them from emission spectra instead.

Theoretical intensity parameters
The theoretical radiative decay rate for the forced electric dipole and magnetic dipole governed transitions, , A rad ed md , is given by =   where e is the elementary charge; 2J + 1 is the degeneracy of the initial state, in this case 5 D 0 , and therefore J = 0. Transitions 5 D 0 → 7 F J with J = 0, 3, and 5 depend on contributions from both electric and magnetic dipole mechanisms and thus are not as easy to calculate. Fortunately, their intensities are very low and, thus, they can be safely disregarded. Transition 5 D 0 → 7 F 1 is the only one which does not have an electric dipole contribution, therefore, is not sensitive to the presence of the ligands around the europium ion, and is relatively small: its magnetic dipole strength is theoretically evaluated as being S md = 96 × 10 -42 esu 2 cm 2 31 . Therefore, for the purpose of this work, we define = ≅ ′ , A A A rad r ad ed rad ed md , restricted to the chemically interesting transitions 5 D 0 → 7 F J with J = 2, 4, and 6. Their corresponding theoretical intensity parameters Ω λ (λ = 2, 4, 6) emerge from the Judd-Ofelt theory 32,33 , depend on the coordination interaction between the lanthanide cation and the ligands, and are given by the following expression: The B λ,t,p terms in eq. (6) are given by the following expression: B t p ed corresponds to the forced electric dipole contribution and λ, , B t p dc corresponds to the dynamic coupling contribution, respectively given by: where Δ E is a constant, approximately given by the energy difference between the barycenters of the ground 4f N and first opposite parity excited state of configuration 4f (N−1) 5d of the europium ion; r 2 = 2.567541 × 10 −17 cm 2 and r 4 = 1.58188 × 10 −33 cm 4 are radial integrals, pre-defined for the europium ion 34 ; θ (t,λ ) are numerical factors for a given lanthanide, estimated by Malta et al. 35 from Hartree-Fock calculations of the radial integrals as being: θ (1,2) = − 0.17; θ (3,2) = 0.345; θ (3,4) = 0.18, θ (5,4) = − 0.24, θ (5,6) = − 0.24, θ (7,6) = 0.24; γ t p are the odd-rank ligand field parameters; (1 − σ λ) is a shielding factor due to the filled 5 s and 5p sub-shells of the lanthanide ion 35 , with σ 2 , σ 4 , and σ 6 being, respectively, 0.6, 0.139, and 0.100 for Eu(III), as previously calculated by Malta and Silva 36 ; f is a Racah tensor operator of rank λ whose values for λ = 2,4,6, are − 1.3660, 1.128, and − 1.270, respectively, for any lanthanide; Γ p t is also a sum over coordinating atoms which further reflects the chemical environment; finally, δ t,λ+1 is a Kronecker delta symbol. All these pre-defined parameters are taken as constants for all europium complexes.
The odd rank ligand field parameters are, in turn, given by: , according to the Simple Overlap Model (SOM) 37,38 , introduces a correction to the crystal field parameters of the point charge electrostatic model (PCEM) 39 , which confers a degree of covalency to the point charge model through the inclusion of parameter ρ , since PCEM only treats the metal-ligand atom bonds as a purely electrostatic phenomenon; g i is the charge factor associated to the lanthanide-ligand atom bond; R i is the lanthanide-ligand atom bond distance; and θ φ ( , ) i are complex conjugate spherical harmonics. The other odd-rank parameter Γ p t , which further reflects the chemical environment is given by: where α i is the polarizability associated to the lanthanide-ligand atom bond. In a recent article 27 , we introduced a protocol to model the charge factors g j of the simple overlap model by electron densities, and the polarizabilities α i of Judd-Ofelt theory by superdelocalizabilities, all obtained by perturbation theory on the semiempirical electronic wavefunction of the complex. A fitting of the theoretical intensity parameters Ω λ is then carried out, which reproduces the experimentally obtained Ω λ exp using only three adjustable constants: Q, D, and C, which must obey the acceptance criterion D/C > 1 leading to a unique adjustment. Whenever D/C ≤ 1, it was shown that the presumed geometry of the coordination polyhedron is not Scientific RepoRts | 6:21204 | DOI: 10.1038/srep21204 seemingly compatible with the experimental intensity parameters and requires improvement, either via calculation by another theoretical model, such as another Sparkle Model 17-21 or RM1 model for lanthanides 22 , or via X-ray crystallographic measurements, etc. The importance of this previous work is that all derived quantities become also uniquely determined for a given complex geometry 27 , including the chemical partition that is being advanced in this article.

Results and Discussion
Partitioning A rad into ligand terms. According to the theory, md . The first term, A rad ed , with even J, is mainly driven by electric dipole transitions. The second term, A rad md , with odd J, is mainly driven by magnetic dipole transitions. Besides, recall that the strengths of the transitions 5 D 0 → 7 F J with J = 0, 3, and 5 are set at zero because of their low values. Likewise, the magnetic dipole driven transition 5 D 0 → 7 F 1 is not sensitive to the ligands and, as a result, is not directly relevant from a chemical point of view. Therefore, the partition will focus on the electric dipole driven transitions with J = 2, 4, 6. Accordingly, we will partition a subset of A rad , we call A rad′ , defined by: rad J rad J exp 2 4 6 exp 0 5 7 the most significant term being the decay rate of the so-called hypersensitive transition 5 D 0 → 7 F 2 , which is highly susceptible to the presence of the ligands. Now, let us turn to compute the λ, , B t p terms from eqs (8)(9)(10)(11).
which can be rewritten as, Accordingly, λ, , B t p are obtained by sums over all coordinating atoms of the ligands of the product of a term, ( ) λ, , K i t p , defined below, which depends only on the lanthanide ion and on the particular coordinating bond as previously described, with a complex conjugate spherical harmonic.
and we can define an auxiliary matrix, , which is a function of only the coordinating atoms of the complex as is an Hermitian matrix because , , *. Therefore, its eigenvalues are all real numbers. Moreover, we will show that ( , ′) is a positive semi-definite matrix and therefore all its eigenvalues are, not only real, but also equal to or greater Scientific RepoRts | 6:21204 | DOI: 10.1038/srep21204 than zero. Q is said to be positive semi-definite if t Qt T is non-negative for every non-zero column vector t of n real numbers. Here t T denotes the transpose of t.
As such, Replacing Q by its expression, B t p terms of eq. (16) can now be computed in terms of the directly coordinating atoms as: Likewise, we can define the efficacy of luminescence matrix ( , ′) where ν λ is the frequency (in cm −1 ) corresponding to the energy gap between the initial 5 D 0 and final 7 F J states. Note that ( , ′) A i i is a real symmetric positive semi-definite matrix, since is also a real symmetric positive semi-definite matrix and the coefficients multiplying Note that the ligand-pair contributions do not contain any atomic contributions, since, by being different ligands, L and L' do not share any directly coordinating atom. Finally, We will use the efficacies of luminescence, or simply efficacies ( , ′) A i i and ( , ′) A L L l in order to interpret, from a chemical point of view, the various influences of the ligands and of their atoms, together with their pair influences, directly on ′ A rad . The elements of both efficacy matrices A and A l are partial decay rates (see eqs (23) and (26)), being expressed in units of a decay rate, usually s −1 . So, in order to obtain the total decay rate in s −1 , just add all elements of either matrix, that is, their grand sums ( ) 1is a column vector with all elements equal to unity.
The chemical partition of A rad′ . The elements of the efficacy matrices A and A l , are often negative, which require an interpretation, which may sometimes be useful, but is certainly less chemically intuitive, of the partition of ′ A rad . For example, if a given complex displays a very low luminescence, that is an ′ A rad close to zero, it is possible that the contribution A l from one of its ligands be + 800 s −1 and the A l of another ligand be − 800 s −1 . Such a situation in which one contribution annihilates the other, renders the role of each of the ligands on the luminescence phenomenon somewhat indiscernible, especially when they are chemically identical.
A chemically more intuitive partition would require only coordinated atom or ligand contributions, always positive, and hence, whenever ′ A rad is zero, they all should be zero. In order to define such a partition, we start by recognizing that matrix Q is Hermitian and positive semi-definite, and, as a consequence, matrix ( , ′) A i i is also Hermitian and positive semi-definite. That implies that their eigenvalues are not only all real, but they are all greater than or equal to zero. Now, define the orthonormal eigenvectors of ( , ′) . In fact, Observe that So, we define the relative contribution of the coordinated atom j to the eigenvalue λ i as We will now obtain a nicer expression for this relative contribution. Since . Thus, U ij 2 is the relative contribution of the coordinated atom j to eigenvalue λ i because these relative contributions are non-negative and sum 1, and thus can be viewed as proportions as we intended. We can now define the vectors with the ligand contributions to the eigenvectors:  In a similar manner, we can define the absolute contribution of coordinated atom j to λ c i i , with Λ ≥ 0 i . The set of coefficients Λ j thus constitutes a partition of ′ A rad in n terms, all positive, each corresponding to each of the n coordinated atoms. These terms reflect how each coordinated atom contributed to make the 5 D 0 → 7 F J less forbidden.
As before, we can aggregate all contributions from each ligand in the complex and define the ligand contributions to ′ A rad in terms of summations over the coordinating atoms k of ligand L as: In summary, the chemical partitions we introduce in this article, ( , ) , Λ j , and Λ L , are from now on available, ready to be interpreted from a myriad of chemical perspectives depending on the system of interest, subject to the creativity of the researcher.
LUMPAC chemical partition implementation. The luminescent software package LUMPAC 12 , since 2013 freely available from http://www.lumpac.pro.br/, is the first state of the art complete software to treat europium luminescence from a theoretical point of view. Recently, the unique adjustment of theoretical intensity parameters developed by our group 27 was implemented in LUMPAC. Now, the chemical partition being advanced in this article has also been implemented and is already available to all users of LUMPAC.
From eqs (10) and (11), clearly geometry makes a profound impact on the calculation of the theoretical intensity parameters Ω 2 , Ω 4 and Ω 6 , and, by extension makes an impact on our partition scheme. Therefore, users must first determine the most stable geometry of the complex of interest via either RM1 22 or any of the Sparkle Models [17][18][19][20][21] in such a manner as to satisfy the binary outcome acceptance conditions for the unique adjustment of theoretical intensity parameters as described in the "LUMPAC implementation" section of ref. 27. Once the adjustment is considered accepted, calculation of the chemical partition follows in a seamless manner.
The partition is first computed per directly coordinated atom, and then subsequently aggregated per ligand by summing up the terms of the directly coordinated atoms of each ligand. It has been proven useful to further aggregate the terms of the ligands into terms for classes of ligands, such as the terms of all ionic ligands and those of all non-ionic ligands.

Interpretation of the chemical partition ligand terms. Interpretation of the ligand terms requires an
understanding of the fact that, according to Laporte rule, the electronic f-f transitions in lanthanide complexes should be forbidden in centrosymmetric molecules, since they conserve parity with respect to the inversion center where the metal is located. In this sense, luminescence happens because the centrosymmetry can be broken by ligands coordinating the lanthanide ion. Since luminescence happens in the europium ion, not at the ligands, the ligand terms of the ′ A rad chemical partition cannot possibly be regarded as ligand contributions to ′ A rad , but rather as measures of the relaxation of the forbidding character of the 5 D 0 → 7 F J transitions (J = 2, 4, 6), conferred by each of the respective ligands to the europium ion. Note that each ligand term of the chemical partition is defined within the distinctive chemical ambiance of the particular complex, and cannot be expected to be transferable from complex to complex.
A complex of this general formula Eu(β -diketonate) 3 (TPPO) 2 may display two possible ligand arrangements: both TPPOs are either adjacent or opposite to each other. All structural data have been computed by either RM1 22 , or, in a single case, by the Sparkle/PM3 18 method. The choice of method followed the QDC acceptance criterion 27 defined in a previous article on the unique adjustment of theoretical intensity parameters 27 . We will now examine the impact of these different geometric arrangements on the chemical partition of ′ A rad . However, since the emission spectra, A rad , Ω 2 and Ω 4 have been measured for these complexes in the opposite TPPO configuration 6,40 , we will use these same values to arrive at the chemical partition for each of the two possible geometrical arrangements for enlightening purposes only. Tables S1 and S2 of the Supplementary Information contain information on the adjustments of the theoretical intensity parameters, unique for each of the geometrical arrangements, and the partition results, aggregated by ligand, for all three complexes considered. Figure 1 shows the chemical partition of the radiative decay rate ′ A rad for each of the ligands coordinated to the metal ion for both cases of adjacent and opposite non-ionic ligands for all three TPPO complexes considered. Figure 1 evidences the chemical nature of the partition because, now, ′ A rad has been sliced into ligand terms that depend on the chemical nature of the ligands, as well as on their collective arrangements around the europium ion.
Scientific RepoRts | 6:21204 | DOI: 10.1038/srep21204 In an environment with three other identical ionic ligands, adjacent non-ionic ligands are less centrosymmetric than opposite ones. So, one would expect that adjacent same ligands should contribute more to the relaxation of the forbidding character of the 5 D 0 → 7 F J transitions (J = 2,4,6) than opposite ones. That is indeed the case for Eu(TTA) 3 (TPPO) 2, where the sum of terms of the adjacent TPPOs is 382 s −1 , whereas for opposite TPPOs it is 122 s −1 . Equivalent numbers for Eu(BTFA) 3 (TPPO) 2 are 422 s −1 and 160 s −1 , and for Eu(DBM) 3 (TPPO) 2 , they are 96 s −1 and 102 s −1 , a more balanced situation which arises seemingly due to the more symmetric and bulky nature of DBM.
Conversely, the β -diketonate ligands are more centrosymmetric when the TPPOs are adjacent (two of them tend to occupy opposite axial-like positions) and therefore they should contribute less to the relaxation of the forbidding character of the 5 D 0 → 7 F J transitions (J = 2, 4, 6). On the other hand, the β -diketonate ligands are less centrosymmetric when the TPPOs are opposite, because in this case they tend to occupy planar trigonal-like positions, in which case they should contribute more to the relaxation of the forbidding character of the 5 D 0 → 7 F J transitions (J = 2, 4, 6). For Eu(TTA) 3 (TPPO) 2, the sum of the three β -diketonate terms for opposite TPPOs is 636 s −1 , whereas for adjacent TPPOs it is 374 s −1 . Equivalent numbers for Eu(BTFA) 3   Applications of the chemical partition. Complexes Eu(TTA) 3 (TPPO) 2 and Eu(BTFA) 3 (TPPO) 2 had their geometries determined by crystallography and deposited in the Cambridge Structural Database, CSD [41][42][43] , with refcodes SABHIM and WIFWIR, respectively. In both cases, the non-ionic ligands appear opposite to each other. Recently, a theoretical determination of the thermodynamic properties of Eu(DBM) 3 (TPPO) 2 also indicated that the opposite TPPO configuration should be the preferred one 40 . This study further extended the analysis for other non-ionic ligands and predicted that Eu(DBM) 3 (DBSO) 2 , and Eu(DBM) 3 (PTSO) 2 should also display opposite non-ionic ligand configurations, where DBSO is dibenzyl sulfoxide and PTSO is p-tolyl sulfoxide. As a consequence, in the present article we assume that all complexes of the general formula Eu(β -diketonate) 3 (L) 2 where L is a non-ionic ligand, will adopt opposite non-ionic ligand configurations. As before, all structural data have been computed by either RM1 22 , or, in a single case, by the Sparkle/PM3 18 method -the choice of method followed the QDC acceptance criterion 27 . Table 1 presents luminescence results for 9 different complexes of the general formula Eu(β -diketonate) 3 (L) 2 , radiative decay rates, both experimental (A rad exp ) and calculated, ( ′ A rad ), the latter one partitioned by ligands and summed up into ionic ligand ( ′ A rad ionic ) and non-ionic ligand ( ′ − A rad non ionic ) terms. Please remember that (A rad exp ) will always be larger than ( ′ A rad ) because A rad exp refers to all 5 D 0 → 7 F J transitions, while ′ A rad only adds up those with J = 2,4,6.
Examination of the average values in Table 1 reveals that the contribution of the non-ionic ligands to the triggering of luminescence decay by the excited europium trivalent ion is much smaller, in average 120 s −1 , than the corresponding contribution of the ionic ligands, 544 s −1 . That could lead to a misunderstanding that the non-ionic ligands, in these cases, are not too relevant to the luminescence phenomenon. However, their modest contribution to ( ′ A rad ) is seemingly due to that fact that they are opposite to each other, and, therefore, in a symmetric configuration with respect to the europium ion, which does not help to relax the Laporte's rule.
Recently, our group introduced a simple strategy to boost three important luminescence properties of complexes of the general formula Eu(β -diketonate) 3 (L) 2 : the quantum yield, Φ , the emission efficiency η , and A rad exp , which was mathematically translated into the following conjecture 6 : where P stands for either Φ , η , or A rad exp and L and L′ are different non-ionic ligands. This conjecture states that mixed non-ionic ligand complexes, Eu(β -diketonate) 3 (L,L′ ), should display larger luminescence properties when compared to the average of the same properties for repeating ligand complexes, Eu(β -diketonate) 3 (L) 2 and Eu(β -diketonate) 3 (L′ ) 2 . This conjecture has already been proven experimentally for all combinations of the non-ionic ligands DBSO, TPPO, and PTSO, for all ternary europium complexes of TTA, BTFA, and DBM. A rad non ionic for the complexes with mixed non-ionic ligands. The role of the non-ionic ligands in the mixed-ligand complexes becomes clearer. Indeed, now the contribution of both different non-ionic ligands, to the triggering of luminescence decay, becomes accentuated to an average of 302 s −1 , up from the average of 120 s −1 in the repeating non-ionic ligand complexes of Table 1. Once again, this behavior can be rationalized in terms of symmetry: the different non-ionic ligands are opposite to each other, rendering the situation considerably more asymmetric, thus much more capable of triggering the luminescent decay of the excited europium ion. On the other hand, the role of the ionic ligands remains, in average, unaffected. Indeed, the average of ′ A rad ionic for the mixed non-ionic ligand complexes is 552 s −1 , whereas in the repeating non-ionic ligand complexes it is 544 s −1 , thus reinforcing the protagonist role of the non-ionic ligands in the luminescence boost. A rad corresponds to the transitions from 5 D 0 to 7 F 2 , 7 F 4 , and 7 F 6 , and is therefore always smaller than A rad exp which, in addition, also includes the transitions to 7 F 0 , 7 F 1 , 7 F 3 , and 7 F 5 . The ′ A rad ionic partition comprises the terms for each of the three identical β -diketonates The ′ − A rad non ionic partition comprises the terms for each of the two identical non-ionic ligands. Geometries were optimized and the chemical partitions were calculated with the RM1 model (except where otherwise indicated). a Geometry was optimized and the chemical partition was calculated with Sparkle/PM3. A rad corresponds to the transitions from 5 D 0 to 7 F 2 , 7 F 4 , and 7 F 6 , and is therefore always smaller than A rad exp which, in addition, also includes the transitions to 7 F 0 , 7 F 1 , 7 F 3 , and 7 F 5 . The ′ A rad ionic partition comprises the terms for each of the three identical β -diketonates The ′ − A rad non ionic partition comprises the terms for each of the two non-ionic ligands. Geometries were optimized and the chemical partitions were calculated with the RM1 model (except where otherwise indicated). a Geometry was optimized and the chemical partition was calculated with Sparkle/RM1. tively, 102 s −1 , and 59 s −1 , leading to complex Eu(DBM) 3 (DBSO,TPPO). Indeed, this complex has the largest A rad exp of 652 s −1 . Furthermore, one can use the same reasoning to arrive at the DBM ternary complex with the next larger A rad exp , where two possibilities exist: Eu(DBM) 3 (PTSO,TPPO) and Eu(DBM) 3 (DBSO,PTSO). The corresponding pair of largest ′ − A rad non ionic that are left are 102 s −1 for Eu(DBM) 3 (TPPO) 2 and 29 s −1 for Eu(DBM) 3 (PTSO) 2 , leading to complex Eu(DBM) 3 (PTSO,TPPO), which, indeed, is the next best complex, with an A rad exp of 572 s −1 . Finally, the last one left is complex Eu(DBM) 3 (DBSO,PTSO), with an A rad exp of 540 s −1 . Undoubtedly, for the case of ternary complexes of DBM, usage of the chemical partition allowed us to perfectly order, in an a priori way, the pair combinations of non-ionic ligands in terms of theirA rad exp , a single outcome out of six possibilities.
Let us now apply the same strategy to the other ternary complexes of TTA and BTFA. The overall situation is present in pictorial format in Fig. 2, which shows that choosing non-ionic ligands by their ′ − A rad non ionic terms in non-ionic repeating ligand complexes also perfectly orders all mixed non-ionic ligand complexes in terms of their A rad exp values for the remaining cases of TTA and BTFA ternary complexes. Overall, the chemical partition predicted, in an a priori manner, a single joint outcome out of 216 possibilities, which, if it were by chance, would have a probability of only 0.46% of occurrence.

Conclusions
For the first time, a molecular global property related to luminescence, the radiative decay rate of europium complexes, is partitioned into ligand contributions. Such a novel approach gives rise to possible chemical interpretations of the effect of each ligand and their interactions on the luminescence phenomenon, allowing for the design of cutting edge compounds with enhanced brightness upon ultra-violet illumination.
As a demonstration of an application made only possible with the chemical partition, we address the case of repeating non-ionic ligand ternary complexes of europium(III) with DBM, TTA, and BTFA. We show that the chemical partition allows us to perfectly order, in an a priori way, the non-obvious pair combinations of non-ionic ligands that led to mixed-ligand compounds with larger values of A rad .
Our new chemical partition has been implemented in the LUMPAC software, which is freely available 12 .