Rationalizing the Thermal Response of Dual‐Center Molecular Thermometers: The Example of an Eu/Tb Coordination Complex

Luminescence thermometry allows the remote monitoring of the temperature and holds the promise to drive the next generation of future nano or micrometric devices. Materials able to sense the temperature are usually based on one or several lanthanide ions allowing a ratiometric measurement. Optimizing the thermometric features is usually achieved through a serendipity approach, but it still appears difficult to accurately predict the sensing performance. Through a combination of experiment and theoretical calculations, the first example of an energy‐driven luminescent molecular thermometer [Tb0.94Eu0.06(bpy)2(NO3)3] (bpy = 2,2′‐bipydine) displaying an exceptional thermal cyclability around room temperature is reported, for which the thermal properties could be theoretically forecasted. This work provides comprehensive guidelines that can be easily extended for any dual‐center thermometer in which energy transfer drives the thermometric performance opening the avenue for the smart engineering of sensing devices.


Introduction
Our days, luminescence thermometry (also called thermographic phosphor thermometry) is in the spotlight of materials science and engineering research representing around 5% of systems [39,45] ) and dual-center examples in which energy transfer drives the thermometric performance. [28,42,[48][49][50][51][52] A theoretical framework for describing ratiometric single-ion Boltzmann-based thermometers was recently established [39] and these systems are one of the very few examples of luminescent primary thermometers, [53] in which the temperature can be predicted without a previous calibration procedure. However, these Boltzmann-based luminescent thermometers are generally ineffective above ≈400 K [38,39] and their relative thermal sensitivity (S r ) rarely exceeds 1%⋅K −1 , which can be a constrain in certain applications. S r is used as a figure of merit to compare the performance of different thermometers [37] since our suggestion in 2012. [3] Thus, dual-center thermometers based on temperature-dependent energy transfer processes are considered the most exciting alternative to circumvent the limitations of single-ion thermometers. However, and despite extensive experimental research in these systems, including the Tb 3+ /Eu 3+ pair, which is by far the most widely reported, a general theory covering the basics of luminescence thermometry through energy transfer (both in single-and dual-center systems) is nearly nonexistent. There are very few exceptions, such as the Ho 3+ -based thermometer reported by van Swieten et al. [52] in which quantitative modeling was presented predicting the output spectrum over a wide range of temperatures (300-873 K) and Ho 3+ concentrations (0.1-30%), or the series of mixed Tb 3+ /Eu 3+ -codoped metal-organic frameworks (MOFs) reported by Trannoy et al. in which a detailed theoretical study on Tb 3+ -to-Eu 3+ energy transfer processes was performed evidencing a notorious agreement with the experimental data. [51] Yet, optimizing the synthetic design of luminescent thermometers with improved performance will require not only to postrationalize the thermometric parameter but, more importantly, to predict it through a concerted theory/experimental approach.
In this sense, Ln 3+ coordination complexes have recently appeared as a fertile field of research in luminescence thermometry, including the very intriguing example of molecular thermometers displaying single-molecule magnet behavior. [21][22][23][24] Molecular chemistry concepts could be efficiently used to control the parameters affecting the thermometric parameters by the careful choice of the building blocks (metal ions, ligands, and, in turn, the intermolecular distances). Yet, one of the general drawbacks of lanthanide molecule-based materials might be associated with their possible lack of thermal stability even in relatively mild conditions (>50-300 °C). [54] This could be mostly related to the presence of interstitial and coordinated solvent molecules, which upon removal might induce a collapse of the crystal structure or a change in the coordination environment of the Ln 3+ ions that alters the emission features and in turn the overall cyclability in the high-temperature range. [54,55] This appears particularly relevant for the thermometers based on lanthanide MOFs for which most examples incorporate solvates (coordinated or interstitial) in their frameworks. [41] As an alternative, discrete mono or polynuclear coordination compounds might afford better control over the coordination environment of the Ln 3+ ions.
Herein, we present the first example in which the thermometric parameter of an energy-driven luminescent thermometer is theoretically predicted showing an astonishing accord with the experimental data. The adopted theoretical methodology is applied to the illustrative example of the [Tb 0.94 Eu 0.06 (bpy) 2 (NO 3 ) 3 ] coordination complex displaying very high thermal sensing reproducibility (>97%). The presented approach can be easily extended for any dual-center thermometer in which energy transfer drives the thermometric performance opening the avenue for the smart engineering of dual-center thermometers instead of the common trial-and-error strategy.

Synthesis, Crystal Structure, and Characterization
To demonstrate our approach, we selected a well-known simple lanthanide complex of formula [Ln(bpy) 2 (NO 3 ) 3 ] (bpy = 2,2′-bipyridine), [56] but for which the thermometric properties were never investigated to our knowledge. This complex presents several advantages such as i) efficient Ln 3+ sensitization by the bpy ligands; [57] ii) chemical stability in the presence of air and water; iii) absence of solvent molecules and iv) easy synthesis in large yields and at the multigram scale.
X-ray diffraction analysis on single-crystals indicates that 1 crystallizes in the orthorhombic Pbcn space group with half a complex in the asymmetric unit. The compound is isostructural to the previously reported pure terbium and europium analogs. [56,63] The calculated Ln 3+ site occupancy of Tb 0.95 Eu 0.05 obtained from the structural refinement is in excellent agreement with the Eu/Tb ratio of 6% found by energy-dispersive spectroscopy (EDS) analysis. The coordination sphere of the Ln 3+ center is constituted by two bpy and three bidentate nitrate moieties giving a coordination number of 10 (Figure 1a). Qualitative analysis of the coordination geometry using the SHAPE software [64] indicates an intermediate geometry between a bicapped square antiprism and a spherocorona (Table S1, Supporting Information). The Ln-N distances, ranging from 2.525(1) to 2.549(4) Å, are slightly longer than the Ln-O ones involving the nitrate moieties (2.455(1)−2.478(1) Å, Table S2, Supporting Information). Analysis of the crystal packing reveals the presence of intermolecular π-stacking interactions between the bpy ligands (Figure 1b), while the shortest Ln 3+ -Ln 3+ distance is equal to 7.696 Å.
Thermogravimetric analysis confirms the absence of solvent molecules and indicates thermal stability of the complex up to 290 °C ( Figure S1, Supporting Information). The powder X-ray diffraction patterns ( Figure S2, Supporting Information), in line with other analyses, such as the magnetic properties ( Figures S3 and S4, Supporting Information), confirm the purity of the sample. www.advopticalmat.de

Photoluminescence
The emission spectra of 1 (12-450 K) reveal the characteristic luminescence of Tb 3+ and Eu 3+ ascribed to the 5 D 4 → 7 F 6-3 and 5 D 0 → 7 F 0-4 transitions, respectively (Figure 2a). As the temperature is increased to 450 K, an increase of the relative intensity of the Eu 3+ intra-4f 6 with respect to that of the Tb 3+ intra-4f 8 is observed, pointing out distinct thermally activated mechanisms for the population and deexcitation of the 5 D 4 and 5 D 0 levels, as detailed below.
The room-temperature excitation spectra of 1 were monitored at the 5 D 4 → 7 F 6 (Tb 3+ ) and 5 D 0 → 7 F 4 (Eu 3+ ) transitions ( Figures S5-S7, Supporting Information). The 5 D 0 → 7 F 4 transition was chosen to avoid overlap with the 5 D 4 → 7 F 3 (Tb 3+ ) transition, thus, enabling the determination of the selective Eu 3+ excitation path. All the excitation spectra are dominated by a broadband in the UV with two main components at around 280 and 330 nm ascribed to the ligand's singlet excited states, [65] and of a series of narrow lines ascribed to the intra-4f 8 and intra-4 6 transitions in the excitation monitored at the 5 D 4 → 7 F 6 (Tb 3+ ) and 5 D 0 → 7 F 4 (Eu 3+ ) transitions, respectively. The negligible intensity of the intra-4f lines points out that the Ln 3+ excited states are mainly populated through ligands sensitization. Noticeably, it is detected the presence of Tb 3+ transitions ( 7 F 6 → 5 D 4-2 , 5 G 6,5 , 5 L 10,9 ) in the excitation spectra selectively monitored within the 5 D 0 → 7 F 4 transition, pointing out a Tb 3+to-Eu 3+ energy transfer. Nonetheless, as the temperature is lowered, the relative intensity of such intra-4f 8 transition decreases being negligible around 175 K (inset of Figure S6, Supporting Information). Therefore, the Tb 3+ -to-Eu 3+ energy transfer is thermally activated (T > 175 K) following the Ln 3+ -to-Ln 3+ energy transfer calculations (see Supporting Information), where the Tb 3+ -to-Eu 3+ energy transfer rate starts to overcome the Eu 3+ -to-Tb 3+ in this range of temperature ( Figure S20, Supporting Information). Although there is indeed an energy transfer between the Tb 3+ and Eu 3+ ions, the long distances between these ions   www.advopticalmat.de (7.7 Å) produce low rates that cannot compete with the Ligandto-Ln 3+ rates that are orders of magnitude higher, as will be discussed later. Therefore, the thermal properties in this specific case do not depend on the Ln 3+ -to-Ln 3+ rates.
The Tb 3+ -to-Eu 3+ energy transfer is related mainly to the increase of the 7 F 1 population (as depicted in Figure S17, Supporting Information) once the main energy transfer pathway involves the 7 F 1 → 5 D 1 (Tb 3+ -to-Eu 3+ energy transfer, pathway 11 in Table S9, Supporting Information), while the other process (Eu 3+ -to-Tb 3+ ) is independent of the 7 F 1 population (Table S10, Supporting Information).
The contribution of the distinct excitation paths can be quantified through the absolute emission quantum yield (Q, Table S3, Supporting Information), whose maximum value is attained under ligands excitation at 330 nm (Q L = 0.66 ± 0.07). Lower values are found when the intra-4f levels are directly excited, pointing out the role of the ligands in the energy transfer processes. Interestingly, when the Eu 3+ is preferentially excited under direct intra-4f 6 excitation ( 5 D 2 , 466 nm), a lower value (Q Eu = 0.08 ± 0.01) is found when compared with that (Q Tb = 0.28 ± 0.03) attained under intra-4f 8 excitation ( 5 D 4 , 489 nm). When excited at 396 nm under a superposition of ligand and 5 L 6 excitation a similar value is also found (Q L,Eu = 0.29 ± 0.03). It is worth mentioning that the theoretical model presented here is also capable to simulate the theoretical emission quantum yields, [66,67] however, this issue is outside of the scope of the present work.
The thermal dependence of the 5 D 4 and 5 D 0 lifetimes was studied by monitoring the emission decay curves of the 5 D 4 → 7 F 5 (Tb 3+ , Figure S8, Supporting Information) and 5 D 0 → 7 F 2 (Eu 3+ , Figure 9, Supporting Information) transitions. We note that the excitation spectra monitored at the 5 D 0 → 7 F 2 transition overlap those monitored at the 5 D 0 → 7 F 4 ( Figure S7, Supporting Information). We, therefore, chose the higher intensity transition to monitor the emission decay without losing selectivity. The emission decay curves are well described by a single exponential function revealing that whereas the 5 D 0 lifetime ( D 5 0 τ ) is nearly independent of the temperature, being D 5 0 τ = 1.12 ± 0.05 ms in the (12-450 K) range, the 5 D 4 ( D 5 4 τ ) lifetime decreases from D 5 4 τ = 1.085 ± 0.005 ms at 12 K to 0.019 ± 0.005 ms at 450 K (Figure 3). Focusing on the 5 D 0 emission decay curves measured within 225−325 K, there is evidence of a rise time (dependent on the temperature, Figure 3a and Figure S9, Supporting Information) well described by: [68] where N 0 is the 5 D 0 population at t = 0 and N 1 is the population at the same instant (t = 0) in an upper feeding level; and τ rs is the 5 D 0 rise time. The best fit to the data points out that the calculated τ rs values are similar to the 5 D 4 lifetime determined independently from selective decay measurements ( Figure 3b). This is evidence that 5 D 4 is the upper feeding level supporting the presence of Tb 3+ -to-Eu 3+ energy transfer above 175 K, as also confirmed by our theoretical Ln 3+ -to-Ln 3+ energy transfer calculations ( Figure S20, Supporting Information). Intriguingly, the temperature where the Tb 3+ -to-Eu 3+ transfer becomes higher than the Eu 3+ -to-Tb 3+ one matches with the observation of the 5 D 0 rise time. Even though the energy transfer from the ligands is much faster than the Ln 3+ -to-Ln 3+ ones, when the excitation source is turned off the 5 D 0 population rises owing to a delayed Tb 3+ -to-Eu 3+ energy transfer process. Besides, we emphasize the independence of the 5 D 0 emission lifetime on the temperature and it is justified by the very low calculated Eu 3+ -to-ligand energy transfer rates ( Figure S20, Supporting Information).

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The fascinating thermal dependence of the optical features of 1, shown in Figure 2, envisages its use as a novel luminescence thermometer and the relative intensities of the 5 D 4 → 7 F 5 (I Tb ) and 5 D 0 → 7 F 2 (I Eu ) transitions were quantified by integration of the emission spectra between 530-565 and 605-635 nm, respectively. We note that the spectral overlap (marked with an asterisk in Figure 2a) between the 5 D 0 → 7 F 2 (Eu 3+ ) and 5 D 4 → 7 F 3 (Tb 3+ ) transitions should be considered to avoid an overestimation of I Eu (Figures S10, S11 and further details in Supporting Information). Figure 2b shows that the I Tb remains approximately constant as the temperature is raised from 12 to 200 K, and decreases at T > 200 K. A distinct behavior is noticed for the I Eu that increases from 12 to 350 K, remaining approximately constant for T > 350 K (Figure 2c). These temperature dependencies in a ratiometric thermometer will be theoretically modeled in the next section.

In Silico Experiments
The time-dependent density functional theory (TD-DFT) calculation, using the optimized structure (Figures S14 and S15, Supporting Information), was performed to obtain the centroid of the triplet T 1 and singlet S 1 states. Their energies (T 1 ≈ 21 400 cm −1 and S 1 ≈ 29 900 cm −1 ) were experimentally measured for the [Gd(bpy) 2 (NO 3 ) 3 ] analog complex. [70,71] The TD-DFT procedure allowed to estimate the donors-acceptors distances (R L , Figure S14, Supporting Information), quantities that are essential for the energy transfer calculations, and this can be obtained from the analysis of the molecular orbitals (MOs) compositions involved in the formation of T 1 and S 1 , as depicted in Figure S15, Supporting Information. It can be noted that the ligands NO 3 − are not involved in the T 1 state. However, the S 1 state is characterized by an electronic density displacement from the occupied MOs at the NO 3 − ligands to the unoccupied at the ancillary ligands bpy. Once both excited states S 1 and T 1 are localized through the bpy ligands (unoccupied MOs in Figure S15, Supporting Information), implying that they have the same value of the donor-acceptor distance R L = 4.12 Å ( Figure S14, Supporting Information).

Intramolecular Energy Transfer Rates
Based on the schematic energy level diagram for both Eu 3+ and Tb 3+ complexes (Figure 4), we aim at visualizing how the intramolecular energy transfer (IET) rates play a fundamental role in the emission process. Since the bpy ligands are responsible for good absorption (pumping rate ϕ, Equation S22, Supporting Information) of the incident light, efficient forward IET rates from S 1 (W S ) and T 1 (W T ) are desirable to populate the Ln 3+ emitting levels ( 5 D 0 and 5 D 4 ), otherwise, no intra-f emission may be observed. Other fundamental quantities displayed in Figure 4 are the backward IET rates ( S b W and T b W ), intersystem crossing S 1 →T 1 rate (W ISC ), multiphonon relaxation from upper Ln 3+ levels to the emitting one (W 3 → 4 ), and decay lifetimes (τ S , τ T , τ).
All IET pathways calculated (Tables S7 and S8, Supporting Information) takes into account the selection rules on the J quantum number (|J-J′| ≤ λ ≤ J+J′ for the dipole-dipole and the dipole-multipole mechanisms, ΔJ = 0, ±1 for the exchange mechanism) [72] and represent the rates obtained at 300 K.  . W ISC is the S 1 →T 1 intersystem crossing rate, ϕ is the pumping rate, W 3 → 4 is the multiphonon relaxation rate from |3〉 to |4〉. The lifetimes τ S , τ T , and τ are regarding the decay of S 1 , T 1 , and emitting levels ( 5 D 0 and 5 D 4 ).

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Considering the non-zero contributions, 60 forward IET pathways (30 for Ligand-to-Eu 3+ and 30 for Ligand-to-Tb 3+ ) were examined. Using the same procedure to calculate all 120 pathways (60 forward and 60 backward IET rates) but varying the temperature, we obtain the summarized data in Table 1.
It can be observed that, for the case of the Tb 3+ complex, only the backward energy transfer from S 1 ( S b W ) changes moderately with the temperature (Table 1). This is related to the ( 7 F 6 → 5 L 6 ) →S 1 (pathway 7, Table S8, Supporting Information) which has an energy difference between the donor and acceptor states (δ) of the order of k B T. This specific pathway is not important for the forward because the S 1 →( 7 F 6 → 5 G 6 ) (pathway 3) is three orders of magnitude higher. From the T 1 channel (W T and T b W ), there is a lack of predominant pathways with δ of the order of k B T (see Table S8, Supporting Information). Therefore, the thermal behavior of the Tb 3+ complex is governed exclusively by its 5 D 4 decay time (τ 5 4 D , Figure 3b). Contrarily, the Eu 3+ complex has almost constant decay time (τ 5 0 D , Figure 3b), showing however significant variations in the IET rates with the temperature increase. This is rationalized taking into account that the population of 7 F 1 is increasing with the temperature [51] (Figure S17, Supporting Information) once the main contributions are associated with pathways in which the 7 F 1 is involved (see pathways 12, 21, and 23 in Table S7, Supporting Information). Besides, pathway 23 (T 1 →( 7 F 1 → 5 D 2 ), Table S7, Supporting Information) has a strong temperature dependence owed to its δ in the order of k B T. Moreover, an unusual behavior like an "energy transfer channel tuning" is also observed for the Eu 3+ which means that the population of the 5 D 0 for low temperatures (T < 125 K) is influenced by the T 1 →Eu 3+ channel (W T ) while for T ≥ 125 K, the 5 D 0 level is more sensitive by the S 1 →Eu 3+ channel (W S ), this is the reason we called energy transfer channel tuning. To support this effect, Figure S18, Supporting Information reveals that the W T and W S have opposite trends when temperature increases. In summary, the uneven thermal dependence on the I Eu behavior can be explained by the relative magnitude of W ISC and W S : i) for T < 125 K, W ISC > W S , preferentially the triplet state is populated and thus energy flows from the lowest energy triplet state to the Eu 3+ ion (T 1 →Eu 3+ ), whereas ii) for T≥ 125 K, W ISC < W S , and thus the singlet state tends to transfer energy directly to the Eu 3+ ion (the S 1 →Eu 3+ energy transfer becomes dominant).

Rationalizing Thermal Dependence of the Integrated Intensities Kinetics Simulations
The kinetics of this system includes an equilibrium between rates of absorption, IET (both forward W and backward W b ) rates, radiative, and non-radiative decay rates. This kinetics can be defined by an appropriate set of ordinary differential equations (ODEs) which comprises the temporal behavior of the populations at each level. However, sometimes it is hard to describe and include the population of all levels of the Ln 3+ ion.
To simplify the setup of the rate equations, groups of levels are represented by |n〉 (with the same indication as in Figure 4) and their respective population as P n . Thus, based on the IET rates and the schematic energy level diagram (Figure 4), a 5-level set of ODEs with initial conditions (when t = 0 s) is given by Eqs.

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The lifetime values of the ligands used to solve the equations above, τ T = 10 −3 s and τ S = 10 −9 s, are typical values found in the literature. [73,74] If we change them to one order of magnitude, for example, the population of the emitting level will decrease (τ S ≈ 10 −10 s) or increase (τ S ≈ 10 −8 ) almost one order of magnitude too. The Ln 3+ emitting level is less sensitive to the τ T , varying less than 1% when we increase/decrease the lifetime. We need to keep in mind that the decay rate of the T 1 is ≈10 3 s −1 (1/τ T ) and the T 1 →Ln 3+ is ≈10 6 s −1 (for the Eu 3+ complex), explaining this weak variation. On contrary, the decay rate of S 1 is 10 9 s −1 (1/τ S ), while the S 1 →Ln 3+ is in the order of 10 5 s −1 (also for the Eu 3+ complex), thus these competitions influence the Ln 3+ emitting level population. However, once we are dealing with the ratio between intensities in two analogs complexes (they must present almost equal values of τ S and τ T ), this difference cancels in the thermometric parameter and consequently in the thermal sensitivity. Except for the intersystem crossing rate which is almost constant with temperature, [75] the τ S , τ T , and W 3→4 might be dependent on the temperature as well. Therefore, as a proof of concept, we made simulations on the ratio between the populations of emitting levels (Tb 3+5 D 4 and Eu 3+5 D 0 ) varying these rates two orders of magnitude each and we found that the thermal behavior remains unchanged ( Figures S22 and S23, Supporting Information and discussion around).
For the Ln 3+ lifetimes τ, the experimental values were used ( Figure 3b). As mentioned before, W ISC is the intersystem crossing rate (S 1 →T 1 ), which is very sensitive to the energy gap between these states, [76,77] and such energy gap of ≈8500 cm −1 may lead to a reasonable value of W ISC ≈ 10 5 s −1 . W 3→4 is the nonradiative energy decay from upper levels of Ln 3+ to the emitting levels ( 5 D 0 and 5 D 4 ), which is in the order of ≈10 6 s −1 for Ln 3+ -chelates. [72] The set of ODEs (Equations (2)-(6)) were numerically solved using the Radau method. [78] As an illustration, Figure S21 Table S11, Supporting Information).

Theoretical and Experimental Thermometry
Once the Eu 3+5 D 0 and Tb 3+5 D 4 emitting levels populations are estimated, we will demonstrate how the behavior of experi-mental thermometric parameter Δ and the relative thermal sensitivity (Equation S1, Supporting Information), can be explained and predicted from the basics of the theory of luminescent energy transfer, through the Judd-Ofelt theory, [79,80] IET theory, [81] and the populational analysis from rate equations. [82] Thus, we can describe the Tb 3+ 5 D 4 → 7 F 5 and Eu 3+ 5 D 0 → 7 F 2 intensities (I Tb and I Eu , respectively) and consequently the thermometric parameter Δ = I Tb /I Eu .
The relative amount of Tb 3+ and Eu 3+ does not substantially impact the kinetics and the emitting level populations. Since each Ln 3+ is located in a single molecular entity [Ln(bpy) 2 (NO 3 ) 3 ], the minimal distance between two Ln 3+ centers will be kept at 7.7 Å. For example, if we consider the composition of 50% Tb 3+ and 50% Eu 3+ , the system [Tb 0.5 Eu 0.5 (bpy) 2 (NO 3 ) 3 ] will present a minimal intermolecular Tb 3+ -Eu 3+ distance of 7.7 Å and the Ln 3+ -to-Ln 3+ rates will undergo a little dependency on the distribution of the ions in the matrix but still produces low Ln 3+ -to-Ln 3+ energy transfer rates (in the order of 10 2 s −1 ). Besides, as discussed before, the kinetics is dominated by the IET (Ligand-to-Ln 3+ ) part which is orders of magnitude higher than the intermolecular energy transfer (Ln 3+ -to-Ln 3+ ), thus the Tb 3+ /Eu 3+ ratio will not significantly change the kinetics and the emitting level populations. The inclusion of such low rates of 10 2 s −1 (Ln 3+ -to-Ln 3+ energy transfer) will not contribute to a significant change of the two emitting levels ( 5 D 4 and 5 D 0 ) populations. Just for illustrating, if we consider the Tb 3+ -to-Eu 3+ rates in a 7-levels set of ODEs, this rate will compete with the backward IET Tb 3+ →T 1 rate which is temperature independent ( = T b W 4.18 × 10 6 s −1 , Table 1) and it populates the T 1 instead of the 5 D 0 level. We should emphasize that this characteristic is exclusive due to the large Tb 3+ -Eu 3+ distance involved. Figure 5 presents the calculated and measured data of Δ together with the corresponding relative thermal sensitivities. It is evident the very nice agreement between calculated and measured data. The maximum S r value, 2.0%⋅K −1 at 262 K (and 1.6%⋅K −1 at 300 K), is comparable to the values reported in the literature for mixtures of Eu 3+ and Tb 3+ complexes [1] and MOFs operating near room temperature [41,51] (Table S4, Supporting Information). Remarkably, the high thermal stability of 1 could be taken as an advantage to investigate the repeatability of the temperature sensing in the room temperature region (where the S r is relatively large) upon thermal cycling. It turns out that 1 exhibits an exceptional 97% repeatability of Δ over 10 temperature cycles between 300 and 325 K ( Figure S13, Supporting Information). Yet, the thermal stability of 1 up to 560 K and the absence of solvate in the crystal lattice suggest it might ODE Initial condition Level (or Levels) t P t W W W P t W P t P 3 (0) = 0 5 D 2 to 5 L 10 (Eu 3 + ), 5 D 3 to 5 F 5 (Tb 3 + ) t P t P t W P t W P t P 4 (0) = 0 5 D 0,1 (Eu 3 + ), 5 D 4 (Tb 3 + ) (6) www.advopticalmat.de also be utilized in a higher temperature range. Moreover, we should emphasize that this is the first time where the thermometric parameters and sensitivity are modeled from theory and it should be a step toward predictions of thermometric properties before the nanothermometers preparation.
In the literature, the sigmoidal-shaped temperature dependence of the integrated intensities is frequently fitted to a semiclassical Mott-Seitz model. [35,51] When a ratio of integrated intensities is considered, it is common to assume that it is dominated by the temperature dependence of one transition presenting the sigmoidal functional form, [51] assuming, thus, that the other transition is virtually temperature independent. It is easy to understand that this is a demanding assumption that generally is not observed in practice. On the contrary, the theoretical approach presented here can model both the thermal quenching of the 5 D 4 → 7 F 5 transition and the thermal rise of the 5 D 0 → 7 F 2 one (that is the thermometric parameter Δ) permitting, thus, to understand how S r can be optimized. Attending to its definition (Equation S1, Supporting Information), to increase S r we must diminish Δ and this can be achieved by decreasing the integrated intensity of the Tb 3+ 5 D 4 → 7 F 5 transition and/or increasing that of the Eu 3+ 5 D 0 → 7 F 2 line. The former is ruled out by the 5 D 4 state lifetime, whereas the latter is controlled by the energy gap between the 7 F 0 and 7 F 1 states and the energy of the singlet and triplet states of the ligand, which must favor the forward IET rate. The energy gap between the 7 F 0 and 7 F 1 states ( Figure S17, Supporting Information) might be experimentally controlled by a change of the ligands as well as by the fine-tuning in the coordination sphere of the lanthanide to reshape the crystal field splitting. The ability to model the temperature dependence of the two integrated intensities opens the possibility to predict S r (T) giving precious hints on the energy transfer processes that regulate the operating temperature range of dual-center Eu/Tb molecular thermometers. The theoretical method presented is also capable to include ligand-to-metal charge transfer states (LMCT) and their interactions with other states (i.e., S 1 , T 1 , and Ln 3+ levels), treating the energy transfer rates involving the LMCT and including it in a rate equation model. [67,72,83] Thus, discernible thermal behavior in which the LMCT plays an important role in Ln 3+ -based molecular thermometers [84,85] could be also modeled.
The guidelines to performing theoretical predictions on thermometric properties could be summarized into four steps: i) structure of the compound, ii) energies of the singlet and triplet states and their distances to the Ln 3+ center (donor-acceptor distances), iii) IET rates (backward and forward), iv) rate equations model to determining the population of the emitting levels. We must emphasize that in the present case the Tb 3+ -to-Eu 3+ energy transfer is not operative due to a large Tb 3+ -Eu 3+ distance (≈7.7 Å). In the opposite case, these rates should be included in step iv) above.

Conclusions
This work provides a comprehensive analysis and rationalization of the mechanisms underpinning the temperature dependence of molecular thermometers. As an illustrative example, we rationalize the thermometric characteristics in a robust and thermally stable coordination complex, [Tb 0.94 Eu 0.06 (bpy) 2 (NO 3 ) 3 ], exhibiting a high thermal sensing reproducibility (>97%) for a molecule-based material. The complex is easily synthesized in large yields, presenting an efficient Ln 3+ sensitization by the bpy ligands together with chemical stability in air and water, and the absence of solvent molecules in the first coordination shell.
In this work, we modeled a dual-center Ln 3+ -based luminescent thermometer through the Judd-Ofelt theory, IET theory, and the populational analysis from rate equations, furnishing a clear picture of the temperature dependence of the luminescence spectrum. To our knowledge, this constitutes the first example in which the functional form of the thermal dependence of the integrated areas could be accurately anticipated, enabling thus the prediction of the thermometric parameter Δ and of the relative thermal sensitivity. Moreover, this approach is a toolbox that can be taken as an advantage to guide the design of novel molecular thermometers based on dual-emissive centers. The required inputs are basic structural  www.advopticalmat.de parameters (e.g., the Ln 3+ -Ln 3+ distance) and the energies of singlet and triplet states of the ligands used. The implementation of these computational procedures such as machine learning and/or data mining could be useful to indicate which systems (or a class of) will give the best responses for operating in the desired range of temperature (e.g., biological and cryogenic).
In clear contrast with the usual trial and error fitting methods, this rational modeling will open an avenue for intelligently engineering optimized dual-center thermometers that could be subsequently integrated into future devices.

Experimental Section
Reagents and Chemicals: All chemical reagents were purchased and used without further purification: Terbium nitrate hexahydrate (ABCR, 99.9% (REO)), Europium nitrate hexahydrate (Alfa Aesar, 99.9% (REO)), 2,2′-bypyridine (Sigma Aldrich, >99%), ethanol absolute (100% PA, VWR). All experiments were carried out under aerobic conditions. Structural and Photophysical Characterization: Elemental analysis was performed with an analyzer Elementar Vario MICRO Cube. Powders were pyrolyzed at 1150 °C and then reduced at 850 °C over hot Copper. Infrared spectra were recorded as KBr disks (1 wt% of the sample) in the range 4000-400 cm −1 on a PerkinElmer Spectrum two spectrophotometer with 8 acquisitions. A background without sample was recorded before the measurements.

Synthesis of [Tb
X-ray powder diffraction patterns were recorded in the 2θ interval 5-60° at room temperature with the PANalytical X'Pert Powder analytical diffractometer mounted in a Debye−Scherrer configuration and equipped with Cu radiation (λ = 1.5418 Å).
Thermogravimetric analyses were obtained with a thermal analyzer STA 409 Luxx (Netzsch) in the range 25-650 °C at a heating speed of 5 °C min −1 .
Energy-dispersive spectroscopy (EDS) analyses were performed on FEI Quanta FEG 200 instrument. The powders were deposited on an adhesive carbon film and analyzed under vacuum. The quantification of the heavy elements was carried out with the INCA software, with a dwell time of 3 µs.
Magnetic susceptibility data were collected with a Quantum Design MPMS-XL SQUID magnetometer working in the range 1.8-350 K with a magnetic field up to 7 Tesla. The AC magnetic susceptibility measurements were carried out in the presence of a 3 Oe oscillating field in zero or applied external DC field.
The photoluminescence measurements were recorded using a Fluorolog3 Horiba Scientific (Model FL3-22) spectroscope, with a modular double grating excitation spectrometer (fitted with a 1200 grooves mm −1 grating blazed at 330 nm) and a TRIAX 320 single emission monochromator TRIAX 320 single emission monochromator (fitted with a 1200 grooves mm −1 grating blazed at 500 nm, reciprocal linear density of 2.6 nm mm −1 ). The excitation source was a 450 W Xe arc lamp. Emission spectra (acquired using the front face mode) were corrected for the spectral response of the monochromators and detector, using the correction spectrum provided by the manufacturer, and the excitation spectra were corrected for the spectral distribution of the lamp intensity recorded using a photodiode reference detector. The time-resolved emission spectra and emission decay curves were acquired with the same instrumentation using a pulsed Xe-Hg lamp (6 µs pulse at half-width and 20-30 µs tail). The temperature was varied using a helium-closed cycle cryostat, a vacuum system (4 × 10 −4 Pa), and an autotuning temperature controller (Lakeshore 330, Lakeshore) with a resistance heater. All the measurements began at least 300 s after the temperature indicated in the temperature controller remained constant, thus ensuring the thermalization of the sample and constant temperature during the measurements.
The emission quantum yield values were measured at room temperature using a system (C9920-02, Hamamatsu) with a 150 W xenon lamp coupled to a monochromator for wavelength discrimination, an integrating sphere as the sample chamber, and a multichannel analyzer for signal detection. Three measurements were made for each sample and the average values obtained were reported with accuracy within 10% according to the manufacturer.
X-Ray Crystallography: The crystal of 1, with dimensions 0.049 mm × 0.093 mm × 0.113 mm, was measured by single-crystal x-ray diffraction using an exposure time of 3.71 h. Frames integration was performed using a narrow-frame algorithm with the Bruker SAINT software package. The integration of the data using an orthorhombic unit cell yielded a total of 73 161 reflections to a maximum θ angle of 45.29° (0.50 Å resolution), of which 9480 were independent (average redundancy 7.717, completeness = 100.0%, R int = 4.09%, R sig = 2.82%) and 6366 (67.15%) were greater than 2σ(F2). The final obtained unit cell parameters: a = 9.0395(3) Å, b = 15.0022(5) Å, c = 16.7187(6) Å, volume = 2267.3(2) Å 3 , were based upon the refinement of the XYZ-centroids of 5700 reflections above 20 σ(I) with 4.855° < 2θ < 84.71°. Data were corrected for absorption effects using the Multi-Scan method (SADABS). The ratio of minimum to maximum apparent transmission was 0.832. The structure solution was solved and refined using the Bruker SHELXTL Software Package, with Z = 4 for the formula unit, Tb 0.95 Eu 0.05 C 20 H 16 N 7 O 9 . The Ln (Eu, Tb) atomic positions, which were located on the 4c Wyckoff sites(1/2,y+1/2,1/4), were refined and the calculated occupations were in agreement with both those expected from the synthesis (Eu 0.05 Tb 0.95 ) and those obtained from EDX characterization (Eu 0.06 Tb 0.94 ).The final anisotropic full-matrix leastsquares refinement on F2 with 175 variables converged at R 1 = 2.92%, for the observed data and w R2 = 6.57% for all data. The goodness-of-fit was 1.110. The largest peak in the final difference electron density synthesis was 1.168 e-Å −3 and the largest hole was −2.342 e-Å −3 with an RMS deviation of 0.312 e-Å −3 . On the basis of the final model, the calculated density was 1.924 g cm −3 and F(000), 1287 e-.
CCDC-2079476 (1) contains the supplementary crystallographic data for this paper. These data were provided free of charge by The Cambridge Crystallographic Data Centre: ccdc.cam.ac.uk/structures. The corresponding CIF files are also available in the Supporting Information.
In Silico Experiments: DFT calculations were used to obtain structural and electronic properties like molecular geometry, symmetry, and molecular orbitals (MOs) compositions of the states involved in the energy transfer process. The geometry optimization was performed using the Gaussian 09 program [86] with B3LYP functional. [87,88] A Pople's basis set 6-31G(d) was employed to nonmetal atoms (hydrogen, carbon, nitrogen, oxygen, fluorine, and silicon) and the Ln 3+ ion was treated with www.advopticalmat.de MWB52 or MWB54 (Ln 3+ = Eu 3+ or Tb 3+ , respectively) basis set. [89] The TD-DFT calculation, using the same functional and basis set, was used to obtain the geometric centroid of the excited singlet (S 1 ) and triplet (T 1 ) states. This procedure allowed estimating the donors-acceptors distances R L , quantities that were essential for the energy transfer calculations.
Energy Transfer Rates: The IET from the excited states S 1 and T 1 to the Ln 3+ ion was calculated considering the dipole-dipole (W d−d , Equation S9, Supporting Information), dipole-multipole (W d−m , Equation S10, Supporting Information) and exchange (W ex , Equation S11, Supporting Information) mechanisms. Thus, the forward energy transfer rates (W S and W T ) were given by the sum of these three mechanisms (Equation S13, Supporting Information). The resonance (or energy mismatch) conditions δ = E L −E Ln (the donor-acceptor energy difference) were included by Malta [81] in an analytical expression (Equation S12, Supporting Information) for the spectral overlap integral, well-examined by Smentek and Kędziorski. [90] The shape of the spectral overlap as a function of the bandwidth at half-height of the ligand's state (γ L ) and the δ conditions can also be found in Moura Jr. et al. [91] The values of the intensity parameters Ω λ (λ = 2, 4, 6) as well the Ln-O and Ln-N overlap integrals ρ [92] for evaluating the shielding factors (1 − σ K ) were calculated using the JOYSpectra program. [93] The backward energy transfer rates (W b ), the energy returned from acceptor (Ln 3+ ) to donor state (S 1 or T 1 ), were obtained by the Boltzmann's factor exp(−|δ|/k B T). See ESI for further detail. Particularly, the thermal behavior of the energy transfer rates can be calculated using Boltzmann's factor. However, in the case of Eu 3+ , the thermally coupled populations of the levels 7 F 0 and 7 F 1 should be considered once their populations were the starting point when the energy transfer occurs. [51] Level Population Analysis: After all rates involved in the complexes were determined, the level population kinetics was described by a set of ordinary differential equations, [72,82,94] where the summations run all levels of the system. P i and P j are the populations of the levels |i〉 and |j〉, the W j→i and W i→j are the energy transfer rates between these states. The appropriate set of rate equations, with their respective initial conditions, were numerically treated using the Radau method [78] in simulations from 0 to 10 ms with a step-size of 10 µs.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.