Modeling the Eu(III)-to-Cr(III) Energy Transfer Rates in Luminescent Bimetallic Complexes

Abstract

There is a growing interest in alternatives to lanthanide ion (Ln(III))-based luminescence sensitizing chromophores for in vivo applications, mainly in optical biological windows. Transition metals (M) are relevant candidates as chromophores as they have high absorption rates and emission bands covering a wide range of visible to near-infrared spectrum. However, despite the importance of theoretical models for the design of M–Ln(III) complexes, few contributions have devoted efforts to elucidating the energy transfer (ET) processes between M and Ln(III) ions. In this context, we adapted the intramolecular energy transfer (IET) to calculate, for the first time, the energy transfer rates for M–Ln(III) complexes. A new model was proposed that considers the assistance of phonons in the calculation of ET rates. As an example, the proposed model can estimate the ET rates between Eu(III) and Cr(III) ions in the [CrEuL₃]⁶⁺ complex (where L = 2-{6-[N,N-diethylcarboxamido]pyridin-2-yl}-1,1′-dimethyl-5,5′-methylene-2′-(5-methylpyridin-2-yl)bis [1H-benzimidazole]). The calculated rates (930–1200 s⁻¹) are in excellent agreement with the experimentally available data (750–1200 s⁻¹) when a phonon-assisted energy transfer process is considered. Thus, this proposed model can be useful to predict and explain photophysical properties driven by the energy transfer between Ln(III) ions and transition metals.

1. Introduction

Lanthanide-based compounds are well established in having a wide range of applications, such as in lighting devices [1,2], medicine and bioimaging [3,4,5,6,7,8], and optical/temperature sensors [9,10,11,12]. However, there is also a great interest in alternative chromophores for lanthanide ion (Ln(III))-based luminescence sensitization for near-infrared applications, mainly in biological optical windows [13,14,15]. Transition metals (M) are often relevant candidates as chromophores because they have high absorption rates and emission bands covering a wide spectral range from visible to infrared [16,17,18]. Among the transition-metal ions, Cr(III) is a promising example with many applications, including in temperature [19,20,21] and pressure [22,23,24,25] sensing. However, despite the importance of theoretical models for the design of M–Ln(III) complexes, only a few contributions have devoted efforts to elucidate and quantify the energy transfer (ET) processes between M and Ln(III) ions [26,27]. Most approaches describing the efficiency of ET between transition metals and lanthanide ions consist of empirical associations from the difference in the relaxation rates of the sensitizer (donor) in the presence and absence of an acceptor [28,29,30]. Additionally, to a lesser extent, these approaches perform estimates based on Förster [31] and Dexter [32] models for dipolar (Coulomb) and exchange interactions [17,26,33].

Förster’s theory was developed by considering the allowed electric dipole transitions between a donor (D) (or sensitizer) and an acceptor (A) (or activator). However, this approach is not adequate for describing the usual interactions involving lanthanide ions because the 4f-4f transitions are, a priori, forbidden by parity. Furthermore, none of the Förster–Dexter approaches consider the Judd–Ofelt theory for 4f-4f intensities, even though this theory (developed a posteriori) was quite successful in the description of intra-configurational 4f-4f transitions [34]. Indeed, approaches established for Ln–Ln [35,36,37,38] and Ligand–Ln [34,39,40] interactions are quite adequate for describing several experimental results [5,41,42,43,44,45]. In this context, there is a gap in theoretical approaches that can quantify ET rates between M–Ln that consider the peculiar properties of both Ln(III) and M centers. Based on Ln–Ln and Ligand–Ln ET approaches, a theoretical framework for describing the ET interactions between M–Ln centers in heterometallic complexes is introduced in this contribution. However, for the M–Ln systems, a new model considering the assistance of phonons in the calculation of ET rates is developed and applied. Estimations of the theoretical ET rates and identification of involved mechanisms are performed for the bimetallic complex [CrEuL₃]⁶⁺ (L = 2-{6-[N,N-diethylcarboxamido]pyridin-2-yl}-1,1′-dimethyl-5,5′-methylene-2′-(5-methylpyridin-2-yl)bis [1H-benzimidazole]) [46]. Comparisons with the experimentally determined rates for [CrEuL₃]⁶⁺ are then discussed.

2. Theoretical Methodology

Based on the intramolecular ET model developed by Malta and collaborators [34,39], the determination of M–Ln rates was performed using Equations (1) and (2). These equations were adapted to consider the dipole strength transitions of the metal M (e.g., Cr(III)) and Ln(III), represented by the $S_{\mathrm{d}}^{\mathrm{M}}$ and $S_{\mathrm{d}}^{\mathrm{Ln}}$.

$$W_{\mathrm{d}-\mathrm{d}}=\frac{2\pi }{\hslash }\frac{S_{\mathrm{d}}^{\mathrm{M}}}{G_{\mathrm{M}}}\frac{S_{\mathrm{d}}^{\mathrm{Ln}}}{R_{\mathrm{d}-\mathrm{f}}^6}F$$

(1)

$$W_{\mathrm{d}-\mathrm{m}}=\frac{2\pi }{\hslash }\frac{S_{\mathrm{d}}^{\mathrm{M}}}{G_{\mathrm{M}}}{\displaystyle \sum }_{\lambda =2,4,6}\frac{S_{\lambda }^{\mathrm{Ln}}}{{\left(R_{\mathrm{d}-\mathrm{f}}^{\lambda +2}\right)}^2}F$$

(2)

where

$$S_{\mathrm{d}}^{\mathrm{Ln}}=\frac{2e^2{\left(1-{\sigma }_1\right)}^2}{\left(2J+1\right)}{\displaystyle \sum }_{\lambda =2,4,6}{\Omega }_{\lambda }^{\mathrm{FED}}{\langle {\psi }^{\prime }{J}^{\prime }\Vert U^{\left(\lambda \right)}\Vert \psi J\rangle }^2$$

(3)

$$S_{\lambda }^{\mathrm{Ln}}=\frac{\left(\lambda +1\right)e^2{\left(1-{\sigma }_{\lambda }\right)}^2}{2J+1}{\langle r^{\lambda }\rangle }^2{\langle f\Vert C^{\left(\lambda \right)}\Vert f\rangle }^2{\langle {\psi }^{\prime }{J}^{\prime }\Vert U^{\left(\lambda \right)}\Vert \psi J\rangle }^2$$

(4)

Equation (1) represents the ET rate for the dipole–dipole mechanism and Equation (2) represents the dipole–multipole mechanism. Equations (3) and (4) define $S_{\mathrm{d}}^{\mathrm{Ln}}$ and $S_{\lambda }^{\mathrm{Ln}}$ that represent, respectively, the equivalent dipole and multipole strengths of the lanthanide ion. $\hslash =h/\left(2\pi \right)$ is the reduced Planck constant, and $2J+1$ and $G_{\mathrm{M}}$ are the degeneracies of the initial states of the Ln(III) ion and the transition metal, respectively. The term $\langle {\psi }^{\prime }{J}^{\prime }\Vert U^{\left(\lambda \right)}\Vert \psi J\rangle$ is the reduced matrix element of the unit tensor operator $U^{\left(\lambda \right)}$, operating between states of the Ln(III) ion [47]. The quantity $\langle r^{\lambda }\rangle$ is the expectation value of the 4f radial wavefunction [48]. The monoelectronic reduced matrix element $\langle f\Vert C^{\left(\lambda \right)}\Vert f\rangle$ is essentially a 3-j symbol [49,50]. The shielding factors, ${\sigma }_1$ and ${\sigma }_2$, can be obtained according to the relation given in references [34,51]. For $\lambda$ = 4 and 6, the values of ${\sigma }_{\lambda }$ can be found in reference [48]. The values of each parameter concerning the Ln(III) ion are already implemented in the JOYSpectra web platform [52]. The symbol ${\Omega }_{\lambda }^{\mathrm{FED}}$ represents the Judd–Ofelt intensity parameter taking into account exclusively the forced electric dipole (FED) contribution. $R_{\mathrm{d}-\mathrm{f}}$ is the distance between the centroid of the donor and acceptor states, considered in the present work as the distance between the two metallic centers (M–Ln).

The $F$ factor that appears in Equations (1) and (2) represents the density of states between the involved transitions and it is associated with the energy mismatch condition $\mathsf{\Delta }=E_{\mathrm{D}}-E_{\mathrm{A}}$ (where $E_{\mathrm{D}}$ and $E_{\mathrm{A}}$ stand for the energy of the donor and acceptor states, respectively). Because the bandwidths of M transitions may vary from the same order of magnitude of Ln(III) transitions (~10² cm⁻¹) to as large as organic ligand bandwidths (>10³ cm⁻¹), the general expression for the $F$-factor [36,53] is calculated as follows:

$$F=\frac{{\rm Y}}{{\left({\gamma }_D^2+{\gamma }_A^2\right)}^{1/2}}\times e^{-{(\Delta /{\gamma }_D)}^2\Gamma }G\left(\mathsf{\Delta },T\right)$$

(5)

where

$$\Gamma =\left[1-\frac{1}{1+{\left({\gamma }_D/{\gamma }_A\right)}^2}\right]\mathit{ln}2,\hspace{1em}\hspace{1em}{\rm Y}={\left(\frac{\mathit{ln}2}{\pi }\right)}^{1/2}\frac{1}{ch}$$

(6)

with ${\gamma }_{\mathrm{D}}$ and ${\gamma }_{\mathrm{A}}$ being, respectively, the bandwidths of the donor and acceptor transitions. $c$ is the speed of light in vacuum and $h$ is the Planck constant.

$G\left(\mathsf{\Delta },T\right)$ is the barrier factor associated with the energy difference between the donor and acceptor transitions ($\mathsf{\Delta }$). For Δ ≥ 0, we have $G\left(\mathsf{\Delta },T\right)=1$, and for Δ < 0, $G\left(\mathsf{\Delta },T\right)=e^{\mathsf{\Delta }/k_{\mathrm{B}}T}$, in which $k_{\mathrm{B}}$ is the Boltzmann constant and $T$ is the temperature.

At low-resonance conditions (i.e., when $\mathsf{\Delta }\gg {\gamma }_{\mathrm{D}}+{\gamma }_{\mathrm{A}}$), the participation of phonons in the energy transfer process should not be ruled out. In fact, it can be crucial for bridging the energy gap, mainly in coordination compounds that may present high-energy vibrational modes (e.g., C=N and C=O stretchings). This process is known as phonon-assisted energy transfer [54]. Consequently, the new approach to calculate this $F$ factor considers the form given by Equation (7), which can be obtained from Equation (6.15) in Ref. [54] by truncating the summation at the first term:

$$F^{\prime }=e^{-\left(s_D+s_A\right)}\left[\frac{{\left(s_D+s_A\right)}^N}{N!}\right]\times F\left(\Delta -N\hslash \overline{\omega }\right)\times \delta \left(N, \Delta /\hslash \overline{\omega }\right)$$

(7)

where $s_{\mathrm{D}}$ and $s_{\mathrm{A}}$ are the Huang–Rhys factors for the donor and acceptor electronic states, and $N$ is the number of phonons with average energy $\hslash \overline{\omega }$ necessary to bridge the energy mismatch $\Delta$. The resonance condition is ensured by the Dirac delta function $\delta \left(N, \Delta /\hslash \overline{\omega }\right)$, while the $F\left(\Delta -N\hslash \overline{\omega }\right)$ function is the value of $F$ (Equation (5)) with $\Delta =0$. Therefore, the contribution of $e^{-{(\mathsf{\Delta }/{\gamma }_{\mathrm{D}})}^2\mathsf{\Gamma }}G\left(\mathsf{\Delta },T\right)$ in Equation (5) is equal to 1.

The dipole strengths $S_{\mathrm{d}}^{\mathrm{M}}$ can be obtained from spectroscopic measurements of the emission lifetime of the transition metal (${\tau }_{\mathrm{M}}$) or of its emission rate ($W_{\mathrm{M}}=1/{\tau }_{\mathrm{M}}$), in which [39],

$$S_d^M\left({\tau }_M\right)=\frac{3\hslash }{32{\pi }^2{\sigma }^3}\frac{1}{{\tau }_M} or S_d^M\left(W_M\right)=\frac{3\hslash }{32{\pi }^2{\sigma }^3}{W}_M{}_{ }$$

(8)

where $\sigma$ is the centroid of the M transition in cm⁻¹. Finally, the energy transfer rate ($W$) is obtained by the sum over $W_{\mathrm{d}-\mathrm{d}}$ and $W_{\mathrm{d}-\mathrm{m}}$ rates.

3. Selected Case

The model was used to determine the ET rates between the Eu(III) and Cr(III) ions in the complex [CrEuL₃]⁶⁺ (L = 2-{6-[N,N-diethylcarboxamido]pyridin-2-yl}-1,1-dimethyl-5,5-methylene-2-(5-methylpyridin-2-yl)bis [1H-benzi-midazole]), in which a characteristic Cr(III) emission is observed when the compound is electronically excited at the Eu(III) ⁵D₀ level, showing the occurrence of Eu(III) → Cr(III) ET path, even though the distance (ca. 9.3 Å) between the ions [46] is considerably large.

The experimental ET rate estimated from the Eu(III) lifetimes is about 1000 s⁻¹, obtained as the difference of the inverse of the donor lifetimes $\left(W=1/{\tau }_{ }-1/{\tau }_0\right)$ in the absence $\left({\tau }_0\right)$ and the presence ($\tau )$ of the acceptor [46].

Figure 1 illustrates the structure of the complex obtained from the complex Crystallographic Information File (CIF). Figure 2 shows the electronic levels of the ions involved in the ET process in the complex [CrEuL₃]⁶⁺.

All spectroscopic data from the literature [46,55] were extracted using the WebPlotDigitizer online tool [56]. From the Eu(III) side, the intensity parameters ${\Omega }_{\lambda }^{\mathrm{FED}}$ were determined utilizing the crystallographic structure of the [CrEuL₃]⁶⁺ complex as the input data in the JOYSpectra web platform [52] and the Eu(III) ⁵D₀ → ⁷F_J (J = 2, 4, and 6) transitions were considered as the donor states.

The theoretical model needs experimental data from the acceptor and donor transitions. Cantuel et al. [46] reported both the emission spectrum of the Eu(III) and the absorption spectrum of the Cr(III) in [CrEuL₃]⁶⁺. However, it is not possible to define the peaks and bandwidths of the Cr(III) states (²E and ²T₁) in the absorption spectrum, the necessary quantities for calculating the $F$ factor (Equation (5)). To circumvent this issue, we employed spectroscopic data from a similar complex available in the literature [55].

Golesorkhi and collaborators [55] reported triple helicate complexes with three metallic centers (two Cr(III) and one Ln(III)), which exhibit similar coordination environments and ligand-field strengths to the [CrEuL₃]⁶⁺ target complex. Comparing the data from the [CrLnCr(bpb-bzimpy)₃]⁹⁺ complexes [55] with the [CrEuL₃]⁶⁺ [46], it is also observed that there is a similarity in the Cr(III) electronic levels. Therefore, the data from the literature [55] were used as a reference for estimating the necessary parameters in the energy transfer rate calculations with Equations (1) and (2).

The theoretical ET rates were estimated with two different procedures: (i) by the summation of the individual ET rates, considering the Cr(III) transitions (²E ← ⁴A₂ and ²T₁ ← ⁴A₂) separately, and and (ii) by considering these two Cr(III) transitions together in an envelope fitted by a Gaussian function, leading to a single ET rate.

Figure 3 shows the Cr(III) ²E, ²T₁ ← ⁴A₂ transitions as well as the envelope including both transitions for the [CrErCr(bpb-bzimpy)₃]⁹⁺ complex [55].

4. Results and Discussion

4.1. Multipolar Energy Transfer Rates

To calculate the dipole–dipole ET rate, it is necessary to obtain the intensity parameters of the forced electric dipole mechanism (${\mathsf{\Omega }}_{\lambda }^{\mathrm{FED}}$, $\lambda =2, 4, 6$) based on the original Judd–Ofelt theory [57,58]. These values were estimated using the crystallographic structure of the complex as the input data in the JOYSpectra web platform [52]. The calculated values, in units of 10⁻²⁰ cm², are ${\mathsf{\Omega }}_{\lambda }^{\mathrm{FED}}=$ 0.269, 0.222, and 0.353 for $\lambda =2, 4, 6$, respectively.

The $F$ factor (Equation (5)) was obtained using the bandwidth ${\gamma }_{\mathrm{D}}\approx$ 200 cm⁻¹ and energy barycenters $E\left({}^{5}D_0\to {}^{7}F_{\lambda }\right)\approx$ 16,166, 14,400, and 12,386 cm⁻¹ for $\lambda =2, 4, 6$, respectively [46,47]. The bandwidths (${\gamma }_{\mathrm{A}}$) of Cr(III) are 236 and 390 cm⁻¹ for the ²E ← ⁴A₂ and ²T₁ ← ⁴A₂ transitions, respectively. The dipole strength $S_{\mathrm{d}}^{\mathrm{M}}\left({\tau }_{\mathrm{M}}\right)\approx$ 2 × 10⁻³⁸ esu²·cm² was estimated from the Cr(III) emission lifetime at room temperature (Equation (8)).

Considering the ET from Eu(III) to individual Cr(III) transitions (²E ← ⁴A₂ and ²T₁ ← ⁴A₂), two set of ET pathways are obtained: Eu(III) [⁵D₀ → ⁷F_J] → Cr(III) [²E ← ⁴A₂] and Eu(III) [⁵D₀ → ⁷F_J] → Cr(III) [²T₁ ← ⁴A₂].

Table 1. Theoretical ET rates at room temperature for the [CrEuL₃]⁶⁺ complex. The donor–acceptor distance is $R_{\mathrm{d}-\mathrm{f}}^{}=9.3\AA$. The values were calculated using $S_{\mathrm{d}}^{\mathrm{M}}\left({\tau }_{\mathrm{M}}\right)=$ 2 × 10⁻³⁸ esu²·cm². $\Delta$ is the energy difference between the donor and acceptor transitions. $W$ is the sum of $W_{\mathrm{d}-\mathrm{d}}$ and $W_{\mathrm{d}-\mathrm{m}}$ in the same pathway, and the total rate $W_{\mathrm{total}}$ is the rate obtained by summing all pathways.

Pathway	Donor	Aceptor	$\Delta$ (cm−1)	$\mathit{F}$ (erg−1)	${\mathit{W}}_{\mathbf{d}-\mathbf{d}}$ (s−1)	${\mathit{W}}_{\mathbf{d}-\mathbf{m}}$ (s−1)	$\mathit{W}$ (s−1)
1	5D0 → 7F2	2E ← 4A2	2748	1 × 10−11	1 × 10−22	3 × 10−20	3 × 10−20
2	5D0 → 7F4	2E ← 4A2	982	7 × 109	4 × 10−2	3 × 10−3	4 × 10−2
3	5D0 → 7F6	2E ← 4A2	−1032	2 × 107	3 × 10−5	5 × 10−10	3 × 10−5
4	5D0 → 7F2	2T1 ← 4A2	2026	2 × 106	7 × 10−6	3 × 10−3	3 × 10−3
5	5D0 → 7F4	2T1 ← 4A2	260	4 × 1012	20	1	21
6	5D0 → 7F6	2T1 ← 4A2	−1754	2 × 104	1 × 10−8	2 × 10−13	1 × 10−8
			${\mathit{W}}_{total}={\displaystyle {\displaystyle \sum }_{\mathit{i}=\mathbf{1}}^{\mathbf{6}}}{\mathit{W}}_{\mathit{i}}=$			21

presents the values of the $F$ factor and the ET rates for each pathway (pathways 1–3 involve the ²E, whilst pathways 4–6 involve the ²T₁ state).

It can be noticed that the donor Eu(III) ⁵D₀ → ⁷F₄ transition is more resonant with the Cr(III) acceptor transitions, mainly with the ²T₁ ← ⁴A₂ one ($\Delta =260$ cm⁻¹, pathway 5 in Table 1), providing high values of $F$ that could result in significant ET rates. However, the rates of pathways 2 and 5 (Table 1) are very low and very dissonant with the experimental ET rate (~1000 s⁻¹). This is because the selection rules on J quantum number, in which ${\langle {}^5\mathrm{D}_0\left|\right|U^{\left(2\right)}\left|\right|{}^7\mathrm{F}_4\rangle }^2=0$, being the dipole–quadrupole interaction equals to zero (first term of the summation in Equation (2)). Consequently, despite being very resonant, the ⁵D₀ → ⁷F₄ transition has only the dipole–hexadecapole (2⁴-pole) contributing to $W_{\mathrm{d}-\mathrm{m}}$, which is orders of magnitude smaller due to the dependence on the donor–acceptor distance, $W_{\mathrm{d}-\mathrm{m}}\propto R_{\mathrm{d}-\mathrm{f}}^{-12}$.

In cases of ligand-to-Ln(III) intramolecular energy transfer, significant ET rates are obtained with a typical energy gap around 2000 cm⁻¹ between the transitions of the ligands and the lanthanide ions, because the bandwidths of the ligand states are of the order of or larger than the energy gap, offering a better resonance situation that results in $F$ factors of ~10¹¹−10¹² erg⁻¹ [59].

For pathways 1 and 4 in Table 1, the bandwidths of both species (donor and acceptor) are smaller than the energy gap $\Delta$. As a result, the F factors are very small and determinants for the negligible ET rates obtained. This situation is different when calculating the Eu(III) → Cr(III) ET rate considering the envelope of ²E ← ⁴A₂ and ²T₁ ← ⁴A₂ transitions (see

Table 2. Theoretical ET rates at room temperature for the [CrEuL₃]⁶⁺ complex considering the acceptor transition as the barycenter of Cr(III) ²E, ²T₁ ← ⁴A₂ transitions (envelope). $R_{\mathrm{d}-\mathrm{f}}^{ }=9.3\AA$. $W$ is the sum of $W_{\mathrm{d}-\mathrm{d}}$ and $W_{\mathrm{d}-\mathrm{m}}$ in the same pathway and the total rate $W_{\mathrm{total}}$ is the sum over of all pathways rates.

Pathway	Donor	$\Delta$ (cm−1)	$\mathit{F}$ (erg−1)	${\mathit{W}}_{\mathbf{d}-\mathbf{d}}$ (s−1)	${\mathit{W}}_{\mathbf{d}-\mathbf{m}}$ (s−1)	$\mathit{W}$ (s−1)
1	5D0 → 7F2	2239	4 × 1010	4 × 10−1	98	98
2	5D0 → 7F4	470	2 × 1012	11	8 × 10−1	11
3	5D0 → 7F6	−1541	2 × 108	2 × 10−4	4 × 10−9	2 × 10−4
				${\mathit{W}}_{total}={\displaystyle {\displaystyle \sum }_{\mathit{i}=\mathbf{1}}^{\mathbf{3}}}{\mathit{W}}_{\mathit{i}}=$		109

) as the entire spectral range in Figure 3. The bandwidth of the envelope has the centroid at 13930 cm⁻¹ with a bandwidth of ~900 cm⁻¹. With a larger Cr(III) bandwidth (about half the energy gap), the F for the ⁵D₀ → ⁷F₂ transition (pathway 1 in Table 2) assumes values of ~10¹⁰ erg⁻¹ and the ET rate for the dipole–multipole mechanism ($W_{\mathrm{d}-\mathrm{m}}$) becomes dominant, although it is less resonant than the ⁵D₀ → ⁷F₄ transition, as displayed for the values of Δ in Table 2. This occurs because the contribution of the dipole–quadrupole mechanism for the ⁵D₀ → ⁷F₂ transition is non-zero due to the selection rule on J quantum number, namely, ${\langle {}^5\mathrm{D}_0\left|\right|U^{\left(2\right)}\left|\right|{}^7\mathrm{F}_4\rangle }^2=0.0032$.

The calculated ET rate considering the envelope of the Cr(III) transitions ($W_{\mathrm{total}}=$ 109 s⁻¹) is one order of magnitude lower than the experimental rate (~1000 s⁻¹). This underestimation is due to a weak resonance between the donor and acceptor transitions (large values of $\mathsf{\Delta }$ for the ⁵D₀ → ⁷F₂ donor transition). Hence, the calculations for the [EuCrL₃]⁶⁺ without considering a phonon-assisted energy transfer process, as displayed in Table 1 and Table 2, are not adequate to produce reasonable values of ET rates.

4.2. Exchange Energy Transfer Rates

The model can also include the contribution of the exchange mechanism in the Eu(III) → Cr(III) ET process. This mechanism strongly depends on the overlap integral between the valence orbitals of the donor and the acceptor ions [32,36]. Thus, for a distance of $R_{\mathrm{d}-\mathrm{f}}=$ 9.3 Å for the [EuCrL₃]⁶⁺ complex, the value of the d–f overlap integral (ρ) is indeed negligible (the highlighted point in Figure 4 indicates $\rho \approx$ 10⁻¹⁰ for $R_{\mathrm{d}-\mathrm{f}}=$ 9.3 Å), and this accounts for the vanishing value of the rate by the exchange mechanism. The Cr(III)–Eu(III) overlap integrals were estimated according to the procedure in Ref. [51].

4.3. Phonon-Assisted Energy Transfer

As the energy transfer process must ensure energy conservation, it is reasonable that the process could be assisted by phonons, bridging the energy gap between the donor and acceptor transitions [60]. In this case, an expression for the $F^{\prime }$ factor (Equation (7)) based on the Miyakawa–Dexter approach [54] was employed, which involves the Huang–Rhys factors associated with the donor and acceptor transitions ($s_{\mathrm{D}}$ and $s_{\mathrm{A}}$, respectively), as well as the average phonon energy ($\hslash \overline{\omega }$) and number of phonons ($N$) involved in the ET process.

The Huang–Rhys factors for lanthanide ions are of the order of $s_{\mathrm{D}}~$10⁻² due to the weak vibronic coupling [61,62,63,64,65]. For transition metals, the Huang–Rhys factors are usually in the intermediate coupling range (values between one and five). Indeed, the transitions involving the ²E and ²T₁ states are in the lower limit of the intermediate vibronic coupling range, presenting values around one [66,67].

To proceed with the phonon-assisted energy transfer rate calculations, the $F^{\prime }$ factor (Equation (7)) was calculated for the typical ranges of the Huang–Rhys factors, considering the vibronic couplings involved. In the case of the Eu(III) donor transitions, the $s_{\mathrm{D}}$ is in the [0.01–0.10] range, while for the Cr(III) acceptor transitions, $s_{\mathrm{A}}$ was considered to be within the [0.1–1.2] range. The average phonon energies were considered to be between 1000 and 1300 cm⁻¹, which matches with the C=N stretching modes of aliphatic amines (1020–1250 cm⁻¹) and aromatic imines (1250–1340 cm⁻¹) [68], indicating that the vibrational modes of the coordinated nitrogen atoms have an important role in the phonon-assisted energy transfer. In addition, this suggests the participation of two phonons in the ET process, which is feasible. These values of the average phonon energies are consistent with the single-configurational coordinate model applied to related triple-helical complexes, Na₃[Ln(dpa)₃]·13H₂O, which employed a phonon energy of 1200 cm⁻¹ [69]. However, this same work employed a Huang–Rhys factor of two to estimate the energy transfer rates, which could be considered too large for Ln(III) complexes.

Figure 5 displays the surface of the ET rates as a function of the Huang–Rhys factors, considering that two phonons are bridging the energy gap. It is noteworthy that the ET rates change only slightly with $s_{\mathrm{D}}$, while they are strongly dependent on the values of $s_{\mathrm{A}}$. Accordingly, to reproduce the experimental ET rate of 1000 s⁻¹, $s_{\mathrm{A}}$ should acquire values close to one, which is consistent with the lower limit of the vibronic couplings of the Cr(III) acceptor transitions [66,67]. To better illustrate these results,

Table 3. Phonon-assisted energy transfer rates ($W$) and their mechanisms’ contributions ($W_{\mathrm{d}-\mathrm{d}}$ and $W_{\mathrm{d}-\mathrm{m}}$). The average phonon energy $\hslash \overline{\mathsf{\omega }}=1120$ cm⁻¹ is considered in a two-phonon process. $s_{\mathrm{D}}$ and $s_{\mathrm{A}}$ are the Huang–Rhys factors for the donor and acceptor, respectively. $F^{\prime }$ is the phonon-assisted spectral overlap factor (Equation (7)).

${\mathit{s}}_{\mathbf{D}}$	${\mathit{s}}_{\mathbf{A}}$	${\mathit{F}}^{\prime }({\mathbf{10}}^{\mathbf{11}}·{\mathbf{erg}}^{-\mathbf{1}})$	${\mathit{W}}_{\mathbf{d}-\mathbf{d}}\left({\mathbf{s}}^{-\mathbf{1}}\right)$	${\mathit{W}}_{\mathbf{d}-\mathbf{m}}\left({\mathbf{s}}^{-\mathbf{1}}\right)$	$\mathit{W}\left({\mathbf{s}}^{-\mathbf{1}}\right)$
0.01	0.85	4.1	3.4	927.6	931.0
0.01	0.90	4.3	3.6	987.9	991.5
0.01	0.95	4.6	3.8	1046.8	1050.6
0.01	1.00	4.8	4.0	1101.2	1105.2
0.04	0.85	4.2	3.5	964.1	967.6
0.04	0.90	4.5	3.8	1023.0	1026.8
0.04	0.95	4.7	4.0	1079.4	1083.4
0.04	1.00	4.9	4.2	1133.0	1137.2
0.08	0.85	4.4	3.7	1011.4	1015.1
0.08	0.90	4.7	3.9	1068.3	1072.2
0.08	0.95	4.9	4.1	1122.5	1126.6
0.08	1.00	5.1	4.3	1174.0	1178.3
0.10	0.85	4.5	3.8	1034.5	1038.3
0.10	0.90	4.8	4.0	1090.3	1094.3
0.10	0.95	5.0	4.2	1143.4	1147.6
0.10	1.00	5.2	4.4	1193.7	1198.1

presents a range of $s_{\mathrm{A}}$ and $s_{\mathrm{D}}$ extracted from the surface in Figure 5, resulting in total rates of around 1000 s⁻¹.

The results in Table 3 emphasize the weak influence of $s_{\mathrm{D}}$ on the energy transfer process. For example, considering fixed values of $s_{\mathrm{A}}$, the ET rates vary between ca. 930 and 1070 s⁻¹ when the $s_{\mathrm{D}}$ increases 10 times (from 0.01 to 0.10). In contrast, when the $s_{\mathrm{A}}$ is slightly increased by 1.18 times (from 0.85 to 1.00), the ET rates significantly vary between 1600 and 1740 s⁻¹. This indicates that the phonon-assisted energy transfer process has a dominant contribution from the coupling of the Cr(III) states with the vibrational modes of the complex. As mentioned, this process should have a significant participation in the N−C vibrational modes in the coordination polyhedron.

In the case of Ln(III)–Ln(III) interaction, the efficiency of the energy transfer has a strong dependence on the resonance of the donor and acceptor states, mainly due to the narrow bandwidths of the transitions [37] and the weak vibronic coupling, expressed by the small Huang–Rhys factors [61,62,63,64,65].

Regarding the ET mechanisms, Ward [26] reported ET rates for M–Ln(III) systems with large intermetallic distances and suggested that the Dexter mechanism should be the most likely ET one (exchange) because the Ln(III) 4f-4f transitions are forbidden and the long M···Ln(III) distances are unfavorable for the dipole–dipole (Förster) mechanism. However, the present work indicates that multipolar mechanisms can be dominant in the ET process.

The model developed here considers the Judd–Ofelt theory, in which the forbidden character of 4f-4f transitions is treated through the forced electric dipole mechanism. Therefore, the model can predict that the multipolar mechanisms play a dominant role in energy transfer for the [CrEuL₃]⁶⁺ complex (see Table 3).

At this point, a comment concerning the cases of Eu(III)−Cr(III), Ln(III)−Ln(III), and organic ligand−Ln(III) non-radiative energy transfer should be made. In the latter case, the bandwidths of the donor (e.g., ligands, ${\gamma }_{\mathrm{D}}$) and the acceptor (e.g., Ln(III), ${\gamma }_{\mathrm{A}}$) are extremely different, with ${\gamma }_{\mathrm{D}}$ being much larger than ${\gamma }_{\mathrm{A}}$, i.e., ${\gamma }_{\mathrm{D}}\gg {\gamma }_{\mathrm{A}}$. Thus, in the evaluation of the energy mismatch factor (or spectral overlap) in the $F$ factor, provided the organic ligand bandwidth is used, the ligand vibrational band structure is encompassed. An approximate form of Equation (5) has been successfully employed for this case. For the ET between lanthanide ions, Ln(III)−Ln(III), the donor and acceptor bandwidths are similar, ${\gamma }_{\mathrm{D}}\approx {\gamma }_{\mathrm{A}}$, and very small due to the very weak vibronic couplings and the features of the 4f–4f transitions. Thus, the complete form of the $F$ factor in Equation (5) has been used. On the other hand, d–d transitions in transition-metal complexes have a considerable vibronic coupling contribution to their intensities, and they are not as large as organic ligand bands. This justifies the use of Equation (7) in terms of the Huang–Rhys factors (in the intermediate or strong vibronic coupling regimes) and an average promoting vibrational frequency.

The assumptions of the present model come from the Judd–Ofelt theory (for the calculation the dipole strengths $S_{\mathrm{d}}^{\mathrm{Ln}}$), which considers that the intra-configurational energy differences, $\Delta E\left(4{\mathrm{f}}^N,4{\mathrm{f}}^{N-1}nl\right)$, are much smaller than the inter-configurational energy differences, $\Delta E\left(4{\mathrm{f}}^N\right)$, i.e., $\Delta E\left(4{\mathrm{f}}^N,4{\mathrm{f}}^{N-1}nl\right)\ll \Delta E\left(4{\mathrm{f}}^N,4{\mathrm{f}}^{N-1}nl\right)$ [70], and from the shielding factors ${\sigma }_1$ and ${\sigma }_2$, whose values are obtained from quantum chemistry calculations for diatomic systems (e.g., Eu–O and Eu–N) [51]. However, even with these approximations, the model can reproduce the values of energy transfer rates in great accordance with the experimental ones.

5. Conclusions

A theoretical model to calculate the ET rates between a lanthanide ion and a transition metal is introduced. The [CrEuL₃]⁶⁺ complex was chosen as a case study and the Eu–Cr energy transfer rates, when considered a phonon-assisted process, agree with the experimental ones. The theoretical rates are around 930–1200 s⁻¹, with a predominance of the dipole–multipole mechanism ($W_{\mathrm{d}-\mathrm{m}}$) for an Eu–Cr distance of 9.3 Å, which is in excellent agreement with the experimental value of ca. 1000 s⁻¹.

The sensitivity of the estimated rates to the Huang–Rhys factors was analyzed within the range of $s_{\mathrm{D}}=$ [0.01 to 0.1] for Eu(III) and $s_{\mathrm{A}}=$ [0.1 to 1.2] for Cr(III), and the results show that the latter, $s_{\mathrm{A}}$, has a strong effect on the ET rates. Indeed, the calculated rates agree with the experimental value when the Huang–Rhys factor for the acceptor (Cr(III)) is within $s_{\mathrm{A}}=$ [0.85 to 1.00], which is consistent with the lower limit of the vibronic couplings in similar Cr(III) complexes. These calculations of the ET rates using the phonon-assisted model considered a two-phonon process to bridge the energy mismatch, which is consistent with C=N vibrational modes ($\hslash \overline{\omega } \approx$ 1000–1200 cm⁻¹) near the metal centers.

The proposed model can be useful for predicting and explaining photophysical properties driven by energy transfer between Ln(III) and transition metals, quantifying the contributions of each mechanism and the factors that may optimize energy transfer rates and efficiencies.

As the first study involving the calculation of theoretical Ln-M ET rates that considers the Judd–Offelt theory and the phonon-assisted process, we hope to motivate more experimental works focused on the determination of energy transfer rates between lanthanides and transition metals for the design of new luminescent materials.