Energy Transfer Mechanism and Quantitative Modeling of Rate from an Antenna to a Lanthanide Ion

The excitation energy transfer (ET) pathway and mechanism from an organic antenna to a lanthanide ion has been the subject of discussion for many decades. In the case of europium (Eu3+), it has been suggested that the transfer originates from the ligand singlet state or a triplet state. Taking the lanthanide complex Eu(TTA)3(H2O)2 as an example, we have investigated the spectra and luminescence kinetics, mainly at room temperature and 77 K, to acquire the necessary experimental data. We put forward an experimental and theoretical approach to measure the energy transfer rates from the antenna to different Eu3+ levels using the Dexter formulation. We find that transfer from the ligand singlet state to Eu3+ may account for the ET pathway, by combined electric dipole–electric dipole (ED–ED) and ED-electric quadrupole (EQ) mechanisms. The contributions from the triplet state by these mechanisms are very small. An independent systems rate equation approach can effectively model the experimental kinetics results. The model utilizes the cooperative processes that take place on the metal ion and ligand and considers S0, S1, and T1 ligand states in addition to 7F0,1, 5D0, 5D1, and 5DJ (=5L6, 5D3, 5D2 combined) Eu3+ states. The triplet exchange ET rate is estimated to be of the order 107 s–1. The observation of a nanosecond risetime for the Eu3+ 5D1 level does not enable the assignment of the ET route or the mechanism. Furthermore, the 5D1 risetime may be contributed by several processes. Observation of its temperature dependence and also that of the ground-state population can supply useful information concerning the mechanism because the change in metal-ion internal conversion rate has a greater effect than changes in singlet or triplet nonradiative rates. A critical comparison is included for the model of Malta employed in the online software LUMPAC and JOYSpectra. The theoretical treatment of the exchange mechanism and its contribution are now being considered.


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Title Page,S S1 EXPERIMENTAL 2 Table S1 Reported bands in the infrared spectrum of Eu(TTA)3(H2O)2 2-3 Figure S1 Thermogravimetric analysis results for Eu(TTA)3(H2O)2 3 S2 COMPUTATION OF MOLECULAR STRUCTURE 4 Table S2 Bond distances in the first coordination sphere of Eu in Eu(TTA)3(H2O)2 4 S3 SINGLET ENERGY AND OSCILLATOR STRENGTH 4 Figure S2 (a) Room temperature emission spectra of HTTA and Eu(TTA)3(H2O)2 in the solid state (b) 77 K emission spectra of La(TTA)3(H2O)2 (blue) and HTTA (magenta); excitation spectra of 550 nm (green) and 580 nm ( Table S3 Selected europium energy levels in materials 7 S4 OSCILLATOR STRENGTHS 7 Figure S3 Room temperature absorption spectrum of EuCl3 . 6H2O dissolved in water 8 Table S4 Measured oscillator strengths of Eu 3+ transitions for various systems 8-9 S5 THE SPECTRAL OVERLAP INTEGRAL 9 Figure S4 Overlap of two Gaussians 10 Figure S5 Spectral overlap of area under two Gaussians 11 S6 SPECTRAL OVERLAP DIAGRAMS 12 Figure S6 Spectral overlap diagrams 12-13 S7 RATE EQUATION MODEL 13 Figure S7 Zero singlet ET. 300 K energy level time profiles following a 5 ns pulse with the parameters in Table 3, columns 3 and 7, except with k, m and g set equal to zero 14 Figure S8 Zero singlet ET, 10 K. This calculation uses the same parameters as in Table 3 columns 4 and 8, with the triplet state energy transfer only involving 7 F0, not 7 F1 15 Figure S9 Zero singlet ET, 10 K. This calculation uses the same parameters as in Table 3  15 S2 columns 4 and 8, and Figure S8, except f = 10 8 s -1 Figure S10 Zero singlet ET, 10 K. This calculation uses the same parameters as in Table 3 column 4 and Figure S8, except f = 10 8 s -1 , a = 10 6 s -1 15 Figure S11 Zero Triplet ET. 300 K energy level time profiles with the parameters in Table 3, columns 3 and 7, except with n and p set equal to zero 16-17 Figure S12 Zero Triplet ET. 10 K energy level time profiles with the parameters in Table 3, columns 4 and 8, except with n and p set equal to zero 18 Figure S13 Zero Triplet ET. 10 K energy level time profiles with the parameters in Table 3, columns 4 and 8, except with n and p set equal to zero and s = 1E7 s -1 19 Figure S14 Calculated 10 K time profiles of levels using the Rosenbrock method, following a 5 ns pulse. The parameters are in columns 4 and 8 of  Table S5 Calculation of spectral overlap integral using direct integration with Maple 2021 and with Eq. S4 for F 21-22 Table S6 Comparison of results for ET rates of Eu(TTA)3(H2O)2 using the software LUMPAC and JOYSpectra 23-24  Figure S1. Thermogravimetric analysis results for Eu(TTA)3(H2O)2 using atmospheres of (a) nitrogen and (b) air. The initial loss of mass commences at 100 o C, up to 120-130 o C, and corresponds to the loss of two water molecules. The X-Ray diffractogram of Eu(TTA)3(H2O)2 is given in Ref. S3. Attempts in the present study to employ X-ray diffraction led to sample decomposition.

S3. SINGLET ENERGY AND OSCILLATOR STRENGTH
We envisage that after ultraviolet excitation, rapid intersystem crossing occurs from the higher singlet states to the lowest one, which then can undergo intersystem crossing and/or direct ET to Eu 3+ . The transfer occurs from the S1 -S0 zero phonon line (ZPL). This energy may be calculated by the SCF method and is generally at several thousand cm -1 lower than that calculated for the corresponding vertical transition by TD-DFT. S5 The room temperature spectra emission spectra of HTTA, La(TTA)3(H2O)2 and Eu(TTA)3(H2O)2 are too broad for estimation of the ZPL energy, Figure S2(a), but it is readily assigned at 25410 cm -1 at 77 K, Figure S2(b).
The room temperature absorption spectra of Eu(TTA)3(H2O)2 in toluene at different concentrations are displayed in Figure S2(c) and are similar to the literature spectra. The singlet oscillator strength can be calculated from absorption, Eq. (2-6) or emission, Eq.
(1) measurements. The absorption spectrum of Eu(TTA)3(H2O)2 in acetonitrile has been reported S1,S2,S6 and our deconvolution of the first broad absorption band in Ref. S7 into two Gaussians gives the oscillator strength of 0.05 for the lower energy one, using Eq. 5. However, our calculations show that there are not two, but about 10 different singlet states in this spectral region and fitting our experimental spectra with multiple Gaussians is not accurate. The calculated TD-DFT oscillator strengths of the lowest singlet state of Eu(TTA)3(H2O)2 in toluene S5 (vertical transition energy 24871 cm -1 , 402 nm) and in the gas phase (vertical transition energy 29867 cm -1 , 335 nm) are 0.0048 and 0.0083, respectively. Figure S2e shows the room temperature excitation spectra of Eu(TTA)3(H2O)2 in toluene and in the solid state. The singlet absorption band is red-shifted and broadened for the solid, where Eu 3+ transitions are observed. The packing of Eu(TTA)3(H2O)2 units increases by a magnitude of 2.2x10 5 from 10 M concentration in a solvent to the solid and this has been taken to indicate greater spin-orbit coupling and hence faster intersystem crossing to the triplet state. S8 The appearance of the europium absorption bands indicates greater competition of internal Eu 3+ excitation with the antenna ET in the solid state, and may involve 4f 6 -4f 6 cross-relaxation processes in the solid state. The measured ratio of the integrated areas ( 7 F0 → 5 D2)/( 7 F0 → 5 D1) [at, in cm -1 : 21552/19052] in the excitation spectrum of solid Eu(TTA)3(H2O)2 at room temperature in Figure S2e is 110, which is far from the nearly equal ratios from the absorption spectra. This may indicate that 5 D0 is more favorably populated in the solid state by excitation into 5 D2, rather than 5 D1, due to the cross-relaxation: 5 D2 + 7 F0 → 5 D0 + 7 F5. There is a firstorder J-selection rule prohibiting 5 D1 → 5 D0 vibrational relaxation, mentioned below.
The singlet emission of Eu(TTA)3(H2O)2 is totally quenched at room temperature when it is dissolved in toluene, but a very weak band with maximum at ~425 nm is observed for the solid ( Figure S2a). This could be due to dissociation, but if it corresponds to Eu(TTA)3(H2O)2 singlet emission, then the singlet ET/intersystem crossing rate is of the same order as the singlet lifetime for Eu(TTA)3(H2O)2 in the solid state at room temperature. The singlet emission of HTTA, with maximum at 441 nm, exhibits monoexponential decay with the lifetime of 1.3 ns at room temperature and 5.1 ns at 77 K ( Figure S2d). The corresponding lifetime for La(TTA)3(H2O)2 at 77 K is 3.50.3 ns ( Figure S2f). From Eq. 1, a 5 ns radiative lifetime is equivalent to an oscillator strength of about 0.19, whereas a 20 ns radiative lifetime would give Pif ~ 0.05. In the case of La(TTA)3(H2O)2, a 20 ns radiative lifetime would represent a quantum efficiency of 18%.  The intensity of these bands can arise from several mechanisms. The 7 F0 -5 D1 and 7 F1 -5 D0 transitions are mainly of magnetic dipole (MD) character. The 7 F0 -5 D0 transition is forbidden under Judd-Ofelt theory, but the breakdown of the closure approximation, J-mixing and the Wybourne-Downer process S17 enable this transition to gain intensity. Furthermore, it can gain intensity via crystal strain to give coincident defect site bands. We measured the oscillator strength for this transition of Eu 3+ in aqueous EuCl3 ( Figure S3) to be 1.7x10 -10 , whereas Hellwege and Kahle S13 gave the magnitude of 2.2x10 -9 and Ferrier et al. S17 gave the value 1.3x10 -8 for Y2SiO5:Eu 3+ . The strongest Eu 3+ transition in our region of interest is 7 F0 -5 L6, with the oscillator strength of the order 10 -6 , which is mainly of ED character. Electric dipole transitions gain intensity by both static and dynamic coupling mechanisms, in addition to the cross-terms of these. S14 The ED intensities depend upon the system geometry, charges and dipolar polarizabilities.

S5. THE SPECTRAL OVERLAP INTEGRAL
This integral is of the form: S10 where the f(E) are the normalized donor emission and acceptor absorption spectra on an energy scale. In our case, the antenna donor (D) has a broad emission spectrum, and the lanthanide acceptor (A) spectrum is a sharp band.
For an example, we model these by two Gaussian functions, y(D) and y(A), where y is given by: Here, y0 is the lower y value; A' is the peak area; Ec is the peak center (unit eV); w = FWHM; The The integral gives the same result as long as the interval extends over the acceptor profile. Plotting the curves gives Figure S4: Figure S4. Overlap of two Gaussians. The singlet donor emission is shown in the inset on an expanded ordinate scale.
where the donor is shown on an expanded ordinate scale in the inset. The spectral overlap area was calculated from the columns y(D) and y(A) in Origin 9 © , S19 with rows having the same abscissa E: and gave the result indicated in the graph, very close to the calculated integral. On the other hand, if we just consider the region where the curves overlap, we have Figure S5, where the area is 47 times smaller. S11 Figure S5. Spectral overlap of area under two Gaussians. S12

S7. RATE EQUATION MODEL
Initial parameter values. The value of Q is arbitrary S20 and its variation within a reasonable range does not affect the conclusions herein. Phosphorescence is quenched at room temperature. The value of parameter b indicates a fluorescence lifetime of 3.3 ns. The intersystem crossing rate (f) in lanthanide complexes is generally taken to be in the range of 10 8 -10 9 s -1 . The larger value is taken because the fluorescence is (mostly?) quenched. We take the room temperature value S21 of 0.2 ms for the 5 D0 lifetime, and the value 0.256 ms at 10 K (Fig. 3a, Ref. S21). The singlet ET rate is the sum of calculated values for S1 → DJ (g), 5 D1 (k) or 5 D0 (m) with 7 F0, 7 F1 at 300 K, and only for 7 F0 at 10 K. The parameter values for g, k, m are similar to those calculated in Table 3 of the manuscript. The triplet ET rate is the sum of T1 → 5 D1 (n) or 5 D0 (p) with 7 F0, 7 F1 at 300 K, and only for 7 F0 at 10 K. The 5 D1 risetime at 300 K or 10 K in Fig. 4b of Ref. S21 was used to calculate rate constant n, considering the level occupations, overlap integrals and degeneracies of 7 F0, 7 F1. Also, we have employed the trial values of n = 1E8 s -1 and 1E9 s -1 , even reducing g = 1E8 s -1 , but then the 5 D1 risetimes are too fast. This puts an upper limit to the exchange contribution to parameter n. The parameter value p is not known but is expected to be smaller than n. The internal conversion parameter s was initially set at 10 8 s -1 . The gap Eu 3+ ( 5 D2 -5 D1) is 2500 cm -1 , which can be spanned by two phonons. However, the most effective nonradiative process cannot just be estimated from the energy gap and the maximum phonon frequency. S22 The energy gap for Pr 3+ ( 3 P0 -1 D2) is 2600-2700 in M2O3:Pr 3+ (M = Y, Lu) with the room temperature multiphonon relaxation rate >10 7 s -1 . S23 Gaps between the energy levels of LiYF4:Nd 3+ of 938-1649 cm -1 are associated with nonradiative decay rates of 10 8 -10 9 s -1 . S22 The parameters d and u were adjusted to give the 5 D1 lifetimes at 300 K and 10 K. The fact that the risetime of 5 D0 equals the lifetime of 5 D1 (Fig. 5, Ref. S21) means that the radiative lifetime of 5 D1 is orders of magnitude longer than the nonradiative relaxation rate. The 5 D1 -5 D0 nonradiative decay is a special case where J = 1 → J = 0 selection rule limits the nonradiative relaxation rate, just as for Pr 3+ ( 3 P1 -3 P0) with the gap of 621 cm -1 but with the slow rate of 3.33x10 5 s -1 . S24 The change in nonradiative relaxation rate of ( 5 D1 -5 D0) from 10 K to 300 K is estimated to be an order of magnitude. S25 Zero energy transfer scenarios at 300 K and 10 K. The zero singlet energy transfer at 300 K gives the following results using a 5 ns pulse. Figure S7. Zero singlet ET. 300 K energy level time profiles following a 5 ns pulse with the parameters in Table 3, columns 3 and 7, except with k, m and g set equal to zero. (a) 5 D0; (b) 5 D1; (c) S1; (d) S0; (e) T1; (f) F0. The population of DJ is zero. Black curves are calculated by Maple 2021 and red curves are fitted by Origin 9 mono-or bi-exponential functions, with lifetimes as indicated. In these and in the following figures, the Counts for F0 and S0 are 0.99999999 or similar.
Zero singlet energy transfer scenario at 10 K. The results are plotted in Figure S8. Figure S8. Zero singlet ET, 10 K. This calculation uses the same parameters as in Table 3 columns 4 and 8, with the triplet state energy transfer only involving 7 F0, not 7 F1. (a) 5 D0; (b) 5 D1; (c) T1. Parameters k, m and g are set equal to zero. Again, a 5 ns pulse was used. Figure S9. Zero singlet ET, 10 K. This calculation uses the same parameters as in Table 3 columns 4 and 8, and Figure S8, except f = 10 8 s -1 . As above a 5 ns pulse was used. (a) 5 D0; (b) 5 D1; (c) T1. The triplet state energy transfer only involves 7 F0, not 7 F1. Black curves are calculated by Maple 2021 and red curves are fitted by Origin 9 mono-or bi-exponential functions, with lifetimes as indicated. Figure S10. Zero singlet ET, 10 K. (a) 5 D1; (b) T1. This calculation uses the same parameters as in Table 3 column 4 and Figure S8, except f = 10 8 s -1 , a = 10 6 s -1 . As in Figures S8 and S9, parameters k, m and g are set equal to zero. A 5 ns pulse was used. The triplet state energy transfer only involves 7 F0, not 7 F1. Black S16 curves are calculated by Maple 2021 and red curves are fitted by Origin 9 mono-or bi-exponential functions, with lifetimes as indicated.
Zero triplet energy transfer scenario, 300 K and 10 K Figure S11. Zero Triplet ET. 300 K energy level time profiles with the parameters in Table 3, columns 3 and 7, except with n and p set equal to zero. A 5 ns pulse was used. (a) 5 D0; (b) 5 D1; (c) DJ; (d) 7 F0; (e) S0; (f) S1; (g) T1. Black curves are calculated by Maple 2021 and red curves fitted by Origin mono-or biexponential functions with lifetimes as indicated. The differential equations were solved by the Rosenbrock (labeled (i)) or Runge-Kutta-Fehlberg (rkf45) (labeled (ii)) algorithm with automatic error estimation using rules of order 4 and 5, as indicated. The Rosenbrock method is more stable but gives an artifact (for example at 3.5 s for 5 D0) which is not present in the rkf45 method. S18 Figure S12. Zero Triplet ET. 10 K energy level time profiles with the parameters in Table 3, columns 4 and 8, except with n and p set equal to zero. A 5 ns pulse was used. (a) 5 D0; (b) 5 D1; (c) T1, ns range; (d) T1, ms range; (e) DJ. Note the different ranges for T1 in (c), (d). Black curves are calculated by Maple 2021 and red curves are fitted by Origin 9 mono-or bi-exponential functions, with lifetimes as indicated. S19 Figure S13. Zero Triplet ET. 10 K energy level time profiles with the parameters in Table 3, columns 4 Table 3 (black  lines). Black curves are calculated by Maple 2021 and red curves are fitted by Origin 9 mono-or biexponential functions, with lifetimes as indicated.

S8. CALCULATIONS USING THE MODEL OF MALTA
A. Comments upon the model. There are some assumptions in Malta's model -necessary to avoid complicated expressions with parameters unrelated to experimental data and some restrictions on the usage of parameters in the LUMPAC S26 and JOYSpectra S27 software. The model employs selection rules based upon J-multiplet terms. By contrast, note that the crystal field levels of each multiplet often have mixed parentage and they are split at a low site symmetry over a range of energy. For example, even in a highly symmetric octahedral environment of Eu 3+ in Cs2NaEuCl6, the 5 L6 multiplet is split by 584 cm -1 . S11 In the theory of Malta, and subsequently in LUMPAC and JOYSpectra, the spectral overlap integral has been represented by F, in units of erg -1 : where L is the full width at half maximum (FWHM) of the broad emission band (rad s -1 ), ℏ is the reduced Planck constant and  (erg) is the energy difference between the band maxima.
For example, as above in Section S6, L = 0.6 eV = 0.6 x 8065.5439 cm -1 = 4839.326 cm -1 ;  = 2.77 -2.14 = 0.63 eV = 5081.293 cm -1 . This value is 4.96 times larger than that calculated from Maple 2021 above. The Table S5 shows calculated values of the overlap integral using direct integration with Maple 2021 and using the formula S4 for F. The ratio of the values obtained is less than 5 for values of  (ligand energy -Eu 3+ energy) ~ 0.7 eV ~ 5646 cm -1 but increases to a factor near 5000 for values of  over 10000 cm -1 . In practice, the singlet emission band maximum is usually lower than 3 eV but both LUMPAC S26 and JOYSpectra S27 use the energy of the vertical transition (and not the zero phonon line) in the absorption spectrum. In this case, the singlet state peak maximum would be about 0.58 eV (4700 cm -1 ) too high in the present case for Eu(TTA)3(H2O)2, leading to considerable error in F. In fact, in our manuscript, the spectral overlap integrals are all small, in the narrow range from 0.17 eV -1 to 2.3 eV -1 (column 3, Table 2). The values calculated by Eq. S4 for F by using the band maxima for donor emission and acceptor absorption are within an order of magnitude except for singlet to 5 D0 ET. Note however that both LUMPAC and JOYSpectra do not use donor emission band maxima but utilize the donor absorption band maxima for calculation and this may be about 8000 cm -1 higher in energy than the emission band maximum. The results then differ considerably.

B. Calculations using LUMPAC and JOYSpectra.
The software LUMPAC S26 utilizes the europium room temperature emission spectrum, the 5 D0 emission lifetime, and an input file with the molecular structure of the complex. The output gives ET rates from the ligand singlet and triplet states to various Eu 3+ states. It is described in a tutorial S28 and in Ref. S20.
Some points concerning the program are that: (i) the ligand singlet transition dipole strength is fixed at 10 -35 esu 2 cm 2 .
(ii) the donor-acceptor spectral overlap function F uses the value of ligand full width at half-maximum (FWHM) equal to 3250 cm -1 (0.40 eV).
(iv) The energy of the singlet vertical absorption transition (instead of emission) is employed in the calculation of F.
(v) The Eu 3+ energy levels and matrix elements are taken from the study of LaF3:Eu 3+ whereas the coordination sphere of Eu 3+ in organic complexes usually comprises nitrogen and/or oxygen atoms.
(vi) The calculation of triplet and singlet state energies is fast using INDO/S-CIS or TD-DFT in ORCA but the results are inaccurate.
(vii) The formulation of the rate equations is not the same as in herein.

S23
(viii) The model of Malta includes the contribution from exchange interaction, but the relevant exchange integral is approximated by that of the dipole moment operator. In particular, the same magnitude is taken for singlet or triplet donors.
When calculating the ET rate from the antenna singlet state to Eu 3+ , both LUMPAC and JOYSpectra S29 require the energy of the S1 → S0 band maximum. The value in LUMPAC is taken from the energy calculation by ZINDO/S for the singlet state with highest oscillator strength, although the state can be changed by manually modifying the file. The value in JOYSpectra may be entered but for the example file of Eu(TTA)3(H2O)2 it has been set at 29900 cm -1 (334 nm). This energy is far above the S1 zero phonon line from which transfer occurs.
The more recent JOYSpectra software S27,S29 overcomes some of these points, in particular in providing the opportunity to adjust the FWHM in the calculation of F. The example input file for Eu(TTA)3(H2O)2 S29 uses S1 and T1 energies 29900 and 20300 cm -1 , respectively, which lead to errors since the values are inaccurate. In particular, ET processes are listed for Eu 3+ levels which are higher in energy than the donor singlet state.
The ET rate calculation was carried out using the coordinates given for Eu(TTA)3(H2O)2 in JOYSpectra. The ORCA energy level calculation in LUMPAC utilizes ZINDO/S and it gave triplet energies commencing at 12755 cm -1 , which is far too low in energy. The normal (auto) calculations in the software choose singlet states which are far above normal excitation wavelengths so that the calculations do not reflect physical situations. For the calculation of ET rates in our case, Table S6, the calculated fourth triplet (19961 cm -1 ) and lowest singlet state (28964 cm -1 ) were chosen as the ligand states in LUMPAC. In JOYSpectra, the ligand states were chosen at S1 25000 cm -1 , T1 19802 cm -1 . Our calculations for the ET rates of Eu(TTA)3(H2O)2 using these energies in the software LUMPAC and JOYSpectra are given in Table  S6. The overwhelming contribution comes from exchange because of the approximation made for its matrix element. S30 Table S6. Comparison of our results for ET rates of Eu(TTA)3(H2O)2 using the software LUMPAC and JOYSpectra with those from Refs. 32,33, as described above. (The mechanisms are given as exexchange; dd -dipole-dipole; dm -dipole-multipole; mp -multipolar). We consider that ET only occurs from the lowest singlet state, not higher ones. Note that the T1 → T0, 7 F0 → 5 D1 rate is 23 times higher than in Table 3 of the manuscript.

C. Calculations employing the model of Malta but with some parameters taken from the present study.
The aim of the following is to show that simple calculations can be made without recourse to software since many matrix elements are zero.
This Section gives our calculations of ET rates for examples of some transitions in Eu(TTA)3(H2O)2 using the model of Malta, with the spectral overlap integral and dipole strength (calculated from oscillator strength) taken from the present study. In conclusion, the ED-ED transfer rates are calculated in the region of 10 5 s -1 but the ED-EQ rate for S1 → S0, 7 F0 → 5 D2 is within an order of magnitude of that calculated in Table 2 of the manuscript for ED-ED transfer.
Reduced spin matrix elements. The squared reduced spin matrix elements of Eu 3+ have been taken by Malta's group from the free ion wavefunctions of Ofelt. S34 The ground state 7 F0 has four SLJ components and other free ion terms have up to ten. Professor Y. Y. Yeung kindly provided to us the free ion compositions, comprising 10 SLJ components for each multiplet term, calculated with 6 decimal places from the fits of the energy levels of Cs2NaEuCl6, LaF3:Eu 3+ and NaCdPO4:Eu 3+ so that we could check if the reduced matrix elements show any dependence upon Eu 3+ coordination. We employed the formula for the matrix element: Where ai represents the product of the compositions of initial and final states, and < || || ′ ′ ′ ′ > = , ′ , ′ , ′ (−1) + + +1 [(2 + 1)(2 ′ + 1) ( + 1)(2 + 1)] 1/2 { ′ 1 } Note the power of (-1). Our results in the Table S7 S7) where: W (s -1 ) = (SL (esu 2 cm 2 ) e 2 (esu 2 ) F (erg -1 )  (cm 2 ))/( h (erg s) R 6 (cm 6 )) and we use the values of the dipole strength SL and F from the present study.
Consider S1 → S0 and 7 F0 -5 D2: The U () matrix elements are given as zero in JOYSpectra so that the respective ET rates are zero. However, the U (2) matrix element squared has been reported as 0.0009 for oxide systems. S36 Noting that Ω 2 = 8.068E-22, and from Table 1, F = 1.335 x 6.241496E11 erg -1 , and correcting for the ground state population, WET ~ 2.01 x 10 5 s -1 .
Consider S1 → S0 and 7 F0 -5 L6: In Table S6, the largest values of Wd-m for the 7 F0 acceptor state are for 5 L6, 5 D4 and 5 G6. We do not consider the latter two because their energies lie above S1 → S0. For 5 L6, the squared U () matrix elements are zero for  = 2,4 and for  = 6 the value is 0.0153. The value of the spectral overlap integral in Table 1 is 1.0735x10 11 erg -1 , so that the calculated multipolar ET rate after correcting for the population of 7 F0, is 2.63x10 4 s -1 , without back-transfer.