On the Role of Spin–Orbit Coupling and State-Crossing Topography in the Nonradiative Decay of Ir(III) Complexes

Abstract

A pillar of our current understanding of the photoluminescence of Ir­(III) complexes is the assumption that the population of triplet metal-centered states determines an efficient nonradiative decay to the ground state minimum. Based on that assumption, the energy separation between the emitting state and the minimum-energy crossing point of the triplet metal-centered and ground states has been employed as a key variable for evaluating the ability of Ir­(III) complexes to decay nonradiatively. We demonstrate that the strong spin–orbit coupling between the triplet metal-centered and ground state of Ir­(III) complexes, together with the sloped topography of their crossing, leads to a significant energy separation between the two states, resulting in a reduced rate of nonradiative ground state recovery. Therefore, we propose that the role of metal-centered states is defined by the tendency of the excited state population to remain trapped in the metal-centered minima.

Ir­(III) transition-metal complexes (Ir3-TMCs), − which are characterized by a d⁶ electronic configuration and strong spin–orbit coupling (SOC), are fundamental for electroluminescent devices and photodynamic therapy (see Sections S1 and S2 of the Supporting Information). Because for both applications slow nonradiative decay is required, much effort has been devoted to determining the main nonradiative processes at play, with the ultimate goal of devising chemical modifications that minimize them. In general, two intramolecular nonradiative decays are considered. − The first is the intersystem crossing (ISC) from the emitting triplet minimum to the ground state (S₀). The second is the population of triplet metal-centered states (³MC), which are expected to efficiently mediate a nonradiative decay to the ground state.

³MC states of Ir3-TMCs are characterized by equilibrium structures in which at least one of the coordinating metal bonds is dissociated (Section S2). This dissociation causes a massive increase of the ground state energy at the ³MC minima and the presence of energetically and geometrically nearby ³MC/S₀ minimum-energy crossing points (MECPs). − It is the existence of such accessible MECPs, together with the large SOC, that leads to the assumption that ³MC states efficiently drive the population back to S₀. Most of the theoretical understanding of the photophysics of Ir3-TMCs is based on the validity of this assumption, relating an increase/decrease in the emission quantum yield with a decrease/increase in the accessibility of ³MC states.

The presumed highly efficient decay of ³MC states has led the scientific community to employ transition state theory (TST) to characterize ³MC-mediated nonradiative decay, treating the ³MC/S₀ MECP as an analogue of the transition state (TS) geometry. − ,, This implies that the probability of decay to S₀ is equal to unity at MECP, as in TST the probability of reaction is unity once the system reaches a TS. Here we demonstrate that for Ir3-TMCs the idea that the ³MC/S₀ MECP mediates an efficient repopulation of the Franck–Condon region cannot be assumed to be in general valid.

As a representative example, we focus on [Ir­(ppy)₂(bpy)]⁺ (where Hppy is 2-phenylpyridine and bpy is 2,2′-bipyridine), which is an archetype of the [Ir­(C^N)₂(N^N)]⁺ cyclometalated Ir­(III) complexes used in electroluminescent applications. , Our results are general and apply to any Ir3-TMCs. The emitting T₁ state of [Ir­(ppy)₂(bpy)]⁺ has metal–ligand charge transfer (MLCT) character and displays an equilibrium structure similar to the S₀ minimum. , From T₁, the processes determining the efficiency of light emission are supposed to start: phosphorescent emission, direct ISC to S₀, and population of the ³MC states with the subsequent transfer back to S₀ via the ³MC/S₀ MECPs. The latter process is a T₁-to-S₀ decay mediated by the ³MC/S₀ MECP, because, even if the nature of the triplet state changes from MLCT to MC, the involved potential energy surface (PES) is always the adiabatic T₁. Two types of ³MC states have been identified in [Ir­(ppy)₂(bpy)]⁺. The axial ³MC_ax state, which is characterized by a minimum structure (³MC_ax)_min displaying the dissociation of one Ir–N_ppy bond, and the equatorial ³MC_eq state, where an Ir–N_bpy bond is broken in its equilibrium geometry (³MC_eq)_min. Both bonds are present in the octahedral structures of the emitting T₁ MLCT and S₀ minima, so the evolution from the former to the latter, through the ³MC/S₀ MECP, implies the stretching of an Ir–N bond up to dissociation and then the reformation of the very same bond (Figure ). Therefore, an Ir–N distance plays the role of the reaction coordinate connecting the T₁ minimum and the ³MC/S₀ MECP. This means that both the initial and final minima (the emitting T₁ and the S₀ minima) are located along the same direction relative to the MECP. Borrowing from the terminology associated with conical intersections (CIs), we can say that the ³MC/S₀ intersection is sloped, with the gradients of two states pointing in the same direction. − The energies, structures, and orbital analysis of the described paths involving the ³MC_ax and ³MC_eq states are present in Section S3. These data have been previously obtained at the B3LYP/def2-SVP CPCM (CH₂Cl₂) level, and now also using the PBE0 functional.

1.

Nonradiative decay of [Ir­(ppy)₂(bpy)]⁺ from the T₁ minimum to the S₀ minimum mediated by (a) the ³MC_ax/S₀ MECP and (b) the ³MC_eq/S₀ MECP.

The topography of the MECP and the value of the SOC between the ³MC and S₀ states are key variables that strongly determine the probability of nonradiative decay. To illustrate this most clearly, we first need to look at the theory behind nonadiabatic events (NAEs) (Section S4). ,− Let us first analyze the case of an NAE between two adiabatic electronic states of pure spin having the same spin multiplicity (i.e., an internal conversion (IC)). Expanding the wave function in the basis of these two spin-pure adiabatic electronic states, it is possible to show that what causes the passage of population from one PES to the other are the nonadiabatic couplings (NACs) (eq S8 and eq ). Since, according to the generalized Hellman–Feynman theorem, the value of the derivative coupling between two states, which is the main component of the corresponding NAC, is inversely proportional to their energy separation (eq S10 and eq ), the closer the states are, the more probable is the NAE, and that is why CIs are key in photochemistry. ,

In the case of considering just the S₀ and first singlet excited state, S₁, the discussed equations will read as follows.

$$i{\mathrm{\psi ̇}}^{{\mathrm{S}}_1}(\mathbf{R},t)=({\mathrm{T̂}}_{\mathrm{R}}+V^{{\mathrm{S}}_1}(\mathbf{R})){\psi }^{{\mathrm{S}}_1}(\mathbf{R},t)+{\mathrm{\Lambda ̂}}^{{\mathrm{S}}_0{\mathrm{S}}_1}{\psi }^{{\mathrm{S}}_0}(\mathbf{R},t)$$ 1

$${\mathrm{\Lambda ̂}}^{{\mathrm{S}}_0{\mathrm{S}}_1}\cong -⟨{\mathrm{S}}_0|{\mathrm{\nabla ̂}}_{\mathrm{R}}|{\mathrm{S}}_1⟩\frac{1}{M}{\nabla }_{\mathbf{R}}$$ 2

$$⟨{\mathrm{S}}_0|{\mathrm{\nabla ̂}}_R|{\mathrm{S}}_1⟩=\frac{⟨{\mathrm{S}}_0|{\mathrm{\nabla ̂}}_{\mathrm{R}}{Ĥ}_{\mathrm{e}\mathrm{l}}|{\mathrm{S}}_1⟩}{V^{{\mathrm{S}}_1}(\mathbf{R})-V^{{\mathrm{S}}_0}(\mathbf{R})}$$ 3

where ψ̇^S₁ (R, t) is the time derivative of the nuclear wave function associated with state S₁, ψ ^S₀ (R, t) is the nuclear wave function associated with S₀, Λ̂ ^S₀S₁ is the NAC between S₀ and S₁, V ^S₁ (R) and V ^S₀ (R) are the S₁ and S₀ PESs, respectively, ⟨S₀|∇̂ _R |S₁⟩ is the derivative coupling between S₀ and S₁, and T̂ _R is the nuclear kinetic energy operator.

Let us now expand our wave function in the basis of two spin-pure adiabatic electronic states having different spin multiplicities, for example, a triplet and a singlet state. Such states are also called spin-diabatic states. − Repeating the previous mathematical manipulations, it now appears that the corresponding NAC is zero due to the orthogonality of the different spin wave functions, and that what couples the two states and drives an NAE is the SOC term (see eq ), assuming it is included in the Hamiltonian. − Considering only the S₀ and T₁ states, the coupling equation can be written as follows.

$$i{\mathrm{\psi ̇}}^{{\mathrm{T}}_1}(\mathbf{R},t)=({\mathrm{T̂}}_{\mathrm{R}}+V^{{\mathrm{T}}_1}(\mathbf{R})){\psi }^{{\mathrm{T}}_1}(\mathbf{R},t)+H_{\mathrm{S}\mathrm{O}\mathrm{C}}^{{\mathrm{S}}_0{\mathrm{T}}_1}{\psi }^{{\mathrm{S}}_0}(\mathbf{R},t)$$ 4

where ψ̇ ^T₁ (R, t) is the time derivative of the nuclear wave function associated with state T₁ and ${Ĥ}_{\mathrm{S}\mathrm{O}\mathrm{C}}^{{\mathrm{S}}_0{\mathrm{T}}_1}$ is the SOC between S₀ and T₁.

Let us again use the two spin-pure adiabatic electronic states of different spin symmetry (a singlet and a triplet) as a basis for the expansion of the wave function, but instead of using them as they are, let us use the combination resulting from the diagonalization of the electronic Hamiltonian including the SOC term (eq ).

$$\left(\begin{array}{ll}{V}^{{\mathrm{S}}_0}& H_{\mathrm{S}\mathrm{O}\mathrm{C}}^{{\mathrm{S}}_0{\mathrm{T}}_1}\\ H_{\mathrm{S}\mathrm{O}\mathrm{C}}^{{\mathrm{S}}_0{\mathrm{T}}_1}& V^{{\mathrm{T}}_1}\end{array}\right)\stackrel{\mathrm{diagonalization}}{\Rightarrow }\left(\begin{array}{ll}{V}^{\mathrm{S}{\mathrm{M}}_0}& 0\\ 0& V^{\mathrm{S}{\mathrm{M}}_1}\end{array}\right)$$ 5

We call the resulting states, no longer of pure spin multiplicity, spin-mixed, or spin-adiabatic states. The energy separation between these spin-adiabatic states depends on the magnitude of the SOC between the original spin-diabatic states. If we expand the wave function in the basis of such spin-adiabatic states, they will be again coupled by NAC, which determines an NAE, now computed over the two spin-mixed states (see eq ). Again, the value of the NAC is inversely proportional to the interstate energy separation; therefore, the closer the spin-mixed states, the more probable the NAE. It must be remembered that the NAC is now different from zero only because of the mixing of the original singlet and triplet states caused by their SOC. The equation coupling the two spin-mixed states SM₀ and SM₁, resulting from the mixing of the original spin-pure states S₀ and T₁, will be as follows.

$$i{\mathrm{\psi ̇}}^{\mathrm{S}{\mathrm{M}}_1}(\mathbf{R},t)=({\mathrm{T̂}}_{\mathrm{R}}+V^{\mathrm{S}{\mathrm{M}}_1}(\mathbf{R})){\psi }^{\mathrm{S}{\mathrm{M}}_1}(\mathbf{R},t)+{\mathrm{\Lambda ̂}}^{\mathrm{S}{\mathrm{M}}_0\mathrm{S}{\mathrm{M}}_1}{\psi }^{\mathrm{S}{\mathrm{M}}_0}(\mathbf{R},t)$$ 6

Spin-pure states and spin-mixed states are two complete basis sets spanning the same space, so the description of Ir3-TMCs using either representation must be equivalent. Let us first describe [Ir­(ppy)₂(bpy)]⁺ using spin-mixed states. At the MECP, the energy separation between the latter states can be estimated as twice the SOC between the corresponding spin-pure states. The computed SOC (Section S5) between the ³MC_ax and S₀ states at the (³MC_ax)_min structure is 3400 cm^–1 (0.42 eV), and for the ³MC_eq and S₀ states at (³MC_eq)_min, it is 1676 cm^–1 (0.21 eV). At the corresponding ³MC_ax/S₀ and ³MC_eq/S₀ MECPs, the splittings of the original spin-pure ³MC and S₀ states when including the SOC are then 0.84 and 0.42 eV, respectively.

It is fundamental to understand that, depending on the relative position of the initial and final minima with respect to MECP, the SOC-induced splitting of the ³MC and S₀ states has a completely different effect on the decay process. When the initial and final minima are located in opposite directions relative to MECP (i.e., when it is possible to define a reaction coordinate along which the reactant minimum, the MECP, and the product minimum are consecutive points), the MECP has a peaked topography. − This case has been the subject of different studies. ,− The mixing induced by the SOC results in two separated spin-adiabatic states, and what in the spin-pure picture was described as an ISC (Figure a, upper panel) now instead corresponds to an adiabatic process, in which the system evolves from one minimum to another along the same spin-adiabatic PES via the TS barrier (Figure a, bottom panel). When instead the initial and final minima are located in the same directions relative to the MECP, the MECP has a sloped topography. − This case − is the one for the T₁-to-S₀ decay mediated by ³MC states in Ir3-TMCs. The mixing induced by the SOC again results in two separated spin-adiabatic states, but what in the spin-pure picture was described as an ISC now still corresponds to an NAE (Figure b). Such an NAE is driven by the NAC between the spin-mixed states that at the original MECP are separated by a value equal to 2 times their SOC. Consequently, it is an NAE whose probability has no reason to be equal to unity at MECP, in turn meaning that the probability of reaching the S₀ minimum has no reason to be high. In our example, according to the computed SOCs, at the ³MC_ax/S₀ and ³MC_eq/S₀ MECPs, the spin-mixed states are separated by as much as 0.84 and 0.42 eV, respectively.

2.

Spin-pure and spin-mixed representations of (a) peaked and (b) sloped MECP-mediated processes.

The same conclusion is also reached by using the spin-pure representation (Figure b, top panel). Once the trajectory propagating on PES1 reaches the MECP, the large SOC results in a high (close to unity) probability of an NAE, but due to the direction of propagation, it is the high-energy branch of PES2 that becomes populated. Once the system runs out of kinetic energy, it goes back to the MECP, where again the large SOC implies a high probability of NAE, now to the low-energy branch of PES1. This results in a very slow population decay to the low-energy branch of PES2, meaning a weak tendency to repopulate the S₀ Franck–Condon region. This conclusion is confirmed by computing the corresponding probability and nonradiative decay rate constant, k _nr(T), at the ³MC_ax/S₀ MECP and ³MC_eq/S₀ MECP of [Ir­(ppy)₂(bpy)]⁺ using the nonadiabatic statistical theory (NAST) (Section S6). , Employing the Landau–Zener equation (eqs S22 and S23), the k _nr(T) values for the nonradiative decay path associated with the ³MC_ax/S₀ and ³MC_eq/S₀ MECPs are 1.81 × 10^–7 and 2.32 × 10^–2 s^–1, respectively. Both values are significantly lower than those predicted by transition state theory (5.37 and 5.50 × 10² s^–1, respectively) and lower than the experimental total nonradiative rate constant (around 10⁶ s^–1). Moreover, the k _nr(T) associated with the ³MC_ax/S₀ is 5 orders of magnitude lower than the rate constant for the ³MC_eq/S₀ MECP. We hypothesize that this difference can be related to the slightly higher energy barrier to reach the ³MC_ax/S₀ MECP than the ³MC_eq/S₀ MECP (Figure S3), but also to the larger SOC at the ³MC_ax/S₀ MECP (3400 cm^–1) than at the ³MC_eq/S₀ MECP (1676 cm^–1). In fact, for the characterized sloped MECPs, a stronger SOC results in a larger energy gap between the spin-adiabatic PESs, leading to a less efficient nonradiative decay. To account for quantum tunneling and the nonlinear behavior of the reaction path, neglected in the Landau–Zener treatment, we computed the k _nr(T) using the Zhu–Nakamura (ZN) transition probability equation for the nonradiative decay path mediated by the ³MC_ax/S₀ MECP. The rate calculated with the ZN transition (2.02 × 10⁴ s^–1) indicates a massive effect of quantum tunneling on the overall nonradiative decay rate from the T₁ state, due to the low MECP barrier and small reduced mass along the reaction coordinate (see Section S6). It is worth noticing that the ZN rate constant, in principle more accurate that the LZ result, does not describe only the probability of nonradiative decay through the MECP (since it includes tunneling). Consequently, the ZN value cannot be taken as a direct measure of the nonradiative decay rate through the MECP, which is the main subject of the present work.

An agreement between the nonradiative decay constant associated with the population of ³MC states and the experimental nonradiative decay constant would constitute an important piece of evidence for the model presented here in which the probability of nonradiative repopulation of the ground state is much lower than unity. Two problems must be faced in order to perform such a test. First, an Ir3-TMC where the ³MC-mediated decay is the main nonradiative decay path must be selected. This is because experimentally only the total nonradiative decay constant can be obtained, so only in such a case is the comparison between the theoretical and experimental values justified. In that sense, the reference [Ir­(ppy)₂(bpy)]⁺ complex is not a good candidate, since the T₁-to-S₀ ISC from the emitting T₁ minimum is probably the most important nonradiative decay path. Second, the test would be valid only if a very accurate energy barrier associated with the process can be provided. Rate constants computed using Arrhenius-like expressions (as in both TST and NAST) are extremely sensitive to the employed energy barrier, where an error as small as 0.1 eV can result in errors of orders of magnitude in the corresponding rates. We are currently evaluating the possibility of performing very computationally intensive RASPT2 calculations on specific Ir3-TMCs to obtain accurate energy barriers and nonradiative decay constants. In the present work, a first RASSCF calculation was performed on the PBE0-optimized ³MC_ax/S₀ MECP, taking as a reference the work of Bokarev et al., followed by the computation of the corresponding spin-mixed states using the RASSI code of OpenMolcas (see Section S7). , From such a calculation, we obtained an SOC between the ³MC_ax and S₀ states of 3028 cm^–1 (0.37 eV), in agreement with the TDDFT value of 3400 cm^–1 (0.42 eV), and a splitting of the spin-mixed states of 0.76 eV, indeed equal to roughly 2 times the SOC value.

Now returning to the spin-mixed representation, one may expect that there are, however, CIs involving the two spin-mixed PESs, where the probability of an NAE is very large. Nevertheless, assuming that the SOC between the ³MC and S₀ states is non-zero, no CIs between these spin-mixed states exist in the S₀ and T₁ two-state model. ,− To understand this, let us recall the condition for having a CI between two pure-spin diabatic states with the same spin. In such a framework (eqs S24 and S25 of Section S8) a CI is encountered when the two states have the same energy (H _el = H _el ) and their coupling is zero (H _el = 0). In our case, the two spin-pure states S₀ and ³MC are our spin-diabatic states, while the corresponding spin-mixed states are the adiabatic states whose CI we are looking for. However, while the first condition (E(S₀) = E(³MC)) can be satisfied (as at the MECP), the condition H _el = 0 (i.e., $H_{\mathrm{S}\mathrm{O}\mathrm{C}}^{{{\mathrm{S}}_0}^3\mathrm{M}\mathrm{C}}=0$ ) is never satisfied, because the S₀ and ³MC states are characterized by a large SOC.

To date, we have ignored the zero-field splitting of the T₁ state into its three sublevels, i.e. T_–1, T₀, and T₁. In a recent work, Wang and Yarkony mathematically describe the transformation from spin-pure to spin-mixed states when considering a model with the S₀ state and the three sublevels composing the T₁ state. They showed that two of the three sublevels have the same energy as the original T₁ state, the energy of the third increases by a value equal to the SOC, and the energy of the spin-mixed S₀ state decreases by a value equal to the SOC (see eqs 7 of ref ). Only the last two spin-mixed states are interacting, which again are energetically separated by a value 2 times their original SOC.

The absence of the CIs does not mean that ³MC states are not involved in the nonradiative decay. ³MC states can still mediate a nonradiative decay in Ir3-TMCs, but according to the presented model, not through a CI, so not in a way as efficient as in CI-mediated NAEs. When ³MC states are accessible and significantly lower in energy than any emitting state, they will play a role, as in the related Ru­(II) d⁶ complexes [Ru­(m-bpy)₃]²⁺ and [Ru­(tm-bpy)₃]²⁺, where their involvement was experimentally proven. We propose that the relevance of a ³MC state is determined by how long Ir3-TMCs remain trapped on the ³MC PES (i.e., in the corresponding ³MC minimum). From the ³MC minimum, the decay back to S₀ through MECP will have a low probability, as shown by our NA-TST calculation (see Section S6). However, if the excited state population remains trapped in the ³MC minimum, the small energy gap with the ground state will efficiently promote a T₁-to-S₀ ISC process, as in the emitting T₁ minima part of the population decay through a T₁-to-S₀ ISC process whose rate constant is normally indicated as k _ISC. , The described small ³MC-S₀ energy separation is caused by the broken coordination bond characterizing the ³MC minima, in turn leading to a massive increase in the S₀ energy. In our example, i.e., [Ir­(ppy)₂(bpy)]⁺, the ³MC-S₀ gaps at the (³MC_ax)_min and (³MC_eq)_min structures are 0.55 and 0.65 eV, respectively. A key difference between the T₁-to-S₀ ISC process operating at the T₁ emitting minima and that at the ³MC equilibrium structures can be predicted. In the former case, the T₁ and S₀ minima are normally nested states and the process will follow the so-called energy gap law, so as the energy separation increases, the T₁-to-S₀ ISC transition probability decreases. In the latter case, the large geometrical difference between the S₀ and ³MC minima can instead lead to the opposite behavior, so as the energy separation increases, the T₁-to-S₀ ISC transition probability increases. , Further studies are required to describe the phenomena.

The scenario exemplified for [Ir­(ppy)₂(bpy)]⁺ is common to any Ir3-TMC, or at least to any Ir3-TMC in which ³MC/S₀ MECPs display similarly large SOC (1500 cm^–1), as is generally the case (see Section S9). Sloped ³MC/S₀ MECPs are indeed present in any Ir3-TMC, as proven in Section S2. Actually, any octahedral low-spin d⁶ TMC would display such sloped ³MC/S₀ MECPs, so in the case of a large SOC between the involved ³MC and S₀ states, again the same conclusions here exemplified for [Ir­(ppy)₂(bpy)]⁺ will be valid. This could be the case for Ru­(II) TMCs, whose nonradiative decay mediated by ³MC states is normally described in a similar way as for Ir­(III) complexes, − and whose ³MC/S₀ MECPs usually display SOC values on the order of 1000 cm^–1.

In summary, we show that the role played by ³MC states in the nonradiative deactivation of Ir3-TMCs deserves reconsideration. Specifically, we showed that it is incorrect to assume that at the ³MC/S₀ MECP the probability of decay back to the ground state minimum is equal to unity, and that, counterintuitively, the larger the SOC between ³MC and S₀ states, the lower such a probability. Moreover, we showed that, assuming a non-zero SOC between ³MC and S₀ states for all geometries, the spin-mixed states resulting from the spin-pure ³MC and S₀ states do not form CIs. Consequently, even if the established strategy of avoiding ³MC population in order to minimize the nonradiative decay can still be valid, we postulate that the role of ³MC states in the nonradiative decay of Ir3-TMCs is not directly related to the ability of reaching the ³MC/S₀ MECPs from the emitting minimum. Instead, we propose that the role of ³MC states is dictated by the tendency of Ir3-TMCs to remain trapped in the ³MC minima, from where, given sufficient time, the nonradiative decay to the ground state will occur by means of a T₁-to-S₀ ISC process. We can then suggest that in order to minimize the nonradiative decay mediated by ³MC states, a key design strategy will be to avoid complexes that in their ³MC minima display stabilizing interactions not present in the emitting structure. Such interactions could in fact determine a sizable barrier from the ³MC to the T₁ minimum and in turn the trapping of the population in the ³MC state.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpclett.5c01776.

• Graphic showing the relevance of Ir­(III) complexes for electroluminescent devices and photodynamic therapy (Section S1); properties of triplet metal-centered states (³MC) of Ir­(III) and Ru­(II) d⁶ transition metals under octahedral geometry (Section S2); DFT (PBE0 and B3LYP) energies, geometries, orbitals, and NTOs that characterized the nonradiative decay through ³MC_ax and ³MC_eq states for complex [Ir­(ppy)₂(bpy)]⁺ (Section S3); derivation of the equations that govern NAEs working with spin-pure states and spin-mixed states (Section S4); computational details to obtain the SOC values at the SOC-TDDFT level of theory for complex [Ir­(ppy)₂(bpy)]⁺ (Section S5); nonradiative decay constants computed using Landau–Zener (LZ) and Zhu–Nakamura (ZN) approximations with the NAST code (Section S6); SOC benchmark at the ³MC_ax/S₀ MECP of complex [Ir­(ppy)₂(bpy)]⁺ computed at the RASSCF (4,2;12,9;4,2)/ANO-RCC-VDZP level of theory (Section S7); required conditions for having a CI in the framework of adiabatic and diabatic states (Section S8); and SOC-TDDFT (PBE0 and B3LYP) values for several Ir­(III) and Ru­(II) ³MC minima (Section S9) (PDF)

• Transparent Peer Review report available (PDF)

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Department of Chemistry, New York University, New York, NY 10003

The authors declare no competing financial interest.