High Intrinsic Phosphorescence Efficiency and Density Functional Theory Modeling of Ru(II)-Bipyridine Complexes with π-Aromatic-Rich Cyclometalated Ligands: Attributions of Spin–Orbit Coupling Perturbation and Efficient Configurational Mixing of Singlet Excited States

Abstract

A series of π-aromatic-rich cyclometalated ruthenium(II)-(2,2′-bipyridine) complexes ([Ru(bpy)₂(π_Ar-CM)]⁺) in which π_Ar-CM is diphenylpyrazine or 1-phenylisoquinoline were prepared. The [Ru(bpy)₂(π_Ar-CM)]⁺ complexes had remarkably high phosphorescence rate constants, k_RAD(p), and the intrinsic phosphorescence efficiencies (ι_em(p) = k_RAD(p)/(ν_em(p))³) of these complexes were found to be twice the magnitudes of simply constructed cyclometalated ruthenium(II) complexes ([Ru(bpy)₂(sc-CM)]⁺), where ν_em(p) is the phosphorescence frequency and sc-CM is 2-phenylpyridine, benzo[h]quinoline, or 2-phenylpyrimidine. Density functional theory (DFT) modeling of the [Ru(bpy)₂(CM)]⁺ complexes indicated numerous singlet metal-to-ligand charge transfers for ¹MLCT-(Ru-bpy) and ¹MLCT-(Ru-CM), excited states in the low-energy absorption band and ¹ππ*-(aromatic ligand) (¹ππ*-L_Ar) excited states in the high-energy band. DFT modeling of these complexes also indicated phosphorescence-emitting state (T_e) configurations with primary MLCT-(Ru-bpy) characteristics. The variation in ι_em(p) for the spin-forbidden T_e (³MLCT-(Ru-bpy)) excited state of the complex system that was examined in this study can be understood through the spin–orbit coupling (SOC)-mediated sum of intensity stealing (∑SOCM-IS) contribution from the primary intensity of the low-energy ¹MLCT states and second-order intensity perturbation from the significant configuration between the low-energy ¹MLCT and high-energy intense ¹ππ*-L_Ar states. In addition, the observation of unusually high ι_em(p) magnitudes for these [Ru(bpy)₂(π_Ar-CM)]⁺ complexes can be attributed to the values for both intensity factors in the ∑SOCM-IS formalism being individually greater than those for [Ru(bpy)₂(sc-CM)]⁺ ions.

Introduction

Since the pioneering work on the photophysical mechanics of the excited states of transition metal-diimine complexes by Demas and Crosby nearly half a century ago,¹⁻³ fundamental investigations related to the excited-state relaxation of low-spin transition metal aromatic-ligand-acceptor chromophores (M-L_Ar) has been of interest in studies on photophysical relaxation of excited states,²⁻⁵ low-energy triplet metal center (³MC) relaxation,⁶⁻⁹ spin–orbit coupling (SOC)¹⁰⁻¹²-mediated configurational mixing between emitting triplet and efficient singlet excited states (SOCM),¹³⁻¹⁶ and photocatalysis,¹⁷⁻¹⁹ where M in M-L_Ar is a transition metal element (ion) with a relatively high atomic number. It should be noted in this respect that the photoinduced relaxation of M-L_Ar-type chromophores has found applications in solar energy conversion,²⁰⁻²⁴ organic light-emitting diodes (OLEDs),²⁵⁻³¹ and photodynamic therapy for the treatment of cancer.³²⁻⁴² The photoinduced excited-state relaxation of molecules in an action model includes^{12−14,43−56} (1) internal conversion (k_ic) between states with the same spin multiplicity and intersystem crossing (k_isc) between states with different spin multiplicities; (2) nonradiative relaxation (k_NRD) of the emitting state directly to the ground state by vibronic coupling; (3) radiative relaxation (k_RAD), including fluorescence (fl) and phosphorescence (p); and (4) other quenching processes (k_q), such as MC excited-state quenching.

The k_RAD formalism for spontaneous luminescence in a molecular system based on Einstein’s expression for atomic fluorescence has appeared in numerous monographs.^12,43,44 The k_RAD of a chromophore model is a function of the average sum of the spectral contributions determined by the observed photonic fluorescence integration in an organic system with two states, namely, the ground state (g) and the excited state (e), and can be expressed by^{12,13,44−46,50,57}

1

where the terms in eq 1 are the transition dipole moment (M_e,g), the fluorescence frequency (ν_em, corresponding to the emission energy hν_em), and C_R = (16π³η³)/(3ε_oc³h), where η is the refractive index, ε_o is the vacuum permittivity, c is the speed of light, and h is Planck’s constant. The experimental k_RAD amplitude is obtained from the emission quantum yield, ϕ_em, and the observed mean of the emission lifetime, τ_obs, where k_RAD = ϕ_em × k_obs and k_obs = 1/τ_obs = k_RAD + k_NRD.

Phosphorescence in an ideal model is fundamentally a spin-forbidden transition, and a theoretical description of this natural phenomenon in a molecular system requires a significant M_e, g contribution from an SOC configuration of the singlet (S_n, n ≥ 1 in a multistate model) and triplet emitting (T_e) states, where M_e, g = α_Te, Sn^SOCM_Sn, S0 and M_e, g = 0 in eq 1 for an ideal phosphorescence (T_e → S₀) transition without SOC perturbation, α_Te, Sn is the SOC mixing coefficient between the T_e and S_n states, and M_Sn, S0 is the transition dipole moment of the S_n excited state.^{12,14,44,50,57,58} The configurational mixing coefficient (α) of the target electronic states in a weak mixing limit, 1 ≫ α² > 0, can be approximated by the term α ≈ H/ΔE,⁵⁹⁻⁶⁴ where H and ΔE are the coupling element and the vertical energy difference between the states, respectively.

The phosphorescence of an M-L_Ar-type chromophore fundamentally occurs through SOC configurations,^{2,14,15,65,66} so the spin–orbit coupling element (H^SOC in α^SOC) of the T_e → S₀ transition is not zero. In early studies,^50,67,68 the H^SOC value was plausibly postulated to increase approximately with the fourth power of the atomic number (Z⁴), which is a so-called heavy atom effect.^12,43 In the present SOC modeling, the differences in the spin angular momenta (S⃗) between the T_e and S_n states and the distance (r) between the nucleus (with effective nuclear charge, Z^eff) and the target electrons must be accompanied by a simultaneous change in the orientation of the orbital angular momenta (L⃗) for a spin–orbit operator, , in the efficient SOC configuration.⁶⁹⁻⁷¹ A useful SOC selection rule for phosphorescence from M-L_Ar-type chromophores was required in this study because the dominant SOC perturbation of the transition moment stealing (α_Te, Sn(T)^SOCM_Sn(T), S0) is from efficient S_n(T) states (S_n in the T_e geometry). In this case, the donor singly occupied molecular orbital (SOMO) contains a different M-dπ orbital from the donor-dπ SOMO of the triplet emitting metal-to-ligand charge-transfer state, ³MLCT (T_e).^14,15,72,73 The SOC-mediated emission intensity stealing due to significant T_e/S_n mixing with a small vertical energy difference, ΔE_Te,Sn = E(S_n)_v − E(T_e)_min, has been widely studied in the early literature.^14,74−76

Thus, the effective transition moment contribution for ι_em(p) in a molecular system is the sum of the SOC-mediated efficient singlet transitions, | ∑ α_Te, Sn(T)^SOCM_Sn(T), S0|², and the weak mixing coefficient (α_Te, Sn(T)) is given by^14,15,67

2

As a result in eq 3, the description of the intrinsic phosphorescence efficiency (ι_em(p)) of an M-L_Ar-type chromophore is obtained by combining eqs 1 and 2 with | ∑ α_Te, Sn(T)^SOCM_Sn(T), S0|².^14,15,73

3

The meaning of ι_em(p) in eq 3 can be expressed as the sum of the SOC-mediated phosphorescence intensity stealing (ΣSOCM-IS) from the efficient singlet excited states, where ν_em(p) corresponds to the phosphorescence energy.

A simple plot can be used to illustrate the ι_em(p) of the phosphorescence transitions of M-L_Ar-type complexes that involve an ³MLCT emitting state (T_e), a singlet ground state (S₀), and efficient SOCM-IS from low-energy (ΔE_Te, Sn) singlet excited states (S_n(T)), as shown in Figure 1.

Figure 1.

Jablonski-type diagram showing the phosphorescence efficiency (ι_em(p) = k_RAD(p)/(ν_em(p))³) through ΣSOCM-IS contributions from efficient SOC-allowed S_n(T) states with T_e coordinates.

Cyclometalated transition metal–polypyridine complexes have been the focus of interesting studies on photophysics and excited-state relaxation,^14,77−85 and these types of complexes have recently been applied in areas of solar energy conversion,⁸⁶⁻⁸⁸ OLEDs,^14,89−91 and photodynamic therapy.^92,93 In previous work, the 77 K phosphorescence parameters of Ru-bpy chromophores for the cyclometalated ruthenium(II)-(2,2′-bipyridine) complexes ([Ru(bpy)₂(CM)]⁺) with a simply constructed cyclometalated (sc-CM) coordination ligand (sc-CM = 2-phenylpyridine (ppy) and benzo[h]quinoline (bhq)) showed unusually higher ι_em(p) values compared to those of typical [Ru(bpy)_3–n(Am)_2n]²⁺ complexes. In addition, in the case of the observed variations in ι_em(p) of the emitting ³MLCT states of the Ru-bpy chromophores for [Ru(bpy)₂(sc-CM)]⁺ and typical [Ru(bpy)_3–n(Am)_2n]²⁺ complexes, a good linear relation with the intensity component values of ∑SOCM-IS was found, as seen in eq 3.⁷³ The photoinduced characteristics of [Ru(bpy)₂(CM)]⁺ complexes with a bidentate π-aromatic-rich CM ligand (π_Ar-CM = diphenylpyrazine (dpz) and 1-phenylisoquinoline (piQ)) in this study are interesting because the observed absorption and photoinduced phosphorescence efficiencies (ι_em(p)) are significantly different from those of [Ru(bpy)₂(sc-CM)]⁺ complexes. The unusually higher ι_em(p) values for the [Ru(bpy)₂(π_Ar-CM)]⁺ complexes compared to the corresponding values for the [Ru(bpy)₂(sc-CM)]⁺ complexes are related to the fundamental principle (eqs 1–3) and the transition intensity perturbation, which is the focus of this investigation. Because studies such as these have been limited in the past, these data should contribute to the future development of and a better understanding of these types of theoretical frameworks. The skeletal structures of several of the ligands, the target [Ru(bpy)₂(CM)]⁺ complexes (a–e) and reference [Ru(bpy)_3–n(Am)_2n]²⁺ ions (1–5), are shown in Figure 2.

Figure 2.

Skeletal structures of the [Ru(bpy)₂(CM)]⁺ and reference [Ru(bpy)_3–n(Am)_2n]²⁺ complexes, where Am = amine ligand and 2 ≥ n ≥ 0.

Experimental Section

Details concerning the commercial starting materials, synthesis procedures, and nuclear magnetic resonance (NMR, ¹H and ¹³C) spectral parameters of the target [Ru(bpy)₂(CM)](PF₆) complexes are shown in Section S1A (page S2); the following compounds: [Ru(bpy)_3–n(Am)_2n](PF₆) (1–5), [Ru(bpy)₂(ppy)](PF₆) (a), [Ru(bpy)₂(bhq)](PF₆) (b), and [Ru(bpy)₂(piQ)](PF₆) (c) in Figure 2 were prepared as described in the literature.^{84,89,94−96} Details regarding the X-ray structure determinations are provided in Section S1B (page S6), and the X-ray structure parameters of [Ru(bpy)₂(dpz)](PF₆)·(CH₂Cl₂) are shown in Figure S1B1 (page S7) and Tables S1B1 and S1B2 (pages S7–S9). Instrumentation details for the electrochemistry, room-temperature (RT) and 87 K absorption spectra, 77 K emission, lifetime measurements, and emission quantum yields (ϕ_(em)) are provided in Section S1C (page S10). The electrochemical results, namely, the E_1/2(Ru^III/II) and E_1/2(bpy^0/–1) values, of the [Ru(bpy)₂(CM)]⁺ ions are shown in Table S2A (page S12), and the photoinduced parameters obtained from experimental observations of the complexes are shown in Table 1 for [Ru(bpy)₂(CM)]⁺ and Table S2B (page S13) for the reference [Ru(bpy)_3–n(Am)_2n]²⁺ complexes. The 87 K absorption spectra of the [Ru(bpy)₂(CM)]⁺ complexes in butyronitrile glasses are shown in Figure S2A (page S14). Computational results concerned about the density functional theory (DFT) modeling of the target complexes are shown in Section S3 (page S14). The DFT calculations performed in the present study used a combination of the B3PW91 functional⁹⁷⁻¹⁰⁰ and the SDDall^101−103 basis set. We used several functionals to calculate the low-energy MLCT electronic absorption envelopes for [Ru(bpy)₂(ppy)]⁺,⁷³ and we ultimately selected the B3PW91 functional for use in the present study; this method has been widely used to calculate absorption spectra of Ru-bpy chromophores that are analogous to the present systems.^{8,9,73,104−107} DFT geometry optimizations for the S₀ and T₁ (T_e) states and time-dependent DFT (TDDFT) calculations for the electronic absorption spectra were performed in an acetonitrile solution simulated by the integral equation formalism polarizable continuum model (IEF-PCM) solvation model.^108−111 Harmonic vibrational frequency analyses were carried out to confirm that all of the optimized structures are minima on the potential energy surface. The electronic structures of the transitions were analyzed using the natural transition orbital (NTO) approach.^112,113 All calculations were performed using the Gaussian 09 program.¹¹⁴

Table 1. Ambient Absorption, 77 K Emission, Emission Decay Constants, and Emission Yield of the Complexes^a.

			77 K emission properties
code	complex	hνmax (abs), cm–1/103, 87 K [298 K]	hνmax (em), cm–1/103	hνave, cm–1/103	τmean (μs)	kobs, μs–1b	ϕem × 103	kRAD, μs–1c	kNRD, μs–1d
a	[Ru(bpy)2(ppy)]+e	18.1 [18.3]	13.48	12.57	3.38	2.96	17 ± 4	0.050 ± 0.009	2.9
b	[Ru(bpy)2(bhq)]+e	18.3 [18.5]	13.68	12.79	4.88	2.05	20 ± 4	0.041 ± 0.008	2.01
c	[Ru(bpy)2(piQ)]+f	18.7 (sh) [18.6]; 19.9 [19.8]	13.58	12.73	3.71	2.70	31 ± 6	0.084 ± 0.014	2.6
d	[Ru(bpy)2(ppm)]+f	18.5 [18.7]	13.86	13.02	5.71	1.75	28 ± 7	0.049 ± 0.012	1.7
e	[Ru(bpy)2(dpz)]+f	17.8 (sh) [18.1]; 20.3 [20.4]	13.97	13.14	5.39	1.86	46 ± 9	0.085 ± 0.017	1.8

^aDominant RT low-energy absorption maxima, hν_max(abs), determined in acetonitrile; emission maxima, hν_max (em); average emission energy (hν_ave) by eq S1b (page S12), ν_ave ≈ ∫ ν_mI_mdν_m/ ∫ I_mdν_m; mean excited-state decay rate constant, k_obs = 1/τ_mean, emission yield, ϕ_em, determined at 77 K in butyronitrile glasses, and sh = shoulder.

^bk_obs = 1/τ_mean.

^ck_RAD = ϕ_em × k_obs.

^dk_NRD = k_obs – k_RAD.

^eRef (73).

^fSpectral parameters in this work.

Results

Experimental Observations

The 87 K absorption spectra of the target [Ru(bpy)₂(CM)]⁺ (a–e) complexes are shown in Figure S2A (page S14), and the 298 K absorption spectra of the reference [Ru(bpy)_3–n(Am)_2n]²⁺ (1, 2, and 4) and target [Ru(bpy)₂(CM)]⁺ (a–e) complexes are shown in Figure 3. The intense low-energy single absorption bands of the three reference complexes in Figure 3 are attributed to the sum of the primary ¹MLCT excited states of the Ru-bpy chromophore.^{1,2,65,115,116} Furthermore, the broad low-energy absorption bands (15,000–22,500 cm^–1, 660–440 nm) of the [Ru(bpy)₂(sc-CM)]⁺ ions (sc-CM = ppy (a) and bhq (b)) are attributed to the sum of the intensities of ¹MLCT excited states of two nonidentical Ru-bpy moieties,^73,84,117 and the shapes of the low-energy absorption bands for the sc-CM = ppm complex in this region are similar to those of the sc-CM = ppy and bhq complexes. The shapes of the low-energy bands in the 298 and 87 K absorption spectra for [Ru(bpy)₂(π_Ar-CM)]⁺ (CM = piQ (c) and dpz (e)) are different from those of the [Ru(bpy)₂(sc-CM)]⁺ ions, and the results of DFT modeling suggest that the first low-energy band in the spectra of the [Ru(bpy)₂(π_Ar-CM)]⁺ complexes includes intense contributions from the ¹MLCT excited states of the Ru-bpy and Ru-CM moieties. The nonidentical absorption curves in the high-energy region (22,500–30,000 cm^–1, 440–333 nm) imply that the absorption transition behaviors for the [Ru(bpy)₂(CM)]⁺ complex series are very different in this work.

Figure 3.

298 K absorption spectra in CH₃CN for the reference complexes, top panel: [Ru(bpy)₃]²⁺ (black solid, 1), [Ru(bpy)₂(en)]²⁺ (gray solid, 2), [Ru(bpy)(en)₂]²⁺ (gray dash, 4), and [Ru(bpy)₂(ppy)]⁺ (dark blue solid in the top and bottom panels, a); 298 K absorption spectra in CH₃CN for target complexes, solid curves in the lower panel: [Ru(bpy)₂(bhq)]⁺ (wine, b), [Ru(bpy)₂(piQ)]⁺ (violet, c), [Ru(bpy)₂(ppm)]⁺ (green, d), and [Ru(bpy)₂(dpz)]⁺ (orange, e). The complex codes are shown in Figure 2.

The 77 K ³MLCT phosphorescence bands in the spectra of the [Ru(bpy)₂(ppy)]⁺ (a) and target reference [Ru(bpy)₃]²⁺ (1), [Ru(bpy)₂(en)]²⁺ (2), and [Ru(bpy)(en)₂]²⁺ (4) ions in the top panel of Figure 4 show recognizable vibronic side bands corresponding to ³MLCT Ru-bpy chromophores, which is attributed to the identified vibronic characteristics of MLCT Ru-bpy chromophores.^{65,73,84,95,116,118−122} The 77 K emission spectra of the [Ru(bpy)₂(CM)]⁺ (a–e) complexes in Figure 4 are attributed to the ³MLCT emitting state (T_e) of a typical Ru-bpy chromophore such as the reference [Ru(bpy)_3–n(Am)_2n]²⁺ (1–5),^73,84 and the observed k_NRD vs the emission energy scale (Figure 5) and a DFT natural transition orbital (NTO) plot of the triplet emitting state (T_e) provided in the section described in the Computational Results further illustrate this point.

Figure 4.

77 K phosphorescence spectra in butyronitrile glasses of the reference [Ru(bpy)_3–n(Am)_2n]²⁺ and target [Ru(bpy)₂(CM)]⁺ complexes. The complex codes are shown in Figure 2.

Figure 5.

Comparison of the variations in k_NRD with emission hν_ave for the complexes shown in Table 1 and Table S2B (page S13). Gray squares denote the reference [Ru(bpy)_3–n(Am)_2n]²⁺ series, and red squares (L_Ar = sc-CM: ppy (a), bhq (b), and ppm (d)) and pink squares (L_Ar = π_Ar-CM: piQ (c) and dpz (e)) denote [Ru(bpy)₂(L_Ar)]⁺ complexes. The gray solid least-squares line of the [Ru(bpy)_3–n(Am)_2n]²⁺ complexes is drawn with a slope of −0.00123 ± 0.00005, an intercept of 31.0 ± 0.6, and an R² value of 0.99.

In principle, the k_NRD (ambient temperature) of the emitting ³MLCT excited state of Ru-diimine complexes can be attributed to the ³MLCT → ³MC internal conversion (k_IC),⁷ but we have no evidence for this assumption from the 77 K k_NRD observations of the Ru-bpy chromophores of the [Ru(bpy)₂(sc-CM)]⁺ (a, b, and d) complexes.^73,84 The 77 K k_NRD relaxation of the lowest-energy ³MLCT-(Ru-bpy) excited state for the typical [Ru(bpy)_3–n(Am)_2n]²⁺ series would be expected to involve only an argument of k_NRD (³MLCT → S₀) ≫ k_IC (³MLCT → ³MC).^{65,95,106,121} Furthermore, the 77 K k_NRD values for the different chromophores can be expressed in terms of a temperature-independent k_NRD formalism based on a single-distortion model, hν_k,^12,44

4

where B ≈ [8π³/(h³ν_khν_ave)]^1/2 and γ_Ek ≈ [ ln (E_em/λ_k) – 1], in which we substituted hν_ave for E_em in eq 4, and the hν_ave scale in Figure 5 is associated with the hν_ave/hν_k scale; H_e,g is a function of SOC; the distorted ν_k frequency in the equation is related to an active distortion mode of the harmonic model whose energy approximates those of the excited state, jhν_k ≈ hν_ave; λ_k is obtained from the Huang–Rhys parameter for this ν_k mode.^116,123 The k_NRD value of the [Ru(bpy)₂(CM)]⁺ complexes is similar to the least-squares fit of the reference [Ru(bpy)_3–n(Am)_2n]²⁺ series in the given emission energy region, and this finding provides empirical evidence in support for the intrinsic characteristic of the ³MLCT-(Ru-bpy) emitting states (T_e) of both series of complexes.^73,84

A plot of the observed phosphorescence efficiency (ι_em(p)) versus emission energy (hν_ave) for the [Ru(bpy)₂(CM)]⁺ and the typical [Ru(bpy)_3–n(Am)_2n]²⁺ series complexes is shown in Figure 6. The 77 K emission quantum yields (ϕ_em) for the [Ru(bpy)₂(CM)]⁺ complexes are in the range within 1–5%, and the ι_em(p) amplitudes of the [Ru(bpy)₂(sc-CM)]⁺ complexes are approximately twofold higher than those of the typical [Ru(bpy)_3–n(Am)_2n]²⁺ series (for sc-CM = ppy, bhq,⁷³ and ppm); in addition, the outstanding ι_em(p) values of the [Ru(bpy)₂(π_Ar-CM)]⁺ complexes (π_Ar-CM = piQ and dpz) are nearly fourfold higher than those of the reference Ru-bpy complexes, i.e., nearly twofold higher than those of the [Ru(bpy)₂(sc-CM)]⁺ complexes at the given energy. These unusually high ι_em(p) amplitudes for the [Ru(bpy)₂(π_Ar-CM)]⁺ complexes are likely derived from the fundamental relations of ι_em(p) and ∑SOCM-IS, based on eq 3.

Figure 6.

Comparison of the variations in ι_em(p) = k_RAD(p)/(ν_ave)³ with hν_ave for the complexes: gray squares denote the reference [Ru(bpy)_3–n(Am)_2n]²⁺ series, red squares denote the [Ru(bpy)₂(sc-CM)]⁺ complexes, and pink squares denote the [Ru(bpy)₂(π_Ar-CM)]⁺ complexes.

Computational Results

The TDDFT-calculated low-energy S_n(X) excited-state (X = S and T at S₀ and T₁ geometries, respectively) parameters, their configurations of the highest occupied molecular orbitals (HOMOs) and the lowest unoccupied molecular orbitals (LUMOs), and orbital plots of the HOMOs and LUMOs for the [Ru(bpy)₂(CM)]⁺ (CM = ppm, piQ, and dpz) ions are shown in Tables S3A1–A3 and Figures S3A1–A3 (pages S15–S19). The DFT model of the target complexes indicates that the first three HOMOs have dπ_∥ (dπ-Ru in HOMO) and two dπ_⊥ (dπ-Ru in HOMO-1 and HOMO-2) orbital characteristics and that the first three LUMOs have π*-bpy (LUMO and LUMO+1) and π*-CM (LUMO+2) characteristics. The TDDFT parameters and NTO plots of S_n(X) and T_n for the target Ru-CM complexes are also shown in Tables S3B1–B3 and Figures SC1–C3 (pages S21–S50), and the donor-NTO component in each low-energy S_n(X) excited state is the primary HOMO (dπ_∥), HOMO-1 (dπ_⊥a), or HOMO-2 (dπ_⊥b), see Tables S3A1–A3 and Figures S3A1–A3 (pages S15–S19); the notations of the correlation dπ orbital are shown in Figure S6B (page S89). Comparisons of the RT-observed and DFT-calculated absorption transitions of the target Ru-CM complexes are shown in Figure 7 and Figure S3B1 (page S20). In the DFT modeling of the [Ru(bpy)₂(CM)]⁺ (CM = ppy) ion in our previous report⁷³ and the target [Ru(bpy)₂(CM)]⁺ (CM = ppm, piQ, and dpz) ions reported here, the notation for the calculated oscillator strength of singlet transitions (lo-f, 0.005 > f; m-f, 0.05 > f > 0.005; and h-f, f > 0.05) was applied in the present text.

Figure 7.

Illustrations of the observed RT absorption (black curve) and DFT (B3PW91)-calculated absorptions (red curve, S₀ geometry) and oscillator strengths (f) of the S_n(X) transitions (X = S and T for singlet and triplet excited states, respectively) of selected complexes are shown in panel (A) ([Ru(bpy)₂(ppm)]⁺) and panel (B) ([Ru(bpy)₂(piQ)]⁺). The log(f) values of the low-energy S₀ → S_n transitions for each complex are plotted in the S₀ geometry (S_n(S), central) and in the T_e geometry (S_n(T), bottom).

Since the phosphorescence of a complex is fundamentally a spin-forbidden transition, the values for the phosphorescence efficiency, ι_em(p), of the emitting state in the formalism of eq 3 should depend on the sum of the SOCM-IS (∑SOCM-IS) intensity from the efficient low-energy singlet excited states (S_n(T)) in a T_e geometry. The difference in low-energy absorption band shapes between the [Ru(bpy)₂(sc-CM)]⁺- and [Ru(bpy)₂(π_Ar-CM)]⁺-type ions in Figure 3 indicates the different MLCT compositions of their low-energy S_n(S) states. Based on these observations and the DFT modeling of the [Ru(bpy)₂(sc-CM)]⁺ ions (sc-CM = ppy⁷³ and ppm), the observed similarities in the absorption band shapes of the complexes are attributed to the similar intensities and energies of the low-energy intense-S_n(S) states. The calculated absorption results for both ions indicate that their observed first lowest-energy absorption band (14,000–22,500 cm^–1) is the highest of the MLCT-(Ru-bpy)-type S_n(S) intensity contribution and includes the intense S_5(S) (h-f) and S_6(S) (h-f) for [Ru(bpy)₂(ppy)]⁺ and S_5(S) (h-f) and S_7(S) (h-f) for [Ru(bpy)₂(ppm)]⁺. Furthermore, the first MLCT-(Ru-CM)-type S_n(S) state has contributions from S_7(S) (lo-f) for [Ru(bpy)₂(ppy)]⁺ and S_6(S) (lo-f) for [Ru(bpy)₂(ppm)]⁺. For both [Ru(bpy)₂(sc-CM)]⁺ complexes, the second low-energy band (22,500–30,000 cm^–1) includes MLCT-(Ru-bpy) and MLCT-(Ru-CM) transitions. In addition, the major absorption bands in the high-energy region (>33,000 cm^–1) are attributed to intraligand ππ* (intra-ππ*) and interligand ππ* (inter-ππ*) transitions. The shapes of the DFT-calculated absorption band for the [Ru(bpy)₂(π_Ar-CM)]⁺-type ions indicate that the first low-energy absorption band (14,000–25,000 cm^–1) can be attributed to the contribution of the intensity from the MLCT excited states of both Ru-bpy- and Ru-CM-type chromophores including (1) intense S_6(S) (h-f, Ru-bpy), S_8(S) (h-f, configurational Ru-bpy and Ru-CM), and S_9(S) (h-f, configurational Ru-bpy and Ru-CM) components for [Ru(bpy)₂(piQ)]⁺ and (2) intense S_6(S) (h-f, Ru-bpy) and S_8(S) (h-f, configurational Ru-bpy and Ru-CM) components for [Ru(bpy)₂(dpz)]⁺. In addition, the first MLCT-(Ru-CM)-type S_n(S) state is presented in S_4(S) (m-f) for [Ru(bpy)₂(piQ)]⁺ and S_4(S) (lo-f) for [Ru(bpy)₂(dpz)]⁺. Furthermore, the absorption for both [Ru(bpy)₂(π_Ar-CM)]⁺ complexes in the 25,000–33,000 cm^–1 region includes MLCT-(Ru-bpy), MLCT-(Ru-CM), intra-ππ* and inter-ππ* transitions, and the absorption in the high-energy region (>33,000 cm^–1) is attributed to intra-ππ* and inter-ππ* transitions. Moreover, S_16(S)/S_17(S) (m-f) for [Ru(bpy)₂(piQ)]⁺ and S_19(S)/S_20(S) (m-f) for [Ru(bpy)₂(dpz)]⁺, corresponding to the first inter-ππ*-type transition, occur at a lower energy than that for the [Ru(bpy)₂(sc-CM)]⁺ ions. The NTO plots and the orbital compositions of the low-energy S_n(S) excited states of the target [Ru(bpy)₂(CM)]⁺ ions are shown in Figures S3C1A–C3A (pages S27–S30, S35–S38, and S42–S45) and Table S5A (page S69).

The calculated parameters for the S_n(T) (in the T_e geometry) excited states and the NTO plots for the [Ru(bpy)₂(CM)]⁺ ions in this work were used for the ∑SOCM-IS formalism (eq 3) in the Discussion section, and no significant difference between the calculated results for the S_n(S) and S_n(T) states was found for each ion; see Tables S3B1–B3 (pages S21–S26) and Figures S3C1A and S3C1B (pages S27–S33) for [Ru(bpy)₂(ppm)]⁺, Figures S3C2A and S3C2B (pages S35–S40) for [Ru(bpy)₂(piQ)]⁺, Figures S3C3A and S3C3B (pages S42–S49) for [Ru(bpy)₂(dpz)]⁺, and our previous work⁷³ concerning [Ru(bpy)₂(ppy)]⁺. The NTOs-S_n(T) compositions of the target complexes should be examined carefully because the phosphorescence efficiencies depend on the ΣSOCM-IS values of the S_n(T) excited states; the detailed orbital configurations for the S_n(T) states of the [Ru(bpy)₂(CM)]⁺ ions are shown in Table S5B (page S71), and more concise results are provided in the Discussion section.

Previous work concerning the DFT modeling of the emitting ³MLCT (T_e) states for typical Ru-bpy complexes predominantly involved MLCT-(Ru-bpy) and some ππ*-bpy components for [Ru(bpy)(NH₃)₄]²⁺,^8,65,73,106 [Ru(bpy)(en)₂]²⁺,¹⁰⁷ and [Ru(bpy)₂(en)]^2+65,73 complexes. The selected low-energy NTO plots of the triplet excited states (T_n) with excited-state energies at the T_e geometry for the target [Ru(bpy)₂(CM)]⁺ ions are shown in Figures S3C1C (page S34), S3C2C (page S41), and S3C3C (page S50). The NTOs of the emitting T_e (T₁) states of [Ru(bpy)₂(CM)]⁺ with T_e coordinates (Figure 8) show donor orbital configurations that are composed primarily of the dπ-Ru component with some contribution from the pπ-CM (for most phenyl groups) and pπ-bpy_(b) moieties and the acceptor orbital, possessing mostly pπ*-bpy_(b) characteristics. We also calculated the Mulliken spin density (SD) values for the T_e excited states at the Ru centers and on the CM and bpy_(b) moieties for the [Ru(bpy)₂(CM)]⁺ series (see Figure 8), and the results show that most of the SD populations of the T_e states are localized on the Ru-bpy_(b) moiety. These results indicate that the emitting T_e excited states of the target [Ru(bpy)₂(CM)]⁺ complexes primarily have ³MLCT-(Ru-bpy) characteristics.

Figure 8.

DFT results with the T_e coordinates for the [Ru(bpy)₂(CM)]⁺ ion: (a) illustrated moiety symbols for the Δ-form target complexes in panel (A) (top); (b) Mulliken orbital population analyses for the NTOs, including donor and acceptor NTOs (central panel (B)) and the calculated spin density composition (SD, bottom panel (C)) of the T_e (T₁) excited state for the Ru center and the CM and bpy_(b) moieties for each ion.

From these results, the following conclusions can be drawn: (a) The lowest-energy absorption band of the [Ru(bpy)₂(CM)]⁺ complexes has a typical MLCT band shape with two observed maxima corresponding to CM = ppy, bhq, and ppm (sc-CM) and an intense, broad band for CM = piQ and dpz (π_Ar-CM). (b) The observed shape of the 77 K emission and the k_NRD amplitudes in the given energy of the target [Ru(bpy)₂(CM)]⁺ complexes indicate the ³MLCT characteristics of the Ru-bpy chromophore.⁷³ (c) The phosphorescence efficiency (ι_em(p)) of the target [Ru(bpy)₂(CM)]⁺ complexes at the specified energy is approximately twofold (for CM = sc-CM) and fourfold (CM = π_Ar-CM) higher on average than those of the typical reference [Ru(bpy)_3–n(Am)_2n]²⁺ ions. (d) DFT modeling of the [Ru(bpy)₂(CM)]⁺ complexes in their ground-state geometries (S₀) indicates that the intensity of the low-energy absorption band has contributions primarily from the MLCT(Ru-bpy)-type excited states for the CM = sc-CM-type complexes, as well as from both efficient MLCT(Ru-bpy)- and MLCT(Ru-CM)-type excited states for the CM = π_Ar-CM-type complexes. The high-energy absorption bands of the [Ru(bpy)₂(CM)]⁺ complexes reported in this work include contributions from the MLCT-type Ru-bpy and Ru-CM excited states and ππ*-type intraligand and interligand excited states. (e) DFT modeling of the NTOs and SD plots of the T_e (T₁) state for each target [Ru(bpy)₂(CM)]⁺ ion shows mostly MLCT-(Ru-bpy)-type characteristics.

Discussion

In a real complex system, the ι_em(p) determined from eq 3 should be obtained from the phosphorescence parameters (see Table 1), and the transition moment term (M_Sn(T), S0) in eq 3 is an oscillator-strength (f) function of the singlet excited state with T_e coordinates:¹²

5

where 1/C_f = 3e²h/8π²m_e, m_e is the mass of an electron, and ν_Sn(T), S0 corresponds to the emission energy of S_n(T) → S₀.

The results of DFT modeling suggest a low calculated reorganization energy (λ_reg(T) ≈ 1590 ± 20 cm^–1; see the definition in Figure S5A, page S68) for the light ground-state distortion in the T_e geometry of the target complexes. The oscillator strength in eq 5 can fundamentally be described by a Gaussian curve (see eq S4m, page S56), but attention needs to be paid to the fundamental quality of the detailed derivation regarding the f_Sn(T), S0 term in eq 5 from the absorption parameters of a complex system with a postulated light ground-state distortion at a T_e geometry. The energy-weighted intensity stealing term, the SOC-mediated energy-weighted absorptivity (SOCEWA),⁷³ instead of f_Sn(T), S0, can then be obtained from the observed absorption parameters; see eqs S4r–S4t (pages S57–S58). The ι_em(p) equation containing the integration of the SOCEWA value (∫SOCEWA dν = ) is related to the absorption spectrum through⁷³

6

where ε_i(ν_i) is the observed absorptivity at ν_i, W_i(ν_i) = (E_Te – ν_i)²ν_i, and C_RAD(p) = (8.60 × 10^–9)(g_Te/g_Sn), where g is a multiplicity weighting factor. The derivation of eq 6 is shown in Section S4B (pages S57–S60). The procedures used for the fitting and parameters for the ∫SOCEWA dν values from the experimental 87 K low-energy absorption curve of the complexes are shown in Figure S4D2 (page S62) and Table S4 (page S65).

The slope of the plot of ι_em(p) versus ∫SOCEWA dν for the typical [Ru(bpy)_3–n(Am)_2n]²⁺ and plain [Ru(bpy)₂(CM)]⁺ complex series suggests that H^SOC ≈ 100 cm^–1 for the Ru-bpy chromophore.⁷³ A plot of ι_em(p) versus ∫SOCEWA dν for the target complexes prepared in this work is presented in Figure 9, and the slope of the least-squares line containing the typical [Ru(bpy)_3–n(Am)_2n]²⁺ ions and [Ru(bpy)₂(sc-CM)]⁺ complexes (CM = ppy, bhq, and ppm) also suggests a similar result of H^SOC ≈ 100 cm^–1 for the Ru-bpy chromophore, where C_RAD(p) = (8.60 × 10^–9)(g_Te/g_Sn) and g_Te/g_Sn = 3 (see eq S4t, page S58). The least-squares line in Figure 9 appears idealized by eq 6, but the issue of why the ι_em(p) values of [Ru(bpy)₂(π_Ar-CM)]⁺ (π_Ar-CM = piQ (c) and dpz (e)) are approximately 1.4–1.5-fold greater than those along the least-squares line in the given ∫SOCEWA dν region is interesting.

Figure 9.

Illustration of the ι_em(p) (k_RAD(p)/(ν_ave)³) dependencies on the integrated SOCEWA (∫SOCEWA dν), based on eq 6, for several Ru-bpy chromophores: [Ru(bpy)₂(CM)]⁺, a–e codes in Figure 1; [Ru(bpy)₃]²⁺, 1; [Ru(bpy)₂(en)]²⁺, 2; [Ru(bpy)₂(NH₃)₂]²⁺, 3; [Ru(bpy)(en)₂]²⁺, 4; [Ru(bpy)(NH₃)₄]²⁺, 5. The least-squares line including the target complexes (1–5 and [Ru(bpy)₂(sc-CM)]⁺ complexes; a, b, and d), with R² = 0.92 and slope = (2.25 ± 0.14) × 10^–4. The ∫SOCEWA dν parameters of the complexes are shown in Table S4 (page S65).

Definition of Donor and Acceptor Orbital Types Leading to SOC Configurational Mixing between the S_n(T) and T_e States

Since the significantly high ι_em(p) amplitudes of the [Ru(bpy)₂(π_Ar-CM)]⁺ complexes (π_Ar-CM = piQ and dpz) in Figure 9 cannot be explained by a simple relationship between the ι_em(p) values and the components of ∑SOCM-IS (by simply observing the ∫SOCEWA dν value) in this work, we needed to return to the fundamental core for this consequence. The simple model of an excited state in a photoinduced transition includes donor and acceptor SOMOs (ϕ_D, D = donor, and ϕ_A, A = acceptor), i.e., S_n = ϕ_D(Sn)ϕ_A(Sn) and T_e = ϕ_D(Te)ϕ_A(Te); ϕ_D and ϕ_A in an MLCT excited state are the metal-dπ and L_Ar-pπ* bases, respectively; the spin–orbit operator in eq 2 can be expanded as Ĥ^SO = Ĥ_D^SO + Ĥ_A for donor and acceptor orbital operations; H_Te, Sn^SOC in eq 2 can be given by

7

The two conditions in eq 7 are attributed to the absence of SOC configurational mixing (type n) between the T_e and S_n(T) states, ϕ_D(Sn) = ϕ_D(Te) with ϕ_A(Sn) = ϕ_A(Te) and ϕ_D(Sn) ≠ ϕ_D(Te) with ϕ_A(Sn) ≠ ϕ_A(Te), and these show both ⟨ϕ_D(Sn)|Ĥ_D^SO|ϕ_D(Te)⟩⟨ϕ_A(Sn)ϕ_A(Te)⟩ = 0 and ⟨ϕ_D(Sn)ϕ_D(Te)⟩⟨ϕ_A(Sn)|Ĥ_A|ϕ_A(Te)⟩ = 0, i.e., H_Te, Sn^SOC = 0 in eq 7.

Two idealized conditions in eq 7 show the logical SOC mixing in the following formalism:

• 1.If ϕ_D(Sn) ≠ ϕ_D(Te) and ϕ_A(Sn) = ϕ_A(Te), then ⟨ϕ_D(Sn)ϕ_D(Te)⟩ = 0 and ⟨ϕ_A(Sn)ϕ_A(Te)⟩ = 1, and then, ⟨ϕ_D(Sn)|Ĥ_D^SO|ϕ_D(Te)⟩ = H_D(Te, Sn) > 0 and ⟨ϕ_A(Sn)|Ĥ_A^SO|ϕ_A(Te)⟩ = 0 in eq 7; thus, ⟨ϕ_D(Sn)|Ĥ_D|ϕ_D(Te)⟩⟨ϕ_A(Sn)ϕ_A(Te)⟩ = H_D(Te, Sn)^SOC and ⟨ϕ_D(Sn)ϕ_D(Te)⟩⟨ϕ_A(Sn)|Ĥ_A|ϕ_A(Te)⟩ = 0. Then, eq 7 can be simplified to H_Te, Sn^SOC ≈ H_D(Te, Sn), which is attributed to donor SOMO-driven SOC configurational mixing (type D^SOC).
• 2.(2) If ϕ_D(Sn) = ϕ_D(Te) and ϕ_A(Sn) ≠ ϕ_A(Te), then ⟨ϕ_D(Sn)ϕ_D(Te)⟩ = 1, ⟨ϕ_A(Sn)ϕ_A(Te)⟩ = 0, ⟨ϕ_D(Sn)|Ĥ_D^SO|ϕ_D(Te)⟩ = 0, and ⟨ϕ_A(Sn)|Ĥ_A|ϕ_A(Te)⟩ = H_A(Te, Sn)^SOC > 0 in eq 7, and then, ⟨ϕ_D(Sn)|Ĥ_D|ϕ_D(Te)⟩⟨ϕ_A(Sn)ϕ_A(Te)⟩ = 0 and ⟨ϕ_D(Sn)ϕ_D(Te)⟩⟨ϕ_A(Sn)|Ĥ_A^SO|ϕ_A(Te)⟩ = H_A(Te, Sn); thus, eq 7 can be expressed as H_Te, Sn^SOC ≈ H_A(Te, Sn), which is considered to be the acceptor SOMO-driven SOC configurational mixing (type A^SOC).

It is also assumed that an S_n(T) state contains more than one component, such as S_n(T) = aS_n(I)(T) + bS_n(II)(T), where a + b ≈ 1 (0.25 < a/b < 4), in the S_4(T) and S_5(T) states of the [Ru(bpy)₂(piQ)]⁺ ion (see the selected NTO plots in the next section; also see Figure S3C2B, pages S39–S40). Moreover, the significant relationship between both the S_n(X)(T) components (X = I or II) of S_n(T) and the T_e states in eq 7 are attributed to the D^SOC- and A^SOC-type-driven SOC configurations, respectively. Consequently, this S_n(T) excited state is considered to contain both characters of D^SOC- and A^SOC-type SOC configurations with the T_e state, an (A + D)^SOC-type-driven form. The formula derivation (eq 7) and its detailed description can be found in the S4 section (pages S52–S54).

Evaluation of the Transition Moment Amplitude in the Phosphorescence Efficiency Formalism (ι_em(p)) by SOC-Mediated Singlet Excited-State Intensity Based on the SOC Principle and DFT Modeling of the Complexes

The Energy-Weighted Oscillator Strength in the ∑SOCM-IS Formalism

To identify the relationship between the ι_em(p) value and the efficient M_Sn(T), S0 contribution of a complex system, a quick plot of ι_em(p) versus ∫SOCEWA dν on the low-temperature experimental observations of the target complexes, as shown in Figure 9, is a plausible relation for the target complexes but cannot explain the significantly high ι_em(p) values of [Ru(bpy)₂(π_Ar-CM)]⁺ ions at the given ∫SOCEWA dν. Because it is difficult to obtain the oscillator-strength value of each singlet excited state from the MLCT band of the experimental absorption spectra of the Ru(II)-L_AR-type complexes, an appropriate parameter from DFT modeling would be expected to assist in the formalism of eq 5. If the different singlet transition moments are approximately orthogonal for simplicity, i.e., M⃗_{Sn(MLCTx, T), S0} · M⃗_{Sn(MLCTy, T), S0} ≈ 0 and S_n(MLCTx, T) ≠ S_n(MLCTy, T) (this statement is described in S6A, pages S83–S87), then the fundamental expression for ι_em(p) in eq 8 based on eqs 3 and 5 can be given by⁷³

8

where , C_i = C_R/C_f (the definition of C_f is shown in eq S4g, page S54), ΔE_{Te – Sn(T)} ≈ E_Sn(T) + λ_reg(T) – hν_em( max ) (see the term definition of ΔE_{Te – Sn(T)} in Figure S5A, page S68), and X is the D^SOC-type (X = D) or A^SOC-type (X = A) mediation operation. The simple ι_em(p) equation focuses on the D^SOC mechanism shown in eqs S4g–S4l, pages S54–S55.

The f_EW term in eq 8 is called the energy-weighted oscillator strength.⁷³ Additionally, if H_{X(Te, Sn(T))}^SOC in eq 8 is assumed to be constant for the SOC-efficient S_n(T) excited states of types D^SOC and A^SOC (this statement for the D^SOC-type configuration is described in the S6B section, page S87), then, ι_em(p) can be represented as

9

Configurations of Natural Transition Orbitals (NTOs) and Parameters of ∑SOCM-IS for the Low-Energy Singlet Excited State in the T_e Coordinates of the Complexes

Fundamentally, the ι_em(p) amplitude is dependent on the efficiency of the vertical SOC configurations between the T_e and the intense low-energy S_n states in the T_e geometry (S_n(T)), and subsequent NTO analyses are based on this logical notion. The plots of the NTOs for vertical S_n(T) transitions (in the T_e geometry) of the [Ru(bpy)(NH₃)₄]²⁺,⁷³ [Ru(bpy)(en)₂]²⁺,¹⁰⁷ [Ru(bpy)₂(en)]²⁺,^73,107 and [Ru(bpy)₂(ppy)]⁺⁷³ complexes were reported in the previous studies, and the orbital compositions of the NTOs of the T_e and low-energy S_n(T) states of [Ru(bpy)₂(en)]²⁺, [Ru(bpy)(en)₂]²⁺, and [Ru(bpy)(NH₃)₄]²⁺ are shown in Table S5C (page S75).

The illustrated NTOs of the low-energy S_n(T) state for the [Ru(bpy)₂(CM)]⁺ complexes are plotted in Figures 10 (CM = ppm) and 11 (CM = piQ). The orbital configurations in the donor and acceptor NTOs of the T_e and low-energy S_n(T) states of the [Ru(bpy)(NH₃)₄]²⁺,⁷³ [Ru(bpy)(en)₂]²⁺,¹⁰⁷ [Ru(bpy)₂(en)]²⁺,⁷³ and [Ru(bpy)₂(CM)]⁺ complexes (CM = ppy⁷³ and ppm) show a simple n-, D^SOC-, and A^SOC-type SOC mediation. The postulated SOC-mediated relationships between the T_e and low-energy S_n(T) states for [Ru(bpy)₂(piQ)]⁺ and [Ru(bpy)₂(dpz)]⁺ complexes show simple n, D^SOC, and A^SOC characteristics and significant (A + D)^SOC characteristics (S_4(T) and S_5(T)), as shown by the low-energy NTOs of S_n(T) for [Ru(bpy)₂(piQ)]⁺ in Figure 11. Information concerning the SOC-mediated relation of the T_e and low-energy S_n(T) states for the target [Ru(bpy)₂(CM)]⁺ complexes is provided in Table 2, and details of the NTO configurations are provided in Table S5B (page S71).

Figure 10.

TDDFT-calculated NTOs of low-energy singlet excited states at the T_e coordinates (S_n(T)) of the [Ru(bpy)₂(ppm)]⁺ ion; the moiety labels for the Δ-form target complexes are shown in Figure 8. The parameters in the panel include the Mulliken orbital population of moieties of the donor and acceptor NTOs, the calculated vertical transition energy (λ for wavelength, nm, with its energy conversion, cm^–1, in parentheses), and the oscillator strength of the target S_n(T) states shown at the bottom (the definition of vertical transition energy is provided in Figure S5A, page S68).

Figure 11.

TDDFT-calculated NTOs of the low-energy singlet excited states (at T_e coordinates, S_n(T)) of the [Ru(bpy)₂(piQ)]⁺ ion; the moiety labels for the Δ-form target complexes are shown in Figure 8. The parameters in the panel include the Mulliken orbital population of moieties of the donor and acceptor NTOs, the calculated vertical transition energy (λ for wavelength, nm, with its energy conversion, cm^–1, in parentheses), and the oscillator strength of the target S_n(T) states shown at the bottom (the definition of vertical transition energy is provided in Figure S5A, page S68).

Table 2. Characteristics of the SOC Mediation Types of Low-Energy Singlet Excited States in the T_e Geometry (S_n(T)) of Target [Ru(bpy)₂(CM)]⁺ Ions^a.

excited state	L	SOC mediation typeb	oscillator strength, f	excited-state energy, cm–1/103	fEW × 1013c
S1(T)	ppy	type n	0.0018	12.64
	ppm	type n	0.0019	13.09
	piQ	type n	0.0020	12.76
	dpz	type n	0.0033	13.33
S2(T)	ppy	type ASOC	0.0015	15.20	0.090
	ppm	type ASOC	0.0014	15.65	0.077
	piQ	type ASOC	0.0015	15.32	0.089
	dpz	type ASOC	0.0016	15.87	0.082
S3(T)	ppy	type DSOC	0.0021	16.37	0.064
	ppm	type DSOC	0.0026	16.92	0.070
	piQ	type DSOC	0.0028	16.31	0.092
	dpz	type DSOC	0.0026	17.39	0.059
S4(T)	ppy	type DSOC	0.0895	17.57	1.57
	ppm	type DSOC	0.0975	17.89	1.713
	piQ	type (A + D)SOC	0.0858	17.51	1.615
	dpz	type (A + D)SOC	0.0881	17.95	1.575
S5(T)	ppy	type n	0.0411	18.62
	ppm	type n	0.0356	19.05
	piQ	type (A + D)SOC	0.0259	17.83	0.427
	dpz	type (A + D)SOC	0.0191	18.18	0.310
S6(T)	ppy	type n	0.0784	20.83
	ppm	type ASOC	0.0014	19.88	0.012
	piQ	type n	0.0370	18.76
	dpz	type n	0.0355	19.34
S7(T)	ppy	type ASOC	0.0038	21.10	0.021
	ppm	type n	0.0823	21.19
	piQ	type n	0.1097	20.58
	dpz	type n	0.0052	19.34

^aNTO configurations based on the DFT modeling are shown in Table S5B (page S71).

^bDefinition of SOC-driven types describe in the first subsection of Discussion.

^cEquation 8.

Component of ∑f_EW(Sn(T)) of ∑SOCM-IS Modeling in Equation 9 Focusing on the Type D^SOC-Driven Contribution

Once again, the ι_em(p) amplitudes of the complexes are based on the 77 K observations and Einstein’s formalism (eq 1). If we assume that the SOC efficiency of the complexes in this work is dominated by the donor SOMO-mediated (type D^SOC) configurational mixing between the S_n(T) and T_e states, the postulation is that H_Te, Sn(T)^SOC ≈ H_{D(Te, Sn(T))} ≫ H_{A(Te, Sn(T))}^SOC in eq 9 and the ι_em(p) values are related only to the component of ∑SOCM-IS, the sum of the D^SOC-type energy-weighted oscillator strengths (∑f_EW(Sn(T))(D^SOC)), in eq 9. The plot of ι_em(p) vs ∑f_EW(Sn(T))(D^SOC) of the complexes (complexes 2, 4, 5, a, and d) in Figure 12 shows good linear relationships, but those of the [Ru(bpy)₂(π_Ar-CM)]⁺ ions, where π_Ar-CM = piQ (c) and dpz (e), are higher than the least-squares line that corresponds to 50% of the given ∑f_EW(Sn(T))(D^SOC) values, which is similar to the result for the plot of ι_em(p) versus ∫SOCEWA dν of the complexes in Figure 9. This phenomenon for the reference ([Ru(bpy)_3–n(Am)_2n]²⁺, 2, 4, and 5) and the [Ru(bpy)₂(sc-CM)]⁺ (sc-CM = ppy (a) and ppm (d)) complexes in Figure 12 is attributed to the primary-type D^SOC-mediated configurational mixing between the S_n(T) and T_e states dominating the ∑SOCM-IS contributions to the ι_em(p) values, but this simple conclusion does not effectively explain the significantly high ι_em(p) values for the [Ru(bpy)₂(π_Ar-CM)]⁺ reported in this work. In summary, the simple ∑SOCM-IS modeling focused on a limited-type D^SOC-mediated contribution does not effectively explain the observation of the remarkably high ι_em(p) amplitudes for the [Ru(bpy)₂(π_Ar-CM)]⁺ complexes in Figure 12.

Figure 12.

Illustration of the postulated dependencies of ι_em(p), k_RAD(p)/(ν_ave)³, versus ∑f_EW(Sn(T))(D^SOC) (based on eq 9, if H_Te, Sn(T)^SOC ≈ H_{D(Te, Sn(T))} ≫ H_{A(Te, Sn(T))}^SOC) for several Ru-bpy chromophores: [Ru(bpy)₂(CM)]⁺, (a, c–e codes in Figure 1); [Ru(bpy)₂(en)]²⁺, 2; [Ru(bpy)(en)₂]²⁺, 4; [Ru(bpy)(NH₃)₄]²⁺, 5. The least-squares line including the target complexes (2, 4, and 5 and [Ru(bpy)₂(sc-CM)]⁺ complexes; a and d) shows R² = 0.92 and slope = (1.27 ± 0.13) × 10⁵.

Presupposed∑f_EW(Sn(T))Amplitudes in Equation 9 Including the Overestimated Type A^SOC-Mediated Contribution

To verify the DFT results, the NTO plots of the low-energy S_n(T) state for [Ru(bpy)_3–n(Am)_2n]²⁺,^{8,65,73,106,107} [Ru(bpy)₂(sc-CM)]⁺ (sc-CM = ppy⁷³ and ppm), and [Ru(bpy)₂(π_Ar-CM)]⁺ (π_Ar-CM = piQ and dpz) show that their acceptor SOMOs include significant dπ-Ru orbital components (2–10%, Ru is assumed to be a heavy atom), and the Ĥ^SOconstruction in eqs 2 and 7 contains angular terms from all atoms, which suggests that the ∑SOCM-IS modeling for ι_em(p) related to the ∑f_EW(Sn(T)) amplitude should possibly consider the ∑f_EW(Sn(T)) containing both D^SOC- and A^SOC-type mediation contributions. The ∑f_EW(Sn(T))(X) values of D^SOC- and A^SOC-type mediation (X = D and A) in eq 9 for the target complexes are shown in Figure S5D (page S77). The f_EW(Sn(T))(X) amplitude for (A + D)^SOC-type S_n(T) is considered to contribute to both the D^SOC- and A^SOC-type mediations, and the ∑f_EW(Sn(T))(X) values of the low-energy S_n(T) states of the complexes in Figure S5D (page S77) are shown in Table 2 and Table S5B,C (pages S71–S77). The ∑f_EW(Sn(T))(A)/∑f_EW(Sn(T))(D) ratios are approximately 0.2–0.3 for the [Ru(bpy)_3–n(Am)_2n]²⁺ and [Ru(bpy)₂(sc-CM)]⁺ complexes and 1.2–1.3 for [Ru(bpy)₂(π_Ar-CM)]⁺ ions. To simply understand the logical relations of ι_em(p) versus the ∑f_EW(Sn(T))(X) for ∑SOCM-IS modeling in eq 9 for the target complexes in this work and to presuppose ∑f_EW(Sn(T))(X) amplitudes including the D^SOC and A^SOC components, two assumptions must be considered:

(1) We postulate that the H_{D(Te, Sn(T))}^SOC and H_{A(Te, Sn(T))} values are constants for the D^SOC- and A^SOC-type mediation relations in eq 9, respectively.

(2) If H_{A(Te, Sn(T))}^SOC = H_{D(Te, Sn(T))}/4, then the H_{A(Te, Sn(T))}^SOC value will be overestimated because most of the orbital population in the acceptor SOMOs is localized on the light atoms of the ligand moiety for the type A^SOC mediation relation in eq 7. As a result, the calculated ∑f_EW(Sn(T)) value from the D^SOC- and A^SOC-type mediation in eq 9 can be estimated.

Based on the two postulations above, eq 9 can be simplified to

10

where ∑f_EW(Sn(T)) = ∑ f_EW(Sn(T))(D) + [ ∑ f_EW(Sn(T))(A)]/16.

In the plot of ι_em(p) versus ∑f_EW(Sn(T)) based on eq 10 for the target complexes shown in Figure 13, the observed ι_em(p) amplitudes for [Ru(bpy)₂(π_Ar-CM)]⁺ (π_Ar-CM = piQ (c) and dpz (d)) are above those of the least-squares line for the reference [Ru(bpy)_3–n(Am)_2n]²⁺ (2, 4, and 5) and [Ru(bpy)₂(sc-CM)]⁺ (a and d) complexes by approximately 30%, and even the ∑f_EW(Sn(T)) = ∑ f_EW(Sn(T))(D) + [ ∑ f_EW(Sn(T))(A)]/16 amplitude overestimates the type A^SOC-mediated ∑f_EW(Sn(T))(A) contribution in eq 10. These results suggest that the intensity term in the ∑SOCM-IS model that contains only D^SOC- and A^SOC-type contributions cannot effectively explain the remarkably high ι_em(p) amplitude of the Ru-bpy chromophore of [Ru(bpy)₂(π_Ar-CM)]⁺ ions compared to those of the reference and [Ru(bpy)₂(sc-CM)]⁺ complexes. A possible solution to this issue is to reconsider the perturbation principle of molecular orbitals and SOC fundamentals. The NTO plots of the low-energy S_n(T) states for the [Ru(bpy)₂(CM)]⁺ ions (see Figures 9 and 10) and [Ru(bpy)(NH₃)₄]²⁺,^8,65,73,106 [Ru(bpy)(en)₂]²⁺,¹⁰⁷ and [Ru(bpy)₂(en)]^2+73,107 complexes show a significant contribution from the π-ligand component in the donor SOMOs related to the configurations between MLCT and ππ*-ligand states, suggesting that the efficient ∑f_EW(Sn(T)) in eq 9 should include the influence of the perturbation of the configuration of low-energy ¹MLCT and high-energy ¹ππ states.

Figure 13.

Illustration of the sum of the energy-weighted oscillator strengths of D^SOC- and A^SOC-type mediation, ∑f_EW(Sn(T))(D) + [ ∑ f_EW(Sn(T))(A)]/16, dependencies of ι_em(p) (k_RAD(p)/(ν_ave)³) based on eq 10 for several Ru-bpy chromophores: [Ru(bpy)₂(CM)]⁺, (a, c–e codes in Figure 1); [Ru(bpy)₂(en)]²⁺, 2; [Ru(bpy)(en)₂]²⁺, 4; [Ru(bpy)(NH₃)₄]²⁺, 5. The least-squares line is based on the parameters of the target complexes (2, 4, 5, and [Ru(bpy)₂(sc-CM)]⁺ complexes; a and d), R² = 0.95 and slope = (1.25 ± 0.13) × 10⁵.

Intensity Perturbation of Configurational Mixing between Singlet Low-Energy MLCT and High-Energy ππ*-L_Ar Excited States in the ∑SOCM-IS Formalism

The unusually high ι_em(p) amplitudes observed for the [Ru(bpy)₂(π_Ar-CM)]⁺ ions (c and e) in Figure 13 (based on eq 10) imply that the influence of the calculated ∑f_EW(Sn(T)) value for the efficient low-energy ¹MLCT excited states of D^SOC- and A^SOC-type contributions should not contain all of the significant intensity contributions from the singlet excited states in the ∑SOCM-IS modeling. Straightforward SOC mixing between T_e and high-energy singlet ππ*-L_Ar states (¹ππ*-L_Ar, L_Ar = aromatic ligand) is not taken into consideration in this work because only light atoms (C, H, and N) are present in the ¹ππ*-L_Ar of the L_Ar moiety; this statement is described in the S6D section (page S89). In more detail, the configurational mixing between low-energy ¹MLCT (S_n) and high-energy ¹ππ*-L_Ar (S_m) states should perturb the M⃗_Sn(T), S0 values in the ι_em(p) formalism, where the selected ¹MLCT (S_n(T)) should focus on D^SOC-type mediation (H_{D(Te, Sn(T))}^SOC ≫ H_{A(Te, Sn(T))} in eq 7) and m > n. The M⃗_Sn(T), S0 term in the ι_em(p) equation (eq 3) should include a linear combination of the S_n(T) and S_m(T) mixing perturbation, M⃗_Sn(T), S0 ≈ M⃗_Sn(T), S0^o + α_{Sm(T), Sn(T)}M⃗_Sm(T), S0| cos θ_m, n|, and the superscript “o” indicates the diabatic transition moment without S_n(T) and S_m(T) mixing, which can be treated in eq 11, where α_{Sm(T), Sn(T)} is a mixing coefficient between S_n(T) and S_m(T). A detailed description of eq 11 is provided in Section S4C (page S59).

11

where the term,

in eq 11 is referred to as the sum of the second-order energy-weighted oscillator strength, ∑(2^nd - f_EW), from the intensity perturbation of the configuration of low-energy ¹MLCT (S_n(T)) and high-energy ¹ππ*-L_Ar (S_m(T)) excited states, and the ΔE_{Sm(T), Sn(T)} is the energy difference between the S_m(T) and S_n(T) states (i.e., E_Sm(T) – E_Sn(T) ≈ ν_Sn(T) – ν_Sm(T)). The formula derivation (eq 11) and its detailed description can be found in the S4C section (pages S59–S61).

If | cos θ_m, n| in eq 11 is a constant for each pair of S_n(T)/S_m(T) mixing perturbations, then this act is only for simplicity in the calculation; the calculated ∑(2^nd - f_EW)/| cos θ_m, n| values of the target complexes in a weak coupling setting (H_{Sm(T), Sn(T)} = 1000 cm^–1 and ΔE_{Sm(T), Sn(T)} > 10,000 cm^–1, α_{Sm(T), Sn(T)} < 0.1) are shown in Tables S5C1–S5C5 (pages S78–S82). Furthermore, the calculated ∑(2^nd - f_EW)/| cos θ_m, n| values of the target complexes in Tables S5C1–S5C5 with intense low-energy S_n(T) (D^SOC-mediated ¹MLCT) and intense high-energy S_m(T) (inter- and intra-¹ππ*-L_Ar, f_Sm(T), S0 > 0.03) states are based on the parameters of DFT modeling. The estimated ∑(2^nd - f_EW)/| cos θ_m, n| values of the [Ru(bpy)₂(sc-CM)]⁺ ions (sc-CM = ppy and ppm) are approximately 1.7 times higher than the corresponding values of the [Ru(bpy)₂(en)]²⁺, and the ∑(2^nd - f_EW)/| cos θ_m, n| values of the [Ru(bpy)₂(π_Ar-CM)]⁺ ions (π_Ar-CM = dpz and piQ) are approximately 1.5-fold greater than those of the [Ru(bpy)₂(sc-CM)]⁺ ions. The above results can be attributed to the following facts (see Tables 5S5C1–S5C5, pages S78–S82): (1) every [Ru(bpy)₂(CM)]⁺ ion has 7–8 intense S_m(T) excited states, and [Ru(bpy)₂(en)]²⁺ has only 3 intense-S_m(T) excited states (f_Sm(T), S0> 0.03); (2) the [Ru(bpy)₂(π_Ar-CM)]⁺ ions (c and e) have relatively lower-energy intense-S_m(T) excited states (ν_Sm(T) in 31,000–35,340 cm^–1) compared to those of [Ru(bpy)₂(sc-CM)]⁺ (a, b, and d; ν_Sm(T) in 34,000–37,340 cm^–1), and ν_Sm(T) is related to the terms of × in ∑(2^nd - f_EW) of eq 11; (3) each of [Ru(bpy)₂(en)]²⁺ and [Ru(bpy)₂(sc-CM)]⁺ ions has only an intense low-energy D^SOC-type ¹MLCT excited state, but each [Ru(bpy)₂(π_Ar-CM)]⁺ ion has two intense low-energy (A + D)^SOC-type ¹MLCT excited states; (4) concerning the above products, the values for the sum of second-order components in ∑(2^nd - f_EW)/| cos θ_m, n| for [Ru(bpy)₂(π_Ar-CM)]⁺ ions are greater than that for the corresponding values of [Ru(bpy)₂(sc-CM)]⁺ ions, and the ∑(2^nd - f_EW)/| cos θ_m, n| values for the [Ru(bpy)₂(sc-CM)]⁺ ions are greater than those of [Ru(bpy)₂(en)]²⁺; (5) the point (2) above should be regarded as the relatively low-energy intense-S_m(T) configurations including a significant ππ*-CM component for [Ru(bpy)₂(π_Ar-CM)]⁺, but the first two low-energy intense-S_m(T) configurations only include the ππ*-bpy component for [Ru(bpy)₂(sc-CM)]⁺. This consequence of the unusually high ι_em(p) value of the [Ru(bpy)₂(π_Ar-CM)]⁺ ion in Figure 12 can be attributed to both significantly high ∑f_EW(Sn(T)) and ∑(2^nd - f_EW) contributions in the ∑SOCM-IS formalism. However, due to the uncertain | cos θ_m, n| magnitude between the efficient low-energy ¹MLCT (S_n) and high-energy ¹ππ*-L_Ar (S_m) transition states, the only plausible ∑(2^nd - f_EW)/| cos θ_m, n| components can be displayed, and the ∑(2^nd - f_EW) value cannot be evaluated in this work.

In summary, recent reports have included the use of computational SOCM-IS modeling to fit the observed singlet-to-triplet absorption and phosphorescence parameters of target M-L_Ar-type complexes, and their modeling is attributed to the primary ΣSOCM-IS contribution from the intense low-energy S_n(T) states.^66,124−130 In this work, the observed ι_em(p) magnitudes of the [Ru(bpy)₂(π_Ar-CM)]⁺ ions were found to be considerably higher than those of the [Ru(bpy)₂(sc-CM)]⁺ and [Ru(bpy)_3–n(Am)_2n]²⁺ complexes at the given ∫SOCEWA dν (Figure 9, based on eq 6) and at the given ∑f_EW(Sn(T))(D) values (Figure 12, based on eq 9), suggesting that the ι_em(p) formalism based on the simple ∑f_EW(Sn(T)) formalism from low-energy ¹MLCT states should not contain all of the significant intensity contributions in the ∑SOCM-IS model. In this work, we consider the ∑(2^nd - f_EW) value obtained from intensity perturbation of the configurations between low-energy ¹MLCT and high-energy ¹ππ-(L_Ar) states, which significantly contributes to the ι_em(p) magnitude in the ∑SOCM-IS formalism (eq 11) for the Ru-bpy phosphorescence chromophore of the [Ru(bpy)₂(CM)]⁺ complex system.

Conclusions

This study represents a continuation of our work directed toward understanding the meaning of the intrinsic phosphorescence efficiency, ι_em(p) = k_RAD(p)/(ν_ave)³, of Ru-bpy phosphorescence chromophores for [Ru(bpy)₂(CM)]⁺ and the reference [Ru(bpy)_3–n(Am)_2n]²⁺ series. The phenomenon of variation in ι_em(p) for complex series is considered to be related to spin–orbit coupling (SOC)-mediated intensity stealing (SOCM-IS) from efficient singlet states, which is based on low-temperature observations, DFT modeling, and basic SOC principles.⁷³ The parameters for the orbital comparison for transition states in DFT modeling helped in the analysis of data concerning the SOCM-IS construct.

The plot in Figure 6 shows that the observed ι_em(p) magnitudes of the Ru-bpy chromophore for the [Ru(bpy)₂(CM)]⁺ ions in the given emission energy region are clearly higher than the corresponding values for [Ru(bpy)_3–n(Am)_2n]²⁺ ions. The ι_em(p) values of the [Ru(bpy)₂(π_Ar-CM)]⁺ ions (π_Ar-CM = dpz and piQ; π-rich cyclometalated ligand) were approximately twofold higher than those of the [Ru(bpy)₂(sc-CM)]⁺ complexes (sc-CM = ppy, bhq, and ppm; a simply constructed cyclometalated ligand).

The observed ι_em(p) variation of the [Ru(bpy)_3–n(Am)_2n]²⁺ and [Ru(bpy)₂(sc-CM)]⁺ complexes can be attributed to the change in the sum of the energy-weighted oscillator strengths from the low-energy S_n(¹MLCT) states of a type D^SOC mediation (∑f_EW(Sn(T))(D), see Figure 12), based on eqs 8 and 9 of the ∑SOCM-IS expression, but the unusually high ι_em(p) magnitudes of the [Ru(bpy)₂(π_Ar-CM)]⁺ ions in the given ∑f_EW(Sn(T))(D) magnitude cannot account for this simple conclusion. However, the significantly high ι_em(p) values for the [Ru(bpy)₂(π_Ar-CM)]⁺ ions are not a simple result of the ∑f_EW(Sn(T)) contribution from the low-energy S_n(¹MLCT) states of D^SOC- and A^SOC-type mediation (see Figures 12 and 13).

Again, the basic intensity factors for the ∑SOCM-IS expression in eqs 8 and 9 do not directly include the straightforward SOC configurations of the T_e and high-energy ¹ππ*-L_Ar states in this work (this statement is expressed in Section S6D, pages S89 and S90, and L_Ar is an aromatic-type ligand), but the second-order intensity perturbation (∑(2^nd - f_EW)) to the ∑SOCM-IS expression from the significant configurations of low-energy ¹MLCT(Ru-bpy) and high-energy ¹ππ*-L_Ar states should be considered. The values for the calculated ∑(2^nd - f_EW) component, ∑(2^nd - f_EW)/| cos θ_m, n|, of the [Ru(bpy)₂(π_Ar-CM)]⁺ complexes were determined to be approximately 1.5-fold higher than the corresponding values for the [Ru(bpy)₂(sc-CM)]⁺ ions.

The current study, which contains experimental low-temperature observations, simple DFT modeling, and elementary parameters based on SOC principles, suggests that the remarkable high ι_em(p) magnitudes of the [Ru(bpy)₂(π_Ar-CM)]⁺ complexes can be attributed to both their ∑f_EW(Sn(T))(D) and ∑(2^nd - f_EW) magnitudes in the case where the ΣSOCM-IS expression is greater than those of the [Ru(bpy)₂(sc-CM)]⁺ complexes.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.2c07276.

• (S1) experimental section containing the materials and syntheses of the complexes, instrumentation, NMR spectra of the target complexes, and single-crystal X-ray data parameters of [Ru(bpy)₂(dpz)](PF₆)·(CH₂Cl₂); (S2) photophysical parameters for the reference [Ru(bpy)_3–n(Am)_2n]²⁺ complexes and electrochemistry and low-temperature absorption spectra of the [Ru(bpy)₂(CM)]⁺ ions; (S3) computational details and results of DFT modeling for the target [Ru(bpy)₂(CM)]⁺ (CM = ppm, piQ, and dpz), including (1) DFT parameters for the compositions of HOMO-to-LUMO transitions in the low-energy S_n excited states, (2) comparison of the observed and DFT-calculated absorption curves of the target [Ru(bpy)₂(CM)]⁺ ions, (3) DFT parameters of the S_n and T_n excited states, and (4) plots of the natural transition orbitals (NTOs) for the S_n(X) and T_n excited states and (4) alpha (α) and beta (β) SOMOs of the T₁(T_e) state for the target complexes; (S4) phosphorescence efficiency expression for the sum of the SOC-mediated intensity stealing (∑SOCM-IS) modeling and data analyses; (S5) parameters of the ∑SOCM-IS modeling based on SOC principles and primary DFT modeling; (S6) some conceptual details for the derivation process of the fundamental equation in Section S4 and manuscript (PDF)

Author Contributions

^† Y.R.C., Y.-T.L., and C.-W.Y. contributed equally.

This work was funded (Y.J.C.) by the Ministry of Science and Technology (Taiwan, ROC) through grant MOST 109-2113-M-030-006.

The authors declare no competing financial interest.