Nature of ultrafast dynamics in the lowest-lying singlet excited state of [Ru(bpy)₃]²⁺

Abstract

This work presents mechanisms to rationalize the nature of ultrafast photochemical and photophysical processes on the first singlet metal–ligand charge transfer state (¹MLCT₁) of the [Ru(bpy)₃]²⁺ complex. The ¹MLCT₁ state is the lowest-lying singlet excited state and the most important intermediate in the early evolution of photoexcited [Ru(bpy)₃]²⁺*. The results obtained from simple but interpretable theoretical models show that the ¹MLCT₁ state can be very quickly formed via both direct photo-excitation and internal conversions and then can efficiently relax to its equilibrium geometry in ca. 5 fs. The interligand electron transfer (ILET) on the potential energy surface of the ¹MLCT₁ state is also extremely fast, with a rate constant of ca. 1.38 × 10¹³ s⁻¹. The ultrafast ILET implies that the excited electron can dynamically delocalize over the three bpy ligands, despite the fact that the excited electron may be localized on either one of the three ligands at the equilibrium geometries of the three symmetric equivalent minima. Since rapid ILET essentially suggests delocalization, the long-standing controversy in inorganic photophysics—whether the excited electron is localized or delocalized—may therefore be calmed down to some extent.

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Introduction

In light harvesting, the photosensitizer is crucially important for transferring the photo-excited electrons to the catalytic center.^1–3 The rapidly growing development of homogeneous photocatalysis mainly relies on the application of photosensitizers, such as organic dyes,^4,5 and transition metal complexes.^6,7 These compounds have attracted widespread attention for their high activity in UV-visible driven photocatalytic reactions, because their excited states are long-lived and effective in energy.^8–10 Ru(II) polypyridine complexes are currently one of the most investigated compounds in coordination chemistry, due to their excellent properties. Moreover, their ligands are modifiable, allowing for the adjustment of the properties of both the ground and excited states.¹¹ Among these compounds, [Ru(bpy)₃]²⁺ is a prototype and has been widely used in many fields, i.e. photovoltaics, photocatalysis and photodynamic therapy, because of its unique photoreactivity.^12–15 After the initial adsorption of a photon of visible light, the molecule is excited to a metal–ligand charge transfer (MLCT) state, in which one electron is excited from one d orbital of the central metal to one ligand's π* orbital compared to the ground state.^16,17 The photoexcited [Ru(bpy)₃]²⁺* in the MLCT states may act as both a better oxidant and a better reductant than in its ground state.¹¹ These MLCT states can participate in intermolecular electron and energy transfer reactions, and the released electron and energy can be captured by acceptors for further reactions.¹⁸ Recently, the electronic structure of the excited states of [Ru(bpy)₃]²⁺ is still a hot topic.^19–22

In particular, whether the excited electron in the lowest MLCT state of the system is localized on a single ligand or delocalized on multiple ligands is still a controversy, and it is relevant to interligand electron transfer (ILET). Moreover, because this issue may determine the intermolecular electron transfer rate,^18,23,24 when the [Ru(bpy)₃]²⁺ complex interacts with acceptors, the ILET also plays an important role in this complex.²³ Solvent dependence of MLCT transition^25,26 and Stark effect spectra²⁷ investigations have shown that the initially prepared ¹MLCT state is highly dipolar,²⁷ which implies that the Franck–Condon (FC) state is localized.²⁵ The time-resolved resonance Raman spectroscopy results imply that the excited electron of the MLCT state is localized on a single bipyridine ligand.^26,28–30 X-ray absorption spectra further supported this viewpoint by analyzing the correlation between the electronic structure and the geometric distortion of the MLCT state.³¹ Moreover, transient absorption anisotropy spectra suggested that the electron is localized immediately after excitation.²³ However, femtosecond time-resolved absorption anisotropy measurements imply that the initial photoinduced MLCT state is delocalized, and followed by localization.³² Furthermore, other time-resolved spectroscopy experiments and time-dependent density functional theory (TDDFT) computations suggest that the excited electron is delocalized over the three ligands, and therefore there is no ILET.^33–37 Besides, several femtosecond time-resolved spectroscopy measurements reveal that the lifetime of the ¹MLCT state of the [Ru(bpy)₃]²⁺ complex is extremely short (about 40 fs),^38,39 and the intersystem crossing (ISC) occurs in only about 30 fs with a quantum yield very much close to unity.^{32,34,39–43} The aforementioned experimental and theoretical works are quite extensive and in-depth; nevertheless, there is still no clear consensus on whether the excited electron in this system is localized or delocalized. By exploring the nature of the electronic structure of the excited states of the system, we aim to provide a comprehensive explanation for the formation and the ILET of the lowest-lying ¹MLCT₁ excited state.

Results and discussion

The theoretical study of this work focuses on the formation and transformation processes of the lowest-lying ¹MLCT₁ state for the [Ru(bpy)₃]²⁺ complex. In photochemical reactions, the lowest-lying singlet excited state in general may form either by direct excitation or by internal conversion (IC). The rate of direct excitation is very fast and often on a timescale of several femtoseconds. Meanwhile, the rate of IC is relatively slower and might be affected by various factors such as intersections of potential energy surfaces (PESs) and the solvent effect, etc. Fig. 1 shows the state correlation diagram for the three lowest-lying ¹MLCT states. Notably, the intersection of energy levels of these states indicates that there is a three-state conical intersection (CI) somewhere along the continuous changing of the molecular geometry from the ground state equilibrium configuration to the minimum energy geometry of the ¹MLCT₁ state with D₃ symmetry constraint. Since the intersection geometry is close to the ground state equilibrium one, the ¹MLCT₁ state has close energies at these geometries. In general, when there is a three-state CI, there will be a two-state CI in its branching space.^44–46 Based on Longuet-Higgins’ geometric phase-change rule,⁴⁷ the phase analysis of the three low-lying ¹MLCT states was carried out around the minimum and transition state geometries of the ¹MLCT₁ state, see Fig. S3 in the ESI† for more details. It demonstrates that the ¹MLCT₁ state itself has no intersection with the other two states, but there is a two-state CI between the ¹MLCT₂ and ¹MLCT₃ states, and the intersection geometry is close to the just mentioned D₃ constrained geometry. Since a CI and its near region is usually a very efficient path for a non-radiative transition, especially when the CI is geometrically close to the FC point, the rate of IC can be quite fast. This is indeed the case here for low-lying ¹MLCT states of [Ru(bpy)₃]²⁺. It is worth mentioning here that it cannot be directly deduced from Fig. 1 whether the initially excited ¹MLCT state is localized or delocalized. However, one can infer that these ¹MLCT states with close energies may convert into each other. Therefore, according to Kasha's rule, these ¹MLCT states can quickly evolve to the S₁ state. Naturally after that, the vibrational relaxation of the S₁ state on its potential energy surface can occur, no matter what geometries the S₁ state is in, whether it is the FC point or the D₃ or other low-symmetry geometries in close proximity to it, regardless of what kinds of electronic structure characteristics the S₁ state has at these geometries, i.e. whether the excited electron is localized or delocalized. Therefore, one may infer that the system would quickly transfer to the lowest-lying ¹MLCT₁ state even if it is initially excited to other ¹MLCT_n states.

Fig. 1 The state correlation diagram for the first three lowest-lying ¹MLCT states between and at the ground state equilibrium geometry and the D₃ symmetry constrained minimum energy geometry of the ¹MLCT₁ state for the [Ru(bpy)₃]²⁺ complex. The reaction coordinates of the starting and ending points, at which the geometries are labelled beneath the coordinates, are set as 0 and 1, respectively. The coordinates of other points are obtained by a linear combination of the two terminal points. The energies marked on the vertical axis are presented relative to the energy of the ¹MLCT₁ state (−1579.36971998 E_h) at the D₃ constrained geometry. All energies were calculated by the TD-PBE0 method. A schematic illustration of the adiabatic states is given in Fig. S2 (ESI†).

Now, we estimate how much time is needed for the ¹MLCT₁ state to reach its equilibrium geometry after the initial excitation by means of a vibrational relaxation. In general, there are so many paths for the vibrational relaxation that it is difficult to accurately calculate the time required. As a rough approximation, the vibrational relaxation in this work is defined as the relaxation of the ¹MLCT₁ state from the FC point to the equilibrium geometry along the most relevant vibrational mode. This is fundamentally different from the vibrational energy relaxation process in which the population distribution of molecules in quantum states of high energy level returns to its thermal equilibrium status, i.e. the Maxwell–Boltzmann distribution. The initially formed ¹MLCT₁ state, whether by direct photo-excitation or by IC from other states, is in higher vibrational states and not at its equilibrium geometry, and so it may relax to the low-lying vibrational states at its equilibrium geometry. The time of vibrational energy relaxation might be influenced by several factors such as vibrational frequency, geometric distortion, solvent effect, etc. Meanwhile, for the vibrational relaxation from the FC point to the ¹MLCT₁ state equilibrium geometry, the main change in the molecular geometry is observed on the bpy ligand where the excited electron resides, as shown in Fig. 2. The largest change comes from the bridged C–C bond length, by ca. 0.051 Å, which is further used as a measure of the vibrational relaxation distance. From Fig. 2, one may also observe that the corresponding Ru–N bond lengths decrease by 0.013 Å after the electron transfer, possibly due to the enhancement of electrostatic attraction between the central metal and the ligand. Presumably, this would further affect the hybridization of the central-metal orbitals on the two Ru–N bonds along the lines and result in a similar degree of elongation in the corresponding bond length. The ground state and the ¹MLCT₁ state are so close in their equilibrium geometries that the latter can be reached in an extremely short time period of about 5 fs only via the relaxation on the most relevant vibrational mode; see ESI† for deduction details. This is also consistent with the fact that the vibrational relaxation from the FC point to the equilibrium geometry just comprises one-fourth of the full vibration period, about 20 fs estimated from the vibrational frequency, 4.98 × 10¹³ s⁻¹. We shall note again that the vibrational relaxation in this work is not the full vibrational energy relaxation which is a more commonly used concept. The full vibrational energy relaxation, including the thermal equilibrium of the MLCT state and the response of the solvent, was found to need a much longer time.⁴⁸ The vibrational relaxation time is even shorter when the initial ¹MLCT₁ state is obtained from an IC process, considering that the three-state CI point is closer to the ¹MLCT₁ state equilibrium geometry than the FC point. Thus, the initially formed ¹MLCT₁ state can reach the equilibrium geometry rapidly through intramolecular vibration relaxation, whether it is obtained by direct excitation or by IC. Apart from the close proximity of the ¹MLCT₁ state equilibrium geometry to the FC point, the fast nuclear vibration on the relevant mode is another factor that accounts for the fast relaxation of the ¹MLCT₁ state. Furthermore, since the ¹MLCT₁ state has three equivalent minima that are symmetrically distributed around the FC point, the vibrational relaxations to these three minima can occur equally along the three directions and shall take the same amount of time. These theoretical results may provide a reasonable explanation for the long-standing controversy in inorganic photophysics regarding whether the excited electron in the initial ¹MLCT state is localized^{23,26,28–32} or delocalized,^32–37 since a slight geometric distortion at the FC point may lead to immediate localization of the excited electron on one bpy ligand.

Fig. 2 Schematic representation of the distortions of the main geometric parameters upon the vibrational relaxation of the ¹MLCT₁ state. The values are given in Å.

In the evolution of the ¹MLCT₁ state, there are several competitive reaction pathways, including ILET, ISC and fluorescence. The ILET reaction occurs remaining on the PES of the ¹MLCT₁ state. To analyze whether the ILET reaction is adiabatic or nonadiabatic, a diabatic representation is applied. The diabatic wave functions are constructed from localized orbitals obtained from a unitary transformation of the CASSCF orbitals by using a similar procedure of Hiberty and coworkers.⁴⁹ The obtained diabatic states exhibit the properties that the excited electron is localized on one ligand, and their electronic structure characteristics are essentially preserved during the whole ILET process. A merit of using the diabatic representation is that the classical valence bond (VB) interpretation,⁵⁰ in particular Shaik's diagram methods,^51–53 might be adopted because for electron transfer, VB structures and diabatic states are highly relevant even though not the same.^54,55 Although the ligands are close in space, we found that the ILET reaction has a significant probability, about 84% from the Landau–Zener formula,^56,57 to occur in a nonadiabatic manner, see ESI† for deduction details. The primary reason for this is that the electronic coupling is small, ca. 0.54 kcal mol⁻¹, reflecting that the interaction between the reactant and the product diabatic states (Φ^d_a and Φ^d_b in Fig. 3) is weak. Because π* molecular orbitals on two ligands are about perpendicular to each other, the electronic coupling is not originated from the direct interaction between the ligand orbitals but mainly from the metal–ligand interaction since a direct exchange interaction is quite operative for the metal center with a hole. It can also be seen from Fig. S4 (ESI†) that the localized ligand orbitals π_la*, π_lb* and π_lc* in fact have a nonnegligible component of d orbitals of the center metal. In terms of the nonadiabaticity of the ILET reaction, the coupling is quite weak. However, its effect on the ILET rate might be significant since it can reduce further the reaction barrier to a quite small value.

Fig. 3 Diagrammatic representation of the diabatic MLCT states. The excited electron is localized on the ligand colored as red. The localized π* orbitals, namely π_la*, π_lb* and π_lc*, are respectively represented by the three p-type orbitals, each of which is perpendicular to its ligand plane. These orbitals are drawn in Fig. S4 (ESI†). Each sub-diagram denominates one diabatic state, namely that (a) , (b) and (c) , where Ω denotes the remaining inner double occupied orbitals. Each diabatic state has a positive phase as the black part of the p-type orbital in each diagramma is in the same direction as the positive axis parallel to it.

The ILET rate constant, k_et, is calculated from the classical transition state formalism,^57–60

k_et = υ_effκ_el exp(−ΔG*/k_BT), (1)

where υ_eff is the effective vibrational frequency, κ_el, which is often termed as the “electronic factor”, is the probability of electron transfer once the nuclear configuration appropriated to the intersection region has been achieved,⁵⁷ and ΔG* is the activation free energy. The k_et is quite large, about 1.38 × 10¹³ s⁻¹, by using the parameters estimated in the ESI.† The fast ILET rate is mainly because of the small reaction barrier and the fast nuclear vibration on the relevant mode. The barrier of ILET is extremely small, highly possible to be barrierless after thermal correction. The effective vibrational frequency (υ_eff) and the electronic factor κ_el have considerable values of 4.98 × 10¹³ s⁻¹ and 0.276, respectively. Thus, the ILET reaction can proceed rapidly on a time scale of 72 fs. As a result of the ultrafast ILET, the excited electron is dynamically delocalized over all three ligands.

Apart from ILET, the evolution of the ¹MLCT₁ state also involves other processes, such as ISC, fluorescence emission and intermolecular electron transfer, etc. Most of these processes can cause the quenching of the ¹MLCT₁ state. Although the fluorescence emission is symmetry allowed, its transition probability is quite small due to a small transition dipole moment. In several elaborately designed experimental works,^32,34,41,61 it was observed that the ISC process⁴¹ with a time constant of about 30 fs is the most competitive to ILET, and the transformation to the lowest-lying ³MLCT state is completed within a time scale of 300 fs.^32,34 Because of the ultrafast ISC together with the long lifetime of the ³MLCT state, [Ru(bpy)₃]²⁺ and its derivative complexes have wide applications in photoelectricity and photocatalysis.^62,63 Meanwhile, the mechanisms of these processes have not been fully exploited and thus thorough investigations and analyses remain necessary in the future.

Conclusions

In summary, this work explains the mechanisms of the rapid formation and transformation of the ¹MLCT₁ state during ultrafast evolution processes of photo-excited [Ru(bpy)₃]²⁺*, by using theoretical computations and wave function analysis. There are several findings that may have theoretical significance. Firstly, the PESs of the low-lying ¹MLCT states have a variety of inter-crossings, such as the three-state and two-state CIs, around the ground state equilibrium geometry. Therefore, according to Kasha's rule, the high-lying ¹MLCT_n states can quickly transfer to the lowest-lying ¹MLCT₁ state via IC processes. Secondly, the initial ¹MLCT₁ state, whether it is formed by direct photo-excitation or by ICs, can reach the three minima of the ¹MLCT₁ state rapidly by intramolecular vibrational relaxation. Thirdly, the ILET of the ¹MLCT₁ state is predominantly nonadiabatic, and it is ultrafast. These theoretical discoveries naturally lead us to an alternative interpretation that the excited electron is dynamically delocalized over all three ligands. This viewpoint of dynamical delocalization may calm down to a certain extent the long-standing controversy over whether the excited electron is localized or delocalized. In brief, the ¹MLCT₁ state plays a vital role as an intermediate state in the early evolution of the photo-excited [Ru(bpy)₃]²⁺*. From the formation to the quenching of the ¹MLCT₁ state, various interesting photochemical and photophysical processes can take place on its PES. Considering the explicit and intuitive explanations for several unusual and intriguing experimental phenomena procured by these theoretical explorations, we expect that this work would provide some valuable information for future experimental studies, such as spectral analysis and measurements of quantum yield.

Computational methods

All geometry optimizations were performed using the PBE0 hybrid functional⁶⁴ with D3 dispersion correction⁶⁵ and def2-TZVP(Ru), 6-31G*(C/H/N) basis sets.^66–69 The effective core potential (ECP) used by the def2-TZVP basis sets for Ru element takes relativistic effects into account. Frequency calculations were carried out at the same theoretical level. In addition, thermodynamic corrections were obtained from TDDFT frequency calculations.

To facilitate the analysis of the wave functions, the complete active space self-consistent field (CASSCF) and the restricted active space configuration interaction (RASCI) calculations were performed.^70,71 The CASSCF computation involves a minimum active space denoted as CASSCF(6,6) of distributing six active electrons on the six active orbitals, including three metal central dπ and three ligand π* orbitals. In RASCI, the active orbitals are partitioned into three sets, viz. in RAS1 orbitals are doubly occupied except for a maximum number of holes; in RAS2 orbitals can take all possible occupations; and in RAS3 orbitals are unoccupied except for a maximum number of electrons. Here, the RAS2 orbital subspace consists of all six active orbitals in CASSCF(6,6), while the RAS1 and RAS3 subspaces include three ligand π orbitals and two metal dσ* orbitals, respectively. Adding these subspaces together gives RASCI(12,11) of distributing twelve electrons in eleven orbitals. The maximum values of holes and electrons, respectively, in RAS1 and RAS3 were set to 2. These active orbitals are given in Fig. S5 (ESI†). All these orbitals are optimized with state-average (SA) CASSCF(6,6), involving ten roots, all of which are ¹MLCT states except for the ground state. Relative energies of the first three ¹MLCT_1–3 states obtained at the RASCI(12,11) and CASSCF(12,11) levels are presented in Table S6 of the ESI.† The results show that RASCI is an acceptable approximation to CASSCF, especially when the latter is difficult to converge in the orbital optimization.

To incorporate the dynamical electronic correlation, the multi-state second-order multiconfigurational perturbation theory calculations based on the SA-CASSCF(6,6) and the RASCI(12,11) reference functions, namely CASPT2^72,73 and RASPT2,^74,75 respectively, were performed. Table S6 of the ESI† reports also CASPT2 computations with SA-CASSCF(12,11) wave functions as a reference. The results show that the latter CASPT2 predicts the same energetic orders as RASPT2 for ¹MLCT states at the ground state geometry.

For single-point energy calculations, several methods were used, including TDDFT, CASSCF, CASPT2, RASCI, and RASPT2. In addition, slightly larger basis sets, namely def2-TZVP(Ru), 6-311G*(C/N) and 6-31G**(H),^68,69 combined with Cholesky decomposition⁷⁶ with a threshold value of 10⁻⁸, were used. The results of TDDFT with the Douglas–Kroll–Hess approach^77,78 shown that the performance of the ECP basis set is consistent with that of Sapporo-DKH3-TZP-2012(Ru)⁷⁹ all-electron basis sets.⁸⁰ All CASPT2 and RASPT2 calculations simultaneously involve four states, i.e. the ground state and the first three ¹MLCT states, and used an imaginary level-shift⁸¹ of 0.1 E_h and an ionization potential-electron affinity (IPEA)⁸² of 1 E_h.

All TDDFT calculations were carried out with the Gaussian 16 package,⁸³ while multiconfigurational calculations were performed with the Open-Molcas package.⁸⁴

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This project has been supported by the Natural Science Foundation of China (no. 22173073).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3cp03806h

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