Ultrafast X-ray spectroscopy as a probe of nonequilibrium dynamics in ruthenium complexes

Chemical Physics 407 (2012) 65–70

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Ultrafast X-ray spectroscopy as a probe of nonequilibrium dynamics in ruthenium complexes Jun Chang, A.J. Fedro, Michel van Veenendaal ⇑ Department of Physics, Northern Illinois University, De Kalb, IL 60115, USA Advanced Photon Source, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA

a r t i c l e

i n f o

Article history: Received 14 June 2012 In final form 30 August 2012 Available online 12 September 2012 Keywords: Spin crossover X-ray spectroscopy Ruthenium complexes

a b s t r a c t The ultrafast intersystem crossing in ruthenium complexes between the singlet and triplet metal-toligand charge-transfer states following photoexcitation is described. The absence of a clear decay mechanism between these states makes it difficult to explain this process within a conventional framework using rate equations based on Fermi’s golden rule. We show that the decay can be mediated by metalcentered (MC) triplet states leading to decay times of the order of several tens of femtoseconds. The calculated stable excited state probability is dominated by the 3MLCT configuration. The detailed nature of this process is clearly reflected in the calculated spectral lineshapes of the time-dependent nonequilibrium X-ray absorption spectroscopy that show a transient crystal-field collapse, dynamic broadenings, and changes in the branching ratio. We demonstrate that ultrafast X-ray spectroscopy is a suitable probe to deliver detailed new insights or discriminate between competing physical scenarios. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Photoexcitation with visible light is crucial for a wide variety of energy conversion processes. For example, in photosynthesis, the absorption of light leads to the production of oxygen from water and converts carbon dioxide into organic compounds [1]. Furthermore, solar cells also rely on the transformation of sun light into an electric current [2,3]. The fast decay processes following the photoexcitation are far from understood both experimentally and theoretically. The recent developments in pulsed X-ray sources have opened the way to obtain chemical-selective and bulk-sensitive information on the nonequilibrium dynamics following the initial excitation [4–9,39]. A typical experiment involves the measurement of the absorption of photons from an X-ray pulse exciting electrons from a core level with a binding energy of 102 –103 eV into the valence states at a delay of 10–100s of femtoseconds (fs) after the photoexcitation. Obviously, an important question is whether such a very high-energy probe can provide information on the quantum entanglement of the states involved in the decay process or if the X-ray absorption can be described by a simple linear combination of the spectra of the respective states weighted by a probability obtained from rate equations. Among the visible-light-responsive materials, Ruthenium (II) complexes have been the subject of intense study for a wide variety of applications [10–27], such as solar energy conversion, pho⇑ Corresponding author at: Department of Physics, Northern Illinois University, De Kalb, IL 60115, USA. E-mail address: [email protected] (M. van Veenendaal). 0301-0104/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chemphys.2012.08.024

todissociation, and photocatalysis. In typical ruthenium complexes, the metal ion is in a close to octahedral surrounding and the d orbitals are split into threefold-degenerate t2g and twofold-degenerate eg orbitals. Due to the large crystal field of ruthenium, the ground state is a low-spin (S ¼ 0) t62g configuration. Because optical selection rules forbid dd excitations, irradiation with visible light predominantly excites electrons from the ruthenium ion into the ligand p states, creating a t 52g L state, where L stands for an electron on the ligands. Since the transition is spin conserving, this state is a singlet metal–ligand charge-transfer (1MLCT) state, see Fig. 1. In principle, the system can relax from this state back into the ground state. Alternatively, there can be an ultrafast intersystem crossing to a 3MLCT. For example, in ½RuII ðbpyÞ3 2þ , where bpy is 2; 20 -bipyridine, this process takes about 300 fs [12,13]. Since in the triplet state, the spin on the ligands is parallel to that of the ruthenium ion, the coupling to the transition metal is strongly reduced with respect to the singlet state. The electron localizes on one of the ligand molecules [13], thereby lowering the symmetry. The triplet state has a lifetime on the order of nanoseconds [20,28]. This metastable excited state is important for efficient charge generation and separation. If the complex is adsorbed on a semiconductor, such as TiO2, the electron can be transferred through the interface and delocalize into the conduction band. The complex recovers by hole injection into a hole-transporting medium. Subsequently, the semiconductor electron and the medium hole are transported to electrodes producing an electric current [2,3]. Surprisingly, despite its obvious relevance to the optimization of energy conversion in dye-sensitized solar cells, this relaxation is not understood.

Energy

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J. Chang et al. / Chemical Physics 407 (2012) 65–70

1

MLCT

1

3

MLCT

2

1

V

3

12

V

3

MC

23

hω

Δ

12

2

3

ε

23

Δ

32

Δx

1

GS ( MC)

x (M-L) II

Fig. 1. Energy level scheme of a Ru -based complex. The system is excited with visible light of energy  hx from the singlet metal-centered (1MC) ground state (GS) into a singlet metal-to-ligand charge-transfer state (1MLCT). It then relaxes to a metastable 3MLCT state. The 3MC states are located close in energy to the MLCT states. On the right, the energy curves are shown as a function of the metal–ligand (M–L) distance. The energy differences and coupling constants are indicated in the figure.

In this paper, we investigate the ultrafast intersystem crossing dynamic process in ruthenium complexes. In the following section, we describe how the singlet to triplet MLCT transition can be mediated by the coupling to triplet metal-centered (MC) states. We show that this leads to the interesting result that the 1MLCT decay time is shorter than the 3MLCT rise time, which is also found experimentally in several studies [14,15]. We then discuss how these effects can be studied using time-dependent nonequilibrium X-ray absorption spectroscopy. We show that this decay mechanism leads to clearly identifiable features in the spectral line shape, such as transient crystal-field collapse, dynamic broadenings, and changes in the branching ratio. We end with the conclusions. 2. Ultrafast intersystem crossing Prototypical examples of intersystem crossing systems are FeII complexes, where the spin–orbit coupling causes a cascade from a low-spin to a high-spin state [18,29–33]. This is generally followed by a slower cascade of the high-spin state back into the ground state. Similar effects are observed in nickel complexes [9]. These phenomena have been explored since the sixties predominantly using optical and Mössbauer spectroscopy [18]. Recent X-ray spectroscopic measurements [9,34–36] on the relatively long-lived transient state have provided more detailed information on the decay process and the changes in the metal–ligand distance. Interestingly, it was claimed [36–40] that the quintet state is populated from the MLCT singlet via the triplet in about 150 fs. It is surprising that 3MLCT state directly relaxes to the 5T2 state bypassing the ligand field triplet states 3T1,2, which are located between the 3MLCT and 5T2. The general understanding of fast intersystem crossing phenomena requires several components. The spin–orbit coupling induces a change of the spin of the system. However since the total angular momentum is conserved, there is also a change in the orbital, for example, a conversion of t 2g into eg electrons. The increase in energy related to the t2g ! eg transition is usually offset by the gain in Hund’s rule exchange energy. At the same time an elongation of the metal–ligand distance of several tenths of an Ångstrom occurs since the change from a t 2g to an eg charge distribution leads to a stronger repulsion between the transition-metal and the ligands. The bond length change leads to a finite overlap between different vibronic levels of the two states, thereby forming a Franck–Condon continuum [41,42]. The electron relaxes into this continuum leading to an exponential decrease of the probability of finding the 1MLCT state [43,44]. Recurrences into the 1MLCT are suppressed by the oscillation damping of the vibronic mode due to the coupling to the surroundings. The explanation works

well for FeII complexes, where the iron ion cascades from the low spin (S ¼ 0) via an intermediate state (S ¼ 1) to the high-spin (S ¼ 2) state [44,45]. In ruthenium complexes, despite a larger spin–orbit coupling, this scenario is no longer valid due to the significantly larger crystal field and the smaller on-site Coulomb exchange. The spin–orbit coupling therefore mainly couples the states inside the t2g manifold. Therefore, only a minimal change in metal–ligand bond length is expected. This has been confirmed experimentally where, for the 3 MLCT state, a bond contraction of only 0.03 Åwith respect to the ground state is observed [25–27]. The expected relaxation constant [45] from state 1 (1MLCT) to state 2 (3MLCT), see Fig. 1, is c12 ¼ 2pF n V 212 =h2 x. The coupling V 12 between the singlet and triplet MLCT states is the spin–orbit coupling of ruthenium, V 12 ¼ 0:12 eV [46]; F n ¼ eg g n =n! is the Franck–Condon factor where n  D12 = hx and g ¼ e12 = hx is the Huang–Rhys factor. The energy difference D12 between the states can be derived from optical spectroscopy. Fluorescence spectra show 1MLCT emission between 500 and 575 nm and 3MLCT features around 620 nm [47]. This corresponds to an energy difference D12 ¼ 0:3 eV. The electron–phonon self-energy difference e / ðDxÞ2 is related to the change in metal–ligand distance Dx. As a result, the change in phonon self-energy e12  ðDxÞ2 should be two orders of magnitude smaller than in typical Fe-complexes. Since this is negligible with respect to D12 ; F n  0 and it is impossible to meet the condition for ultrafast decay that the energy gap is near the electron–phonon self-energy difference [45]. The lifetime is therefore expected to be much longer than picoseconds (ps). As a consequence, the system should oscillate between the singlet and triplet states via the spin– orbit interaction. On these grounds, it was already noticed [21,48] that a discussion in terms of singlet and triplet states is probably erroneous due to the large spin–orbit coupling with respect to the singlet–triplet energy gap. However, the latter theoretical picture is completely contrary to the experimentally-observed ultrafast decay between spin-labeled states with a near unit quantum yield in Ru-complexes [12–15,17,19,20,23,28]. In almost all literature, a discussion of the microscopic mechanism underlying the singlet-to-triplet MLCT decay is generally avoided. The above dilemma can be solved by considering additional states that will give rise to vibronic excitations necessary to dissipate the energy in the decay process. From experiments [21,22,24,47,49], it is known that 1,3MLCT states (labelled states 1 and 2, respectively) are close in energy to 3MC (state 3) states with a t52g eg configuration, see Fig. 1. There is no direct coupling V 13 between the 1MLCT and 3MC either by the spin–orbit coupling or electron hopping. In pure octahedral symmetry, there is also no coupling V 23 between the 3MLCT and the 3MC, since the eg states would not couple to the ligand p states. However, since most Ru-complexes have a lower symmetry [50,51], a small hybridization between the two triplet states should be present depending on the amount of distortion. The weak hopping between the ligands p or p and metal ion’s eg orbitals is of the order of 0.1 eV [52,53]. Here, we take a coupling of V 23 ¼ 0:08 eV. The difference in metal–ligand bond length between the metal-centered and MLCT states is several tenths of an Å [51], corresponding to a change of electron–phonon self-energy of the order of several hundred meV’s [36,54] and we take e13 ¼ e23 ¼ 0:4 eV. Each of the states is coupled to the stretching mode with energy  hx. Although there are many phonon modes in real systems, our model with a single system phonon mode can be approximately mapped from a general spin-boson model for open quantum systems [55–57]. We take the typical Ru-ligand stretching mode [58] of h  x ffi 30 meV. In addition, the characteristic mode is damped due to the coupling to the surroundings. We include a damping C of the internal stretching mode due to the solvent reorganization [59]. Although the typical intramolecular vibrational relaxation

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J. Chang et al. / Chemical Physics 407 (2012) 65–70

X   Ei ni þ hxay a þ ki ni ay þ a ;

ð1Þ

i

where ni gives the occupation of state i and ay is the step operator for the vibrational mode. This Hamiltonian can be diagonalized with P pffiffiffiffi a unitary transformation Hs ¼ eS Hs eS , with S ¼ i ni g i ðay  aÞ and g i ¼ ei = hx with ei ¼ k2i =ð hxÞ, giving,

Hs ¼

X ðEi  ei Þni þ hxay a:

ð2Þ

i

with eigenstates jwin i for states i and n the occupation of the phonon mode. We define the energy difference between the states after   diagonalization Dij ¼ ðEi  ei Þ  Ej  ej . In addition, since a spatial translation of the coordinates shifts all the ki by a constant in Hs , only the relative change in coupling is of importance. Therefore, it is useful to define the electron–phonon self-energy difference  eij ¼ ki  kj 2 =ðhxÞ between two different states (as shown in right panel of Fig. 1). An interaction coupling different states that cannot be diagonalized with the electron–phonon term is the spin–orbit coupling between different spin states or hopping between same spin states,

X HI ¼ V ij ðcyi cj þ h:c:Þ;

dPin ðtÞ ¼ 2nCP in ðtÞ þ 2ðn þ 1ÞCPi;nþ1 ; dt

1

ih

djwðtÞi ¼ ðH0 þ iDÞjwðtÞi; dt

in

dt

c

0.05 −0.15

Δ =−0.15 eV 32

0.5

P (t)

1

MLCT

P (t)

3

MLCT

P (t)

3

MC

2 3

i

P (t)

0 1

b

Δ =0.05 eV 32

0.5

0 1

a Δ =0.35 eV 32

0

P where Pk ðtÞ ¼ jck ðtÞj2 with jwðtÞi ¼ k ck ðtÞjwk i. The dissipation does not necessarily have to be diagonal. The eigenvectors jwin i of Hs are selected as the basis. The vibrational cooling by the bath can be taken into account by the dissipative Schrödinger equation in Eq. (4) with H0 substituted by Hs þ HI and HI ¼ eS HI eS . We still need the detailed time evolution formula for P in ðtÞ. For state i with occupation number n of the phonon

MLCT

1

0

2

3

Δ = 32 0.35 eV

MC

0.35 eV

0 1

ð3Þ

ð5Þ

MLCT

0.05

where H0 is the Hamiltonian of the system and D describes the effective environmental dissipation given by

jwin ihwin j:

3

1

0.5

0.5

D¼

phonon

−0.15

ð4Þ

h X d ln Pin ðtÞ

T

d

ij

where V ij is the coupling constant and cyi cj causes a particle-conserving transition between states j and i. This ‘‘local’’ system is considered part of a larger system such as a molecule in solution or a solid. The latter constitute the effective surroundings that can dissipate energy from the local system. In order to obtain an irreversible decay between the singlet and triplet MLCT states, it is essential to include the damping of the oscillation of the local vibronic mode. This damping is a result of intramolecular energy redistribution due to the coupling to environment or bath. In recent work [44,45], we developed a dissipative Schrödinger equation for the system wavefunction jwðtÞi that essentially adds a dissipative term to the Hamiltonian given by

ð6Þ

 V 2 = where P in ðtÞ ¼ jhwin jwðtÞij2 , C ¼ pq h is the relaxation constant,  is the effective environmental phonon density of states where q and V is the interaction between the local system and the environment. Since fast intersystem crossings are experimentally known to be almost temperature independent, we take temperature T ¼ 0. The decay path depends strongly on the relative energy position of the 3MC with respect to the MLCT states. Let us first consider the situation where the 3MC is higher in energy than the MLCT states, see Fig. 2a. In this case, the 3MC does not play a role in the coupling between the singlet and triplet MLCT states. We observe an oscillation in the probability of finding the MLCT due to the spin–orbit coupling. This situation dramatically changes when the 3MC state is located between the MLCT states, see Fig. 2b with D32 ¼ 0:05 eV (D31 ¼ 0:25 eV). Ultrafast decay occurs, and after about 300 fs the average populations of the system are becoming stable. The population of the 3MLCT state increases to about 85% at 600 fs. Fitting the curves using kinetic rate equations [45],

0

Hs ¼

mode, on the one hand, the vibrational coupling to the surroundings relaxes a state with n phonons to a n  1 phonon state by the emission of phonons. On the other hand, the probability of the n-phonon state increases due to the decay of the state with n þ 1 phonons [44,61]. This gives a change in the probability of the n-phonon state

x/x

time is on the picosecond time scale, in ultrafast dye intermolecular electron transfer processes, the damping time ð2CÞ1 can be less than 100 fs [60]. Note that the decay time of a particular state is not directly related to the phonon damping C. However the role of dissipation is to prevent a recurrence of probability into the original photoexcited state [44]. The 1MLCT nonradiative decay time can be as low as 40 fs [14,15] and no recurrences of the initial photoexcited state are observed. This implies that the vibronic damping time is of the same order. Our numerical calculations are not sensitive to a change in ð2CÞ1 from 20 fs to 60 fs. In the following calculations, we take ð2CÞ1 ¼ 40 fs. In describing the singlet to triplet MLCT decay in the [RuII(bpy)3]2+ complexes, we consider 1,3MLCT and 3MC three excited states couple to the vibrational/phonon mode of the surrounding ligands, via the Hamiltonian

100

200

300

400

500

600

t (fs) Fig. 2. Time evolution of the photoexcited state probabilities and metal–ligand distances. The populations of the 1MLCT (P 1 ðtÞ, blue), the 3MLCT (P 2 ðtÞ, red), and the 3 MC (P 3 ðtÞ, black) as a function of time for different values of the energy difference D32 between the 3MC and 3MLCT states, see inset in (d). (a) D32 ¼ 0:35 eV; (b) D32 ¼ 0:05 eV; (c) D32 ¼ 0:15 eV; (d) The bond length change with respect to the ground state x normalized to the bond elongation of the 3MC state (x0 ¼ 0:4 Å) [51]. The oscillation period of the vibronic mode is T phonon ¼138 fs. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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J. Chang et al. / Chemical Physics 407 (2012) 65–70

60 fs for 1MLCT decay and 160 fs for 3MLCT rise time constants are obtained. We have also been able to find shorter relaxation times when the value of e12  D12 . 45 fs and 120 fs fitting values are more close to experimental 30–40 fs and 110 fs [14,15]. It is worthy to note that the 1MLCT decay time is less than the 3MLCT population time due to the involvement of the 3MC state. Although, at first sight, the time dependence of the probabilities resembles a cascading from state 1 (1MLCT) to 2 (3MLCT) via state 3 (3MC), it is important to remember that in the model there is no direct coupling between state 1 and 3 (V 13 ¼ 0). Therefore, in a classical rate equa2 tion, the expected decay time T 13 ¼  h x=ð2pF n V 213 Þ ! 1. A detailed analysis of the evolution of the state occupations with time shows that the initial 1MLCT state couples to the lowest vibrational level of the 3MLCT via the spin–orbit coupling. Minor distortions from octahedral symmetry allow a hopping V 23 of the electron from the ligand p orbitals into the ruthenium eg orbital, populating the 3MC state. Due to the large change in electron– phonon self-energy e23 , the lowest vibrational level of the 3MLCT couples to the Franck–Condon continuum of 3MC states like the Fano effects [44]. On the other hand, the hopping to the lowest vibrational 3MC state is strongly reduced due to the Franck– Condon factors. Therefore, the hopping between the 3MLCT and 3 MC does not conserve phonons. This allows the system to decrease its energy via vibrational cooling due to intramolecular energy redistributions. The probability in the 3MC vibrational levels can however also couple back again to the Franck–Condon continuum of the 3MLCT, again leading to the creation of excited phonon states that are subject to vibrational cooling. Therefore, the interplay between the 3MLCT and the 3MC state leading to the creation of excited phonons allows the local system to dissipate energy via the damping of the Ru-ligand bond-length oscillations. The consequence is a reduction of the recurrence of the 1MLCT state. In the relaxed state ( J 500 fs), the contribution of the 3MC is small after reaching a maximum of close to 50% around 100 fs. The contribution of the 1MLCT is of the order of 10%, justifying the traditional designation of the final state of the decay in Rucomplexes as a triplet. In addition, let us consider the case that the 3MC state is lower in energy than the 3MLCT state. For D32 ¼ 0:15 eV, we observe a cascading process from the 1MLCT via the 3MLCT to the 3MC state, see Fig. 2c. The 3MC reaches an occupation of approximately 92% at about 600 fs. In a real system, the 3MC state will decay back to the ground state (not included in the calculation). The involvement of the 3MC is clearly reflected in the evolution of the metal–ligand bond length as a function of time, see Fig. 2d. The calculated bond length elongation is strongly related to the probability of state 3, see Fig. 2a–c. For the situation, where the 3 MC state is located between the MLCT state (D32 ¼ 0:05 eV), the bond length initially increases and reaches a maximum change of about 0.16 Åaround 100 fs. Subsequently, the length decreases as the occupation of the 3MLCT state increases. At about 600 fs, the average bond elongation is only 0.024 Ålarger with respect to the MLCT states. This agrees with X-ray absorption near-edge structure measurements on the transient metastable state [25] taken 70 ps after the photo-excitation that show a minor bond length change with respect to measurements in the absence of a photoexcitation. This rules out a strong contribution to the metastable state of the 3 MC, which is expected to have a significantly larger bond length due to the presence of electron density in the eg orbitals. This behavior should be contrasted with the situation where the 3MC state is lower in energy than the 3MLCT (D32 ¼ 0:15 eV). Here, the bond length increases by several tenths of an Å, as the 3MC is occupied. On the other hand, if the 3MC state is higher in energy than that of the 1MLCT (D32 ¼ 0:35 eV), the bond length is also almost the same as that of the MLCT states.

3. Dynamic X-ray absorption spectra We now address the question whether dynamic X-ray absorption is sensitive enough to test the proposed mechanism. Fig. 3a and b show the evolution of the X-ray spectral line shape as it goes through the singlet to triplet MLCT decay. For calculational details, see the Methods section. Since not all ruthenium ions will be excited by the laser pulse, one can only measure the difference with respect to the ground state spectrum, see Fig. 3c. The spectral line shape of Ru-complexes is relatively simple. There are two clearly separated edges as a result of the large 2p core-hole spin–orbit coupling. For the spectrum starting from the ground state, each edge is almost a single line consisting of excitations into the empty eg states, see Fig. 3c. Some small intensity at 3–5 eV above the edge 6 is observed. This results from the coupling between the 2p5 4d and 5 2p5 4d L configurations. After photoexcitation, an additional peak is observed below the main line related to the hole density in the t 2g states resulting from the transfer of an electron from the ruthenium ion to the ligands. The main line also broadens due to the presence of splittings related to the higher-order terms in the 2p–4d and 4d–4d Coulomb interaction. The spectrum of the 3MLCT in the long-lived transient state has been measured earlier [25] and serves as a boundary condition for our calculations. The focus here is the study of the dynamic X-ray spectral line shape, i.e. the changes that occur during the decay from the photoexcited state into the transient state. We consider the situation where the 3MC state is located between the MLCT states and study whether the X-ray spectra provide information on the mechanism underlying the decay from the singlet to triplet MLCT. A very clear change is observed in the low-energy side feature around 2836– 2838 eV, about 2–4 eV below the main line. There is a significant shift towards higher energy in the first 150 fs after the photoexcitation. This is due to the strong dependence of the crystal field [ð10DqÞ0 ¼ ðr0 =r0 Þ5 10Dq] and the p hybridization [V 023 ¼ ðr 0 =r 0 Þ3:5 V 23 ] on the elongation of the metal-to-ligand distance r with respect to its ground-state value r 0 . This, in combination with the presence of spectral weight from the 3MC states, leads to a crystal-field ‘‘collapse’’ around 150–200 fs. Although, the 3MC contribution becomes significantly smaller for t > 200 fs, the energy separation between the low-energy feature and the main line never fully recovers its value at t ¼ 0. This is indicative of the presence of the 3MLCT state. It is important to realize that the total crystal field has two components: the point charge contribution 10Dq, which almost returns to its original value when the 3MC population decreases, and an additional part due to the p hybridization V 23 . The latter component disappears in the 3MLCT state, since the electron on the ligands no longer couples to the ruthenium due to their parallel spins. Additional information is obtained from the main line. First, the main line shows a significant broadening, see Fig. 3c, due to the fact that more Coulomb multiplets can be accessed from the MLCT states compared to the ground state and from the presence of the 3MC state in the nonequilibrium dynamics. For t < 50 fs, this causes increased intensity at the low- and high-energy side of the main line in combination with a steep drop in intensity at the main line. We also notice that, around 100 fs, the main line intensity of the excited states moves below the ground state main line, leading to increased intensity around 2839 eV in Fig. 3a and b. This is the region with the largest metal–ligand elongation, see Fig. 2d, and therefore the smallest crystal field. When the crystal field recovers for t > 150 fs, this intensity shifts towards higher energies and reappears above the main line around 2841–2843 eV. Another spectral feature that is indicative of the involvement of the spin–orbit coupling in the decay process is the change in the intensity ratio of the L3 and L2 edges, also known as the branching

69

a

c

300

250

Intensity (arb.un.)

J. Chang et al. / Chemical Physics 407 (2012) 65–70

time (fs)

200

L

2835

L

before excitation t=160 fs t=280 fs

3

2840

b

2845 2970 Energy (eV)

2975

2

2980

150 0 50

100

100 150

50

200

2830

250

Energy (eV) 0

2845

2835

2840

300

time (fs)

2845 2840

Energy (eV)

Fig. 3. Time-dependence of the X-ray absorption spectral line shape of photoexcited RuII-based complexes. (a), (b) The difference between the nonequilbrium X-ray absorption spectrum at time t after the photoexcitation and the spectrum starting from the ground state at the Ru L3 edge. The same color scheme is used. (c) The XAS spectra at the L3 and L2 edges before and at t ¼ 160 and 280 fs after the photoexcitation.

ratio. For the ground-state spectrum, see Fig. 3c, this ratio is 2, which is close to the statistical value related to the degeneracies of 4:2 for the j ¼ 32 ; 12 values of the 2p level. However, the branching ratio BR ¼ IL3 =IL2 is very sensitive to changes in the expectation value of the angular part of the spin–orbit coupling hL  Si of the initial state [62]. When defining r ¼ hL  Si=hnh i, where hnh i is the number of holes on the ruthenium ion, the branching ratio is given by BR ¼ ð2 þ rÞ=ð1  rÞ. Since r ffi 0 in the ground state, the branching ratio is close to statistical. However, when an electron is transferred from the ruthenium to the ligands, the spin–orbit coupling can couple the different t2g states. The coupling between the singlet and triplet MLCT states is a direct result of the alignment of the spin and orbital by the spin–orbit coupling. However, when the spin of the ruthenium ion is flipped, it directly affects the angular momentum, since the total moment has to be conserved. This increases hL  Si, which manifests itself directly in an increase of the branching ratio. We calculate an increase in the ratio from 2 in the ground state to 3 at 300 fs after the photoexcitation. This is larger than the experimentally observed value in the metastable state [28,25]. However, these experiments were performed 50 ps after the photoexcitation, and spin–orbit coupling effects can be further quenched by additional interactions and delocalization effects. In the simulation of the X-ray absorption spectral line shape, we take into account the detailed nature of the states by calculating them for a cluster with a ruthenium ion octahedrally surrounded 6 by ligands. For the ground state, we take into account 4d and 5 4d L configurations, where L stands for an electron in the p li7 6 gands. The final state is given by 2p4d and 2p4d L configurations, where 2p is the 2p core hole. The Hamiltonian [63] includes the 4d and 2p spin–orbit coupling, an octahedral crystal field of 3.8 eV in the ground state, and multiplet Coulomb interaction between 4d electrons and between the 4d electrons and the 2p core hole. Coulomb and spin–orbit coupling parameters are calculated within the Hartree–Fock limit. With an energy difference between the 6 5 lowest 4d and 4d L configurations of 1 eV and a p hybridization of 0.25 eV, an energy difference of 2.6 eV between the ground state and the 1MLCT is obtained. At every time step, the ground state is adjusted so that the proper mixture of 1MLCT, 3MLCT, and 3MC states is obtained. The hybridization and crystal field are adjusted to account for differences in metal–ligand distances, see the main

text. The spectrum is calculated in the dipole limit using Fermi’s golden rule. It is clear that the time-dependent lineshapes cannot be simply described by a linear superposition of the absorption spectra for different states involved in the decay. This shows that optical pump/X-ray probe spectroscopy not only measures the population densities, but is also sensitive to the entanglement of the different states.

4. Discussion Understanding the singlet to triplet MLCT decay in ruthenium complexes is of importance because of its role in the optimization of, for example, dye-sensitized solar cells [2,3], since the transfer of electrons from the ruthenium complex into the semiconductor depends strongly on the singlet to triplet MLCT decay time [64]. Adjusting the energy of the 3MC state between the singlet and triplet MLCT can effectively improve the injection efficiency. Several aspects provide interesting experimental tests of the described mechanism. First, although the metastable state reached after several hundreds of fs shows little change in the metal–ligand bond length, there is an increase in bond length of the order of 0.15– 0.2 Å around 100 fs following the photoexcitation. The observation of this dynamic response is in the reach of state-of-the-art optical pump/X-ray probe techniques [36]. Second, the coupling between the 3MLCT and 3MC states is a result of finite hopping matrix elements due to distortions that break the octahedral symmetry. The dependence of the decay time in different geometries can provide additional information on the detailed nature of the decay mechanism. The entanglement of the different states is reflected in dynamic X-ray absorption spectroscopy using pulsed X-ray sources. Clear signatures of the underlying decay process manifest themselves in the calculated dynamic X-ray absorption spectra through strong changes in crystal-field parameters and branching ratios. This proves the viability of performing time-dependent X-ray absorption experiments using short-pulsed X-ray sources. In conclusion, we have demonstrated that the solution phase decay from singlet to triplet MLCT states in RuII complexes can be mediated by the presence of 3MC states. The dependence of the mechanism on the distortion from octahedral symmetry and the energy

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