Vibronic Theory of Ultrafast Intersystem Crossing Dynamics in a Single

Letter pubs.acs.org/JPCL

Vibronic Theory of Ultrafast Intersystem Crossing Dynamics in a Single Spin-Crossover Molecule at Finite Temperature beyond the Born−Oppenheimer Approximation Nikolay Klinduhov†,‡ and Kamel Boukheddaden*,† †

Groupe d’Etude de la Matière Condensée, Université de Versailles, CNRS UMR 8635, 45 Avenue des Etats-Unis, 78035 Versailles Cedex, France ‡ Institute of Technical Acoustics, National Academy of Sciences of Belarus, 13 Lyudnikova st, 210023 Vitebsk, Belarus S Supporting Information *

ABSTRACT: Quantum density matrix theory is carried out to study the ultrafast dynamics of the photoinduced state in a spin-crossover (SC) molecule interacting with a heat bath. The investigations are realized at finite temperature and beyond the usual Born−Oppenheimer (BO) approach. We found that the SC molecule experiences in the photoexcited state (PES) a huge internal pressure, estimated at several gigapascals, partly released in an “explosive” way within ∼100 fs, causing large bond length oscillations, which dampen in the picosecond time scale because of internal conversion processes. During this regime, the BO approximation is not valid. Depending on the tunneling strength, the ultrafast relaxation may proceed through the thermodynamic metastable high-spin state or prevent it. Interestingly, we demonstrate that final relaxation toward the low-spin state always follows a local equilibrium pathway, where the BO approach is valid. Our formulation reconciles the nonequilibrium and the equilibrium properties of this fascinating phenomenon and opens the way to quantum studies on cluster molecules.

P

(change in the metal−ligand distances and angular distortions of the ligand, essentially) accompanying the electronic relaxation create an elastic energy barrier which stabilizes the photoinduced HS metastable state at low temperature. This problem benefited recently of nice time-resolved14−17 femtosecond X-ray absorption, combined with optical spectroscopy measurements, which revealed that the photoswitching involves first molecular breathing vibrations of the ligand which dampen rapidly to the benefit of molecular bending vibrations. Theoretical investigations18,19 have also considered the quantum description of this phenomenon at very low temperatures within the BO approximation. Here, we provide a detailed study of the quantum dynamics of the multichannel relaxation of the photoinduced state, described in the frame of vibronic theory,20 beyond the BO approximation, whose validity is revisited when going from nonequilibrium properties to asymptotic thermodynamic equilibrium. We restrict our study to the isolated molecule in contact with a heat bath. One may legitimately ask about the benefit of going beyond the BO approximation in the description of the quantum dynamics of a vibronic system. First, it is important to recall that the BO approximation does not allow the accurate reproduction of the

ulsed ultrafast photoinduced processes are observed in a wide range of physical phenomena both for gases and at the condensed phase. Among them, one can quote (not an exhaustive list) the photoinduced ultrafast electron-transfer,1,2 femtosecond photomagnetism,3,4 ultrafast photochemical reactions,5,6 etc. Some of them are single-site processes, whereas others belong to cooperative ultrafast phenomena also called domino effect,7 such as photoinduced structural phase transitions.8 Fe(II)-based spin-crossover (SC) materials converting thermally from the diamagnetic low-spin (LS, S = 0, t62ge0g) to the paramagnetic high-spin (HS, S = 2, t42ge2g) states are model systems of photoswitching9−12 between these two states. A crucial issue for this work is the preparation of the initial excited state of the molecule. According to experimental data, the very short (femtoseconds) pump pulse promotes the spin-crossover molecule from the ground LS state into an excited metal-toligand charge-transfer (quintet MLCT) state via a Franck− Condon transition, from which the system rapidly decays in a nonadiabatic way through multiple excited-state channels (including ligand-field states)13 coupled with coherent vibrational motions and internal conversion. During this complex process, electronic and vibrational states are strongly interlinked, and in most of the cases, BO approximation fails. We simulate here this complex mechanism with the simple Franck−Condon transition in a double-well potential (see Figure 1). The local structural molecular reorganization © 2016 American Chemical Society

Received: January 4, 2016 Accepted: February 2, 2016 Published: February 2, 2016 722

DOI: 10.1021/acs.jpclett.6b00014 J. Phys. Chem. Lett. 2016, 7, 722−727

Letter

The Journal of Physical Chemistry Letters

Figure 1. (a) Adiabatic potential energies versus the molecular deformation with the exact eigenvalues (dotted lines) and their corresponding nuclear probability densities at T = 3 K for J = 20 meV. The vertical arrow indicates the schematic Franck−Condon process. (b) Thermal variation of the average magnetization, ⟨σ⟩, showing a gradual spin-crossover transition.

eigenvalues, denoted Ei, are displayed in Figure 1a (dotted lines). The associated vibronic eigenstates, which have the form

excited eigenvalues and eigenstates of a quantum system when there is an active vibronic coupling between these states, as for SC systems. This implies that a change in the nuclei configuration from Q to Q + ΔQ (Q is a configuration coordinate) causes electronic transitions. Depending on the type of treatment (BO or exact quantum resolution), the system will explore a different pathway in the space coordinates, leading to different internal dynamics including in the intersystem crossing region. Thus, strong vibronic interaction and/or strong tunneling mixing lead to the breakdown of the BO approach. Here, we refer the reader to relevant reviews treating the problem of the dynamics of the photoinduced cis− trans transition21 beyond the BO limit and to useful fundamental works based on the application of stochastic Monte Carlo methods for solving nonadiabatic quantum correlated systems at finite temperature.22,23 In the present work, the SC molecule is described as a vibronic oscillator whose Hamiltonian is24−26 Hmol =

kQ 2 ω 2P 2 + + ( Δ − SQ ) σ z + Jσ x 2k 2

|φi⟩ =

∑ (cniLS|χnLS ⟩|LS⟩ + cniHS|χnHS ⟩|HS⟩)

(2)

n

lead to the corresponding computed (see Supporting Information) nuclear probability densities depicted in Figure 1a. This allows us to derive (see Supporting Information) the thermal dependence of the fictitious magnetization, ⟨σ⟩, displayed in Figure 1b for J = 20 meV, showing a gradual 2δ spin conversion at the transition temperature, Teq = k ln g ∼ B

120 K. The thermal dependence of ⟨Q⟩ is given in Figure S1a of the Supporting Information. To help the reader to obtain a comprehensive view and deeper insights into the quantum resolution of this problem, we provide, in the Supporting Information, the BO solution of Hamiltonian 1 that we compare in Figure S1b to the exact quantum resolution. Next, we define the reservoir (thermal bath) as a set of independent harmonic oscillators, whose Hamiltonian is

(1)

Hph =

∑ ℏwqbq+bq (3)

q

where ω2p2/2k is the kinetic energy, Q the molecular distortion, 1 and k the harmonic elastic constant; Δ = δ − 2 kBT ln g is the effective ligand field energy splitting, accounting for the energy gap, δ, and entropic contribution of the HS/LS degeneracy ratio, g.27,28 The parameter S is the vibronic coupling at the S LS = −k origin of the existence of two equilibrium positions, Q eq

where bq and b+q are the usual annihilation and creation phonon operators and wq is the frequency of the qth normal mode. Restricting ourselves to the bilinear contributions in different coordinates, the molecule−bath Hamiltonian is Hmol−ph =

∑ (gqbq+a + gq*bqa+)

(4)

q

S

HS and Q eq = k , for the bond length distances (see Figure 1a), and J is the tunneling contribution originating from high-order spin−orbit coupling. σz and σx are the usual Pauli matrices of a two-states fictitious spin, with eigenvalues (respectively eigenstates) for σz, −1 and +1 (respectively, |LS⟩ and |HS⟩). The calculations are performed using realistic parameter values: LS k = 7 eV/Å2 and S = 0.8eV/Å leading to ΔQeq = QHS eq − Qeq ∼ 0.2 Å, δ ≅ 20 meV (∼200 K), ln g = 4.6, J ∼ 10−30 meV, and ℏω ∼ 30 meV. Hamiltonian 1, whose matrix elements are given in section 1 of the Supporting Information, is solved exactly beyond the BO HS approximation in the 2 × ∞ basis, (|χLS n (Q)⟩⊗|LS⟩, |χn (Q)⟩⊗| LS HS HS⟩), where |χn (Q)⟩ and |χn (Q)⟩ are the respective wave functions of the LS and HS displaced harmonic oscillators. The

where (a + a+) ∼ Q. The Hamiltonian Hmol−ph is rewritten in the diagonal basis (eq 2) of the isolated molecule, and treating the molecular−bath interaction as a perturbation, we could apply the usual Redfield theory for the density matrix.29,30 The density matrix of the system, ρ (already traced over bath q variables), is written in the interaction representation ⎛ iH t ⎞ ⎛ iH t ⎞ ρI (t ) = exp⎜ 0 ⎟ ρ(t )exp⎜ − 0 ⎟ ⎝ ℏ ⎠ ⎝ ℏ ⎠ ⎛ it = exp⎜⎜ ⎝ℏ

⎛ ⎞ ⎟ ∑ Ej|φj⟩⟨φj|⎟ ρ(t )exp⎜⎜− it j ⎝ ℏ ⎠

⎞ ∑ Ej|φj⟩⟨φj|⎟⎟ j ⎠ (5)

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Figure 2. Time dependence of the quantum averages of molecular distortion, ⟨Q⟩ (left panel), and fictitious spin, ⟨σ⟩ (middle panel), for T = 3 K for two values of the tunneling constant: (a) J = 2 meV and (b) J = 20 meV. The horizontal blue lines plotted in the ⟨Q⟩ and ⟨σ⟩ panels are the thermal equilibrium values. The insets sketch the enlarged views of the dynamics at short time scale. Right panel: corresponding phase portraits ⟨σ⟩ versus ⟨Q⟩ showing the behavior of the system in its phase space.

the excited HS state.16,32 This fast first process is omitted in this work. Concretely, the initial excited state, |Ψ0⟩, is built from the vibronic ground state, |φ0⟩ (see eq 2), as

leading after mathematical developments (detailed in section 2 of the Supporting Information) to the system of differential equations for the molecular density matrix elements31 dρmm dt dρmj dt

=

∑ (γmkρkk

|Ψ0⟩ =

− γkmρmm )

n

k≠i

= = −(iωmj + Γmj)ρmj ,

in which, ωmk =

Em − Ek ℏ

m≠j

∑ (cnLS0 |χnLS ⟩ + cnHS0 |χnHS ⟩)|HS⟩ ∑ (∑ cniHScmLS0S(m , n) + cniHScnHS0 )|φi⟩ in

(6)

m

where S(n,m) are the usual Franck−Condon integrals, given by HS S(n,m) = ⟨χLS n |χm ⟩. At very low temperature, the initial density matrix is written as ρ0 = |Ψ0⟩⟨Ψ0|, and its expression at finite temperature is given in section 4 of the Supporting Information. We have solved the set of nonlinear differential equations of motion (eq 6) for γ ∼ 1.5 ps−1, using Euler’s method (see section 5 of the Supporting Information for the technical details) for different tunneling interaction, J, in the low- and relatively high-temperature regimes. At each time, we compute the quantum averages of ⟨σ⟩ and ⟨Q⟩ and that of the elastic energy. Figure 2 illustrates the case of the lowtemperature limit, corresponding to kBT ≪ ℏω. Because of the initial density matrix, the system starts at point A, (⟨Q⟩ ∼ QLS eq ≈ − 0.1 Å and ⟨σ⟩ ∼ +1), from which it evolves, in a very short period (less than 100 fs), toward point B, expanding suddenly so as to exceed the equilibrium value of the HS distortion (QHS eq ≈ 0.1 Å), while the electronic state, (i.e., ⟨σ⟩ value), experiences a very weak change of less than 1%

(Em, is the mth eigenvalue of 1

Hamiltonian 1) and Γmj = 2 ∑k (γkm + γkj). The expressions of γmk, the damping factors, (detailed in section 3 of the Supporting Information) depend on the bath spectral function and are given by γmk = γ[akm2N (ωmk) + amk2(N (ωkm) + 1)], in which γ is the relaxation (damping) rate part and 1 N(ω) = ℏω/kBT . e −1 A crucial issue of this work is the assumption that the optical pulse duration (some femtoseconds) is short compared to the time scales of vibrations (picoseconds), which supports the vertical transition picture of Figure 1a. Moreover, the Franck− Condon transition is considered here between the electronic LS and HS states of the SC molecule, while in real cases it occurs between the ground state and the metal-to-ligand chargetransfer state (both LS), from which it decays very quickly to 724


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Figure 3. Time dependence of the internal pressure, P (in GPa), for T = 3 K for two values of the tunneling constant: J = 2 meV (left panel) and J = 20 meV (right panel).

Figure 4. Blue curves: instantaneous elastic energy versus the instantaneous average displacement ⟨Q⟩ for T = 3 K showing the relaxation pathway in the adiabatic potential energy diagrams (black curves) for different values of the tunneling term, J (from left to right): 2, 20, and 50 meV.

(respectively 25%) for J = 2 meV (respectively J = 20 meV). The case J = 50 meV is presented in Figure S2. This first event is shorter for stronger tunneling constant J because it involves the strength of the coupling between the two electronic states. Next, between B and C, coherent vibrational motions take place because of internal energy transfer between the electronic and the vibrational degrees of freedom so as to reach the point C, located in the intersystem crossing region (see Figure 3). In this regime, whose duration is around 1 ps, the molecular vibrations dampen as a result of the coupling with the heat bath. This coupling, which was silent in the first regime, is activated by the interference between the vibrations and thermal exchange rate time windows. From point C, both “order parameters” ⟨Q⟩ and ⟨σz⟩ decay on time following stretched exponentials toward their thermal equilibrium values (blue horizontal line). The phase portraits of Figure 2, representing the system’s behavior in its phase space, also reveal the above-mentioned three dynamical regimes: (i) a very rapid molecular transition to the HS state for ⟨Q⟩ between A to B followed by (ii) a nonlinear dependence of ⟨σz⟩ versus ⟨Q⟩ between B and C. Here, ⟨Q⟩ oscillates with damping, while ⟨σz⟩ may oscillate or not, depending on the strength of the tunneling term. Finally from C to D, a linear dependence is obtained between ⟨σz⟩(t) and ⟨Q⟩(t), interpreted as being due to the occurrence of the BO regime where both quantities track

each other. This overall multiscale relaxation process can be approached through the time dependence of the initial internal stress (internal pressure) generated by the Franck−Condon process on the bond length, arising from the frustration caused by the incompatibility between the initial electronic state (HS, ⟨σz⟩ ∼ 1) and the vibrational state (⟨Q⟩ = QLS eq ). For a weak tunneling, the instantaneous stress is estimated as,

(

S

)

F = −k ⟨Q ⟩ − k ⟨σ z⟩

and vanishes at equilibrium when

S ⟨σ z⟩ k

⟨Q ⟩ = (see Figure 1a). Assuming the SC molecule is a hard sphere, its radius changes as R(t) = R0 + ⟨Q⟩(t), where R0 (= 2 Å) is the average radius between the LS and the HS states. F The internal pressure, evaluated as P(t ) = 4πR2 (given in

Figure 3 for J = 2 and 20 meV), reaches nominal values of 6−8 GPa in the first regime. This value is in excellent agreement with experimental findings of Brillouin scattering, atomic force microscopy, and spectroscopic ellipsometry studies.33−35 Interestingly, when the system enters the BO regime in point C, the stress vanishes (i.e., P(t) ∼ 0), whatever the value of the tunneling coupling, J, as displayed in Figure 3. To better illustrate the intersystem crossing and its intimate relation with the strength of the tunneling coupling, we computed the quantum average of the total potential energy, ⟨Epot⟩(t) during the low-temperature relaxation process, drawn 725



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as a phase portrait, ⟨Epot⟩(t) versus ⟨Q⟩(t), on the energy potential diagrams (compare Figure 4). The corresponding high-temperature case is presented in Figures S3 and S4. We see clearly from these figures that the previously admitted36−38 idea that the relaxation path goes through the bottom of the HS state as a long-lived metastable state is true only for weak tunneling coupling and low temperature, where the BO regime takes place in a short time scale. Indeed, for J = 2 meV, the relaxation process in the Epot−Q portrait shows coherent nuclear motions around a fixed average value of ⟨Q⟩ = QHS eq . Although it is not shown here, the system ends its excursion in the LS state after very long time simulations. For J = 20 and 50 meV, the onset of the BO regime is longer because the molecular deformation and the spin state are hardly separated (see Figure S2) and the BO description fails. As a result, the system skips the pure HS state during its evolution. In summary, we proposed a vibronic microscopic description for the nonadiabatic relaxation of the photoexcited spincrossover molecule in contact with a heat bath that we solved exactly in the full quantum treatment, beyond the usually admitted BO approximation. We performed this study as a function of temperature and tunneling mixing, which plays a key role in the validity of the adiabatic description. We found in the low-temperature region the existence of three relaxation regimes during the intersystem crossing process, which are governed by the strength of the tunneling coupling. Interesting extensions of the present model relate with the existence of two active deformation coordinates in the course of the relaxation process,14 whose interplay will open new relaxation channels in a higher-dimension phase space and the consideration of a high-energy MLCT state in the PES process. The case of two or more interacting molecules is also a challenging problem in view of the study of photoinduced phase transitions and collective domino effect. These extensions can be carried out using new promising methods enabling the study of nonadiabatic effects at finite temperature based on quantum Monte Carlo calculations at the variational level.22,23 These stochastic techniques are indeed more appropriate to the large Hilbert space arising from the study of cooperative systems.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.6b00014. Additional computational methods and data (PDF)

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Letter

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by Université de Versailles SaintQuentin, Université Paris-Saclay, Centre National de la Recherche Scientifique and the “Agence Nationale de la Recherche” (ANR Project BISTA-MAT: ANR-12-BS07-003001). Financial support was provided by CNRS, Versailles University, and ANR-BISTAMAT program (ANR-12-BS070030). N.K. thanks Versailles University and CNRS for financial support as an invited researcher. 726


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Vibronic Theory of Ultrafast Intersystem Crossing Dynamics in a Single

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