Quantum Mechanical Wave Packet Dynamics at a Conical Intersection

Jan 11, 2016 - In addition, it is desirable to identify the signatures of a CI not only via the ultrafast character of the radiationless electronic de...
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Quantum Mechanical Wave Packet Dynamics at a Conical Intersection with Strong Vibrational Dissipation Hong-Guang Duan†,‡,§ and Michael Thorwart*,†,§ †

I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany Max Planck-Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany § The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany ‡

S Supporting Information *

ABSTRACT: We derive a reduced model for the nonadiabatic quantum dynamics of an electronic wave packet moving through a conical intersection in the presence of strong vibrational damping. Starting from the dissipative two-state two-model model, we transform the tuning and the coupling mode to the bath. The resulting quantum two-state model with two highly structured environments is solved numerically exactly in the regime of strong vibrational damping. We find negative cross peaks in the ultrafast optical 2D spectra as clear signatures of the conical intersection. They arise from secondary excitations of the wave packet after having passed through the photophysical energy funnel. This feature is in agreement with recent transient absorption measurements of rhodopsin.

C

onical intersections (CIs) describe ultrafast radiationless transitions of electronic wave packets in polyatomic molecules. Within the Born−Oppenheimer approximation, the nucleus of an atom moves on adiabatic potential energy surfaces (PESs) formed by well-separated electronic quantum mechanical states. When two PESs become degenerate, a CI is formed at this singular point. In the vicinity of the CI, a strong nonadiabatic coupling between the electronic states prevails, the Born−Oppenheimer approximation breaks down, and a strongly mixed quantum dynamics of the electron and the nucleus occurs.1−4 The degeneracy of the PESs enables the radiationless transition of the wave packet between two PESs and induces ultrafast electronic and vibrational relaxation.5,6 It is nowadays well established that CIs widely exist in the photophysics and photochemistry of polyatomic molecules, for instance, in carotenoid,7 DNA bases,8,9 cyanine dyes, and rhodopsin.10−12 Yet revealing the details of the nonadiabatic quantum dynamics in the vicinity of a CI is, in general, still challenging because a CI usually involves an enormous number of electronic and nuclear degree of freedoms. Advanced timedependent quantum wave packet as well as quantum-classical numerical simulations provide the microscopic mechanisms for the ultrafast dynamics around a CI.13−20 For a minimal modeling of a CI, at least two 2D PESs are required.21 In this two-state two-mode model, the coupling mode in one dimension refers to the interaction between two PESs (see Figure 1), while the tuning mode in the second dimension contains the shift of two PESs. The latter enables the optical transitions.16 To clearly observe the nonadiabatic dynamics around the CI, the barrier at the transition point of the wave packet has to separate two regions clearly, which requires a rather large shift of the two PESs. In addition, long simulations © XXXX American Chemical Society

Figure 1. Conical intersection of two PES illustrated by coupling (Q1) and tuning mode (Q2) for the ground and excited electronic states in the adiabatic basis, respectively.

times are prerequisite. For all of this, a large vibrational eigenbasis is needed to obtain converged numerical results. For instance, more than 20 basis functions were necessary in ref 22 and lead to a 1200-dimensional Hamiltonian matrix. These requirements hamper all efforts of an exact study around a CI, also under the influence of an ubiquitous dissipative environment. Hence, an efficient theoretical description of the nonadiabatic quantum dynamics around a CI still remains challenging. In addition, it is desirable to identify the signatures of a CI not only via the ultrafast character of the radiationless Received: December 16, 2015 Accepted: January 11, 2016

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DOI: 10.1021/acs.jpclett.5b02793 J. Phys. Chem. Lett. 2016, 7, 382−386

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The Journal of Physical Chemistry Letters electronic decay but also via robust spectroscopic information. Available optical techniques monitor state populations10,23 or identify characteristic traces in transient vibrational spectra.24−28 Recently, a transient spectroscopic technique that is only sensitive to electronic coherence has been introduced.28 It measures the frequency-resolved stimulated Raman scattering of a probe pulse as a function of the time delay with respect to the pump pulse. Clearly separated cross-correlation peaks in optical 2D spectra have also been reported.22,29,30 We formulate a reduced model to describe the nonadiabatic quantum dynamics around a CI, which allows for an efficient numerically exact treatment of a CI in the presence of strong vibrational relaxation. The latter occurs, for example, in the 2D spectrum in the deep UV of para-terphenyl in ethanol.31 The key to this is the transformation of both the tuning and coupling mode into the reservoir and to treat their role as part of the non-Markovian environment. Exploiting the performance of numerically exact tools to describe the resulting nonMarkovian dissipative dynamics, we calculate 2D electronic spectra to analyze the electronic dynamics around a CI for the particularly difficult case of strong vibrational damping. To identify unique features associated with CI, we compare our results to those of a model without a CI. Interestingly, we find an off-diagonal branch of the spectral signal with negative amplitude that clearly identifies the CI. Its origin is traced back to excited-state absorption processes which connect the local minimum of the ground state (g′ in Figure 1) with the excited state at the position e′. The time evolution of the negative cross peaks in the 2D spectra is unique in the dynamics of an electronic wave packet in the presence of strong vibrational relaxation moving in the region of a CI. We start with two-state two-mode model in which a molecule is assumed to possess a spectroscopically accessible CI between two PES associated with the ground (|g⟩) and excited (|e⟩) electronic state. In addition, we include a dissipative interaction of the standard harmonic environment. The total Hamiltonian can be written as H = Hmol + Henv. The molecular Hamiltonian is given by Hmol = Hg + He, with Hg = | g⟩(h1 − ϵ/2)⟨g| and He = |e⟩(h2 + ϵ/2)⟨e| + (|e⟩V⟨g| + h.c.). Here ϵ is the energy gap between the ground and the excited state and ℏ = 1. The vibrational Hamiltonians h1, h2 are associated with the ground and excited electronic states, respectively. They include two vibrational modes, the coupling mode characterized by the reaction coordinate Q1 and the tuning mode described by the reaction coordinate Q2. They are 1 given by hg = 2 ∑i = 1,2 Ωi(Pi2 + Q i2) and h1 = hg − κQ2, h2 =

Henv =∑ [ α

pα2 2mα

+

2 cαQ 1 ⎞ mα ωα2 ⎛ ⎟ ⎜xα + 2 ⎝ mα ωα2 ⎠

2 dαQ 2 ⎞ Mανα2 ⎛ ⎟] ⎜y + + + 2Mα 2 ⎝α Mανα2 ⎠

qα2

(1)

Here the momenta of the bath oscillators are denoted as pα and qα, while their coordinates, masses, and frequencies are denoted by xα, mα, ωα and yα, Mα, να. The respective coupling constants are cα and dα. The baths are characterized by the spectral densities J1(ω) = J2 (ω) =

π 2

∑α

dα2 Mανα

π 2

∑α

cα2 δ(ω mαωα

− ωα) and

δ(ω − να). Throughout this work, we

assume that both the tuning and coupling mode experience fluctuations with an Ohmic spectral distribution according to J1/2(ω) = η1/2ω. Here, η1/2 are the damping strengths for the coupling and tuning mode, respectively. The transition dipole moment is defined within the Condon approximation as μ = | e⟩⟨g| + |g⟩⟨e|. A well-separated PES crossing can be induced by a large shift Δ = 2κ/Ω2 between the minima of the |g⟩ and |e⟩-PES. A large number of vibrational states is required to obtain converged numerical results.22 This is possible in the regime of a weakly damped vibrational dynamics within the Born−Markov approximation. Numerically exact methods such as, for example, the quasiadiabatic propagator path integral33,34 or the hierarchy equation of motion (HEOM)35,36 technique, are not applicable due to the large vibrational Hilbert space, especially when the vibrational damping is strong. However, a numerically exact treatment becomes possible if the two modes Q1 and Q2 are transformed into the bath and treated as effective modes with their full non-Markovian dynamics.34,37 The unitary transformation of ref 37 is readily generalized to the two-mode case and yields the total Hamiltonian HM =

⎡ p′ 2 α

ϵ σz − σx ∑ cα′ xα′ + 2 α

∑⎢

− σz ∑ dα′ yα′ +

∑⎢

α

α

⎢⎣ 2mα′ ⎡ q ′2 α

α

⎢⎣ 2Mα′

+

mα′ ωα′ 2 2 ⎤ xα′ ⎥ 2 ⎥⎦

+

Mα′ να′ 2 2 ⎤ y′ ⎥ 2 α ⎥⎦ (2)

Here, σz = |g⟩⟨g| − |e⟩⟨e| and σx = |g⟩⟨e| + |e⟩⟨g| are Pauli matrices. The resulting effective spectral densities follow as eff J1/2 (ω) = λ1/2

hg + κQ2, where Ωi=1,2 are the frequencies of the harmonic coupling and tuning mode, respectively. Moreover, κ is the vibronic coupling strength. The electronic coupling between the two PES is assumed to linearly depend on Q1, such that V = ΛQ1 with the electronic coupling strength Λ. Furthermore, we assume that all relevant interactions between the two electronic PES are captured by the coupling mode, which is explicitly included. Thus, the bath only couples vibrational states within the same electronic PES, and we assume that the two vibrational modes are coupled to their own linear bath according to the Hamiltonian32

2 γ1/2 Ω1/2 ω 2 2 (Ω1/2 − ω 2)2 + γ1/2 ω2

(3) 1

where λ1 = Λ2/(2Ω1) and λ 2 = κ 2/(2Ω 2) = 2 Ω 2(Δ/2)2 are the reorganization energies for the coupling and tuning modes, respectively. They represent two effective structured harmonic reservoirs for the electronic dynamics, when both are assumed to initially be in thermal equilibrium at the same temperature ; . In this effective picture characterized by eq 2, the coupling strength between the ground and excited PES can be tuned by the reorganization energy of bath 1. The shift between the ground and the excited PES is encoded in the magnitude of the reorganization energy of bath 2.38 We have assumed the shift Δ to be large, a large reorganization energy for tuning mode 383

DOI: 10.1021/acs.jpclett.5b02793 J. Phys. Chem. Lett. 2016, 7, 382−386

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The Journal of Physical Chemistry Letters

branch. Interestingly, the negative off-diagonal peak around ωt = 1000 cm−1 reaches its maximum magnitude at T = 500 fs. The negative branch decays rapidly with increasing waiting time and become almost invisible at T = 1000 fs. The negative peak is generated by a secondary excitation that lifts the wave packet from the minimum of ground state PES (g′) to the excited state PES (e′), as indicated by the blue arrow in Figure 1. This excited-state absorption process can exist only when the initially excited wave packet (indicated in red in Figure 1) has moved from the Franck−Condon region on the excited-state PES through the CI downhill; see the purple arrow A in Figure 1. This can only occur in the presence of a CI, and the negative cross peaks can thus serve as a unique identifier of it. More quantitatively, the energy gap between those two states, which is associated with this transition, can be estimated in the adiabatic basis as ∼1000 cm −1 with the momentary configuration of PES. This quantitatively fits with the transition frequency of the negative peak along ωt ≈ 1000 cm−1. A comparison with the model without a CI is helpful to verify this picture. This is realized by setting λ′1 = 0 while keeping the strong vibrational dissipation with λ′2 = 150π cm−1 and γ′2 = 20 cm−1. The analogue sequence of 2D spectra is shown in the right column of Figure 2 (labeled as ’no CI’). It shows the typical relaxation dynamics in which negative peak are completely missing. Besides, the maxima of the 2D spectra decay much slower than in the case with a CI being present. Clearly, the wave packet no longer can relax in a radiationless process to the electronic ground state PES through a funneling of the CI. The magnitudes of selected peaks are traced for increasing waiting times to quantitatively compare both cases. The results are shown in Figure 3. We compare the maximal and minimal

results and leads to the system-bath interaction of eq 2, valid in the strong coupling region. The structured environment is characterized by the two localized modes, which induce two spectral peaks at the frequencies Ωi with the widths given by γi. In the present work, we are interested in the most difficult case when the vibrational relaxation of the two modes is overdamped. Hence, in the limit γ1/2 ≫ Ω1/2, we obtain eff, ∞ ′ J1/2 (ω) = λ1/2

′ ω γ1/2 ′2 ω 2 + γ1/2

(4)

The effective reorganization energies and damping constants in the overdamped limit follow as λ′1/2 = λ1/2 and γ′1/2 = Ω21/2/γ1/2. On the basis of the effective Hamiltonian of eq 2 of a quantum two-level system coupled to a nonstandard structured environment, we have calculated the resulting non-Markovian dynamics by employing the numerically exact HEOM approach.36 In particular, we are interested in the spectroscopic traces of a CI in 2D electronic spectra.39 Therefore, we evaluate the total 2D electronic spectra (rephasing plus nonrephasing part) by Fourier transforming the third-order nonlinear response function; see the Supporting Information (SI) for further details.40 For the calculation, we have set the electronic gap in the Franck−Condon region to ϵ = 1000 cm−1 after having performed the usual rotating wave approximation. The bath parameters are chosen as λ1′ = 150 π cm−1, γ1′ = 150 cm−1 and λ2′ = 150 π cm−1, γ2′ = 20 cm−1, and the temperature was set to ; = 300 K. In Figure 2, we show a collection of selected 2D electronic spectra for different waiting times T (left column). For T = 300 fs, we observe a split peak with one positive and one negative

Figure 3. Extremal peak heights in the 2D spectra as a function of the waiting time. The red symbols (labeled “CI max” and “CI min”) show the magnitude of the peaks at (ωτ = 1240 cm−1, ωt = 1240 cm−1) and (ωτ = 1240 cm−1, ωt = 880 cm−1) in the 2D spectra for the case with a CI, respectively. Moreover, the blue symbols (labeled “no CI”) show the magnitude of the peak at (ωτ = 1200 cm−1, ωt = 1200 cm−1) of the spectra for the case without a CI.

peak heights of the case with a CI to the maximal peak height of the case without a CI. Clearly, for the model with CI, the extremal peaks of the 2D spectra decay very fast and reach zero at 1500 fs (left ordinate in Figure 3). In turn, in the case without a CI, such a rapid decay is not present. Instead, we find still a larger maximum of ∼0.28 (in arbitrary units) for the longest waiting time of 1500 fs considered (right ordinate in Figure 3).

Figure 2. Selected 2D electronic spectra calculated for the effective quantum two-level model with the structured environment formed by the two baths. The left column shows 2D spectra calculated in the presence of the CI, while the right column shows the results in absence of the CI with λ′1 = 0. The 2D electronic spectra are normalized separately according to their maximal peak in the 2D spectrum at T = 300 fs. Here kcm−1= 1000 cm−1. 384

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A further confirmation can be obtained from the timeevolved transition absorption spectrum shown in Figure 4. Two

Letter

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b02793. Calculation of the 2D electronic spectrum, convergence of the HEOM for strong dissipation, population dynamics, and parameters for the potential energy surface. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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Figure 4. Transient absorption spectrum as a function of the waiting time in the presence of a CI. It shows two clearly separated bands (positive and negative) centered at 1600 and 850 cm−1, respectively.

ACKNOWLEDGMENTS We acknowledge financial support from the Max Planck-Society and the excellence cluster ”The Hamburg Center for Ultrafast Imaging - Structure, Dynamics and Control of Matter at the Atomic Scale” funded by the Deutsche Forschungsgemeinschaft. H-G. D. acknowledges generous funding provided by the Joachim Herz-Stiftung (Hamburg) within the PIER Fellowship Program.

bands show up with positive and negative amplitudes. The time scales on which the dynamics of the wave packet around the CI occurs can be clearly distinguished by considering the dynamics associated with the positive and negative bands. After the initial photoexcitation, a part of the wave packet moves directly through the CI and reaches the ground-state PES (arrow B in Figure 1). This process contributes significantly to the positive bands associated with the fast relaxation. In addition, the remaining part of the wave packet moves through the CI to the PES minimum (arrow A in Figure 1). It can be identified with the starting point of the negative bands at ∼200 fs. This process reaches its maximum at ∼500 fs and is followed by a fast relaxation. This is caused by the secondary excitation and the backward motion to ground state PES through the CI. The entire relaxation process is completed within ∼1.2 ps. Interestingly, this feature is in perfect agreement with recent transient absorption measurements of rhodopsin.10 In conclusion, we have established a reduced model for the nonadiabatic quantum dynamics of an electronic wave packet in the region of a conical intersection. It is obtained from the wellknown two-state two-mode model by transforming the two harmonic potential energy surfaces to the harmonic bath. The resulting quantum mechanical two-level model with two highly structured harmonic baths can be solved by advanced numerically exact non-Markovian techniques, such as the hierarchy equation-of-motion approach. This greatly facilitates the numerical efforts. Most importantly, it allows us to tackle the notoriously difficult case of strong vibrational damping. The signatures of the conical intersection show up in form of branches with negative peaks in optical 2D spectra. They clearly can be traced back to secondary excitations of wave packets, which have moved through the conical intersection. Their ultrafast time scale is also revealed by the time dependence of the cross peaks in the 2D spectra. Finally, we note that negative amplitude cross peak could, in principle, also arise when an extremely large shift between the ground- and the excited-state potential energy surface exists. For this, the Huang−Rhys factor must be larger than ϵ/Ω1,2. For any realistic molecules, this factor will be unrealistically large.

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REFERENCES

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