Elsevier

Chemical Physics

Volume 515, 14 November 2018, Pages 21-27
Chemical Physics

Signature of the geometric phase in the wave packet dynamics on hypersurfaces

https://doi.org/10.1016/j.chemphys.2018.09.016Get rights and content

Abstract

We study the nonadiabatic dynamics of an electronic wave packet on two coupled potential energy surfaces. We focus on the difference between two configurations, the presence of a conical intersection (CI) and an avoided crossing (AC). We project the time-dependent dynamics onto the tuning and the coupling modes. For weak intersurface coupling, no significant difference appears. However, significant differences in the deactivation of the wave packet arise for increasing intersurface coupling. Most striking is a strong destructive interference of the two pathways of the wave packet which differ by a factor of π when moving around the CI. This yields to a vanishing of the wave packet in the CI configuration, which does not exist with an avoided crossing. By this, it is straightforward to identify the geometric phase in the wave packet dynamics and to use it to distinguish a CI from an AC configuration.

Introduction

A potential energy surface (PES) describes the energy of electronic states of a molecular system as the function of one or more effective coordinates of the atomic nuclei. It can serve as a conceptual tool for the analysis of the molecular geometry and the pathways which a chemical reaction follows [1]. With varying the atomic coordinates, two PESs can come very close to each other or even cross. At this singular point, the nonadiabatic coupling of two electronic states is non-vanishing and can give rise to a very rapid transfer between the two surfaces. The resulting conical intersection (CI) is the geometric point at which two PESs are degenerate and a strong nonadiabatic coupling pervails. This strongly mixes the quantum dynamics of the electrons and the nuclei, implying that the Born-Oppenheimer approximation breaks down [2], [3]. The degeneracy of the molecular PESs permits an ultrafast radiationless transition of the wave packet between the electronic states [4], [5], [6], [7], [8]. A different situation occurs when two PESs show an avoided crossing (AC) of energy surfaces (or levels): Two PESs come close to each other but do not cross. In the vicinity of the AC, a nonadiabatic coupling exists, which enables the wave packet to tunnel between the PES. It is nowadays well established that CIs and ACs widely exist in the photophysics and photochemistry of photoactive polymolecular molecules [9], [10], [11].

Due to the strong vibronic coupling, the wave-packet dynamics in the vicinity of a CI can be monitored by its associated vibrational coherence of the reaction coordinates by using coherent spectroscopic techniques. Among them, the femtosecond stimulated Raman spectroscopy (FSRS) is a powerful tool to detect the time evolution of the vibrational coherence [15]. For instance, the FSRS of rhodopsin has been used to study the isomerization process, which was estimated to occurs within 200 fs or less [10]. More recently, due to higher temporal resolution, the isomerization rate was refined to be below the timescale of 50 fs [12]. The photoisomerization of stibene has also been measured by the FSRS and the structural changes have been revealed by the gradual shift of the torsional mode [16]. Moreover, two-dimensional (2D) time-resolved stimulated Raman spectroscopy has been applied to measure the anharmonicity of the vibrational coherence of charge-transfer modes of a dimer [17]. Furthermore, transient absorption spectroscopy is another diagnostic tool to monitor the dynamics of a wave packet passing through the CI. To find the signature of the CI in the spectrum, the cis-trans isomerization of rhodopsin has been studied by transient absorption spectroscopy and the existence of the CI was revealed by the ultrafast isomerization rate and the associated frequency-shift from the stimulated emission of the excited state [9]. Moreover, in the transient absorption spectrum, the vibrational coherence has been examined both in the reactant and the product states to study the wave packet dynamics in the vicinity of the CI. Evidence of the CI in the isomerization [18], [19] and singlet fission [20] was also provided by monitoring the coherent transfer of a wave packet along intra- and intermolecular vibrational coordinates. In addition, the nonadiabatic dynamics of the nuclear basis has been measured by 2D electronic spectroscopy in the UV region [13] and the signature of the CI was uncovered by the excited state absorption after the wave packet has passed through the CI [14].

In a theoretical description, the essentials of the dynamics of the CI can be captured by a two-state two-mode model [21]. It has been proposed to use it for the modeling of the isomerization dynamics of rhodopsin [22], [23]. To monitor the nonadiabatic dynamics, the motion of the wave packet on the PESs has been projected onto the effective reaction coordinates and dissipation of the electronic wave packet is commonly treated in terms of a quantum master equation [24]. This model has been further simplified by applying a unitary transformation of the effective reaction coordinates into the harmonic bath and by solving the resulting non-Markovian dynamics by a numerically exact approach [25], [26]. The ultrafast appearance of the excited state absorption was observed and revealed as evidence of the existence of the conical intersection. Moreover, the associated vibrational coherence of the tuning and coupling modes were studied and it was found that the vibrational coherence of the tuning mode passing through the CI and the coherence of the coupling mode are both strongly affected by the CI. To capture the signature of the CI, attosecond Raman spectroscopy has been theoretically proposed to detect the transient electronic coherence of wave packet in the vicinity of the CI [27], [28]. Furthermore, diffraction has also been proposed to catch the signature of the CI by directly monitoring the motion of the molecular structure [29]. A further reduction of the complexity of the model has been suggested in terms of a two-state two-path model [30]. Here, interference effects occur between two pairs of quasi-classical paths on two PESs which can traverse the region around a CI in different configurations.

A CI not only allows for nonadiabatic transitions between potential energy surfaces, but also induces a geometrical phase to the electronic wave packet when it moves around the CI [31], [32], [33] and enhances the non-adiabatic transition probability for a wave-packet part that experiences a central collision with the CI [33]. Accurate theoretical modeling of the wave packet dynamics has to take the geometric phase into account, which is possible even for mixed quantum-classical approaches [34], [35]. While the geometric phase occurs also in higher-dimensional models with many nuclear modes, its significance is reduced due to the reduced intersurface coupling or the reduced non-rotational wave packet component due to the increased phase space [36]. In general, the unique identification of the geometric phase in the wave packet dynamics remains an interesting problem [33].

In principle, the ultrafast radiationless deactivation of the wave packet can also be induced by the AC. For instance, the retinal isomerization in rhodopsin has been successfully described by a model on the basis of an AC [37]. Yet, the aforementioned experiments and theoretical works on the basis of monitoring the excited state absorption [9], [13], analyzing vibrational coherence [19], [20] and detecting the electronic coherence [27], [38] are not sufficient to uniquely distinguish the CI from the AC constellation and a direct evidence of the CI and the AC is still lacking.

Motivated by this situation, we study the molecular wave-packet dynamics in the vicinity of a CI and compare it to the one of a configuration with an AC. To monitor the nonadiabatic dynamics, we directly visualize the motion of the wave packet by projecting it onto the reaction coordinates given by the tuning and the coupling mode. The associated vibrational coherence of the reaction coordinates is detected and it does not show a significant difference between the AC and the CI configuration when the intersurface electronic coupling between the two excited state surfaces is weak. In both cases, we find that the vibrational coherence of the tuning mode is not affected while the wave packet passes through the CI. However, the vibrations of the coupling mode are strongly damped at the CI. With increasing of the intersurface coupling strength between the two PESs, we find a clear evidence of the geometric phase resulting from the presence of the CI. In particular, the two pathways of the wave packet differ by a phase factor of π and thus lead to destructive interference in the wave-packet dynamics along the coupling mode, which is absent in the AC configuration. This result emphasizes the importance and the possibility of finding a clear signature of the geometric phase which can be used to uniquely demonstrate the existence of the CI [39] and to distinguish it from an AC configuration.

Section snippets

Theoretical modeling

We start from a quantum two-state model composed of two electronic states |e1 and |e2. Two vibrational modes, the tuning (Qt) and the coupling (Qc) mode, interact with the two electronic states with the vibronic coupling strength, κ. The molecule is assumed to be coupled to a dissipative bath of harmonic oscillators and the total Hamiltonian can be written as H=Hmol+Henv. The system Hamiltonian is given by Hmol=He1+He2, where He1=|e1(h1-/2)e1| and He2=|e2(h2+/2)e2|+(|e1Ve2|+h.c.),

Weak coupling of the potential surfaces

To compare the wave-packet dynamics in the configurations with a CI or an AC, we start by considering regime of weak coupling of the two PESs by the coupling mode (Qc). For this, we choose Λ=200 cm−1. On the basis of the assigned parameters discussed in the previous section, the PESs of the CI and AC configurations are calculated according to the model Hamiltonian and are shown in Fig. 1. In (a) and (b), the PESs of the CI case are plotted along tuning and coupling modes, respectively. The

Strong coupling of the potential surfaces

The difference in the wave packet dynamics between the CI and the AC configuration becomes more pronounced when we increase the electronic coupling Λ between the two PESs. Tuning Λ actually changes the effective potential barrier between the points “B” and “C” on the PESs of the electronic excited state |e1. For this, we choose increasing values of the coupling strength from Λ=200 cm−1 to 800 cm−1. The specific forms of the resulting PESs for these choices of Λ are shown in the Supplementary

Conclusions

In conclusion, we have studied the nonadiabatic dynamics of a molecular electronic wave packet on two coupled excited state potential energy surfaces in the vicinity of a conical intersection and of an avoided crossing of the surfaces. Aiming to reveal the difference between both configurations and, in particular, to identify the signature of the geometric phase which a wave packet acquires when moving around a conical intersection, we project the wave packet onto the tuning and coupling mode

Acknowledgments

This work was supported by the Max Planck Society and the Excellence Cluster “The Hamburg Center for Ultrafast Imaging – Structure, Dynamics and Control of Matter at the Atomic Scale” of the Deutsche Forschungsgemeinschaft. Mr. D-L Qi thanks the Fund of ECNU for Overseas and Domestic Academic Visits for support.

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    These authors contributed equally to this work.

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