Coherence-Controlled Nonadiabatic Dynamics via State-Space

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Coherence-controlled Nonadiabatic Dynamics via State-space Decomposition: A Consistent Way to Incorporate Ehrenfest and Born-Oppenheimer-like Treatments of Nuclear Motion Guohua Tao J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b01857 • Publication Date (Web): 17 Oct 2016 Downloaded from http://pubs.acs.org on October 19, 2016

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The Journal of Physical Chemistry Letters

Coherence-controlled Nonadiabatic Dynamics via State-space Decomposition: A Consistent Way to Incorporate Ehrenfest and Born-Oppenheimer-like Treatments of Nuclear Motion Guohua Tao*

School of Advanced Materials, Peking University Shenzhen Graduate School, Shenzhen, China 518055

Shenzhen Key Laboratory of New Energy Materials by Design, Peking University, Shenzhen, China 518055

* Corresponding author: [email protected]

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ABSTRACT

Accurately describing nuclear motion is crucial in electronically nonadiabatic dynamics simulations. In this work, a coherence-controlled (CC) approach is proposed based on the mapping between the classical state space and the full electronic matrix and that between the decomposed state space and different nuclear dynamics that allows nuclear motion to properly follow either Ehrenfest dynamics in the coherence domain or Born-Oppenheimer-like dynamics in the single-state domain in a consistent manner. This new method is applied to several benchmark models involving nonadiabatic transitions in two-state or three-state systems, and the obtained results are in excellent agreement with exact quantum calculations. As a generalization of the recently developed symmetrical quasiclassical approach and the augmented image (AI) version of the multi-state trajectory approach, the proposed method is extremely efficient and numerically stable. Therefore, it has great potential for implementation in nonadiabatic molecular dynamics simulations for realistic complex systems, such as materials and biological molecules.

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Table of Contents

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Electronically nonadiabatic processes involve coupled electronic-nuclear motion that goes beyond the well-known Born-Oppenheimer approximation.1 A variety of theoretical methods2-6 have been proposed for simulating the dynamics evolving on multiple potential energy surfaces. Among these methods, mixed quantum-classical (MQC) approaches, such as Ehrenfest7 and surface hopping,8 are widely applied because of their simplicity and potential success in the modeling of realistic complex molecular systems. The shortcomings of these MQC methods have also been noted; for example, quantum coherence is not properly described. Indeed, the mean-field Ehrenfest method overestimates the electronic coherence.9 Additionally, nuclear dynamics are always located on an averaged potential energy surface, whereas surface hopping overestimates the nuclear coherence by assuming a sudden local state switch for classical particles and, therefore, omitting the Frank–Condon factor10 and potentially altering the coupled electron coherence. More recently, substantial progress has been made toward improving MQC approaches and increasing their practicality.4-6 Clearly, nonadiabatic dynamics require a consistent treatment of both electronic and nuclear degrees of freedom (DOFs) to be accurately described. One rigorous and practical approach for satisfying this requirement is to write the coupled electronic-nuclear Hamiltonian as suggested by Meyer and Miller (MM):11,12

ℋ , , , = + ∑ + ∑ ∑ ,

(1)

where µ is the reduced mass; x, p, Q, and P are the coordinates and momenta for the electronic and nuclear DOFs; Vkk and Vkl are the Hamiltonian matrix elements; 4 / 20


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and ≡ x + p − γ and ≡ x x + p p are the population of state k and

the coherence between state k and state l, respectively. Here, γ may be taken as a zero-point energy parameter. Eq. 1 creates a mapping between the discrete quantum states and continuous classical electronic phase space, and the electronic population and coherence can be determined using classical electronic variables. Recently, a symmetrical quasiclassical (SQC) approach13-18 was developed that applies a windowing function to the initial and final states symmetrically in the classical state space, substantially improving the accuracy of the predicted electronic transition probability. Furthermore, Miller and Cotton19 suggested that electronic coherence may be obtained by counting the trajectories that pass though the region centered at the midpoint between the corresponding states. Alternatively, here, we propose a different scheme for the mapping of the full electronic density matrix to the classical state space that leads to a consistent treatment of the nuclear motions in different domains, i.e., Ehrenfest in the coherence domain and Born-Oppenheimer-like in the single-state domain. This method facilitates obtaining an efficient and accurate description of the coupled electronic-nuclear dynamics. Our scheme to unify the mean-field and single-state dynamics arises naturally from the state-space decomposition and does not require additional perturbative treatment of the coupled electronic-nuclear dynamics, in contrast to many previous attempts, such as setting a switching criterion based on the Ehrenfest’s accuracy20 or the inclusion of additional electronic decoherence terms.9,21 Fig. 1 illustrates our approach for a two-state system where h = n , and h1 and 5 / 20



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h2 are the population variables for states 1 and 2, respectively. The windowing functions for the occupied and unoccupied states are 1 − γ " h#$$ " 1 + γ and −γ " h%#$$ " γ, respectively. Therefore, the squares centered at (1,0) and (0,1) in the state space in Fig. 1 represent the single-state regions corresponding to the system in on state 1 (blue) or on state 2 (golden), respectively. The off-diagonal electronic coherence is mapped to the square regions shown in green and centered at (0,0) and (1,1) with a width γc. Here, we set γ=γc= 0.5; thus, all square domains (Dij) are connected, form a filled space, and are formally mapped to the full electronic matrix, i.e., &

D D

ρ D ( ↔ *ρ D

ρ ρ ,.

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Figure 1. State-space decomposition for a two-state model. Colored regions representing single-state domains for states 1 (D11, blue) and 2 (D22, golden), and the coherence domain (D12, green) are mapped to Born-Oppenheimer-like and Ehrenfest dynamics, respectively. Here, the window width γ=γc=0.5. Red arrows schematically illustrate different paths for representative initial conditions that conserve the total electronic population. In addition to evaluating the full electronic matrix based on the MM model and specific windowing functions, this seemingly redundant mapping may be used to obtain useful physical insights, and we believe that this application is of even greater importance. According to Eq. 1 and the corresponding Hamiltonian dynamics, the total electronic population is conserved; i.e., ∑ n = constant. Without loss of generality, taking a two-state system as an example, we have h1(t)+h2(t) = h(0), which corresponds to the straight lines in the two-dimensional (2-D) plane. Therefore, a trajectory in the state space must resemble these red arrows, depending on the initial conditions h(0). For example, the trajectories represented by the solid, dashed, and dashed-dotted arrows correspond to transiting directly from state 1 to state 2 via the critical point (0.5,0.5), staying within the original state, or the general trajectories of nonadiabatic transitions passing through the coherence regions, respectively. When a trajectory enters the coherence regions, the system is assigned neither to state 1 nor to state 2 and, therefore, does not contribute to the state population in the standard SQC approach. Consequently, the choice of γ< 0.5 may result in a transition latency, at least at very short times, because a gap always exists between the initial and final states for all trajectories. Once the state space is decomposed into the single-state domains and the off-diagonal coherence domains, it naturally provides a controlled scheme for the 7 / 20



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propagation of nuclear dynamics in different ways. For the trajectory in the single-state region, when the quantization condition invoked by the windowing function is applied to the Hamiltonian dynamics according to Eq. 1, assuming that the system is in the occupied state l=occ (i.e., n#$$ → 1, and n → 0 for all k ≠ occ), the nuclear force is Born-Oppenheimer-like and is given by

Coherence-Controlled Nonadiabatic Dynamics via State-Space

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