Femtosecond Quantum Dynamics of Excited-State Evolution of Halide Perovskites: Quantum Chaos of Molecular Cations

The excited-state quantum dynamics of the organic cation in hybrid perovskites are investigated using the time-dependent density functional theory (TDDFT). The time-dependent non-adiabatic bond fluctuation behaviors reveal that the energy relaxation follows different pathways depending on the chemical bonding characteristics and energy transfer modes within the cation molecule, which can fundamentally affect its photostability. For the ammonium-group-containing cations, such as methylammonium (MA) or ethylammonium (EA), local vibrational modes survive for a long time. However, as their lowest unoccupied molecular orbital (LUMO) having  * characters, the amidinium-group-containing cations, such as formamidinium (FA) or guanidinium (GA), efficiently dissipate deposited energy via chaotic intramolecular vibrational energy redistribution (IVR). The distinct dynamic behaviors of A-site molecular cations are closely related to the quantum ergodicity, which can bring enhanced photochemical stability of FA and GA compared to MA and EA. Our theoretical investigation reveals the quantum chaos origin of better light stability of FA-based perovskites and serves the future research direction of the A-site engineering for better solar cells and light-emitting devices.

Methylammonium (MA) cation was the first adopted cation molecule to make solar cells by the Ethylammonium (EA) and guanidinium (GA) cations have been tried even in early stage 16,17 , and other bigger cation molecules have also shown promising behaviors for the development of lowdimensional perovskites and 2D-3D hybrid perovskite structures 18,19 .
Organic cations are located at the A-site of ABX3 type perovskite crystal structures. A-site cations play crucial roles in deciding the stabilities and carrier transport characteristics. FA has been considered a more robust molecular cation because empirically FAPbI3 has shown better thermal and photostability than MA [20][21][22] . FA exhibits a smaller dipole moment which can improve the stability against moisture 23 . Recently, there have been experimental reports presenting the enhanced stability by incorporating GA or EA cations as additives or dopants 24,25 .
Photostability can be the fundamental bottleneck for the commercialization of hybrid perovskites. The stability under illumination needs further progress than that of dark conditions despite recent fast improvements in the long term stability of perovskite solar cells 26  cations can be dissociated into CH3NH2 and hydrogen by UV light irradiation 27 . These authors have proposed a scenario that the photodissociation of MA is induced by the electron excitation from the valence band to the antibonding orbitals of MA molecules based on their gas effusion experimental data 27 . Similar photodissociation studies have been reported for various experimental conditions 28,29 , and even in the case of layered perovskites 30 .
It is also found that A-site molecular cations can affect the dynamics of excited carriers as well.
Chen et al. recently reported that hot carrier (HC) cooling has cation-dependence, and the A-site FA shows the fastest HC cooling among FA, MA, and Cesium cations 30 . It was proposed that a slow charge-carrier cooling can give the chance to extract hot-carrier, enhancing photovoltaic efficiency 31 .
These findings show that excited-state dynamics of A-site cations have a close relationship with photostability and HC cooling. However, most theoretical studies on the dynamics of the excited carrier have focused on the interaction between the electron and inorganic-lattice phonon through time-domain density functional theory, where the ionic motion is limited in an equilibrium state [32][33][34] . The effects of carrier injection into A-site cation molecules will introduce transient electronion non-equilibrium dynamics over attosecond-femtosecond time scales, but has not been clearly examined although the direct evidence was proven by the Nickel et al.'s experiment 27 . Such an excited electron dynamics study is beyond the capacity of usual DFT simulations and has not been clearly examined yet to elucidate the underlying mechanisms of excited electron-ion dynamics of A-site molecules in the hybrid perovskites.
This study aims to investigate the impacts of excited-state evolution with attosecond resolution for various A-site molecules for the femtosecond ranges through the state-of-the-art time-dependent density functional theory (TDDFT) calculations 35,36 . Calculated dynamic behaviors are closely correlated with the photostability and hot carrier dynamics of halide perovskites. From the fundamental analysis of excited electron-ion quantum dynamics, quantum chaos [37][38][39] and quantum ergodicity 40 are found to play prominent roles in determining the photostability of hybrid perovskites with different A-site cations.

Results
Model description. Fig. 1a represents an ideal cubo-octahedral lattice structure of ABX3 type perovskites. A distinct feature of organic-inorganic hybrid perovskites is that A-site cation can be made of various molecules. It is well known that the valence band maximum (VBM) state and conduction band minimum (CBM) state mainly consist of I 5p and Pb 6p orbitals, respectively. (3) the LUMO is occupied by an electron that travels in the CB of an inorganic cell after photoexcitation; (4) we cannot exclude the possibility of excitation from filled trap states within band gap such as Ii, IMA, and Pb dimer induced by VI 42,43 . The excitation is accompanied by creating a highly delocalized hole in the valence band, which, however, hardly influences the Asite molecule electronic structure due to the delocalized wavefunctions in periodic solids. Therefore, the molecular systems with an additional electron-MAI -1 , EAI -1 , FAI -1 , and GAI -1are selected to model the light-induced degradation originating from the A-site cation instability (after a model validation by comparing with A-site molecules in crystalline structure as shown in Supplementary Notes 1). During electron-ion dynamics simulations, the position of an iodine atom keeps unchanged representing the rigidity of I ions in the lattice structure. The A-I molecule model is a reasonable model approximation in which atoms forming A-site molecules are relatively free to move within the framework of iodine atoms forming strong covalent bonds to maintain the perovskite structure.
Our non-adiabatic dynamics simulation begins with the electron capture into the LUMO state of A-site molecule because the LUMO-occupied state most likely governs the pertinent molecular dynamics. Even if we start with an electron in higher energy LUMO+α state of A-site molecule, the electron rapidly comes down to the LUMO according to Kasha's rule, and the molecule readily reaches the local electronic ground state. Ionic energy deposited during this initial relaxation process will be redistributed in accordance with the LUMO-occupied electronic state. When subsequent ionic energy deposition can occur in any form, the stored energy or any temporary localized vibration will be redistributed following the LUMO-occupied electronic state as well.
Accordingly, for an extensively long time, the LUMO character plays a prominent role in molecular dynamics. and FAI -1 are more prominent than those of EAI -1 and GAI -1 .
The N-H bond vibration dynamics can elucidate a potential origin of the lattice instability in halide perovskites. It has been commonly accepted that the kinetic path of intrinsic degradation of MAPbI3 starts from the deprotonation of methylammonium cation 44,45 . The strong and localized N-H bond vibrational mode of MAI -1 may induce the bond breaking and dissociate CH3NH3I into CH3NH2 + HI. Although the bond breaking occurs in reality, it is intractable to directly simulate this dissociation process because the pertinent reaction barrier is too high to overcome with a simple electron attachment process. To computationally reproduce this rare event with attosecond resolution, it is necessary to deal with a vast number of samples with different initial ionic kinetic energies for a very long time. Therefore, we will discuss dynamic evidence differentiating the molecular dissociation process instead of directly simulating the event. indicate results using BO approximation, where 0.05 fs was chosen for the time step after the convergence test (Fig. S4).
To grasp the origin of different energy relaxation mechanisms for different cations, the projected density of states (PDOS) and wavefunctions of the anion HOMO state are shown in Fig. 3. Note that the anion HOMO states of the A-I molecules with an additional electron (that is the chemical MA and EA belonging to the AMG, the LUMO has a significant contribution from the I 5p orbital (Fig. 3a,c and Fig. S5a,c). This originates from the strong interaction between the A-site cation and I, namely a hydrogen bond that is formed by stabilizing the * antibonding orbital of the hydrogen bond donor. Comparing anion HOMOs of excited states (Fig. 3a,c) with LUMOs of ground states (Fig. S5a,c), it turns out that this character survives even after electron attachment.
Consequently, the strong intermolecular hydrogen bond stabilizes a particular N-H bond.
Furthermore, the C-N bond is hardly influenced by the electron attachment according to the LUMO character; thus, the overall molecular structure remains intact. As a result, the stretching vibration For the A-site cations belonging to the ADG, FA has a LUMO mostly localized on the * antibonding orbital of the ADG (Fig. S5b). This is observed in the anion HOMO of the FAI -1 as well (Fig. 3b). For GA, its LUMO calculated using the zero-kelvin structure of GAI shown in Fig.   S5 looks different from the * antibonding orbital although it exhibits originally the * antibonding orbital without I (see Fig. S6 for LUMOs of A-site cations without I). During dynamics after the electron attachment, the anion HOMO of GAI -1 is converted into its original character, the * antibonding orbital of the ADG, within 50 fs (Fig. S7). In contrast, the conversion does not occur without the additional electron (Fig. S8). Thus, the electron attachment to ADG-containing cations makes the C-N -bond flexible, which opens up new vibrational mixing channels by enabling intramolecular rotation and bending vibration [48][49][50] . Consequently, a rapid IVR following the conventional RRKM theory is allowed for FA and GA.  The different energy dissipation from carrier excitation can be the source of distinct electronionic coupling behaviors. Figure S9 presents the ion temperature variation for considered A-I molecules during simulation time. The ion temperature of ADG increases significantly faster and higher than that of AMG, which can be attributed to the aforementioned the π * antibonding state of C-N bonds of ADG. Our calculation results agree with the previous experimental report, in which FA-based perovskites show faster HC cooling than MA-based or Cs-based one 31 .
Quantum ergodicity of MA and FA. This modeling study found that quantum ergodicity provides an insight into the distinct dynamics of A-site molecules in the two different groups. In the remaining of this paper, we discuss a consequence of non-adiabatic dynamics; then the quantum ergodicity of MA and FA is investigated in terms of energy level spectra.
We note that the TDDFT-based Ehrenfest dynamics result in different dynamics from the Born-Oppenheimer (BO) approximation owing to the limitation of adiabatic dynamics in describing the electron attached systems (Fig. 2b). Non-adiabatic dynamics can appear when the initial forces on ions are nonzero even if dynamics begin with the electronic ground state 51 . Further analysis of orbital occupation as a function of time provides additional information about the deviation from the BO approximation and the fast IVR in MAI -1 and FAI -1 (Fig. 4). These populations are obtained by projecting time-evolving Kohn-Sham (KS) orbitals into KS eigenstates, which are calculated using diagonalization with respect to the ion geometry and electron density at a given time. See spin-down orbital is excited into the 14th orbital through the non-adiabatic dynamics, and this accounts for the discrepancy between the Ehrenfest dynamics and the BO approximation (Fig. 2b).
Occupying the 14th spin-down orbital, which still possesses the * antibonding character albeit distorted, imposes anharmonic vibrational coupling, which facilitates IVR 37   As aforementioned, slow IVR associated with non-RRKM dynamics is responsible for fast unimolecular dissociation due to non-ergodic behavior 47  Wigner-Dyson statistics in FAI -1 is a consequence of level repulsion induced by symmetry breaking, and the IVR process is faster in the molecules of lower symmetry 55 . Accordingly, the electronic quantum chaos turns out to play an essential role in the ergodicity of ion dynamics, responsible for the fast IVR. This study also presents that the energy dissipation process from excited states of A-site cation molecules is closely relevant to the quantum chaos and ergodicity. The energy level spectrum analysis based on quantum chaos reveals that the electronic quantum chaos of ADG plays an essential role in the ergodicity of ion dynamics, responsible for the fast IVR.
This work suggests the theoretical background of the reason why FA-based perovskites can have better photostability than MA-based ones. Our in-depth analysis presents that quantum ergodicity is the descriptor for the intrinsic photostability of organic-inorganic hybrid perovskites. Moreover, our calculation enlightens the importance of the chemical bonding nature of A-site molecule for the stability and carrier dynamics under the light irradiation.

Methods
TDDFT modeling. The real-time TDDFT 35,36,56,57 in the framework of Ehrenfest dynamics 58,59 is employed to carry out non-adiabatic ab initio molecular dynamics study. The approach has been implemented in the real-space code, Octopus 60-63 . We use the set of the Hartwigsen-Goedecker-Hutter (HGH) 64 pseudopotentials for local density approximation (LDA) 65 , and the adiabatic approximation 36 is employed for the exchange-correlation (XC) functional in TDDFT. The spacing between the points in the real-space mesh is set to 0.18 Å, and the time step Δt for integration is set to 0.0023 ħ/eV. We have confirmed the total energy conservation for simulation time to verify computational stability and accuracy. Coupled electron-ion dynamics driven by electron attachment is simulated as follows. First, with a neutral charge, geometry is optimized until the maximum force is smaller than 0.05 eV/Å. Then, we manipulate occupation numbers of orbitals to add an electron into the lowest unoccupied molecular orbital (LUMO) of the original system, which then turns into the highest occupied molecular orbital (HOMO) of the modified system. This is referred to as anion HOMO. Once electronic minimization is performed with the original geometry, initial forces are exerted on ions. Finally, we let the system evolve in time with the initial forces based on the TDDFT-based Ehrenfest dynamics. contributed to idea development, discussion, and preparation of the manuscript.

Additional information
Supplementary Information accompanies this paper at https://.