Revealing Ultrafast Population Transfer between Nearly Degenerate Electronic States

The response of a molecule to photoexcitation is governed by the coupling of its electronic states. However, if the energetic spacing between the electronically excited states at the Franck–Condon window becomes sufficiently small, it is infeasible to selectively excite and monitor individual states with conventional time-resolved spectroscopy, preventing insight into the energy transfer and relaxation dynamics of the molecule. Here, we demonstrate how the combination of time-resolved spectroscopy and extensive surface hopping dynamics simulations with a global fit approach on individually excited ensembles overcomes this limitation and resolves the dynamics in the n3p Rydberg states in acetone. Photoelectron transients of the three closely spaced states n3px, n3py, and n3pz are used to validate the theoretical results, which in turn allow retrieving a comprehensive kinetic model describing the mutual interactions of these states for the first time.

T he mechanistic understanding of the initial processes in light-induced molecular excited-state dynamics is both of fundamental interest and essential for the development of various technological applications, for example efficient lightharvesting systems 1 or molecular machines. 2 The first steps of these dynamics, which proceed in the femto-to picosecond range, are often accompanied by nonadiabatic population transfer between the excited states and can therefore be observed in real time with ultrafast time-resolved spectroscopy. 3 Such spectroscopy, in conjunction with state-of-the-art excited-state computational chemistry methods has revealed general concepts for understanding excited-state interactions in molecules 4−7 by using static 8,9 as well as dynamic 10,11 approaches. Nevertheless, following the nonadiabatic dynamics of molecules becomes rather challenging when the energy differences between excited electronic states within the Franck−Condon window becomes small. Although in polyatomic molecules the density of states is generally large, the situation can be particularly demanding in transition metal complexes, 12,13 extended biomolecules, 14 or within the Rydberg manifolds of small molecules, where not only higher-lying states in the Rydberg series are close in energy, 15 but also occasionally low-lying states, 16 especially if they have the same principal quantum number (e.g., n3p x , n3p y , n3p z ). Spectroscopic observation of the dynamical behavior of the individual states is then often buried under the "averaged" population decay of all states within the particular energetic region. These experimental difficulties have two reasons. First, the energetic proximity of the states in combination with the spectral width of sufficiently short laser pulses does not allow exciting single electronic states, but rather creates a complex distribution of population in multiple states. Second, simple sequential or parallel decay models, which can be routinely applied in data analysis via global fitting routines in simpler situations, 17−20 cannot accommodate complex relaxation pathways and state mixing arising from multiple couplings. These two issues obscure the nonadiabatic processes occurring within a dense set of states, requiring specialized approaches to obtain a comprehensive mechanistic understanding of the energy transfer in certain molecules.
In this work, we demonstrate that the combination of timeresolved photoelectron spectroscopy (TRPES) 21,22 and surface hopping simulations with the SHARC method 11 is able to fully characterize the dynamics in the regime of multiple nearly degenerate electronic states. Specifically, we are resolving the population transfer dynamics in the n3p Rydberg manifold of the acetone molecule after two-photon excitation. The photophysical and photochemical processes involving acetone have been in the spotlight since the early days of femtochemistry, 23−25 with a recent renaissance 15,26−30 motivated by the importance of acetone as the simplest aliphatic ketone. The difficulty of resolving light-induced processes in acetone derives from its electronic structure, characterized by series of Rydberg states that are strongly coupled to valence states, with the consequence of complex nonadiabatic relaxation dynamics where the electronic population is cascading down the ladder of Rydberg states. 15,28,30−32 As shown schematically in Figure 1a, the n3p manifold, energetically located at the lower end of the Rydberg series, consists of the three states n3p x (A 2 , 7.34 eV), n3p y (A 1 , 7.40 eV), and n3p z (B 2 , 7.45 eV), 16 lying within about 100 meV in the Franck−Condon region of the ground state. This manifold of states was investigated in time-resolved experiments and is considered to play an important role in the population transfer to lower states. 26,27,29 The energetic proximity of the three n3p states leads to significant vibrational coupling andtogether with the presence of symmetry-dependent couplings with the ππ* (A 1 ) valence statea complex interaction among all these states. 33−37 As a consequence, the dynamics of the individual n3p states within this dense region has evaded detailed observation to date, despite the importance of the n3p manifold as the bottleneck for the radiationless deactivation pathways of the higher-lying Rydberg states.
Here, we solve this problem and follow the transient population of the acetone n3p Rydberg states after excitation in terms of the individual nonadiabatic transition time scales between the four most important states: n3p x , n3p y , n3p z , and ππ*. We note here that it was previously shown in the literature 29,32 that also other dark states (nπ* and n3s) might be involved in the dynamics. Hence, in the kinetic model fits, our label "ππ*" collectively includes also those dark states, although the ππ*, nπ*, and n3s states are explicitly present in both the experiments and the simulations. The transition time scales between the four states are summarized in Figure 1b, which shows the six inter-Rydberg time constants (τ xy , τ yx , τ xz , τ zx , τ yz , and τ zy ) and the three Rydberg decay time constants (τ xπ , τ yπ , and τ zπ ) considered here. We describe the population dynamics by the following system of differential equations representing a unimolecular first-order kinetic model:  i k j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j y { z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z (2) and g(t)P ⃗ 0 is the source term that populates the states (containing the temporal profile of the laser excitation g(t) and a state-dependent prefactor P ⃗ 0 ). The main goal of our study is to assign the numeric values of the nine time constants and to investigate their dependence on the excitation energy. This task can be achieved only by a combination of experiment and simulation. The experiment can observe only "effective" time constants τ m , which are related to the eigenvalues λ m of the coupling matrix M by τ m = λ m −1 (in the following, m enumerates the eigenvalues of M). These effective time constants correspond to the so-called decay associated spectra (DAS), 17−20 discussed in detail in section S1 in the Supporting Information. As we do not consider the decay of the ππ* state here, one eigenvalue of M is always zero (i.e., one effective time constant is infinite), leaving in principle three time constants that could be obtained from the experiment. In contrast, appropriate nonadiabatic dynamics simulations can directly access the electronic populations of each individual state, what allows obtaining all nine individual time constants from a suitable global fit. The effective time constants derived from the simulated coupling matrix can then be compared to the experimental ones to verify the accuracy of the simulations.
In the TRPES experiments, isolated acetone molecules are excited into the n3p manifold by two-photon pump excitation (320 to 336 nm) and the transient population is probed after a variable time delay via one-photon probe photoionization (402 nm). The time-resolved photoelectron spectrum (see section S2 in the Supporting Information for details on the energy calibration) for 333 nm excitation is exemplarily shown in Figure 2 (see section S3 in the Supporting Information for results at other excitation wavelengths). In the spectrum, the n3p x , n3p y and n3p z states are observed as separated photoelectron bands. The photoelectron kinetic energies (probe photon energy plus state energy minus ionization potential, see section S2 in the Supporting Information) of the three bands are centered at approximately 0.75, 0.80, and 0.84 eV, respectively. The narrow Franck−Condon envelope of the three bands indicates Rydberg state ionization to the ionic ground state with a Δν = 0 propensity rule. The n3p y and n3p z states are simultaneously populated at t = 0 fs and show similar signal increase and decrease. In contrast, the n3p x signal behaves differently, building up later and decreasing much more slowly.
where Θ(t) is the Heaviside step function; τ m = λ m −1 are the effective time constants, and DAS m (E) are the decay-associated spectra (see section S1 in the Supporting Information). By varying the number of decay-associated spectra and time constants contained in the global fit, we find that m = {1, 2} (fit shown in Figure 2b) is sufficient and m = {1, 2, 3} does not increase the fit quality. This finding reflects the apparent similarity of the n3p y and n3p z transient signals. Hence, it appears that two of the three relevant effective time constants are very similar and that two time constants are sufficient to describe the time-resolved photoelectron spectrum. The two corresponding DAS (Figure 2c) confirm that n3p y and n3p z behave very similar, as given by the two peaks of the DAS corresponding to τ 2 . Consequently, we can assign the larger constant τ 1 to the decay of n3p x and the smaller constant τ 2 to the parallel decay of the n3p y and n3p z states. Unfortunately, the DAS does not allow uncovering more details of the complex population transfer between the different n3p states other than a population transfer from n3p y and n3p z to n3p x , which is indicated by the negative DAS amplitude of the fast time constant.
The dependence of the effective time constants on the excitation energy is plotted in Figure 2d. Whereas τ 2 seems to remain constant at about 150 fs, τ 1 is strongly energydependent and decreases from about 1800 to 700 fs upon increasing the pump energy. At the highest excitation energy (7.75 eV), our values of τ 1 show very good agreement with the results of Huẗer and Temps. 27 Moreover, time constants reported by these authors for higher energies show a consistent behavior in the energy dependence. Unfortunately, their energy resolution did not allow separating the different n3p states, so that they did not report decay constants that could be compared to our τ 2 values.
As the experiment cannot obtain the individual matrix elements of M, we performed nonadiabatic dynamics simulations using the surface hopping approach SHARC. 11,38 The potential energy surfaces were described employing a linear vibronic coupling (LVC) model 39 composed of 49 diabatic electronic states, including all states from the S 0 ground state to the ππ* state, which is the S 48 at the Franck−Condon geometry. We propagated (see section S4 in the Supporting Information) three independent ensembles of about 1000 trajectories each, in which the initial electronic wave function was the pure n3p x , n3p y , or n3p z state. For each ensemble we monitored the population of the n3p x , n3p y , n3p z , and ππ* states, resulting in a set of 12 population transients. This strategy of simulating the decay of each initial state separately is essential, because there are nine independent time constants to fit and employing only four population transients (from a single ensemble) would lead to severe overfitting.
Fitted population transients are presented in Figure 3. Panels a−c show the temporal evolution of the 12 population transients together with the fitting functions; the kinetic model fits the data excellently, with almost no systematic deviations. The obtained time constants, which are shown in Figure 3d together with their error estimates (obtained by bootstrapping), are quite interesting for the following three reasons: (i) The six inter-Rydberg time constants are relatively similar, with all constants ranging from 390 to 530 fs except τ xy (700 (ii) Population transfer toward energetically lower states proceeds faster (τ zx , τ zy , and τ yx ) than vice versa (τ xz , τ yz , and τ xy ). (iii) According to the three Rydberg decay constants, the n3p y state decays the fastest among the n3p states (5 and 13 times faster than n3p z and n3p x , respectively), as previously suggested in the literature. 33−37 The time constants for the n3p z and n3p x states have large errors because the simulation time (1000 fs) is smaller than these estimated time constants. Especially the time constant for n3p x is extremely large and imprecise, which indicates that this time constant is not necessary to describe the population transients. As expected, the fastest decay occurs from the n3p y state to the ππ* state, because both states have the same symmetry (A 1 ) and it is sufficient to accumulate energy in one of the totally symmetric modes that lead to an avoided crossing of these states. This situation is different from the decay between states of different symmetry, which requires accumulated energy in at least two normal modes: a totally symmetric tuning mode that makes the diabatic energies equal and a nonsymmetric coupling mode of the correct irreducible representation that induces some coupling between two states. The requirement of two activated normal modes explains why all population transfer processes between states of different symmetry are slower than the samesymmetry n3p y → ππ* transfer. In general, the magnitude of the time constants suggests that in acetone the n3p Rydberg states readily interconvert among each other because of their energetic closeness and the presence of coupling modes of all required symmetries. In order to obtain information about the energy dependence of the relaxation time constants, we group all trajectories into 11 subensembles based on their individual excitation energies. All fitted time constants are presented in section S5 in the Supporting Information. The energy intervals and the corresponding effective time constants obtained are shown in Figure 3e. The agreement of the energy dependence of the computed effective time constants with its experimental counterpart is very good. The simulated values shown in Figure 3e confirm that there is one strongly energy-dependent effective time constant (τ 1 ), being about 1600 fs for the lowest energies and about 500 fs for the highest ones. Additionally, there are two almost identical effective time constants (τ 2 and τ 3 ) in the range of approximately 80−200 fs, only weakly depending on energy.
In conclusion, we demonstrate how a combination of TRPES with extensive surface hopping simulations is able to reveal the details of nonadiabatic dynamics in energetically dense sets of states. Our synergistic approach allows disentangling the interconversion time scales among the three n3p states (n3p x , n3p y , and n3p z ) and the decay time scales to the dark ππ* state in acetone. This work constitutes the first study that resolves the highly nonsequential population flow among these nearly degenerate states. An important key of accessing the individual time constants was the simulation of multiple independent ensembles with complementary initial conditions, as otherwise the global fit is severely under-determined. Despite the known limitations of trajectory surface-hopping methods, 40−42 the results of the simulationsvalidated through their good agreement with the experimental effective time constants and energy dependence, see also a simulated TRPES spectrum in Figure S20 in the Supporting Informationshow that the n3p y is most strongly coupled to the ππ* state and exhibits the fastest decay, as expected from symmetry arguments. The n3p y also acts as the gateway state for the deactivation of the n3p x and n3p z states, as indicated by the fact that the n3p x/z → n3p y → ππ* route is faster than the direct n3p x/z → ππ* routes. The present results   43 where a small retarding field was used to increase the electron kinetic energy resolution. High-purity acetone was introduced into the chamber as background gas with a partial pressure of about 4 × 10 −6 mbar. Computational Methods. Potential energy surfaces of all electronic states of acetone were represented with a linear vibronic coupling (LVC) model 39,44 including all 24 vibrational degrees of freedom and 49 electronic states. This large number of states is necessary in order to include the important ππ* state at the reference geometry (S 48 ). The reference harmonic potential was obtained from an optimization plus frequency calculation for the n3s state at the SOS-ADC(2) level of theory. 45 Parameters for the description of the excitedstate potentials were obtained from calculations at the same level of theory. The basis set was a combination of cc-pVTZ 46 for C and O, cc-pVDZ 46 for H, and an additional 10s8p6d4f Rydberg basis set 47 at O. All electronic structure calculations were carried out with Turbomole 7.0. 48 Initial conditions were sampled from the Wigner distribution of the SOS-ADC(2) harmonic oscillator of the ground state, generating 1000 initial geometries and velocities. Three independent sets of trajectories were simulated from these, starting in the n3p x , n3p y , or n3p z diabatic state. The trajectories were propagated with SHARC2.0 11,38 for 1000 fs using a 0.5 fs time step; the electronic coefficients were propagated with a 0.02 fs step using the local diabatization approach. 49 We applied an energy-based decoherence correction 50 and rescaled the full momentum vector during a hop.
The results were analyzed in terms of the diabatic state populations, to which the kinetic model was fitted. Additional analysis was carried out by dividing the set of trajectories into different energy windows and fitting their populations independently.
See the Supporting Information for additional computational details.