# AI-enhanced on-the-fly simulation of nonlinear time-resolved spectra

Sebastian V. Pios  Zhejiang Laboratory, Hangzhou 311100, China  Maxim F. Gelin
School of Science, Hangzhou Dianzi University, Hangzhou 310018, China  Arif
Ullah  School of Physics and Optoelectronic Engineering, Anhui University,
Hefei 230601, China  Pavlo O. Dral  State Key Laboratory of Physical Chemistry
of Solid Surfaces, College of Chemistry and Chemical Engineering, Fujian
Provincial Key Laboratory of Theoretical and Computational Chemistry, and
Innovation Laboratory for Sciences and Technologies of Energy Materials of
Fujian Province (IKKEM), Xiamen University, Xiamen, Fujian 361005, China
<EMAIL_ADDRESS>Lipeng Chen  Zhejiang Laboratory, Hangzhou 311100, China
<EMAIL_ADDRESS>
###### Abstract

Time-resolved spectroscopy is an important tool for unraveling the minute
details of structural changes of molecules of biological and technological
significance. The nonlinear femtosecond signals detected for such systems must
be interpreted, but it is a challenging task for which theoretical simulations
are often indispensable. Accurate simulations of transient-absorption or two-
dimensional electronic spectra are, however, computationally very expensive,
prohibiting the wider adoption of existing first-principles methods. Here, we
report an AI-enhanced protocol to drastically reduce the computational cost of
simulating nonlinear time-resolved electronic spectra which makes such
simulations affordable for polyatomic molecules of increasing size. The
protocol is based on doorway-window approach for the on-the-fly surface-
hopping simulations. We show its applicability for the prototypical molecule
of pyrazine for which it produces spectra with high precision with respect to
ab initio reference while cutting the computational cost by at least 95%
compared to pure first-principles simulations.

The rapid development of ultrafast laser techniques with capabilities to
control femtosecond and even attosecond pulse durations has largely shaped our
knowledge of the photoinduced reactions in polyatomic molecules. In
particular, experiments characterizing ultrafast nonadiabatic photophysical
and photochemical processes through conical intersections of electronic states
of the system upon interaction with light have gained increasing attention as
they are critical for a variety of applications in renewable energy, chemical
synthesis, and bioimaging 1, 2, 3, 4, 5, 6. For example, broad-band pump-probe
spectroscopy 7 and heterodyne-detected transient-grating spectroscopy 8
measurements identified key vibrational modes that drive the primary conical-
intersections mediated photoisomerization reaction in rhodopsin, while
transient-absorption pump-probe spectroscopy 9 and impulsive two- and three-
pulse vibrational spectroscopy 10 experiments clarified the mechanistic role
of conical intersections in singlet fission in rubrene and tri-
isopropylsilyly-ethynyl-pentacene (TIPS-pentacene), respectively. Two-
dimensional electronic spectroscopy 11, 12, which emerged as a powerful tool
for monitoring energy transfer in multi-excitonic complexes in the early 2000s
13, 14, 15, 16, was also used to comprehensively characterize wavepacket
dynamics at conical intersections, for example, in $\alpha$-terpinene 17,
pyrene 18, silicon 2,3-naphthalocyanine bis(trihexylsilyloxide) (SiNc) 19 and
DNA nucleobases 20.

However, the interpretation of time-resolved signals in terms of nonadiabatic
dynamical processes is far from straightforward, and adequate theory is
indispensable to decipher the intrinsic system dynamics from the detected
spectroscopic responses. Nowadays, ab initio simulations of spectroscopic
responses are quickly establishing themselves, and a number of powerful
methods have been developed to calculate femtosecond nonlinear spectroscopic
signals of excitonic systems21, 22, 23, 24, 25, 26, 27 and polyatomic
chromophores28, 29, 30, 31, 32 ab initio on-the-fly. The majority of ab initio
on-the-fly protocols for the simulation of nonlinear spectroscopic signals
hinges upon mixed quantum-classical approaches in which the electrons are
treated quantum-mechanically and the nuclei classically 33, 34, 6. The most
widely employed mixed quantum-classical method for the description of
nonadiabatic dynamics is the trajectory surface hopping method 34, 6. In this
method, classical trajectories are propagated on a single PES and nonadiabatic
transitions between adiabatic PESs are treated via instantaneous switches, the
so-called “hops”. Many algorithms have been proposed to evaluate the hopping
probability, each with its advantages and disadvantages 33, 34, 35. One of
them, which we use in this work (see SI for details), is based on the
Landau–Zener formalism36, 37 that has an advantage of not suffering from over-
coherence. The transition probability in this formalism can be estimated
relatively easily based on the energy difference between adjacent adiabatic
electronic states and its second derivative with respect to time, thus
avoiding the calculation of rapidly varying and possibly divergent
nonadiabatic couplings. One limitation introduced using the Landau–Zener
formalism is the restriction that no more than two states may be close in
energy for it to be reliable.38

Usually, the simulation of nonlinear time-resolved spectroscopic signals by ab
initio methods requires a computationally tedious evaluation of third-order
response functions 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32. However,
such signals can be extracted from the dynamical simulations using
computationally much more efficient doorway-window approach 39. It is
applicable if the laser pulses are well separated in time, i.e., are non-
overlapping. The on-the-fly doorway-window simulation protocol27, 40, 41
adopted in this study is based on several additional approximations, the most
significant of which is the so-called classical Condon approximation (see SI
for details).

Despite the progress in developing theoretical framework for simulating time-
resolved spectra with the on-the-fly approaches, their application potential
is severely limited due to the high computational cost. As is becoming
increasingly common in such situations, machine learning (ML) tools developed
for AI research can be applied to overcome the cost limitations 42. Successful
examples in the related fields include creating surrogate ML models for
excited-state properties43, 44, which can be used to increase the precision of
linear absorption spectra 45, 46, uncertainty quantification 47, predict two-
photon absorption cross sections 48, and perform nonadiabatic molecular
dynamics of molecular systems49, 50, 51, 52, 53. Beyond linear spectra, ML
methods were also successfully used in the interpretation of nonlinear
spectroscopic signals, i.e., for the reconstruction of certain facets of
system dynamics from experimental transient-absorption pump-probe 54, 55 and
2D electronic spectra 56, 57, 58, 59, as well as for “denoising” experimental
signals 60. A few studies invoked ML for the evaluation of nonlinear signals,
i.e., for predicting 2D electronic spectra of proteins61, 62 and extracting
relevant parameters such as orientations of transition dipole moments from
these spectra 61. The application of ML for speeding up on-the-fly simulation
of time-resolved spectra has remained an unexplored avenue.

In this study we introduce an approach that leverages machine learning (ML) to
conduct direct, real-time trajectory simulations of the nonlinear time- and
frequency-resolved spectra of polyatomic chromophores in the gas phase. Our
adopted approach combines the well-established doorway-window protocol, used
for evaluating spectroscopic signals, with ML-accelerated on-the-fly surface-
hopping simulations of nonadiabatic molecular dynamics. To demonstrate the
efficacy of our approach, we apply it to the simulation of stimulated emission
contributions in transient-absorption pump-probe and two-dimensional (2D)
electronic spectra of pyrazine. Pyrazine serves as a quintessential molecular
system, known for its multiple conical intersections of the lower-lying
excited electronic states. The photophysics of pyrazine has been extensively
explored through ab initio quantum-chemistry calculations 63, 64, dynamic
simulations 65, 66, femtosecond photoelectron spectroscopy measurements 67,
68, 69, and calculations of nonlinear spectroscopic signals 40, 41, 70, 71,
72, 73.

Our ML approach yields highly accurate spectra of pyrazine while significantly
reducing computational costs compared to conventional (non-ML) ab initio
surface-hopping simulations. To achieve this, we developed a robust protocol
that involves the creation of ML surrogate models which learn adiabatic state
energies, forces, and oscillator strengths. The robustness of these models is
ensured by using a well-sampled training set that covers relevant regions on
PESs. Since it is practically impossible to guarantee that ML can confidently
reproduce all regions of high-dimensional PES, particularly those near
critical conical intersections, we employ uncertainty quantification to
identify and address problematic areas. In the following, we outline the
adopted protocol using pyrazine as an example, where we consider the first
five singlet adiabatic states S0–S4. Our findings demonstrate that this
protocol can slash the computational cost by up to 95%.

The process of sampling training data is of utmost importance. It must strike
a balance between ensuring model robustness and computational affordability.
In the case of pyrazine, our training set was generated from 50 Landau–Zener
surface hopping trajectories, propagated for 100 fs with a time step of 0.5
fs, using the ab initio method ADC(2) 74, 75. While the number and lengths of
trajectories are smaller than what is needed for precise spectra (as we will
show later), they suffice to cover relevant PES regions. From these
trajectories, we sampled 5,151 training points, capturing snapshots at
approximately 1 fs intervals (50% of the data set), while the remaining points
were held back for testing. The training set is further divided into sub-
training and validation sets in an 80:20 ratio, consisting of 4,120 and 1,031
data points, respectively.

The choice of the ML model architecture is another critical factor, and in
this work, we opt for ANI-type models due to their high accuracy and
efficiency in obtaining energies, gradients, and oscillator strengths within
short training and prediction times (see SI for details) 76.

The ML test errors for pyrazine, given in Table 1, are in the range of the
errors reported for slower type of ML models with more complex architecture in
similar tasks 51. The data follow the known trend 43, 44 of increasing error
for higher excited states. Encouragingly, the error for the oscillator
strengths is also rather low with ANI-type models.

Table 1: The root-mean-square errors (RMSEs) and mean absolute errors (MAEs) in the energies [eV] and gradients [eV$/\text{\r{A}}$] for their respective state as well as the RMSEs and MAEs in the oscillator strengths [a.u.] for the transition from the electronic ground-state to the corresponding excited state. | RMSE | MAE
---|---|---
| GS | S1 | S2 | S3 | S4 | GS | S1 | S2 | S3 | S4
Energies | 0.0076 | 0.0110 | 0.0139 | 0.0176 | 0.0315 | 0.0051 | 0.0082 | 0.0106 | 0.0130 | 0.0196
Gradients | - | 0.0422 | 0.0833 | 0.1059 | 0.1493 | - | 0.0247 | 0.0443 | 0.0540 | 0.0833
| RMSE | MAE
Osc. str. | - | 0.0012 | 0.0053 | 0.0068 | 0.0101 | - | 0.0005 | 0.0022 | 0.0030 | 0.0042

The accuracy of the ultrafast dynamics hinges on how well the PESs in the
vicinity of conical intersections are described. Since these regions have a
rather complex topography which is challenging for learning methods, we
decided to circumvent the difficult learning problem by switching to the ab
initio method for calculating excited-state properties when the energy gap of
adjacent electronic states is smaller than 0.1 eV as done previously 77, 50.
In addition, we monitor the uncertainty of the ML predictions by evaluating
the differences of energies and gradients generated from a pair of models for
each adiabatic state, as is common for related query-by-committee strategies
78. If the deviation exceeds 0.17 eV for energies or 0.17 eV$/$Å for energy
gradients, we recalculate the properties for the current step with the ab
initio method. This threshold was chosen with respect to the largest RMSE of
the gradients, as especially for the S4 the error increases, potentially
triggering an unnecessary number of ab initio calculations. This mitigation
strategy, depicted in Figure 1, greatly enhanced the robustness of the
simulations, while the number of required ab initio calculations only slightly
grew by about 1.6%. We propagated an increasing number of on-the-fly dynamics
trajectories for 200 fs based on the ML surrogate models with the
aforementioned mitigation strategies. We stopped running trajectories once the
resulting spectra were converged (Figure 1). The convergence of the spectrum
was judged on the mean absolute deviation of normalized intensities between
the current and previous set of trajectories. Each set contains 100 more
trajectories than the previous set. The criterion to consider the spectrum as
converged was set to a mean absolute deviation of $10^{-4}$. For pyrazine, a
total of 600 trajectories was sufficient to obtain a well-converged time-
resolved stimulated emission spectrum $I(T,\omega_{\mathrm{pr}})$, which is
plotted as a function of the population time $T$ and the probe-pulse carrier
frequency $\omega_{\mathrm{pr}}$ in Figure 2a (see Figure S2 in Supporting
Information for spectrum convergence). Excellent agreement between the ML
(Figure 2a) and the reference ab initio (Figure 2b) spectra in the entire
time-frequency domain demonstrates the robustness and accuracy of our
approach. Importantly, the spectrum simulated with 600 trajectories shows a
marked improvement over the spectrum obtained from 50 trajectories used to
furnish the training set (Figure 2c). Not just the intensities significantly
improved but, remarkably, the 200-fs ML spectrum is able to accurately capture
intricate features well beyond the 100-fs range of the training trajectories.
This proves that the precision and range of the ab initio spectrum can be
greatly improved with little additional cost by ML.

ML training on small number of short ab initio trajectoriesSample initial
conditions from the phase-space probability distributionUse ML model pairs to
obtain quantities y Uncertainty quantification and small-gap detection, i.e.,
$\Delta y\leq y_{\rm th}$ and $\Delta$E $\geq$ Eth? Triggerab
initiocalculationsAccept results from ML model pairsPropagate trajectory Has
tmax been reached? Spectrum convergence? End simulationYesNoNoYesNoYes

Figure 1: Flowchart depicting the steps in obtaining the time-resolved
spectrum as proposed in this work. $\Delta y$ denotes the difference in the
outputs of ML models in each model pair which is used for uncertainty
quantification. These differences are quantified for both energies and energy
gradients for each electronic state and checked whether they exceed the
thresholds $y_{\rm th}$. Furthermore, we check whether the absolute energy
difference between the populated and its adjacent states, $\Delta$E, exceeds
the threshold Eth.

The spectrum $I(T,\omega_{\mathrm{pr}})$ (Figures 2a, b) mirrors the
wavepacket motion in the coupled excited electronic states S1–S4, where
$\omega_{\mathrm{pr}}$ monitors position(s) of the wavepacket(s) at time $T$.
This spectrum, which gives a more detailed view of the conical-intersections
mediated dynamics than the excited-state adiabatic populations (see SI, Figure
S1) can be interpreted as follows. $I(T,\omega_{\mathrm{pr}})$ shows how a
Gaussian wavepacket created by the short pump pulse in the bright B2u state
($T\approx 0$, $\hbar\omega_{\mathrm{pr}}\approx 5.2$ eV) moves ballistically
driven by the slope of the PES towards the B2u–B3u conical intersection, which
is manifested as the bifurcation point in the spectrum around $T\approx 20$ fs
and $\hbar\omega_{\mathrm{pr}}\approx 4.2$ eV. The revival of the spectrum at
$T\approx 35$ fs and $\hbar\omega_{\mathrm{pr}}\approx 5.0$ eV signifies the
wavepacket movement to the region of its initial excitation. Due to the
predominantly irreversible dynamics at the conical intersection, large
portions of the wavepacket remain trapped in the coupled lower-lying B3u and
Au states, which possess low oscillator strengths 64 and therefore have
relatively weak contributions to the spectrum. $I(T,\omega_{\mathrm{pr}})$
exhibits maxima of decreasing intensity at $T\approx$ 85, 140 and 195 fs,
which are caused by partial revivals of the wavepacket on the complex
landscape of PESs of the coupled B2u, B3u, and Au states. The bifurcation of
the spectrum $I(T,\omega_{\mathrm{pr}})$ at $T=20$ fs as well as its
subsequent $\approx 50$ fs periodic revivals are corroborated experimentally
68, 69 and by fully-quantum simulations of the models of reduced
dimensionality 70, 71.

Figure 2:  Integral time-resolved stimulated-emission spectrum
$I(T,\omega_{\mathrm{pr}})$ of pyrazine as a function of the population time
$T$ and the carrier frequency $\omega_{\mathrm{pr}}$ of the probe pulse. (a).
The converged spectrum calculated with 600 ML-accelerated trajectories. (b).
Reference spectrum calculated with 600 pure ab-initio trajectories. (c).
Spectrum calculated from 50 pure ab-initio trajectories (100 fs), which were
used for the construction of the ML training set. Figure 3: Non-rephasing
stimulated-emission 2D electronic spectra
$I_{\mathrm{2D}}(\omega_{\tau},T,\omega_{t})$ of pyrazine as a function of the
excitation frequency $\omega_{\tau}$ and the detection frequency $\omega_{t}$
at population times $T=0$ (a), 20 (b), 35 (c), 60 fs (d) obtained with ML-
accelerated doorway-window method. For each $T$, the intensities of the
spectra are scaled for better visibility.

The non-rephasing stimulated emission 2D spectra
$I_{\mathrm{2D}}(\omega_{\tau},T,\omega_{t})$ at several population times $T$
obtained with the ML-accelerated doorway-window method are shown in Figure 3.
These spectra complement and refine the information on the photoinduced
dynamics of pyrazine extracted from the pump-probe spectra. Namely, 2D spectra
show how the wavepacket excited with photon energy $\hbar\omega_{\tau}$ and
detected with photon energy $\hbar\omega_{t}$ evolves during the population
time $T$. The single peak around $\hbar\omega_{\tau}=\hbar\omega_{t}\approx
5.2$ eV at $T=0$ fs in panel (a) is the signature of the initially-excited
bright B2u state. The peak in panel (b) illustrates that the wavepacket
created around $\hbar\omega_{\tau}\approx 5.2$ eV at $T=0$ fs moves to
$\hbar\omega_{t}\approx 4$ eV at $T=20$ fs, indicating the motion through the
B2u–B3u conical intersection. The peak in panel (c) mirrors the partial return
of the wavepacket to its initial location at $T=35$ fs. Panel (d) shows a
progression of peaks elongated around $\hbar\omega_{t}$. These peaks reveal
splitting of the wavepacket at $T=60$ fs among several areas around the flat
PES minimum of the coupled Au and B3u states. The stimulated emission 2D
spectra presented in Figure 3 are in excellent agreement with those obtained
by the reference 600-trajectory surface-hopping simulations 41 and fully-
quantum reduced-dimensional model calculations 70, 71. The spectrum quality is
also highly dependent on the number of trajectories used (see Figure S3 in
Supporting Information).

In summary, we implemented a ML-enhanced on-the-fly doorway-window protocol
for the simulation of nonlinear time- and frequency-resolved spectroscopic
signals. The protocol is demonstrated for the computation of the stimulated-
emission contribution to transient-absorption pump-probe and 2D electronic
spectra, showing a drastic (95%) reduction of the necessary ab initio
electronic structure calculations for the prototypical molecule of pyrazine. A
number puts this into perspective: 600 trajectories comprise a total of
251,135 time steps (accounting for convergence of the hopping probability) for
which calculations of quantum chemical properties have to be done. Our ML
approach allows to reduce the number of required ab initio calculations to
only 14,352 evaluations to generate the training data and ensure the
robustness of dynamics propagation. The switches between the predictions of
the ML models and ab initio calculations also conserved the total energy of
the system, preventing nonphysical events from occurring. The simulated
stimulated emission signals of pyrazine are in excellent agreement with the
available experimental and theoretical data, which unequivocally validates the
accuracy of the developed methodology. The latter opens up the possibility to
investigate larger molecular systems, as the prediction time of the ML model
scales linearly with the system size. After the inclusion of appropriate
higher-lying excited electronic states, our methodology can straightforwardly
be applied to the simulation of any nonlinear spectroscopic signal detected
with short laser pulses and on-the-fly doorway-window protocols.

S.V.P. and L.P.C. acknowledge support from the Key Research Project of
Zhejiang Lab (No. 2021PE0AC02). M. F. G. acknowledges support from the
National Natural Science Foundation of China (No. 22373028). P.O.D.
acknowledges funding by the National Natural Science Foundation of China (No.
22003051) and funding via the Outstanding Youth Scholars (Overseas, 2021), the
Fundamental Research Funds for the Central Universities (No. 20720210092), and
via the Lab project of the State Key Laboratory of Physical Chemistry of Solid
Surfaces as well as support by Science and Technology Projects of Innovation
Laboratory for Sciences and Technologies of Energy Materials of Fujian
Province (IKKEM) (No: RD2022070103). The authors also acknowledge Fuchun Ge
and Lina Zhang for discussions.

##  1 Data availability statement

The training and test data as well as the trained ML models are available at
https://github.com/psebastianzjl/pyrazine_stimulated_emission.

##  2 Code availability statement

All modified codes from other program packages as well as the codes for
predicting the stimulated emission spectra are available at
https://github.com/psebastianzjl/pyrazine_stimulated_emission.

##  3 Competing interests

The authors declare that they have no competing interests.

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