# The Photochemistry of Rydberg-Excited Cyclobutanone: Photoinduced Processes
and Ground State Dynamics

J. Eng  Chemistry, School of Natural and Environmental Sciences, Newcastle
University, Newcastle upon Tyne, NE1 7RU, UK<EMAIL_ADDRESS>C. D.
Rankine  Department of Chemistry, University of York, York, YO10 5DD, UK  T.
J. Penfold<EMAIL_ADDRESS>Chemistry, School of Natural and
Environmental Sciences, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK

###### Abstract

Owing to ring-strain, cyclic ketones exhibit complex excited-state dynamics
with multiple competing photochemical channels active on the ultrafast
timescale. While the excited-state dynamics of cyclobutanone after
$\pi^{\ast}\leftarrow n$ excitation into the lowest-energy excited singlet
state (S1) has been extensively studied, the dynamics following 3$s\leftarrow
n$ excitation into the higher-lying singlet Rydberg (S2) state are less well
understood. Herein, we couple quantum and excited-state trajectory surface-
hopping molecular dynamics simulations to study the relaxation of
cyclobutanone following 3s$\leftarrow n$ excitation and to predict the
ultrafast electron diffraction scattering signal that we anticipate to arise
from the relaxation dynamics that we observe. Our simulations indicate that
relaxation from the initially-populated singlet Rydberg state occurs on the
hundreds-of-femtosecond to picosecond timescale consistent with the symmetry-
forbidden nature of the state-to-state transition involved. Once cyclobutanone
has relaxed non-radiatively to the electronic ground state (S0), the
vibrationally hot molecules have sufficient energy to form multiple
fragmentory products on the electronic ground-state surface including C2H4 \+
CH2CO (C2; 20%), and C3H6 \+ CO (C3; 2.5%). We discuss the limitations of our
simulations, how these may influence the outcome of the excited-state dynamics
we observe, and – ultimately – the predictive power of the simulated
experimental observable.

##  I Introduction

Ketones, i.e. organic compounds containing a carbonyl group, are among the
simplest chromophores owing to their small size and low density of valence
excited states. Consequently, their photochemistry has been intensively
studied for several decades.Norrish and Bamford (1936); Turro  _et al._ (1972)
Cyclic ketones form an important sub-class of these systems; despite their
apparently equivalent simplicity, their photochemistry is often comparatively
complex and features activity across multiple competing photochemical channels
most commonly characterised by Norrish Type I processes. Here, upon excitation
of an electron from the non-bonding orbital ($n$) on the carbonyl oxygen atom
to the antibonding ($\pi^{\ast}$) molecular orbital of the carbonyl group, a
carbon-carbon bond adjacent to the carbonyl (i.e. the carbon-carbon bond to
the $\alpha$ carbon, Figure 1) cleaves, opening up the possibility of the
formation of a variety of fragmentory products on the electronic ground-state
surface after non-radiative relaxation. Amongst cyclic ketones, cyclobutanone
has particular photochemistry owing to the high degree of ring strain in the
cyclobutane ring (arising as a consequence of the small ring size).Kao  _et
al._ (2020) The particular photochemistry of cyclobutanone has attracted
significant interest from both an experimental Denschlag and Lee (1967);
Campbell and Schlag (1967); Morton and Turro (1973); Hemminger and Lee (1972);
Lee and Lee (1969); Turro and Southam (1967); Kao  _et al._ (2020); Diau,
Kötting, and Zewail (2001); Kuhlman, Sølling, and Møller (2012); Zhang,
Chiang, and Laane (1994); Lee, Hemminger, and Rusbult (1971); Whitlock and
Duncan (1971); Trentelman  _et al._ (1990); Larsen, Stephansen, and Sølling
(2017) and theoretical Xia  _et al._ (2015); Liu and Fang (2016); Hopkinson,
Lee-Ruff, and Lien (1988); Momicchioli, Baraldi, and Di Lonardo (1975); Dalton
and Turro (1970); Chen and Ye (2004); Kuhlman  _et al._ (2012, 2013a)
perspective.

Figure 1: Schematic of cyclobutanone and the potential photoproducts
identified in previous work.

Many of these previous studies have aimed at elucidating the photochemistry
and, in particular, the excited-state dynamics arising from excitation at the
weak S${}_{1}\leftarrow$S0 absorption band (ca. 330-240nm) associated with the
symmetry-forbidden $\pi^{\ast}\leftarrow n$ transition. Early ultrafast
spectroscopy carried out by Diau et al.Diau, Kötting, and Zewail (2001)
reported that, upon excitation at 307 nm (i.e. slightly above energy of the
$\pi^{\ast}\leftarrow n$ transition by ca. 2 kcal.mol-1) $\alpha$-cleavage
occurs over a timescale of ca. 5 ps, driven initially by the C=O out-of-plane
wagging, cyclobutane ring puckering, and C=O stretching modes. Following
$\alpha$-cleavage, a channel along which an S1/S0 conical intersection (CI) is
encountered and via which non-radiative relaxation to the electronic ground
state can proceed, cyclobutanone can undergo substantial structural changes
that lead to the production of either i) a vibrationally-hot S0 species, ii) a
diradical intermediate that fragments to yield C2H4 \+ CH2CO (C2 products), or
iii) the formation of C3H6 \+ CO (C3 products). The proposed fragmentation
mechanism is consistent with the observed wavelength dependence of the C3:C2
product ratios;Denschlag and Lee (1967); Lee and Lee (1969); Lee, Hemminger,
and Rusbult (1971); Hemminger and Lee (1972); Breuer and Lee (1971) this ratio
is reported to be 0.5 on excitation at 313 nm, and increases to 0.8 at 248 nm
and 1.0 at 200 nm.Lee, Hemminger, and Rusbult (1971); Tang and Lee (1976) At
wavelengths longer than 315 nm, there is a marked increase in the C3:C2
product ratio; it is reported to be 2.0 at 326 nm and as high as 7.0 at 344
nm.Hemminger and Lee (1972); Tang and Lee (1976) The increase is indicative of
an alternative mechanism in the S1 state which becomes operational at longer
wavelengths and it is proposed to involve photochemistry on the triplet
manifold Tang and Lee (1976) as there is insufficient energy available to
overcome the barrier encountered along the $\alpha$-cleavage channel and reach
the S1/S0 CI. This mechanistic picture is largely maintained in solution as
demonstrate by Kao et al.Kao  _et al._ (2020) who observed Norrish Type I
$\alpha$-cleavage on the sub-picosecond timescale, a consequence of excess
vibrational energy available from UV-photon absorption from the electronic
ground state. Kao et al.Kao  _et al._ (2020) proposed that the observed
wavelength dependence arose from the need to overcome a barrier on the S1
surface (e.g. the barrier encountered along the $\alpha$-cleavage channel). In
addition, while triplet states have be invoked in the Norrish mechanism,Lee,
Hemminger, and Rusbult (1971) Kao et al.Kao  _et al._ (2020) reported that,
even with sufficient excess vibrational energy available, triplet states may
still play a mechanistic role although only in indirect channels active on
timescales longer than 500 ps.

The second absorption band of cyclobutanone occurs between 206 to 182 nm and
has been assigned to the $3s\leftarrow n$ transition into the second
electronically-excited singlet state (S2); a state that exhibits Rydberg
character.Whitlock and Duncan (1971) Trentelman et al.Trentelman  _et al._
(1990) have studied the 193 nm photolysis of cyclobutanone and have reported
that 57% of the electronically-excited-state species form the C3 products, 30%
form the C2 products in the electronically excited state, and 13% form the C2
products in the (hot) electronic ground state. Importantly, the observations
of Trentelman et al.Trentelman  _et al._ (1990) suggest that the formation of
the C3 products is potentially a slow process as it requires intersystem
crossing (ISC) onto the triplet manifold. However, the proposed photochemical
mechanism assumes an S1 intermediate that survives long enough to undergo
thermalisation and, therefore, produce a statistical partitioning of products;
this assumption does not necessarily hold given the speed at which S0
$\leftarrow$ S1 relaxation is expected to take place. Kuhlman et al.Kuhlman,
Sølling, and Møller (2012); Kuhlman  _et al._ (2013b) have studied the 200 nm
photolysis of cyclobutanone, comparatively, and reported conversion from the
S2 to the S1 associated with two time constants of ca. 350 and ca. 750 fs.
Using time-resolved mass spectrometry (TR-MS), the authors were able to
resolve two fragments: the parent, and a fragment with a mass-to-charge ratio
$m/z=42$, corresponding to H2C–C=O, both exhibiting similar electronically-
excited-state decay constants. The authors also reported a blue-shift in the
photoelectron spectrum, arising from a coherent oscillatory motion assigned to
a low frequency ring puckering mode with a frequency of 35 cm-1, promoted by
removal of an electron from an oxygen $n$ orbital which leads to a relaxation
of the sp2 hybridization of the carbonyl carbon due to mixing with components
of the bonding orbitals in the carbonyl group as well as a relaxation of the
adjacent C-C bonds.

From the perspective of theory, Xia et al.Xia  _et al._ (2015) performed a
quantum chemistry investigation optimized minima, transition states, MECIs,
and relaxed two-dimensional S1 and T1 potential energy surfaces using high-
level quantum chemical calculations. On the basis of these calculations, Xia
et al. proposed that ring opening predominately occurs in the S1 state through
an accessible S1/S0 MECI. Liu et al.Liu and Fang (2016) extended this work by
performing dynamics simulations within the ab initio multiple spawning
(AIMS)Ben-Nun, Quenneville, and Martínez (2000); Curchod and Martínez (2018)
framework following S1 photoexcitation. They observed that relaxation occurred
primarily through the S1/S0 MECI associated with fission of the first
$\alpha$-C-C bond, which fell into the region with the $\beta$-C-C bond is
$\sim$1.6 Å, and the $\alpha$-C-C bond between 2.5-3.5 Å. In comparison they
found that passage through intersections associated with the formation of the
C2 products, occurred when the $\beta$-C-C bond was around 4 Å, but this
constitutes only 15%. They reported an excited state lifetime in the S1 state
of ca. 500 fs and, on this basis, proposed that non-ergodic behavior was the
driver of the change in C3:C2 branching ratio as a function of excitation
wavelength. In contrast, Kuhlman et al.Kuhlman  _et al._ (2013a) developed a
four-state, five-dimensional model Hamiltonian and carried out quantum
dynamics to study the S1 $\leftarrow$ S2 relaxation dynamics. They concluded
that S1 $\leftarrow$ S2 relaxation involves specific nuclear modes included in
the model Hamiltonian (including the cyclobutane ring-puckering, carbonyl out-
of-plane deformation, symmetric and asymmetric C-CO-C stretches, and carbonyl
stretching modes) that couple the S2 and S1 states and promote population
transfer on the picosecond timescale. While informative, these potentials are
built within a harmonic normal mode representation and are unable to address
satisfactorily large amplitude nuclear motions associated with the formation
of photoproducts.

Despite the concerted efforts of experiment and theory, there remain a
significant number of open questions regarding the photochemistry and
photochemical dynamics of cyclobutanone which evolve post-photoexcitation to
the S2 Rydberg ($3s\leftarrow n$) state. Nonetheless, the emergence of modern
light sources Filippetto  _et al._ (2022); Pellegrini, Marinelli, and Reiche
(2016) is facilitating the study of ultrafast structural dynamics using both
X-rays and electronsChergui and Zewail (2009); Yang  _et al._ (2020); Katayama
_et al._ (2023); Minitti  _et al._ (2015) with ever-increasing spatial and
temporal resolution and providing the potential for new and increasingly
detailed insights into complex photochemical processes such as these. These
developments are bringing into focus a crucial question (which is the focus of
the present Special Issue to which this Article contributes): how accurate are
modern nonadiabatic excited-state molecular dynamics simulations really?

In this Article, we combine quantum- and excited-state trajectory surface-
hopping molecular dynamics simulations at the LR-TDDFT(PBE0) and ADC(2) levels
of theory to explore the relaxation dynamics of cyclobutanone post-
photoexcitation to the S2 Rydberg ($3s\leftarrow n$) state. Our excited-state
molecular dynamics simulations are subsequently used predict the scattering
signals for an ultrafast electron diffraction experiment in reciprocal and
real space to establish how the complex photochemical dynamics of
cyclobutanone might manifest as an experimental observable. In such an
ultrafast electron diffraction experiment, free cyclobutanone molecules
introduced in vacuo will be photoexcited at 200 nm [i.e. into the S2 Rydberg
($3s\leftarrow n$) state] and probed via electron scattering at a sequence of
temporal delays to image directly the evolving excited-state structural
dynamics.

##  II Theory and Computational Details

###  II.1 Quantum Chemistry

All quantum chemical calculations were carried out using Turbomole (v7.4).tur
; Ahlrichs  _et al._ (1989) The quantum chemical calculations on the
electronic ground and excited states used density functional theory (DFT) and
linear-response time-dependent DFT (LR-TDDFT), respectively, with the PBE0
functionalAdamo and Barone (1999) and the aug-cc-pVDZ basis set.Dunning
(1989); Kendall, Dunning, and Harrison (1992) The Tamm-DancoffHirata and Head-
Gordon (1999) approximation (TDA) was used throughout. Vibrational analysis
were performed to confirm the absence of imaginary frequencies at the ground
state minimum. Minimum-energy conical intersections (MECIs) were optimised at
the LR-TDDFT(PBE0) level using Turbomole (v7.4) coupled with an external
(penalty-function-based) optimiser.Rankine . Linear reaction channels
(produced via linear interpolation in internal coordinates; LIIC) between
critical geometries and these CIs are available in the SI alongside benchmarks
of the electronic structure and basis set.

Additional quantum chemical calculations were carried out at critical
geometries using the second-order algebraic diagrammatic construction scheme
[ADC(2)] and n-electron valence state perturbation theory (NEVPT2). All ADC(2)
calculations were carried out using Turbomole (v7.4);tur ; Ahlrichs  _et al._
(1989) all NEVPT2 calculations were carried out using ORCA.Neese (2011, 2017)

###  II.2 Vibronic Coupling Hamiltonian and Quantum Dynamics

The model Hamiltonian developed is based upon the vibronic coupling
approximationKöppel, Domcke, and Cederbaum (2004); Cederbaum, Köppel, and
Domcke (2009) and is expressed as:

$\mathbf{H}=\left(T_{N}+V_{0}\right)\mathbf{1}+\mathbf{W}^{dia.}+\mathbf{W}^{SOC}$
(1)

The electronic diabatic Hamiltonian elements, $W_{n,n}$, are obtained by
expanding $\mathbf{W}-V_{0}\mathbf{1}$ the diabatic potential as a Taylor
series around a reference nuclear geometry, $Q_{0}$, here taken as the Franck-
Condon (e.g. S0 minimum-energy) geometry. $V_{0}$ is a reference potential,
here defined as a set of harmonic potentials with vibrational frequencies
$\omega_{i}$ along dimensionless normal coordinates $Q_{i}$. In this case, the
Hamiltonian elements are expressed as:

$W_{n,m}-V_{o}\delta_{nm}=\epsilon_{n}\delta_{nm}+\sum_{i}^{3N-6}\left.\frac{\partial
W_{n,m}}{\partial
Q_{i}}\right|_{Q_{0}}Q_{i}+\frac{1}{2!}\sum_{i,j}^{3N-6}\left.\frac{\partial^{2}W_{n,m}}{\partial
Q_{i}\partial
Q_{j}}\right|_{Q_{0}}Q_{i}Q_{j}+\frac{1}{3!}\left.\frac{\partial^{3}W_{n,m}}{\partial
Q_{i}\partial Q_{j}\partial Q_{k}}\right|_{Q_{0}}Q_{i}Q_{j}Q_{k}+\cdots$ (2)

where $\delta_{nm}$ is the Kronecker delta. $Q_{i}$ denotes the $3N-6$
dimensionless normal coordinates related to the normal modes of vibration
where $N$ is the number of atoms. $V_{0}$, under the harmonic approximation,
is expressed as:

$V_{0}=\frac{1}{2}\omega_{i}Q_{i}^{2}$ (3)

and the kinetic energy operator, $\hat{T}_{N}$, takes the form:

$\hat{T_{N}}=-\sum_{i}\frac{1}{2}\omega_{i}\frac{\partial^{2}}{\partial
Q_{i}^{2}}$ (4)

Due to the anharmonicity of the potential energy surface, the Taylor series
expansion is carried out up to fourth order.Penfold and Worth (2009); Penfold
_et al._ (2012) Obtaining the expansion coefficients for the Hamiltonian is
simplified for the present system via use of symmetry; the minimum-energy S0
geometry of cyclobutanone is Cs symmetric and, consequently, first- and
second-order couplings are only allowed if the following selection rules are
satisfied:

$\Gamma_{n}\otimes\Gamma_{Q_{a}}\otimes\Gamma_{n}\supset A^{\prime}$ (5)
$\Gamma_{n}\otimes\Gamma_{Q_{a}}\Gamma_{Q_{b}}\otimes\Gamma_{n}\supset
A^{\prime},$ (6)

All expansion coefficients were obtained using a development version of the
in-house-developed VCMakerPenfold and Eng (2023); Pope  _et al._ (2024)
software, available at Ref. Eng, J. .

Quantum dynamics simulations over the multi-dimensional potential energy
surface(s) were carried out using the multiconfigurational time-dependent
Hartree (MCTDH)Beck  _et al._ (2000); Meyer, Gatti, and Worth (2009) approach
as implemented in Quantics.Worth  _et al._ The initial wavefunction for the
electronic ground state was built using one-dimensional harmonic oscillator
functions with zero initial momentum and vertically excited into the S2 state
at the Franck-Condon geometry ($Q$ = 0). The complete computational details
are provided in the SI and ensured convergence for the 2 ps of dynamics
presented.

###  II.3 Excited-State Trajectory Surface-Hopping Molecular Dynamics

Ab initio excited-state trajectory surface-hopping molecular dynamics were
performed using Newton-X (v2.4).Barbatti  _et al._ (2013, 2007) The potential
energy surfaces, derivatives, and respective couplings were computed on-the-
fly using Turbomole (v7.4)tur ; Ahlrichs  _et al._ (1989) at two separate
levels of theory [LR-TDDFT(PBE0) and ADC(2)] in two separate sets of
simulations. The trajectories were propagated using the velocity Verlet
algorithmVerlet (1967); Swope  _et al._ (1982) for a maximum of 5 ps
($t_{\mathrm{max}}=5000$ fs) with a time step of 0.5 fs ($dt=0.5$ fs). The
state-to-state transitions were simulated using the Hammes-Schiffer-and-Tully
fewest-switches algorithmHammes-Schiffer and Tully (1994) with the state-to-
state coupling estimated via the time-dependent Baeck-AnT. do Casal  _et al._
(2021) coupling scheme. The excited-state trajectory surface-hopping molecular
dynamics simulations were initiated by simulated vertical projection of a set
of initial conditions, generated according to a Wigner distribution with a
temperature of 100K, from the electronic ground state (S0) into the S2 Rydberg
($3s\leftarrow n$) state.

To avoid instabilities at/around the S1/S0 crossing seam, S0 $\leftarrow$ S1
surface hops were forced when the S1/S0 energy gap fulfilled the criterion
$\Delta{E}_{\mathrm{{S_{1}-S_{0}}}}<0.1$ eV.

###  II.4 Electron Diffraction Simulations

Throughout this work, we present the ultrafast electron diffraction scattering
signal as a modified electron scattering intensity ($sM(\mathbf{s})$) as
computed under the independent atom model (IAM). This presentation is used to
enhance the oscillations in the scattering signal associated with the
molecular interference terms and suppress the rapid drop-off in scattering
signal intensity as a function of $s$ from the elastic scattering amplitude.
The modified electron scattering intensity is given by:

$sM(\mathbf{s})=\dfrac{I_{\mathrm{mol.}}(\mathbf{s})}{I_{\mathrm{at.}}(\mathbf{s})}\mathbf{s}$
(7)

where $\mathbf{s}$ is the momentum transfer, or scattering, vector.
$I_{\mathrm{at.}}(\mathbf{s})$ is the atomic scattering term, given by:

$I_{\mathrm{at.}}(\mathbf{s})=\sum_{i}^{N}|f_{i}(\mathbf{s})|^{2}$ (8)

and $I_{\mathrm{mol.}}(\mathbf{s})$ is the molecular scattering term,
expressed as a sum of interference terms, and given by:

$I_{\mathrm{mol.}}(\mathbf{s})=\sum_{i}^{N}\sum_{j\neq
i}^{N}|f_{i}(\mathbf{s})||f_{j}(\mathbf{s})|\dfrac{sin(\mathbf{s}r_{ij})}{\mathbf{s}r_{ij}}$
(9)

where $f_{i}(\mathbf{s})$ and $f_{j}(\mathbf{s})$ are the elastic scattering
amplitudes for atoms $i$ and $j$, respectively, and $r_{ij}$ is the
internuclear distance between atoms $i$ and $j$.

$sM(\mathbf{s})$ can be transformed into a pair-distribution function (PDF;
i.e. from reciprocal space into real space) using a sine transform:

$\mathrm{PDF}(\mathbf{r})=\int_{0}^{s_{\mathrm{max}}}sM(\mathbf{s})\sin(\mathbf{sr})e^{-k\mathbf{s}^{2}}ds$
(10)

where $s_{max}$ is the maximum momentum transfer in the data, $r$ is the
internuclear distance between atom pairs, and $k$ is a damping factor used to
drive down smoothly the contribution(s) at high $s$ to zero. A damping factor
of $k=0.03$ was used throughout.

The focus of this work is upon the ultrafast excited-state dynamics of
cyclobutanone and, accordingly, the ultrafast electron diffraction scattering
signal is presented as a transient (i.e. as an excited state $-$ ground state
’difference’) signal where appropriate under the nuclear ensemble model. The
ground-state signal used to generate the transient is predicted from ground-
state molecular dynamics, i.e. a nuclear ensemble of equilibrated
configurations pre-photoexcitation. No scaling for, e.g., excitation
percentage/photolysis yield has been carried out.

##  III Results

###  III.1 Characterizing Critical Points on the Ground- and Electronically-
Excited Potential Energy Surfaces

Table 1 shows the energies of the low-lying singlet (Sn; $n=[1..5]$) and
triplet (Tn; $n=[1..5]$) electronically-excited states of cyclobutanone.
Figure S1 shows the molecular orbitals involved in the electronic transitions.
The symmetry point group of cyclobutanone (Cs at the S0 minimum-energy
geometry) is used to characterise the symmetries of the electronically-excited
states (an important facet of the generation and interpretation of the model
Hamiltonian used in the following section).

At the S0 minimum-energy geometry, the S1 and S2 excited states are located
4.21 and 5.99 eV, respectively, above the electronic ground state at the
ADC(2)/aug-cc-pVTZ level of theory, and at 4.33 and 6.10 eV, respectively,
above the electronic ground state at the LR-TDDFT(PBE0)/aug-cc-pVTZ level of
theory. Both levels of theory give good agreement with the experimental
absorption spectrum recorded by Diau et al.,Diau, Kötting, and Zewail (2001)
in which absorption bands are observed at ca. 4.2 and ca. 6.2 eV. At all
levels of theory the S1 and S2 states have $\pi^{\ast}\leftarrow n$ (LUMO
$\leftarrow$ HOMO) and $3s\leftarrow n$ (LUMO+1 $\leftarrow$ HOMO) character,
respectively. The Rydberg character of the S2 state requires a larger basis
set to describe accurately its energy and electronic structure, while – in
contrast – the valence S1 state exhibits little to no dependence on the basis
set (Table S1). Both the S2 and S1 states are of A" symmetry; direct first-
order vibronic coupling between these two states is consequently forbidden,
and this can be expected to slow down the rate of internal conversion between
the two states.

The ADC(2) calculations clearly give very good agreement with experimental
observations at the Franck-Condon geometry. However, previous work has
highlighted a limitation of this approach for studying non-radiative pathways
of carbonyl-containing molecules Marsili, Prlj, and Curchod (2021) as it
predicts an artificial S1/S0 crossing along C=O vibrational arising from a
n$\rightarrow\pi^{\ast}$ which is too shallow combined with a ground state
which destabilises too rapidly. To address this, we calculate the potential
along the C=O stretching normal mode, which is shown in Figure S2 calculated
using LR-TDDFT(PBE0), ADC(2) and NEVPT2(12,12) levels of theory. This shows
good agreement between the LR-TDDFT(PBE0) and NEVPT2(12,12) simulations, and
although the ground state and S1 surfaces become closer, they do not cross. In
contrast, as expected from ref. Marsili, Prlj, and Curchod (2021), there is a
clear crossing between the two surfaces in ADC(2), which corresponds to the
C=O bond length of $\sim$1.6 Å. This low lying accessible crossing point is
likely to distort the excited state dynamics crossing from the S1 to ground
state surface and indeed such dynamics are presented in the supplementary
material.

State | ADC(2) | LR-TDDFT(PBE0)
---|---|---
| Energy / eV | Character (Irrep.) | Energy / eV | Character (Irrep.)
GS | 0.00 | — (A’) | 0.00 | — (A’)
S1 | 4.21 | H$\rightarrow$L (A") | 4.33 | H$\rightarrow$L (A")
S2 | 5.99 | H$\rightarrow$L+1 (A") | 6.10 | H$\rightarrow$L+1 (A")
S3 | 6.53 | H$\rightarrow$L+3 (A") | 6.67 | H$\rightarrow$L+3 (A")
S4 | 6.64 | H$\rightarrow$L+2 (A’) | 6.77 | H$\rightarrow$L+2 (A’)
S5 | 6.68 | H$\rightarrow$L+4 (A") | 6.78 | H$\rightarrow$L+4 (A")
T1 | 3.87 | H$\rightarrow$L (A") | 3.73 | H$\rightarrow$L (A")
T2 | 5.95 | H-1$\rightarrow$L (A’) | 5.98 | H-1$\rightarrow$L (A’)
|  | H-4$\rightarrow$L |  | H-4$\rightarrow$L
T3 | 6.32 | H$\rightarrow$L+1 (A") | 6.02 | H$\rightarrow$L+1 (A")
T4 | 6.50 | H$\rightarrow$L+3 (A") | 6.61 | H$\rightarrow$L+3 (A")
T5 | 6.62 | H$\rightarrow$L+2 (A’) | 6.63 | H$\rightarrow$L+2 (A’)
Table 1: Summary of electronic energies, characters, and symmetries of the
ground- (GS) and electronically-excited (Sn/Tn; $n=[1..5]$) states of
cyclobutanone evaluated at the S0 minimum-energy geometry and at the
ADC(2)/aug-cc-pVTZ and LR-TDDFT(PBE0)/aug-cc-pVDZ levels of theory.
Comparative tables for evaluations at the S1\- (Table S3) and S2-state (Table
S4) minimum-energy geometries are presented in the SI. The relevant molecular
orbitals are shown in the SI. The HOMO is designated as H; the LUMO is
designated as L.

The S1 state minimum-energy geometry is reached from the Franck-Condon point
via an out-of-plane puckering and slight elongation of the C=O bond (1.21 to
1.26 Å); the C-C bonds in the cyclobutane ring remain almost unchanged. These
structural changes destabilise the electronic ground state (Table S3); it is
increased in energy by 1.13 eV relative to the S0 minimum-energy geometry at
the LR-TDDFT(PBE0)/aug-cc-pVDZ level of theory. In contrast, the S1
($\pi^{\ast}\leftarrow n$) state is stabilised, decreasing the energy gap with
the S0 state while increasing the gap with the higher-lying singlet states
(Sn; $n>1$). The predicted emission energy from the S1 minimum-energy geometry
is ca. 3.9 eV, a value that is in excellent agreement with the emission
spectrum recorded by Lee et al.Lee, Shortridge Jr, and Rusbult (1971) The
broad band observed in the absorption spectrum recorded by Diau et al.Diau,
Kötting, and Zewail (2001) is consistent with a short-lived electronically-
excited state, however, which is indicative of competitive photochemical
channels, e.g. non-radiative decay on the femto/picosecond timescale through
accessible CIs as discussed by Liu et al.Liu and Fang (2016)

The S2 state minimum energy geometry (Table S4), in contrast, is reached from
the Franck-Condon point via contraction of the C=O bond (1.21 to 1.16 Å) and
slight elongation of the C-C bonds in the cyclobutane ring. The largest
influence on energy observed is a destabilisation of the ground state, which
increases in energy by 0.51 eV with a similar (0.58 eV) increase observed for
the S1 state. The vibronic structure observed in the absorption spectrum for
this state exhibits a distinct vibronic structure Diau, Kötting, and Zewail
(2001) indicating both a longer lived excitation state and dominant
vibrational modes activated upon excitation, which will be discussed in the
following section.

Previous studies, e.g. Kao et al.,Kao  _et al._ (2020) have hypothesised as to
the potential role of triplet states in the photochemistry of cyclobutanone;
this is our motivation for presenting these states (Tn; $n=[1..5]$) in Table 1
and the relevant state-to-state spin-orbit couplings (SOCs) in the SI (Table
S7). At the Franck-Condon geometry, the lowest-energy electronically-excited
triplet state (T1) is located at 3.73 eV above the electronic ground state at
the LR-TDDFT(PBE0)/aug-cc-pVDZ level of theory and is of the same
$\pi^{\ast}\leftarrow n$ character as the S1 state; consequently, S1/T1 SOC
will be formally forbidden.Penfold and Worth (2010); Penfold  _et al._ (2018)
The T2 and T3 states are near-degenerate and located a little under the S2
(6.10 eV) at 5.98 and 6.02 eV, respectively, above the electronic ground state
at the LR-TDDFT(PBE0)/aug-cc-pVDZ level of theory. The SOC between these
triplet states and the low-lying singlet electronically-excited states is
generally small (Table S7), suggesting that the triplet states are unlikely to
play a significant role in the early-time (e.g. $<$2 ps) photochemistry
focused upon in the present work.

Figure 2: Key geometries of cyclobutanone: (a) the S0 minimum-energy geometry,
(b) the (symmetry-broken) S2/S1 MECI, and the S1/S0 (c) Cα-cleavage, (d) Cβ-
cleavage, and (e) concerted Cα-/Cβ-cleavage MECIs. All geometries were
optimised at the LR-TDDFT(PBE0)/aug-cc-pVDZ level of theory.

Figure 2b shows the structure of an S2/S1 MECI, while Figures 2c-e show the
structures of three S1/S0 MECI. Cartesian coordinates are given in the SI.
Potential energy surface(s) between the Franck-Condon point and each of the
S1/S0 MECI were calculated via linear interpolation in internal coordinates
(LIIC) at the LR-TDDFT(PBE0) level are also given in the SI.

The S2/S1 MECI (Figure 2b) is located at 5.85 eV above the S0 minimum-energy
geometry, i.e. ca. 0.5 eV below the $3s\leftarrow n$ excitation energy, at the
LR-TDDFT(PBE0)/aug-cc-pVDZ level of theory. Its structure is similar to the
Franck-Condon geometry although it features a symmetry-breaking distortion of
the cyclobutane ring (seen clearly on inspection of the position of the
hydrogen atoms in Figure 2b). The potential energy surface between the Franck-
Condon geometry and this S2/S1 MECI is barrierless (Figure S3) which would
suggest that S2/S1 internal conversion should be (ultra)fast in the absence of
any additional considerations, however the symmetry of the two states at the
Franck-Condon geometry is such that the interstate coupling is forbidden; even
at the distorted S2/S1 MECI geometry, it is weak and results in a predictably
longer lifetime for the S2 state.

The three S1/S0 MECI (Figure 2c-e) obtained are in close qualitative agreement
(geometrically and energetically) with those located at the CASPT2 level of
theory and reported in Ref. Liu and Fang (2016). The first is located along
the $\alpha$-cleavage channel, the second along the $\beta$-cleavage channel,
and the third along a concerted $\alpha$/$\beta$-cleavage channel. The
energetic ordering of the three S1/S0 MECIs (3.40, 2.56 and 5.0 eV,
respectively, above the S0 minimum-energy geometry at the LR-TDDFT(PBE0)/aug-
cc-pVDZ level of theory) follows qualitatively the trend observed for the
three S1/S0 MECIs in Ref. Liu and Fang (2016), although it is important to
note that the single-reference nature of LR-TDDFT(PBE0) favours charged rather
than biradical dissociation along the $\alpha$\- and $\beta$-cleavage channels
and, furthermore, renders it unable to describe properly the
topology/dimensionality of the S1/S0 crossing seam. Although all of the S1/S0
MECIs are energetically accessible (i.e. they and their barriers are submerged
relative to the ca. 6.2 eV excitation energy, the description of the potential
energy surface around the S1/S0 MECIs is likely to be problematic for LR-
TDDFT(PBE0) and it is quite possible that this might influence the
photoproduct production by influencing the internal conversion dynamics
through the MECI and, subsequently, the motion of the trajectory/wavepacket on
the electronic ground state potential energy surface.

Overcoming these aforementioned limitations could be achieved using a
multireference (active space) method, e.g. CASSCF/CASPT2 or NEVPT2, for the
excited-state molecular dynamics simulations. However, the performance of
these methods is greatly dependent on the choice of active space; an
appropriate active space should be large enough to incorporate all of the
orbitals required over all of the nuclear configurations explored in the
excited-state molecular dynamics simulations while not too large so as to
render the simulations computationally costly to the point of intractability.
We found an active space of eight electrons in eight orbitals [e.g.
NEVPT2(8,8)] unstable with respect to orbital rotation(s) at some distorted
cyclobutane geometries, while a larger active space of twelve electrons in
twelve orbitals [e.g. NEVPT2(12,12)] was too computationally expensive to
carry out practicably excited-state molecular dynamics simulations.
Consequently, in the present work, we have carried out our excited-state
molecular dynamics simulations using LR-TDDFT(PBE0) and ADC(2), keeping in
mind the aforementioned (although well-understood) limitations and their
potential impact on the dynamics (which we discuss in detail). However, we
note within the context of the present challenge, other contributors have
performed excited-state molecular dynamics simulations with CASSCF based on an
eight-electron-in-eleven-orbital active spaceDaría  _et al._ (2024) and
extended multistate CASPT2 (XMS-CASPT2) based on an eight-electron-in-eight-
orbital active space.Janoš  _et al._ (2024)

###  III.2 Early-Time Dynamics Using a Spin-Vibronic Coupling Hamiltonian

To i) identify possible photochemical (fragmentation) channels, ii) clarify
the potential involvement of triplet states, and iii) establish a mechanism
for internal conversion from the initially-excited S2 ($3s\leftarrow n$) state
to the lowest-energy singlet electronically-excited state (S1) at early times,
we employ a model Hamiltonian and carry out quantum dynamics simulations as
described above. The model Hamiltonian comprises the electronic ground state
(S0) and nine electronically-excited states [four singlets (Sn; $n=[1..4]$)
and five triplets (Tn; $n=[1..5]$)]. The inclusion of singlet states higher in
energy than the S2 (e.g. Sn; $n>2$) is motivated by the absence of vibronic
coupling between the S1 and S2 states, both of which are of A" symmetry (Table
1); here, coupling to higher-lying singlet states of, e.g., alternative
symmetries offers the potential of second-order population transfer channels
comparable to those identified in other systems where direct population
transfer channels are weak or otherwise absent.Thompson, Eng, and Penfold
(2018); Eng  _et al._ (2020); Gu  _et al._ (2022)

Figure 3: Theoretical (black) and experimental (grey) S2 $\leftarrow$ S0
($3s\leftarrow n$) absorption spectrum. The theoretical spectrum is shown
shifted by $\Delta{E}=+0.14$ eV. The experimental spectrum was recorded by
Diau et al. and digitised from Ref. Diau, Kötting, and Zewail (2001)

In nuclear configurational space, the model Hamiltonian incorporates eight
degrees of vibrational freedom: $\nu_{1}$, $\nu_{7}$, $\nu_{10}$, $\nu_{11}$,
$\nu_{12}$, $\nu_{13}$, $\nu_{15}$, and $\nu_{21}$, which were identified by
the magnitude of their first-order couplings and the symmetries of the
vibrational modes. The degrees of vibrational freedom included the model
Hamiltonian comprise cyclobutane ring puckering ($\nu_{1}$ and $\nu_{12}$),
symmetric and anti-symmetric cyclobutane ring breathing ($\nu_{7}$ and
$\nu_{10}$), cyclobutane ring deformation ($\nu_{11}$, $\nu_{13}$ and
$\nu_{15}$), and the C=O stretching mode ($\nu_{21}$), consistent with
previous works.Kuhlman, Sølling, and Møller (2012) To assess the approximate
accuracy of our model Hamiltonian, Figure 3 compares the experimental Diau,
Kötting, and Zewail (2001) and theoretical absorption spectrum for the S2
state. There is excellent agreement between the two, indicating that our model
Hamiltonian describes accuracy the (local) potential energy surface(s). The
absorption spectrum features a distinct vibrational progression with
structured peaks separated by ca. 0.1 eV consistent with the C=O out-of-plane
wagging and cyclobutane ring puckering vibrational modes previously identified
Baba and Hanazaki (1984) and arising in the present model via the inclusion of
$\nu_{12}$.

Figure 4: Population kinetics obtained from quantum dynamics simulations over
600 fs post-photoexcitation into the S2 ($3s\leftarrow n$) state. The S2 state
population is shown in black; the S1 state population is shown in blue; the
triplet (Tn; $n=[1..5]$) state populations are shown (collectively) in green.

Figure 4 shows the transfer of population from the S2 state over 600 fs post-
photoexcitation into the S2 ($3s\leftarrow n$) state as computed via quantum
dynamics. The S1 $\leftarrow$ S2 population transfer occurs at a rate of ca.
$1.67\times 10^{-12}$ s-1 with the S1 state reaching peak population within
ca. 350 fs and is qualitatively consistent with the decay of the peak
associated with the S2 state in the photoelectron spectrum recorded by Kuhlman
et al.Kuhlman, Sølling, and Møller (2012) (the authors report a biexpotential
fit yielding time constants of ca. 350 and 750 fs). The S1 $\leftarrow$ S2
population transfer in the model Hamiltonian obtains intensity via mixing with
the higher-lying singlet electronically-excited states, especially the S4
state (the lowest-lying singlet electronically-excited state of A’ character).
This occurs most prominently along $\nu_{11}$. While this mode is responsible
for the state-to-state coupling, it is $\nu_{1}$, $\nu_{12}$, $\nu_{13}$, and
$\nu_{21}$ which exhibit the largest-amplitude nuclear oscillations and
electronically-excited-state structural changes which are likely to be
observed in the ultrafast electron diffraction experiment. The population
oscillations observed between the S2 and S1 states are associated with the
overcoherence of the reduced model Hamiltonian and are, in any case, faster
than the temporal resolution of the ultrafast electron diffraction
experiments.

S0 $\leftarrow$ S1 population transfer is not included in the present model
Hamiltonian. The high energy of the S2 state results in an exceptionally hot
wavefunction on the electronic ground state that is difficult to converge
under the framework of the quantum dynamics simulations. In addition, previous
works – and our own quantum chemical calculations – have identified S1/S0 MECI
at highly distorted geometries beyond the limits of the normal model
representation on which the quantum dynamics simulations are predicated (the
representation is only valid to small distortions from the equilibrium, e.g.
S0 minimum-energy, geometry). Consequently, to describe more completely the
excited-state relaxation dynamics, we explore excited-state molecular dynamics
simulations operating in unconstrained nuclear configurational space through
the trajectory surface-hopping approach.

###  III.3 Excited-State Relaxation Dynamics in Unconstrained Nuclear
Configuration Space Using Trajectory Surface-Hopping Dynamics

Figure 5 shows the populations of the S2, S1 and S0 states over the first 2 ps
post-photoexcitation into the S2 ($3s\leftarrow n$) state as obtained from 289
on-the-fly surface-hopping trajectories propagated at LR-TDDFT(PBE0)/aug-cc-
pVDZ level of theory. A similar set of trajectory surface-hopping dynamics
propagated at the ADC(2)/aug-cc-pVDZ level of theory are presented and
analysed in the SI and are of comparative interest.

Figure 5: Population kinetics obtained from 289 on-the-fly surface-hopping
trajectories over 2 ps post-photoexcitation into the S2 ($3s\leftarrow n$)
state. The S2 state population is shown in black; the S1 state population is
shown in blue; the S0 state population is shown in gray.

Figure 5 shows S1 $\leftarrow$ S2 population transfer with a decay constant of
ca. 356 fs. This is slightly faster than observed in the quantum dynamics
simulations, i.e. 50% of population decays from S2 in 225 fs, which is due to
the inclusion of the ground state to which the wavepacket can rapidly decay.
This decay constant is in close agreement with the fastest time-constant
extracted from a photoelectron spectroscopic study,Kuhlman, Sølling, and
Møller (2012), but we do not see any dynamics associated with the $\sim$700 fs
component reported. Interestingly, this slower component is in close agreement
with the population kinetics for the dynamics performed using potentials
calculated at ADC(2) level of theory, shown in the supporting information.

Figure 6: Bar chart showing the the number of times that each normal mode
features in the top six largest displacements for an (electronically-excited-
state) trajectory. The red bars represent those normal modes included in the
model Hamiltonian.

While the time scales between the quantum dynamics and TSH simulations
suggests similar dynamics, further analysis is required to assess this in more
detail. To achieve this, we transform the first 500 fs of excited state
molecular dynamics from Cartesian coordinates into a normal mode
representation similar to ref. Capano  _et al._ (2017). Figure 6 shows the
normal modes active during the TSH simulations as a bar chart of the number of
times each normal mode features in the top 6 of the largest displacements
during a trajectories excited state dynamics. The red bars represent those
normal modes included in the model Hamiltonian. This shows close agreement
between the modes which appear most frequently in the TSH simulations and
those included in the quantum dynamics. The most notable exception is
$\nu_{5}$ which exhibits larger amplitude motion due to the flat nature of the
potential, but does not act as either tuning or coupling modes and therefore
do not strongly influence the excited state dynamics.

Figure 7: Average Cα and Cβ bond lengths at (a) S1 $\leftarrow$ S2 and (b) S0
$\leftarrow$ S1 surface-hopping events.

Figure 7 shows average Cα and Cβ bond lengths for the structures where each
trajectory hops from the S2-S1 (a) and S1-S0 (b) states. For the former, there
is a clear cluster around 1.5-1.6 Å consistent with the ground state structure
and therefore close to the Franck-Condon geometry, as expected from the
optimised S2/S1 discussed above. In contrast, for the S1-S0 hopping
geometries, there is a significant change, with the majority of hops occurring
for Cα bond lengths $>$2.2 Å. Using the geometries provided in the supporting
information, the Cα CI occurs when the Cα bond length is 2.35 Å, with very
little correspond change along the Cβ bond. This suggests, in agreement with
previous work, that crossing from the S1-S0 occurs primarily at the CI
exhibiting a Cα bond break.

Figure 8: Fractional population of the photoproducts obtained from the 289
trajectories. The black trace shows cyclobutanone, the grey trace shows the C2
products (C2H4 \+ CH2CO), the green trace shows the C3 products (C3H6 \+ CO),
and the blue trace shows ring-opened structures.

Figure 8 shows the fractional population of the photoproducts formed from the
TSH trajectories. This indicates $\sim$20% of the excited states form the C2H4
\+ CH2CO, i.e. decay via the C2 channel, while 2.5% forms the C3 products. The
ring-open species are formed, but are very short-lived and either contribute
forming either the C2 or C3 products or undergo bond reformation to form
vibrationally excited ground state cyclobutanone. The formation of C2
comparable, but lower than other excited state dynamics simulations performed
at a higher level of theory reported in ref. Janoš  _et al._ (2024) (34%) and
with Trentelman et al. Trentelman  _et al._ (1990) who reported 43% of yield
experimentally. The major discrepancy in our simulations occurs for the C3
channels, which is $>$60% in these previous works. The near-absence of the C3
channel is associated with the multi-reference character of the potential in
this region and the bias of single reference methods for charged rather than
biradical bond breaking. While this does not significantly increase the
energies of the Cα and Cβ CIs (see Figure S4), it does increase the energy of
the double bond breaking CI, making the formation of the C3 channel
challenging. To assess this we also perform dynamics using trajectories in the
T1 state, performed using unrestricted Kohn-Sham facilitating the description
of biradical character. These were initiated at random from trajectories
populating the S1 state. Importantly, these show a much higher formation of C3
photoproducts (C3: 53%, C2: 5% and ring-open: 17%) consistent with previous
experiments Trentelman  _et al._ (1990).

Figure 9: Transient ($\Delta I/I$) scattering (a) without and (b) with 150 fs
(FWHM) temporal broadening. Transient PDF (c) without and (d) with 150 fs
(FWHM) temporal broadening. The ground-state (pre-photoexcitation) signal used
to generate the transient signal was obtained from the trajectory surface-
hopping dynamics initial conditions, i.e. the nuclear ensemble representing
the state of the system at $t=0$. All plots were produced using the 289
2000-fs trajectories simulated at the LR-TDDFT(PBE0)/aug-cc-pVDZ level of
theory.

###  III.4 Electron Diffraction Simulations

Figure 9 shows the time resolved electron diffraction simulations arising
after photoexcitation of cyclobutanone into the S2 state. Figure 9a show the
electron diffraction scattering signal as calculated, while Figure 9b is
convolved along the temporal axis with a Gaussian kernel (FWHM = 150 fs) to
reproduce the effect(s) of the finite temporal resolution of the proposed
electron diffraction experiment. Figures 9c and d show time-resolved pair-
distribution function (PDFs) maps with and without temporal broadening,
respectively, and were produced via sine transformation of the modified
scattering intensity maps in Figures 9a and b, respectively.

The modified scattering intensity maps (Figures 9a and b) show two strong
negative (ca. 1 and 9 Å-1) and two positive (ca. 2.7 and 7.5 Å-1) features but
do not reveal the richness of the dynamics that reflect the complex
photochemistry of cyclobutanone, in part due to the incoherent/stochastic
nature of the photochemical processes taking place.

Figure 10: Initial ($t=0$ fs; black) and final ($t=2000$ fs; grey) PDFs and
the difference PDF (red) calculated using the 289 2000 fs surface hopping
trajectories simulated using potentials are LR-TDDFT(PBE0) level of theory.

A deeper understanding of the structural changes can be established from
Figures 9c and d, which show the time-resolved PDF. For clarity, Figure 10
shows the PDF($t=0$ fs), PDF($t=2000$ fs), and the $\Delta$PDF($t=2000$
fs).The PDF acquired for the 289 initial conditions exhibits 3 peaks at
$\sim$1.5 Å, $\sim$2.5 Å and $\sim$3.0 Å. The first corresponds to the
neighing C-C and C=O distances, the second corresponds to C-C distance on the
opposite side of the cycle and the final peak corresponds to the C-O distances
which are not directly bonded. The transient indicates primarily a loss of the
first two peaks associated with dissociation and the formation of the C2
products. We note here that despite the aforementioned differences in the
photoproduct formation, the transient scattering and PDFs are very similar to
those in ref. Janoš  _et al._ (2024). This highlights the challenge in
disentangling the exact photoproduct formation of these systems due to the
overlapping bands.

##  IV Discussion and Conclusions

In this work, we have carried out quantum and excited-state trajectory
surface-hopping molecular dynamics simulations to study the electronically-
excited-state relaxation mechanisms and electronic ground-state dynamics of
cyclobutanone post-photoexcitation into the S2 Rydberg ($3s\leftarrow n$)
state. Our focus has been upon translating these simulations to predict the
experimental observables associated with the ultrafast electron diffraction
experiments and ultimately to answer the question: Can excited state dynamics
simulations be predictive? However, even for small molecules such as
cyclobutanone, certain approximations in the underlying computational chemical
methods are required which will influence the outcome of such simulations: we
highlight this in comparison between the present work and other works related
to the same challenge.Miao, Diemer, and Mitrić (2024); Suchan  _et al._
(2024); Daría  _et al._ (2024); Janoš  _et al._ (2024) Consequently, in this
section we discuss the relaxation mechanism observed in our simulations as
well as potential sources of error and how we expect that these will influence
interpretation of the experimental observables.

Within this challenge, the objective has been to translate excited state
dynamics into experimental observables. The importance of this cannot be
understated. In many cases, collaboration between experimental and theoretical
studies focus upon the comparison of quantities that are easy to calculate,
such as electronic state population kinetics. These kinetics are then compared
to experimentally extracted timescales and agreement is taken as accuracy of
the simulations. However, as shown in this work the LR-TDDFT (356 fs) and XMS-
CASPT2 Janoš  _et al._ (2024) (335 fs) dynamics provide very similar decay
kinetics, but different predictions of photoproducts which would influence the
experimental signal. While slightly slower, the excited state dynamics
performed using ADC(2) potentials also occurs on a comparable timescale
($\sim$700 fs), but owing to the artificial crossing along the C=O bond
stretch Marsili, Prlj, and Curchod (2021), the excited state decay occurs via
a completely different mechanism.

Our simulations indicate that after excitation of the $3s\leftarrow n$ Rydberg
state the system relaxes within 1-2 ps to form a broad range of photoproducts.
Decay of this initially excited S2 state occurs with a time-constant of
$\sim$350 fs. This is in good agreement with the fastest kinetics reported
from previous time-resolved photoelectron experiments by Kuhlman et al.
Kuhlman, Sølling, and Møller (2012). However, we note that ref. Kuhlman,
Sølling, and Møller (2012) also reports a strong contribution from a slower
time component, $\sim$750 fs. This is not observed within the LR-TDDFT
population kinetics, but is in very close agreement with the ADC(2) kinetics
presented in the supporting information. The exact origin for this difference
between LR-TDDFT and ADC(2) is unclear, however analysis of the hopping
geometries, indicates a flatter potential in the case of the latter, which
permits a slightly wider spread of the trajectories in nuclear configuration
space. Importantly, in both cases despite the small nuclear displacement
required to reach the crossing point, the internal conversion from S2 to S1 is
comparatively slow due to the symmetry forbidden nature of the transition.

Once populated, the S1 undergoes a large structural distortion, primarily
along the Cα bond, consistent with previous work Diau, Kötting, and Zewail
(2001); Liu and Fang (2016). This drives population to be rapidly transferred
into the electronic ground state to form very vibrationally hot species. The
fast nature of the population transfer from the S1 to the ground state means
that population of the S1 doesn’t exceed $\sim$30%. The excited state
molecular dynamics consider only the dynamics within the singlet manifold. To
assess the potential influence of the intersystem crossing and the triplet
states, our quantum dynamics include the low lying excited triplet states.
These simulations indicate negligible amount of intersystem crossing into the
triplet manifold leading us to conclude that this channel will be unable to
compete with internal conversion rates found here.

Although the excited state dynamics and high energy of excitation leads to
some highly distorted geometries, our simulations point to the formation of
the photoproducts being determined as the trajectory passes through the CI
between the first electronically excited state and the ground state via a
ring-opened intermediate. This is consistent with the conclusions from ref.
Diau, Kötting, and Zewail (2001) who demonstrated that motion away from the CI
branching space leads to all of the observed photoproducts. Herein lies the
most significant approximation within our work, as neither of the single
reference method used will capture the biradical nature of the photoproducts.
Indeed, although all of the CI identified Liu and Fang (2016) in previous work
have been found within the LR-TDDFT(PBE0) framework and exist at accessible
energies, i.e. below the excitation energy, our simulations show a much lower
fraction of photoproduct formation than previous theoretical Miao, Diemer, and
Mitrić (2024); Suchan  _et al._ (2024); Daría  _et al._ (2024); Janoš  _et
al._ (2024) and experimental Trentelman  _et al._ (1990) works. This appears
to most strongly affect the C3 photoproducts, which only forms in 2.5% of the
trajectories. In contrast, trajectories in the T1 state, performed using
unrestricted Kohn-Sham facilitating the description of biradical character,
initiated at random from trajectories populating the S1 state, show a much
higher formation of C3 photoproducts (C3: 53%, C2: 5% and ring-open: 17%)
consistent with previous experiments Trentelman  _et al._ (1990). The motion
through the CI and therefore the potential shape in this region is likely to
be critical in determining the branching ratio of the photoproducts. Here it
may not only be a limitation of single reference methods but a also condition
of the excited state dynamics used. As stated in the methods section, our
present dynamics attempts to avoid instabilities in the multi-configurational
region, near the degeneracy by enforcing a hop to the ground state when the
S1-S0 energy gap became $\Delta\mathrm{E_{S_{1}-S_{0}}}<0.1\mathrm{eV}$. While
this avoids the explicit motion through the CI, the enforced earlier
transition may also promote populated transfer closer the cyclobutanone
structure encouraging reformation of vibrationally hot cyclobutanone, rather
than the photoproducts.

From these excited state molecular dynamics simulations, the ultrafast
electron diffraction observable shows distinct changes and by studying the
time-resolved PDF, these are largely associated with a loss in intensity for
interactions at 1.5 and 2.5 Å, arising from dissociation. Despite the rich
dynamics and the distinct changes observed, the time-resolved scattering
curves show very little distinct dynamics largely associated with the
incoherent nature of the dynamics and the comparatively low temporal
resolution (150 fs).

Finally, a logical question would be ask if the limitations discussed above
can be overcome within the present framework, i.e. without adopting a multi-
reference wavefunction method which would prove challenging for larger
systems. Here, it is important to stress that while the relative yields of
photoproducts formed appears somewhat at odds with previous works, all major
reported products are generated, i.e. the full nuclear configuration space has
been sampled. Excited state simulations have previously been used to simulate
the experimental observables associated with structurally sensitive techniques
of electron Figueira Nunes  _et al._ (2023) and X-ray diffraction Minitti  _et
al._ (2015). Importantly in both of these works the outcomes of the
trajectory-based dynamics were used as a basis to fit to experimental data and
deliver an interpretation. While, the use of a fit means that such an approach
may not be classed as fully predictive, in both cases an excellent agreement
between experiment and theory was achieved and providing deep insight into the
dynamics observed.

## Acknowledgements

This research made use of the Rocket High Performance Computing service at
Newcastle University and the Viking High Performance Computing service at the
University of York. We also acknowledge the COSMOS Programme Grant
(EP/X026973/1). T. J. P would like to thank the EPSRC for an Open Fellowship
(EP/W008009/1).

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