# Using a multistate Mapping Approach to Surface Hopping to predict the
Ultrafast Electron Diffraction signal of gas-phase cyclobutanone

Lewis Hutton<EMAIL_ADDRESS>Physical and Theoretical Chemistry
Laboratory, Department of Chemistry, University of Oxford, Oxford OX1 3QZ,
United Kingdom.  Andrés Moreno Carrascosa  Physical and Theoretical Chemistry
Laboratory, Department of Chemistry, University of Oxford, Oxford OX1 3QZ,
United Kingdom.  Andrew W. Prentice  Institute of Chemical Sciences, School of
Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS,
United Kingdom.  Mats Simmermacher  Physical and Theoretical Chemistry
Laboratory, Department of Chemistry, University of Oxford, Oxford OX1 3QZ,
United Kingdom.  Johan E. Runeson  Physical and Theoretical Chemistry
Laboratory, Department of Chemistry, University of Oxford, Oxford OX1 3QZ,
United Kingdom.  Martin J. Paterson  Institute of Chemical Sciences, School of
Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS,
United Kingdom.  Adam Kirrander<EMAIL_ADDRESS>Physical and
Theoretical Chemistry Laboratory, Department of Chemistry, University of
Oxford, Oxford OX1 3QZ, United Kingdom.

###### Abstract

Using the recently developed multistate mapping approach to surface hopping
(multistate MASH) method combined with SA(3)-CASSCF(12,12)/aug-cc-pVDZ
electronic structure calculations, the gas-phase isotropic ultrafast electron
diffraction (UED) of cyclobutanone is predicted and analyzed. After excitation
into the n-3s Rydberg state (S2), cyclobutanone can relax through two S2/S1
conical intersections, one characterized by compression of the CO bond, the
other by dissociation of the $\mathrm{\alpha}$-CC bond. Subsequent transfer
into the ground state (S0) is then achieved via two additional S1/S0 conical
intersections that lead to three reaction pathways: $\mathrm{\alpha}$ ring-
opening, ethene/ketene production, and CO liberation. The isotropic gas-phase
UED signal is predicted from the multistate MASH simulations, allowing for a
direct comparison to experimental data. This work, which is a contribution to
the cyclobutanone prediction challenge, facilitates the identification of the
main photoproducts in the UED signal and thereby emphasizes the importance of
dynamics simulations for the interpretation of ultrafast experiments.

††preprint: AIP/123-QED

##  I  Introduction

The photochemistry of cyclic ketones has been studied extensively.Benson and
Kistiakowsky (1942); Lee, Shortridge, and Rusbult (1971); Baba and Hanazaki
(1984); Tang and Lee (1976); Harrison  _et al._ (1980); Causley and Russell
(1980) Despite their small size, they have a rich photochemistry with multiple
competing pathways that include dissociation, fluorescence, and intersystem
crossing.Kao  _et al._ (2020); Hemminger and Lee (1972) The relative
importance of the different pathways is closely linked to the size of the
organic ring and the associated ring strain.Kao  _et al._ (2020) The
photodissociation of cyclic ketones has been of particular interest ever since
the pioneering work of Norrish.Saltmarsh and Norrish (1935)

A target of recurring interest in this class of molecules is cyclobutanone,
(CH2)3CO.Benson and Kistiakowsky (1942); Denschlag and Lee (1968); Lee and Lee
(1969); Lee, Shortridge, and Rusbult (1971); Diau, Kötting, and Zewail (2001);
Kao  _et al._ (2020) Early experimental studies identified the photoproducts
of cyclobutanone as propylene (cyclopropane) and carbon monoxide (C_3H_6 + CO)
or ethylene and ketene (CH2CH2 + CH2CO). The corresponding quantum yields were
found to be 40% and 60%,Benson and Kistiakowsky (1942) although these quantum
yields were later shown to have a strong dependence on the excitation
wavelength.Hemminger and Lee (1972) Recent experimental work includes
ultrafast transient absorption spectroscopy on cyclobutanone in solution.Kao
_et al._ (2020) Upon excitation to the S1 (n$\pi^{*}$) state using UV pulses
in the range 255–312 nm UV pulses, Kao _et al._ found that singlet
dissociation pathways are dominant at shorter wavelengths, predominantly via
Norrish Type-1 $\alpha$ cleavage, with minor contributions from ketene
formation. At higher excitation energies, such as 200 nm, the n-3s Rydberg
state comes into play.Drury-Lessard and Moule (1978); O’Toole  _et al._
(1991); Whitlock and Duncan (1971) In this regime, using time-resolved mass
spectrometry and photoelectron spectroscopy, Kuhlman _et al._ found that a
ring puckering mode effectively couples the S2 and S1 states, allowing for
rapid internal conversion.Kuhlman, Sølling, and Møller (2012)

The experimental studies have been accompanied by theoretical work. Electronic
structure calculations using complete active space self-consistent field,
CASSCF(10,8), and multistate complete active space second-order perturbation
theory, MS-CASPT2(10,8), predicted a small barrier for ring-opening on the S1
state, with the possibility of minor contributions from barrierless
dissociation in the T1 state.Xia  _et al._ (2015) Building on this,
simulations of cyclobutanone by _ab initio_ multiple spawning (AIMS) found a
ring-opening mechanism to be dominant with a small portion of trajectories
producing CH_2=CH_2 + CH_2CO and CH_2CH_2 + CH2=CO fragmentation.Liu and Fang
(2016) We also note that Kuhlman _et al._ constructed a five-dimensional
linear-vibronic Hamiltonian (LVH) model to simulate the S2/S1 decay in
cyclobutanone using the multiconfigurational time-dependent Hartree (MCTDH)
method, which succeeded in replicating the short time constants of 0.95 ps.
Kuhlman  _et al._ (2012)

The current paper is motivated the prediction challenge based on an UED
experiment carried out at the SLAC MeV-UED facility.Martínez  _et al._
Ultrafast electron diffraction has emerged as a powerful experimental
technique for observing molecular dynamics in the recent decade,Ishikawa  _et
al._ (2015); Yang  _et al._ (2015, 2018); Wolf  _et al._ (2019); Razmus  _et
al._ (2022) alongside ultrafast x-ray scattering. Minitti  _et al._ (2015);
Ruddock  _et al._ (2019a, b); Gabalski  _et al._ (2022) Notably, both
techniques have made advances in extracting information that extends beyond
structural dynamics.Yang  _et al._ (2020); Yong  _et al._ (2020) We use this
challenge as an opportunity to benchmark the electronic structure calculations
for cyclobutanone. This comparison includes non-standard selected-CI
electronic structure methods that overcome the issue of large active space
selection, constitute a black-box alternative to complete active space (CAS)
methods and provide complementary benchmarks.Prentice, Coe, and Paterson
(2023a); Jacquemin  _et al._ (2023) The nonadiabatic simulations are carried
out using the newly developed multistate MASH method.Runeson and Manolopoulos
(2023) Furthermore, we use the simulations to predict the UED signals and
analyze these to determine the main contributions to the scattering.

##  II Theoretical methods

###  II.1  Nonadiabatic Molecular dynamics

Many computational methods have been developed to simulate nonadiabatic
dynamics over the past decades. Some, such as MCTDH Meyer, Manthe, and
Cederbaum (1990); Beck  _et al._ (2000) and Multi-Layer MCTDH,Wang and Ziegler
(2005) can provide numerically exact results. However, these methods require
high-accuracy precomputed potential energy surfaces (PESs), which, combined
with the exponential scaling of quantum mechanics, imposes severe limitations
on the number of degrees of freedom that can be treated. To circumvent this
problem, the nuclear wavefunction can be represented as a linear combination
of traveling Gaussian basis functions or classically-guided trajectories, and
the electronic structure quantities needed can be calculated _on-the-fly_ ,
also known as direct dynamics. Such approaches can significantly reduce the
phase space explored and therefore the computational cost per degree of
freedom. _On-the-fly_ methods include examples where the nuclei are propagated
using equations of motion derived from the variational principle, notably
variational multiconfigurational Gaussian (vMCG),Worth, Robb, and Burghardt
(2004); Lasorne  _et al._ (2006); Richings  _et al._ (2015) or
semiclassically, which includes methods such as full or _ab initio_ multiple
spawning (FMS/AIMS)Curchod and Martínez (2018); Martínez, Ben-Nun, and Levine
(1997); Martinez, Ben-Nun, and Levine (1996); Martínez and Levine (1997) and
multiconfigurational Ehrenfest (MCE),Shalashilin (2009); Saita and Shalashilin
(2012); Makhov  _et al._ (2017) or classically, such as trajectory surface
hopping (TSH).Tully (1990) In addition to the above methods, mapping methods
treat the electronic and nuclear degrees of freedom in an equal manner.
Examples include the Meyer-Miller-Stock-Thoss (MMST) mapping Hamiltonian,
symmetrical quasi-classical windowing, and generalized spin mapping.Meyer and
Miller (1979); Cotton and Miller (2013); Miller and Cotton (2016); Runeson and
Richardson (2020) Ideas from this last category informed the development of
the surface-hopping variant used in this paper (see Section II.1.2 below). For
an in-depth overview of different methods for nonadiabatic dynamics, the
reader is directed towards the edited volume in reference 53.

####  II.1.1  Trajectory surface hopping

For nonadiabatic molecular dynamics, TSH is perhaps the most commonly employed
approach to support the interpretation of experiments. Ruckenbauer  _et al._
(2016); Bellshaw  _et al._ (2017); Squibb  _et al._ (2018); Polyak  _et al._
(2019); Chakraborty  _et al._ (2020); Merritt, Jacquemin, and Vacher (2021);
Hutton and Curchod (2022); Borne  _et al._ (2024) In TSH, an ensemble of
individual trajectories is used to represent the propagation of the nuclear
wavepacket. In each trajectory, the nuclei are treated classically and the
electronic amplitudes are used to predict hops to other electronic states. For
a full discussion of TSH please see reference 62, while we continue by
highlighting the key aspects of the fewest switches surface hopping (FSSH)
variant of TSH.

Each trajectory in FSSH is initialized by projecting a ground-state
vibrational distribution into the excited state of interest. We consider pure
excitation, i.e. $c_{a}=1$, where $c_{a}$ is the electronic wavefunction
coefficient for the active state $a$. The nuclear coordinates,
$\bar{\mathbf{R}}$, are propagated using classical equations of motion,

$\frac{d^{2}\mathbf{\bar{R}_{\alpha}}}{dt^{2}}=-\frac{1}{M_{\alpha}}\nabla_{\alpha}E_{a},$
(1)

for each atom $\alpha$, with $t$ the time, $M_{\alpha}$ the mass of atom
$\alpha$, and $\nabla_{\alpha}E_{a}$ the gradient (force) on the _active_
electronic adiabatic state. The integration is typically carried out using a
standard velocity-Verlet scheme, though other propagators may be able to
achieve better convergence.Blanes and Moan (2002) The forces are normally
obtained from electronic structure calculations of adiabatic states. In
parallel with the nuclear propagation, one must integrate the electronic
coefficients, $c=(c_{1},\ldots,c_{N})$, for the $N$ electronic states
considered. In the adiabatic representation, the probability
$P^{\mathrm{FSSH}}_{a\rightarrow b}$ of a hop from state $a$ to state $b$, is
evaluated according to,

$P^{\mathrm{FSSH}}_{a\rightarrow b}=\mathrm{max}\Bigr{[}0,\frac{2\Delta
t}{|c_{a}|^{2}}\operatorname{Re}(\mathbf{d}_{ab}\cdot\mathbf{\bar{v}}c_{b}c_{a}^{*})\Bigr{]}\text{,}$
(2)

where $\mathbf{d}_{ab}$ is the nonadiabatic coupling vector between states $a$
and $b$, $\mathbf{\bar{v}}$ is the classical nuclear velocity, $c_{a}$ and
$c_{b}$ are the electronic coefficients for states $a$ and $b$, and $\Delta t$
is the timestep.Crespo-Otero and Barbatti (2018) In some cases, an overlap
scheme is employed to avoid the expensive calculations of nonadiabatic
couplings.Hammes-Schiffer and Tully (1994); Plasser  _et al._ (2016) At each
time step, $P^{\mathrm{FSSH}}_{a\rightarrow b}$ is evaluated and compared to a
random number, resulting in either acceptance or rejection of the hop. If a
hop is accepted but the energy in the system is insufficient for it to occur,
the hop is rejected and the nuclear velocities are reflected. This scenario is
referred to as a frustrated hop. For hops that do take place, the active state
is updated ($a\rightarrow b$), the velocities are modified to conserve total
energy, and the trajectories continue to be propagated using the forces on the
new active state.

The algorithm presented above is robust, but also causes one of the most well-
known shortcomings of TSH, the so-called overcoherence issue. A number of,
more or less ad hoc, decoherence correction schemes have been developed in
response.Granucci, Persico, and Zoccante (2010); Subotnik  _et al._ (2016) The
method presented next aims to alleviate the need for such _ad hoc_
corrections.

####  II.1.2  Multistate mapping approach to surface hopping

Recently, a new version of TSH was devised by Mannouch and Richardson, known
as "mapping approach to surface hopping" (MASH).Mannouch and Richardson (2023)
MASH combines elements of mapping methods with TSH to attempt to get the
advantages of both. Hence, in contrast to the stochastic algorithm in FSSH,
MASH uses a deterministic algorithm to evaluate the active state,

$P^{\mathrm{MASH}}_{a\rightarrow b}=\begin{cases}1,&\text{if
}|c_{b}|^{2}>|c_{a}|^{2}\\\ 0,&\text{otherwise.}\\\ \end{cases}$ (3)

In other words, MASH defines the active state to be the state with the largest
value of $|c_{i}|^{2}$, alleviating the need for _ad hoc_ decoherence
corrections and improving accuracy when compared to the original FSSH
procedure.Mannouch and Richardson (2023) Originally, MASH was formulated for
two-state systems, but MASH was then generalized to any number of states by
Runeson and Manolopoulos, giving rise to what we hereon refer to as multistate
MASH.Runeson and Manolopoulos (2023); Runeson, Fay, and Manolopoulos (2024)

To start trajectories, multistate MASH determines the nuclear initial
conditions in the same manner as TSH, mapping the ground-state wavefunction
onto a classical phase space. However, following other mapping approaches,
MASH also considers the electronic coefficients $c$ as a phase space variable.
Therefore, rather than describing the initial state as pure, such as is done
in FSSH, multistate MASH represents the initial state as a distribution over
all values of $c=(c_{1},\ldots,c_{N})$. In practice, this is done by random
assignment of the complex-valued coefficients, $c_{i}=x_{i}+\mathrm{i}y_{i}$,
for each electronic state from Gaussian distributions with zero mean and
standard deviation one for both the real and imaginary components. The
resulting set of coefficients is normalized such that
$1=\sum_{i=1}^{N}|c_{i}|^{2}$. Coefficients where the initial state of
interest does _not_ have the largest absolute value are resampled until
suitable sets are found. It is worth noting that other choices of initial
distributions are possible, but have been found to yield similar results in
most cases. Runeson and Manolopoulos (2023) Therefore, we use this method of
selecting initial conditions due to its ease of implementation.

To measure populations, several different estimators have been proposed in the
literature. In this paper we measure the active state, corresponding to an
estimator of the form

$\Theta_{n}(t)=\begin{cases}1,&\text{if }P_{n}(t)>P_{m}(t)\text{
}\forall\text{ }m\neq n\\\ 0,&\text{otherwise}\end{cases}$ (4)

With this choice of estimator and the initial distribution described above,
the ensemble-averaged population transfer from state $i$ to state $n$ is

$P_{i\rightarrow n}(t)=\langle\Theta_{i}(0)\Theta_{n}(t)\rangle$ (5)

In the two-state case, Mannouch and Richardson have shown that this
prescription overestimates the rate of transfer compared to Landau-Zener
theory. It does, however, maintain a physical population estimate with a low
number of trajectories. Note, that this approach is different from what is
used in both MASH and multistate MASH. Other alternatives can be seen in the
SI.

One potential side-effect of multistate MASH is hopping between states that
are not coupled. It is possible to nevertheless accept all hops, regardless of
the coupling between the states, which was the approach taken in reference 34.
However, due to the risk of unphysical behavior, we introduce for all hops an
energy criterion, such that $|E_{a}-E_{b}|<0.055$ Hartree ($\approx$ 1.5 eV)
to prevent erroneous hops between uncoupled states. Hops that do not meet this
criterion are treated on the same footing as reflected hops, with the total
velocity of all atoms reflected. This direction of rescaling differs from
previous work with MASH and is chosen due to simplicity.

###  II.2 Ultrafast electron diffraction (UED)

We simulate electron diffraction using the independent atom model (IAM),
originally devised by Debye.Debye (1930); Bewilogua (1932); Kirrander and
Weber (2017); Simmermacher, Weber, and Kirrander (2023). This method,
extensively used in the analysis of scattering and crystallography
experiments, approximates the electron scattering probability,
$\big{|}Z(\bm{s},\bar{\bm{R}})\big{|}^{2}$, as a coherent sum of
tabulatedPrince (2006) form factors, $f_{N}^{\mathrm{e}}(s)$,

$\displaystyle\big{|}Z(\bm{s},\bar{\bm{R}})\big{|}^{2}=\sum_{A}^{N_{\mathrm{at}}}\sum_{B}^{N_{\mathrm{at}}}\
f_{A}^{\mathrm{e}}(s)\ f_{B}^{\mathrm{e}}(s)\
e^{\mathrm{i}\bm{s}\cdot\bm{R}_{AB}}\text{,}$ (6)

where $\bm{s}$ is the momentum transfer vector, $\bar{\bm{R}}$ the molecular
geometry, $\bm{R}_{AB}$ the distance vector between atoms $A$ and $B$, and
$N_{\mathrm{at}}$ the number of atoms in the molecule. Note that
$\big{|}Z(\bm{s},\bar{\bm{R}})\big{|}^{2}$ is given in units of the Rutherford
cross-section (see e.g. reference 74 for further details). The electron
scattering atomic form factors, $f_{A}^{\mathrm{e}}(s)$, are defined as,

$\displaystyle f_{A}^{\mathrm{e}}(s)=f_{A}^{\mathrm{x}}(s)-Z_{A}\text{,}$ (7)

where $f_{A}^{\mathrm{x}}(s)$ are the tabulated form factors for x-rays and
$Z_{A}$ the nuclear charges.

We calculate the rotationally averaged signal accounting for all possible
molecular orientations in a thermal ensemble. This yields the isotropic
elastic electron scattering probability, $\mathcal{S}_{0}(s,\bar{\bm{R}})$,
which takes the form,

$\displaystyle\mathcal{S}_{0}(s,\bar{\bm{R}})=\sum_{A}^{N_{\mathrm{at}}}\sum_{B}^{N_{\mathrm{at}}}\
f_{A}^{\mathrm{e}}(s)\ f_{B}^{\mathrm{e}}(s)\ j_{0}(sR_{AB})\text{,}$ (8)

where $s=|\bm{s}|$ is the norm of the momentum transfer vector and
$R_{AB}=|\bm{R}_{AB}|$ is the distance between atoms $A$ and $B$, and where we
have introduced the zeroth-order spherical Bessel function $j_{0}(sR_{AB})$
defined as,

$\displaystyle j_{0}(sR_{AB})=\frac{\sin{(sR_{AB})}}{sR_{AB}}\text{.}$ (9)

In the experiment, not only the elastic component of electron scattering is
measured but also the inelastic component. While the IAM cannot describe
individual inelastic transitions, the total inelastic component can be
approximated by an incoherent sum of _atomic_ inelastic scattering functions
$S_{A}(s)$. This yields the total isotropic scattering,
$\mathcal{S}_{t}(s,\bar{\bm{R}})$, as the sum of the elastic and inelastic
contributions,

$\displaystyle\mathcal{S}_{t}(s,\bar{\bm{R}})=\mathcal{S}_{0}(s,\bar{\bm{R}})+\sum_{A}^{N_{\mathrm{at}}}S_{A}(s)\text{.}$
(10)

We note that further improvements in the scattering signal are
straightforward. For instance, alignment effects resulting from the linear
polarization of the pump laser can be accounted forSimmermacher, Kirrander,
and Henriksen (2020) and corrections to the form factors due to relativistic
effects can be made,Salvat, Jablonski, and Powell (2005) however for the
latter we note that this effect is comparatively minor for the electron
energies at the SLAC MeV-UED source. It is also possible to calculate the
scattering cross-sections from ab initio electronic wavefunctions, which makes
it possible to account accurately for the effects on scattering from chemical
bonding, electron correlation, and change in the electronic state due to
excitation by the pump pulse. Carrascosa  _et al._ (2019); Zotev  _et al._
(2020); Carrascosa  _et al._ (2022)

##  III Computional details

A ground-state minimum energy structure of cyclobutanone was obtained using
state-averaged CASSCF,Roos, Taylor, and Sigbahn (1980)
SA(3)-CASSCF(12,12)/aug-cc-pVDZ, i.e. state-averaging over the three lowest
singlet states, an active space with 12 electrons in 12 orbitals with the aug-
cc-pVDZ Cartesian basis.Woon and Dunning (1993) Further details are given in
the supplementary information. All electronic structure calculations were done
in OpenMolcas version v23.02-10-gb7266214b.Aquilante  _et al._ (2020); Li
Manni  _et al._ (2023) The ground-state minimum energy structure was subjected
to a frequency calculation yielding no imaginary frequencies, confirming the
structure to be a true minimum. Further excited state structures were
optimized using SA(3)-CASSCF(12,12) to obtain the S2 minimum, two S2/S1
minimum energy conical intersections (MECIs), and two S1/S0 MECIs. Selected
structures were subjected to linear interpolation in internal coordinates
(LIIC) to yield a progression of intermediate structures. Potential energy
curves along the LIICs were calculated using SA(3)-CASSCF(12,12) and extended
multistate complete active space second-order perturbation theory, XMS-CASPT2,
with the same (12,12) active space as for the CASSCF calculations (see
supplementary information).Shiozaki  _et al._ (2011) For stability, an
imaginary shift of 0.5 a.u. was applied for all XMS-CASPT2
calculations.Forsberg and Malmqvist (1997)

For further comparison, the potential energy curves were also calculated using
a modified version of adaptive configuration interaction (ACI) and Monte-Carlo
configuration interaction (MCCI). Schriber and Evangelista (2016, 2017);
Prentice, Coe, and Paterson (2023b) The state-averaged MCCI calculations were
performed using MCCI V4Tong  _et al._ (2000); Coe and Paterson (2013) with the
SA error-controlled calculations performed using a locally modified
development version of GeneralSCI 1.0.Prentice, Coe, and Paterson (2023c)
Further details are given in the supplementary information.

The calculated ground state frequencies were used to generate a Wigner
distribution from which 1000 geometries were sampled. At each geometry,
excitation energies and oscillator strengths were calculated using
SA(3)-CASSCF(12,12). The photoabsorption cross-section was constructed using
the nuclear ensemble approach (NEA), convoluted by Lorentzian functions with a
phenomenological broadening parameter of 0.05 eV.Crespo-Otero and Barbatti
(2012) Initial conditions for the multistate MASH were sampled from this
Wigner distribution using an implicit laser pulse of width 0.2 eV, centered at
6.2 eV. All trajectories were initiated from the bright S2 (n-3s Rydberg)
state using multistate MASH (see section II.1.2 for more details) Runeson and
Manolopoulos (2023) implemented in a development version of SHARC.Mai,
Marquetand, and González (2018) The electronic structure calculations were
carried out using SA(3)-CASSCF(12,12)/aug-cc-pVDZ with all three electronic
states available in dynamics. The trajectories were propagated using a 0.5 fs
time step, with 25 substeps, for 300 fs using a local diabatization scheme.
For trajectories in which the total energy changes by more than 0.25 eV, the
propagation is stopped and only the time points with the correct total energy
are used in the analysis. Additionally, we note that a subset of trajectories
are ’trapped’. This occurs when the trajectories enter a region where a
reflected hop happens but the reflected hop fails to remove the trajectory
from the ’forbidden’ region. Such trapped trajectories are only included in
the analysis until the point that they get trapped. For a summary of the
trajectories removed from the analysis, see the supplementary information.
Populations of the ensemble of trajectories are estimated using the multistate
Populations were estimated according to Section II.1.2. Finally, rotationally
averaged total scattering diffraction patterns were calculated using the
geometries from the multistate MASH trajectories and the independent atom
model (see Section II.2) for the initial 200 fs of dynamics.

##  IV Results and discussion

###  IV.1 Excited states of cyclobutanone

The character of the two lowest-lying S1 and S2 electronic states of
cyclobutanone has previously been identified as n$\pi^{*}$ and n-3s,
respectively.Kuhlman, Sølling, and Møller (2012); Drury-Lessard and Moule
(1978) The next set of excited states are the higher-lying 3p Rydberg states,
which appear a full 0.7 eV above the n-3s Rydberg state.O’Toole  _et al._
(1991) Triplet states have been shown to be important for the photodynamics
upon excitation with long wavelengths.Kao  _et al._ (2020) However, due to the
short excited state lifetime of cyclobutanone following excitation into the S2
state, triplet states are less likely to make a significant contribution to
the current photodynamics. For these reasons, we have chosen to focus our
attention on an accurate description of the two lowest-lying singlet states S1
and S2.

A summary of the low-lying excited singlet states available to cyclobutanone
at the equilibrium geometry of cyclobutanone is shown in Table 1. Both
SA(3)-CASSCF(12,12) and XMS-CASPT2(12,12) yield an S1 state with n$\pi^{*}$
character at the Franck-Condon (FC) geometry and an S2 state with n-3s Rydberg
character. Furthermore, the excitation energies for SA(3)-CASSCF(12,12) are in
excellent agreement with XMS-CASPT2(12,12), with SA(3)-CASSCF(12,12) energies
showing a constant difference of 0.04 eV for both excited states. Excitation
energies for SA(3)-ACI and SA(3)-MCCI can also be found in Table 1. These
methods give a good description of the low-lying singlet states of
cyclobutanone with excitation energies deviating by $\sim$0.05 eV for the
valence n$\pi^{*}$ state and $\sim$0.1 eV for the n-3s Rydberg state compared
to the XMS-CASPT2(12,12) benchmark. In the ACI simulation, a large orbital
window, corresponding to a CAS-CI(12,36), has also been tested. The large
orbital window gives results analogous to the ones obtained with
CASSCF(12,12), providing further support for this choice of active space.

Table 1:  Table of excitation energies in eV for the lowest two excited electronic states of cyclobutanone at the S0 minimum energy geometry optimized using SA(3)-CASSCF(12,12)/aug-cc-pVDZ. State characters are given in parentheses along with excitation energies. Method | S1 | S2
---|---|---
SA(3)-CASSCF(12,12) | 4.44 (n$\pi^{*}$) | 6.24 (n-3s)
XMS-CASPT2(12,12) | 4.40 (n$\pi^{*}$) | 6.20 (n-3s)
SA(3)-ACI | 4.43 (n$\pi^{*}$) | 6.31 (n-3s)
SA(3)-MCCI | 4.45 (n$\pi^{*}$) | 6.30 (n-3s)

A theoretical photoabsorption cross-section of cyclobutanone calculated using
SA(3)-CASSCF(12,12) is shown in Fig. 1, allowing for a direct comparison with
experimental absorption spectra. Good agreement can be observed between the
two spectra, further validating our electronic structure method along with
confirming the assignment of states using the insets in Fig. 1.

Figure 1: Photoabsoprtion cross-section of gas-phase cyclobutanone calculated
using the SA(3)-CASSCF(12,12)/aug-cc-pVDZ, the NEA and Wigner sampling of 1000
geometries (purple line). Experimental data is reproduced from reference 95
(blue line). Orbitals for the S1 and S2 states are shown as an inset.

###  IV.2 Photochemical reaction pathways and benchmarking

After the photophysics of cyclobutanone has been studied in the FC region, the
different deactivation pathways available upon photoexcitation can be
characterized. A series of potentially important optimized geometries for
these pathways are shown in Fig. 2, optimized using SA(3)-CASSCF(12,12)/aug-
cc-pVDZ. The S0 minimum energy geometry (S0, upper left) can be characterized
by a single carbon atom breaking the planar symmetry of the molecule, along
with a C=O bond length of 1.19 Å and an $\mathrm{\alpha}$-CC bond of 1.58 Å.
When the molecule is excited by a 200 nm pulse, the S2 excited electronic
state is populated at the FC region and can potentially relax into the S2
minimum. The S2 minimum energy geometry (S2, upper center) displays subtle
changes in bond lengths, along with a planarization of the carbon ring. Two
S2/S1 minimum energy conical intersections (MECIs) were successfully
optimized, with the S2/S1 MECI (S2/S1, lower left) already reported
previously.Kuhlman  _et al._ (2012) This MECI displays a rather severe
compression of the C=O bond. The second S2/S1 MECI (S2/S1, upper right)
exhibits a dissociated $\mathrm{\alpha}$-CC bond alongside a compressed C=O
bond. Two further MECIs are included in Fig. 2, corresponding to MECIs between
the ground and first excited state. Both these S1/S0 MECIs have been reported
previously by Liu _et al._.Liu and Fang (2016) The first of these MECIs
(S1/S0, lower center), termed CI-3 in reference 18, breaks the ring structure
to produce CH_2=CH_2 and O=CCH_2. The second S1/S0 MECI (S1/S0, lower right),
termed CI-1 in reference 18, is closely related to the second S2/S1 MECI
(S2/S1, upper right), with an dissociated $\mathrm{\alpha}$-CC bond. However,
in contrast to the S2/S1 MECI, the S1/S0 MECI contains a slightly stretched
C=O bond relative to the S0 geometry. It is obvious from this static
perspective of excited state geometries, that cyclobutanone possesses a rich
photochemistry. However, the relative importance of each of these structures
is still unknown, as are the final products of each conical intersection (CI).

Figure 2: Six critical geometries are shown for gas-phase cyclobutanone. The
S0 minimum (upper left), S2 minimum (upper center), a dissociative S2/S1 MECI
(upper right), another S2/S1 MECI (lower left), a ring breaking S1/S0 MECI
(lower center), and a ring-opening S1/S0 (lower right). All structures shown
are optimized using SA(3)-CASSCF(12,12)/aug-cc-pVDZ.

To gain a better understanding of the PESs of cyclobutanone and to benchmark
SA(3)-CASSCF(12,12) away from the FC region, an LIIC was done on some of the
key geometries shown in Fig. 2. Four structures were selected: the S0 minimum
(Fig. 2, S0, upper left), S2 minimum (Fig. 2, S2, upper center), the
dissociative S2/S1 MECI (Fig. 2, S2/S1, upper right) and finally the
$\mathrm{\alpha}$-CC bond dissociation S1/S0 MECI (Fig. 2, S1/S0, upper
right).

Figure 3: A linear interpolation in internal coordinates from the S0 minimum
to S2 minimum to the dissociative S2/S1 MECI ending with the ring opening
S1/S0 MECI. Inset structures correspond to the optimized structures used along
the LIIC coordinate. Excitation energies for the three lowest-lying singlet
states are given using both SA(3)-CASSSCF(12,12)/aug-cc-pVDZ (dotted line) and
XMS-CASPT2(12,12)/aug-cc-pVDZ (solid line).

Upon photoexcitation to the S2 state, cyclobutanone in the S0 minimum
structure can relax to the closeby S2 minimum via a planarization of the
carbon ring (see green line in panel 1 of Fig. 3). Here, cyclobutanone
undergoes a dissociation of an $\mathrm{\alpha}$-CC bond resulting in a
barrier height of 0.21 eV for SA(3)-CASSCF(12,12), somewhat below the 0.49 eV
predicted by XMS-CASPT(12,12), with the barrier heights given as energy
differences relative the S2 minimum. Note that absolute values for transition
state barriers in LIICs should be treated with care as they are known to be
overestimated. Regardless, SA(3)-CASSCF(12,12) is likely to estimate a faster
rate of decay from S2 to S1 as a result of this smaller reaction barrier. A
steep drop in energy is observed for both methods after the transition barrier
leading down to the S2/S1 MECI (see Fig. 2, upper right and inset of Fig. 3).
This S2/S1 MECI is peaked and has a single pathway according to the criteria
set out in reference 96. This allows for efficient funneling onto the S1 state
(see blue line in Fig. 3) where a barrier-less decay is observed to the
structurally similar S1/S0 MECI (see Fig. 2) where the S1/S0 MECI is observed
with a peaked bifurcating branching space.

From Fig. 3 it can be deduced that there is excellent agreement between
SA(3)-CASSCF(12,12) and XMS-CASPT2(12,12), with slight differences observed
for the approximate transition barrier height. Nevertheless, overall the
SA(3)-CASSCF(12,12) results offer an excellent comprise between speed and
accuracy for the on-the-fly nonadiabatic dynamics of gas-phase cyclobutanone.
Additional LIICs for the same geometries can be found in SI (Figures 3 and 4)
for both SA(3)-ACI and SA(3)-MCCI. These further support our assessment. We
also note that this indicates that good agreement can be obtained with methods
that potentially offer a more black-box approach to the calculation of
multireference wavefunctions than the CAS family of methods.

###  IV.3 Dynamics

A total of $229$ trajectories were initiated on the S2 state. Trajectories
were removed from the statistics if either a change in total energy of $0.25$
eV or more was observed or if consecutive reflected hops indicated that the
trajectory was trapped in a ’forbidden’ region of the PES. This resulted in a
significant number of discarded trajectories. For a full summary of discarded
trajectories see the SI (Figure 2). To briefly summarize, after $50$ fs the
number of trajectories had dropped to $146$, and by $200$ fs only $18$
trajectories of the initial $229$ trajectories had survived, while at $300$ fs
this was down to $10$ trajectories. At $300$ fs a total of $71$ trajectories
had been discarded due to consecutive reflected hops, while $148$ trajectories
had been discarded due to issues with the energy conservation. We acknowledge
that this could potentially cause artifacts in the simulations, especially if
certain processes are systematically removed or enhanced by the removal of
trajectories.

Figure 4: Adiabatic populations for the three lowest-lying singlet states,
S0-2, given by the multistate MASH population estimator (Section II.1.2) given
as a function of time and taking all trajectories into account.

The adiabatic populations for the three electronic states S0-2 used in our
simulations are given in Fig. 4 for the initial $200$ fs of the dynamics. A
rapid decay of the S2 state (green line) can be observed after $\sim$8 fs.
This decay is correlated with a population increase on S1 (teal line),
indicating that fast initial dynamics on S2 rapidly reaches an S2/S1 CI. For
the early time dynamics ($t<30$ fs), the ground-state can be seen to be
inaccessible from the S2 state (see panel 1 in Fig. 3). After $30$ fs, there
is a steady increase in the ground-state S0 (purple line) population, showing
passage through the S1/S0 CI and therefore formation of the photoproducts
linked to each CI (Fig. 2). After $100$ fs, we observe a decreased rate of
population transfer from S2 to S1, resulting in depletion of S1 and an overall
decrease in the population of the two excited states matched by the continued
increase in the population of the ground state. After $\sim 175$ fs, the
population transfer is largely over, although errors in this region will be
high due to the small number of trajectories. The populations observed in Fig.
4 indicate ultrafast non-radiative deactivation pathways for cyclobutanone,
there is then likely the possibility of further non-equilibrium ground-state
chemistry as a result of very hot molecules in the ground state following the
decay dynamics.

As shown previously, the deactivation pathways available to cyclobutanone
imply the cleavage of one or two bonds in the ring structure. To identify
these different pathways, the bond lengths for all multistate MASH
trajectories are plotted, separated into $\mathrm{\alpha}$ and
$\mathrm{\beta}$-CC bonds according to the inset in Fig. 5. Four distinct
outcomes are observed:

  * _(i)_

There is no bond breaking (purple), and the molecule remains in the FC region.
These are characterized by short $\mathrm{\alpha}$ and $\mathrm{\beta}$-CC
bond lengths ($R<2.2$ Å). It is important to note that the trajectories
included in this group do not progress over 90 fs.

  * _(ii)_

There exists a single $\mathrm{\alpha}$-CC bond breaking (teal). This pathway
gives rise to a ring-opened structure i.e. CH_2CH_2CH_2CO. This is indicated
by an elongation of an $\mathrm{\alpha}$-CC bond, but due to the carbon
backbone still being intact, the distance between the two carbon atoms is
limited to $R\sim 4.5$ Å. This pathway is a result of passing through the
S1/S0 CI (upper right in Fig. 2).

  * _(iii)_

Two $\mathrm{\alpha}$-CC bonds break to liberate CO, where there is a large
increase in $\mathrm{\alpha}$-CC bonds to values above 4.5 Å (blue). Attempts
to optimize an MECI using SA(3)-CASSCF(12,12) involving CO dissociation were
unsuccessful, however, trajectories were observed with CO dissociation after
having decayed to the ground state via the $\mathrm{\alpha}$-CC dissociation
S1/S0 CI (lower right, Fig. 2). This means the CO dissociation takes place on
the ground state rapidly after passing through a CI. It is worth noting for
option _iii)_ a single trajectory also displays ground state dissociation of
CH_2CH_2CH_2 into CH_2CH_2 and CH_2, which can be seen in Fig. 5 as a blue
line displaying increases in both $\mathrm{\alpha}$ and $\mathrm{\beta}$-CC
bonds.

  * _(iv)_

An $\mathrm{\alpha}$-CC bond and $\mathrm{\beta}$-CC bond breaking (pale
green) showing the stepwise ring-breaking mechanism where ethene and ketene
are produced. Here, an $\mathrm{\alpha}$-CC bond breaks first, followed by a
$\mathrm{\beta}$-CC bond breaking. This can be linked to the S1/S0 MECI in
Fig. 2 (S1/S0, lower center).

Figure 5: A 2D projection of all CC bond lengths across all trajectories. The
CC bonds are partitioned according to the labeling on the inset. Each
trajectory is clustered based on the final geometry of the trajectory,
resulting in four channels: _i)_ FC, _ii)_ ring opening, _iii)_ CO production
and _iv)_ ethene production. See the main text for more details.

###  IV.4 UED

The nonadiabatic dynamics of cyclobutanone presented in the previous section
has served as the basis for calculating the rotationally averaged total
electron scattering signal as a function of time, using the independent atom
model (IAM) as described in the theory section. To correctly account for the
time evolution of the molecular geometries we use a percent difference in the
form,

$\%S_{t}(s,t)=100\times\frac{S_{t}(s,t)-S_{t}(s,0)}{S_{t}(s,0)},$ (11)

where $S_{t}(s,t)$ is the total electron scattering probability at time $t$
and $S_{t}(s,0)$ is the total electron scattering probability at $t=0$ (i.e.
before the pump). In Fig. 6, the time-resolved UED pattern for cyclobutanone
is shown as a function of the amplitude of the momentum transfer vector $s$.
Due to the rapid loss of trajectories during the simulations, discussed
earlier, we only show the UED signal for the first 200 fs.

Figure 6: Gas-phase UED pattern shown with percent difference as a function of
time for cyclobutanone. All active trajectories are taken into account via the
process described in section III. Dashed horizontal lines represent the key
features of the UED pattern. Figure 7: Six static UED percent difference
signals for key optimized geometries of gas-phase cyclobutanone and the static
signal from an example structure obtained from our dynamics simulations
showing CO dissociation (blue) and an average of the trajectories where the
considered structure is the final product (green). Insets show the structure
used to calculate the static signal: a) S2 minimum, b) S2/S1 compression, c)
S2/S1 $\mathrm{\alpha}$ cleavage, d) S1/S0 $\mathrm{\alpha}$ cleavage, e)
S2/S1 ethene production and finally f) CO production. Vertical dashed lines
match the horizontal dashed lines shown in Figure 6, showing key features in
the overall UED.

UED cross-section percent differences shown in Fig. 6 present four distinct
features at $s\approx$ 1.25, 2.4, 4.6, and 7.4 Å-1. These features constitute
a fingerprint of the structural dynamics taking place in cyclobutanone,
therefore, they can be interpreted with the help of additional static signals
for the key geometries discussed in Sections IV.2 (purple line in Fig. 7). A
static signal for a fixed time (80 fs for panels a and b, and 150 fs for
panels c and f) is also shown in Fig. 7 for each pathway (green lines),
showing some broadening of the peaks due to vibrational dispersion, along with
subtle shifts of peak maxima. In addition, the electron scattering cross-
sections are calculated for the four individual outcomes using groups of
trajectories in the dynamics simulation (_i,ii,iii,iv_ in Section IV.2) and
shown in Fig. 8, to aid the interpretation of the results. Due to the
significant similarities between the photoproducts and the high degree of
symmetry in cyclobutanone, there is a large number of overlapping features in
all static and individual cluster signals (see Figures 7 and 8). Based on
these similarities, we postulate that the strong negative signal at $\sim$1.25
Å-1 and the strong positive feature at $\sim$2.4 Å-1, common to all possible
photoproducts, are related to the breaking of one or two
$\mathrm{\alpha}$-bonds to produce open-ring (_i_), CO and CH_2CH_2CH_2
(_ii_), or ethene and ketene (_iii_). These two features, which reach a
maximum at $t\sim 50$ fs, are then related to the electronic state population
transferring through the S2/S1 CI and subsequently through the S1/S0 CI (see
Figures 7d), 7e) and 7f)). This timescale is consistent with what is observed
in the populations of S1 and the S0 ground state as shown in Fig. 4, but
significantly faster than that observed in reference 19. It is possible that
our simulations underestimate the time in S2 due to a smaller barrier, as
discussed in the context of Fig. 3 for SA(3)-CASSCF(12,12) compared to the
matching XMS-CASPT2, resulting in somewhat fast decay and product formation.

The other UED feature common to the total signal (Fig. 6) and all individual
pathways (Fig. 8) is an oscillatory positive signal at $s\approx 7.4$ Å-1. For
the initial 50 fs of simulation, in this $s$ region, one can see strong
coherent motions with an oscillatory period of $\sim 15$ fs, indicating a
possible coherent vibrational motion. All bonds between heavy atoms (C and O)
are plotted for an exemplary trajectory in the SI (Fig. 6). Only the CO bond
length has a period that matches these oscillations, suggesting the carbonyl
stretch may be responsible for this feature.

An additional feature can be observed at $s\sim 4.6$ Å-1 with weak positive
intensity for the initial $80$ fs of the time-resolved UED, becoming weakly
negative after. This positive weak feature ($4-6$ %) only shows in those
pathways where $\beta$-bond breaking does not occur (Figures 8a) and 8c)) and
it is negative in the one where this happens (Fig. 8b)). When looking at the
static signals in Fig. 7, one can see that the CO dissociation channel (Fig.
7f)) yields a positive signal at $s\sim 4.6$ Å-1, and that this feature is
also present in the ring-open structures (Fig. 7c) and 7d)) at $s\sim 4.9$
Å-1. Therefore, one can relate the sign and strength of this feature to the
separation between the two carbons forming the $\beta$ bonds in the molecule
(see inset in Fig. 5). The change of sign for this feature at $t\approx 80$
fs, together with the great influence path _iv)_ has on the dynamics, suggests
the ethene and ketene formation can be directly observed by looking at this
region of the time-resolved UED signal.

It is important to note that the FC group presents oscillatory behavior in all
of the aforementioned UED features. These oscillations can be explained by
taking into account that the molecule is vibrationally excited (i.e. hot) when
the FC region is accessed, giving rise to high-amplitude motions. This
behavior is not seen in the averaged signal due to the small contribution of
this feature to the overall dynamics.

Figure 8: Percent difference UED patterns for each cluster of trajectories
reported in Fig. 5. Subplots show the clustered UED for: a)
$\mathrm{\alpha}$-CC cleavage, b) ethene production, c) CO production, and d)
FC. Clusters that do not contain data until the target 200 fs endpoint are
truncated at the final data point.

##  V Conclusions

In summary, motivated by an experiment at the SLAC MeV-UED facility and the
associated prediction challenge, we have applied multistate MASH and
SA(3)-CASSCF(12,12)/aug-cc-pVDZ to identify the gas-phase photodynamics of
cyclobutanone after photoexcitation by 200 nm pump pulses into the n-3s
Rydberg state. Mechanistic details have been identified using both a static
analysis and nonadiabatic dynamics, finding a fast decay from the S2 state
into the S1 via two CIs. A further two CIs allow for similarly fast access to
the ground state. This results in the same set of photoproducts previously
observed, namely $\mathrm{\alpha}$ ring-opening and ethene+ketene production,
the former of which can further dissociate on the ground electronic state to
liberate CO.

For direct comparison with experimental results we have also predicted the
gas-phase isotropic UED signal of cyclobutanone (see Fig. 6). We find that
there is significant overlap between several of the products meaning that the
UED signal contains multiple features. However, several distinct features
indicate the formation of reaction products and loss of cyclobutanone in the
equilibrium geometry. The timescale of the reaction can also be inferred from
the UED signals, which, unsurprisingly, closely matches that of our simulated
reaction dynamics. However, we do note that the reaction, in our simulations,
proceeds significantly faster than previous observations.Kuhlman  _et al._
(2012); Kuhlman, Sølling, and Møller (2012)

Significant caveats remain for our predictions. The instability of the active
space means that we could be potentially blind to certain reaction channels.
Additionally, the value obtained for each channel’s quantum yield is
susceptible to significant errors from the choice of electronic structure
method and the associated instabilities. This does highlight the need for
robust ’black-box’ electronic structure methods compatible with on-the-fly
nonadiabatic dynamics simulations. Methods such as ADC(2) or TDDFT already
exist, but are not universally applicable as they have well-documented flaws.
A potential way forward, could be the selected CI family of methods,Tong  _et
al._ (2000); Coe and Paterson (2013) some benchmarking of which has been
already been included in our results. Further development and benchmarking of
these methods is required to confirm whether they indeed offer accurate
electronic structure in a more robust, black-box fashion. Also, it is
important to add that the truncation of trajectories due to uncoupled hops
affects the statistics of the dynamics, and therefore needs to be refined in
future work. Finally, the multistate MASH used in the simulations is still
being developed, and we anticipate that refinement of the algorithm will
improve the results further.

##  VI Acknowledgements

The authors would like to thank the organizers of the cyclobutanone prediction
challenge for arranging this special edition. AK, MJP, LH, and AP acknowledge
funding from the Leverhulme Trust (RPG-2020-208), and AK, AMC and MJP further
acknowledge EPSRC EP/V006819. AK also acknowledges EPSRC EP/V049240 and grant
DE-SC0020276 from the US Department of Energy. JER was funded by a mobility
fellowship from the Swiss National Science Foundation under Award
P500PN_206641/1.

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