# Molecular Frame Photoelectron Angular Distributions in Polyatomic Molecules
from Lab Frame Coherent Rotational Wavepacket Evolution

Margaret Gregory  Department of Chemistry and Physics, University of Mary
Washington, 1301 College Avenue, Fredericksburg VA, 22401.  Paul Hockett
National Research Council of Canada, 100 Sussex Drive, Ottawa, ON, K1A 0R6,
Canada  Albert Stolow  National Research Council of Canada, 100 Sussex Drive,
Ottawa, ON, K1A 0R6, Canada  Department of Physics, University of Ottawa, 150
Louis Pasteur, Ottawa, ON, K1N 6N5, Canada  Department of Chemistry and
Biomolecular Sciences, University of Ottawa, 150 Louis Pasteur, Ottawa, ON,
K1N 6N5, Canada  Joint Centre for Extreme Photonics, NRC-University of Ottawa,
ON, K1A 0R6, Canada  Varun Makhija<EMAIL_ADDRESS>Department of Chemistry
and Physics, University of Mary Washington, 1301 College Avenue,
Fredericksburg VA, 22401.

(August 27, 2024)

###### Abstract

A theory and nethod for a matrix-based reconstruction of Molecular Frame (MF)
photoelectron angular distributions (MFPADs) from Laboratory Frame (LF)
measurements (LFPADs) is developed and basic applications are explored. As
with prior studies of MF reconstruction, the experimental side of this
protocol is based upon time-resolved LF measurements, in which a rotational
wavepacket is prepared and probed as a function of time via photoionization,
followed by a numerical reconstruction routine. In contrast to other
methodologies, the protocol presented here does not require determination of
the photoionization matrix elements, and consequently takes a relatively
simple numerical form of a matrix equation. Significantly, this simplicity
allows the successful reconstruction of MFPADs for polyatomic molecules. We
numerically demostrate this scheme for two realistic molecular photoionization
cases: $N_{2}$ and $C_{2}H_{4}$. This new technique is expected to be
generally applicable to a broad range of MF reconstruction problems involving
photoionization of polyatomic molecules.

##  1 Introduction

Photoelectron Angular Distributions (PADs) from isolated molecules provide a
wealth of information about the state of the molecule from which the electron
is ejected. In the case of ionization of a stationary, ground state molecule
the outgoing electron experiences the molecular potential which imprints an
orbital angular momentum dependent phase shift on its scattering wavefunction.
The relative phases of the various angular momentum channels (or partial
waves) which comprise the scattering wavefunction determine the shape of the
PAD. One may therefore infer both electronic and vibrational structural
information from the PAD [1, 2, 3, 4, 5, 6, 7]. This is equally true of
excited molecular electronic states, whose properties may be directly inferred
from PADs [8, 9, 10, 11]. The information content of the measurement is
greatly enhanced if Molecular Frame PADs (MFPADs) can be measured. In such
cases, the partial wave content of the scattering wavefunction is greatly
incresed, since there is no orientational averaging, thus making available
additional information on the molecular and electronic structure [12, 4, 11,
5, 13]. For example, if MFPADs can be measured for a number of laser
polarization directions relative to the molecular axes, a quantum tomography
of the final state (the ejected electron plus the ion) is possible in some
cases by retrieval of the relative phase of each partial wave channel. This is
traditionally termed a ‘complete experiment’ [14, 15, 16, 17, 18, 19, 20, 21].
Recent experiments and related studies have also demonstrated direct
measurement of molecular structure from such an experiment [6, 22, 23, 7, 24].
These features of MFPADs may also be applied to the case of a time-evolving
excited state - i.e. molecular wavepacket dynamics. In such cases both MFPADs
and LFPADs can track the time varying electronic character of the excited
state, thus allowing measurement of time-resolved electronic-vibrational
dynamics, including their non-adiabatic coupling [4, 11, 25, 26, 22, 23, 27].
MFPADs, as in the static case, provide a greater depth of information and
potentially directly imprint the electronic character of the wavepacket onto
the photoelectron observable [25, 11]. For a general review of the
fundamentals of PADs see [28, 29].

It is therefore of considerable interest to develop methods that will allow
the measurement of MFPADs in non-linear polyatomic molecules, the primary goal
of this manuscript. There are several experimental approaches which can be
used to determine MFPADs. Conceptually these rely on either (i) transforming
to the MF via vector correlations in the LF, or (ii) alignling molecular axes
in the LF to some degree. The first is based on kinematically complete (6D)
determination of the 3D recoil momentum vectors of both photoelectrons and
associated ionic photofragments in coincidence [15, 30, 31, 14, 24, 12, 32,
33, 34]. A fundamental restriction of this approach is that it applies only to
dissociative photoionization 111Or requires a post-ionization dissociation
laser pulse to achieve this, as used in Coulomb explosion imaging (CEI) type
measurements.. Nevertheless, this approach was used to study ultrafast
nonadiabatic photodissociation of a polyatomic molecule [35]. Combined with
the axial recoil approximation, this permits transformation of the LFPAD into
the recoil frame which is often very close to that of the MF. The axial recoil
approximation, however, is very restrictive in that it requires fragments to
be emitted axially along the bond direction, such as in favourable cases
containing carbon-halogen bonds. Unfortunately, molecules in excited states
generally bend as they dissociate, thus obviating broader use of the axial
recoil approximation [36]. Importantly, the general approach presented here
depends on neither dissociative photoionization processes nor the axial recoil
approximation.

The second approach to MFPADs is via laser-aligning or orienting molecular
ensembles prior to photoionization, for which numerous experimental techniques
have been developed, and are thoroughly reviewed in [37] and [38]. Briefly,
ensembles of linear molecules with a high degree of laser-induced alignment
are readily achievable under field-free conditions [39, 40, 21], while
orientation of molecules lacking inversion symmetry is much harder to achieve
[41, 42, 43, 44, 45, 46]. In addition, the experimentally achievable degree of
ground state alignment directly impacts the ability to extract MFPADs [47].
For non-linear polyatomic molecules, aligning or orienting all three body-
fixed (MF) axes under field-free conditions has also achieved for some cases,
but still presents a significant hurdle to MF measurements in general [48, 49,
50, 51]. Even in cases where all axes can be orientated, an arbitrary
molecular orientation cannot be accessed since all rotational degrees of
freedom cannot be independently controlled. Finally, for the extension of
these techniques to excited state vibrionic wavepacket studies, one must keep
in mind that the coherent MF axis distributions constructed for the molecular
ground state are modified in the excited state due to the pump transition
dipole matrix elements which themselves have polarization sensitive selection
rules. In general, one cannot expect that the excited state axis distribution
moments, following pump pulse absorption, will be the same as those for the MF
ground state. This presents a further challenge to the use of selected
rotational revivals for the ‘fixed-in-space’ approach to MFPADs experiments.

To overcome these hurdles, an alternative route to the MF is available via
time domain Rotational Wavepacket (RWP) measurements, specifically the
Orientation Resolution through Rotational Coherence Spectroscopy (ORRCS)
method [36, 52, 53, 21, 54, 55, 56, 57]. In this approach, a RWP is launched
by an impulsive nonresonant femtosecond laser pulse (the pump pulse) in a
thermal molecular gas. Rather than trying to simultaneously fix all three axes
in space at a specific time delay, this method generates a coherent ground
state rotational wavepacket (RWP). The RWP is then photoionized by a delayed
femtosecond probe pulse, and the signal of interest is measured as a function
of the time delay. This time dependent signal provides sufficient information
in some cases to permit reconstruction of the signal for a perfectly oriented
molecule, at an arbitrary orientation with respect to the LF. The ORRCS
approach has been demonstrated for the cases of strong field ionization of
linear [52] and non-linear polyatomic molecules [54, 55, 53], strong field
photodissociation of linear molecules [36], molecular high harmonic generation
spectroscopy [40, 58, 59], and photoionization matrix element retrieval (hence
MFPAD reconstruction) for one-photon ionization of a homonuclear diatomic
molecule. [21].

Here we show that the ORRCS approach applies to the reconstruction of MFPADs
for the photoionization of polyatomic molecules, without any requirement for
dissociative photoionization, or the more stringent requirements of complete
photoionization matrix element retrieval (as recently demonstrated for a
homonuclear diatomic in Ref. [21]). A summary of the general approach we
develop is as follows. We first reduce the problem to a system of linear
equations

$\mathbf{C}^{mol}=\mathbf{\hat{G}}\mathbf{C}^{lab},$ (1)

by applying symmetry considerations. Here $\mathbf{C}^{mol}$ and
$\mathbf{C}^{lab}$ are vectors in the molecular and lab frames respectively,
and $\mathbf{\hat{G}}$ is a known matrix. Components of the vector
$\mathbf{C}^{lab}$ can be directly extracted from time resolved measurements
of LFPADs from one-photon ionization of an evolving RWP. Provided that a
sufficient number of components are extracted, we can uniquely determine
$\mathbf{C}^{mol}$ by solving Eq. 1, which is directly related to the MFPADs
for various polarization geometries. The structure of matrix
$\mathbf{\hat{G}}$ determines the number of $\mathbf{C}^{mol}$ that can be
retrieved, which then determines for which polarization geometries the MFPAD
can be reconstructed. We detail this route to the MFPADs, as follows. In
section 2, we outline the theory that leads to an equation like Eq. 1. This is
further divided into subsections. In section 2.1.1, we briefly discuss the
formalism used to describe the ground state RWP. This leads to section 2.1.2,
in which we discuss the formalism of photoionization from a RWP. In section
2.2 we descirbe the related formalism for ionization in the MF. This then
leads to section 2.3, in which we relate photoinzation in the MF and LF and
invoke symmetry srguments to derive the desired Eq. 28 \- an equation of the
form of Eq. 1 \- for inversion symmetric molecules having both a horizontal
and vertical mirror plane. In section 3, we demonstrate the reconstruction of
MFPADs, for various polarizations, for photoionization of $N_{2}$ ($D_{\infty
h}$ point group symmetry) and the asymmetric top $C_{2}H_{4}$ ($D_{2h}$ point
group symmetry) to their ground cationic electronic states. Finally, in
section 4 we provide a summary of our work, potential future avenues of
investigation and suggested experimental applications.

##  2 Theory

In the method proposed here MFPADs, are reconstructed using LFPADs measured
during the time evolution of a RWP. First, a non-resonant, femtosecond
duration, linearly polarized intense pump pulse intersects a molecular gas,
exciting via stimulated Raman Scattering several rotational states in the
ground electronic state [38, 37]. Next, a time delayed probe pulse with photon
energy greater than the molecular ionization potential ionizes the molecule
and the momentum distribution of the ejected photoelectrons is measured. This
measurement is repeated at several time delays, resulting in a time sequence
of LFPADs. These LFPADs serve as raw material for the reconstruction described
here. We begin discussing the theoretical formalism needed to describe this
experiment, based on the formalism developed by Underwood and coworkers [60,
51, 11]. In the following, we first discuss the description of the rotational
wavepacket (Sec. 2.1.1), followed by the description of the ionization at each
time delay (Sec. 2.1.2). In the subsequent subsection (Sec. 2.2) we discuss
photoionization in the MF, and finally derive the connection between the
LFPADs and MFPADs that allow us to go from the experiment to the MFPAD (Sec.
2.3), which is the main result of this work. For further background to the
current work, interested readers can find more detailed introductory material
on the topic of ionization from aligned molecular ensembles in, for example,
Refs. [8, 11, 28], and references therein. Readers interested purely in the
applications of the protocol may wish to skip the derivations and theoretical
discussion, and proceed directly to Sec. 3, which illustrates, numerically, MF
reconstruction demonstrated here for two cases.

The overarching aim of this section is to reduce the LF-MF connection to a
solvable system of linear equations which of the form of Eq. 1. Here
$\mathbf{C}^{mol}$ and $\mathbf{C}^{lab}$ are vectors in the molecular and lab
frames respectively, and $\mathbf{\hat{G}}$ is a known matrix. Components of
the vector $\mathbf{C}^{lab}$ can be directly extracted from time resolved
measurements of LFPADs from one-photon ionization of an evolving RWP. Provided
that a sufficient number of components are extracted, one can uniquely
determine $\mathbf{C}^{mol}$ by solving Eq. 1, should such a solution prove to
exist.

###  2.1 Ionization from the Rotational Wavepacket

####  2.1.1 Describing the Rotational Dynamics

In our formalism, the evolution of the excited rotational wavepacket can be
described by the time and orientation angle dependent molecular axis
distribution [60, 51, 11],

$P(t,\phi,\theta,\chi)=\sum_{KQS}A^{K}_{QS}(t)D^{K*}_{QS}(\phi,\theta,\chi).$
(2)

This equation is a multipole expansion of the probability -
$P(t,\phi,\theta,\chi)$ \- that, at a time $t$ after the excitation, the
molecule will have an orientation described by the Euler angles
$(\phi,\theta,\chi)$. These angles serve to define the orientation of a rigid
body in three dimensions [61], with respect to a space-fixed (LF) coordinate
system. The space-fixed coordinate system is labelled by $(X,Y,Z)$, and a
body-fixed (MF) coordinate system by $(x,y,z)$, in the standard manner [62].
In the space-fixed frame the $Z$-axis is defined by the laser polarization and
the $Y$-axis by the laser propagation direction. The molecular axes define the
molecular frame $(x,y,z)$, and are chosen according to molecular geometry,
with $z$ usually corresponding to the axis of highest symmetry. The Wigner D
matrix elements $D^{K}_{QS}(\phi,\theta,\chi)$ are a basis on the space of
orientations [63] and facilitate the expansion. It may be noted that, were the
third Euler angle $\chi$ set to zero (as would be the case for linear
molecules), the Wigner D matrix elements reduce to the well known spherical
harmonics [62]. In the more general case of a 3D rigid rotor, the third angle
is required to describe the orientation of the rotor about $z$-axis.

Note that the time dependence of the wavepacket is fully described by the
functions $A^{K}_{QS}(t)$. These multipole moments of the axis distribution,
or axis distributions moments (ADMs), are exactly analogous to the spherical
multipole moments of any electrostatic charge distribution [64], with the
fundamental difference being the inclusion of an additional index needed due
to the dependence on a third angle - note the three expansion indices $K,Q$
and $S$, rather than the usual indices $l$ and $m$ of the (speherical
harmonic) multipole expansion. The index $K$ takes integer values and
corresponds to $l$ (degree or rank of the operator, often corresponding
physically to total angular momentum) in the standard multipole expansion; the
remaining indicies define components (order of the operator, or projection
terms) within the ($K$ or $l$) subspace. For instance, $K=1$ is a dipole
moment, and if the coefficient $A^{1}_{00}$ dominates the expansion at a
particular time, then the axis distribution strongly resembles $\cos\theta$ at
that time and so on. The indices $Q$ and $S$ each run from $-K$ to $K$, by
analogy with $m$ which runs from $-l$ to $l$, and describe the axis
distribution as a function of the angles $\phi$ and $\chi$ respectively. Also
analagous to $l$, $K$ admits the physical interpretation of angular momenta
associated with the rotational wavepacket. The maximum value of $K$ can be
thought of as the maximum angular momentum transferred to the molecule by the
pump222Strictly speaking, this is the maximum angular momentum present in the
axis distribution of the LF ensemble, defined to include all rotational state
contributions, see Ref. [11] for details. and, analogous to $m$, $Q$ is its
projection on the LF $Z$ axis. The new index $S$ is the projection of $K$ on
the molecular $z$ axis, and is conjugate to the angle $\chi$ that tracks the
rotation of the molecular body around the $z$ axis.

Finally we note the restrictions on the values of $K$, $Q$ and $S$ which are
imposed by symmetry. For a wavepacket excited by a linearly polarized field,
the molecular axis distribution must be independent of $\phi$ in order to
preserve cylindrical symmetry, thus rendering $Q=0$. Further, $K$ must be even
to preserve inversion symmetry. Although the equations herein are general, we
usually restrict discussion to this special (but common) case, and drop the
additional subscript where it is unnecessary. (This inversion and
cylindrically-symmetric case is generally termed molecular alignment, and is
conceptually distinct from molecular orientation, which also allows non-
cylindrically symmetric cases with up-down asymmetry, hence additionally
yielding odd $K$ and $Q\neq 0$ terms.) The restrictions on $S$ depend on the
point group symmetry of the molecule, and are dealt with individually for the
specific cases analyzed here (Sec. 3).

####  2.1.2 Describing the Ionization Step

Excitation of a RWP by a pump pulse is followed by ionization by a time
delayed probe pulse. The measured, time resolved LFPADs are also typically
described by a multipole expansion as follows,

$\sigma(\epsilon,t,\theta_{e},\phi_{e})=\sum_{LM}\beta_{LM}(\epsilon,t)Y^{L}_{M}(\theta_{e},\phi_{e}),$
(3)

where $\epsilon$ is the kinetic energy of the electron and $\theta_{e}$ and
$\phi_{e}$ its polar and azimuthal ejection angles. The anisotropy parameters
$\beta_{LM}(\epsilon,t)$ are now the time dependent multipole moments of the
LFPADs, which fully characterize their variation as a result of the evolving
rotational wavepacket. These therefore must contain the ADMs described above,
and $L$ and $M$ must be related to $K$ and $S$ (recall that $Q=0$). The
relation between the ADMs and the anisotropy for perturbative one-photon
ionization was derived under the dipole approximation in [11]. The resulting
expression may be compactly expressed as,

$\displaystyle\beta_{LM}(\epsilon,t)=\sum_{KS}C^{LM}_{KS}(\epsilon)A^{K}_{0S}(t)$
(4) $\displaystyle
C^{LM}_{KS}(\epsilon)=\sum_{\zeta\zeta^{\prime}}D_{\zeta}(\epsilon)D^{*}_{\zeta^{\prime}}(\epsilon)\Gamma^{\zeta\zeta^{\prime}LM}_{K0S}.$
(5)

where we have explicitly set $Q=0$ (i.e. cylindrical symmetry). The daunting
number of subscripts and superscripts in these equations (resulting from
several tensor products of irreducible representations of $SO(3)$ [11])
primarily represent the angular momenta involved in the problem. We first note
that, much like in Eq. 2, the time dependence of the $\beta_{LM}(\epsilon,t)$
is entirely contained in the ADMs as expected. The energy dependent
coefficients $C_{KS}^{LM}(\epsilon)$ couple $K$ and $S$ to the angular
momentum of the ejected electron $L$ and its projection $M$ on the molecular
$z$ axis. The coupling is facilitated by a coupling parameter we designate
$\Gamma^{\zeta\zeta^{\prime}LM}_{KQS}$, which is summed over possible
irreducible components of the electric field tensor - essentially arising as
the square of the ionizing photon’s angular momentum [11], thus defining an
effective angular momentum coupling into the observables - $P=0,1$ or $2$ and
their projections on the molecular axis, $\Delta q=q^{\prime}-q$, where $q$
and $q^{\prime}$ are spherical components of the electronic dipole moment
operator as discussed below. The full form of
$\Gamma^{\zeta\zeta^{\prime}}_{KQS}$, built from a number of Wigner 3j
Symbols, is provided in Eq. 29 in the appendix. Physically, it is of note that
the terms arise from the square of the photoionization matrix elements, and
include various couplings which ultimately contribute to any experimental
observables; these can be treated as effective angular momentum couplings of
the derived quatities. For example, the coupling between the electric field
tensor rank $P$ and the ADMs with rank $K$ (hence the axis distribution
$P(t,\phi,\theta,\chi)$, see Eq. 2), results in a term with allowed angular
momentum values ranging from $L=K-P$ to $L=K+P$ and molecular frame
projections $S+\Delta q$. This term therefore provides selection rules for
$L$, and sets limits on the observable spatial anisotropy in the LF,
$L_{max}$, which can be no higher than that defined by the anistoropy of the
electric field tensor coupled to the molecular axis distribution (additional
selection rules may pertain, see below and Eq. 29).

The remaining label $\zeta$ pertains to basis or channel functions used to
represent the photoionization process in the formalism in [11]. $\zeta$ labels
consist of three parts: $\zeta^{+}=\nu_{\alpha}^{+}$, $\alpha^{+}$ labels
vibrational and electronic states of the ionic core, $\zeta_{dip}=q=-1$,$0$ or
$1$, the spherical component of the electronic dipole moment operator which
facilitates ionization, and $\zeta_{f}$=$\varGamma$,$\mu$,$h$,$l$ labels the
basis functions used to construct the wave function of the ionized electron.
This wavefunction, defined exclusively in the MF, appears in expressions for
both the LF and MFPADs, and thus is the key to linking the two. In order to
interpret the label $\zeta_{f}$, we need to analyze the form of this
wavefunction, which is written as,

$\displaystyle\phi(\mathbf{r^{\prime}_{e}},\epsilon;\mathbf{R})=\sum_{\varGamma\mu
hl}X^{\varGamma\mu*}_{hl}(\theta^{\prime}_{e},\phi^{\prime}_{e})\psi_{\varGamma\mu
hl}(r^{\prime}_{e},\epsilon;\mathbf{R}),$ (6) $\displaystyle
X^{\varGamma\mu}_{hl}(\theta^{\prime}_{e},\phi^{\prime}_{e})=\sum_{\lambda}b^{\varGamma\mu}_{hl\lambda}Y_{l\lambda}(\theta^{\prime}_{e},\phi^{\prime}_{e}).$
(7)

$\mathbf{R}$ represents the nuclear coordinates,
$r^{\prime}_{e},\theta^{\prime}_{e},\phi^{\prime}_{e}$ are coordinates of the
electron in the MF, and $\epsilon$ its energy. Further,
$X^{\varGamma\mu}_{hl}(\theta^{\prime}_{e},\phi^{\prime}_{e})$ are symmetry
adapted spherical harmonics - a superposition of spherical harmonic functions
- $Y_{l\lambda}(\theta^{\prime}_{e},\phi^{\prime}_{e})$ \- which form a basis
for the representation $\varGamma$ of the molecular point group. $\mu$ and $h$
are corresponding symmetry labels which replace the projection of $l$ on the
$z$-axis $\lambda$ when cylindrical symmetry is lost, while accounting for any
remaining degeneracy. For a linear molecule, $\lambda$ is a conserved quantity
and these labels vanish. Given this equation, we can interpret $l$ as the
orbital angular momentum of the ejected electron in the molecular frame. This
is, however, not a good quantum number in the region of the nascent ion core
(i.e., not a conserved quantity), since the molecular potential experienced by
the outgoing electron is not spherically symmetric. Physically, angular
momentum can be exchanged with the potential during ionization/electron
scattering. Furthermore, it is also the case that the initial bound state
orbital is usually described by a superposition of many $l$ terms. It is of
particular note that this results, in general, in more complex scattering
patterns than would pertain in the simplest case: the final state is not
restricted by a $\Delta l=\pm 1$ selection rule as might be intuitively
expected for a simple model of single electron-photon coupling under spherical
symmetry, as is often applicable in atomic photoionization problems from
states with well-defined initial orbital angular momentum (see, for instance,
Refs. [65, 4, 28] for further discussion). In the observable, the mixing of
pairs of channels labeled by $l$ and $l^{\prime}$, leads to the lab frame
angular momentum $L$ with values
$|l-l^{\prime}|,|l-l^{\prime}+1|,...,|l+l^{\prime}-1|,|l+l^{\prime}|$. The
imperative function performed by $\Gamma^{\zeta\zeta^{\prime}LM}_{K0S}$ is to
simultaneously couple $K$, $S$, $l$ and $l^{\prime}$ into a single lab frame
angular momentum $L$. It performs the same function with respect to the
relevant projections of each of these angular momenta. This connection between
the scattering channels and the LF angular momentum will be imperative in the
following section in which a connection between the LF and MFPADs is derived.

With this, we can finally identify the quantities $D_{\zeta}(\epsilon)$ in Eq.
5 as the matrix elements of irreducible components of the MF electronic dipole
operator, between the ground electronic state and a continuum channel function
labeled by $\zeta$. We note that it is the set of products
$D_{\zeta}(\epsilon)D_{\zeta^{\prime}}(\epsilon)^{*}$ that appear in the
equation. We designate these as $d_{\zeta\zeta^{\prime}}$, and rewrite Eq. 5
as,

$C^{LM}_{KS}(\epsilon)=\sum_{\zeta\zeta^{\prime}}d_{\zeta\zeta^{\prime}}(\epsilon)\Gamma^{\zeta\zeta^{\prime}LM}_{K0S}.$
(8)

As shown in the next section, these coefficients $C^{LM}_{KS}$ represent the
coefficients $C^{lab}$ in Eq. 1, which can be extracted by measurements of
time varying LFPADs from a rotational wavepacket. The ADMs in Eq. 4 can be
accurately simulated for the electronic ground in the rigid rotor
approximation, provided the experimental conditions are well known [56, 66,
21]. Therefore, the coefficients $C^{LM}_{KS}$ can be extracted _uniquely_
from a measurement of the $\beta_{LM}(\epsilon,t)$ by linear regression under
appropriate experimental conditions. This procedure, constituting the method
of ORRCS, has been demonstrated and analyzed by several other authors for
different molecules and experimental conditions, in several contexts including
one-photon ionization [36, 52, 53, 21, 54, 55, 56]. Therefore, for the
remainder of the discussion of the reconstruction protocol developed herein,
we will start with the assumption that the $C^{LM}_{KS}$ have been reliably
extracted from an experimental measurement. As detailed in the previous
publications, the number of coefficients extractable depends on the number of
rotational states excited by the pump pulse, since a larger spread of
coherently excited $J$ states enhances the contribution of the higher $K$ ADMs
[11, 21]. In the following we identify the molecular frame coefficients
$C^{mol}$, and derive the equivalent of Eq. 1 which will allow reconstruction
of MFPADs using the extracted coefficients $C^{LM}_{KS}$.

###  2.2 Ionization in the Molecular Frame

In the MF, the PAD depends on energy, the MF ejection angles of the electron
$\theta^{\prime}_{e}$, and $\phi^{\prime}_{e}$ and the relative orientation
angles between the molecule and the laser polarization, $(\phi,\theta,\chi)$.
The latter are the standard Euler angles as in Eq. 2, describing the
orientation of a rigid body in the LF, and map to angles of the polarization
vector in the MF. For linearly polarized light, $\theta$ and $\chi$ are the
spherical polar and azimuthal angles of the polarization vector in the MF
[66]. Here again we continue to follow the formalism developed in [11]. The
MFPAD can be expanded in a basis set of spherical harmonics, giving the
equivalent to Eq. 3 as follows,

$\sigma(\epsilon,\phi,\theta,\chi,\theta^{\prime}_{e},\phi^{\prime}_{e})=\sum_{LM}\beta_{LM}(\epsilon,\phi,\theta,\chi)Y^{L}_{M}(\theta^{\prime}_{e},\phi^{\prime}_{e})$
(9)

The MF equivalent of Eq. 4 and Eq. 5 can be written as,

$\displaystyle\beta_{LM}(\epsilon,\phi,\theta,\chi)=\sum_{PR\Delta
q}C^{LM}_{PR}(\epsilon,\Delta q)D^{P}_{R\Delta q}(\phi,\theta,\chi)$ (10)
$\displaystyle C^{LM}_{PR}(\epsilon,\Delta
q)=\sum_{\zeta\zeta^{\prime}}d_{\zeta\zeta^{\prime}}(\epsilon)\Gamma^{\zeta\zeta^{\prime}LM}_{PR\Delta
q}$ (11)

Note that the value of $\Delta q$ is fixed in the sum over $\zeta$ and
$\zeta^{\prime}$ in Eq. 11, instead of taking all allowed values as in Eq. 4
(see Appendix for further details).

In the MF, $R$ is the projection of $P$ on the polarization axis, thus it can
take the values $R=-P...P$. For linearly polarized light, $R=0$. Additionally,
$\Delta q$ is the projection of $P$ on the MF axis. As in Eq. 2, the
$D^{P}_{R\Delta q}(\phi,\theta,\chi)$ in Eq. 9 are elements of the Wigner D
matrix, which are analytically known irreducible representations of $SO(3)$.
Thus, if the $C^{LM}_{PR}(\epsilon,\Delta q)$ can be somehow determined from
the experimental measurement of $C^{LM}_{KS}$ (Eqs. 5, 8), the
$\beta_{LM}(\epsilon,\phi,\theta,\chi)$ can be determined providing the MFPAD,
Eq. 9. The $D^{P}_{R\Delta q}(\phi,\theta,\chi)$, and by extension the angular
momentum of the ionizing photon determines the orientation angle dependence of
the MFPADs. The shape for a particular orientation, characterized by the
anisotropy parameters $\beta_{LM}(\epsilon,\phi,\theta,\chi)$, is determined
solely by channel mixing driven by the molecular potential experienced by the
ejected electron. This channel mixing is codified in Eq. 11 by both the
ionization dipole product $d_{\zeta\zeta^{\prime}}(\epsilon)$, which are the
same as in Eq. 8 and the MF coupling parameter
$\Gamma^{\zeta\zeta^{\prime}LM}_{PR\Delta q}$. In this case, the allowed
values are determined by
$L=|l-l^{\prime}|,|l-l^{\prime}+1|,...,|l+l^{\prime}-1|,|l+l^{\prime}|$, by a
given pair of partial wave channels $l$ and $l^{\prime}$. Note that, unlike in
the LF where $L$ must be additionally determined by the sum of the ionizing
photon angular momentum $P$ and the rotational wavepacket angular momentum
$K$, there is no additional restriction on $L$ in MF. Physically, this is
simply due to the lack of any restriction on the observable spatial anisotropy
in the MF, while the LF result corresponds, in essence, to the averaging of
the MF result over all molecular axis alignments, as defined by the angular
momentum coupling detailed in Sec. 2.1.2. As discussed in the introduction,
the MFPAD directly conveys information on molecular structure and symmetry.

The coefficients $C^{LM}_{PR}(\epsilon,\Delta q)$ in Eq. 11 are the
coefficients $C^{mol}$ in Eq. 1 that we would like to be able to extract from
experimental data, specifically from the coefficients $C^{LM}_{KS}$ in Eq. 8.
In the following section we show that an equation like Eq. 1 exists, allowing
this extraction for inversion symmetric molecules with a vertical and
horizontal mirror plane.

###  2.3 From the Lab Frame to the Molecular Frame

To make a connection between the $C^{LM}_{KS}$ in the LF and the
$C^{LM}_{PR}$, we must return to the experimentally relevant equation, Eq. 4,
for the measureable $\beta_{LM}(\epsilon,t)$, from which the coefficients
$C^{LM}_{KS}$ can be extracted. These are then subject to Eq. 8 which we
restate here,

$C^{LM}_{KS}(\epsilon)=\sum_{\zeta\zeta^{\prime}}d_{\zeta\zeta^{\prime}}(\epsilon)\Gamma^{\zeta\zeta^{\prime}LM}_{K0S}.$
(12)

This equation relates the $C^{LM}_{KS}$ to products of ionization transition
dipole moments $d_{\zeta\zeta^{\prime}}$ from the ground electronic state into
a pair of scattering channels $\zeta$ and $\zeta^{\prime}$ . We note that
these are complex in general, and to emphasize this we write the
$d_{\zeta\zeta^{\prime}}$ in terms of their amplitudes and phases as
$d_{\zeta\zeta^{\prime}}=|d_{\zeta\zeta^{\prime}}|e^{i\phi_{\zeta\zeta^{\prime}}}$.
Noting that $d_{\zeta^{\prime}\zeta}=d^{*}_{\zeta\zeta^{\prime}}$, Eq. 8 can
be written as

$C^{LM}_{KS}=\sum_{\zeta}d_{\zeta\zeta}\Gamma^{\zeta\zeta
LM}_{KS}+\sum_{\zeta\zeta^{\prime}<\zeta}|d_{\zeta\zeta^{\prime}}|\left(\Gamma^{\zeta\zeta^{\prime}LM}_{K0S}e^{i\phi_{\zeta\zeta^{\prime}}}+\Gamma^{\zeta^{\prime}\zeta
LM}_{K0S}e^{-i\phi_{\zeta\zeta^{\prime}}}\right).$ (13)

Further, the following general relation can be derived by permutation of
$\zeta$ and $\zeta^{\prime}$ (cf. Appendix):

$\Gamma^{\zeta^{\prime}\zeta
LM}_{K0S}=(-1)^{K-S}(\Gamma^{\zeta\zeta^{\prime}LM}_{K0-S})^{*}$ (14)

Eq. 13 can then be recast as,

$C^{LM}_{KS}=\sum_{\zeta}d_{\zeta\zeta}\Gamma^{\zeta\zeta
LM}_{K0S}+\sum_{\zeta\zeta^{\prime}<\zeta}|d_{\zeta\zeta^{\prime}}|\left(\Gamma^{\zeta\zeta^{\prime}LM}_{K0S}e^{i\phi_{\zeta\zeta^{\prime}}}+(-1)^{K-S}(\Gamma^{\zeta\zeta^{\prime}LM}_{K0-S})^{*}e^{-i\phi_{\zeta\zeta^{\prime}}}\right).$
(15)

Provided that a sufficient number of $C^{LM}_{KS}$ are extracted from an
experimental measurement using Eq. 4, one may attempt to extract several
$|d_{\zeta\zeta^{\prime}}|$ and their associated phases using this equation.
We note, however, that in general a unique solution to this problem is not
attainable, since Eq. 13 represents a more difficult version of the restricted
phase problem encountered in coherent diffractive imaging [67] or ultrafast
pulse measurement [68]. In the usual restricted phase problem, the magnitudes
of a set of complex numbers are directly measured and the relative phases
between them are extracted by applying appropriate restrictions to the data
during the extraction. Here, we cannot directly measure the magnitudes
$|d_{\zeta\zeta^{\prime}}|$, and must instead extract both these and their
associated phases from a measurement of the $C^{LM}_{KS}$. Solving this
problem constitutes a complete molecular photoionization experiment, for which
a unique solution has been attained in few select cases - not including non-
linear polyatomic molecules - by applying restrictions due to molecular
symmetry combined with careful numerical and statistical analysis [16, 17, 18,
19, 20, 21]. Here we instead tackle the reduced questions: What information
_can_ we uniquely determine from a measurement of the $C^{LM}_{KS}$, and is
this information sufficient to construct the MFPAD for photoionization by
linearly polarized light? Note that, as discussed above, the MFPAD still
directly provides information on molecular structure and symmetry since the
channel mixing which determines its shape is driven by the molecular potential
experienced by the ejected electron. In addition, the ionization dipole matrix
elements $D_{\zeta}(\epsilon)$ that determine the MFPADs are themselves
dependent on the nuclear configuration [16]. A means to retrieve MFPADs
without the burden of a ”complete” photoionization methodology - including the
challenge of a full matrix element (phase) retrieval procedure [29] \- is
therefore desirable.

To tackle the above question, we now turn our attention to Eq. 4 for the
measured $\beta_{LM}(\epsilon,t)$ and investigate the connection between
$d_{\zeta\zeta^{\prime}}$ and experimental data. As discussed in previous
articles on the rotational dynamics of asymmetric tops, see Refs. [66, 69, 70,
71] for details, inversion symmetry in the LF and the presence of a vertical
mirror plane in the MF ensures that only ADMs of even $K$ and $S$ are excited,
that the ADMs are real and that $A^{K}_{0S}(t)=A^{K}_{0-S}(t)$. Eq. 4 then
becomes,

$\beta_{L0}(\epsilon,t)=\sum_{KS=0}C^{L0}_{K0}(\epsilon)A^{K}_{00}(t)+\sum_{KS>0}\left(C^{L0}_{KS}(\epsilon)+C^{L0}_{K-S}(\epsilon)\right)A^{K}_{0S}(t).$
(16)

Here we set $M=0$ as required for cylindrical symmetry to be maintined by
photoionization using linearly polarized light. By measurement of the
$\beta_{L0}(\epsilon,t)$ for a fixed photoelectron energy, the
$\left(C^{L0}_{KS}(\epsilon)+C^{L0}_{K-S}(\epsilon)\right)$ can be determined
uniquely by linear regression if the experimental parameters are known and the
$A^{K}_{0S}(t)$ can be accurately simulated [36, 54]. Using Eq. 14, the
general relationship for $\Gamma^{\zeta^{\prime}\zeta LM}_{K0S}$, and Eq. 13
for $C^{LM}_{KS}$ we get,

$\frac{1}{2}\left(C^{L0}_{KS}+C^{L0}_{K-S}\right)=\sum_{\zeta}x_{\zeta\zeta}\Gamma^{\zeta\zeta
L0}_{K0S}+\sum_{\zeta\zeta^{\prime}<\zeta}x_{\zeta\zeta^{\prime}}\left(\Gamma^{\zeta\zeta^{\prime}L0}_{K0S}+\Gamma^{\zeta^{\prime}\zeta
L0}_{K0S}\right).$ (17)

Now defining
$\overline{C}^{L0}_{KS}=\frac{1}{2}\left(C^{L0}_{KS}+C^{L0}_{K-S}\right)$, Eq.
17 becomes:

$\overline{C}^{L0}_{KS}=\sum_{\zeta}x_{\zeta\zeta}\Gamma^{\zeta\zeta
L0}_{K0S}+\sum_{\zeta\zeta^{\prime}<\zeta}x_{\zeta\zeta^{\prime}}\left(\Gamma^{\zeta\zeta^{\prime}L0}_{K0S}+\Gamma^{\zeta^{\prime}\zeta
L0}_{K0S}\right).$ (18)

The left hand side of this equation is a measurable number, and the right hand
side is linear in the _real_ parameters $x_{\zeta\zeta}$ and
$x_{\zeta\zeta^{\prime}}=|d_{\zeta\zeta^{\prime}}|\cos\phi_{\zeta\zeta^{\prime}}$.
Thus, Eq 18 can be reduced to a simple matrix equation,

$\mathbf{\overline{C}}^{L0}_{KS}=\mathbf{\Gamma}^{\zeta\zeta^{\prime}L0}_{K0S}\mathbf{x}_{\zeta\zeta^{\prime}}.$
(19)

Solving this equation for $\mathbf{x}_{\zeta\zeta^{\prime}}$ results in the
following expression,

$\mathbf{x}_{\zeta\zeta^{\prime}}=(\mathbf{\Gamma}^{\zeta\zeta^{\prime}L0}_{K0S})^{+}\mathbf{\overline{C}}^{L0}_{K0S}$
(20)

Designated by $+$, we use the numerical Moore-Penrose inverse of
$\mathbf{\Gamma}^{\zeta\zeta^{\prime}L0}_{K0S}$, found by reduced singular
value decomposition of $\mathbf{\Gamma}^{\zeta\zeta^{\prime}L0}_{K0S}$ [72].
Eq. 19 is a single mathematical statement of the information content of the
measurement. It tells us which pairs molecular frame scattering channels
$\zeta$ and $\zeta^{\prime}$ interfere to produce a LF anisotropy parameter
$\beta_{L0}$. Eq. 20 inverts this equation to provide the smallest vector
magnitude, least-squares solution for $\mathbf{x}_{\zeta\zeta^{\prime}}$ for
an experimentally measured $\mathbf{\overline{C}}^{L0}_{K0S}$ vector [72]. We
note here that this is by no means a unique inversion, since in general the
number of interfering scattering channels will far outnumber the coefficients
$\overline{C}^{L0}_{K0S}$. However, this does not matter provided that we can
find a similar matrix equation linking $\mathbf{x}_{\zeta\zeta^{\prime}}$ to
the MF coefficients $C^{LM}_{PR}(\epsilon,\Delta q)$. We can then substitute
for $\mathbf{x}_{\zeta\zeta^{\prime}}$ using Eq. 20, eliminating
$\mathbf{x}_{\zeta\zeta^{\prime}}$ from the problem.

For this purpose, we now examine Eq. 11,

$C^{LM}_{PR}(\epsilon,\Delta
q)=\sum_{\zeta\zeta^{\prime}}d_{\zeta\zeta^{\prime}}(\epsilon)\Gamma^{\zeta\zeta^{\prime}LM}_{PR\Delta
q}$ (21)

for the desired coefficient $C^{LM}_{PR}(\epsilon,\Delta q)$ in the MF. Again,
we write the $d_{\zeta\zeta^{\prime}}$ in terms of their amplitudes and phases
as
$d_{\zeta\zeta^{\prime}}=|d_{\zeta\zeta^{\prime}}|e^{i\phi_{\zeta\zeta^{\prime}}}$.
Noting that $d_{\zeta^{\prime}\zeta}=d^{*}_{\zeta\zeta^{\prime}}$, Eq. 11 can
be written as

$C^{LM}_{PR}(\epsilon,\Delta q)=\sum_{\zeta}d_{\zeta\zeta}\Gamma^{\zeta\zeta
LM}_{PR\Delta
q}+\sum_{\zeta\zeta^{\prime}<\zeta}|d_{\zeta\zeta^{\prime}}|\left(\Gamma^{\zeta\zeta^{\prime}LM}_{PR\Delta
q}e^{i\phi_{\zeta\zeta^{\prime}}}+\Gamma^{\zeta^{\prime}\zeta LM}_{PR\Delta
q}e^{-i\phi_{\zeta\zeta^{\prime}}}\right).$ (22)

Using a general relation, derived from the behavior of
$\Gamma^{\zeta\zeta^{\prime}LM}_{PR\Delta q}$ under permutation of the indices
$\zeta\zeta^{\prime}$ (cf. Appendix):

$\Gamma^{\zeta^{\prime}\zeta LM}_{PR\Delta q}=(-1)^{\Delta
q-M}(\Gamma^{\zeta\zeta^{\prime}L-M}_{PR-\Delta q})^{*}$ (23)

Eq. 22 can be expressed as,

$C^{LM}_{PR}(\epsilon,\Delta q)=\sum_{\zeta}d_{\zeta\zeta}\Gamma^{\zeta\zeta
LM}_{PR\Delta
q}+\sum_{\zeta\zeta^{\prime}<\zeta}|d_{\zeta\zeta^{\prime}}|\left(\Gamma^{\zeta\zeta^{\prime}LM}_{PR\Delta
q}e^{i\phi_{\zeta\zeta^{\prime}}}+(-1)^{\Delta
q-M}(\Gamma^{\zeta\zeta^{\prime}L-M}_{PR-\Delta
q})^{*}e^{-i\phi_{\zeta\zeta^{\prime}}}\right).$ (24)

This is the MF equivalent of Eq. 15. We now seek a relation similar to Eq. 19,
which relates the $C^{LM}_{PR}(\epsilon,\Delta q)$ to the parameters
$x_{\zeta\zeta^{\prime}}$. To progress from Eq 15 to Eq. 19, we required that
the molecule have a mirror plane. Similarly, to progress from Eq. 22, we
restrict ourselves to inversion symmetric molecules with a horizontal plane.
We are thus restricted to molecules with $D_{nh}$ point group symmetry.

For such molecules, we find that (cf. Appendix)

$C^{LM}_{P0}(\epsilon,\Delta q)=(-1)^{M-\Delta q}C^{L-M}_{P0}(\epsilon,-\Delta
q).$ (25)

This equation holds for any inversion symmetric molecule with a horizontal
mirror plane, ionized by linearly polarized light (thus $R=0$). Using Eq. 24
and the identity given in Eq. 25, we add $\pm\Delta q$ terms, and simplify to
get:

$C^{LM}_{P0}(\epsilon,\Delta q)=\sum_{\zeta}x_{\zeta\zeta}\Gamma^{\zeta\zeta
LM}_{P0\Delta
q}+\sum_{\zeta\zeta^{\prime}<\zeta}x_{\zeta\zeta^{\prime}}(\Gamma^{\zeta\zeta^{\prime}LM}_{P0\Delta
q}+\Gamma^{\zeta^{\prime}\zeta LM}_{P0\Delta q})$ (26)

Writing Eq. 26 as a matrix equation gives,

$\mathbf{C}^{LM}_{P0}(\epsilon,\Delta
q)=\mathbf{\Gamma}^{\zeta\zeta^{\prime}LM}_{P0\Delta
q}\mathbf{x}_{\zeta\zeta^{\prime}}.$ (27)

And finally using Eq. 20 for $\mathbf{x}_{\zeta\zeta^{\prime}}$ in Eq. 27
gives,

$\mathbf{C}^{LM}_{P0}(\epsilon,\Delta q)=\mathbf{\hat{G}}^{LMP\Delta
q}_{L^{\prime}0KS}\mathbf{\overline{C}}^{L^{\prime}0}_{KS}.$ (28)

This equation is the key result of this manuscript, of the desired form (Eq.
1) connecting the experimentally accessible coefficients in the vector
$\mathbf{\overline{C}}^{L^{\prime}0}_{KS}$ to the set of coeffecients
$\mathbf{C}^{LM}_{P0}(\epsilon,\Delta q)$ which can be used to construct the
MFPAD (the entire reconstruction procedure is summarized at the beginning of
the next section with reference to Fig. 1). We denote
$\mathbf{\hat{G}}^{LMP\Delta
q}_{L^{\prime}0KS}=\mathbf{\Gamma}^{\zeta\zeta^{\prime}LM}_{P0\Delta
q}(\mathbf{\Gamma}^{\zeta\zeta^{\prime}L^{\prime}0}_{K0S})^{+}$, again using
the Moore-Penrose inverse. $\mathbf{\hat{G}}^{LMP\Delta q}_{L^{\prime}0KS}$ is
a known matrix connecting the LF to the MF for any inversion symmetric
molecule containing a horizontal and vertical mirror plane, rotationally
excited by a linearly polarized pump-pulse. The elements of
$\mathbf{\hat{G}}^{LMP\Delta q}_{L^{\prime}0KS}$ couple a particular
$L^{\prime}$ ($M^{\prime}=0$ due to cylindrical symmetry) in the LF to a $LM$
in the MF, automatically finding sets of interfering ionization channels
$\zeta$ and $\zeta^{\prime}$ which contribute to both a particular
$\beta_{L^{\prime}0}$ in the LF and a $\beta_{LM}$ in the MF. Thus, non-zero
rows in $\mathbf{\hat{G}}^{LMP\Delta q}_{L^{\prime}0KS}$, corresponding to a
set of $LMP\Delta q$, indicate retrievable sets of
$C^{LM}_{P0}(\epsilon,\Delta q)$. If $\mathbf{\hat{G}}^{LMP\Delta
q}_{L^{\prime}0KS}$ is of full rank (in other words, if the set of linear
equations Eq. 28 are consistent, or the problem is not ill-posed), then all
such $C^{LM}_{P0}(\epsilon,\Delta q)$ can be uniquely determined as can,
therefore, the corresponding $\beta_{LM}$ in the MF and thus the MFPADs. If
not, then a least squares solution of Eq. 28 can be obtained [72], and the
resulting MFPADs compared with calculated MFPADs to determine the reliability
of the solution, or compared with the constraint of the experimental
observables. In the next section, we numerically demonstrate the
reconstruction procedure using Eq. 28 for the $D_{\infty h}$ molecule $N_{2}$
and the $D_{2h}$ molecule $C_{2}H_{4}$.

##  3 Molecular Frame Reconstruction

Figure 1: A flow chart summarizing the MFPAD reconstruction method starting
from experimental data. Measured time resolved LFPADs are the starting point,
from which LF time dependent anisotropy parameters
$\beta_{L^{\prime}0}(\epsilon,t)$ can be determined by linear regression.
Following this, the LF coefficients $C^{L^{\prime}0}_{K0S}$ can be uniquely
determined also by linear regression. The crucial next step, highlighted in
red in the flow chart, connects the
$\overline{C}^{L^{\prime}0}_{KS}=1/2(C^{L^{\prime}0}_{K0S}+C^{L^{\prime}0}_{K-S})$
to the MF coefficients $C^{LM}_{P0}(\epsilon,\Delta q)$ via the matrix
$\mathbf{\hat{G}}^{LMP\Delta q}_{L^{\prime}0KS}$. These coefficients then
allow the reconstruction of the MFPADs
$\sigma(\epsilon,\theta,\chi;\theta^{\prime}_{e},\phi^{\prime}_{e})$.

With Eq. 28 in hand, we can outline an MFPAD reconstruction procedure for
molecules with $D_{nh}$ point group symmetry. We summarize this procedure
starting from the experimentally measured, time-resolved LFPADs from an
evolving rotational wavepacket excited by a linearly polarized, non-resonant
pump-pulse, in Fig. 1. The measured LFPAD from a evolving RWP is described by
Eq. 3, displayed in the top box of Fig. 1 for ionization by linearly polatized
light ($M^{\prime}=0$). This data can be used to extract the LF parameters
$\beta_{L}^{\prime}0(\epsilon,t)$ at a particular electron kinetic energy
$\epsilon$. The time resolved anisotropy parameters $\beta_{L^{\prime}0}(t)$
determined from these data can then be fit to uniquely determine the LF
coefficients $C^{L^{\prime}0}_{KS}$ using Eq. 5, as shown in the second box in
Fig. 1. This procedure has been experimentally demonstrated in previous
studies [36, 21]. In lieu of experimental data, here we directly calculate the
$C^{L^{\prime}0}_{KS}$ using Eq. 8,
$C^{L^{\prime}0}_{KS}=\sum_{\zeta\zeta^{\prime}}d_{\zeta\zeta^{\prime}}\Gamma^{\zeta\zeta^{\prime}L^{\prime}0}_{K0S}$.
To do so, we need the product of ionization dipole matrix elements
$d_{\zeta\zeta^{\prime}}=D_{\zeta}D^{*}_{\zeta^{\prime}}$, which we calculate
using the ePolyscat ionization code [73, 74] in order to provide physically
realistic values. Therefore our numerical demonstrations start with using
these calculated $C^{L^{\prime}0}_{KS}(\epsilon)$ in Eq. 28, shown in the
center red box in Fig. 1. Using these we retrieve the MF coefficients
$C^{LM}_{P0}(\epsilon,\Delta q)$. Then moving further down the flowchart in
the figure, we can use Eq. 10 and Eq. 9 shown in the bottom two boxes to
construct the MFPADs
$\sigma(\epsilon,\theta,\chi;\theta_{e}^{\prime},\phi_{e}^{\prime})$ for
ionization of the molecule by linearly polarized light.

We can also calculate the MFPADs directly from the dipole matrix elements
using Eq. 11, Eq. 23 and Eq. 10. We can then compare these directly calculated
MFPADs to those reconstructed from the LF coefficients $C^{L^{\prime}0}_{KS}$
using Eq. 28 in the central red box of Fig. 1 to test the reliability of the
method. To use Eq. 28, we also need to calculate the numerical matrix
$\mathbf{\hat{G}}^{LMP\Delta q}_{L^{\prime}0KS}$, which we can do using the
analytical equations Eq. 29 and Eq. 32 (cf. Appendix) for the LF and MF
coupling parameters $\Gamma^{\zeta\zeta^{\prime}L0}_{K0S}$ and
$\Gamma^{\zeta\zeta^{\prime}LM}_{P0\Delta q}$. This matrix in fact determines
which $C^{LM}_{P0}(\epsilon,\Delta q)$ can be determined, and in turn which
MFPADs can be determined. As discussed below, in some cases
$\mathbf{\hat{G}}^{LMP\Delta q}_{L^{\prime}0KS}$ has rows that are entirely
zero, indicating that the particular LF $L^{\prime}$ and MF $L$ do not share
any common pairs of photoionzation channels $\zeta$,$\zeta^{\prime}$. This
prevents the extraction of the corresponding $C^{LM}_{P0}(\epsilon,\Delta q)$
and therefore the corresponding $\beta_{LM}$ in the MF. This limits the MFPADs
that can be extracted using this method.

###  3.1 Results for $D_{\infty h}$ Symmetry

A molecule with $D_{\infty h}$ symmetry in the ground state constitutes a
linear rigid rotor. For such a molecule, $S=0$ and we adhere to the convention
$\chi=0$ [62]. Additionally, the MFPAD is independent of the azimuthal
orientation angle $\phi$ due to cylindrical symmetry, which we set to zero.
Here, we attempt to retrieve MFPADs for ionization of $N_{2}$ from its neutral
ground electronic state, producing photoelectrons with $\epsilon=$7.6 eV by
one-photon ionization. This choice is based on previous work [21] in which a
complete experiment was performed for the same continuum channels at this
photoelectron energy, using a RWP in the ground state.

To numerically demonstrate our methodology, we first calculated the LF
coefficients $C^{L^{\prime}0}_{K0}(\epsilon)$ from values of the ionization
dipole matrix elements - $D_{\zeta}(\epsilon)$ \- calculated using ePolyscat
code at $\epsilon=7.6$ eV [73, 74]. We provide these values for each
ionization channel in Table 1 along with quantum numbers that specify each
channel, and the corresponding values for $b^{\varGamma\mu}_{hl\lambda}$ for
each channel (cf. Eq. 7). We limited $l$ to a maximum value of $3$, resulting
in values of $L$ up to $L=6$. The same information can be used to calculate
the MF coeffients $C^{LM}_{P0}(\epsilon,\Delta q)$. The resulting MFPADs at
$\epsilon=7.6$ eV for $\theta=0^{\circ},45^{\circ}$ and $90^{\circ}$ are shown
in Fig. 3 (a),(b), and (c) respectively. (We note that adding an additional
partial wave channel with $l=5$ changes the calculated MFPADs by at most 3%.)

Symmetry | $l$ | $\lambda$ | $b^{\varGamma\mu}_{hl\lambda}$ | $q$ | $D_{\zeta}$
---|---|---|---|---|---
$\sigma_{u}$ | $1$ | $0$ | $1$ | $0$ | $1.9200+1.5840i$
$\sigma_{u}$ | $3$ | $0$ | $1$ | $0$ | $-2.4750+1.0362i$
$\pi_{u}$ | $1$ | $\mp 1$ | $0.7071$ | $\pm 1$ | $-1.1605-0.4746i$
$\pi_{u}$ | $3$ | $\mp 1$ | $0.7071$ | $\pm 1$ | $1.5866-0.2253i$
Table 1: Symmetrized matrix elements for $N_{2}$, a molecule with $D_{\infty
h}$ symmetry, for the ionizing transition from the neutral ground state
$N_{2}(X^{1}\Sigma^{+}_{g})$ to the ground ionic state
$N^{+}_{2}(X^{2}\Sigma^{+}_{g})$. For this transition, the ejected electron
can only be found in the $\sigma_{u}$ or $\pi_{u}$ state.

We then carried out the retrieval by first calculating
$\mathbf{\Gamma}^{\zeta\zeta^{\prime}LM}_{P0\Delta q}$ and
$\mathbf{\Gamma}^{\zeta\zeta^{\prime}L^{\prime}0}_{K00}$, and using them to
build $\mathbf{\hat{G}}^{L0P\Delta q}_{L^{\prime}0K0}$. Since $S=0$ for a
linear molecule,
$\mathbf{\overline{C}}^{L^{\prime}0}_{K0}=\mathbf{C}^{L^{\prime}0}_{K0}$ in
Eq. 28. Figure 2 shows the matrix $\mathbf{\hat{G}}^{LMP\Delta
q}_{L^{\prime}0K0}$ with corresponding ${LMP\Delta q}$ and
${L^{\prime}M^{\prime}K}$ ($M^{\prime}=0$ due to cylindrical symmetry as
previously discussed) values, for allowed terms up to $l=3$. From the figure,
it is seen that for $N_{2}$, rows corresponding to $M=-\Delta q=1$ in
$\mathbf{\hat{G}}^{LMP\Delta q}_{L^{\prime}0K0}$ are entirely zero.

Figure 2: Values of $\mathbf{\hat{G}}^{LMP\Delta q}_{L^{\prime}M^{\prime}KS}$
matrix elements for $N_{2}$, for the ionizing transition from the ground state
$N_{2}(X^{1}\Sigma^{+}_{g})$ to the ground ionic state
$N^{+}_{2}(X^{2}\Sigma^{+}_{g})$. See text for details.

This indicates that there are no equations linking these MF
$C^{LM}_{P0}(\epsilon,\Delta q)$ to any LF $C^{L^{\prime}0}_{K0}(\epsilon)$,
rendering them irretrievable. Removing these leaves eleven
$C^{L^{\prime}0}_{K0}(\epsilon)$ and eleven $C^{LM}_{P0}(\epsilon,\Delta q)$,
implying a consistent system of liner equations, Eq. 28, with one independent
linear equation for each unknown $C^{LM}_{P0}(\epsilon,\Delta q)$. It is
therefore possible to retrieve these eleven $C^{LM}_{P0}(\epsilon,\Delta q)$
uniquely. With these we expect to only retrieve the MFPADs at
$\theta=0^{\circ}$ and $90^{\circ}$ since $\Delta q=1$ coefficients do not
contribute at either orientation due to the following properties of the Wigner
D matrix elements: $D^{P}_{0\Delta q}(0,0,0)=\delta_{0\Delta q}$ and
$D^{P}_{01}(0,\pi/2,0)=0$. Within these constraints, Eq. 28, highlighted in
the red box in Fig. 1, was used to retrieve the $C^{LM}_{P0}(\epsilon,\Delta
q)$ staring with calculated LF coefficients $C^{L^{\prime}0}_{K0}(\epsilon)$.
Eq. 10 and Eq. 9, in the bottom two boxes in Fig. 1, were then used to
construct the retrieved MFPADs. Fig. 3 (d),(e), and (f) show the resulting
MFPADs for light polarized along the $z$-axis, at $\theta=45^{\circ}$ with
respect to the $z$ axes, and along the $x$ axis respectively (note that here
any polarisation direction in the $(x,y)$ plane is equivalent). As is evident,
we are only able to exactly reproduce the MFPADs for light polarized along the
$z$ and $x$ axes from the LF $C^{L^{\prime}0}_{K0}$, as expected. Physically,
it is interesting to note that the missing information in this case
corresponds to the phase difference (hence interference) between the
$\sigma_{u}$ and $\pi_{u}$ continua, which is restricted due to the symmetry
of the problem. The reconstruction in this case can actually be recognised as
the symmetrized result in $D_{\infty h}$, in which cylindrical symmetry about
the $z$-axis is imposed, and the result is averaged over all orientations of
the $x$-axis. As previously mentioned, this is _less_ information than
available from a complete experiment, which has been performed for this case
in [21], permitting _all_ MFPADs to be retrieved. However, no similar
retrieval has ever been demonstrated for a polyatomic molecule, to which our
methodology is easily extended, as shown by the results in the next section.
It is also worth noting once more that similar “information content” effects
and issues are found in traditional complete experiments, in which certain
terms may be irretrievable due to fundamental and/or experimentally-imposed
symmetry restrictions [29].

Figure 3: Calculated ((a),(b),(c)) and retrieved ((d),(e),(f)) MFPADs for
$N_{2}$ with light polarized (red arrows) along the $z$ axis, at
$\theta=45^{\circ}$ with respect to the $z$ axes, and along the $x$ axis
respectively. The MF is shown to the lef, with $x$, $y$ and $z$ axes
corresponding to those in the MFPAD plots.

###  3.2 Results for $D_{2h}$ Symmetry

Symmetry | $l$ | $\lambda$ | $b^{\varGamma\mu}_{hl\lambda}$ | $q$ | $D_{\zeta}$
---|---|---|---|---|---
$A_{g}$ | $0$ | $0$ | $1$ | $\mp 1$ | $\pm(1.4827-1.8389i)$
$A_{g}$ | $2$ | $0$ | $1$ | $\pm 1$ | $\pm(1.7666+1.1582i)$
$A_{g}$ | $2$ | $[-2,2]$ | $[0.7071,0.7071]$ | $\mp 1$ | $\pm(2.6519+1.3927i)$
$B_{1g}$ | $2$ | $[-2,2]$ | $[-0.7071,0.7071]$ | $\pm 1$ | $2.1671+1.7278i$
$B_{2g}$ | $2$ | $[-1,1]$ | $[-0.7071,0.7071]$ | $0$ | $-1.1444+0.0000i$
Table 2: Symmetrized matrix elements for $C_{2}H_{4}$, a molecule with
$D_{2h}$ symmetry, for ionizing transition from the ground state of
$C_{2}H_{4}$ (${}^{1}A_{g}$) to the ground state (${}^{2}B_{3u}$) of
$C_{2}H_{4}^{+}$. For this transition, the ejected electron can be found in
the $A_{g}$, $B_{1g}$, or $B_{2g}$ states.

We choose $C_{2}H_{4}$ as an example of a molecule with $D_{2h}$ symmetry and
demonstrate the retrieval of MFPADs for the $X^{1}A_{g}\rightarrow
X^{2}B_{3u}$ ionization channel at a photoelectron energy of 1 eV. For
$D_{2h}$, $S$ is even and the MFPADs are $\chi$ dependent, but still
independent of the azimuthal angle $\phi$, as only linearly polarized light is
considered, implying that $M^{\prime}=0$ in the LF. For $C_{2}H_{4}$, we find
by numerical calculation that $C^{L^{\prime}0}_{KS}$=$C^{L^{\prime}0}_{K-S}$
and as a result that $\overline{C}^{L^{\prime}0}_{KS}=C^{L^{\prime}0}_{KS}$.
Calculations were carried out using a max $l$ value of $l=2$. Table 2 shows
the possible symmetries of the ejected electron and the corresponding values
of $b^{\varGamma\mu}_{hl\lambda}$. In this table, the corresponding values of
$D_{\zeta}(\epsilon)$ were calculated using ePolyscat with $\epsilon=1$ eV.
This results in eighteen $C^{L^{\prime}0}_{KS}$ and twenty-seven
$C^{LM}_{P0}(\epsilon,\Delta q)$, and no zero rows in the
$\mathbf{\hat{G}}^{LMP\Delta q}_{L^{\prime}0KS}$. If the system of linear
equations represented by Eq. 28 were consistent, the
$C^{LM}_{P0}(\epsilon,\Delta q)$ would be uniquely retrievable. However, since
there is a larger number of $C^{LM}_{P0}(\epsilon,\Delta q)$ than
$C^{L^{\prime}0}_{KS}$, we cannot uniquely determine the
$C^{LM}_{P0}(\epsilon,\Delta q)$. Nonetheless, solving Eq. 28 by the Moore-
Penrose inverse provides a least-squares fit of the $C^{L^{\prime}0}_{KS}$
such that the retrieved vector $\mathbf{C}^{LM}_{P0}(\epsilon,\Delta q)$ has
the smallest possible norm [72]. This then still allows the construction of
retrieved MFPADs which can be checked against MFPADs calculated directly from
the ionization dipole matrix elements. In figures 4 and 5 we compare the
calculated MFPADs to the retrieved MFPADs for different polarization
geometries in the MF. Figure 4 shows the MFPADs for light polarized along the
$x$, $y$, and $z$ molecular axes. It is noted that for $\theta=0^{\circ}$, the
MFPADs do not depend on the value of $\chi$. In Figure 5, we plot the MFPADs
for light polarized in the $xy$ and $zy$ planes. Though small differences are
evident, in each case it is seen that the retrieved MFPADs closely resemble
the calculated MFPADs of the same orientation.

Figure 4: Calculated ((a),(b),(c)) and retrieved ((d),(e),(f)) MFPADs for
$C_{2}H_{4}$ for light polarized (red arrows) along the $z$, $x$, and $y$
axes, respectively. The MF is shown to the left, for the ground state
molecular geometry. The $x$, $y$ and $z$ axes correspond to those in the MFPAD
plots. Figure 5: Calculated ((a),(b)) and retrieved ((c),(d)) MFPADs for
$C_{2}H_{4}$ for light polarized (red arrows) in the $xy$ plane and $zy$
plane, respectively. The MF is shown to the left, for the ground state
molecular geometry. The $x$, $y$ and $z$ axes correspond to those in the MFPAD
plots.

The limit of $l=2$ here, restricting $L$ to 4, is intended to limit $K$ to a
potentially experimentally achievable maximum value of 6 [36, 54]. The width
of the rotational wavepacket excited by the pump pulse, quantified by its
angular momentum $K$, therefore limits the resolution of the retrieved MFPAD.
We note that adding an additional partial wave channel with $l=4$ does not
qualitatively affect the shape of the MFPADs, the most significant effect
being at the node of the $x$ polarized PAD. A potential route to improving the
retrieval would be to break symmetry in the LF by using an elliptically
polarized pump pulse. Breaking cylindrical symmetry increases the number of
$C^{L^{\prime}M^{\prime}}_{KQS}$, since $M^{\prime}$ and $Q$ are no longer
zero in the LF, while leaving the number of $C^{LM}_{P0}(\epsilon,\Delta q)$
unchanged in the MF. This could provide a better least-squares retrieval or
potentially a consistent set of equations in Eq. 28, resulting in a unique
solution for $\mathbf{C}^{LM}_{P0}(\epsilon,\Delta q)$. However, the form of
the $\Gamma^{\zeta\zeta^{\prime}L^{\prime}M^{\prime}}_{K0S}$ and therefore
$\mathbf{\hat{G}}^{LMP\Delta q}_{L^{\prime}M^{\prime}KS}$ must be re-derived
for this case, with $Q\neq 0$ and $M\neq 0$. Again, it is of note that similar
considerations and restrictions pertain, in general, to traditional complete
experiment methodologies, and the precise details are also case-specific [29].

##  4 Conclusion and Outlook

Time-resolved MFPADs are a powerful but challenging probe of valence
electronic dynamics during ultrafast molecular processes.We described here a
route to reconstructing MFPADs for a polyatomic molecule which avoids the
burden of first determining the photoionization matrix elements. We employ, as
raw material for the reconstruction, information from time resolved LF
measurements of PADs from a RWP prepared in the ground electronic state of the
molecule. Such a measurement provides the coefficients, the $\mathbf{C}^{lab}$
in Eq. 1, a matrix equation which can potentially be used to retrieve the set
of coefficients $\mathbf{C}^{mol}$, used to reconstruct the MFPADs. For
inversion symmetric molecules containing a horizontal and vertical mirror
plane ($D_{nh}$ point group symmetry) which are photoionized by linearly
polarized light, we derived the matrix equation Eq. 28 \- equivalent to Eq. 1
\- shown in the central red box in Fig. 1 by systematic application of
symmetry arguments. The coefficients $C^{L^{\prime}0}_{KS}$ represent the
measureable coefficients $C^{lab}$, and solving for the coefficients
${C}^{LM}_{P\Delta q}$ provides the MFPADs, importantly with no requirement
for the ‘complete’ determination of the complex dipole photoionization matrix
elements. We numerically demonstrate this reconstruction procedure for two
molecules $N_{2}$ and $C_{2}H_{4}$, having point group symmetry $D_{\infty h}$
and $D_{2h}$, respectively. The results for $C_{2}H_{4}$ in Figs 4 and 5 show
that this method can faithfully reconstruct the MFPADs for a polyatomic
molecule. In a future direction, we will examine the sensitivity of our MFPADs
reconstruction procedure to noise in the experimental data, an important issue
when dealing with real experimental data. To this end, we note that several
numerical methods are available to obtain a least-squares solution to Eq. 28
(red box in Fig. 1) even for an inconsistent system of equations [75, 76, 77].

We expect this method to be particularly useful for polyatomic molecules.
Recent studies have shown that a molecular ensemble needs to be nearly
perfectly aligned before a significant difference can be observed between PADs
from aligned and isotropic ensembles [47]. This, in general, is very difficult
to achieve under field free conditions, particularly for all three axes of a
polyatomic molecule. In contrast, the method presented here does not
necessarily demand a high degree of alignment, but rather requires only that
large enough values of $K$ be excited - up to $K=8$ for $N_{2}$ and $K=6$ for
$C_{2}H_{4}$ for the reconstructions presented here - corresponding to a broad
wavepacket of rotational states. The degree of alignment is then determined by
the time evolution of the relative phases of these states under field free
conditions. Since linear molecules have harmonically spaced rotational energy
levels, high degrees of alignment automatically ensue and repeat periodically.
The rotational level spacings for polyatomic molecules, on the other hand, are
not necessarily harmonic, leading to complex, aperiodic rotational dynamics
from a broad rotational wavepacket, not necessarily providing a high degree of
alignment. Nevertheless, recent experimental studies indicate that the
$\mathbf{C}^{lab}$ coefficients can be faithfully retrieved from such a
wavepacket, given sufficient signal-to-noise in the data [36, 54]. Therefore,
we conclude that the fundamental requirements for the experimental realization
of our MFPADs reconstruction method are: (i) a low initial rotational
temperature, (ii) a pump-pulse (or pulse sequence [21, 49, 40, 78, 79, 39, 80,
69]) which produces a broad rotational wavepacket, and (iii) high fidelity
data. We have also noted that a potential route to improving our
reconstruction method would be to break cylindrical symmetry in the LF through
use of an elliptically polarized pump pulse or time-separated cross-polarized
‘kick’ pulses. This would preclude the use of standard photoelectron imaging
techniques such as (2D) VMI which commonly use algorithms requiring
cylindrical symmetry. However, we note that a variety of 3D VMI approaches
have now been developed [81, 82, 83], also in development in our lab, which
circumvent this experimental symmetry restriction.

We hope that these results will benefit the study of photoionization dynamics
of ground state polyatomic molecules - particularly in regions of the
continuum containing resonances, which are particularly sensitive to molecular
orientation [12, 84, 85, 86, 40, 87]. We also anticipate that this approach
will benefit the study of excited state dynamics. In both cases MFPADs are a
valuable source of information, and the results presented here allow this
information to be readily accessed.

_Acknowledgments_
VM and PH thank Vinod Kumarappan for valuable discussions. VM and MG
acknowledge the Summer Science Institute at University of Mary Washington for
funding. AS thanks the NSERC Discovery Grants program and the NRC-uOttawa JCEP
for financial support.

##  5 Appendix

###  5.1 Derivation of Permutation identities for the Coupling Parameters

The $\Gamma^{\zeta\zeta^{\prime}LM}_{K0S}$ parameter, first seen in Eq. 5 in
section 2.1.2, is expanded below:

$\begin{split}\Gamma^{\zeta\zeta^{\prime}LM}_{K0S}=\sum_{P}\sqrt{2L+1}(-1)^{K}(-1)^{P}(2P+1)\begin{pmatrix}1&1&P\\\
0&0&0\end{pmatrix}\begin{pmatrix}P&K&L\\\ M&0&M\end{pmatrix}(-1)^{q}\\\
\begin{pmatrix}1&1&P\\\
q&-q^{\prime}&q^{\prime}-q\end{pmatrix}\begin{pmatrix}P&K&L\\\
q^{\prime}-q&-S&S+q-q^{\prime}\end{pmatrix}\sqrt{(2l+1)(2l^{\prime}+1)}\begin{pmatrix}l&l^{\prime}&L\\\
0&0&0\end{pmatrix}B^{\varGamma\mu
hl}_{\varGamma^{\prime}\mu^{\prime}h^{\prime}l^{\prime}}(L,S,q,q^{\prime}),\end{split}$
(29)

In Eq. 29, the term $B^{\varGamma\mu
hl}_{\varGamma^{\prime}\mu^{\prime}h^{\prime}l^{\prime}}(L,S,q,q^{\prime})$
contains the sum over $\lambda$ and $\lambda^{\prime}$.

$B^{\varGamma\mu
lh}_{\varGamma^{\prime}\mu^{\prime}l^{\prime}h^{\prime}}(L,S,q,q^{\prime})=\sum_{\lambda\lambda^{\prime}}(-1)^{\lambda^{\prime}}\begin{pmatrix}l&l^{\prime}&L\\\
-\lambda&\lambda^{\prime}&S+q-q^{\prime}\end{pmatrix}b^{\varGamma\mu}_{hl\lambda}b^{\varGamma^{\prime}\mu^{\prime}*}_{h^{\prime}l^{\prime}\lambda^{\prime}}.$
(30)

It is from the Wigner 3j symbols contained in Eqs. 29 and 30 that the LF
selection rules presented in Section 2.1.2 are determined. Using the behavior
of the Wigner 3j symbol in Eq. 30 under odd permutation of the columns, time
reversal, and that its second row sums to zero, the following identity can be
determined:

$B^{\varGamma^{\prime}\mu^{\prime}l^{\prime}h^{\prime}}_{\varGamma\mu
lh}(L,S,q^{\prime},q)=(-1)^{-S-q^{\prime}+q}B^{\varGamma\mu
lh*}_{\varGamma^{\prime}\mu^{\prime}l^{\prime}h^{\prime}}(L,-S,q,q^{\prime})$
(31)

This and the permutation identities for the 3j symbols lead to the general
relation given in Eq. 14 in section 2.3.

The parameter $\Gamma^{\zeta\zeta^{\prime}LM}_{P\Delta q}$ introduced in Eq. 5
can be written out as,

$\begin{split}\Gamma^{\zeta\zeta^{\prime}LM}_{P\Delta
q}=\sqrt{2L+1}(2P+1)(-1)^{q^{\prime}}\begin{pmatrix}1&1&P\\\
q&-q^{\prime}&\Delta q\end{pmatrix}\begin{pmatrix}1&1&P\\\
0&0&0\end{pmatrix}\\\
\sqrt{(2l+1)(2l^{\prime}+1)}\begin{pmatrix}l&l^{\prime}&L\\\
0&0&0\end{pmatrix}B^{\varGamma\mu
lh}_{\varGamma^{\prime}\mu^{\prime}l^{\prime}h^{\prime}}(L,M),\end{split}$
(32)

In this case, the term $B^{\varGamma\mu
lh}_{\varGamma^{\prime}\mu^{\prime}l^{\prime}h^{\prime}}(L,M)$ contains the
sum over $\lambda$ and $\lambda^{\prime}$:

$B^{\varGamma\mu
lh}_{\varGamma^{\prime}\mu^{\prime}l^{\prime}h^{\prime}}(L,M)=\sum_{\lambda\lambda^{\prime}}(-1)^{\lambda}\begin{pmatrix}l&l^{\prime}&L\\\
\lambda&-\lambda^{\prime}&M\end{pmatrix}b^{\varGamma\mu}_{hl\lambda}b^{\varGamma^{\prime}\mu^{\prime}*}_{h^{\prime}l^{\prime}\lambda^{\prime}}.$
(33)

The MF selection rules discussed is Section 2.2 originate from the Wigner 3j
symbols given in Eqs. 32 and 33. Like with Eq. 30, we use Eq. 33, the behavior
of the Wigner 3j symbol under odd permutation of the columns, time reversal,
and that its second row sums to zero to determine:

$B^{\varGamma^{\prime}\mu^{\prime}l^{\prime}h^{\prime}}_{\varGamma\mu
lh}(L,M)=(-1)^{M}B^{\varGamma\mu
lh*}_{\varGamma^{\prime}\mu^{\prime}l^{\prime}h^{\prime}}(L,-M)$ (34)

Again, this and and the permutation identities for the 3j symbols lead to the
general relation given in Eq. 23.

###  5.2 Molecular Frame Symmetry Arguments

In the MF, for all inversion symmetric molecules with a horizontal mirror
plane,

$\sigma(\epsilon,\theta^{\prime}_{e},\phi^{\prime}_{e};\phi,\theta,\chi)=\sigma(\epsilon,\pi-\theta^{\prime}_{e},\pi-\phi^{\prime}_{e};\phi,\theta,\chi)$
(35)

This symmetry, with the relation that
$Y_{LM}(\pi-\theta^{\prime}_{e},\pi-\phi^{\prime}_{e})=(-1)^{L-M}Y_{L-M}(\theta^{\prime}_{e},\phi^{\prime}_{e})$,
leads to the following relation between the $\beta_{LM}$’s,

$\beta_{LM}(\epsilon,\phi,\theta,\chi)=(-1)^{L-M}\beta_{L-M}(\epsilon,\phi,\theta,\chi).$
(36)

Substituting this, and the relation and the relation
$Y_{L-M}=(-1)^{M}Y_{LM}^{*}$ in Eq. 23, leads to the further determination
that the $\beta_{LM}(\epsilon,\phi,\theta,\chi)$ must be real since
$\sigma(\epsilon,\theta^{\prime}_{e},\phi^{\prime}_{e};\phi,\theta,\chi)$ is
necessarily real.

Further, for molecules with inversion symmetry, inversion is equivalent to the
rotation $\phi\rightarrow\phi+\pi$, $\theta\rightarrow\pi-\theta$ and
$\chi\rightarrow\pi-\chi$ of an arbitrarily oriented molecule, followed by
reflection of the through the molecular plane. The
$\beta_{LM}(\epsilon,\phi,\theta,\chi)$ must remain invariant under this
rotation. The Wigner rotation matrix elements in Eq. 10 behave as
$D^{P}_{R\Delta q}(\phi+\pi,\pi-\theta,\pi-\chi)=(-1)^{P-\Delta
q}D^{P}_{R-\Delta q}(\phi,\theta,\chi)$, leading to the following equation for
the $C^{LM}_{PR}(\epsilon,\Delta q)$

$C^{LM}_{PR}(\epsilon,\Delta q)=(-1)^{P+M-\Delta
q}C^{L-M}_{PR}(\epsilon,-\Delta q),$ (37)

after taking Eq. 36 into account. For linearly polarized ionizing light, $P$ =
0 or 2, and $R=0$ lead to the identity

$C^{LM}_{P0}(\epsilon,\Delta q)=(-1)^{M-\Delta q}C^{L-M}_{P0}(\epsilon,-\Delta
q).$ (38)

As given in Eq. 25.

_Data Availability Statement_
The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * [1] Howard D Cohen and U Fano.  Interference in the photo-ionization of molecules.  Physical Review, 150(1):30, 1966.
  * [2] Dan Dill.  Fixed-molecule photoelectron angular distributions.  The Journal of Chemical Physics, 65(3):1130–1133, 1976.
  * [3] J Cooper and RN Zare.  Atomic collision processes.  In Lectures in Theoretical Physics, volume 11, page 317. Gordon & Breach New York, 1969.
  * [4] Katharine L Reid.  Photoelectron angular distributions.  Annual review of physical chemistry, 54(1):397–424, 2003.
  * [5] Katharine L. Reid.  Photoelectron angular distributions: Developments in applications to isolated molecular systems.  Molecular Physics, 110(3):131–147, 2012.
  * [6] Joshua Brown Williams, CS Trevisan, MS Schöffler, T Jahnke, I Bocharova, H Kim, B Ulrich, R Wallauer, F Sturm, TN Rescigno, et al.  Imaging polyatomic molecules in three dimensions using molecular frame photoelectron angular distributions.  Physical review letters, 108(23):233002, 2012.
  * [7] Rajesh Kumar Kushawaha, Minna Patanen, Renaud Guillemin, Loic Journel, Catalin Miron, Marc Simon, Maria Novella Piancastelli, C Skates, and Piero Decleva.  From double-slit interference to structural information in simple hydrocarbons.  Proceedings of the National Academy of Sciences, 110(38):15201–15206, 2013.
  * [8] Tamar Seideman.  Time-resolved photoelectron angular distributions as a probe of coupled polyatomic dynamics.  Physical Review A, 64(4):042504, 2001.
  * [9] Daniel M Neumark.  Time-resolved photoelectron spectroscopy of molecules and clusters.  Annual review of physical chemistry, 52(1):255–277, 2001.
  * [10] Albert Stolow, Arthur E Bragg, and Daniel M Neumark.  Femtosecond time-resolved photoelectron spectroscopy.  Chemical reviews, 104(4):1719–1758, 2004.
  * [11] Albert Stolow and Jonathan G Underwood.  Time-resolved photoelectron spectroscopy of nonadiabatic dynamics in polyatomic molecules.  Advances in chemical physics, 139:497–584, 2008.
  * [12] E Shigemasa, J Adachi, M Oura, and A Yagishita.  Angular distributions of 1 s $\sigma$ photoelectrons from fixed-in-space n 2 molecules.  Physical review letters, 74(3):359, 1995.
  * [13] Akira Yagishita.  Photoelectron angular distributions from single oriented molecules: Past, present and future.  Journal of Electron Spectroscopy and Related Phenomena, 200:247–256, 2015.
  * [14] Oliver Geßner, Y Hikosaka, Björn Zimmermann, Andreas Hempelmann, RR Lucchese, JHD Eland, P-M Guyon, and Uwe Becker.  4 $\sigma$\- 1 inner valence photoionization dynamics of no derived from photoelectron-photoion angular correlations.  Physical review letters, 88(19):193002, 2002.
  * [15] RR Lucchese, A Lafosse, JC Brenot, PM Guyon, JC Houver, M Lebech, G Raseev, and D Dowek.  Polar and azimuthal dependence of the molecular frame photoelectron angular distributions of spatially oriented linear molecules.  Physical Review A, 65(2):020702, 2002.
  * [16] Paul Hockett, Michael Staniforth, Katharine L Reid, and Dave Townsend.  Rotationally resolved photoelectron angular distributions from a nonlinear polyatomic molecule.  Physical review letters, 102(25):253002, 2009.
  * [17] Katharine L Reid, David J Leahy, and Richard N Zare.  Complete description of molecular photoionization from circular dichroism of rotationally resolved photoelectron angular distributions.  Physical review letters, 68(24):3527, 1992.
  * [18] David J Leahy, Katharine L Reid, and Richard N Zare.  Complete description of two-photon (1+ 1’) ionization of no deduced from rotationally resolved photoelectron angular distributions.  The Journal of chemical physics, 95(3):1757–1767, 1991.
  * [19] M Lebech, JC Houver, A Lafosse, D Dowek, C Alcaraz, L Nahon, and Robert R Lucchese.  Complete description of linear molecule photoionization achieved by vector correlations using the light of a single circular polarization.  The Journal of chemical physics, 118(21):9653–9663, 2003.
  * [20] Ying Tang, Yoshi-Ichi Suzuki, Takuya Horio, and Toshinori Suzuki.  Molecular frame image restoration and partial wave analysis of photoionization dynamics of no by time-energy mapping of photoelectron angular distribution.  Physical review letters, 104(7):073002, 2010.
  * [21] Claude Marceau, Varun Makhija, Dominique Platzer, A Yu Naumov, PB Corkum, Albert Stolow, DM Villeneuve, and Paul Hockett.  Molecular frame reconstruction using time-domain photoionization interferometry.  Physical Review Letters, 119(8):083401, 2017.
  * [22] TN Rescigno, Nicolas Douguet, and AE Orel.  Imaging molecular isomerization using molecular-frame photoelectron angular distributions.  Journal of Physics B: Atomic, Molecular and Optical Physics, 45(19):194001, 2012.
  * [23] N Douguet, TN Rescigno, and AE Orel.  Carbon-k-shell molecular-frame photoelectron angular distributions in the photoisomerization of neutral ethylene.  Physical Review A, 88(1):013412, 2013.
  * [24] A Menssen, CS Trevisan, MS Schöffler, T Jahnke, I Bocharova, F Sturm, N Gehrken, B Gaire, H Gassert, S Zeller, et al.  Molecular frame photoelectron angular distributions for core ionization of ethane, carbon tetrafluoride and 1, 1-difluoroethylene.  Journal of Physics B: Atomic, Molecular and Optical Physics, 49(5):055203, 2016.
  * [25] Christer Z Bisgaard, Owen J Clarkin, Guorong Wu, Anthony MD Lee, Oliver Geßner, Carl C Hayden, and Albert Stolow.  Time-resolved molecular frame dynamics of fixed-in-space cs2 molecules.  Science, 323(5920):1464–1468, 2009.
  * [26] Guorong Wu, Paul Hockett, and Albert Stolow.  Time-resolved photoelectron spectroscopy: from wavepackets to observables.  Physical Chemistry Chemical Physics, 13(41):18447–18467, 2011.
  * [27] Varun Makhija, Kevin Veyrinas, Andrey E Boguslavskiy, Ruaridh Forbes, Iain Wilkinson, Rune Lausten, Simon P Neville, Stephen T Pratt, Michael S Schuurman, and Albert Stolow.  Ultrafast molecular frame electronic coherences from lab frame scattering anisotropies.  Journal of Physics B: Atomic, Molecular and Optical Physics, 53(11):114001, 2020.
  * [28] Paul Hockett.  Quantum Metrology with Photoelectrons, Volume 1: Foundations.  IOP Publishing, 2018.
  * [29] Paul Hockett.  Quantum Metrology with Photoelectrons, Volume 2: Applications and Advances.  IOP Publishing, 2018.
  * [30] AV Golovin, F Heiser, CJK Quayle, P Morin, M Simon, O Gessner, P-M Guyon, and U Becker.  Observation of site-specific electron emission in the decay of superexcited o 2.  Physical review letters, 79(23):4554, 1997.
  * [31] K Ito, J-I Adachi, Y Hikosaka, S Motoki, K Soejima, A Yagishita, G Raseev, and NA Cherepkov.  Photoelectron angular distributions from o k shell of oriented co molecules: A critical comparison between theory and experiment.  Physical review letters, 85(1):46, 2000.
  * [32] F Heiser, O Gessner, J Viefhaus, K Wieliczek, R Hentges, and U Becker.  Demonstration of strong forward-backward asymmetry in the c 1 s photoelectron angular distribution from oriented co molecules.  Physical review letters, 79(13):2435, 1997.
  * [33] M Takahashi, JP Cave, and JHD Eland.  Velocity imaging photoionization coincidence apparatus for the study of angular correlations between electrons and fragment ions.  Review of Scientific Instruments, 71(3):1337–1344, 2000.
  * [34] Peter Downie and Ivan Powis.  Molecule-frame photoelectron angular distributions from oriented cf 3 i molecules.  Physical review letters, 82(14):2864, 1999.
  * [35] Oliver Gessner, AMD Lee, James P Shaffer, Hanna Reisler, Sergey V Levchenko, Anna I Krylov, Jonathan G Underwood, H Shi, Allan LL East, DM Wardlaw, et al.  Femtosecond multidimensional imaging of a molecular dissociation.  Science, 311(5758):219–222, 2006.
  * [36] Huynh Van Sa Lam, Suresh Yarlagadda, Anbu Venkatachalam, Tomthin Nganba Wangjam, Rajesh K Kushawaha, Chuan Cheng, Peter Svihra, Andrei Nomerotski, Thomas Weinacht, Daniel Rolles, et al.  Angle-dependent strong-field ionization and fragmentation of carbon dioxide measured using rotational wave packets.  Physical Review A, 102(4):043119, 2020.
  * [37] Christiane P Koch, Mikhail Lemeshko, and Dominique Sugny.  Quantum control of molecular rotation.  Reviews of Modern Physics, 91(3):035005, 2019.
  * [38] Henrik Stapelfeldt and Tamar Seideman.  Colloquium: Aligning molecules with strong laser pulses.  Reviews of Modern Physics, 75(2):543, 2003.
  * [39] James P Cryan, Philip H Bucksbaum, and Ryan N Coffee.  Field-free alignment in repetitively kicked nitrogen gas.  Physical Review A, 80(6):063412, 2009.
  * [40] Xiaoming Ren, Varun Makhija, Anh-Thu Le, Jan Troß, Sudipta Mondal, Cheng Jin, Vinod Kumarappan, and Carlos Trallero-Herrero.  Measuring the angle-dependent photoionization cross section of nitrogen using high-harmonic generation.  Physical Review A, 88(4):043421, 2013.
  * [41] Sharly Fleischer, Yan Zhou, Robert W Field, and Keith A Nelson.  Molecular orientation and alignment by intense single-cycle thz pulses.  Physical Review Letters, 107(16):163603, 2011.
  * [42] KN Egodapitiya, Sha Li, and RR Jones.  Terahertz-induced field-free orientation of rotationally excited molecules.  Physical Review Letters, 112(10):103002, 2014.
  * [43] S De, Irina Znakovskaya, D Ray, F Anis, Nora G Johnson, IA Bocharova, M Magrakvelidze, BD Esry, CL Cocke, IV Litvinyuk, et al.  Field-free orientation of co molecules by femtosecond two-color laser fields.  Physical Review Letters, 103(15):153002, 2009.
  * [44] Ran Damari, Shimshon Kallush, and Sharly Fleischer.  Rotational control of asymmetric molecules: dipole-versus polarizability-driven rotational dynamics.  Physical review letters, 117(10):103001, 2016.
  * [45] Keita Oda, Masafumi Hita, Shinichirou Minemoto, and Hirofumi Sakai.  All-optical molecular orientation.  Physical review letters, 104(21):213901, 2010.
  * [46] Kenta Kitano, Nobuhisa Ishii, Natsuki Kanda, Yoshiyuki Matsumoto, Teruto Kanai, Makoto Kuwata-Gonokami, and Jiro Itatani.  Orientation of jet-cooled polar molecules with an intense single-cycle thz pulse.  Physical Review A, 88(6):061405, 2013.
  * [47] Katharine L Reid.  Accessing the molecular frame through strong-field alignment of distributions of gas phase molecules.  Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376(2115):20170158, 2018.
  * [48] Kang Lin, Ilia Tutunnikov, Junjie Qiang, Junyang Ma, Qiying Song, Qinying Ji, Wenbin Zhang, Hanxiao Li, Fenghao Sun, Xiaochun Gong, et al.  All-optical field-free three-dimensional orientation of asymmetric-top molecules.  Nature communications, 9(1):1–9, 2018.
  * [49] Xiaoming Ren, Varun Makhija, and Vinod Kumarappan.  Multipulse three-dimensional alignment of asymmetric top molecules.  Physical Review Letters, 112(17):173602, 2014.
  * [50] Kevin F Lee, DM Villeneuve, PB Corkum, Albert Stolow, and Jonathan G Underwood.  Field-free three-dimensional alignment of polyatomic molecules.  Physical review letters, 97(17):173001, 2006.
  * [51] Jonathan G Underwood, Benjamin J Sussman, and Albert Stolow.  Field-free three dimensional molecular axis alignment.  Physical review letters, 94(14):143002, 2005.
  * [52] Péter Sándor, Adonay Sissay, François Mauger, Mark W Gordon, TT Gorman, TD Scarborough, Mette B Gaarde, Kenneth Lopata, KJ Schafer, and RR Jones.  Angle-dependent strong-field ionization of halomethanes.  The Journal of chemical physics, 151(19):194308, 2019.
  * [53] Péter Sándor, Adonay Sissay, François Mauger, Paul M Abanador, Timothy T Gorman, Timothy D Scarborough, Mette B Gaarde, Kenneth Lopata, Kenneth J Schafer, and Robert R Jones.  Angle dependence of strong-field single and double ionization of carbonyl sulfide.  Physical Review A, 98(4):043425, 2018.
  * [54] Varun Makhija, Xiaoming Ren, Drue Gockel, Anh-Thu Le, and Vinod Kumarappan.  Orientation resolution through rotational coherence spectroscopy.  arXiv preprint arXiv:1611.06476, 2016.
  * [55] Jochen Mikosch, Andrey E Boguslavskiy, Iain Wilkinson, Michael Spanner, Serguei Patchkovskii, and Albert Stolow.  Channel-and angle-resolved above threshold ionization in the molecular frame.  Physical Review Letters, 110(2):023004, 2013.
  * [56] Jochen Mikosch, Christer Z Bisgaard, Andrey E Boguslavskiy, Iain Wilkinson, and Albert Stolow.  The quantitative determination of laser-induced molecular axis alignment.  The Journal of chemical physics, 139(2):024304, 2013.
  * [57] S Ramakrishna and Tamar Seideman.  On the information content of time-and angle-resolved photoelectron spectroscopy.  Journal of Physics B: Atomic, Molecular and Optical Physics, 45(19):194012, 2012.
  * [58] Xu Wang, Anh-Thu Le, Zhaoyan Zhou, Hui Wei, and CD Lin.  Theory of retrieving orientation-resolved molecular information using time-domain rotational coherence spectroscopy.  Physical Review A, 96(2):023424, 2017.
  * [59] Bincheng Wang, Yanqing He, Xi Zhao, Lixin He, Pengfei Lan, Peixiang Lu, and CD Lin.  Retrieval of full angular-and energy-dependent complex transition dipoles in the molecular frame from laser-induced high-order harmonic signals with aligned molecules.  Physical Review A, 101(6):063417, 2020.
  * [60] Jonathan G Underwood and Katharine L Reid.  Time-resolved photoelectron angular distributions as a probe of intramolecular dynamics: Connecting the molecular frame and the laboratory frame.  The Journal of Chemical Physics, 113(3):1067–1074, 2000.
  * [61] Herbert Goldstein, Charles Poole, and John Safko.  Classical mechanics.  Pearson, 2001.
  * [62] Richard N Zare.  Angular momentum, volume 33.  Wiley, New York, 1988.
  * [63] Jun John Sakurai and Jim J Napolitano.  Modern quantum mechanics, revised edition.  Pearson, 2010.
  * [64] John David Jackson.  Classical electrodynamics.  Wiley, New York, 1998.
  * [65] U. Fano and Dan Dill.  Angular Momentum Transfer in the Theory of Angular Distributions.  Physical Review A, 6(1):185–192, July 1972.
  * [66] Varun Makhija, Xiaoming Ren, and Vinod Kumarappan.  Metric for three-dimensional alignment of molecules.  Physical Review A, 85(3):033425, 2012.
  * [67] Garry Taylor.  The phase problem.  Acta Crystallographica Section D: Biological Crystallography, 59(11):1881–1890, 2003.
  * [68] Rick Trebino.  Frequency-resolved optical gating: the measurement of ultrashort laser pulses.  Springer Science & Business Media, 2012.
  * [69] Stefan Pabst, Phay J Ho, and Robin Santra.  Computational studies of x-ray scattering from three-dimensionally-aligned asymmetric-top molecules.  Physical Review A, 81(4):043425, 2010.
  * [70] A Rouzée, S Guérin, O Faucher, and B Lavorel.  Field-free molecular alignment of asymmetric top molecules using elliptically polarized laser pulses.  Physical Review A, 77(4):043412, 2008.
  * [71] Tamar Seideman.  Revival structure of aligned rotational wave packets.  Physical Review Letters, 83(24):4971, 1999.
  * [72] M Planitz.  Inconsistent systems of linear equations.  The Mathematical Gazette, 63(425):181–185, 1979.
  * [73] Franco A Gianturco, Robert R Lucchese, and Nico Sanna.  Calculation of low-energy elastic cross sections for electron-cf4 scattering.  The Journal of Chemical Physics, 100(9):6464–6471, 1994.
  * [74] Alexandra PP Natalense and Robert R Lucchese.  Cross section and asymmetry parameter calculation for sulfur 1s photoionization of sf 6.  The Journal of chemical physics, 111(12):5344–5348, 1999.
  * [75] Stephen Boyd, Neal Parikh, and Eric Chu.  Distributed optimization and statistical learning via the alternating direction method of multipliers.  Now Publishers Inc, 2011.
  * [76] S Li, F Cropp, K Kabra, TJ Lane, G Wetzstein, P Musumeci, and D Ratner.  Electron ghost imaging.  Physical review letters, 121(11):114801, 2018.
  * [77] Taran Driver, Siqi Li, Elio G Champenois, Joseph Duris, Daniel Ratner, Thomas J Lane, Philipp Rosenberger, Andre Al-Haddad, Vitali Averbukh, Toby Barnard, et al.  Attosecond transient absorption spooktroscopy: a ghost imaging approach to ultrafast absorption spectroscopy.  Physical Chemistry Chemical Physics, 22(5):2704–2712, 2020.
  * [78] Monika Leibscher, I Sh Averbukh, and Herschel Rabitz.  Enhanced molecular alignment by short laser pulses.  Physical Review A, 69(1):013402, 2004.
  * [79] Christer Z Bisgaard, Mikael D Poulsen, Emmanuel Péronne, Simon S Viftrup, and Henrik Stapelfeldt.  Observation of enhanced field-free molecular alignment by two laser pulses.  Physical review letters, 92(17):173004, 2004.
  * [80] James P Cryan, James M Glownia, Jakob Andreasson, Ali Belkacem, Nora Berrah, Cosmin I Blaga, Christoph Bostedt, John Bozek, Christian Buth, Louis F DiMauro, et al.  Auger electron angular distribution of double core-hole states in the molecular reference frame.  Physical review letters, 105(8):083004, 2010.
  * [81] M. Wollenhaupt, M. Krug, J. Köhler, T. Bayer, C. Sarpe-Tudoran, and T. Baumert.  Three-dimensional tomographic reconstruction of ultrashort free electron wave packets.  Applied Physics B, 95(4):647–651, April 2009.
  * [82] Suk Kyoung Lee, Yun Fei Lin, Steven Lingenfelter, Lin Fan, Alexander H Winney, and Wen Li.  Communication: Time-and space-sliced velocity map electron imaging, 2014\.
  * [83] Duke A Debrah, Gabriel A Stewart, Gihan Basnayake, Andrei Nomerotski, Peter Svihra, Suk Kyoung Lee, and Wen Li.  Developing a camera-based 3d momentum imaging system capable of 1 mhits/s.  Review of Scientific Instruments, 91(2):023316, 2020.
  * [84] Susan M Bellm, Julia A Davies, Paul T Whiteside, Jingwei Guo, Ivan Powis, and Katharine L Reid.  An unusual $\pi$* shape resonance in the near-threshold photoionization of s 1 para-difluorobenzene.  The Journal of chemical physics, 122(22):224306, 2005.
  * [85] Julien B Bertrand, Hans Jakob Wörner, Paul Hockett, David M Villeneuve, and Paul B Corkum.  Revealing the cooper minimum of n 2 by molecular frame high-harmonic spectroscopy.  Physical review letters, 109(14):143001, 2012.
  * [86] Michael Staniforth, Steven Daly, Katharine L Reid, and Ivan Powis.  A generic $\pi$* shape resonance observed in energy-dependent photoelectron angular distributions from two-colour, resonant multiphoton ionization of difluorobenzene isomers.  The Journal of chemical physics, 139(6):064304, 2013.
  * [87] Andrei Kamalov, Anna L Wang, Philip H Bucksbaum, Daniel J Haxton, and James P Cryan.  Electron correlation effects in attosecond photoionization of co 2.  Physical Review A, 102(2):023118, 2020.