# A Laboratory Frame Density Matrix for Ultrafast Quantum Molecular Dynamics

Margaret Gregory Current Address: Department of Earth, Atmospheric, and
Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts
Avenue, Cambridge, MA 02139, USA. Department of Chemistry and Physics,
University of Mary Washington, 1301 College Avenue, Fredericksburg VA, 22401.
Simon Neville  National Research Council Canada, 100 Sussex Drive, Ottawa, ON,
K1A 0R6, Canada  Michael Schuurman  National Research Council Canada, 100
Sussex Drive, Ottawa, ON, K1A 0R6, Canada  Department of Chemistry and
Biomolecular Sciences, University of Ottawa, 150 Louis Pasteur, Ottawa, ON,
K1N 6N5, Canada  Varun Makhija<EMAIL_ADDRESS>Department of Chemistry and
Physics, University of Mary Washington, 1301 College Avenue, Fredericksburg
VA, 22401.

(August 28, 2024)

###### Abstract

In most cases the ultrafast dynamics of resonantly excited molecules are
considered, and almost always computed in the molecular frame, while
experiments are carried out in the laboratory frame. Here we provide a
formalism in terms of a lab frame density matrix which connects quantum
dynamics in the molecular frame to those in the laboratory frame, providing a
transparent link between computation and measurement. The formalism reveals
that in any such experiment, the molecular frame dynamics vary for molecules
in different orientations and that certain coherences which are potentially
experimentally accessible are rejected by the orientation-averaged reduced
vibronic density matrix. Instead, Molecular Angular Distribution Moments
(MADMs) are introduced as a more accurate representation of experimentally
accessible information. Furthermore, the formalism provides a clear definition
of a molecular frame quantum tomography, and specifies the requirements to
perform such a measurement enabling the experimental imaging of molecular
frame vibronic dynamics. Successful completion of such a measurement fully
characterizes the molecular frame quantum dynamics for a molecule at any
orientation in the laboratory frame.

##  1 Introduction

Discerning the dynamics of resonantly excited isolated molecules is a
challenging many body quantum problem, which has spurred the development of
numerous computational and experimental methods in chemistry and physics [1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 17, 19, 20, 21,
22, 23, 24]. The combination of these powerful experimental and theoretical
techniques can provide reduced dimensionality models that explain
spectroscopic observations. Typically, studies of femtosecond and attosecond
dynamics focus on vibrational and/or electronic dynamics and ignore the
rotational degrees of freedom since rotation occurs on much longer timescales
[25, 26, 27, 28, 29, 30]. Vibrational and electronic motion are also of more
immediate interest, since their coupling drives the transfer of energy in
molecular systems [1, 7, 8]. In particular, the development of attosecond
laser pulses provides the opportunity to directly measure electronic
coherences prior to dephasing [31, 32, 33, 34, 25, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45], generating much recent interest in directly imaging charge
transfer dynamics [26, 28, 29, 30]. However, electronic and vibrational
coordinates are defined in the molecular frame, and experiments always occur
in the laboratory frame. We may greatly enhance the probability of a
particular molecular orientation for some molecules using laser-alignment
techniques [46, 47], but quantum mechanics precludes the possibility of
producing a fixed orientation [48, 49]. In other words, considering the
vibronic (vibrational-electronic) dynamics in a fixed molecule is an
approximation. We may then ask what, if any, are the measurable consequences
of this approximation? In particular, does this approximation impact the
efficacy of theoretical proposals for measuring coherent molecular dynamics?
Quantum mechanically, ignoring rotation amounts to neglecting the manifold of
very closely spaced total angular momentum eigenstates that lie on each
vibronic state. The bandwidth of ultrashort laser pulses invariably exceeds
this energy gap, exciting a large superposition of angular momentum states.
Here, we develop a density matrix-based framework for vibronic dynamics that
accounts for this superposition, thus including the rotational degrees of
freedom. This framework can be considered a straight forward extension of that
developed by Fano to study the alignment and orientation of electrons in
excited atoms, which has also been applied for this purpose to diatomic
molecules [50, 51].

Even though rotational dynamics do not occur on femto and attosecond
timescales, we find that molecules at different orientations in the lab frame
will exhibit different vibronic dynamics. Additionally, certain coherences
that may appear in a fixed molecule calculation can be explicitly rejected by
the reduced vibronic density matrix (with rotations traced out or,
equivalently, orientation-averaged). This may be particularly relevant for
experiments that target specific density matrix elements, which is sometimes
possible by two-dimensional spectroscopy [6]. Further, as we discuss in
detail, these missing coherences can potentially be probed experimentally
making the orientation averaged vibronic density matrix an inaccurate
representation of such experiments. Finally, since molecules at different
orientations undergo different dynamics, we show that it is potentially
possible to extract density matrix elements from an experiment in which
vibronic dynamics are initiated from initially _un-oriented_ molecules. In
particular cases it may be possible to extract all relevant density matrix
elements experimentally, thus completely determining the molecular frame
vibronic dynamics for a molecule at any orientation in the lab frame. This
therefore constitutes a molecular frame quantum tomography.

We derive and describe the density matrix framework in section 2 below. A core
component of the framework are the Angular Momentum Coherence Operators
(AMCOs) introduced in section 2.2. These are an extension of the state
multipoles introduced by Fano [50, 51], however we do not endeavour to explore
the tensor nature of these operators in detail. The AMCOs then facilitate the
decomposition of a lab frame density matrix derived in section 2.3 into time-
depdendent Molecular Angular Distribution Moments (MADMs), which encode the
orientation dependent vibronic dynamics of the molecule in the lab frame. The
MADMs are also the quantities that provide a direct connection between the
density matrix and measurement. In section 4 this connection is explored in
detail for time resolved photoelectron angular distributions from a resonantly
excited molecule. Making a sufficient number of measurements such that all
relevant MADMs can be determined leads to a molecular frame quantum
tomography. Finally, we conclude by providing a summary and discuss some
potential applications of our work.

##  2 Theoretical Framework

###  2.1 Preliminaries

We first consider vibrational and electronic dynamics in the molecular frame.
These are typically described in a Born-Oppenheimer (BO) basis of electronic
and vibrational states, which we label $\ket{\alpha}$ and $\ket{v_{\alpha}}$,
respectively. A broadband light pulse can excite a superposition of these
states. Typically, it is assumed that the excitation occurs out of a single
state, allowing the resulting wavepacket to be written as a pure state,

$\ket{\Psi_{mol}(t)}=\sum_{\alpha v_{\alpha}}C_{\alpha
v_{\alpha}}(t)\ket{\alpha}\ket{v_{\alpha}}.$ (1)

Ignoring rotation implicitly restricts us to the molecular frame, hence the
state label $\ket{\Psi_{mol}(t)}$. The resulting molecular wavefunction can be
found by projecting onto the electronic positions and vibrational modes basis,
which we label $\ket{\textbf{r}}$ and $\ket{\textbf{Q}}$, respectively. Here
$\textbf{r}=\\{r_{i}|i=1,2,\dots,\\}$ represents the set of electronic
position vectors in the molecular frame, and
$\textbf{Q}=\\{q_{1},q_{2},q_{3},\dots\\}$ the set of vibrational coordinates.
Thus, the molecular probability distribution can be expressed as follows,

$|\Psi_{mol}(t,\textbf{r},\textbf{Q})|^{2}=|\bra{\textbf{r},\textbf{Q}}\ket{\Psi_{mol}(t)}|^{2}=\sum_{\alpha\alpha^{\prime}v_{\alpha}v^{\prime}_{\alpha^{\prime}}}\Phi_{\alpha}(\textbf{r})\Phi^{*}_{\alpha^{\prime}}(\textbf{r})\psi_{v_{\alpha}}(\textbf{Q})\psi^{*}_{v^{\prime}_{\alpha^{\prime}}}(\textbf{Q})C_{\alpha,v_{\alpha}}(t)C^{*}_{\alpha^{\prime},v^{\prime}_{\alpha^{\prime}}}(t)$
(2)

where $\Phi_{\alpha}(\textbf{r})=\bra{\textbf{r}}\ket{\alpha}$ is the
electronic wavefunction for electronic state $\ket{\alpha}$, with
$\absolutevalue{\Phi_{\alpha}(\textbf{r})}^{2}$ the probability density of
finding an electron located at r in the state $\ket{\alpha}$. Similarly,
$\psi_{v_{\alpha}}(\textbf{Q})=\bra{\textbf{Q}}\ket{v_{\alpha}}$ is the
vibrational wavefunction for vibrational state $\ket{v_{\alpha}}$, with
$\absolutevalue{\psi_{v_{\alpha}}(\textbf{Q})}^{2}$ the probability density of
locating the nuclei at $Q$ in the state $\ket{v_{\alpha}}$. Rewriting Eq. 2 in
terms of the density matrix and separating it into diagonal and off-diagonal
terms:

$\begin{split}|\Psi_{mol}(t,\textbf{r},\textbf{Q})|^{2}=\sum_{\alpha,v_{\alpha}}\rho(\alpha,v_{\alpha},t)P_{\alpha,v_{\alpha}}(\textbf{r},\textbf{Q})+\\\
\sum_{\alpha\neq\alpha^{\prime}}\sum_{v_{\alpha}\neq
v^{\prime}_{\alpha^{\prime}}}\rho(\alpha,v_{\alpha},\alpha^{\prime},v^{\prime}_{\alpha^{\prime}},t)\Phi_{\alpha}(\textbf{r})\Phi^{*}_{\alpha^{\prime}}(\textbf{r})\psi_{v_{\alpha}}(\textbf{Q})\psi^{*}_{v^{\prime}_{\alpha^{\prime}}}(\textbf{Q})\end{split}$
(3)

In the equation above,
$\rho(\alpha,v_{\alpha},t)=\absolutevalue{C_{\alpha,v_{\alpha}}(t)}^{2}$ and
$\rho(\alpha,v_{\alpha},\alpha^{\prime},v^{\prime}_{\alpha^{\prime}},t)=C_{\alpha,v_{\alpha}}(t)C^{*}_{\alpha^{\prime},v^{\prime}_{\alpha^{\prime}}}(t)$
are the diagonal and off-diagonal elements of the density matrix.
Additionally, $P_{\alpha
v_{\alpha}}(\textbf{r},\textbf{Q})=\absolutevalue{\Phi_{\alpha}(\textbf{r})}^{2}\absolutevalue{\psi_{v_{\alpha}}(\textbf{Q})}^{2}$
is the probability density of locating the electrons at r and the nuclei at
$Q$ in the Born-Oppenheimer State $\ket{\alpha v_{\alpha}}$. Thus, the values
of $\rho(\alpha,v_{\alpha},t)$ give the contribution of a particular BO State
to the total molecular probability density in the molecular frame, and are the
so-called populations. The term
$\Phi_{\alpha}(\textbf{r})\Phi^{*}_{\alpha^{\prime}}(\textbf{r})$ gives the
spatial overlap between $\Phi_{\alpha}(\textbf{r})$, and
$\Phi_{\alpha^{\prime}}(\textbf{r})$ and
$\psi_{v_{\alpha}}(\textbf{Q})\psi^{*}_{v^{\prime}_{\alpha^{\prime}}}(\textbf{Q})$
describes the spatial overlap between $\psi_{v_{\alpha}}(\textbf{Q})$ and
$\psi_{v^{\prime}_{\alpha^{\prime}}}(\textbf{Q})$. Hence, the coherences
$\rho(\alpha,v_{\alpha},\alpha^{\prime},v^{\prime}_{\alpha^{\prime}},t)$
indicate to what degree the interference between two states contributes to the
total probability density. This molecular frame density matrix, or some
variant thereof, is typically used to compute observables and interpret
experimental data.

In subsequent sections we develop the tools needed to derive an equivalent,
laboratory frame density matrix which fully characterizes the dynamics in any
given experiment. This is done by including rotation and considering the total
angular momentum of the molecule. To do so, the angular momenta associated
with all states excited by the light pulse need to be correctly coupled. For
this purpose, we introduce Angular Momentum Coherence Operators in the next
subsection.

###  2.2 Angular Momentum Coherence Operators

We begin our analysis by considering the irreducible representations of the
rotation group SO(3), the Wigner matrix elements $D^{j}_{mk}$ [52, 53]. Here
$j$ specifies the representation, with $m$ and $k$ running from $-j$ to $j$
facilitating a rotation between basis functions within a representation [52,
53],

$\left|jm\right\rangle=\sum_{k}D^{j*}_{mk}(\mathbf{\Omega})\left|jk\right\rangle,$
(4)

$\mathbf{\Omega}$ specifying the rotation angles, here taken to be the Euler
angles $\mathbf{\Omega}=\\{\phi,\theta,\chi\\}$. The properties of the
$D^{j}_{mk}$ make them suitable basis functions for molecular rotation. Of
particular note is that the irreducible representations $j_{1}$ and $j_{2}$
are easily combined to generate irreducible representations $j_{3}$ [52, 53],

$D^{j_{1}}_{m_{1}k_{1}}(\Omega)D^{j_{2}}_{m_{2}k_{2}}(\Omega)=\sum_{j_{3}}(2j_{3}+1)\left(\begin{array}[]{c
c c}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}&m_{3}\end{array}\right)\left(\begin{array}[]{c c
c}j_{1}&j_{2}&j_{3}\\\
k_{1}&k_{2}&k_{3}\end{array}\right)D^{j^{*}_{3}}_{m_{3}k_{3}}(\Omega)$ (5)

where the large parenthesis are the Wigner three-j symbols. This enables
straightforward coupling of angular momenta such that
$j_{3}=|j_{1}-j_{2}|,|j_{1}-j_{2}+1|,\dots,|j_{1}+j_{2}-1|,|j_{1}+j_{2}|$,
$m_{3}=m_{1}+m_{2}$ and $k_{3}=k_{1}+k_{2}$. Furthermore the eigenfunctions,
$\left|JKM\right\rangle$, of the rigid symmetric top Hamiltonian are also
given by the Wigner matrix elements [52],

$\left\langle\mathbf{\Omega}|JKM\right\rangle=\sqrt{\frac{2J+1}{8\pi^{2}}}D^{J*}_{MK}(\mathbf{\Omega}).$
(6)

Here $J$ is the total angular momentum, $K$ its projection on a chosen
molecular axis and $M$ on the space fixed axis. The asymmetric top Hamiltonian
mixes states of different $K$, but $J$ is conserved. In fact no intramolecular
interaction can mix states of different $J$ since the rotation group SO(3) is
a symmetry group of the full molecular Hamiltonian - the total angular
momentum $J$ conserved. With these considerations it is appropriate to specify
orthogonal molecular basis functions
$\left|n\right\rangle\left|J_{n}K_{n}M_{n}\right\rangle$, with
$n={\alpha,v_{\alpha}}$ labeling molecular frame vibronic BO basis states used
above. While intramolecular interactions conserve angular momentum, a
polarized laser pulse necessarily excites a coherent superposition of $J_{n}$
states. These can easily be coupled as prescribed by Eq. 5 to give the total
angular momentum of a wavepacket of states. This motivates definition of the
following tensor operators, indexed by the total angular momentum and its
projections,

$\begin{split}&A^{K}_{QS}=\frac{2K+1}{8\pi^{2}}\sum_{J_{n}K_{n}M_{n}}\sum_{J^{\prime}_{n^{\prime}}K^{\prime}_{n^{\prime}}M^{\prime}_{n^{\prime}}}\sum_{nn^{\prime}}\sqrt{(2J_{n}+1)(2J^{\prime}_{n^{\prime}}+1)}(-1)^{M_{n}-K_{n}}\\\
&\times\left(\begin{array}[]{ccc}J_{n}&J^{\prime}_{n\prime}&K\\\
-M_{n}&M^{\prime}_{n^{\prime}}&Q\end{array}\right)\left(\begin{array}[]{ccc}J_{n}&J^{\prime}_{n^{\prime}}&K\\\
-K_{n}&K^{\prime}_{n^{\prime}}&S\end{array}\right)\left|n^{\prime}\right\rangle\left|J^{\prime}_{n^{\prime}}K^{\prime}_{n^{\prime}}M^{\prime}_{n^{\prime}}\right\rangle\left\langle
J_{n}K_{n}M_{n}\right|\left\langle n\right|.\end{split}$ (7)

Here
$K=|J_{n}-J^{\prime}_{n^{\prime}}|,|J_{n}-J_{n^{\prime}}+1|,\dots,|J_{n}+J_{n^{\prime}}-1|,|J_{n}+J_{n^{\prime}}|$
is the total wavepacket angular momentum which results from coupling the
angular momenta in the excited wavepacket. We adhere to the unfortunate
convention where $K$ is the total wavepacket angular momentum while $K_{n}$ is
the projection of $J_{n}$ onto the molecular axis. $Q$ and $S$ are the
projections of $K$ onto the space fixed and molecular axes respectively [54].
Note that these are constructed from differences of the projection angular
momenta - $Q=M^{\prime}_{n^{\prime}}-M_{n}$ and
$S=K^{\prime}_{n^{\prime}}-K_{n}$ \- rather than the sum as in Eq. 5. This is
done so that the set of operators $A^{K}_{QS}$ project out quantum beats
between states of different angular momentum, which in turn correspond to a
time varying anisotropy as shown below. We refer to the $A^{K}_{QS}$ as
Angular Momentum Coherence Operators (AMCOs). They are defined in a manner
similar to the state multipole operators $T^{K}_{Q}$ introduced by Fano and
used to analyze anisotropy in coherent distributions of $M_{n}$ states. As we
will show below the AMCOs, as defined above, are an important part of
calculating the matrix elements of the density operator and understanding the
origins and behavior of coherences in the laboratory frame.

###  2.3 The Laboratory Frame Density Matrix

Matrix elements of the density operator, like the molecular wavefunction, can
be used to determine all observable properties of a molecule [51]. In the
$\ket{n}\ket{J_{n}K_{n}M_{n}}$ basis, the density matrix has elements

$\bra{J^{\prime}_{n^{\prime}}K^{\prime}_{n^{\prime}}M^{\prime}_{n^{\prime}}n^{\prime}}\hat{\rho}(t)\ket{nJ_{n}K_{n}M_{n}}=\rho_{\omega_{n}\omega^{\prime}_{n^{\prime}}}(n,n^{\prime},t)$
(8)

with $\omega_{n}=\\{J_{n}K_{n}M_{n}\\}$ and $n=\alpha,v_{\alpha}$. This
constitutes a density matrix in the laboratory frame, in contrast to the
molecular frame quantity in Eq. 3. We may write the density operator in terms
of the density matrix elements
$\rho_{\omega_{n}\omega^{\prime}_{n^{\prime}}}(n,n^{\prime},t)$, as follows,

$\hat{\rho}(t)=\sum_{\alpha\alpha^{\prime}v_{\alpha}v^{\prime}_{\alpha^{\prime}}}\sum_{J_{n}K_{n}M_{n}}\sum_{J^{\prime}_{n^{\prime}}K^{\prime}_{n^{\prime}}M^{\prime}_{n^{\prime}}}\rho_{\omega_{n}\omega^{\prime}_{n^{\prime}}}(n,n^{\prime},t)\ket{\alpha}\ket{v_{\alpha}}\ket{J_{n}K_{n}M_{n}}\bra{J^{\prime}_{n^{\prime}}K^{\prime}_{n^{\prime}}M^{\prime}_{n^{\prime}}}\bra{\alpha^{\prime}}\bra{v^{\prime}_{\alpha^{\prime}}}$
(9)

This equation is consistent with Eq. 8 since the BO states and
$\bra{J^{\prime}_{n^{\prime}}K^{\prime}_{n^{\prime}}M^{\prime}_{n^{\prime}}}\ket{J_{n}K_{n}M_{n}}=\delta_{J^{\prime}_{n^{\prime}}J_{n}}\delta_{K^{\prime}_{n^{\prime}}K_{n}}\delta_{M^{\prime}_{n^{\prime}}M_{n}}$
are both orthogonal. For a molecule with orientation given by the Euler Angles
$\phi$, $\theta$, and $\chi$, in a nuclear configuration Q the probability of
finding the electrons at r is given by the diagonal elements of
$\hat{\rho}(t)$ in the
$\ket{\textbf{Q}}\ket{\textbf{r}}\ket{\phi,\theta,\chi}$ basis -
$\bra{\textbf{Q},\textbf{r};\phi,\theta,\chi}\hat{\rho}(t)\ket{\textbf{Q},\textbf{r};\phi,\theta,\chi}$.

$\begin{split}P(\textbf{Q},\textbf{r},\mathbf{\Omega},t)=\sum_{\alpha\alpha^{\prime}v_{\alpha}v^{\prime}_{\alpha^{\prime}}}\sum_{\omega_{n}\omega^{\prime}_{n^{\prime}}}\rho_{\omega_{n}\omega^{\prime}_{n^{\prime}}}(n,n^{\prime},t)\Phi_{\alpha}(\textbf{r})\Phi^{*}_{\alpha^{\prime}}(\textbf{r})\psi_{v_{\alpha}}(\textbf{Q})\psi^{*}_{v^{\prime}_{\alpha^{\prime}}}(\textbf{Q})\\\
\times\frac{\sqrt{(2J_{n}+1)(2J^{\prime}_{n^{\prime}}+1)}}{8\pi^{2}}(-1)^{M_{n}-K_{n}}D^{J_{n}}_{-M_{n}-K_{n}}(\mathbf{\Omega})D^{J^{\prime}_{n^{\prime}}}_{M^{\prime}_{n^{\prime}}K^{\prime}_{n^{\prime}}}(\mathbf{\Omega})\end{split}$
(10)

Here $P(\textbf{Q},\textbf{r},\mathbf{\Omega},t)$ is the probability of
finding a molecule oriented at $\mathbf{\Omega}$ with a configuration of
nuclei located at Q, and electrons located at r. Using Eq. 5 to couple angular
momenta gives

$\begin{split}P(\textbf{Q},\textbf{r},\mathbf{\Omega},t)=\sum_{\alpha\alpha^{\prime}v_{\alpha}v^{\prime}_{\alpha^{\prime}}}\Phi_{\alpha}(\textbf{r})\Phi^{*}_{\alpha^{\prime}}(\textbf{r})\psi_{v_{\alpha}}(\textbf{Q})\psi^{*}_{v^{\prime}_{\alpha^{\prime}}}(\textbf{Q})\\\
\times\sum_{\omega_{n}\omega^{\prime}_{n^{\prime}}}\rho_{\omega_{n}\omega^{\prime}_{n^{\prime}}}(n,n^{\prime},t)\frac{\sqrt{(2J+1)(2J^{\prime}_{n^{\prime}}+1)}}{8\pi^{2}}(-1)^{M_{n}-K_{n}}\sum_{KQS}(2K+1)\\\
\times\begin{pmatrix}J_{n}&J^{\prime}_{n^{\prime}}&K\\\
-M_{n}&M^{\prime}_{n^{\prime}}&Q\end{pmatrix}\begin{pmatrix}J_{n}&J^{\prime}_{n^{\prime}}&K\\\
-K_{n}&K^{\prime}_{n^{\prime}}&S\end{pmatrix}D^{K*}_{QS}(\mathbf{\Omega})\end{split}$
(11)

Rewriting this equation with

$\begin{split}\bra{\alpha^{\prime}v^{\prime}_{\alpha^{\prime}}\omega^{\prime}_{n^{\prime}}}\hat{A}^{K}_{QS}\ket{\alpha
v_{\alpha}\omega_{n}}=\frac{\sqrt{(2J+1)(2J^{\prime}_{n^{\prime}}+1)}}{8\pi^{2}}(-1)^{M_{n}-K_{n}}\\\
\times(2K+1)\begin{pmatrix}J_{n}&J^{\prime}_{n^{\prime}}&K\\\
-M_{n}&M^{\prime}_{n^{\prime}}&Q\end{pmatrix}\begin{pmatrix}J_{n}&J^{\prime}_{n^{\prime}}&K\\\
-K_{n}&K^{\prime}_{n^{\prime}}&S\end{pmatrix}\end{split}$ (12)

which are the matrix elements of the AMCOs, Eq. 7, in the BO basis, results
in:

$\begin{split}P(\textbf{Q},\textbf{r},\mathbf{\Omega},t)=\sum_{KQS}\sum_{\alpha\alpha^{\prime}v_{\alpha}v^{\prime}_{\alpha^{\prime}}}\Phi_{\alpha}(\textbf{r})\Phi^{*}_{\alpha^{\prime}}(\textbf{r})\psi_{v_{\alpha}}(\textbf{Q})\psi^{*}_{v^{\prime}_{\alpha^{\prime}}}(\textbf{Q})\\\
\times\sum_{\omega_{n}\omega^{\prime}_{n^{\prime}}}\rho_{\omega_{n}\omega^{\prime}_{n^{\prime}}}(n,n^{\prime},t)\bra{\alpha^{\prime}v^{\prime}_{\alpha^{\prime}}\omega^{\prime}_{n^{\prime}}}\hat{A}^{K}_{QS}\ket{\alpha
v_{\alpha}\omega_{n}}D^{K*}_{QS}(\mathbf{\Omega})\end{split}$ (13)

Defining the time-depdendent coefficients as:

$A^{K}_{QS}(\alpha,\alpha^{\prime},v_{\alpha},v^{\prime}_{\alpha^{\prime}};t)=\sum_{\omega_{n}\omega^{\prime}_{n^{\prime}}}\rho_{\omega_{n}\omega^{\prime}_{n^{\prime}}}(n,n^{\prime},t)\bra{\alpha^{\prime}v^{\prime}_{\alpha^{\prime}}\omega^{\prime}_{n^{\prime}}}\hat{A}^{K}_{QS}\ket{\alpha
v_{\alpha}\omega_{n}},$ (14)

we get

$\begin{split}P(\textbf{Q},\textbf{r},\mathbf{\Omega},t)=\sum_{KQS}\sum_{\alpha\alpha^{\prime}v_{\alpha}v^{\prime}_{\alpha^{\prime}}}\Phi_{\alpha}(\textbf{r})\Phi^{*}_{\alpha^{\prime}}(\textbf{r})\psi_{v_{\alpha}}(\textbf{Q})\psi^{*}_{v^{\prime}_{\alpha^{\prime}}}(\textbf{Q})\\\
\times
A^{K}_{QS}(\alpha,\alpha^{\prime},v_{\alpha},v^{\prime}_{\alpha^{\prime}};t)D^{K*}_{QS}(\mathbf{\Omega})\end{split}$
(15)

Finally, we separate the diagonal terms
$v_{\alpha}=v^{\prime}_{\alpha^{\prime}}$ and $\alpha=\alpha^{\prime}$ from
the off-diagonal terms $v_{\alpha}\neq v^{\prime}_{\alpha^{\prime}}$ and
$\alpha\neq\alpha^{\prime}$ and consolidate the time and orientation-angle
dependent coefficients to give:

$\begin{split}P(\textbf{Q},\textbf{r},\mathbf{\Omega},t)=\sum_{\alpha,v_{\alpha}}\rho_{\alpha
v_{\alpha}}(\mathbf{\Omega},t)P_{\alpha,v_{\alpha}}(\textbf{r},\textbf{Q})+\\\
\sum_{\alpha\neq\alpha}\sum_{v_{\alpha}\neq
v^{\prime}_{\alpha^{\prime}}}\rho_{\alpha\alpha^{\prime}v_{\alpha}v^{\prime}_{\alpha^{\prime}}}(\mathbf{\Omega},t)\Phi_{\alpha}(\textbf{r})\Phi^{*}_{\alpha^{\prime}}(\textbf{r})\psi_{v_{\alpha}}(\textbf{Q})\psi^{*}_{v^{\prime}_{\alpha^{\prime}}}(\textbf{Q})\end{split}$
(16)

We can compare this with the molecular frame probability distribution Eq. 3,
which we reproduce here:

$\begin{split}|\Psi_{mol}(t,\textbf{r},Q)|^{2}=\sum_{\alpha,v_{\alpha}}\rho(\alpha,v_{\alpha},t)P_{\alpha,v_{\alpha}}(\textbf{r},\textbf{Q})+\\\
\sum_{\alpha\neq\alpha^{\prime}}\sum_{v_{\alpha}\neq
v^{\prime}_{\alpha^{\prime}}}\rho(\alpha,v_{\alpha},\alpha^{\prime},v^{\prime}_{\alpha^{\prime}},t)\Phi_{\alpha}(\textbf{r})\Phi^{*}_{\alpha^{\prime}}(\textbf{r})\psi_{v_{\alpha}}(\textbf{Q})\psi^{*}_{v^{\prime}_{\alpha^{\prime}}}(\textbf{Q})\end{split}$
(17)

We can thus identify
$\rho_{\alpha\alpha^{\prime}v_{\alpha}v^{\prime}_{\alpha^{\prime}}}(\mathbf{\Omega},t)$
as the off-diagonal density matrix elements in the laboratory frame. Since
these off diagonal matrix elements are dependent on orientation, coherences in
the laboratory frame are also orientation dependent. Note in both Eq. 16 and
Eq. 17, the off-diagonal elements are coefficients of molecular frame
electronic and vibrational wave functions; therefore, they track coherences
between molecular frame vibrational and electronic states. This term is
partially responsible for the time-depdendent nature of the probability
distribution and controls which coherences in the molecular frame contribute
to the laboratory frame density and at what angle and time. Similarly, we can
identify $\rho_{\alpha v_{\alpha}}(\mathbf{\Omega},t)$ as the diagonal density
matrix elements. The diagonal matrix elements give the probability of finding
the molecule at an orientation $\mathbf{\Omega}$, in a BO state $\ket{\alpha
v_{\alpha}}$ as a function of time.

The lab frame density matrix elements in Eq. 16 contain the functions
$A^{K}_{QS}(\alpha,\alpha^{\prime},v_{\alpha},v^{\prime}_{\alpha^{\prime}};t)$
from Eq. 14 in a multipole expansion,

$\rho_{\alpha
v_{\alpha}}(\mathbf{\Omega},t)=\sum_{KQS}A^{K}_{QS}(\alpha,v_{\alpha};t)D^{K*}_{QS}(\mathbf{\Omega})$
(18)
$\rho_{\alpha\alpha^{\prime}v_{\alpha}v^{\prime}_{\alpha^{\prime}}}(\mathbf{\Omega},t)=\sum_{KQS}A^{K}_{QS}(\alpha,\alpha^{\prime},v_{\alpha},v^{\prime}_{\alpha^{\prime}};t)D^{K*}_{QS}(\mathbf{\Omega})$
(19)

The $A^{K}_{QS}(\alpha,v_{\alpha};t)$ and
$A^{K}_{QS}(\alpha,\alpha^{\prime},v_{\alpha},v^{\prime}_{\alpha^{\prime}};t)$
are therefore multipole moments of the diagonal and off diagonal density
matrix elements, respectively. Collectively, we will refer to these as the
Molecular Angular Distribution Moments (MADM). The off diagonal elements
$A^{K}_{QS}(\alpha,\alpha^{\prime},v_{\alpha},v^{\prime}_{\alpha^{\prime}};t)$
determine the time varying orientation-dependent shape of the vibronic
coherences in the laboratory frame, which we refer to as Electronic Angular
Distribution Moments (EADMs) [55]. The diagonal matrix elements
$A^{K}_{QS}(\alpha,v_{\alpha};t)$ determine the time varying probability
distribution of molecular orientations in a particular vibronic state, and are
called the Axis Distribution Moments (ADMs) [54]. These collectively dictate
the orientation-dependent shape of the all density matrix elements in the lab
frame. For example, if $K=1$ is the dominant term in the sum, the density
matrix element will resemble a dipole. If the matrix element in question is
diagonal, molecules in that particular vibronic state will be oriented and the
$A^{1}_{QS}(\alpha,v_{\alpha};t)$ will track this orientation over time. If it
is an off-diagonal matrix element, there will be a significant difference in
the vibronic dynamics depending on the orientation of the molecule.
Additionally, this difference will vary with time on a time scale determined
by the energy gap between the coherently excited, molecular frame vibronic
states. This time variation is encoded by
$A^{1}_{QS}(\alpha,\alpha^{\prime},v_{\alpha},v^{\prime}_{\alpha^{\prime}};t)$.
A familiar example is the macroscopic dipole responsible for resonant
absorption induced by coherent excitation out of the ground state. Similarly,
if $K=2$ is dominant then the matrix elements will resemble a quadrupole, and
so on. In general, as indicated by Eq. 18 and Eq. 19, the density matrix
elements will be some combination of these different moments. These density
matrix elements in turn determine the full-dimensional laboratory frame
probability distribution of the molecule, as given by Eq. 16.

The general conclusion we can draw from Eq. 18 and Eq. 19 is that the
molecular frame dynamics will differ for molecules orientated at different
angles in the lab frame, and the MADMs determine these dynamics. The lab frame
density matrix elements thus provide a direct connection between the molecular
frame and lab frame, and fully determine the dynamics in the molecular frame
at any orientation. Eq. 18 and Eq. 19 indicate that a number of MADMs need to
be calculated in order to determine the lab frame density matrix. In the
following subsection we derive selection rules for the MADMs, restricting the
sum over $K$ in these equations. We also provide relationships between the
MADMs that further reduce the number that need to be computed.

####  2.3.1 MADM selection Rules and relations

Angular momentum selection rules can render most MADMs zero. It is instructive
to begin with the MADMs that result by one photon excitation from an
isotropic, rotation-less ground state with $J=0$, $\ket{000}$. These selection
rules will also apply to one photon excitation from a mixed state representing
an isotropic distribution of molecules with zero average angular momentum and
zero average projection of angular momentum on any axis - the typical initial
condition for most ultrafast dynamics experiments. One photon selection rules
restrict excited-state angular momenta to $J_{n}=1$,$K_{n}=0,\pm 1$ and
$M_{n}=0,\pm 1$. In the specific case of excitation by linearly polarized
light $M_{n}=0$. We can now use these selection rules to determine which MADMs
are non-zero upon excitation by examining the AMCOs defined by Eq. 7. Matrix
elements of the AMCOs then determine the MADMs $A^{K}_{QS}(n,n^{\prime};t)$,
as evident from Eq. 14. From the Wigner 3j symbols in Eq. 7, we determine
allowed values of $K=0,1,2$ and $S=0,\pm 1,\pm 2$. $Q=0$ for excitation by
linearly polarized light, or $Q=0,\pm 1$ in general. We can further separate
these into selection rules for coherences between excited states, and between
the ground and excited state. For coherences between the ground ($n=1$) and
excited state, $J_{1}=0$ and $J^{\prime}_{n^{\prime}}=1$ are the only allowed
excited-state angular momenta. This renders the first 3j symbol in Eq. 7 zero
unless $K=1$. In other words, coherences between the ground and excited states
are always oriented, and this orientation varies in time as
$A^{1}_{0S}(1,n^{\prime};t)$. This physically manifests as an oscillating
dipole, as previously discussed. Similarly, the first 3j symbol restricts
$K=0,2$ for coherences between excited states. In other words these coherences
will always be aligned, but never oriented, with an alignment varying in time
with $A^{2}_{0S}(n,n^{\prime};t)$. Note that the MADM
$A^{0}_{00}(n,n^{\prime};t)$ is an isotropic moment and does not induce
orientation or alignment (as discussed below, this is the reduced vibronic
density matrix). This is the only moment allowed by Eq. 7 for the ground state
population, while excited populations admit $K=0$ and $K=2$ moments. This
essentially indicates that molecules in the excited state may be aligned but
not oriented. Based on the Wigner-Ekart Theorem [52, 53], for $N$-photon
excitation from an isotropic ground state, we can simply extend values of $K$
up to $2N$, with ground state - excited coherences only admitting the $K=N$
moment, and excited state coherences and populations only admitting even $K$.

These selection rules significantly limit the number of MADMs that need to be
calculated to determine the density matrix. The computational burden can be
further reduced by determining relationships between some of the MADMs. Since
the Wigner Matrix Elements $D^{K}_{QS}(\mathbf{\Omega})$ are orthogonal [52]
the MADMs can be written as

$A^{K}_{QS}(n,n^{\prime};t)=\frac{2K+1}{8\pi^{2}}\int
d\mathbf{\Omega}D^{K}_{QS}(\mathbf{\Omega})\rho_{nn^{\prime}}(\mathbf{\Omega},t).$
(20)

Using
$\rho_{n^{\prime}n}(\mathbf{\Omega},t)=\rho_{nn^{\prime}}^{*}(\mathbf{\Omega},t)$
and $D^{K}_{-Q-S}=(-1)^{Q-S}D^{K*}_{QS}$, we find the relation

$A^{K}_{-Q-S}(n^{\prime},n;t)=(-1)^{Q-S}A^{K*}_{QS}(n,n^{\prime};t).$ (21)

This along with the selection rules limits the number MADMs that need to be
calculated in any particular case. However, Eq. 14 indicates that the full
rovibronic time-depdendent density matrix,
$\rho_{\omega_{n},\omega^{\prime}_{n^{\prime}}}(n,n^{\prime};t)$, is needed to
compute the MADMs. For most cases this is not computationally feasible [48].
As previously discussed, most computations of vibronic dynamics occur in the
molecular frame, for a molecule at a fixed orientation. In the following
subsection we discuss computation in this fixed-molecule approximation.

####  2.3.2 The fixed molecule approximation and the reduced vibronic density
matrix

Calculations carried out in the molecular frame assume that the molecule is in
an eigenstate of the orientation angles $\ket{\mathbf{\Omega}}$. We may expand
this in the $\ket{JKM}$ basis,

$\ket{\mathbf{\Omega}}=\sum_{JKM}\ket{JKM}\innerproduct{JKM}{\mathbf{\Omega}},$
(22)

from which it is evident that an infinite superposition of $\ket{JKM}$ states
is required to construct $\ket{\mathbf{\Omega}}$. This is not practically
achievable. We note here that while quantum mechanics precludes the
preparation of $\ket{\mathbf{\Omega}}$ states, it allows _projection_ of these
onto a prepared state, $\left<\Omega|\psi\right>$, which constitutes a
measurement of a quantity at fixed $\mathbf{\Omega}$. Coincidence detection of
dissociative photoionization in the axial recoil approximation can pick a
particular orientation from an underlying distribution, and is an example of
such measurements [56]. On the other hand, impulsive alignment experiments aim
to prepare a state approximating $\ket{\mathbf{\Omega}}$ by laser-excitation
of a broad superposition of $\ket{JKM}$ states. In general, such distributions
are difficult to achieve for polyatomic molecules under laser-field free
conditions [47]. Furthermore, the Pauli Exclusion Principle obviates the
existence of such approximate orientation eigenstates for certain molecules
[49]. As discussed in in section 4 below on measurable quantities and quantum
tomography, the lab frame density matrix and the MADMs serve as an improved,
direct comparison with experimental data. These may be computed using the
fixed molecule approximation as well.

This is achieved by recognizing that the matrix elements
$\rho_{nn^{\prime}}(\mathbf{\Omega},t)$ are in fact matrix elements of the
density operator in the $\ket{\mathbf{\Omega}n}$ basis, diagonal in
$\ket{\mathbf{\Omega}}$,
$\bra{\mathbf{\Omega}n}\hat{\rho}(t)\ket{n^{\prime}\mathbf{\Omega}}$. A
derivation is presented in Appendix B. We note that there are also density
matrix elements which are off-diagonal in $\ket{\mathbf{\Omega}}$. These
cannot be written in terms of the MADMs but are evidently not needed for the
determination of vibronic dynamics (cf. Eq 16). Given this, the standard
vibronic calculations can be carried out to determine the vibronic (or
electronic, depending on the problem) density matrix $\rho_{n,n^{\prime}}(t)$
at different orientation angles $\mathbf{\Omega}$ with respect to the exciting
pump pulse to yield $\rho_{n,n^{\prime}}(\mathbf{\Omega},t)$. While this is
still an increased computational burden over a single molecular frame
calculation, since only moments up to $K=2$ are excited by one photon
relatively low orientation-angle resolution will suffice. The integral in Eq.
20 can then be carried out to compute the MADMs to serve as a basis for
comparison with experiment (cf section 4).

With the identification
$\rho_{nn^{\prime}}(\mathbf{\Omega},t)=\bra{\mathbf{\Omega}n}\hat{\rho}(t)\ket{n^{\prime}\mathbf{\Omega}}$,
we have the normalization condition $Tr\\{\hat{\rho}\\}=1$, or

$\sum_{n}\int{\rho_{nn}(\mathbf{\Omega},t)d\mathbf{\Omega}}=1.$ (23)

This along with Eq. 18 and the integral of the Wigner Matrix Elements [52]
provides the following normalization for the MADMs,

$\sum_{n}A^{0}_{00}(n,n;t)=\frac{1}{8\pi^{2}}.$ (24)

Additionally, integrating Eq. 18 and 19 over all orientations provides the
experimentally accessible reduced vibronic density matrix,

$\rho_{nn^{\prime}}(t)=8\pi^{2}A^{0}_{00}(n,n^{\prime};t).$ (25)

Thus, the orientation-averaged reduced vibronic density matrix includes only
coherences that have a non-zero isotropic MADM in the lab frame
($A^{0}_{00}(n,n^{\prime};t)\neq 0$). A direct consequence of this result and
the selections rules from the previous sub-section is that the coherences
between the ground and excited state are _not_ included in the reduced
vibronic density matrix for single photon excitation from an isotropic
ensemble, since such coherences only have a $K=1$ MADM. Any experiment
designed to specifically target the reduced vibronic density matrix will not
be sensitive to these coherences, if the experiment is initiated from an
isotropic ensemble. On the another hand, as discussed in [55] and section 4,
experiments that measure the anisotropy in the distribution of a particle
scattered off the excited molecule will potentially be directly sensitive to a
number of MADMs. Thus the orientation averaged reduced vibronic density matrix
does not accurately represent the information available from such experiments.

While the fixed molecule approximation is useful, particularly for expensive
vibronic calculations, a calculation which treats the rotational states
explicitly is preferable yielding the vibronic dynamics at any orientation
angle from Eq. 16, and equivalently the full lab frame molecular probability
distribution. In particular, this approximate method fails in cases in which
rotational and electronic degrees of freedom are coupled [48, 55]. In the
following section we present a numerical calculation of only electronic
dynamics within the rigid rotor and BO approximations that demonstrates the
concepts presented thus far, and verifies some of the theoretical results.

##  3 A Numerical Example

A rigorous calculation of
$\rho_{\alpha\alpha^{\prime}v_{\alpha}v^{\prime}_{\alpha^{\prime}}}(\mathbf{\Omega},t)$
in Eq. 16 demands considering all molecular degrees of freedom. Since this is
impractical in most scenarios, here we neglect vibrational degrees of freedom
in order to demonstrate some of the general features of
$\rho_{\alpha\alpha^{\prime}v_{\alpha}v^{\prime}_{\alpha^{\prime}}}(\mathbf{\Omega},t)$
discussed above. We consider a superposition of molecular eigenstates which
can be excited by a broad band laser pulse:

$\ket{\Psi(t)}=\sum_{J_{\alpha}M_{\alpha}\tau_{\alpha}\alpha}a^{\alpha}_{J_{\alpha}M_{\alpha}\tau_{\alpha}}\exp\left(-iE^{\alpha}_{J_{\alpha}M_{\alpha}\tau_{\alpha}}t\right)\ket{\alpha}\ket{J_{\alpha}M_{\alpha}\tau_{\alpha}}.$
(26)

Here we assume that the BO states
$\ket{\alpha}\ket{J_{n}M_{\alpha}\tau_{\alpha}}$ are eigenstates of the
Hamiltonian with energies
$E^{\alpha}_{J_{\alpha}M_{\alpha}\tau_{\alpha}}=E_{\alpha}+E_{J_{\alpha}M_{\alpha}\tau_{\alpha}}$,
$E_{\alpha}$ and $E_{J_{\alpha}M_{\alpha}\tau_{\alpha}}$ being the electronic
and rotational energies, respectively. This level of calculation is useful in
situations where only electronic dynamics are of interest, such as in
attosecond charge migration. That is, when considering only very short times
after the pulse, during which the nuclei can be satisfactorily approximated as
frozen. The states $\ket{J_{\alpha}M_{\alpha}\tau_{\alpha}}$ are eigenstates
of the asymmetric top Hamiltonian [52], and can be determined in the symmetric
top basis,

$\ket{J_{\alpha}M_{\alpha}\tau_{\alpha}}=\sum_{K_{\alpha}}C^{J_{\alpha}M_{\alpha}\tau_{\alpha}}_{K_{\alpha}}\ket{J_{\alpha}K_{\alpha}M_{\alpha}}$
(27)

given a set of rotational constants $A=1/2I_{aa}$, $B=1/2I_{bb}$ and
$C=1/2I_{aa}$, where $I_{jj}$ are diagonal components of the moment of inertia
tensor with $I_{cc}>I_{bb}>I_{aa}$. Since we assume a frozen geometry, we take
these to be the same for all electronic states $\ket{\alpha}$ in the
superposition. Doing so gives each electronic state the same set of rotational
eigenstates. We also note here that, unlike $K_{\alpha}$, $\tau_{\alpha}$ is
not an angular momentum projection on the molecular frame. This projection is
not a conserved quantity for an asymmetric top. Rather, $\tau_{\alpha}$ is an
index used to keep track of the set of energy eignestates with the same
$J_{\alpha}$ and $M_{\alpha}$ [52].

The system chosen here is 4-amino-4′-nitrostilbene excited to the manifold of
three closely-spaced bright $\pi\pi^{*}$ states centred around 4 eV. The
required electronic energies $E_{\alpha}$ and transition dipole moments
$\mu_{0\alpha,i}=\bra{\alpha}\hat{\mu}_{i}\ket{0}$ were computed at the pruned
combined density functional theory and multireference configuration
interaction (p-DFT/MRCI)[57] in conjunction with the def2-TZVP basis. These
values are shown in Table 1. The inertia tensor $I_{jj}$ of the ground
electronic state was calculated at the DFT level of theory using the TZVP
basis and B3LYP exchange correlation functional. The resulting principal
moments of inertia are $I_{aa}=821.87$ amu$\cdot$Bohr2, $I_{bb}=15275.20$
amu$\cdot$Bohr2 and $I_{cc}=16079.53$ amu$\cdot$Bohr2, and the $z$-axis is
identified with the $a$-axis, the $x$-axis with the $b$-axis and the $y$-axis
with the $c$-axis. The molecular geometry with the molecular frame axes
labelled is depicted in the inset in Fig. 1. The inertia tensor is used to
compute the rotational energies $E_{J_{\alpha}M_{\alpha}\tau_{\alpha}}$, which
are considered to be the same for all electronic states. These are determined
for the asymmetric top Hamiltonian following the procedure outlined in [52].

$\alpha$ | State |  $E_{\alpha}$ (eV) |  $\mu_{0\alpha,z}$ (au) |  $\mu_{0\alpha,x}$ (au) |  $\mu_{0\alpha,y}$ (au)
---|---|---|---|---|---
2 | $S_{4}(\pi\pi^{*})$ | 3.87 | 1.07552 | 0.10603 | -0.01137
3 | $S_{5}(\pi\pi^{*})$ | 4.04 | 0.53848 | 0.18773 | 0.01271
4 | $S_{6}(\pi\pi^{*})$ | 4.08 | -0.84664 | 0.32543 | -0.01292
Table 1: Excitation energies, $E_{\alpha}$, and transition dipole matrix
elements, $\mu_{0\alpha,i}=\bra{\alpha}\hat{\mu}_{i}\ket{0}$, for the states
included in the 4-amino-4′-nitrostilbene calculations. All quantities were
calculated at the p-DFT/MRCI level of theory with the def2-TZVP basis set.

We use first order perturbation theory, implying a one-photon process, in a
linearly polarized electric field to determine the amplitudes
$a^{\alpha}_{J_{\alpha}M_{\alpha}\tau_{\alpha}}$ in Eq. 26,

$a^{\alpha}_{J_{\alpha}M_{\alpha}\tau_{\alpha}}\propto\bra{J_{\alpha}M_{\alpha}\tau_{\alpha}}\bra{\alpha}\mu_{Z}\ket{0}\ket{J_{in}M_{in}\tau_{in}}$
(28)

where the overall multiplicative factor is determined by the bandwidth of the
pulse. We ignore this factor, hence assuming a flat-top bandwidth across the
set of states in table 1. In this limit, the excitation pulse is much shorter
than the timescale of electronic quantum dynamics which, in turn, is
determined by the spacing between electronic states involved. Eq. 28 implies
that the system starts in the ground state
$\ket{J_{in}M_{in}\tau_{in}}\ket{0}$ and is excited to the state
$\ket{J_{\alpha}M_{\alpha}\tau_{\alpha}}\ket{\alpha}$.

The transition dipole moments in Table 1 are in the molecular frame while
$\mu_{Z}$ in Eq. 28 is in the lab frame. In order to transform Eq. 28 into the
molecular frame, we convert $\mu_{Z}$ into a spherical tensor [52]. In the
spherical basis, $\mu_{Z}=\mu^{1}_{0}$ and can be expressed in terms of the
molecular frame spherical dipole moments $\mu^{1}_{q}$:

$[\mu^{1}_{0}]^{LF}=\sum_{q}D^{1*}_{0q}(\mathbf{\Omega})\mu^{1}_{q}$ (29)

Here, $\mu^{1}_{q}$ are the spherical components of the dipole moment in the
molecular frame with values of $q$ ranging from -1 to 1. The irreducible
spherical components of the molecular frame dipole moment are as follows [52]:

$\mu^{1}_{-1}=\frac{1}{\sqrt{2}}(\mu_{x}-i\mu_{y})$ (30)
$\mu^{1}_{1}=-\frac{1}{\sqrt{2}}(\mu_{x}+i\mu_{y})$ (31)
$\mu^{1}_{0}=\mu_{z}.$ (32)

Using Eq. 29 in Eq. 28 gives the following expression for
$a^{\alpha}_{J_{\alpha}M_{\alpha}\tau_{\alpha}}$.

$a^{\alpha}_{J_{\alpha}M_{\alpha}\tau_{\alpha}}=\sum_{q}\bra{J_{\alpha}M_{\alpha}\tau_{\alpha}}D^{1*}_{0q}\ket{J_{in}M_{in}\tau_{in}}\bra{\alpha}\mu^{1}_{q}\ket{0},$
(33)

which, with the use of Eq. 27, can be expressed as a sum over $q,K_{in}$ and
$K_{\alpha}$:

$a^{\alpha}_{J_{\alpha}M_{\alpha}\tau_{\alpha}}=\sum_{q}\sum_{K_{in}K_{\alpha}}C^{J_{in}M_{in}\tau_{in}}_{K_{in}}C^{J_{\alpha}M_{n}\tau_{n}*}_{K_{\alpha}}\bra{J_{\alpha}K_{\alpha}M_{\alpha}}D^{1*}_{0q}\ket{J_{in}K_{in}M_{in}}\bra{\alpha}\mu^{1}_{q}\ket{0}.$
(34)

The set of dipole matrix elements, calculated at the p-DFT/MRCI/def2-TZVP
level of theory, are given in table 1. The matrix elements
$\bra{J_{\alpha}K_{\alpha}M_{\alpha}}D^{1*}_{0q}\ket{J_{in}K_{in}M_{in}}$ are
straightforward to determine analytically in terms of Wigner 3j symbols
[52].This finally gives

$\begin{split}a^{\alpha}_{J_{\alpha}M_{\alpha}\tau_{\alpha}}=\sum_{q}\sum_{K_{in}K_{\alpha}}C^{J_{in}M_{in}\tau_{in}}_{K_{in}}C^{J_{\alpha}M_{n}\tau_{\alpha}*}_{K_{\alpha}}\sqrt{(2J_{\alpha}+1)(2J_{in}+1)}(-1)^{q}(-1)^{K_{in}-M_{in}}\times\\\
\begin{pmatrix}J_{in}&1&J_{\alpha}\\\
-M_{in}&0&M_{\alpha}\end{pmatrix}\begin{pmatrix}J_{in}&1&J_{\alpha}\\\
-K_{in}&-q&K_{\alpha}\end{pmatrix}\bra{\alpha}\mu^{1}_{q}\ket{0}\end{split}$
(35)

In order to calculate the MADMs Eq. 26 must be rewritten in the
$\ket{J_{\alpha}K_{\alpha}M_{\alpha}}$ basis:

$\ket{\Psi(t)}=\sum_{J_{\alpha}K_{\alpha}M_{\alpha}\alpha}\left(\sum_{\tau_{\alpha}}a^{\alpha}_{J_{\alpha}M_{\alpha}\tau_{\alpha}}C^{J_{\alpha}M_{\alpha}\tau_{\alpha}}_{K_{\alpha}}\exp\left(-iE^{\alpha}_{J_{\alpha}M_{\alpha}\tau_{\alpha}}t\right)\right)\ket{\alpha}\ket{J_{\alpha}K_{\alpha}M_{\alpha}}$
(36)

The coefficient in parenthesis provides the amplitude in the
$\ket{J_{\alpha}K_{\alpha}M_{\alpha}}$ basis,

$x_{J_{\alpha}K_{\alpha}M_{\alpha}}(t)=\sum_{\tau_{\alpha}}a^{\alpha}_{J_{\alpha}M_{\alpha}\tau_{\alpha}}C^{J_{\alpha}M_{\alpha}\tau_{\alpha}}_{K}\exp(-iE^{\alpha}_{J_{\alpha}M_{\alpha}\tau_{\alpha}}t)$
(37)

leading to the density matrix elements:

$x_{J_{\alpha}K_{n}M_{\alpha}}(t)x^{*}_{J^{\prime}_{\alpha^{\prime}}K^{\prime}_{\alpha^{\prime}}M_{\alpha}}(t)=\rho_{{\omega}_{\alpha}{\omega}^{\prime}_{\alpha^{\prime}}}(\alpha,\alpha^{\prime};t)$
(38)

The MADMs can then be determined using Eq. 14. In an experiment the initial
condition is typically a thermal distribution of rotational states in the
ground vibronic state. To simulate this we repeat the calculation for a number
of initial rotational states $\ket{J_{in}M_{in}\tau_{in}}$, and Boltzmann
average over the resulting set of MADMs. The necessary number of initial
states can be determined by summing the Boltzmann probability distribution
over energy. Once the statistical probability reaches $.99$, the sum is
terminated, the last state in the sum being the highest energy initial state
considered. Thermally averaged MADMs at a temperature of 0.1 K, corresponding
to 84 initial rotational states, are used in Eq. 18 and Eq. 19 to calculate
the lab frame density matrix elements.

Figure 1: The ground state geometry of 4-amino-4′-nitrostilbene, with the
Euler angles $\theta$ and $\chi$ labeled with respect to the lab frame
$Z$-axis.The molecular frame coordinate system is also shown in the inset.
Figure 2: (a)-(c) Initial orientation-dependent off-diagonal density matrix
elements between the ground and excited states,
$\rho_{12}(\theta,\chi,0),\rho_{13}(\theta,\chi,0),\rho_{14}(\theta,\chi,0)$.
(d)-(f) Initial orientation-dependent off-diagonal density matrix elements
between excited states,
$\rho_{23}(\theta,\chi,0),\rho_{24}(\theta,\chi,0),\rho_{34}(\theta,\chi,0)$.

Fig. 1 shows the geometry of 4-amino-4′-nitrostilbene as well as the relevant
orientation angles $\mathbf{\Omega}=\\{\theta,\chi\\}$ with respect to the
laser polarization. All quantities are independent of the azimuthal angle
$\phi$ due to cylindrical symmetry. In order to illustrate the analytical
results derived above, the calculated
$\rho_{\alpha\alpha^{\prime}}(\mathbf{\Omega},t)$ are plotted at $t=0$ as a
function of $\theta$ and $\chi$ in Fig. 2. We show only the real parts, as the
density matrix elements are all real at $t=0$. The top row, Figs. 2 (a) to
(c), show coherences between the ground and excited states. Evidently, these
coherences are oriented - their values for molecules oriented at $\theta,\chi$
and $\theta+\pi,\chi+\pi$ are different thus breaking inversion symmetry in
the lab frame. This is anticipated by the selection rule $K=1$ for one-photon
excited states. Conversely, the coherences in the bottom row are aligned and
do not break inversion symmetry in the lab frame, in accordance with the
selection rule $K=0,2$. Fig. 3(a) shows the relevant MADMs for the coherence
between the ground and first excited state (labelled 2 in Table 1). Only the
MADMs $A^{1}_{0S}(1,2;t)$ are non-zero, representing the oscillating dipole
induced by the excitation. Note that this decays over time due to spontaneous
emission, not included in our calculation. Similarly, in Fig. 3(b) we see that
the MADMs with $K=1$ are zero for the coherence between the excited states
labeled $2$ and $3$, and MADMs $A^{0}_{00}(2,3;t)$ and $A^{2}_{0S}(2,3;t)$ are
non-zero.

Figure 3: (a) MADMs contributing to the ground state-excited state density
matrix element $\rho_{12}(\theta,\chi,t)$. The MADM $A^{2}_{00}(1,2;t)$ is
always zero in agreement with the selection rules detailed in the previous
section, as are all the MADMs $A^{2}_{0S}(1,2;t)$ and $A^{0}_{00}(1,2;t)$ (not
shown).(b) Some of the MADMs contributing to the excited states density matrix
element $\rho_{23}(\theta,\chi,t)$. The MADM $A^{1}_{00}(1,2;t)$ is always
zero in agreement with the selection rules detailed in the previous section,
as are all the MADMs $A^{1}_{0S}(1,2;t)$ (not shown). For all density matrix
elements MADMs with $K>2$ are zero in agreement with the selection rules.

To elucidate the orientation dependent electronic dynamics, we plot the time-
depdendent coherences between the ground state and excited states in Fig. 4.
Note that we are treating these as eigenstates, rendering the populations
constant. Further, Fig. 3(b) indicates that in this case coherences between
excited states evolve on a much slower time scale, as a result of the much
smaller energy spacing. While our calculations within the rigid body
approximation do not include vibration, we note that including the vibrational
degrees of freedom will drive the components of the wave packet in different
electronic states apart, eventually destroying the coherence. In general, the
timescale for decoherence will be sensitive to the differential topology of
the potential energy surfaces involved. Based on such consideration, recent
studies indicate that the coherences could decay significantly within a range
of time windows, anywhere between 2 and 10 fs [58, 35, 59, 28]. We therefore
do not consider the slow excited state - excited state coherences, like that
in Fig. 3(b), any further. In Fig. 4, we focus on the early time dynamics,
during which the excited state-excited state coherences are essentially
constant. Comparing the phase relation of the set of coherences for a molecule
oriented at $\theta=0$, $\chi=0$ to one at $\theta=\pi$, $\chi=0$, we see that
we can expect very similar dynamics in both cases, but with a relative time
delay of approximately $0.5$ fs between the two since each coherence is
shifted in phase by $\pi$. On the other hand, we can expect very different
dynamics for a molecule at $\theta=\pi/2$, $\chi=0$, since the phase relation
between the coherences is completely different. This illustrates the general
point that the molecular frame dynamics occurring in an experiment will vary
for molecules in different orientations. While we have held the angle $\chi$
fixed here, Fig. 2 shows that the coherences also vary with $\chi$ indicating
that the molecular frame dynamics will vary accordingly.

Figure 4: ground state-excited state density matrix elements
$\rho_{12}(\theta,\chi,t)$, $\rho_{13}(\theta,\chi,t)$ and
$\rho_{14}(\theta,\chi,t)$ at orientations {$\theta=0,\chi=0$},
{$\theta=\pi,\chi=0$} and {$\theta=\pi/2,\chi=0$}, respectively. Figure 5:
Time evolution of the one-electron difference density $\Delta
p(\boldsymbol{r}_{1},\mathbf{\Omega},t)$ for various orientation angles
$\mathbf{\Omega}=\\{\theta,\chi=0\\}$ and times $t$. The regions colored white
correspond to a net-positive change, i.e., accumulation, of electron density
relative to the ground electronic state. Those colored blue correspond to net-
negative change, i.e., depletion, of electron density.

In Fig. 5, we show the time evolution of the one-electron difference density
$\Delta p(\boldsymbol{r}_{1},\mathbf{\Omega},t)$ for various orientation
angles $\mathbf{\Omega}=\\{\theta,\chi=0\\}$. $\Delta
p(\boldsymbol{r}_{1},\mathbf{\Omega},t)$ is here defined as the difference of
the wave packet and ground state electronic probability distributions
integrated over all but one electronic degree of freedom,

$\Delta p(\boldsymbol{r}_{1},\mathbf{\Omega},t)=N\int
d\boldsymbol{r}_{2}\cdots
d\boldsymbol{r}_{N}\left[P(\boldsymbol{r}_{1},\dots,\boldsymbol{r}_{N},\mathbf{\Omega},t)-P_{0}(\boldsymbol{r}_{1},\dots,\boldsymbol{r}_{N})\right].$
(39)

The details of these calculations are given in Appendix C. The expected phase
difference between molecules oriented at $\theta=0$ and $\theta=\pi$ is
evident. Additionally, molecules at $\theta=\pi/2$ exhibit entirely different
dynamics. Furthermore, these dynamics are driven entirely by the set of ground
state-excited state coherences shown in Fig. 4, which are in turn determined
solely by the the MADMs $A^{1}_{0S}(1,\alpha^{\prime};t)$. Since the
orientation-averaged reduced vibronic density matrix is
$8\pi^{2}A^{0}_{00}(\alpha,\alpha^{\prime};t)$ (cf Eq. 25), these dynamics are
completely invisible to an experiment designed to specifically measure the
reduced vibronic density matrix elements but may, in general, be accessible
experimentally. In fact, the variety of dynamics evident in Fig. 5 indicates
that a great deal of information is potentially contained in a lab frame
experiment on un-oriented molecules, but this information is difficult to
extract. The next section discusses the possibility of extracting this
information, and the limitations associated with such an extraction.

##  4 Measurements and Quantum Tomography

The MADMs shown in Eq. 18 and Eq. 19 to be the time-depdendent components of
the lab frame density matrix provide a direct connection between the density
matrix and measurements. Determination of this density matrix from measurement
then allows the expectation value of any _molecular frame_ operator $\hat{O}$
to be determined for a molecule at any orientation angle,

$\left<\hat{O}\right>(\mathbf{\Omega},t)=\sum_{nm}\rho_{nm}(\mathbf{\Omega},t)\bra{m}\hat{O}\ket{n}.$
(40)

Since $\hat{O}$ is defined in the molecular frame, it is exclusively a
function of vibronic coordinates and momenta. Therefore it acts as the unit
operator on $\ket{\mathbf{\Omega}}$, and density matrix elements that are off-
diagonal in $\ket{\mathbf{\Omega}}$ are not needed to calculate its
expectation value. Note that these are needed to determine expectation values
of lab frame operators, such as the total angular momentum. As previously
discussed, these cannot be written in terms of the MADMs. Determination of the
lab frame density matrix also provides the full molecular probability density
through Eq. 16, and therefore the time varying molecular frame vibronic
probability density for a molecule at any orientation angle. We can therefore
refer to the determination of all the relevant MADMs as a molecular frame
quantum tomography. To perform this tomography we must consider the link
between specific measurements and the MADMs. In particular, time resolved
measurements of photoelectron angular distributions from a molecular
wavepacket contain the MADMs. This has been shown theoretically [54, 60, 61]
and recently demonstrated experimentally for an electronic wavepacket in the
$B^{1}E^{\prime\prime}$ state of ammonia [55]. This relationship has
previously been exploited to determine molecular frame photoelectron angular
distributions using a rotational wavepacket in the ground state of isolated
molecules [62, 63]. In a similar study, the angular distributions of
diffracted electron and x-ray pulses from a ground state rotational wavepacket
were recently shown to contain the ADMs, and a method to extract the molecular
geometry using this relationship for electron diffraction was numerically
demonstrated [64]. Recent ultrafast X-ray diffraction measurements from an
excited state wavepacket imply a direct connection between the X-ray angular
distribution and the MADMs [65], though this has yet to be shown analytically.
A direct connection, such as that now known to exist for time resolved
photoelectron angular distributions, provides a potential route to determining
all relevant lab frame density matrix elements from experiment constituting a
molecular frame quantum tomography. We discuss this possibility in more detail
below.

Time resolved photoelectron angular distributions are typically analyzed by
multipole expansion as follows,

$\sigma(\epsilon,t,\theta_{e},\phi_{e})=\sum_{LM}\beta_{LM}(\epsilon,t)Y^{L}_{M}(\theta_{e},\phi_{e}),$
(41)

where $\epsilon$ is the kinetic energy of the electron and $\theta_{e}$ and
$\phi_{e}$ its polar and azimuthal ejection angles. The time-depdendent
anisotropy parameters $\beta_{LM}(\epsilon,t)$ can then be expressed in terms
of the MADMs [54, 66, 67, 55, 62],

$\displaystyle\beta_{LM}(\epsilon,t)=\sum_{nn^{\prime}}\sum_{KQS}C^{LM}_{KQS}(n,n^{\prime};\epsilon)A^{K}_{QS}(n,n^{\prime};t)$
(42) $\displaystyle
C^{LM}_{KQS}(n,n^{\prime};\epsilon)=\sum_{\zeta\zeta^{\prime}}D^{n}_{\zeta}(\epsilon)D^{n^{\prime}*}_{\zeta^{\prime}}(\epsilon)\Gamma^{\zeta\zeta^{\prime}LM}_{KQS}.$
(43)

Eq. 42 separates the excited state dynamics - fully characterized by the MADMs
- from the photoionization dynamics characterized by the coefficients
$C^{LM}_{KQS}(n,n^{\prime};\epsilon)$. These coefficients contain the
ionization dipole matrix elements $D^{n}_{\zeta}(\epsilon)$ and
$D^{n^{\prime}*}_{\zeta^{\prime}}(\epsilon)$ between the pair of bound states
$n$ and $n^{\prime}$ into the set of continuum channels $\zeta$ and
$\zeta^{\prime}$ accessible from each of the bound states. In other words,
$C^{LM}_{KQS}(n,n^{\prime};\epsilon)$ is zero unless both bound states $n$ and
$n^{\prime}$ ionize to the same final state producing an electron of energy
$\epsilon$. The analytical coupling factor
$\Gamma^{\zeta\zeta^{\prime}LM}_{KQS}$ ensures that total angular momentum is
conserved during the photoionization process. It has been previously studied
in detail [62], and we need not discuss it further here. Since a set of
$\beta_{LM}(\epsilon,t)$ are typically measured in a time resolved experiment,
if the $C^{LM}_{KQS}(n,n^{\prime};\epsilon)$ are known the MADMs can
potentially be determined from Eq. 42, which are a set of linear equations in
the MADMs at each time. Eq. 18 and Eq. 19 then provide the lab frame density
matrix.

A number of issues need to be carefully considered here. Firstly, experimental
determination of the
$C^{LM}_{KQS}(n,n^{\prime};\epsilon)$, while possible is highly non-trivial.
This requires a so-called complete experiment - determination of the
ionization dipole matrix elements (typically in the partial wave basis) -
accomplished for a only a handful of molecules [68, 69, 70, 71, 72, 73, 74,
75, 63]. While a number of methods have been applied to this end, it has been
most reliably accomplished for an excited state of a polyatomic molecule by
rotationally resolved, resonance enhanced multiphoton ionization (REMPI)
through the excited state of interest [73, 74, 75]. Even in cases where the
ionization dipole matrix elements are known, an additional challenge is posed
by Koopman’s Correlations to the continuum [54]. Often, a single bound state
is correlated with a single ionic state. While these correlations have proved
invaluable for tracking populations of electronic states [76, 54], they render
the $C^{LM}_{KQS}(n,n^{\prime};\epsilon)$ zero when $n\neq n^{\prime}$
removing off-diagonal MADMs, and therefore any direct information about
coherences from the measurement. Finally, in most cases there will not be a
sufficient number of equations to uniquely determine all relevant MADMs needed
to construct the density matrix. For one-photon excitation by a linearly
polarized pulse followed by one-photon ionization by an elliptically polarized
pulse $\beta_{LM}$ with $L=0,2$ and $4$ are allowed by conservation of angular
momentum, with $M=0$, $M=0,\pm 2$ and $M=0,\pm 2$ and $\pm 4$ respectively.
Additionally $\beta_{L-M}=(-1)^{M}\beta^{*}_{LM}$. This provides a total of
only 6 independent equations. Even for one-photon excitation of only a single
excited state, the selection rules for, and relations between the MADMs result
in 4 independent MADMs two of which are complex, leaving 6 unknowns in the
absence of any additional symmetry. Adding an additional excited state adds
additional non-zero MADMs, but does not increase the number of unique
$\beta_{LM}$ rendering the problem ill-posed.

Nonetheless, in certain cases all of these difficulties can potentially be
overcome by symmetry, and limiting the number of excited states. One such case
is the previously mentioned $B^{1}E^{\prime\prime}$ state of of ammonia. This
is nominally a doubly degenerate electronic state, in which degeneracy is
broken by Jahn-Teller distortion and the Electronic Coriolis coupling [77]. In
a recent experiment with linearly polarized ionizing light, it was shown that
the various MADMs are separately projected into the anisotropy parameters
$\beta_{00}$, $\beta_{20}$ and $\beta_{40}$, the coherence between the near-
degenerate states being detected in $\beta_{40}$ [55]. The Koopman’s
Correlations fail in this case and both electronic states ionize to the ground
ionic state ejecting energetically degenerate photoelectrons. Further,
symmetry of the electronic states imposes additional selection rules resulting
in the excitation of only three unique MADMs, all of which are real [55]. We
note that since the excitation pulse used in the experiment was significantly
longer than the timescale of the ground state-excited state coherence, this
coherence is washed out by the pulse. A complete experiment has also been
performed through this state by REMPI [75], therefore the
$C^{LM}_{KQS}(n,n^{\prime};\epsilon)$ can be determined. This then provides
three independent equations from the combination of the time and frequency
resolved experiments, resulting in sufficient information to determine the
three unknown MADMs and thus the lab frame density matrix as a function of
time. We are currently working towards this goal.

##  5 Summary and Outlook

We have developed a density matrix formalism for quantum molecular dynamics in
terms of Molecular Angular Distribution Moments (MADMs) which elucidates the
transformation from the molecular frame to the lab frame. This connection is
clearly provided by Eq. 16, the vibronic density matrix for a molecule at a
particular orientation. Importantly, we find that the lab frame orientation of
certain coherences - such as that between the ground and resonantly excited
state - removes them from the experimentally accessible reduced vibronic
density matrix. Nonetheless, these coherences may be accessible by measurement
indicating that the orientation-averaged reduced vibronic density matrix is
not an accurate representation of experimentally accessible information. The
MADMs can therefore serve as an improved comparison with experimental data.
Further, since the lab frame vibronic density matrix Eq. 18 and Eq. 19 is
orientation dependent, in any given experiment initiated from an isotropic
ensemble a distribution of vibronic dynamics will occur (cf. Fig. 5). This
fact can be exploited to extract the MADMs from an experiment, and therefore
the density matrix elements for any orientation angle, constituting a
molecular frame tomography. This serves as the raw data to produce time
resolved probability densities, such as those in Fig. 5, for a molecule at any
orientation angle providing an experimental route to imaging molecular frame
vibronic dynamics. Additionally, the ability to perform such a tomography is
also the necessary first step toward the detailed experimental study of
fundamental quantum mechanical phenomena in individual molecules, such as
entanglement. Recent measurements have shown that exploiting attosecond laser
pulses makes such phenomena accessible on the timescales of electronic motion
[78, 79]. We anticipate that quantum tomography will be a vital ingredient for
further studies in this direction, as has been the case for such
investigations with optical systems [80].

##  Appendix A Derivation of the Wigner D-Matrix Matrix Elements

Using the resolution of the identity in the Euler Angle basis
$1=\int{}\ket{\mathbf{\Omega}}\bra{\mathbf{\Omega}}d\mathbf{\Omega}$,

$\bra{J_{n}K_{n}M_{n}}D^{1*}_{0q}\ket{J_{in}K_{in}M_{in}}=\int{}\bra{J_{n}K_{n}M_{n}}\ket{\mathbf{\Omega}}D^{1*}_{0q}\bra{\mathbf{\Omega}}\ket{J_{in}K_{in}M_{in}}d\mathbf{\Omega}$
(A.1)

Then using Eq. 6 for the symmetric top wavefunction
$\bra{\mathbf{\Omega}}\ket{JKM}$ gives

$\bra{J_{n}K_{n}M_{n}}D^{1*}_{0q}\ket{J_{in}K_{in}M_{in}}=\int{}\frac{\sqrt{(2J_{n}+1)(2J_{in}+1)}}{8\pi^{2}}D^{J_{n}}_{M_{n}K_{n}}D^{1*}_{0q}D^{J_{in}*}_{M_{in}K_{in}}d\mathbf{\Omega}$
(A.2)

Further simplification to this equation can be achieved by using the
relationship $D^{j*}_{m^{\prime}m}=(-1)^{m-m^{\prime}}D^{j}_{-m^{\prime}-m}$

$\begin{split}\bra{J_{n}K_{n}M_{n}}D^{1*}_{0q}\ket{J_{in}K_{in}M_{in}}=\frac{\sqrt{(2J_{n}+1)(2J_{in}+1)}}{8\pi^{2}}(-1)^{q-0}(-1)^{K_{in}-M_{in}}\times\\\
\int{}D^{J_{n}}_{M_{n}K_{n}}D^{1}_{0-q}D^{J_{in}}_{-M_{in}-K_{in}}d\mathbf{\Omega}\end{split}$
(A.3)

The solution to this integral is well known and given in [52]. Thus,
$\bra{J_{n}K_{n}M_{n}}D^{1*}_{0q}\ket{J_{in}K_{in}M_{in}}$ can be expressed in
terms of a set of Wigner 3j symbols,

$\begin{split}\bra{J_{n}K_{n}M_{n}}D^{1*}_{0q}\ket{J_{in}K_{in}M_{in}}=\sqrt{(2J_{n}+1)(2J_{in}+1)}(-1)^{q}(-1)^{K_{in}-M_{in}}\times\\\
\begin{pmatrix}J_{in}&1&J_{n}\\\
-M_{in}&0&M_{n}\end{pmatrix}\begin{pmatrix}J_{in}&1&J_{n}\\\
-K_{in}&-q&K_{n}\end{pmatrix}\end{split}$ (A.4)

##  Appendix B Alternative Derivation of the MADMs

We consider
$\left\langle\mathbf{\Omega}n\right|\rho(t)\left|n\mathbf{\Omega}\right\rangle$,
which is the probability of finding a molecule in vibronic state $n$ at the
orientation $\mathbf{\Omega}={\phi,\theta,\chi}$. Using Eq. 5 and 6 in Eq. 9,
we find

$\begin{split}&\left\langle\mathbf{\Omega}n\right|\hat{\rho}(t)\left|n\mathbf{\Omega}\right\rangle=\sum_{KQS}\frac{(2K+1)}{8\pi^{2}}\sum_{J_{n}K_{n}M_{n}}\sum_{J^{\prime}_{n}K^{\prime}_{n}M^{\prime}_{n}}\rho_{{\omega}_{n}{\omega}^{\prime}_{n^{\prime}}}(t)\\\
&\times\sqrt{(2J_{n}+1)(2J_{n^{\prime}}+1)}(-1)^{(M_{n}-K_{n})}\left(\begin{array}[]{c
c c}J_{n}&J^{\prime}_{n}&K\\\
-M_{n}&M^{\prime}_{n}&Q\end{array}\right)\left(\begin{array}[]{c c
c}J_{n}&J^{\prime}_{n}&K\\\
-K_{n}&K^{\prime}_{n}&S\end{array}\right)D^{K*}_{QS}(\mathbf{\Omega}).\end{split}$
(B.1)

The matrix elements of the AMCOs are easily found using Eq. 7. Comparing the
matrix element
$\left\langle
J^{\prime}_{n^{\prime}}K^{\prime}_{n^{\prime}}M^{\prime}_{n^{\prime}}n^{\prime}\right|A^{K}_{QS}\left|nJ_{n}K_{n}M_{n}\right\rangle$
to the second line of the above equation for
$\left\langle\mathbf{\Omega}n\right|\rho(t)\left|n\mathbf{\Omega}\right\rangle$
gives,

$\left\langle\mathbf{\Omega}n\right|\hat{\rho}(t)\left|n\mathbf{\Omega}\right\rangle=\sum_{KQS}D^{K*}_{QS}(\mathbf{\Omega})\sum_{J_{n}K_{n}M_{n}}\sum_{J^{\prime}_{n}K^{\prime}_{n}M^{\prime}_{n}}\rho_{{\omega}_{n}{\omega}^{\prime}_{n}}(t)\left\langle
J^{\prime}_{n}K^{\prime}_{n}M^{\prime}_{n}n\right|A^{K}_{QS}\left|nJ_{n}K_{n}M_{n}\right\rangle.$
(B.2)

The double sum over rotational states above is the definition of the ADMs from
Eq. 14,

$\left\langle\mathbf{\Omega}n\right|\hat{\rho}(t)\left|n\mathbf{\Omega}\right\rangle=\sum_{KQS}A^{K}_{QS}(n,n,t)D^{K*}_{QS}(\mathbf{\Omega}),$
(B.3)

Which is identical to Eq. 18, the diagonal elements of the lab frame density
matrix. An expression identical to Eq. 19 for
$\left\langle\mathbf{\Omega}n^{\prime}\right|\hat{\rho}(t)\left|n\mathbf{\Omega}\right\rangle$
can be derived following the same procedure giving the EADMs and off-diagonal
elements of the lab frame density matrix.

##  Appendix C Calculation of one-electron difference densities

The orientation- and time-dependent one-electron difference density $\Delta
p(\boldsymbol{r}_{1},\mathbf{\Omega},t)$ (Eqaution 39) may be expressed in
terms of the molecular frame electronic wave functions
$\Phi_{\alpha}(\boldsymbol{r}_{1},\dots,\boldsymbol{r}_{N})$ and density
matrix elements $\rho_{\alpha\alpha^{\prime}}(\mathbf{\Omega},t)$ as

$\Delta
p(\boldsymbol{r}_{1},\mathbf{\Omega},t)=\sum_{\alpha\alpha^{\prime}}\rho_{\alpha\alpha^{\prime}}(\mathbf{\Omega},t)p_{\alpha\alpha^{\prime}}(\boldsymbol{r}_{1})-p_{00}(\boldsymbol{r}_{1}),$
(C.1)

where $p_{\alpha\alpha^{\prime}}(\boldsymbol{r}_{1})$ denotes the one-electron
reduced (transition) densities,

$p_{\alpha\alpha^{\prime}}(\boldsymbol{r}_{1})=N\int d\boldsymbol{r}_{2}\cdots
d\boldsymbol{r}_{N}\Phi_{\alpha}^{*}(\boldsymbol{r}_{1},\dots,\boldsymbol{r}_{N})\Phi_{\alpha^{\prime}}(\boldsymbol{r}_{1},\dots,\boldsymbol{r}_{N}).$
(C.2)

In practice, the one-electron reduced (transition) densities are expanded in
terms of a single-particle basis $\\{\varphi_{p}(\boldsymbol{r}_{1})\\}$,

$p_{\alpha\alpha^{\prime}}(\boldsymbol{r}_{1})=\sum_{pq}d_{pq}^{(\alpha,\alpha^{\prime})}\varphi_{p}^{*}(\boldsymbol{r}_{1})\varphi_{q}(\boldsymbol{r}_{1}),$
(C.3)

where the one-electron reduced (transition) density matrices
$d_{pq}^{(\alpha,\alpha^{\prime})}$ are defined as

$d_{pq}^{(\alpha,\alpha^{\prime})}=\left\langle\Phi_{\alpha}\middle|\hat{a}_{p}^{\dagger}\hat{a}_{q}\middle|\Phi_{\alpha^{\prime}}\right\rangle,$
(C.4)

with $\hat{a}_{p}^{\dagger}$ ($\hat{a}_{p}$) denoting the elementary fermionic
creation (annihilation) operators corresponding to the single-particle basis
$\\{\varphi_{p}(\boldsymbol{r}_{1})\\}$. The
$\boldsymbol{d}^{(\alpha,\alpha^{\prime})}$ were computed at the p-DFT/MRCI
level of theory using the def2-TZVP basis.

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