# Towards Reconstructing the Molecular Frame for Photoionzation of Polyatomic Molecules from Time Domain Measurements

## Introduction

Measurements in the molecular frame help elucidate the fundamental physics of molecules. The physics of molecules in turn inform the nature of interaction between a number of quantum particles, important for both the understanding of larger systems and that of fundamental quantum phenomena. Measurements made in the laboratory frame from randomly oriented molecules, or a thermal mixture of numerous angular momentum states, result in a loss of information due to incoherent averaging over the orientations, or equivalently, the rotational states. Methods to overcome this either coherently drive a large number of rotational states, or select the ground rotational state from an ensemble. Either scenerio results in spatially anisotropic distribution, with the former approaching the molecular frame as the distribution of rotational states broadens. This is the preferred method in ultrafast physics, since the time resolution allows for measurements at the moment of sharpest alignment. The latter method is typically applied for precision spectroscopic measurements. Recent advances in laser cooling of di- and tr-atomic linear molecules has enabled coherent selection of the ground rotational state, allowing for precise spectroscopic measurements of dissociative molecular states.

## Ionization from Molecular Wavepackets

### The Genral Formulation

We closely follow the general formulation for photoionzation from a evolving molecular wavepacket developed in (Stolow 2008). The time evolving PAD is expanded in a basis set of spherical harmonics,

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

where $$\epsilon$$ is the kinetic energy of the electron and $$\theta_{e}$$ and $$\phi_{e}$$ its polar and azimuthal ejection angles. A right handed Cartesian coordinate system is implied with the ionizing light polarized in the $$Z$$-$$X$$ plane and propagating along the $$Y$$-axis. Body fixed coordinate axes are labeled $$xyz$$, in order of increasing polarizability. In the dipole approximation it can be shown that $$\beta_{L,M}(\epsilon,t)$$ is given by the following expression,

$$\label{eq:beta} \label{eq:beta}\beta_{LM}(\epsilon,t)=\sum_{\zeta\zeta^{\prime}}\sum_{n_{\alpha}n_{\alpha}^{\prime}}D^{n_{\alpha}}_{\zeta}(\epsilon)D^{n_{\alpha}^{\prime}*}_{\zeta^{\prime}}(\epsilon)\left(\sum_{PKQS}\gamma^{\zeta\zeta^{\prime}LM}_{PKQS}A^{K}_{QS}(n_{\alpha},n_{\alpha}^{\prime};t)\right).\\$$

$$n_{\alpha}$$ and $$\zeta$$ represent sets of quantum numbers labeling vibronic Born-Oppenheimer basis functions used to construct the bound and continuum states respectively. $$n_{\alpha}$$=$$\nu_{\alpha}$$,$$\alpha$$, where $$\nu_{\alpha}$$ labels vibrational quanta and $$\alpha$$ the electronic basis state. $$\zeta$$ labels scattering channels and thus consists 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 dipole moment which facilitates ionization, and $$\zeta_{f}$$=$$\Gamma$$,$$\mu$$,$$h$$,$$l$$ labeling basis functions used to construct the wave function of the ionized electron. In the molecular frame this is expressed as

\begin{aligned} \phi(\mathbf{r^{\prime}_{e}},\epsilon;\mathbf{R})=\sum_{\Gamma\mu hl}X^{\Gamma\mu*}_{hl}(\theta^{\prime}_{e},\phi^{\prime}_{e})\psi_{\Gamma\mu hl}(r^{\prime}_{e},\epsilon;\mathbf{R}), \\ X^{\Gamma\mu*}_{hl}(\theta^{\prime}_{e},\phi^{\prime}_{e})=\sum_{l\lambda}b^{\Gamma\mu}_{hl\lambda}Y_{l\lambda}(\theta^{\prime}_{e},\phi^{\prime}_{e}).\\ \end{aligned}

$$\mathbf{R}$$ represent the nuclear coordinates, $$r^{\prime}_{e},\theta^{\prime}_{e},\phi^{\prime}_{e}$$ are coordinates of the electron in the molecular frame, and $$\epsilon$$ its energy. Further, $$X^{\Gamma\mu}_{hl}(\theta^{\prime}_{e},\phi^{\prime}_{e})$$ are symmetry adapted harmonics - a superposition of spherical harmonic functions-$$Y_{l\lambda}(\theta^{\prime}_{e},\phi^{\prime}_{e})$$-that form a basis for the representation $$\Gamma$$ of the molecular point group. $$\mu$$ and $$h$$ are corresponding symmetry labels that replace $$l$$ and its projection on the $$z$$-axis $$\lambda$$ when spherical and cylidrical symmetry are lost. The scattering amplitude for scattering out of an initial state $$n_{\alpha}$$, into a channel $$\zeta$$ is given by $$D^{n_{\alpha}}_{\zeta}(\epsilon)$$, the matrix element of the electronic dipole operator connecting the two states. These are also assumed to contain the amplitudes of the BO basis functions in cases where the BO approximation fails.

The part of Eq. \ref{eq:beta} in parenthesis contains all the angular momentum coupling in, and between the neutral and channel functions. It also entirely determines the time dependence of the PADs. Here we rewrite this as a separate entity,

$$\label{eq:chADMs} \label{eq:chADMs}A^{\zeta\zeta^{\prime}LM}_{n_{\alpha}n_{\alpha}^{\prime}}(t)=\sum_{PKQS}\gamma^{\zeta\zeta^{\prime}}_{PKQS}A^{K}_{QS}(n_{\alpha},n_{\alpha}^{\prime};t).\\$$

Based on considerations of symmetry and conservation of angular momentum, the channel resolved coherence functions (CRCFs) $$A^{\zeta,\zeta^{\prime}}_{n_{\alpha},n_{\alpha}^{\prime}}$$ provide a measure of the sensitivity of a particular scattering channel $$\zeta$$ to a particular quantum beat between component states $$n_{\alpha}$$,$$n_{\alpha}^{\prime}$$ of the neutral wavepacket. Additionally, off-diagonal CRCFs with $$\zeta\neq\zeta^{\prime}$$ quantify the sensitivity of pairs of interfering channels to a quantum beat. Each CRCF is then weighted by the appropriate pair of scattering dipole matrix element to provide its contribution to the measured $$\beta$$ parameter, as in Eq. \ref{eq:beta}. The CRCFs contain a geometric factor $$\gamma^{\zeta\zeta^{\prime}LM}_{PKQS}$$ and the axis distribution moments (ADMs) $$A^{K}_{QS}(t,n_{\alpha},n_{\alpha}^{\prime})$$. $$P$$, $$Q$$, $$K$$, and $$S$$ are angular momentum quantum numbers. $$P=0$$,$$1$$ or $$2$$ results from aggregating the angular momentum transferred from the ionizing photon to the electron. $$K$$ results from adding component rotational of the neutral wavepacket, $$J_{\alpha}$$, and $$S$$ and $$Q$$ from the addition of the projection of $$J_{\alpha}$$ on the $$z$$ and $$Z$$ axis respectively. The factor $$\gamma^{\zeta,\zeta^{\prime}LM}_{PKQS}$$, the explicit expression for which is provided in Appendix A., appropriately couples $$P$$, $$K$$, $$Q$$, $$S$$, $$l$$, $$\lambda$$ and $$q$$ into $$L$$ and $$M$$. The diagonal ADMs ($$n_{\alpha}=n_{\alpha}^{\prime}$$) quantify the molecular axis distribution in a particular vibronic state, while the off-diagonal AMDs quantify the coherence between angular momentum coherence between different vibronic states. These are sums over the state multipoles $$\left<T(n_{\alpha},n_{\alpha}^{\prime})_{KQ}\right>$$ that describe angular momentum coherences in the wavepacket in terms of an effect alignment.