Glossary

Glossary#

MF#: Molecular frame (MF) - coordinate system referenced to the molecule, usually with the z-axis corresponding to the highest symmetry axis. See Sect. 3.3.3 for further details.
LF#: Laboratory or lab frame (LF) - coordinate system referenced to the laboratory frame, usually with the z-axis corresponding to the laser field polarization. For circularly or elliptically polarized light the propagation direction is conventionally used for the z-axis. In some cases a different z-axis may be chosen, e.g. as defined by a detector. See Sect. 3.3.3 for further details.
AF#: Aligned frame (AF) - coordinate system referenced to molecular alignment axis or axes. For 1D alignment, the z-axis usually corresponds to the alignment field polarization, and hence may be identical to the standard LF definition, although is usually reserved for use in cases where there is some molecular alignment. For the limiting case of an isotropic distribution, the AF and (traditional) LF are identical. For high degrees of (3D) alignment the AF may approach the MF in the ideal case, although will usually be limited by the symmetry of the system. See Sect. 3.3.3 for further details.
PADs#: Photoelectron angular distributions (PADs), often with a prefix denoting the reference frame, e.g. LFPADs, MFPADs (sometimes also hypenated, e.g. LF-PADs). Usage is often synonymous with the associated anisotropy paramters (or “betas”).
anisotropy paramters#: Expansion parameters \(\beta_{L,M}\) for an expansion in spherical harmonics (or similar basis sets of angular momentum functions in polar coordinates), e.g. Eq. (3.37). Often referred to simply as “beta parameters”, and may be dependent on various properties, e.g. \(\beta_{L,M}(\epsilon,t...)\). Herein upper-case \(L,M\) usually refer to observables or the general case, whilst lower-case \((l,m)\) usually refer specifically to the photoelectron wavefunction partial waves (see partial-wave expansion), and \((l,\lambda)\) usually denote these terms referenced specifically to the molecular frame.
ADMs#: Expansion parameters \(A_{Q,S}^{K}(t)\) for describing a molecular ensemble alignment described as a set of axis distribution moments, usually expanded as Wigner rotation matrix element, spherical harmonics or Legendre polynomial functions. See Sect. 3.5 for details.
axis distribution moments#: See ADMs.
MS#: Molecular symmetry group. Symmetry group classification of a molecule, isomorphic to the point group in rigid molecules. See Bunker and Jensen [89] for discussion.
PG#: Point group. Symmetry group classification of a molecule, strictly only applicable to rigid systems. See MS for more general case.
HOMO#: Highest occupied molecular orbital. Short-hand for the outermost (highest energy) valence orbital, also often used in the form HOMO-n to number lower-lying orbitals in reverse energetic order, e.g. HOMO-1 for the penultimate valence orbital.
VMI#: Velocity-map imaging. Experimental technique for measuring energy and angle-resolved photoelectron “images”.
RWP#: Rotational wavepacket. A purely rotational wavepacket (superposition of rotational eigenstates) in a molecular system, typically created via cascaded Raman interaction with a (relatively) strong IR pulse (\(>10^{12}\)~Wcm\(^{-2}\)). The resulting time-dependent molecular axis distribution can be described by a set of ADMs.
partial-wave expansion#: General term for an expansion of a wavefunction in a spherical-wave basis in scattering theory, typically spherical harmonics \(Y_{l,m}\), where the spherical harmonics are the partial wave basis set, and specific \(\psi_{l,m}\) terms can be referred to as partial waves - see for example Refs. [145, 146, 147]. Note conventional use of lower-case \(l,m\) for these components, whilst upper-case \(L,M\) are usually used for labelling harmonics pertaining to observable quantities (see anisotropy paramters).
partial-waves#: See partial-wave expansion.
channel functions#: Geometric (angular-momentum) coupling parameters in the tensor formulation of photoionzation, denoted by \(\varUpsilon_{L,M}^{u,\zeta\zeta'}\) herein. See Eq. (3.13). These can be regarded as an alternative form of the more traditional geometric coupling parameters (Eq. (3.9)). See also geometric coupling parameters.
geometric coupling parameters#: Geometric (angular-momentum) coupling parameters in photoionization, comprising all angular-momentum coupling terms. Denoted \(\gamma_{l,m}\) herein (Eq. (3.6)), and \(\gamma_{\alpha\alpha_{+}l\lambda ml'\lambda'm'}\) for the coherent square of these terms (Eq. (3.9)). See also channel functions.
radial matrix elements#: General term for the radial part of the ionization matrix elements, after separation into radial and angular parts. Although this type of separation may be applied in many cases, herein this term always refers specifically to the radial (or reduced) photoionization dipole matrix elements. These are denoted herein as \(\mathbf{r}_{k,l,m}\) (see Eqs. (3.6), (3.7)), and also appear as \(\mathbb{I}^{\zeta\zeta'}\) for the coherent square of these terms in the channel functions (tensor) form, see Eq. (3.13). These complex matrix elements are the unknowns to be determined in quantum metrology with photoelectons fitting or reconstruction problems.
symmetrized harmonics#: A basis set of spherical harmonics expanded/defined for a given point-group symmetry. See Sect. 3.6.2, particularly Eq. (3.38), for details. Other symmetrized functions may assume such a basis set, or explicitly incorporate symmetry parameters/weightings directly, as is the case for symmetrized radial matrix elements which incorporate symmetry parameters \(b_{hl\lambda}^{\Gamma\mu}\) into the value of the matrix elements.
frame rotation#: General term for the rotation of one frame of reference (corresponding to a set of \(x,y,z\) axes), e.g. as defined by an electric field vector in the laboratory, to another, e.g. as defined by a molecular axis. Usually specified herein in terms of a set of Euler angles \(R_{\hat{n}}=\{\chi,\Theta,\Phi\}\), and can also be schematically denoted as, e.g., \((x,y,z)\leftarrow(x',y',z')\).
Euler angles#: A set of angles \(R_{\hat{n}}=\{\chi,\Theta,\Phi\}\) defining a frame rotation. For discussion see Wikipedia[148] and Zare, Chpt. 3 [83].
Wigner rotation matrix elements#: A function defining a basis set for rotations in three-dimensions (\(SO(3)\)). Also known as “Wigner D-matrix elements”. For discussion see Wikipedia[149] and Zare, Chpt. 3 [83].
molecular alignment#: General term for describing the case where molecules have some alignment in the LF, as compared to an isotropic distribution. Defined herein terms of molecular axis distribution moments and associated parameters \(A_{Q,S}^{K}(t)\). See Sect. 3.5 for details.
bootstrap retrieval protocol#: General term of a retrieval method which determines successively more complex properties of a system in multiple steps, each of which builds on the previous step and adds complexity. Herein, used for the “generalised bootstrapping” method for radial matrix elements retrieval from photoionzation data. For general useage, see wikipedia’s Bootstrapping page.