Methods¶

07/02/20

Each notebook on ePSdata shows various aspects of the results from an ePolyScat computation. The main sections of the notebooks are self-describing. Some additional details on the methods used are given here.

For more infomation on the ePolyScat results, see the ePS manual, and refs. therein.
Additional computations make use of ePSproc codes, and the ePS matrix elements, to compute values such as \(\beta_{L,M}\) for molecular and lab frames.
For more information on plotting methods, see the `ep.lmPlot demo <https://epsproc.readthedocs.io/en/latest/demos/ePSproc_demo_matE_plotting_Feb2020.html>`__.

Formalism¶

ePolyScat matrix elements:

\begin{equation} I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)=\langle\Psi_{i}^{p_{i},\mu_{i}}|\hat{d_{\mu}}|\Psi_{f}^{p_{f},\mu_{f}}\varphi_{klm}^{(-)}\rangle\label{eq:I} \end{equation}

MFPADs are given by:

\begin{equation} T_{\mu_{0}}^{p_{i}\mu_{i},p_{f}\mu_{f}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})=\sum_{l,m,\mu}I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)Y_{lm}^{*}(\theta_{\hat{k}},\phi_{\hat{k}})D_{\mu,\mu_{0}}^{1}(R_{\hat{n}})\label{eq:TMF} \end{equation}

\begin{equation} I_{\mu_{0}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})=\frac{4\pi^{2}E}{cg_{p_{i}}}\sum_{\mu_{i},\mu_{f}}|T_{\mu_{0}}^{p_{i}\mu_{i},p_{f}\mu_{f}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})|^{2}\label{eq:MFPAD} \end{equation}

In this formalism:

\(I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)\) is the radial part of the dipole matrix element, determined from the initial and final state electronic wavefunctions \(\Psi_{i}^{p_{i},\mu_{i}}\)and \(\Psi_{f}^{p_{f},\mu_{f}}\), photoelectron wavefunction \(\varphi_{klm}^{(-)}\) and dipole operator \(\hat{d_{\mu}}\). Here the wavefunctions are indexed by irreducible representation (i.e. symmetry) by the labels \(p_{i}\) and \(p_{f}\), with components \(\mu_{i}\) and \(\mu_{f}\) respectively; \(l,m\) are angular momentum components, \(\mu\) is the projection of the polarization into the MF (from a value \(\mu_{0}\) in the LF). Each energy and irreducible representation corresponds to a calculation in ePolyScat.
\(T_{\mu_{0}}^{p_{i}\mu_{i},p_{f}\mu_{f}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})\) is the full matrix element (expanded in polar coordinates) in the MF, where \(\hat{k}\) denotes the direction of the photoelectron \(\mathbf{k}\)-vector, and \(\hat{n}\) the direction of the polarization vector \(\mathbf{n}\) of the ionizing light. Note that the summation over components \(\{l,m,\mu\}\) is coherent, and hence phase sensitive.
\(Y_{lm}^{*}(\theta_{\hat{k}},\phi_{\hat{k}})\) is a spherical harmonic.
\(D_{\mu,\mu_{0}}^{1}(R_{\hat{n}})\) is a Wigner rotation matrix element, with a set of Euler angles \(R_{\hat{n}}=(\phi_{\hat{n}},\theta_{\hat{n}},\chi_{\hat{n}})\), which rotates/projects the polarization into the MF .
\(I_{\mu_{0}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})\) is the final (observable) MFPAD, for a polarization \(\mu_{0}\) and summed over all symmetry components of the initial and final states, \(\mu_{i}\) and \(\mu_{f}\). Note that this sum can be expressed as an incoherent summation, since these components are (by definition) orthogonal.
\(g_{p_{i}}\) is the degeneracy of the state \(p_{i}\).

See: Toffoli, D., Lucchese, R. R., Lebech, M., Houver, J. C., & Dowek, D. (2007). Molecular frame and recoil frame photoelectron angular distributions from dissociative photoionization of NO2. The Journal of Chemical Physics, 126(5), 054307. https://doi.org/10.1063/1.2432124

From the matrix elements other quantities can be computed. The \(\beta_{L,M}\) parameters (MF) are defined as:

:nbsphinx-math:`begin{eqnarray} beta_{L,-M}^{mu_{i},mu_{f}} & = & sum_{l,m,mu}sum_{l’,m’,mu’}(-1)^{M}(-1)^{m}(-1)^{(mu’-mu_{0})}left(frac{(2l+1)(2l’+1)(2L+1)}{4pi}right)^{1/2}left(begin{array}{ccc} l & l’ & L\ 0 & 0 & 0 end{array}right)left(begin{array}{ccc} l & l’ & L\ -m & m’ & -M end{array}right)nonumber \

& times & sum_{P,R’,R}(2P+1)(-1)^{(R’-R)}left(begin{array}{ccc}

1 & 1 & P\ mu & -mu’ & R’ end{array}right)left(begin{array}{ccc} 1 & 1 & P\ mu_{0} & -mu_{0} & R end{array}right)D_{-R’,-R}^{P}(R_{hat{n}})I_{l,m,mu}^{p_{i}mu_{i},p_{f}mu_{f}}(E)I_{l’,m’,mu’}^{p_{i}mu_{i},p_{f}mu_{f}*}(E) end{eqnarray}`

See the ePSproc docs on ``ep.mfblm()` <https://epsproc.readthedocs.io/en/latest/modules/epsproc.MFBLM.html>`__ for further details, and this demo notebook.