# 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.

## Formalism¶

ePolyScat matrix elements:

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

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

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

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.