# Geometric methods summary

19/08/21

This notebook summarises the current state of the geometric tensor methods, these are currently used for MF and AF calculations

## Formalism

The equations for the $$\beta_{LM}$$ parameters, in the molecular and lab frames (MF & AF respectively), can be written in terms of geometric tensor paramters.

For the MF, denoted $$\beta_{LM}$$ (full development notes here):

\begin{eqnarray} \beta_{L,-M}^{\mu_{i},\mu_{f}}(E) & = & (-1)^{M}\sum_{P,R',R}(2P+1)^{\frac{1}{2}}{E_{P-R}(\hat{e};\mu_{0})}\sum_{l,m,\mu}\sum_{l',m',\mu'}(-1)^{(\mu'-\mu_{0})}{\Lambda_{R',R}(R_{\hat{n}};\mu,P,R,R')B_{L,-M}(l,l',m,m')}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}

For the AF, denoted $$\bar{\beta}_{LM}$$ (full development notes here):

\begin{eqnarray} \bar{\beta}_{L,-M}^{\mu_{i},\mu_{f}}(E,t) & =(-1)^{M} & \sum_{P,R',R}{[P]^{\frac{1}{2}}}{E_{P-R}(\hat{e};\mu_{0})}\sum_{l,m,\mu}\sum_{l',m',\mu'}(-1)^{(\mu'-\mu_{0})}{\Lambda_{R'}(\mu,P,R')B_{L,S-R'}(l,l',m,m')}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)\sum_{K,Q,S}\Delta_{L,M}(K,Q,S)A_{Q,S}^{K}(t)\label{eq:BLM-tidy-prod-2} \end{eqnarray}

Where $$I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)$$ are the (radial) dipole ionization matrix elements, as a function of energy $$E$$, obtained from an ePolyScat (or other) calculation, defined by a set of partial-waves $$\{l,m\}$$, for polarizations $$\mu$$ and channels (symmetries) labelled by initial and final state indexes $${p_{i}\mu_{i},p_{f}\mu_{f}}$$.

In both cases a set of geometric tensor terms are required, defined below. Note that, in this case, time-dependence arises purely from the $$A_{Q,S}^{K}(t)$$ terms in the AF case, and the electric field term currently describes only the photon angular momentum coupling, time-dependent/shaped fields are not yet supported (as of v1.3.0, but will be soon). Similarly, a time-dependent initial state (e.g. vibrational wavepacket) could also describe a time-dependent MF case, but is currently not included here.

### Electric field term

$$E_{PR}(\hat{e})=[e\otimes e^{*}]_{R}^{P}=[P]^{\frac{1}{2}}\sum_{p}(-1)^{R}\left(\begin{array}{ccc} 1 & 1 & P\\ p & R-p & -R \end{array}\right)e_{p}e_{R-p}^{*}\label{eq:EPR-defn-1}$$

Where $$e_{p}$$ and $$e_{R-p}$$ define the field strengths for the polarizations $$p$$ and $$R-p$$, which are coupled into the spherical tensor $$E_{PR}$$.

Note this currently describes only the photon angular momentum coupling, time-dependent/shaped fields are not yet supported (as of v1.3.0, but will be soon).

### $$B_{L,M}$$ term

The coupling of the partial wave pairs, $$|l,m\rangle$$ and $$|l',m'\rangle$$, into the observable set of $$\{L,M\}$$ is defined by a tensor contraction with two 3j terms.

$$B_{L,M}=(-1)^{m}\left(\frac{(2l+1)(2l'+1)(2L+1)}{4\pi}\right)^{1/2}\left(\begin{array}{ccc} l & l' & L\\ 0 & 0 & 0 \end{array}\right)\left(\begin{array}{ccc} l & l' & L\\ -m & m' & M \end{array}\right)$$

Note for the AF case $$B_{L,S-R'}(l,l',m,m')$$ instead of $$B_{L,-M}(l,l',m,m')$$ for MF case. This allows for all MF projections to contribute (rather than a single specified polarization geometry).

### $$\Lambda$$ Term

Define MF projection term, $$\Lambda_{R',R}(R_{\hat{n}})$$:

$$\Lambda_{R',R}(R_{\hat{n}})=(-1)^{(R')}\left(\begin{array}{ccc} 1 & 1 & P\\ \mu & -\mu' & R' \end{array}\right)D_{-R',-R}^{P}(R_{\hat{n}})$$

This is similar to the $$E_{PR}$$ term, and essentially rotates it into the MF by a set of rotations (Euler angles) defined by $$R_{\hat{n}}$$.

For the AF case, a simplified form is used (since there is no specific orientation/rotation into the MF, and the relations are defined by the molecular axis distribution):

$$\bar{\Lambda}_{R'}=(-1)^{(R')}\left(\begin{array}{ccc} 1 & 1 & P\\ \mu & -\mu' & R' \end{array}\right)\equiv\Lambda_{R',R'}(R_{\hat{n}}=0)$$

### Alignment term

$$\Delta_{L,M}(K,Q,S)=(2K+1)^{1/2}(-1)^{K+Q}\left(\begin{array}{ccc} P & K & L\\ R & -Q & -M \end{array}\right)\left(\begin{array}{ccc} P & K & L\\ R' & -S & S-R' \end{array}\right)$$

The axis distribution moments (ADMs) define the LF in this case, and are given above as a set of parameters $$A_{Q,S}^{K}(t)$$; the coupling between the LF and MF is, effectively, defined by the final term in the AF:

$$\sum_{K,Q,S}\Delta_{L,M}(K,Q,S)A_{Q,S}^{K}(t)$$

Refs for the full AF-PAD formalism above:

1. Reid, Katharine L., and Jonathan G. Underwood. “Extracting Molecular Axis Alignment from Photoelectron Angular Distributions.” The Journal of Chemical Physics 112, no. 8 (2000): 3643. https://doi.org/10.1063/1.480517.

2. Underwood, Jonathan G., and Katharine L. Reid. “Time-Resolved Photoelectron Angular Distributions as a Probe of Intramolecular Dynamics: Connecting the Molecular Frame and the Laboratory Frame.” The Journal of Chemical Physics 113, no. 3 (2000): 1067. https://doi.org/10.1063/1.481918.

3. Stolow, Albert, and Jonathan G. Underwood. “Time-Resolved Photoelectron Spectroscopy of Non-Adiabatic Dynamics in Polyatomic Molecules.” In Advances in Chemical Physics, edited by Stuart A. Rice, 139:497–584. Advances in Chemical Physics. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2008. https://doi.org/10.1002/9780470259498.ch6.

Where [3] has the version as per the full form above (full asymmetric top alignment distribution expansion).

## Numerical implementation

In the current version (v1.3.0, 17/08/21), the above formalism is implemented in the modules geomFunc.mfblmGeom and geomFunc.afblmGeom, defined as a tensor product and making use of direct multiplication of Xarrays corresponding to each term in the eqns. (See geometric method dev pt 1 for more details of the component tensors, source functions are in primarily in the geomFunc.geomCalc module.)

This is easy to follow, and relatively fast, although can be RAM heavy for large sets of matrix elements (despite the relatively sparse arrays). Parallelization is not yet implemented, but should be fairly easy using Xarray’s GroupBy functionality. Fast functions are used on the back-end, although there is likely some significant speed-ups to be had by making use of GPU methods here (and in the tensor handling) - this will be necessary for larger problems, e.g. fitting data.

Additionally, a set of phase convention flags can be set for the calculations. This allows quick testing of phase convention choices and the effects of various phase terms.

The tensor codes have been checked against a few MF/LF results, and so far seem robust.

TODO: more test cases and continuous testing.

## Additional background & development materials

Previous developmental tests for: