\(\beta_{L,M}\) calculations demo

19/09/19

Source notebook on Github.

Basic IO

* pyevtk not found, VTK export not available.

* ePSproc readMatEle(): scanning files for DumpIdy segments (matrix elements)

* Scanning dir
D:codegithubePSprocdata
Found 2 .out file(s)


* Reading ePS output file:  D:codegithubePSprocdatan2_3sg_0.1-50.1eV_A2.inp.out
Expecting 51 energy points.
Expecting 2 symmetries.
Expecting 102 dumpIdy segments.
Found 102 dumpIdy segments (sets of matrix elements).

Processing segments to Xarrays...
Processed 102 sets of matrix elements (0 blank)

* Reading ePS output file:  D:codegithubePSprocdatano2_demo_ePS.out
Expecting 1 energy points.
Expecting 3 symmetries.
Expecting 3 dumpIdy segments.
Found 3 dumpIdy segments (sets of matrix elements).

Processing segments to Xarrays...
Processed 3 sets of matrix elements (0 blank)

Formalism

The \(\beta_{L,M}\) parameters are defined as:

\[\begin{split}\begin{eqnarray} \beta_{L,-M}^{\mu_{i},\mu_{f}} & = & \sum_{l,m,\mu}\sum_{l',m',\mu'}(-1)^{M}(-1)^{m}(-1)^{(\mu'-\mu_{0})}\left(\frac{(2l+1)(2l'+1)(2L+1)}{4\pi}\right)^{1/2}\left(\begin{array}{ccc} l & l' & L\\ 0 & 0 & 0 \end{array}\right)\left(\begin{array}{ccc} l & l' & L\\ -m & m' & -M \end{array}\right)\nonumber \\ & \times & \sum_{P,R',R}(2P+1)(-1)^{(R'-R)}\left(\begin{array}{ccc} 1 & 1 & P\\ \mu & -\mu' & R' \end{array}\right)\left(\begin{array}{ccc} 1 & 1 & P\\ \mu_{0} & -\mu_{0} & R \end{array}\right)D_{-R',-R}^{P}(R_{\hat{n}})I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)I_{l',m',\mu'}^{p_{i}\mu_{i},p_{f}\mu_{f}*}(E) \end{eqnarray}\end{split}\]

Calculations use ep.mfblm(), which will calculate all values at each energy point for the supplied dataset. This may take a while in some cases due to multiple nested sums - this code will be parallelised in future.

\(N_2\) mutli-E

Calculate \(\beta_{LM}\) as function of energy.

Calculating MFBLMs for 81 pairs... Eke = 0.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 1.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 2.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 3.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 4.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 5.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 6.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 7.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 8.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 9.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 10.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 11.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 12.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 13.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 14.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 15.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 16.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 17.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 18.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 19.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 20.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 21.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 22.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 23.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 24.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 25.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 26.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 27.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 28.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 29.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 30.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 31.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 32.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 33.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 34.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 35.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 36.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 37.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 38.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 39.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 40.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 41.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 42.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 43.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 44.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 45.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 46.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 47.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 48.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 49.1 eV, eAngs = ([0, 0, 0])
Calculating MFBLMs for 81 pairs... Eke = 50.1 eV, eAngs = ([0, 0, 0])
Elapsed time = 42.335490226745605 seconds, for 51 energy points.

<xarray.DataArray (Euler: 1, Eke: 51, BLM: 121)>
array([[[2.301322 +0.j, 0.       +0.j, ...,      nan+nanj,      nan+nanj],
        [2.382333 +0.j, 0.       +0.j, ...,      nan+nanj,      nan+nanj],
        ...,
        [0.092948 +0.j, 0.       +0.j, ..., 0.       +0.j, 0.       +0.j],
        [0.095428 +0.j, 0.       +0.j, ..., 0.       +0.j, 0.       +0.j]]])
Coordinates:
  * Euler    (Euler) MultiIndex
  - P        (Euler) int64 0
  - T        (Euler) int64 0
  - C        (Euler) int64 0
  * BLM      (BLM) MultiIndex
  - l        (BLM) int64 0 1 1 1 2 2 2 2 2 3 3 ... 10 10 10 10 10 10 10 10 10 10
  - m        (BLM) int64 0 -1 0 1 -2 -1 0 1 2 -3 -2 ... 0 1 2 3 4 5 6 7 8 9 10
  * Eke      (Eke) float64 0.1 1.1 2.1 3.1 4.1 5.1 ... 46.1 47.1 48.1 49.1 50.1
Attributes:
    Lmax:      11
    Targ:      SG
    QNs:       ['m', 'l', 'mu', 'ip', 'it', 'Value']
    dataType:  BLM
    file:      n2_3sg_0.1-50.1eV_A2.inp.out
    fileBase:  D:codegithubePSprocdata
    thres:     0.0001
    sumDims:   ('l', 'm', 'mu', 'Cont', 'Targ', 'Total', 'it')
    selDims:   [('Type', 'L')]

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../_images/output_11_1.png 
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../_images/output_12_1.png

MFPADs: calculate & plot from \(\beta_{LM}\)

Plotting with mpl
../_images/output_15_1.png 
[<Figure size 432x288 with 1 Axes>]

N2 test data, MFPADs vs E

<xarray.plot.facetgrid.FacetGrid at 0x1fd4b91f048>
../_images/output_16_2.png 
Plotting with mpl
Data dims: ('Euler', 'Eke', 'Theta', 'Phi'), subplots on Eke
../_images/output_17_1.png ../_images/output_17_2.png ../_images/output_17_3.png ../_images/output_17_4.png ../_images/output_17_5.png 
[<Figure size 432x288 with 1 Axes>,
 <Figure size 432x288 with 1 Axes>,
 <Figure size 432x288 with 1 Axes>,
 <Figure size 432x288 with 1 Axes>,
 <Figure size 432x288 with 1 Axes>]

Plotting with pl

\(NO_2\) (x,y,z) polarizations

Benchmark results, single energy, multiple polarization geometries.

Calculate

Calculating MFBLMs for 12544 pairs... Eke = 0.81 eV, eAngs = ([0. 0. 0.])
Elapsed time = 124.42453789710999 seconds
Calculating MFBLMs for 12544 pairs... Eke = 0.81 eV, eAngs = ([0.         1.57079633 0.        ])
Elapsed time = 126.24425601959229 seconds
Calculating MFBLMs for 12544 pairs... Eke = 0.81 eV, eAngs = ([1.57079633 1.57079633 0.        ])
Elapsed time = 125.88551211357117 seconds

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../_images/output_23_1.png

MFPADs

Plotting with mpl
Data dims: ('Euler', 'Eke', 'Theta', 'Phi'), subplots on Euler
../_images/output_25_1.png ../_images/output_25_2.png ../_images/output_25_3.png 
[<Figure size 432x288 with 1 Axes>,
 <Figure size 432x288 with 1 Axes>,
 <Figure size 432x288 with 1 Axes>]

Benchmark vs. “direct” results

Compare against results from ep.mfpad, see main “ePSproc demo notebook” for calculation details.

MFPADs for test NO2 dataset (single energy, (z,x,y) pol states)
Plotting with mpl
../_images/output_28_1.png 
Plotting with mpl
../_images/output_28_3.png 
Plotting with mpl
../_images/output_28_5.png

(Visual comparison looks OK, still some benchmarks to to for numerical re/im comparisons, which currently show some differences.)