Compute Wigner 3j and Clebsch-Gordan coefficients

# Calculation of Wigner symbols and related constants

This package computes Wigner 3j coefficients and Clebsch-Gordan coefficients in pure Rust. The calculation is based on the prime factorization of the different factorials involved in the coefficients, keeping the values in a rational root form (sign * \sqrt{s / n}) for as long as possible. This idea for the algorithm is described in:

H. T. Johansson and C. Forssén, SIAM Journal on Scientific Compututing 38 (2016) 376-384

This implementation takes a lot of inspiration from the WignerSymbols Julia implementation (and even started as a direct translation of it), many thanks to them! This package is available under the same license as the Julia package.

## Usage

### From python

pip install wigners


And then call one of the exported function:

import wigners

w3j = wigners.wigner_3j(j1, j2, j3, m1, m2, m3)

cg = wigners.clebsch_gordan(j1, m1, j2, m1, j3, m3)

# full array of Clebsch-Gordan coefficients, computed in parallel
cg_array = wigners.clebsch_gordan_array(ji, j2, j3)

# we have an internal cache for recently computed CG coefficients, if you
# need to clean it up you can use this function
wigners.clear_wigner_3j_cache()


### From rust

Add this crate to your Cargo.toml dependencies section:

wigners = "0.3"


And then call one of the exported function:

let w3j = wigners::wigner_3j(j1, j2, j3, m1, m2, m3);

let cg = wigners::clebsch_gordan(j1, m1, j2, m1, j3, m3);

wigners::clear_wigner_3j_cache();


## Limitations

Only Wigner 3j symbols for full integers (no half-integers) are implemented, since that's the only part I need for my own work.

6j and 9j symbols can also be computed with this approach; and support for half-integers should be feasible as well. I'm open to pull-request implementing these!

## Benchmarks

This benchmark measure the time to compute all possible Wigner 3j symbols up to a fixed maximal angular momentum; clearing up any cached values from previous angular momentum before starting the loop. In pseudo code, the benchmark looks like this:

if cached_wigner_3j:
clear_wigner_3j_cache()

# only measure the time taken by the loop
start = time.now()
for j1 in range(max_angular):
for j2 in range(max_angular):
for j3 in range(max_angular):
for m1 in range(-j1, j1 + 1):
for m2 in range(-j2, j2 + 1):
for m3 in range(-j3, j3 + 1):
w3j = wigner_3j(j1, j2, j3, m1, m2, m3)

elapsed = start - time.now()


Here are the results on an Apple M1 Max (10 cores) CPU:

angular momentum wigners (this) wigner-symbols v0.5 WignerSymbols.jl v2.0 wigxjpf v1.11 sympy v1.11
4 0.190 ms 7.50 ms 2.58 ms 0.228 ms 28.7 ms
8 4.46 ms 227 ms 47.0 ms 7.36 ms 1.36 s
12 34.0 ms 1.94 s 434 ms 66.2 ms 23.1 s
16 156 ms 9.34 s 1.98 s 333 ms /
20 531 ms / 6.35 s 1.21 s /

A second set of benchmarks checks computing Wigner symbols for large j, with the corresponding m varying from -10 to 10, i.e. in pseudo code:

if cached_wigner_3j:
clear_wigner_3j_cache()

# only measure the time taken by the loop
start = time.now()
for m1 in range(-10, 10 + 1):
for m2 in range(-10, 10 + 1):
for m3 in range(-10, 10 + 1):
w3j = wigner_3j(j1, j2, j3, m1, m2, m3)

elapsed = start - time.now()

(j1, j2, j3) wigners (this) wigner-symbols v0.5 WignerSymbols.jl v2.0 wigxjpf v1.11 sympy v1.11
(300, 100, 250) 38.7 ms 16.5 ms 32.9 ms 7.60 ms 2.31 s

To run the benchmarks yourself on your own machine, execute the following commands:

cd benchmarks
cargo bench # this gives the results for wigners, wigner-symbols and wigxjpf

python sympy-bench.py # this gives the results for sympy

julia wigner-symbol.jl # this gives the results for WignerSymbols.jl


## Comparison to wigner-symbols

There is another Rust implementation of wigner symbols: the wigner-symbols crate. wigner-symbols also implements 6j and 9j symbols, but it was not usable for my case since it relies on rug for arbitrary precision integers and through it on the GMP library. The GMP library might be problematic for you for one of these reason:

• it is relatively slow (see the benchmarks above)
• it is written in C and C++; and as such is hard to cross-compile or compile to WASM;
• it does not support the MSVC compiler on windows, only the GNU compilers

As you can see in the benchmarks above, this usage of GMP becomes an advantage for large j, where the algorithm used in this crate does not scale as well.

## Project details

### Source Distribution

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