wigners 0.2.1

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 function:

from  wigners import wigner_3j, clebsch_gordan

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

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

From rust

Add this crate to your Cargo.toml dependencies section:

wigners = "0.2"

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);

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 Intel Core i7-4870HQ CPU:

angular momentum	wigners (this)	wigner-symbols v0.5	WignerSymbols.jl v2.0	wigxjpf v1.11	sympy v1.9
4	0.470 ms	28.2 ms	3.08 ms	0.478 ms	83.8 ms
8	9.84 ms	867 ms	66.2 ms	14.5 ms	3.50 s
12	73.4 ms	7.35 s	698 ms	122 ms	64.2 s
16	342 ms	36.2 s	3.20 s	624 ms	/
20	1.14 s	/	11.1 s	2.18 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 distributed under LGPL (this crate is distributed under Apache/MIT);
• 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

However, while this crate should be able to compute winger 3j coefficients up to relatively high angular momentum, it does not use arbitrary precision integers and might fail for very high value. This crate was validated up to l=100, which is more than enough for my use case.

License

This crate is distributed under both the MIT license and the Apache 2.0 license.