Calculate CG coefficient, Racah coefficient, Wigner 3j, 6j, 9j coefficient, Moshinsky bracket, etc. Some formula please see CGcoefficient.jl.
This is a head-only library, and only one file WignerSymbol.hpp is needed.
#include "WignerSymbol.hpp"
using namespace util;
int djmax = 21;
wigner_init(djmax, "2bjmax", 6);
double x = wigner_6j(dj1, dj2, dj3, dj4, dj5, dj6);For quite large quantum number, the package will give wrong answer, since it use float number arithmetic. Please see wigner-benchmark for the error estimate and performance benchmark.
// reserve binomial table
void wigner_init(int num, std::string type, int rank);
// fast access binomial table, it may return 0 for very large `n`
double fast_binomial(int n, int k);
// CG coefficient
double CG(int dj1, int dj2, int dj3, int dm1, int dm2, int dm3);
// CG coefficient for two spin-1/2, equivalent to `CG(1, 1, 2*S, ds1, ds2, ds1+ds2)`, and faster
double CGspin(int dm1, int dm2, int S);
// <S12,M12|1/2,m1;1/2,m2><S,M|S12,M12;1/2,m3>
double CG3spin(int dm1, int dm2, int dm3, int S12, int dS);
// CG coefficient with m1 == m2 == m3 == 0
double CG0(int j1, int j2, int j3);
// Wigner 3j symbol
double wigner_3j(int dj1, int dj2, int dj3, int dm1, int dm2, int dm3);
// Wigner 6j symbol
double wigner_6j(int dj1, int dj2, int dj3, int dj4, int dj5, int dj6);
// Racah coefficient
double Racah(int dj1, int dj2, int dj3, int dj4, int dj5, int dj6);
// Wigner 9j symbol
double wigner_9j(int dj1, int dj2, int dj3, int dj4, int dj5, int dj6, int dj7, int dj8, int dj9);
// normalized Wigner 9j symbol
double wigner_norm9j(int dj1, int dj2, int dj3, int dj4, int dj5, int dj6, int dj7, int dj8, int dj9);
// LS-coupling to jj-coupling transformation coefficient
double lsjj(int l1, int l2, int dj1, int dj2, int L, int S, int J);
// Wigner d-function <j,m1|exp(i*beta*jy)|j,m2>
double dfunc(int dj, int dm1, int dm2, double beta);
// Moshinsky bracket，Ref: Buck et al. Nuc. Phys. A 600 (1996) 387-402
double Moshinsky(int N, int L, int n, int l, int n1, int l1, int n2, int l2, int lambda, double tan_beta = 1.0);Becase the angular momentum qunatum number can be half integers, people often use double of the exact quantum number as arguments. In this library, we also use the same convention. However, this library contains some other functions like Moshinsky, which only needs orbital quantum number, using doubled arguments is not needed.
So we take such convention: the arguments starts with d-character means double of the quantum number, while other arguments represent just the equivalent quantum number.
So if you want to calculate <10|1/2,1/2;1/2,-1/2>, you should call like this,
double x = CG(1, 1, 2, 1, -1, 0);
double y = CGspin(1, -1, 1); // we use `S`, not `dS`, because `S` can only be `0, 1`We calculate the Wigner Symbols with binomials, and we will store some binomials first, then when we need one binomial, we just load it. In this package, the binomial function is only valid in the stored range. If you call a binomial function out of the range, it just gives you 0.
The binomial table is stored in the WignerSymbols class。 We define a globle variable wigner of the class to serve for all the functions. (The inline variable needs the c++17 feature.)
inline WignerSymbols wigner;
inline void wigner_init(int num, std::string type, int rank) { wigner.reserve(num, type, rank); }When constructing the WignerSymbols object, it will store binomials from binomial(0, 0) to binomial(67, 33) (All can be exactly represented with a uint64_t). And one can use wigner_init function to extend the binomial table.
The wigner_init actually extends the maximum n for binomial(n, k). To help users to make sure which nmax is safe for all of the following calculations, it defines several modes.
wigner_init(21, "2bjmax", 6);This means the maximum single particle angular momentum is 21/2, and thus the maximum two-body coupled angular momentum is 21, and the rank = 6 means you only need to calculate CG and/or 6j symbols, you don't need to calculate 9j symbols.
The rank can only be 3, 6, 9, which respectively means wigner_3j & CG, wigner_6j & Racah and wigner_9j level calculation.
The "2bjmax" mode means your calculation only consider two-body coupling, and no three-body coupling. This mode assumes that in all these coefficients, at least one of the angular momentun is a single particle angular momentum, thus in this example no larger than 21/2. With this assumption, "2bjmax" mode will use less memory than "Jmax" mode.
"Jmax" means the global maximum angular momentum, for every parameters.
wigner_init(21, "Jmax", 9);This means in all the calculations, Jmax = 21, and we calculate upto 9j symbols.
In the "Jmax" mode, it is always safe with out any assumption. Even having three-body coupling, you just need to use the maximum three body coupled angular momentum as Jmax, although it will cost more memory.
Actually, the memory used for store the binomials is not very large. For example, in the "Jmax" mode, and Jmax = 200 for 9j calculations, the memory cost is just 2MB.
The "nmax" mode directly set nmax of the binomial table, and the rank parameter is ignored. This maybe useful when you only want to calculate binomials using this library.
The following table shows the exact nmax setted in different condition. See Estimate-the-capacity for details.
| Calculate range | CG & 3j | 6j & Racah | 9j | |
|---|---|---|---|---|
meaning of type |
type\rank |
3 | 6 | 9 |
| max angular momentum | "Jmax" |
3*Jmax+1 |
4*Jmax+1 |
5*Jmax+1 |
| max two-body coupled angular momentum | "2bjmax" |
2*jmax+1 |
3*jmax+1 |
4*jmax+1 |
| max binomial | "nmax" |
nmax |
namx |
nmax |
The wigner_init function is not thread safe. So you shuld not call winger_init function dymanically in a multi-threading program. The correct way to use this package is find the maximum angular momentum quantum number in you system, and call wigner_init at the beginning of the code, and then don't call it any more.
- T. Engeland and M. Hjorth-Jensen, the Oslo-FCI code. https://github.com/ManyBodyPhysics/CENS.
- A. N. Moskalev D. A. Varshalovich and V. K. Khersonskii, Quantum theory of angular momentum.