Angular provides functions and classes for calculating and storing angular factors (3,6,9-J symbols etc.) More...

Classes
class	CkTable
	Lookup table for C^k and 3j symbols (special m=1/2, q=0 case) More...

class	SixJTable
	Lookup table for Wigner 6J symbols. More...

Functions
constexpr int	jindex_to_twoj (int jindex)
	Converts jindex to 2*j [helper function].

constexpr int	twoj_to_jindex (int twoj)
	Converts 2*j to jindex {1/2, 3/2, 5/2} -> {0, 1, 2} [helper function].

constexpr int	kappa_to_jindex (int ka)
	Converts kappa to jindex {-1, 1, -2} -> {0, 0, 1} [helper function].

template<class A >
int	twojk (const A &a)

template<class A , class B , class C , class D , class E , class F >
double	SixJ (const A &a, const B &b, const C &c, const D &d, const E &e, const F &f)
	"Magic" 6j, for integers and DiracSpinors. Pass int (for k), or DiracSpinor for j. e.g.: (F,F,k,F,F,k) -> {j,j,k,j,j,k}. Do NOT *2! Do NOT pass j or 2j directly!

constexpr int	l_k (int ka)
	returns l given kappa

constexpr int	twoj_k (int ka)
	returns 2j given kappa

constexpr int	parity_k (int ka)
	returns parity [(-1)^l] given kappa

constexpr int	parity_l (int l)
	returns parity [(-1)^l] given l

constexpr int	l_tilde_k (int ka)
	"Complimentary" l := (2j-l) = l+/-1, for j=l+/-1/2

constexpr int	kappa_twojl (int twoj, int l)
	returns kappa, given 2j and l

constexpr int	kappa_twojpi (const int twoj, const int pi)
	returns kappa, given 2*j and parity (+/-1),

constexpr bool	zeroQ (double x, double eps=1.0e-10)
	Checks if a double is within eps of 0.0 [eps=1.0e-10 by default].

constexpr std::uint64_t	kappa_to_kindex (int ka)
	returns kappa_index, given kappa

constexpr int	kindex_to_kappa (std::uint64_t i)
	Returns kappa, given kappa_index.

constexpr int	kindex_to_twoj (std::uint64_t i)
	returns 2j, given kappa_index

constexpr int	kindex_to_l (std::uint64_t i)
	returns l, given kappa_index

constexpr std::uint64_t	states_below_n (int n)
	Returns number of possible states below given n (i.e., n' < n)

constexpr std::uint64_t	nk_to_index (int n, int k)
	Returns nk_index, given {n, kappa}.

std::pair< int, int >	index_to_nk (std::uint64_t nk_index)
	Returns {n, kappa} given nk_index.

std::pair< int, std::uint64_t >	nkindex_to_n_kindex (std::uint64_t nk_index)
	Returns {n, kappa_index} given nk_index.

int	nkindex_to_kappa (std::uint64_t nk_index)
	Returns kappa, given nk_index.

int	nkindex_to_twoj (std::uint64_t nk_index)
	Returns 2*j, given nk_index.

int	nkindex_to_l (std::uint64_t nk_index)
	Returns l, given nk_index.

constexpr bool	evenQ (int a)
	Returns true if a is even - for integer values.

constexpr bool	evenQ_2 (int two_a)
	Returns true if a (an int) is even, given 2*a (true if two_a/2 is even)

constexpr int	neg1pow (int a)
	Evaluates (-1)^{a} (for integer a)

constexpr int	neg1pow_2 (int two_a)
	Evaluates (-1)^{two_a/2} (for integer a; two_a is even)

constexpr int	parity (int la, int lb, int k)
	Parity rule. Returns 1 only if la+lb+k is even.

constexpr int	triangle (int j1, int j2, int J)
	Returns 1 if triangle rule is satisfied. nb: works with j OR twoj!

constexpr int	sumsToZero (int m1, int m2, int m3)
	Checks if three ints sum to zero, returns 1 if they do.

double	threej_2 (int two_j1, int two_j2, int two_j3, int two_m1, int two_m2, int two_m3)
	Calculates wigner 3j symbol. Takes in 2j (or 2l) - intput is integer.

double	special_threej_2 (int two_j1, int two_j2, int two_k)
	Special (common) 3js case: (j1 j2 k, -0.5, 0.5, 0)

double	cg_2 (int two_j1, int two_m1, int two_j2, int two_m2, int two_J, int two_M)
	Clebsh-Gordon coeficient <j1 m1, j2 m2 \| J M> [takes 2*j, as int].

bool	sixj_zeroQ (int a, int b, int c, int d, int e, int f)
	Checks triangle conditions for 6j symbols (for 2*j)

bool	sixjTriads (std::optional< int > a, std::optional< int > b, std::optional< int > c, std::optional< int > d, std::optional< int > e, std::optional< int > f)
	Checks if a 6j symbol is valid - each input is optional.

double	sixj_2 (int two_j1, int two_j2, int two_j3, int two_j4, int two_j5, int two_j6)
	6j symbol {j1 j2 j3 \ j4 j5 j6} - [takes 2*j as int]

double	ninej_2 (int two_j1, int two_j2, int two_j3, int two_j4, int two_j5, int two_j6, int two_j7, int two_j8, int two_j9)
	9j symbol {j1 j2 j3 \ j4 j5 j6 \ j7 j8 j9} [takes 2*j as int]

double	Ck_kk (int k, int ka, int kb)
	Reduced (relativistic) angular ME: <ka\|\|C^k\|\|kb> [takes k and kappa].

double	Ck_kk_mmq (int k, int ka, int kb, int twoma, int twomb, int twoq)
	Full C^k_q matrix element - rarely used.

bool	Ck_kk_SR (int k, int ka, int kb)
	Ck selection rule only. Returns false if C^k=0, true if non-zero.

double	tildeCk_kk (int k, int ka, int kb)
	tildeCk_kk = (-1)^{ja+1/2}*Ck_kk

std::pair< int, int >	kminmax_Ck (int ka, int kb)
	Returns [k_min, k_kmax] for C^k (min/max non-zero k, given kappa_a, kappa_b) Accounts for parity.

double	S_kk (int ka, int kb)
	Reduced spin angular ME: (for spin 1/2): <ka\|\|S\|\|kb>

Detailed Description

Angular provides functions and classes for calculating and storing angular factors (3,6,9-J symbols etc.)

Provides functions to:

Calculate wigner 3,6,9-J symbols + Clebsh-Gordon coefs etc..
Also provied look-up tables, which are faster than calculating symbols on-the-fly
Wrapper functions to calculate wigner 3,6,9-J symbols.
Uses GSL: https://www.gnu.org/software/gsl/doc/html/specfunc.html?highlight=3j#coupling-coefficients

The general equations are defined:

\begin{align} \frac{1}{r_{12}} &= \sum_{kq} \frac{r_<^k}{r_>^{k+1}}(-1)^q C^k_{-q}(\hat{n}_1)C^k_{q}(\hat{n}_2)\\ C^k_{q} &\equiv \sqrt{\frac{4\pi}{2k+1}} Y_{kq}(\hat{n}), \end{align}

and

\begin{align} C^k_{ab} &\equiv \langle{\kappa_a}||C^k||{\kappa_b}\rangle \equiv (-1)^{j_a+1/2} \widetilde C^k_{ab} ,\\ &= (-1)^{j_a+1/2}\sqrt{[j_a][j_b]} \begin{pmatrix} {j_a} & {j_b} & {k} \\ {-1/2} & {1/2} &{0} \end{pmatrix} \pi(l_a+l_b+k). \end{align}

Function Documentation

◆ jindex_to_twoj()

constexpr int Angular::jindex_to_twoj ( int jindex )

constexpr

Converts jindex to 2*j [helper function].

◆ twoj_to_jindex()

constexpr int Angular::twoj_to_jindex ( int twoj )

constexpr

Converts 2*j to jindex {1/2, 3/2, 5/2} -> {0, 1, 2} [helper function].

◆ kappa_to_jindex()

constexpr int Angular::kappa_to_jindex ( int ka )

constexpr

Converts kappa to jindex {-1, 1, -2} -> {0, 0, 1} [helper function].

◆ SixJ()

template<class A , class B , class C , class D , class E , class F >

double Angular::SixJ	(	const A &	a,
		const B &	b,
		const C &	c,
		const D &	d,
		const E &	e,
		const F &	f
	)

"Magic" 6j, for integers and DiracSpinors. Pass int (for k), or DiracSpinor for j. e.g.: (F,F,k,F,F,k) -> {j,j,k,j,j,k}. Do NOT *2! Do NOT pass j or 2j directly!

◆ l_k()

constexpr int Angular::l_k ( int ka )

constexpr

returns l given kappa

◆ twoj_k()

constexpr int Angular::twoj_k ( int ka )

constexpr

returns 2j given kappa

◆ parity_k()

constexpr int Angular::parity_k ( int ka )

constexpr

returns parity [(-1)^l] given kappa

◆ parity_l()

constexpr int Angular::parity_l ( int l )

constexpr

returns parity [(-1)^l] given l

◆ l_tilde_k()

constexpr int Angular::l_tilde_k ( int ka )

constexpr

"Complimentary" l := (2j-l) = l+/-1, for j=l+/-1/2

◆ kappa_twojl()

constexpr int Angular::kappa_twojl	(	int	twoj,
		int	l
	)

constexpr

returns kappa, given 2j and l

◆ kappa_twojpi()

constexpr int Angular::kappa_twojpi	(	const int	twoj,
		const int	pi
	)

constexpr

returns kappa, given 2*j and parity (+/-1),

◆ zeroQ()

constexpr bool Angular::zeroQ	(	double	x,
		double	eps = `1.0e-10`
	)

inlineconstexpr

Checks if a double is within eps of 0.0 [eps=1.0e-10 by default].

◆ kappa_to_kindex()

constexpr std::uint64_t Angular::kappa_to_kindex ( int ka )

constexpr

returns kappa_index, given kappa

Kappa Index: For easy array access, define 1-to-1 index for each kappa:

kappa	-1	1	-2	2	-3	3	-4	4	...
index	0	1	2	3	4	5	6	7	...

\[ i_k(\kappa)= \begin{cases} -2\kappa-2, & \kappa<0, \\ 2\kappa-1, & \kappa>0 . \end{cases} \]

\[ \kappa(i_k)= \begin{cases} -\left(\frac{i_k}{2}+1\right), & i_k\ \text{even},\\ \frac{i_k+1}{2}, & i_k\ \text{odd}. \end{cases} \]

◆ kindex_to_kappa()

constexpr int Angular::kindex_to_kappa ( std::uint64_t i )

constexpr

Returns kappa, given kappa_index.

see Angular::kappa_to_kindex()

◆ kindex_to_twoj()

constexpr int Angular::kindex_to_twoj ( std::uint64_t i )

constexpr

returns 2j, given kappa_index

see Angular::kappa_to_kindex()

◆ kindex_to_l()

constexpr int Angular::kindex_to_l ( std::uint64_t i )

constexpr

returns l, given kappa_index

see Angular::kappa_to_kindex()

◆ states_below_n()

constexpr std::uint64_t Angular::states_below_n ( int n )

constexpr

Returns number of possible states below given n (i.e., n' < n)

This could be made more "compact" with a maximum l. In practice, we go to large n, but never very large l.

Warning

n must be >1 (no non-standard states)

Cannot overflow; converts to std::uint64_t.

◆ nk_to_index()

constexpr std::uint64_t Angular::nk_to_index	(	int	n,
		int	k
	)

constexpr

Returns nk_index, given {n, kappa}.

For convenient array access we define a one-to-one mapping between the pair \((n,\kappa)\) and a non-negative index:

\[ i_{nk}(n,\kappa) = (n-1)^2 + i_k(\kappa), \]

where \(i_k(\kappa)\) is the kappa index defined by Angular::kappa_to_kindex()

Equivalently,

\[ i_{nk}(n,\kappa) = n^2 - 2n + 1 + i_k(\kappa). \]

Since

\[ n^2 - 2n + 1 = (n-1)^2 = \texttt{states\_below\_n}(n), \]

this counts the number of states with principal quantum number \(n' < n\), plus the index within the \(n\) shell.

Note: This indexing is only valid for physical bound-state quantum numbers, n >= 1, and cannot in general be used for arbitrary basis states.

Warning: To safely convert the returned index (uint64_t) to int, one must have n <= 46340.; To safely convert the returned index to Coulomb::nkIndex (uint16_t), one must have n <= 256.

◆ index_to_nk()

std::pair< int, int > Angular::index_to_nk ( std::uint64_t nk_index )

inline

Returns {n, kappa} given nk_index.

Inverse of Angular::nk_to_index().

◆ nkindex_to_n_kindex()

std::pair< int, std::uint64_t > Angular::nkindex_to_n_kindex ( std::uint64_t nk_index )

inline

Returns {n, kappa_index} given nk_index.

Inverse of Angular::nk_to_index().

◆ nkindex_to_kappa()

int Angular::nkindex_to_kappa ( std::uint64_t nk_index )

inline

Returns kappa, given nk_index.

See Angular::nk_to_index().

◆ nkindex_to_twoj()

int Angular::nkindex_to_twoj ( std::uint64_t nk_index )

inline

Returns 2*j, given nk_index.

See Angular::nk_to_index().

◆ nkindex_to_l()

int Angular::nkindex_to_l ( std::uint64_t nk_index )

inline

Returns l, given nk_index.

See Angular::nk_to_index().

◆ evenQ()

constexpr bool Angular::evenQ ( int a )

constexpr

Returns true if a is even - for integer values.

◆ evenQ_2()

constexpr bool Angular::evenQ_2 ( int two_a )

constexpr

Returns true if a (an int) is even, given 2*a (true if two_a/2 is even)

◆ neg1pow()

constexpr int Angular::neg1pow ( int a )

constexpr

Evaluates (-1)^{a} (for integer a)

◆ neg1pow_2()

constexpr int Angular::neg1pow_2 ( int two_a )

constexpr

Evaluates (-1)^{two_a/2} (for integer a; two_a is even)

◆ parity()

constexpr int Angular::parity	(	int	la,
		int	lb,
		int	k
	)

constexpr

Parity rule. Returns 1 only if la+lb+k is even.

◆ triangle()

constexpr int Angular::triangle	(	int	j1,
		int	j2,
		int	J
	)

constexpr

Returns 1 if triangle rule is satisfied. nb: works with j OR twoj!

◆ sumsToZero()

constexpr int Angular::sumsToZero	(	int	m1,
		int	m2,
		int	m3
	)

constexpr

Checks if three ints sum to zero, returns 1 if they do.

◆ threej_2()

double Angular::threej_2	(	int	two_j1,
		int	two_j2,
		int	two_j3,
		int	two_m1,
		int	two_m2,
		int	two_m3
	)

inline

Calculates wigner 3j symbol. Takes in 2*j (or 2*l) - intput is integer.

\[ \begin{pmatrix}j1&j2&j3\\m1&m2&m3\end{pmatrix} \]

◆ special_threej_2()

double Angular::special_threej_2	(	int	two_j1,
		int	two_j2,
		int	two_k
	)

inline

Special (common) 3js case: (j1 j2 k, -0.5, 0.5, 0)

\[ \begin{pmatrix}j1&j2&k\\-1/2&1/2&0\end{pmatrix} \]

◆ cg_2()

double Angular::cg_2	(	int	two_j1,
		int	two_m1,
		int	two_j2,
		int	two_m2,
		int	two_J,
		int	two_M
	)

inline

Clebsh-Gordon coeficient <j1 m1, j2 m2 | J M> [takes 2*j, as int].

◆ sixj_zeroQ()

bool Angular::sixj_zeroQ	(	int	a,
		int	b,
		int	c,
		int	d,
		int	e,
		int	f
	)

inline

Checks triangle conditions for 6j symbols (for 2*j)

◆ sixjTriads()

bool Angular::sixjTriads	(	std::optional< int >	a,
		std::optional< int >	b,
		std::optional< int >	c,
		std::optional< int >	d,
		std::optional< int >	e,
		std::optional< int >	f
	)

inline

Checks if a 6j symbol is valid - each input is optional.

i.e., sixjTriads(a,b,c,{},{},{}) will check if any 6J symbol of form {a b c \ * * *} is valid (* can be any value)

◆ sixj_2()

double Angular::sixj_2	(	int	two_j1,
		int	two_j2,
		int	two_j3,
		int	two_j4,
		int	two_j5,
		int	two_j6
	)

inline

6j symbol {j1 j2 j3 \ j4 j5 j6} - [takes 2*j as int]

◆ ninej_2()

double Angular::ninej_2	(	int	two_j1,
		int	two_j2,
		int	two_j3,
		int	two_j4,
		int	two_j5,
		int	two_j6,
		int	two_j7,
		int	two_j8,
		int	two_j9
	)

inline

9j symbol {j1 j2 j3 \ j4 j5 j6 \ j7 j8 j9} [takes 2*j as int]

◆ Ck_kk()

double Angular::Ck_kk	(	int	k,
		int	ka,
		int	kb
	)

inline

Reduced (relativistic) angular ME: <ka||C^k||kb> [takes k and kappa].

◆ Ck_kk_mmq()

double Angular::Ck_kk_mmq	(	int	k,
		int	ka,
		int	kb,
		int	twoma,
		int	twomb,
		int	twoq
	)

inline

Full C^k_q matrix element - rarely used.

◆ Ck_kk_SR()

bool Angular::Ck_kk_SR	(	int	k,
		int	ka,
		int	kb
	)

inline

Ck selection rule only. Returns false if C^k=0, true if non-zero.

◆ tildeCk_kk()

double Angular::tildeCk_kk	(	int	k,
		int	ka,
		int	kb
	)

inline

tildeCk_kk = (-1)^{ja+1/2}*Ck_kk

◆ kminmax_Ck()

std::pair< int, int > Angular::kminmax_Ck	(	int	ka,
		int	kb
	)

inline

Returns [k_min, k_kmax] for C^k (min/max non-zero k, given kappa_a, kappa_b) Accounts for parity.

◆ S_kk()

double Angular::S_kk	(	int	ka,
		int	kb
	)

inline

Reduced spin angular ME: (for spin 1/2): <ka||S||kb>

Reduced spin angular ME: (for spin 1/2!) <ka||S||kb> = d(la,lb) * (-1)^{ja+la+3/2} * Sqrt([ja]jb) * sjs{ja, 1, jb, 1/2, la, 1/2} Special 6j case: sjs{ja, 1, jb, 1/2, la, 1/2} = 0.5 (-1)^{ka+kb} * Sqrt(abs[{(ka-1)^2 - kb^2}/{3ka(1+2ka)}]) triangle rule for j! At least ~least 20% faster

Classes

Functions

Detailed Description

Function Documentation

◆ jindex_to_twoj()

◆ twoj_to_jindex()

◆ kappa_to_jindex()

◆ SixJ()

◆ l_k()

◆ twoj_k()

◆ parity_k()

◆ parity_l()

◆ l_tilde_k()

◆ kappa_twojl()

◆ kappa_twojpi()

◆ zeroQ()

◆ kappa_to_kindex()

◆ kindex_to_kappa()

◆ kindex_to_twoj()

◆ kindex_to_l()

◆ states_below_n()

◆ nk_to_index()

◆ index_to_nk()

◆ nkindex_to_n_kindex()

◆ nkindex_to_kappa()

◆ nkindex_to_twoj()

◆ nkindex_to_l()

◆ evenQ()

◆ evenQ_2()

◆ neg1pow()

◆ neg1pow_2()

◆ parity()

◆ triangle()

◆ sumsToZero()

◆ threej_2()

◆ special_threej_2()

◆ cg_2()

◆ sixj_zeroQ()

◆ sixjTriads()

◆ sixj_2()

◆ ninej_2()

◆ Ck_kk()

◆ Ck_kk_mmq()

◆ Ck_kk_SR()

◆ tildeCk_kk()

◆ kminmax_Ck()

◆ S_kk()