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ampsci
c++ program for high-precision atomic structure calculations of single-valence systems
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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 | twoj (int jindex) |
| Converts jindex to 2*j [helper function]. | |
| constexpr int | jindex (int twoj) |
| Converts 2*j to jindex {1/2, 3/2, 5/2} -> {0, 1, 2} [helper function]. | |
| constexpr int | jindex_kappa (int ka) |
| Converts kappa to jindex {-1, 1, -2} -> {0, 0, 1} [helper function]. | |
| 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 int | indexFromKappa (int ka) |
| returns kappa_index, given kappa More... | |
| constexpr int | kappaFromIndex (int i) |
| Returns kappa, given kappa_index. | |
| constexpr int | twojFromIndex (int i) |
| returns 2j, given kappa_index | |
| constexpr int | lFromIndex (int i) |
| returns l, given kappa_index | |
| constexpr int | states_below_n (int n) |
| Returns number of possible states below given n. | |
| constexpr int | nk_to_index (int n, int k) |
| return nk_index given {n, kappa}: nk_index(n,k) := n^2 - 2n + 1 + kappa_index More... | |
| std::pair< int, int > | index_to_nk (int index) |
| return {n, kappa} given nk_index: | |
| int | nkindex_to_kappa (int index) |
| Returns kappa, given nk_index. | |
| int | nkindex_to_twoj (int index) |
| Returns 2*j, given nk_index. | |
| int | nkindex_to_l (int 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 2*j (or 2*l) - intput is integer. More... | |
| 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) More... | |
| 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. More... | |
| 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> More... | |
Angular provides functions and classes for calculating and storing angular factors (3,6,9-J symbols etc.)
Provides functions to:
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}
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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 ... kappa(i) = (-1,i+1)*(int(i/2)+1)
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constexpr |
return nk_index given {n, kappa}: nk_index(n,k) := n^2 - 2n + 1 + kappa_index
nk_index: For easy array access, define 1-to-1 index for each {n, kappa}: nk_index(n,k) := n^2 - 2n + 1 + kappa_index. nb: n^2 - 2n + 1 = states_below_n - number of possible states with n'<n. Note: ONLY valid for n >= 1 (i.e., cannot be used for general basis states)
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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
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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)
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inline |
Special (common) 3js case: (j1 j2 k, -0.5, 0.5, 0)
\[ \begin{pmatrix}j1&j2&k\\-1/2&1/2&0\end{pmatrix} \]
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inline |
Calculates wigner 3j symbol. Takes in 2*j (or 2*l) - intput is integer.
\[ \begin{pmatrix}j1&j2&j3\\m1&m2&m3\end{pmatrix} \]
1.9.1