AngularMomentumOperators

AngularMomentumOperators(NF, IndexUp, IndexDn) returns a table with the angular momentum, multipole, Slater and Casimir operators of a single shell. It adds no functionality: every entry can also be created one at a time with NewOperator(). It only shortens scripts, by replacing the block of thirty NewOperator calls that a multiplet calculation normally starts with by a single line.

Example.Quanty

NF = 10
IndexDn = {0,2,4,6,8}
IndexUp = {1,3,5,7,9}
 
F0, F2, F4 = 5.0, 10.0, 6.0
zeta, Bz   = 0.1, 0.01
 
d = AngularMomentumOperators(NF, IndexUp, IndexDn)
 
H = F0*d.SlaterF[0] + F2*d.SlaterF[2] + F4*d.SlaterF[4] + zeta*d.ldots + Bz*(2*d.Sz + d.Lz)

All operators are created in the basis of spherical harmonics and rotated from there with a single rotation matrix, so a Slater integral $F^k$ keeps its meaning whichever basis is asked for.

• NF : Integer, the number of fermionic modes in the one particle basis.
• IndexUp : A list of indices. For the bases “Y”, “Z” and “K” the $2l+1$ orbitals with spin up, ordered from $m=-l$ to $m=l$. For the basis “jjz” the $2l$ states of $j=l-1/2$, ordered from $j_z=-j$ to $j_z=j$.
• IndexDn : A list of indices. For the bases “Y”, “Z” and “K” the $2l+1$ orbitals with spin down, ordered from $m=-l$ to $m=l$. For the basis “jjz” the $2l+2$ states of $j=l+1/2$, ordered from $j_z=-j$ to $j_z=j$.
• Possible options

  • “Basis” : a string, one of “Y” (the default), “Z”, “K” or “jjz”. The same values and the same index conventions as the “Basis” option of NewOperator().

The angular momentum $l$ of the shell is deduced from the length of the index lists. If no options are given the basis is taken to be “jjz” when IndexDn is two longer than IndexUp, and “Y” otherwise.

• A table with the following entries.

Every operator carries a Name, so that PrintExpectationValues() prints readable column headers without any further work.

The Coulomb repulsion within the shell is split into the contributions of each angular momentum transfer $k$, which run over the even values from $0$ to $2l$. SlaterF[k] is the operator that multiplies $F^k$, so that \begin{eqnarray} U = \sum_{k=0,2,\ldots}^{2l} F^k \, \mathrm{SlaterF}[k] \end{eqnarray} is the same operator as NewOperator(“U”, NF, IndexUp, IndexDn, {F0, F2, …}). The index of the list is $k$ itself, not a counter, so the list of a $d$ shell has the entries SlaterF[0], SlaterF[2] and SlaterF[4]. The Slater integrals themselves are obtained from GetSlaterIntegrals().

Racah's unit tensor $u^{(k)}_q$ of a shell of angular momentum $l$ is the one particle operator, summed over spin, whose reduced matrix element is one: \begin{eqnarray} u^{(k)}_q = \sum_{m_1 = -l}^{l} \sum_{m_2 = -l}^{l} \sum_{\sigma} (-1)^{l-m_1} \begin{pmatrix} l & k & l \\ -m_1 & q & m_2 \end{pmatrix} a^{\dagger}_{m_1,\sigma} a^{\phantom{\dagger}}_{m_2,\sigma}. \end{eqnarray} The table entry GG[k] is the scalar product of two of them, \begin{eqnarray} \mathrm{GG}[k] = u^{(k)} \cdot u^{(k)} = \sum_{q=-k}^{k} (-1)^q \, u^{(k)}_q \, u^{(k)}_{-q}. \end{eqnarray} In this normalisation the $k=1$ one is the square of the orbital angular momentum up to a factor, \begin{eqnarray} L^2 = l(l+1)(2l+1) \, \mathrm{GG}[1], \end{eqnarray} which is the easiest way to check which convention a given table is using.

For odd $k$ the operators $u^{(k)}$ generate the group $SO(2l+1)$, and the $k=1$ one alone generates $SO(3)$. The sum of GG[k] over the odd $k$ that generate a group is the Casimir operator of that group, and these are returned under Casimir, keyed by the name of the group:

For an $f$ shell this is Racah's chain $SO(3) \subset G_2 \subset SO(7)$, the standard way of labelling the terms of $f^n$ that $L$ and $S$ alone do not separate. Note that any scaling of a Casimir operator is allowed in a fit, and the conventions in the literature differ, which is why the $k$ resolved GG are returned next to the named combinations. For the $k=1$ one in particular it is usual to fit $\alpha L^2$ rather than $\alpha' \, \mathrm{Casimir.SO3}$, and $L^2 = l(l+1)(2l+1)\,\mathrm{Casimir.SO3}$ relates the two.

On the jjz basis the table carries, in addition to SlaterF, the entry CoulombJJ, in which each pair of $j$ shells has its own set of Slater integrals:

If the radial wave functions of $j=l-1/2$ and $j=l+1/2$ are the same, these reduce to the SlaterF of the shell. If they are not, which is the point of solving a Dirac equation rather than a Schrödinger equation, CoulombJJ is the more general parametrisation and SlaterF is an approximation to it. The extra parameters are usually neglected; having both available makes it possible to check how much that costs.

The seventeen terms of $f^3$. The two $^2H$ terms have the same $S$ and the same $L$ and are told apart only by the Casimir operators of $G_2$ and $SO(7)$; the same holds for the two $^2F$ terms and for the $^4S$ and $^4F$ terms, which the Coulomb interaction happens to leave degenerate at these parameters.

AngularMomentumOperators.Quanty

NF = 14
IndexDn = {0,2,4,6,8,10,12}
IndexUp = {1,3,5,7,9,11,13}
 
f = AngularMomentumOperators(NF, IndexUp, IndexDn)
 
-- the last two terms are tiny, they are there only to lift the accidental
-- degeneracy between terms that the Coulomb interaction puts at the same energy
H = 1.0*f.SlaterF[2] + 0.6*f.SlaterF[4] + 0.4*f.SlaterF[6]
  + 1E-7*f.Casimir.G2 + 1E-9*f.Casimir.SO7
 
psiList = Eigensystem(H, {NF, 0, {"11111111111111", 3, 3}}, 364, {{"DenseBorder", 400}})
 
-- keep one state per multiplet, so that each term shows up once
psiTerm = {}
Eprev = nil
for i = 1, #psiList do
  local E = psiList[i] * H * psiList[i]
  if (Eprev == nil) or (math.abs(E - Eprev) > 1E-8) then
    psiTerm[#psiTerm + 1] = psiList[i]
    Eprev = E
  end
end
 
PrintExpectationValues(psiTerm, {f.Ssqr, f.Lsqr, f.Casimir.SO3, f.Casimir.G2, f.Casimir.SO7}, H)

AngularMomentumOperators.out

         E       Ssqr    Lsqr    C SO3   C G2    C SO7
 1      -0.3786  3.75    42      0.5     0.7879  0.9784
 2      -0.2471  0.75    30      0.3571  0.8555  1.205
 3      -0.2345  3.75    0       0       0       0.8571
 4      -0.2345  3.75    12      0.1429  0.2857  0.8571
 5      -0.1996  0.75    20      0.2381  0.6856  1.1661
 6      -0.1752  0.75    56      0.6667  1.1212  1.4069
 7      -0.1522  3.75    20      0.2381  0.5974  0.7879
 8      -0.1277  0.75    6       0.0714  0.4762  1.0952
 9      -0.1276  0.75    2       0.0238  0.381   1.0952
 10     -0.0421  0.75    42      0.5     0.7879  1.4069
 11     -0.0357  0.75    72      0.8571  1.2597  1.5455
 12     -0.0081  3.75    6       0.0714  0.4762  0.6667
 13      0.0062  0.75    30      0.3571  0.664   1.3144
 14      0.0166  0.75    6       0.0714  0.6883  0.974
 15      0.0816  0.75    12      0.1429  0.5533  0.7803
 16      0.2215  0.75    20      0.2381  0.7214  1.1456
 17      0.4741  0.75    12      0.1429  0.4727  0.6743