AeF-hyperfine-structure: Alkaline-Earth Monofluoride molecular hyperfine structure calculator

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This code here is a testbed for computing the hyperfine structure of $^{138}BaF$ molecules in their $^2\Sigma$ electronic and vibrational ground states. This is based on PRA 98, 032513 (2018) using

Under these circumstances, the state of each molecule can be described using three coupled angular momenta:

• $\vec{I}$: total nuclear spin ($I = \frac{1}{2}$ always since $^{138}Ba$ has $I=\frac{1}{2}$ and $^{19}F$ has $I=0$)
• $\vec{S}$: total electron spin ($S = \frac{1}{2}$ in the electronic ground state)
• $\vec{N}$: molecular rotational angular momentum ($n\in\mathbb{Z}$)

The total angular momentum of the molecule is denoted $\vec{F}=\vec{I}+\vec{S}+\vec{N}$ with quantum numbers $f$ and $m_f$. Since there are three angular momenta, there are several possible bases

J-basis

One possible basis couples $S$ and $N$ first to form $\vec{J} = \vec{N} + \vec{S}$ ($j=n\pm\frac{1}{2}$), and then couples I to $J$ to form $F$. This will be used as the "default" basis since $S$ and $N$ couple more strongly to each other than to $I$.

For a given $n\neq0$, the quantum number $j$ has two possible values $j^+(n) = n + \frac{1}{2}$ and $j^-(n) = n - \frac{1}{2}$. For $n=0$, only $j^+(0)=\frac{1}{2}$ is defined.

Similarly, $f$ has two possible values for a given $j$, $f^+(j) = j + \frac{1}{2}$ and $f^-(j) = j - \frac{1}{2}$. Note that $j$ is always a half-integer, so both $f^\pm$ are defined for all $j$.

Thus, in the $J$-basis a molecular state can be described as $\ket{i(sn)jfm_f}$. This can be abbreviated to $\ket{njfm_f}$ since $i$ and $s$ are always $\frac{1}{2}$.

G-basis

Under certain circumstances, it is useful to couple $I$ and $S$ first instead of $S$ and $N$. In these circumstances, a new angular momentum $\vec{G} = \vec{I} + \vec{S}$ is used instead. It has quantum number $g = 0, 1$ with the usual singlet-triplet set.

These two bases are related by the wigner 6j symbol -- $$ \braket{i(sn)jf}{(is)gnf} = \xi'()$$

3M-basis

There is also an uncoupled basis $\ket{im_i,sm_s,nm_n}$. Since $f$ is not diagonal in this basis, this is only really useful in evaluating the "tensor" part of the hyperfine Hamiltonian.

Hamiltonian in free space

The effective Hamiltonian in vacuum (with possible electric fields can be described as the sum of three parts: a rotational Hamiltonian, a Stark shift, and a hyperfine shift: $$ H = H_{rot} + H_{st} + H_{hfs} $$

Rotational Hamiltonian

The rotational Hamiltonian is $$ H_{rot} = BN^2 - DN^4 + \gamma\vec{N}\cdot\vec{S} + \delta N^2 \vec{N}\cdot\vec{S}$$ Note that this is diagonal in the $J-basis$, and

J-basis index

It is often useful to assign a natural number index to each element of the most frequently used basis. For example, this makes it easy to efficiently represent operators as n-d matricies.

References

1. PRA 98, 032513 (2018) (EDM3 proposal paper)
2. J. Chem. Phys. 105, 7412 (1996).

Licensing information

AeF-hyperfine-structure is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

AeF-hyperfine-structure is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with AeF-hyperfine-structure. If not, see https://www.gnu.org/licenses/.