# Trying to reproduce the energy levels of a molecule from a paper

• A
• BillKet
In summary, the conversation discusses a specific question about reproducing a figure from a paper about the PV experiment performed on BaF. The speaker describes their method of obtaining the eigenstates and the diagonal elements for the Hamiltonian. They mention the terms that matter in the off-diagonal elements and their inclusion in the plot. Another person confirms the accuracy of the plot and notes some small differences that can be explained by omitted terms. The conversation ends with a question about proposed experiments for measuring CP-violation using large magnetic fields.

#### BillKet

Hello! This is quite a specific question, so if anyone knows the details I would really appreciate your help (@Twigg ?). I am trying to reproduce figure 1 from this paper (it's for the PV experiment performed on BaF). While I am getting quite close to it, the levels don't fully match (I am attaching below the plot I obtained). What I did was to get the eigenstates of ##H = H_0 + H_Z## from equations 1 and 2. I first got the diagonal element. For ##H_0## these are:

$$BN(N+1)+\gamma m_N m_S + b m_I m_S + \frac{c}{3}m_I m_S$$

where ##m_{I,S,N}## are the projections along the z axis (defined by the magnetic field) of the I, S and N operators (I am ignoring the ##DN^4## as that is negligible for the ##N=0,1## states we are interested in). For the ##H_z##, the diagonal terms are, given that B is along z:

$$-g_\perp\mu_Bm_S B - \frac{1}{3}(g_\parallel-g_\perp)\mu_Bm_SB- g_I\mu_N m_I B - g_{rot}\mu_Nm_N B$$

where I used from the cited literature: ##B = 6473.9588##, ##\gamma = 80.923##, ##b = 66.25## and ##c = 8.2233## (all in MHz). In terms of off-diagonal elements, there are none connecting ##N=1## to ##N=0##, as they have different parities. Within a given N manifold, the terms that matter are the ones connecting states with the same value of ##m_S##, as different values of ##m_S## are suppressed by about ##\frac{\gamma}{B}##. The only terms able to do that come from ##c(I\cdot n)(S \cdot n)## and are given by ##c\frac{\sqrt{2}}{10}m_S## and they are included in the plot below. All the other terms shouldn't matter as they are too small and I think I included all the relevant term. But I can't seem to reproduce their plots. Did I miss any term or miss-calculated something? Any insight would be really appreciated. Thank you!

BillKet said:
But I can't seem to reproduce their plots.
Looks pretty good to me? Am I missing something? The only differences I can see are at the 1% level or so. This is consistent with ##\gamma/B \approx 0.013##.

Also, can you say which unperturbed state is which plot line?

Twigg said:
Looks pretty good to me? Am I missing something? The only differences I can see are at the 1% level or so. This is consistent with ##\gamma/B \approx 0.013##.

Also, can you say which unperturbed state is which plot line?
The levels in my plot should be in the same order, except for the blue-dotted one in my plot, which doesn't appear at all in theirs (not sure if they willingly decide to not plot it). Also the splitting between the continuous blue and dotted red is much bigger in my case, than in theirs.

BillKet said:
Also the splitting between the continuous blue and dotted red is much bigger in my case, than in theirs.
It seems different by a few 10's of MHz. Let's say 20 MHz, to be generous. 20 MHz / 6550MHz = 0.3%, consistent with the terms you omitted. I think you nailed it?

BillKet
Twigg said:
It seems different by a few 10's of MHz. Let's say 20 MHz, to be generous. 20 MHz / 6550MHz = 0.3%, consistent with the terms you omitted. I think you nailed it?
I didn't realize the difference was so small . Not fully related to this, but do you know of any proposed experiments that aim to measure CP-violation using large magnetic fields?

BillKet said:
Not fully related to this, but do you know of any proposed experiments that aim to measure CP-violation using large magnetic fields?
I don't, sorry. Sounds hard, from an experimental point of view.