Breit with RPA [probably no issue]

[benroberts999]

For hyperfine: possibly linked to diagram method?

[benroberts999]

Some of the Breit corrections to Hyperfine constants differ significantly from Derevianko 10.1103/PhysRevA.65.012106. The Breit corrections at Hartree-Fock level agree perfectly.

This is in contrast to E1, where Breit corrections at HF level match exactly, and agree very well at RPA level. Also, Breit corrections to E1 match exactly with diagram and TDHF method for RPA.

The difference is at most a factor of two for the RPA correction part; the result is only noticable when there is large cancellation between HF and RPA contributions, so in the end, doesn't seem like major issue.

HF	ampsci	Dzuba	Derevianko
6s+	0.011	0.010	0.011
6p-	-0.680	-0.680	-0.680
6p+	-0.062	-0.062	-0.060
5d-	0.099	0.099	0.099
5d+	0.053	0.053
HF+RPA	ampsci	Dzuba	Derevianko
6s+	4.383	4.010	4.111
6p-	0.012	-0.270	-0.29
6p+	0.114	0.072	0.04
5d-	0.266	0.233	0.199
5d+	-0.169	-0.183
RPA only	ampsci	Dzuba	Derevianko
6s+	4.372	4.000	4.1
6p-	0.692	0.410	0.39
6p+	0.176	0.135	0.1
5d-	0.168	0.134	0.1
5d+	-0.221	-0.236

Full code outputs: rpa-breit.txt

[benroberts999]

Note: get exactly the same Breit corrections with diagram and "TDHF Basis" methods

• This essentially means issue is not in RPA equations, which are implemented very differently between these two methods
• Possibly issue in Breit equations; or, possibly no issue at all!
• "Basis" method is equivalent to TDHF, but instead of solving TDHF equations, expands solutions using a basis
  • nb: "basis" method is extremely slow, and should only be used for testing

	HF+RPA		RPA Only
	Diagram	Basis	Diagram	Basis
6s+	4.383	4.383	4.372	4.372
6p-	0.012	0.015	0.692	0.694
6p+	0.114	0.115	0.176	0.177
5d-	0.266	0.269	0.168	0.170
5d+	-0.169	-0.167	-0.221	-0.220

Full code input/output: rpa-breit-basis.txt

I'll leave this issue up for now as a note that this probably requires extra testing, comparison with other calculations etc. But essentially, it seems there is not an error in ampsci

[benroberts999]

As an extra note - also less than perfect agreement for two-body Breit correction to energies at second-order compared to Derevianko. This correction is complicated, and depends strongly on Brueckner convergence, and overall correction still agrees well. Therefore, not really major issue, but something that would be good to check:

(Nb: it's not non-linear in Breit terms which make the difference, which can be easily check by scaling Breit potential by small number)

Breit corrections at Hartree-Fock and Brueckner levels, accounting for one-body (1) and two-body (2) Breit contributions. Derevianko results 10.1103/PhysRevA.65.012106 in italics.

	HF		BO(1)		BO(2)		Total
6s_1/2	3.2	3.2	-6.1	-5.0	-0.3	-0.8	-3.2	-2.6
7s_1/2	1.1	1.1	-1.0	-1.1	-0.1	-0.3	0.02	-0.26
6p_1/2	7.5	7.5	-0.2	-0.1	-0.4	-0.3	6.9	7.1
7p_1/2	2.7	2.7	-0.1	-0.1	-0.1	-0.1	2.5	2.5
6p_3/2	2.9	2.9	-2.3	-1.8	0.02	-0.25	0.59	0.84
7p_3/2	1.0	1.0	-0.6	-0.6	0.01	-0.09	0.44	0.38
5d_3/2	-10.2	-10.2	-15.8	-12	-1.2	-0.4	-27.1	-22
5d_5/2	-11.7	-11.8	-19.2	-14	0.4	-0.3	-30.6	-26

Full code output: 240125-Breit-BO2.txt