Benchmarking with HFODD

To demonstrate that the isospin-invariant formalism has been properly implemented, we provide a detailed comparison between the HFODD [27] and HFBTHO frameworks. This benchmarking is meaningful as the two pnEDF codes were developed independently and have fairly different structures. In particular, the HF equations in HFODD [28,29] are solved in three-dimensional Cartesian basis while HFBTHO employs the two-dimensional cylindrical basis.

Calculations were performed for the $A=40$, $T\simeq8$ deformed IASs with SkM* EDF parametrization using the s.p. basis space of $N_{\rm sh} = 10$. The oscillator length was assumed to be $b$ = 1.697626fm. The mass constant in Eq. (5) was fixed at $\hbar^2/2m = 20.73$MeVfm$^2$. As far as integration is concerned, we used $N_{\rm GH} = 26$ Gauss-Hermite nodes for each Cartesian coordinate in HFODD, whereas in HFBTHO, the numbers of Gauss-Hermite ($\rho$-direction) and Gauss-Laguerre ($z$-direction) nodes were assumed to be equal: $N_{\rm GH} = N_{\rm GL}=40$. In addition, in HFBTHO, the number of Gauss-Legendre nodes used in the integration of the direct Coulomb field was set to 80, and the Coulomb length scale was taken to be $L=50$fm. This set of parameters was recommended as a default value in the latest version of the HFBTHO, as it provides a sufficient precision on the direct Coulomb energy [24]. Without Coulomb, the isocranking frequency was set to $\lambda ' = 27.092394$MeV and $\lambda_{\text{off}} = 0$MeV. With Coulomb, we took the values $\lambda ' =28.613615$MeV and $\lambda_{\text{off}} = -6.010741$MeV.

Table: Benchmarking of HFODD with HFBTHO for the deformed $A=40, T\simeq 8$ IASs using SkM* EDF ( $N_{\rm sh} = 10$), and $\theta '=0^\circ$ ($^{40}$Mg). Other parameters are: $\lambda ' = 27.092394$MeV, $\lambda _{\rm off} = 0$MeV (without Coulomb), and $\lambda ' =28.613615$MeV, $\lambda _{\rm off} = -6.010741$MeV (with Coulomb). Shown are various contributions to the total binding energy $E_{\rm tot}$ (in MeV), r.m.s. radii (in fm), expectation values of $T^2, T_z$, and $T_x$, and the quadrupole deformation $\beta _2$. The digits which do not coincide in HFODD and HFBTHO are marked in bold.

	HFODD	HFBTHO	HFODD	HFBTHO
	Without Coulomb		With Coulomb
$E_{\rm tot}$	-303.42519	-303.42520	-276.47641	-276.47643
$E_{\rm kin}^{({\rm n})}$	498.448466	498.448464	495.53929	495.53930
$E_{\rm kin}^{({\rm p})}$	175.371764	175.371762	171.30205	171.30206
$E_{\rm pot}$	-977.24542	-977.24543	-970.09923	-970.09926
$E_{\rm SO}$	-34.357903	-34.357905	-33.184812	-33.184816
$E_{\rm Cou}^{({\rm dir})}$			30.920704	30.920697
$E_{\rm Cou}^{({\rm exc})}$			-4.139228	-4.139228
$r_{\rm rms}^{({\rm n})}$	3.697718	3.697718	3.709975	3.709975
$r_{\rm rms}^{({\rm p})}$	3.176356	3.176356	3.217587	3.217587
$T^2$	72.022743	72.022743	72.023123	72.023123
$T_z$	8.000000	8.000000	8.000000	8.000000
$T_x$	0.000000	0.000000	0.000000	0.000000
$\beta _2$	0.304201	0.304201	0.311518	0.311518

Table 2: Similar to Table 1 but for $\theta '=90^\circ$ ($^{40}$Ca).

	HFODD	HFBTHO	HFODD	HFBTHO
	Without Coulomb		With Coulomb
$E_{\rm tot}$	-303.42519	-303.42520	-234.429	-234.419
$E_{\rm kin}^{({\rm n})}$	336.910115	336.910113	333.68	333.71
$E_{\rm kin}^{({\rm p})}$	336.910115	336.910113	318.704	318.713
$E_{\rm pot}$	-977.245420	-977.24543	-954.79	-954.83
$E_{\rm SO}$	-34.357903	-34.357905	-31.49	-31.51
$E_{\rm Cou}^{({\rm dir})}$			75.105	75.106
$E_{\rm Cou}^{({\rm exc})}$			-7.12522	-7.12525
$r_{\rm rms}^{({\rm n})}$	3.549360	3.549360	3.5930	3.5928
$r_{\rm rms}^{({\rm p})}$	3.549360	3.549360	3.63519	3.63512
$T^2$	72.022743	72.022743	72.149	72.155
$T_z$	0.000000	0.000000	0.15649	0.15652
$T_x$	8.000000	8.000000	8.0051	8.0054
$\beta _2$	0.304201	0.304201	0.3184	0.3182

The benchmarking results for the deformed case are shown in Tables 1 and 2 for $\theta '=0^\circ$ and $90^\circ$, respectively. In the absence of the Coulomb term, the difference in the total energy $E_{\rm tot}$ is less than 20eV, and the total isospin $\langle\,\hat{T}^2 \,\rangle$ agrees up to the sixth decimal place. With the inclusion of the Coulomb term, the agreement is slightly reduced but is still excellent. A comparison between HFBTHO and HFODD was also performed for the spherical $A=48$, $T\simeq4$ IASs and the results are presented in Table 3 for $\theta '=90^\circ$ where the differences between the two codes are largest. In this case, it is found that deviation in the total energy is about 20eV.

It is to be noted that both HFODD and HFBTHO use the same number of basis harmonic oscillator states. Moreover, as it has been demonstrated previously [24], using a sufficient number of quadrature points in HFBTHO and HFODD, the results of both solvers agree with a high accuracy of several eV. The differences between the two codes with the Coulomb potential turned on, can be traced back to different techniques used to compute the direct Coulomb field: the solver HFODD, uses a more accurate Green's function approach. The benchmark examples discussed in this section demonstrate that the p-n mixing has been implemented correctly in both codes. Presently, we are in process of implementing the cylindrical Green's function treatment of the Coulomb potential into HFBTHO and it is expected that the agreement between the two codes will further improve.

Table: Similar to in Table 1 but for the spherical $A=48, T\simeq 4$ IASs and $\theta '=90^\circ$ ($^{48}$Cr). Other parameters are: $\lambda ' = 11.0$MeV, $\lambda _{\rm off} = 0$MeV (without Coulomb), and $\lambda ' =12.0$MeV, $\lambda _{\rm off} = -8.0$MeV (with Coulomb).

	HFODD	HFBTHO	HFODD	HFBTHO
	Without Coulomb		With Coulomb
$E_{\rm tot}$	-491.243706	-491.243724	-389.8454	-389.8438
$E_{\rm kin}^{({\rm n})}$	422.93152	422.93158	415.69	415.71
$E_{\rm kin}^{({\rm p})}$	422.93152	422.93143	404.673	404.679
$E_{\rm pot}$	-1337.10675	-1337.10673	-1310.44	-1310.46
$E_{\rm SO}$	-36.736418	-36.736417	-34.1216	-34.1240
$E_{\rm Cou}^{({\rm dir})}$			109.3774	109.3781
$E_{\rm Cou}^{({\rm exc})}$			-9.14576	-9.14581
$r_{\rm rms}^{({\rm n})}$	3.497939	3.497940	3.52633	3.52628
$r_{\rm rms}^{({\rm p})}$	3.497939	3.497939	3.57787	3.57784
$T^2$	20.037818	20.037818	20.0756	20.0771
$T_z$	0.000000	0.000003	-0.012676	-0.012663
$T_x$	4.000000	4.000000	4.00264	4.00278