# Impact of the molecular resonances on the
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion reaction rate

Yasutaka Taniguchi  Masaaki Kimura

###### Abstract

The properties of the low-energy ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$
molecular resonances, which potentially enhance the fusion reaction rate at
low temperatures, have been investigated by a full-microscopic nuclear model
employing various nuclear energy density functionals. We show that some
density functionals plausibly describe the observed high-spin
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ molecular resonances and predict many
$0^{+}$ and $2^{+}$ resonances at low energies, which enhance the reaction
rate. We also discuss how the uncertainty in the nuclear energy density
functionals propagates to that of the reaction rate.

###### keywords:

${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion reaction rate , X-ray superburst
, Microscopic nuclear model

††journal: Physics Letters B

[kosen]organization=Department of Information Engineering, National Institute
of Technology (KOSEN), Kagawa College,addressline=551 Kohda, Takuma-cho,
city=Mitoyo, postcode=769-1192, state=Kagawa, country=Japan

[riken]organization=RIKEN Nishina Center,addressline=2-1 Hirosawa, city=Wako,
postcode=351-0198, state=Saitama, country=Japan

##  1 Introduction

The ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion reaction plays a vital role
in explosive phenomena in the universe, such as X-ray superbursts (XRSBs) [1,
2], type Ia supernovae [3, 4], and the evolution of massive stars [5]. Its
reaction rate at low temperatures has been of great interest for many years.
Recently, direct measurements have reached to the energies as low as $E\simeq
2$ MeV [6, 7, 8]. In addition, an indirect measurement by the Trojan Horse
Method (THM) [9] has reported that resonances below 2 MeV might significantly
enhance the reaction rate. However, the reaction rate at low temperatures
still remains inconclusive due to the uncertainties in the measurements.

Consequently, the reaction rates extrapolated to low temperatures are used in
the simulations of astrophysical phenomena. The standard estimation by
Caughlan and Fowler (CF88) [10] assumes a constant modified astrophysical
spectroscopic factor ($S^{\ast}$-factor), whereas the hindrance model [11]
asserted its suppression at low temperatures. Contrary, Cooper et al. [12]
discussed an increased reaction rate at low temperatures. Thus, there are ad-
hoc variations of the reaction rate, and hence, it is crucial to impose a
constraint by the nuclear model calculations. In particular, it is an
essential task for nuclear theories to investigate whether the resonances
increase the reaction rate.

In our previous work [13], on the basis of the antisymmetrized molecular
dynamics (AMD) calculations, we discussed the low-energy resonances which
affect the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion reaction. Here, we
conduct extensive research, and report the low-energy resonances which have
the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ molecule-like structure and
significantly affect the reaction rage. Employing several nuclear density
functionals, we assess the constraint on the reaction rate imposed by the
nuclear theory.

##  2 Theoretical Model

The description of the low-energy resonances requires an accurate description
of various reaction channels and rearrangement of nucleons. To meet this
requirement, we employ the generator coordinate method (GCM) [14, 15] which
describes a resonance by a superposition of the wave functions for various
channels,

$\displaystyle\Psi^{J}=\sum_{ciK}f_{ciK}P^{J}_{MK}\Psi_{c}(d_{i})+\sum_{iK}g_{iK}P^{J}_{MK}\Psi_{\rm
Mg}(\beta_{i}),$ (1)

where $P^{J}_{MK}$ is the angular momentum projector, and $\Psi_{c}(d_{i})$ is
the channel wave function where the subscript $c$ denotes the channels
relevant to the fusion reaction, i.e., ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$,
$\alpha+{}^{20}\mathrm{Ne}$, and $p+{}^{23}\mathrm{Na}$ channels. The inter-
nuclear distance $d_{i}$ ranges from 0.5 to 8 fm for the
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ , from 0.5 to 9 fm for the
$\alpha+{}^{20}\mathrm{Ne}$, and from 0.5 to 7 fm for the
$p+{}^{23}\mathrm{Na}$, with the common intervals of 0.5 fm. In addition, the
wave functions of the compound nucleus ${}^{24}{\rm Mg}$, denoted by
$\Psi_{\rm Mg}(\beta_{i})$, are also superposed to describe the coupling
between the entrance and exit channels and the compound states. The resonance
energy and the coefficients $f_{ciK}$ and $g_{iK}$ are determined by the
diagonalization of the Hamiltonian defined by the density functionals. We have
employed several parameter sets of the density functionals, the Gogny
parameter sets (D1S [16] and D1M* [17]) and Skyrme parameter sets (SkM* [18],
SLy4 [19, 20], and SIII [21]).

Each channel wave function is the Slater determinant of the nucleon wave
packets projected to positive parity,

$\Psi=\frac{1+P_{r}}{2}{\mathcal{A}}\\{\varphi_{1}\cdots\varphi_{A}\\},\\\ $
(2)

where $P_{r}$ is the parity operator, and each nucleon is described by
Gaussian wave packet [22],

$\varphi_{p}=\prod_{\sigma=x,y,z}e^{-\nu_{\sigma}\left(r_{\sigma}-Z_{p\sigma}\right)^{2}}\left(a_{p}\ket{\uparrow}+b_{p}\ket{\downarrow}\right)\eta_{p},$
(3)

where isospin $\eta_{p}$ is fixed to either proton or neutron for all
nucleons. The parameter $\bm{Z}_{p}$ represents the mean position and momentum
of each nucleon. By controlling the parameter $\bm{Z}_{p}$, the rearrangement
of nucleons can be described in an unified manner. In the practical
calculations of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$,
$\alpha+{}^{20}\mathrm{Ne}$, and $p+{}^{23}\mathrm{Na}$ channels,
$\bm{Z}_{p}$, spin ($a_{p}$ and $b_{p}$), and width parameters $\bm{\nu}$ are
determined by the energy variation with the constraint on the inter-nuclear
distance $d_{i}$ [23]. Through the variational calculations, the rotation and
polarization of nuclei are naturally described depending on the inter-nuclear
distance. The wave functions for the compound nucleus ${}^{24}{\rm Mg}$ are
calculated by the energy variation with the constraint on the quadrupole
deformation parameter $\beta$ [24].

Figure 1:  (a) The energies of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$
molecular resonances calculated by using the D1M* and SkM* functionals with
only the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ channel . The lines connect the
states with strong E2 transitions. The observed data [25, 26, 27, 28, 29, 30,
31, 32, 33, 34] are also shown for comparison. (b) The energies of the
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ molecular resonances calculated by using
the D1M* functional without (white bars) and with (blue bars) the channel
coupling. The lengths of the blue bars indicate the overlap of the wave
functions with the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ channel.

##  3 Results and Discussions

Figure 1(a) shows the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ molecular bands
calculated with only the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ channel,
neglecting the coupling with the exit channels and compound states. For
comparison, we show the results obtained by using the D1M* and SkM*
functionals. Both density functionals yield two molecular bands, and the lower
one qualitatively reproduces the observed resonances [25, 26, 27, 28, 29, 30,
31, 32, 33, 34] from low to high spin. The D1M* yields lower resonance
energies than the SkM*, and this trend is common to other parameter sets,
i.e., the Gogny functionals always yield lower resonance energies than the
Skyrme functionals. This may be due to the finite-range two-body interaction
of the Gogny functionals, which gives stronger attraction between two nuclei
at large distances. As discussed later, this makes the qualitative difference
in the low-temperature reaction rate.

Now, we discuss the channel coupling effects. Figure 1(b) shows how the
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ molecular states are fragmented into
many resonances by the channel coupling in the case of the D1M* functional.
The white bars show the energies of the $J^{\pi}=0^{+}$ and $2^{+}$ resonances
obtained with only the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ channel, whereas
the blue bars show the resonances obtained by including all channels. The
lengths of blue bars indicate the magnitude of the overlap between the wave
functions obtained by including all channels and those obtained by including
only the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ channel (white bars). It is
clear that the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ molecular states are
fragmented into many resonances due to the channel coupling. Note that some of
the resonances are located down to 1 MeV within the Gamow window of the XRSBs.
In contrast, Skyrme functionals do not yield the resonances as low as 1 MeV,
because the energies of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ molecular
states without the channel coupling is too high.

We apply the $R$-matrix theory [35] and the Breit-Wigner formula [36] to
evaluate resonance parameters and $S^{\ast}$-factors. We calculate the reduced
width amplitude (RWA) [37, 38] which is the overlap between the decay channel
and resonance wave functions,

$\displaystyle a^{2}y_{l}^{A_{1}+A_{2}}(a)$
$\displaystyle=\sqrt{\left(\begin{array}[]{c}24\\\ A_{1}\\\
\end{array}\right)}\Braket{\delta(r-a)[\Phi_{A_{1}}\Phi_{A_{2}}Y_{l}(\hat{r})]^{J}_{M}}{\Psi^{J\pi}_{M}},$
(6)

where the wave function for the decay channel is composed of nuclei with
masses $A_{1}$ and $A_{2}$ separated by the distance $a$ with the orbital
angular momentum $l$. We consider six decay channels,
$\alpha+{}^{20}\mathrm{Ne}(0^{+}_{1})$,
$\alpha+{}^{20}\mathrm{Ne^{*}}(2^{+}_{1})$,
$\alpha+{}^{20}\mathrm{Ne}^{\ast}(4_{1}^{+})$,
$p+{}^{23}\mathrm{Na}(3/2^{+}_{1})$, $p+{}^{23}\mathrm{Na^{*}}(5/2^{+}_{1})$,
and $p+{}^{23}\mathrm{Na}^{\ast}(7/2_{1}^{+})$, which are denoted by
$\alpha_{0}$, $\alpha_{1}$, $\alpha_{2}$, $p_{0}$, $p_{1}$, and $p_{2}$,
respectively. We also calculated $\alpha_{3}$ and $\alpha_{4}$ channels and
found that they are negligible. The wave functions of $\alpha$,
${}^{20}\mathrm{Ne}$, and ${}^{23}\mathrm{Na}$ are also calculated by AMD. In
Table 1, we list the RWAs of the resonances as the ratio to the Wigner limit,

$\theta^{2}_{A_{1}+A_{2},l}(a)=\frac{a}{3}\left|ay_{l}^{A_{1}+A_{2}}(a)\right|^{2},$
(7)

where the channel radius $a$ is chosen to connect the RWAs to the Coulomb wave
function smoothly. In principle, if the channel radius $a$ is sufficiently
large so that nuclear force and channel coupling are negligible, the decay
width calculated by Eq. (6) does not depend on the choice of channel radius.
However, because we performed numerical calculations within a finite spatial
size, the decay widths and the reaction rates calculated from them depend on
the channel radius. In order to assess this uncertainty, we also made the
calculations by artificially changing the channel radius by 1 fm. We found
that the reaction rates change by, at most, a factor of two. Thus, the impact
of this uncertainty on astrophysical applications is relatively minor,
although it cannot be ignored from a nuclear physics perspective. The partial
decay width $\Gamma_{A_{1}+A_{2},l}$ is calculated from the RWA,

$\Gamma_{A_{1}+A_{2},l}=P_{\mathrm{C}}^{(l)}(a)\frac{3\hbar^{2}}{2\mu
a^{2}}\theta^{2}_{A_{1}+A_{2},l}(a).$ (8)

$P_{\mathrm{C}}^{(l)}$ is the Coulomb penetration factor,

$P_{\mathrm{C}}^{(l)}=\frac{2ka}{F_{l}^{2}(ka)+G_{l}^{2}(ka)},$ (9)

where $k$ is the wave number, $F_{l}$ and $G_{l}$ are the regular and
irregular Coulomb functions, respectively, and $\mu$ is the reduced mass.

To evaluate the $S^{\ast}$-factor, we calculate the
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion cross section, assuming that each
resonance is narrow and isolated. In this case, the cross section at the
center-of-mass energy $E$ is given by the Breit-Wigner formula,

$\sigma(E)=\frac{\pi\hbar^{2}(2J+1)}{\mu
E}\frac{\Gamma_{0}\Gamma}{\left(E-E_{\mathrm{R}}\right)^{2}+\Gamma^{2}/4},$
(10)

where $J$ and $E_{\mathrm{R}}$ denote the spin and energy of a resonance,
respectively, and $\Gamma_{0}$ is the partial width of the entrance channel
(${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$). The total width $\Gamma$ is estimated
by a sum over the relevant channels,

$\Gamma=\Gamma_{a_{0}}+\Gamma_{a_{1}}+\Gamma_{a_{2}}+\Gamma_{p_{0}}+\Gamma_{p_{1}}+\Gamma_{p_{2}}.$
(11)

We note that this approximation may bring about additional uncertainty in the
reaction rates. For instance, as seen in the results of D1M* listed in Table
1, the $2^{+}$ resonances at 2.10 and 2.45 MeV are close in energy and
overlapping. In such cases, the shape of the cross section can deviate from
that given by Eq. (8) due to the interference of the resonances. In principle,
such resonance interference should be properly considered, but due to
limitations in the framework of this study, it is neglected.

Given the cross section, the $S^{\ast}$-factor is defined as,

$S^{\ast}(E)=E\sigma(E)/G^{\ast}(E),$ (12)

with the modified Gamow parameter
$G^{\ast}=\exp(2\pi\eta+0.46~{}\mathrm{MeV}^{-1}E)$, and the Sommerfeld
parameter $\eta=36/137\sqrt{\mu c^{2}/2E}$ [39].

Table 1:  The properties of the $J^{\pi}=0^{+}$ and $2^{+}$ resonances obtained by using the D1M* and SkM* density functionals. Only the resonances with energies ($E_{R}$) lower than 4 MeV are listed. $\Gamma$ is a total decay width, and $\theta^{2}_{\rm C}$ is a dimensionless reduced width amplitude for the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ channel in percent at channel radius $a=6$ fm (asterisked numbers are for $a=7$ fm). Branching ratios larger than $10^{-2}$ are also listed. $M(\mathrm{IS}\lambda\uparrow)$ is the matrix element of the isoscalar monopole (quadrupole) transitions from the ground state of ${}^{24}{\rm Mg}$ to the $J^{\pi}=0^{+}$ ($2^{+}$) resonances. DF | $J^{\pi}$ |  $E_{\mathrm{R}}$ [MeV] |  $\Gamma$ [MeV] |  $\theta^{2}_{\rm C}$ [%] | branching ratio |  $M(\mathrm{IS}\lambda\uparrow)$ [W.u.]
---|---|---|---|---|---|---
|  |  |  |  | $\alpha_{0}$ | $\alpha_{1}$ | $\alpha_{2}$ | $p_{0}$ | $p_{1}$ | $p_{2}$ |
D1M* | $2^{+}$ | 0.40 | 0.14 | 0.6* | .31 | .24 |  | .19 | .25 |  | 1.37
| $0^{+}$ | 0.93 | 0.47 | 4.8 | .93 | .04 |  | .01 |  |  | 0.39
| $2^{+}$ | 1.06 | 0.10 | 1.1* | .54 | .03 |  | .27 | .14 |  | 2.34
| $2^{+}$ | 2.10 | 0.45 | 0.6* | .40 | .40 |  | .02 | .15 |  | 3.89
| $0^{+}$ | 2.33 | 0.63 | 8.8 | .58 | .38 |  |  | .03 |  | 0.74
| $2^{+}$ | 2.45 | 0.18 | 0.2* | .08 | .67 |  | .09 | .13 |  | 0.96
| $2^{+}$ | 3.21 | 0.67 | 5.8 | .42 | .46 | .01 | .03 | .06 | .01 | 3.68
| $0^{+}$ | 3.84 | 0.73 | 8.4 | .26 | .65 | .02 | .01 | .04 |  | 1.07
| $2^{+}$ | 4.29 | 0.79 | 7.8 | .30 | .31 | .16 | .12 | .07 | .01 | 1.90
SkM* | $2^{+}$ | 2.42 | 0.36 | 1.2* | .01 | .13 |  | .53 | .27 | .05 | 3.23
| $2^{+}$ | 3.30 | 0.38 | 1.8 | .20 | .20 |  | .20 | .36 | .01 | 1.42
| $0^{+}$ | 3.49 | 0.32 | 11.1 | .65 |  |  | .06 | .27 |  | 0.94
| $2^{+}$ | 4.27 | 2.41 | 0.5* |  |  |  | .68 | .26 | .04 | 1.83
Figure 2:  ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion S*-factors obtained by
using Skyrme and Gogny density functionals. The experimental data [40, 6, 8,
7], the evaluations by CF88 [10], and the hindrance model [11] are also shown.

Figure 2 shows the calculated $S^{\ast}$-factors from all density functionals
in comparison with the observed data and extrapolations. The resonance
contributions appear as peak structures in the $S^{\ast}$-factor. In the
$E>2.5$ MeV region, all density functionals yield the $S^{\ast}$-factors with
the same order of magnitude, which are roughly consistent with the observed
data, but slightly lower than the CF88. On the other hand, there is striking
difference in the low-energy region with $E\lesssim 2$ MeV. The
$S^{\ast}$-factors obtained from the Gogny D1M* and D1S functionals yield
prominent peaks at 1 and 1.8 MeV, which enhance the $S^{\ast}$-factor
comparable with CF88. In contrast, the Skyrme functionals predict no
significant peaks in this region, and their $S^{\ast}$-factors are closer to
the hindrance model.

This remarkable difference in the $S^{\ast}$-factors originates in the low-
energy $0^{+}$ and $2^{+}$ resonances, which are listed in Table 1 for the
D1M* and SkM* functionals. For example, a notable peak at 1 MeV in the
$S^{\ast}$-factor obtained by the D1M* is created by the $0^{+}$ and $2^{+}$
resonances at 0.93 and 1.06 MeV, whereas they do not exist in the SkM*
results. We remark that the contributions from the $2^{+}$ resonances are
important. Since the peak height of the $S^{\ast}$-factor created by a single
resonance is proportional to

$(2J+1)\frac{P_{\mathrm{C}}^{(l)}(a)}{G^{\ast}(E_{\mathrm{R}})}\frac{\theta^{2}_{\rm
C}(a)}{\Gamma},$ (13)

the $2^{+}$ resonance contributions are amplified by a factor of $2J+1$ if
other factors are the same order of magnitude. This is case for the $2^{+}$
resonance at 1.06 MeV, and it manifests itself as a prominent peak at 1 MeV.

Thus, the deep sub-barrier resonances with $J^{\pi}=0^{+}$ and $2^{+}$
potentially have large impact on the reaction rate, and hence, it is of utmost
importance to experimentally search for them. We have suggested identifying
the $0^{+}$ resonances using their strong isoscalar monopole transitions as a
probe [13]. This approach was successfully applied in an $\alpha$-inelastic
scattering experiment [41], leading to the discovery of a $0^{+}$ resonance at
1.38 MeV. Interestingly, this resonance is located between the $0^{+}$
resonances obtained from the D1M* and D1S functionals, which are located at
0.93 and 1.64 MeV, respectively. Therefore, the reality may lie somewhere
between the predictions of D1M* and D1S. Similarly, we suggest exploring the
$2^{+}$ resonances, which potentially have greater impact on the reaction
rate, by means of the isoscalar quadrupole transitions. As shown in Table 1,
the transition strengths of the $2^{+}$ resonances are greater than the
Weisskopf unit, thereby demonstrating the feasibility of this approach.

We also comment on the partial decay widths of the $\alpha_{0}$ and $p_{0}$
channels. In the recent particle-$\gamma$ coincidence measurements [6, 8, 7],
these widths could not be measured. Therefore, they were estimated by assuming
a linear dependence of the branching ratio on the energy [40]. Since we do not
find such clear linearity in our results, it would be desirable to directly
measure them. The $\alpha$-inelastic scattering experiments mentioned above,
in principle, can measure the partial widths.

Finally, we discuss the nuclear reaction rate $N_{\mathrm{A}}\braket{\sigma
v}$ which is defined as a function of temperature $T$,

$\displaystyle N_{\mathrm{A}}\braket{\sigma
v}=\sqrt{\frac{8}{\pi\mu}}\frac{N_{\mathrm{A}}}{(k_{\mathrm{B}}T)^{3/2}}\int_{0}^{\infty}E\sigma(E)e^{-\frac{E}{k_{\mathrm{B}}T}}\mathrm{d}E,$
(14)

where $N_{\mathrm{A}}$ and $k_{\mathrm{B}}$ denote the Avogadro and Boltzmann
constants, respectively.

Figure 3:  The ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion reaction rates
relative to CF88 obtained by using the Gogny and Skyrme functionals. The upper
scale shows the Gamow peak energies in MeV. The reaction rates obtained from
the THM experiment [9] and the hindrance model [11] are also shown.

Figure 3 shows the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion reaction rates
relative to CF88, which exhibits the same trend as in the $S^{*}$-factor. At
higher temperatures, all density functionals consistently yield the reaction
rate slightly smaller than CF88. In contrast, at lower temperature $T<1$ GK,
the Gogny and Skyrme functionals predict different behavior. The former give
the reaction rates slightly smaller than CF88, whereas the latter give much
hindered rates comparable with the hindrance model. We consider that the
region enclosed by the reaction rates from the Gogny and Skyrme functionals is
a constraint on the reaction rates imposed by the nuclear density functionals.
The THM reaction rate is significantly higher than this, whereas the hindrance
model approximates its lower limit. We also remark that, very recently, Dohi
et al. [42] performed simulations of XRSB using these reaction rates (THM,
CF88 and D1S) and found differences in the results that are sufficient to
distinguish the THM rate from the others.

##  4 Summary

Using a full-microscopic nuclear model with various nuclear density
functionals, we have assessed the fusion reaction rate of
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ at stellar temperatures. The reaction
rates obtained from Gogny functionals are close to the CF88 estimation. In
contrast, Skyrme functionals predict lower reaction rates, similar to the
hindrance model. We interpret the intermediate region between the reaction
rate obtained from the Gogny and Skyrme functionals as the constraints imposed
by the nuclear density functionals.

The difference between the Gogny and Skyrme functionals owes to the
$J^{\pi}=0^{+}$ and $2^{+}$ resonances at deep sub-barrier energies. In the
case of the Gogny functionals, ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ molecular
resonances appear approximately 1 MeV due to its coupling with other channels,
leading to a substantial enhancement of the reaction rate. On the other hand,
in the case of the Skyrme functionals, such resonances appear slightly higher
energies, and do not contribute to the reaction rate at low temperatures.

Hence, exploring the $J^{\pi}=0^{+}$ and $2^{+}$ resonances at deep sub-
barrier energy is a crucial step toward reducing the uncertainties in the
reaction rate. For this purpose, we propose the measurements of the isoscalar
transitions associated with these resonances. In particular, the $2^{+}$
resonances, which potentially have great impact on the reaction rate, can be
identified from their enhanced isoscalar dipole transitions.

We thank Dr. Nishimura and Dr. Dohi for helpful discussions. This work was
partly supported by the RCNP Collaboration Research Network program as the
project number COREnet-039. This research used computational resources of
Wisteria/BDEC-01 Odyssey (the University of Tokyo), provided by the
Multidisciplinary Cooperative Research Program in the Center for Computational
Sciences, University of Tsukuba. This work was financially supported by the
JSPS KAKENHI Grant Nos. 22K03610 and 22H01214.

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