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11institutetext:  Graduate School of Science and Technology, University of
Tsukuba, Tsukuba, 305-8571, Japan 22institutetext: KEK Theory Center,
Institute of Particle and Nuclear Studies, High Energy Accelerator Reserach
Organization, Tsukuba 305-0801, Japan 33institutetext: Center for
Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan
44institutetext: Faculty of Pure and Applied Sciences, University of Tsukuba,
Tsukuba 305-8571, Japan 55institutetext: Facility for Rare Isotope Beams,
Michigan State University, East Lansing, Michigan 48824, USA 66institutetext:
RIKEN Nishina Center, Wako 351-0198, Japan

# Collective-subspace requantization for sub-barrier fusion reactions:
Inertial functions for collective motions

Chisato Ruike 11 Kai Wen 2233 Nobuo Hinohara 334455 Takashi Nakatsukasa 334466

###### Abstract

The adiabatic self-consistent collective coordinate (ASCC) method is used to
determine the optimum reaction path and to calculate the potential and the
inertial functions of the reaction model. The properties of the inertial
functions are investigated with the ASCC method, in comparison with those of
the cranking formulae. In addition, the properties of the pair rotation are
investigated in the BCS pair model. The moments of inertia for rotation in
both the real and the gauge spaces may decrease as the deformation develops.

##  1 Introduction

Time-dependent density functional theory (TDDFT) is a standard tool for
studies of a variety of heavy-ion reactions Neg82 ; Sim12 ; NMMY16 , such as
nuclear fusion above the Coulomb barrier energy GN12-P ; Was15 ; EN15-P ,
induced fission reaction BMRS16 , quasi-fission reaction UOS16 , and multi-
nucleon transfer reactions Sim10 ; SY16 . It also provides an important method
in studies of nuclear response, namely the (quasiparticle) random-phase
approximation ((Q)RPA) in the linear regime. In addition, the properties of
the giant resonances have been extensively investigated with TDDFT real-time
simulations NY05 ; Eba10 ; SBMR11 ; SL13-2 . A universal description of nuclei
in the entire mass region and a unified microscopic description of nuclear
structure and reaction are great features of the TDDFT method.

On the other hand, the current TDDFT models of nuclei have a problem in a
proper account of fluctuation effects. The problems are especially serious in
transitional regions where the nuclear shape fluctuates and is not well-
defined. In nuclear reaction studies, a prominent weak point is evident for
the sub-barrier fusion and the spontaneous fission reactions, which never
takes place in the straightforward TDDFT calculations. The theory lacks a part
of important quantum fluctuation associated with the nuclear shape. In order
to recover the missing quantum fluctuation, theories beyond the mean field are
required.

The most popular method of improving the mean-field theory is the generator
coordinate method (GCM) RS80 ; Sch19 . The GCM is a quantum mechanical
approach and is capable of describing the quantum fluctuations along the
chosen generator coordinates. Increasing the number of generator coordinates,
in principle, the solution converges to the exact eigenstates of the
Hamiltonian. However, this is not the case for the energy density functionals.
Although many of the nuclear energy density functionals are based on density-
dependent effective interactions, the exact ground state of the Hamiltonian is
not physical. Therefore, we must carefully choose a small number of generator
coordinates to avoid such unphysical states. This also prevents us from
utilizing a variational method to determine the generator coordinates SONY06 ;
Fuk13 . The GCM has both numerical and theoretical problems NMMY16 ,
especially when we use the effective interactions which depend on a fractional
power of density AER01 ; DSNR07 ; Dug09 .

Our strategy for a solution to these problems is the requantization of
collective degrees of freedom which are selected according to the TDDFT
dynamics. Similar approaches have been adopted in studies of low-lying
excitation spectra and in the fission dynamics, e.g., the five-dimensional
collective Hamiltonian (5DCH) method instead of the GCM on the
$(\beta,\gamma)$ plane KB67 ; Del10 ; NMMY16  and the collective model of
spontaneous fission Sad20 . However, in these models, the shape variables are
chosen by hand and the inertia parameters are approximated by the cranking
formula Ing54 ; Ker61 . We aim to improve the models in terms of the
microscopic point of view.

We determine a collective subspace which is decoupled from the other intrinsic
degrees of freedom. For this purpose, we use the adiabatic self-consistent
collective coordinate (ASCC) method MNM00 ; HNMM07 ; HNMM09 ; Nak12 ; NMMY16 .
The solution of the ASCC equations provides a set of canonical variables
$(q,p)$ to specify the one-body density $\hat{\rho}(q,p)$, and the generators
of collective variables $(\hat{Q}(q),\hat{P}(q))$. The collective potential is
given by the energy values at $p=0$, $V(q)=E[\rho(q,p=0)]$, and the expansion
with respect to the $p$ provides the kinetic energy with the inertial masses,
$M^{-1}(q)=\partial^{2}E/\partial p^{2}|_{p=0}$. This leads to the collective
Hamiltonian

$H=\frac{p^{2}}{2M(q)}+V(q).$ (1)

Since the scale of the coordinate $q$ is arbitrary, we can adjust it so as to
make the collective inertia constant, $M(q)=\mu$. In the study of the sub-
barrier nuclear fusion, we expect that the coordinate $q$ should be
proportional to the distance between the projectile and the target nuclei,
$r$, when two nuclei are far apart. Thus, it is intuitive to write down the
collective Hamiltonian in the relative coordinate $r$ and its conjugate
momentum $p_{r}=p(dq/dr)$. It should be noted that the collective inertia
$M(r)$, given by $\\{M(r)\\}^{-1}=(dr/dq)^{2}\mu^{-1}$, is no longer constant,
especially at $r\lesssim R_{\rm touch}$ where $R_{\rm touch}$ is a touching
distance of two nuclei.

Assuming the axial symmetry with respect to the axis of the relative
coordinate $\mathbf{r}$ between the projectile and the target, two angles to
fix the direction of $\mathbf{r}$ describe the rotational motion, and their
moments of inertia are calculated solving the Thouless-Valatin equation TV62
at each point on the collective subspace $r$. Eventually, the following
collective Hamiltonian is obtained.

$H=\frac{p_{r}^{2}}{2M(r)}+\frac{L^{2}}{2\mathscr{I}(r)}+V(r),$ (2)

where $L^{2}$ is the square of the angular momentum produced by the rotation
of the coordinate $\mathbf{r}$. In the end, the canonical quantization is
performed, replacing $p_{r}$ by $-i\partial/\partial q$ ($\hbar=1$) with the
Pauli’s prescription and $L^{2}$ by the angular momentum operator
$\hat{L}^{2}$. It should be noted that the point transformation from $q$ to
$r$ does not change the physics, as far as the inertial functions and the
conjugate momentum are properly defined.

In this paper, we focus our discussion on the inertial functions for various
collective motions. In Sec. 2, we briefly review the ASCC method and present
methods of calculating the inertial functions in Sec. 3. We especially,
describe the derivation of the cranking formulae based on the TDDFT equation
of motion, which can clarify the missing correlations in the cranking
formulae. In Sec. 5, we show a somewhat surprising property of the moment of
inertia for the pair rotation. The conclusion and the perspectives are given
in Sec. 6.

##  2 Collective subspace

Using the energy density functional $E[\rho]$ and the local (normal-mode)
generator $Q(q)$, the collective subspace for the low-energy collective motion
is defined by

$\biggr{[}h[\rho(q)]-\lambda Q(q),\rho(q)\biggr{]}=0,$ (3)

where $h_{ij}\equiv\delta E/\delta\rho_{ji}$. The Lagrange multiplier
$\lambda$ corresponds to the gradient of the energy, $\lambda=\partial
E[\rho(q)]/\partial q$. The generators, $(Q(q),P(q))$, are given by a solution
of the moving-frame random-phase approximation (MF-RPA).

$\left\\{A(q)+B(q)\right\\}\left\\{A(q)-B(q)\right\\}Q(q)=\omega^{2}Q(q),\quad\quad
P(q)=M(q)\left\\{A(q)-B(q)\right\\}Q(q),$ (4)

where the matrices $A(q)$ and $B(q)$ are given by

$A_{php^{\prime}h^{\prime}}(q)=(\epsilon_{p}-\epsilon_{h})\delta_{pp^{\prime}}\delta_{hh^{\prime}}+\frac{\partial
h_{ph}}{\partial\rho_{p^{\prime}h^{\prime}}},\quad\quad
B_{php^{\prime}h^{\prime}}(q)=\frac{\partial
h_{ph}}{\partial\rho_{h^{\prime}p^{\prime}}}.$ (5)

The MF-RPA equation determines only the particle-hole and hole-particle matrix
elements of the generators, $(Q(q),P(q))$. In this paper, we assume that their
particle-particle and hole-hole matrix elements vanish. The single-particle
energies $\epsilon_{i}$ in Eq. (5) are the eigenvalues of $h-\lambda Q(q)$.
The derivatives in Eq. (5) are calculated according to the finite-amplitude
method (FAM) NIY07 ; AN11 ; Sto11 , especially using the m-FAM scheme in Ref.
AN13 .

The MF-RPA equation (3) cannot fix the magnitude of $Q(q)$. Since these
equations are scale invariant concerning $Q(q)\rightarrow\alpha Q(q)$,
$P(q)\rightarrow\alpha^{-1}P(q)$, and $M(q)\rightarrow\alpha^{-2}M(q)$, we can
arbitrarily normalize $Q(q)$. In actual calculations, we set $M(q)$ a constant
value, $M(q)=\mu$, to determine the normalization of $Q(q)$ according to the
weakly canonicity condition,
$\mathrm{Tr}\left(\left[Q(q),P(q)\right]\rho(q)\right)=i$.

The ASCC equations consist of Eqs. (3) and (4) which should be solved self-
consistently. Among solutions of Eq. (4), we select a normal mode of the
lowest energy keeping the axial symmetry. Note that $\omega^{2}$ can be
negative. The numerical computation produces a series of density
$\\{\rho(q_{0}),\rho(q_{1}),\cdots\\}$ on the discretized collective
coordinate $q_{n}=q_{0}+n\times\Delta q$. This defines the collective
subspace, namely the reaction path in the present study. The potential in Eq.
(1), $V(q)=E[\rho(q)]$, then, transformed into $V(r)=V(q(r))$ in Eq. (2).

##  3 Inertial functions

In this paper, we focus our discussion on the properties of collective
inertias and their effect on the nuclear dynamics.

###  3.1 ASCC inertial functions

Since the ASCC method provides a decoupled collective subspace, the inertial
mass tensor should be block-diagonal: no off-diagonal elements between the
collective subspace and the rest. The value of the constant mass $\mu$
determines the scale of the coordinate $q$, then, the inertial function $M(r)$
in Eq. (2) is given by

$M(r)=\mu\left(\frac{dq}{dr}\right)^{2},\quad\quad\left(M_{ij}(r)=\sum_{kl}\frac{\partial
q^{k}}{\partial r_{i}}\mu_{kl}\frac{\partial q^{l}}{\partial r_{j}}\right).$
(6)

The second equation in the bracket corresponds to the case that the collective
subspace is multi-dimensional, in general.

The rotational modes correspond to the zero-frequency solutions of Eq. (4).
For the axially symmetric states, we should obtain two independent solutions
corresponding to the rotation around the axes perpendicular to the symmetry
axis. Since these modes violate the axial symmetry ($K\neq 0$), they are
exactly decoupled from the adopted axially symmetric mode ($K=0$),
$(Q(q),P(q))$ in Sec. 2.

Although the rotational moments of inertia can be deduced from the zero-
frequency solutions of Eq. (4), they can be even more easily obtained from the
calculation of the strength function at the zero frequency Hin15 . We use this
method to calculate the moments of inertia $\mathscr{I}(r)$ in Eq. (2). This
is equivalent to solutions of the Thouless-Valatin equations TV62 ; RS80  with
the moving-frame Hamiltonian.

###  3.2 Cranking mass

The most popular method to estimate the collective masses is the cranking
formula Ing54 ; Ker61 . It has been utilized in a variety of applications.
However, it has been known for many years that the collective masses of the
cranking formula have serious problems with missing dynamical residual
effects.

####  3.2.1 Translational motion

In this section, first, we present a derivation of the cranking formula for
the translational motion, and demonstrate the importance of the residual
effects.

In the TDDFT, a nucleus in the ground state is given by the stationary
condition for the density $\rho$.

$\left[h_{0},\rho_{0}\right]=0,$ (7)

where $\rho_{0}$ is the one-body density of the ground state. $h_{0}$ is the
Kohn-Sham (KS) Hamiltonian at the density $\rho_{0}$ which is represented by
the KS orbitals $\ket{\phi_{i}}$, defined by the KS equation
$h_{0}\ket{\phi_{i}}=\epsilon_{i}\ket{\phi_{i}}$, as

$\rho_{0}=\sum_{h=1}^{A}\ket{\phi_{h}}\bra{\phi_{h}}.$ (8)

The center of mass of $\rho_{0}$ is arbitrary and chosen at the origin. Let us
consider a nucleus in the ground state, moving in the $x$ direction with a
constant velocity $v$. The density is given as

$\rho=e^{imv\hat{x}}e^{-ix(t)\hat{p}}\rho_{0}e^{ix(t)\hat{p}}e^{-imv\hat{x}},$
(9)

where $m$ is the nucleon mass. The expectation values of the center of mass
$X$ and the total momentum $P$ are

$\displaystyle
X=\frac{1}{A}\mathrm{Tr}\left(\hat{x}\rho\right)=\frac{1}{A}\mathrm{Tr}\left(\hat{x}\rho_{0}\right)+\frac{x(t)}{A}\mathrm{Tr}\left(\rho_{0}\right)=x(t),$
(10) $\displaystyle
P=\mathrm{Tr}\left(\hat{p}\rho\right)=\mathrm{Tr}\left(\hat{p}\rho_{0}\right)+mv\mathrm{Tr}\left(\rho_{0}\right)=Amv.$
(11)

Here, we use the commutation relation $[\hat{x},\hat{p}]=i$ and the fact that
the trace of a product is commutable. Equation (11) indicates that the mass
for the translational motion is nothing but the total mass $Am$. These
equations, (10) and (11), are valid at any values of $x$ and $v=\dot{x}$.
Assuming $x$ and $v$ are small, we may linearize the density variation as
$\delta\rho=x\delta\rho^{(x)}+v\delta\rho^{(v)}$ with
$\delta\rho^{(x)}=-i[\hat{p},\rho_{0}]$ and
$\delta\rho^{(v)}=im[\hat{x},\rho_{0}]$. It is easy to verify that
$\rho=\rho_{0}+\delta\rho$ produces the same results as Eqs. (10) and (11).
Note that the particle-particle and hole-hole matrix elements of $\delta\rho$
vanish,
$\bra{\phi_{p}}\delta\rho\ket{\phi_{p^{\prime}}}=\bra{\phi_{h}}\delta\rho\ket{\phi_{h^{\prime}}}=0$.

The density of Eq. (9) should satisfy the TDDFT equation
$i\dot{\rho}=\left[h[\rho],\rho\right]$. Now, we linearize the equation as

$i\frac{\partial}{\partial
t}\left(x\delta\rho^{(x)}+v\delta\rho^{(v)}\right)=\left[h_{0},x\delta\rho^{(x)}+v\delta\rho^{(v)}\right]+\left[x\delta
h^{(x)}+v\delta h^{(v)},\rho_{0}\right],$ (12)

where $\delta h=x\delta h^{(x)}+v\delta h^{(v)}$ is the residual field induced
by the density variation, $\delta h^{(i)}=\partial
h/\partial\rho|_{\rho_{0}}\cdot\delta\rho^{(i)}$ with $i=x,v$. The term
proportional to $x$, $x\delta h^{(x)}$, represents a change in the KS
potential for a shift of the center of mass, that is present when the
translational symmetry is violated in $h_{0}$. The term proportional to $v$,
$v\delta h^{(v)}$, is induced by the velocity of the nucleus, which appears
when the KS potential has the velocity dependence. The left-hand side of Eq.
(12) is

$\mathrm{LHS}=i\frac{\partial}{\partial
t}\left(x\delta\rho^{(x)}+v\delta\rho^{(v)}\right)=iv\delta\rho^{(x)}=v\left[\hat{p},\rho_{0}\right],$
(13)

and the right-hand side is

$\mathrm{RHS}=\left[h_{0},\delta\rho\right]+\left[\delta
h,\rho_{0}\right]=v\left[h_{0},\delta\rho^{(v)}\right]+v\left[\delta
h^{(v)},\rho_{0}\right].$ (14)

Here, we use the condition that the density (9) at any center-of-mass
coordinate with $v=0$ must satisfy the stationary equation (7). The particle-
hole and hole-particle matrix elements of Eq. (12) lead to

$\displaystyle\bra{\phi_{p}}\hat{p}\ket{\phi_{h}}=\left(\epsilon_{p}-\epsilon_{h}\right)\bra{\phi_{p}}\delta\rho^{(v)}\ket{\phi_{h}}+\bra{\phi_{p}}\delta
h^{(v)}\ket{\phi_{h}},$ (15)
$\displaystyle\bra{\phi_{h}}\hat{p}\ket{\phi_{p}}=\left(\epsilon_{p}-\epsilon_{h}\right)\bra{\phi_{h}}\delta\rho^{(v)}\ket{\phi_{p}}+\bra{\phi_{h}}\delta
h^{(v)}\ket{\phi_{p}}.$ (16)

The expectation value of the total momentum is calculated as

$P=\mathrm{Tr}\left(\hat{p}\delta\rho\right)=v\mathrm{Tr}\left(\hat{p}\delta\rho^{(v)}\right)=v\sum_{p,h}\frac{\bra{\phi_{p}}\hat{p}\ket{\phi_{h}}\bra{\phi_{h}}\left(\hat{p}-\delta
h^{(v)}\right)\ket{\phi_{p}}+\mathrm{c.c.}}{\epsilon_{p}-\epsilon_{h}},$ (17)

which should be equal to $Amv$, as we have shown in Eq. (11).

The cranking formula is obtained by the approximation to neglect the residual
field $\delta h^{(v)}$ in Eq. (17), that results in the mass of the
translational motion,

$M=\frac{dP}{dv}=2\sum_{p,h}\frac{\left|\bra{\phi_{p}}\hat{p}\ket{\phi_{h}}\right|^{2}}{\epsilon_{p}-\epsilon_{h}}.$
(18)

In general, the cranking formula (18) produces $M\neq Am$. This is due to the
violation of the Galilean invariance of the static KS Hamiltonian $h_{0}$. In
fact, if $h_{0}$ satisfies the relation,

$i\left[h_{0},\hat{x}\right]=\frac{\hat{p}}{m},$ (19)

then, we can rewrite the quantity

$\left[\hat{p},\rho_{0}\right]=im\left[\left[h_{0},\hat{x}\right],\rho_{0}\right]=im\left[h_{0},\left[\hat{x},\rho_{0}\right]\right]=\left[h_{0},\delta\rho^{(v)}\right],$
(20)

where we use Eq. (7). This guarantees that the second term on the right-hand
side of the TDDFT equation (14) vanishes. See Eqs. (13) and (14). Thus, the
cranking formula gives the exact total mass, $M=Am$. In other words, when Eq.
(19) is violated for $h_{0}$, the residual field $\delta h^{(v)}$ is necessary
to restore the Galilean invariance of the energy density functional $E[\rho]$.

####  3.2.2 Rotational motion

The arguments for the translational motion can be directly applicable to the
motion of the uniform rotation. When the ground-state solution of the KS
equation violates the rotational symmetry, the rotational degrees of freedom
emerge from the orientation of the deformed intrinsic state. Assuming the
rotation around the $x$ axis with a constant angular velocity
$\omega=\dot{\theta}$, the density is given, similar to Eq. (9), by

$\rho=e^{iI\omega\hat{\theta}}e^{-i\theta(t)\hat{j}_{x}}\rho_{0}e^{i\theta(t)\hat{j}_{x}}e^{-iI\omega\hat{\theta}},$
(21)

where the angle operator $\hat{\theta}$ is a conjugate variable to the angular
momentum $\hat{j}_{x}$, namely $[\hat{\theta},\hat{j}_{x}]=i$.
$I=\mathscr{I}_{x}/A$ is linked to the moment of inertia of the nucleus,
$\mathscr{I}_{x}$. Here, we only assume the existence of the angle operator
$\hat{\theta}$ and the moment of inertia $\mathscr{I}$, but do not need the
exact expression of those quantities111 This is different from the case of
translational motion. We explicitly know both the center-of-mass coordinate,
the total linear momentum, and the exact value of the total mass in Sec.
3.2.1. . In addition, in contrast to the translational case in Sec. 3.2.1, the
decoupling between the rotational and the intrinsic motions is not exact.
Therefore, strictly speaking, Eq. (21) is true only when $\omega=0$, and the
density $\rho_{0}$ in Eq. (21) should change at a non-zero value of $\omega$.
In this section, we assume a small value of $\omega$ and neglect the
$\omega$-dependence of the intrinsic structure.

The rest of the derivation is the same as those in Sec. 3.2.1. Neglecting the
residual fields, $\delta h^{(\omega)}$, the cranking formula for the moment of
inertia is given by

$\mathscr{I}_{x}=\frac{dJ}{d\omega}=2\sum_{p,h}\frac{\left|\bra{\phi_{p}}\hat{j}_{x}\ket{\phi_{h}}\right|^{2}}{\epsilon_{p}-\epsilon_{h}}.$
(22)

In the rotational case, there is no exact relation analogous to the Galilean
invariance. Nevertheless, the condition in analogy to Eq. (19) results in the
justification of the neglect of the residual effect. Therefore, the cranking
formula (18) is valid (only) when the KS potential has no velocity dependence.
In general, the residual terms associated with $\delta h^{(\omega)}$
proportional to $\omega$ (terms of $\delta h^{(v)}$ in Eq. (17)) is necessary
to give a proper value for the moment of inertia.

####  3.2.3 Collective motion on the reaction path

Instead of determining the collective subspace by solving Eq. (3), the
generator $\hat{Q}(q)$ is most commonly replaced by a constraining one-body
operator given by hand, $\hat{C}$. The constrained KS density is obtained by

$\left[h_{\lambda},\rho_{\lambda}\right]=0,\quad\quad h_{\lambda}\equiv
h[\rho_{\lambda}]-\lambda\hat{C}$ (23)

where $\lambda$ is the Lagrange multiplier. We may change the label $\lambda$
into the collective coordinate, for instance, the distance between two
colliding nuclei $r$. The relation between $\lambda$ and $R$ is given by
$r=\mathrm{Tr}(\hat{r}\rho_{\lambda})$. The choice of the operator $\hat{r}$
may not be unique, but a possible choice is given in Ref. WN22 . The density
is represented by the KS orbitals $\ket{\phi_{i}(r)}$ which are solutions of
$h_{\lambda}\ket{\phi_{i}(r)}=\epsilon_{i}(r)\ket{\phi_{i}(r)}$.

The cranking formula of the inertial mass for the collective motion with
respect to the coordinate $r$ can be given, in analogy to Eqs. (18) and (22),
as

$M^{\rm np}(r)=2\sum_{p,h}\frac{|\bra{\phi_{p}(r)}\partial/\partial
r\ket{\phi_{h}(r)}|^{2}}{\epsilon_{p}(r)-\epsilon_{h}(r)}.$ (24)

The formula (24) is often called “non-perturbative cranking formula.” This is
why we put the superscript for $M^{\rm np}$. A further approximation can give
“perturbative cranking formula” WN22 , denoted as $M^{\rm p}$ in this paper,

$M^{\rm
p}=\frac{1}{2}S_{3}(\hat{C},\hat{C})\left\\{S_{1}(\hat{r},\hat{C})\right\\}^{-2},$
(25)

with

$S_{k}(\hat{A},\hat{B})\equiv\frac{1}{2}\sum_{p,h}\frac{\bra{\phi_{h}}\hat{A}\ket{\phi_{p}}\bra{\phi_{p}}\hat{B}\ket{\phi_{h}}+\mbox{c.c.}}{\left(\epsilon_{p}-\epsilon_{h}\right)^{k}}=S_{k}(\hat{B},\hat{A}),$
(26)

where $\hat{A}$ and $\hat{B}$ are the one-body operators. In the case of
$\hat{A}=\hat{B}$, $S_{k}(\hat{A},\hat{A})$ corresponds to the $k$th inverse
energy-weighted sum-rule value, $m_{k}(\hat{A})\equiv S_{k}(\hat{A},\hat{A})$.
In the case of $\hat{C}=\hat{r}$, the perturbative cranking formula has become
an even simpler form

$M^{\rm p}=\frac{1}{2}m_{3}(\hat{r})\left\\{m_{1}(\hat{r})\right\\}^{-2},$
(27)

which has been extensively utilized in the literature.

##  4 Results

###  4.1 Inertial functions for sub-barrier nuclear reactions

The energy density functional is slightly modified from $E_{\rm BKN}[\rho]$ of
Ref. BKN76  to allow the KS potential to be velocity-dependent.

$E[\rho]=E_{\rm BKN}[\rho]+B_{3}\int
d\mathbf{r}\left\\{\rho(\mathbf{r})\tau(\mathbf{r})-\mathbf{j}^{2}(\mathbf{r})\right\\},$
(28)

where $\rho(\mathbf{r})$ is the isoscalar density, $\tau(\mathbf{r})$ is the
isoscalar kinetic density, and $\mathbf{j}(\mathbf{r})$ is the isoscalar
current density. The positive value of the parameter $B_{3}$ produces the
effective mass smaller than the bare nucleon mass, $m^{*}/m<1$.

Most of the numerical results are shown in the paper WN22 . Here, we
recapitulate the summary of the results. The ASCC inertial functions
calculated according to Eq. (6) reproduce

  1. 1.

the total mass $Am$ for the translation,

  2. 2.

the reduced mass, $\mu_{\rm red}=A_{1}A_{2}m/(A_{1}+A_{2})$, for the relative
motion at large $r$ ($r\gg R_{\rm touch}$),

  3. 3.

the reduced value of the moment of inertia, $\mathscr{I}_{\rm
red}\equiv\mu_{\rm red}r^{2}$ , at large $r$.

On the other hand, the cranking formulae, Eqs. (18), (22), (24), and (25),
fail to reproduce these quantities. Instead, the cranking formulae
approximately produce

  1. 1.

the total effective mass, $A\langle m^{*}\rangle$ for the translational
motion,

  2. 2.

the reduced effective mass $\mu^{*}_{\rm red}=A_{1}A_{2}\langle
m^{*}\rangle/(A_{1}+A_{2})$ at large $r$

  3. 3.

the moments of inertia $\mu^{*}_{\rm red}r^{2}$ at large $r$

These results apparently indicate the failure of the cranking formulae and the
superiority of the ASCC masses over the cranking masses.

We have also found an interesting property of the moments of inertia, opposite
to our naive intuition: the calculated values decrease in the vicinity of the
ground state as a function of the relative distance $r$, or equivalently as a
function of the nuclear deformation $\beta$.

$\frac{d\mathscr{I}}{dr}<0,\quad\quad\frac{d\mathscr{I}}{d\beta}<0.$ (29)

This is due to the property that the moment of inertia is close to the rigid-
body value at the ground state (without pairing),
$\mathscr{I}\approx\mathscr{I}_{\rm rig}$, then, immediately drops to the
reduced value $\mathscr{I}_{\rm red}$ once two nuclei are apart, $r\gtrsim
R_{\rm touch}$. This is a consequence of the quantum mechanical requirement.
The total moment of inertia for two separated nuclei can be decomposed as

$\mathscr{I}(r)=\mathscr{I}_{1}+\mathscr{I}_{2}+\mathscr{I}_{\rm red}(r),$
(30)

where $\mathscr{I}_{i}$ ($i=1,2$) is the moment of inertia of the nucleus $i$
concerning its own center of mass. When the two nuclei are both spherical, the
quantum mechanics requires $\mathscr{I}_{1}=\mathscr{I}_{2}=0$ and produces
the reduced value, $\mathscr{I}(r)=\mathscr{I}_{\rm red}$ at $r\gtrsim R_{\rm
touch}$.

For more details and possible effects on the fusion cross sections, the reader
is referred to Ref. WN22 .

##  5 Moments of inertia for pair rotation

In a somewhat different context, we have recently found a property, similar to
Eq. (29), that the moments of inertia of the pair rotation decreases as a
function of the pairing gap (deformation in the gauge space) $\Delta$.

$\frac{\partial\mathscr{I}_{\rm pair}}{\partial\Delta}<0,$ (31)

where the moment of inertia is calculated, using the ground-state energies
with different neutron (proton) numbers, as

$\frac{1}{\mathscr{I}_{\rm pair}}=\frac{E(N+2)-2E(N)+E(N-2)}{4}.$ (32)

This behavior can be qualitatively understood as follows, using the BCS
treatment of the pair model. The particle number $N$ is written as

$N(\lambda,\Delta)=\int_{\lambda-\Lambda}^{\lambda+\Lambda}\rho(\epsilon)g(\epsilon-\lambda,\Delta)d\epsilon,\quad\quad
g(\tilde{\epsilon},\Delta)\equiv\frac{1}{2}\left(1-\frac{\tilde{\epsilon}}{\sqrt{\tilde{\epsilon}^{2}+\Delta^{2}}}\right),$
(33)

where $\rho(\epsilon)$ is the single-particle level density and $\lambda$ is
the chemical potential. We truncate the model space for the pairing
correlations to include the single-particle levels in the section of
$2\Lambda$ centered around the chemical potential. The moment of inertia is
given by

$\mathscr{I}_{\rm pair}=\frac{\partial
N}{\partial\lambda}=\int_{\lambda-\Lambda}^{\lambda+\Lambda}\rho(\epsilon)\frac{\partial
g(\epsilon-\lambda,\Delta)}{\partial\lambda}d\epsilon=-\int_{-\Lambda}^{\Lambda}\rho(\tilde{\epsilon}+\lambda)\frac{\partial
g(\tilde{\epsilon},\Delta)}{\partial\tilde{\epsilon}}d\tilde{\epsilon}.$ (34)

Here, we keep the model space invariant when we differentiate the equation
with respect to $\lambda$. Then, its derivative with respect to $\Delta$ is

$\frac{\partial\mathscr{I}_{\rm
pair}}{\partial\Delta}=\frac{\partial^{2}N}{\partial\lambda\partial\Delta}=-\int_{-\Lambda}^{\Lambda}d\tilde{\epsilon}\rho(\tilde{\epsilon}+\lambda)\frac{\partial^{2}g(\tilde{\epsilon},\Delta)}{\partial\tilde{\epsilon}\partial\Delta}=-\frac{1}{2\Delta}\int_{-\Lambda/\Delta}^{\Lambda/\Delta}\rho(\lambda+x\Delta)\frac{1-2x^{2}}{(x^{2}+1)^{5/2}}dx.$
(35)

In the energy interval of $2\Lambda$, we approximate the level density in the
linear order as

$\rho(\lambda+x\Delta)\approx\rho(\lambda)+x\Delta\rho^{\prime}(\lambda),$
(36)

which is inserted into Eq. (35) to obtain

$\frac{\partial\mathscr{I}_{\rm
pair}}{\partial\Delta}=-\frac{\rho(\lambda)}{\Delta}I(\Lambda/\Delta),$ (37)

and

$I(\eta)\equiv\int_{0}^{\eta}dx\frac{1-2x^{2}}{(x^{2}+1)^{5/2}}=\frac{\eta}{(\eta^{2}+1)^{3/2}}.$
(38)

For $\eta=\Lambda/\Delta>0$, it is trivial to have $I(\eta)>0$. Thus, Eq. (31)
is proved.

##  6 Conclusion and future perspectives

Using the ASCC method, we have microscopically derived a collective subspace
(reaction path) for the sub-barrier fusion reaction. In addition, the method
provides the potential and the inertial functions, which leads to the
collective Hamiltonian of the reaction model. In this paper, we particularly
focus our discussion on properties of the inertial functions.

The ASCC inertial functions perfectly agree with the correct values in the
asymptotic region, such as the total mass for the translational motion, the
reduced mass for the relative motion at large $r$, and the reduced moments of
inertia at large $r$. On the other hand, the cranking formulae fail to
reproduce these asymptotic values. This indicates the superiority of the ASCC
method over the cranking formula.

The moment of inertia at the ground state is approximately equal to that of
the rigid-body value, but surprisingly, it decreases as the deformation
increases. We may expect that the isotropic velocity distribution at the
equilibrium is distorted to be anisotropic, leading to a transition from the
“rigid body” to the “irrotational flow.” Since the moment of inertia for the
irrotational fluid vanishes for spherical systems, the two spherical nuclei,
whose mass numbers $A_{1}$ and $A_{2}$, at distance $r$, produce the moment of
inertia, $\mu_{\rm red}r^{2}$. This is well reproduced by ASCC moments of
inertia, but not for the cranking formula.

Further studies on the collective masses are desired, particularly effects of
pairing and beyond the mean field. We have shown that the recovery of the
Galilean invariance violated in the KS potential is essential to produce the
collective inertia. Since the pair field is known to break the Galilean
invariance BM75 , it is of great interest to investigate residual effects of
the pair fields.

We have also studied properties of the pair rotation in the gauge space, then,
have found that the moment of inertia also decreases as the pair deformation
increases. This property can be explained by the BCS pair model. This may
indicate a prominent difference in the effect of the deformation between the
spatial rotation and the pair rotation. Further investigation is under
progress.

Acknowledgments

This work is supported in part by JSPS KAKENHI Grant Nos. 18H01209, 20K14458,
20K03964, 23H01167, 23K25864, and by JST ERATO Grant No. JPMJER2304. This
research in part used computational resources provided by the
Multidisciplinary Cooperative Research Program in the Center for Computational
Sciences, University of Tsukuba.

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