Multiconfigurational time-dependent density functional theory for atomic nuclei: technical and numerical aspects

Abstract

The nuclear time-dependent density functional theory (TDDFT) is a tool of choice for describing various dynamical phenomena in atomic nuclei. In a recent study, we reported an extension of the framework – the multiconfigurational TDDFT (MC-TDDFT) model – that takes into account quantum fluctuations in the collective space by mixing several TDDFT trajectories. In this article, we focus on technical and numerical aspects of the model. We outline the properties of the time-dependent variational principle that is employed to obtain the equation of motion for the mixing function. Furthermore, we discuss evaluation of various ingredients of the equation of motion, including the Hamiltonian kernel, norm kernel, and kernels with explicit time derivatives. We detail the numerical methods for resolving the equation of motion and outline the major assumptions underpinning the model. A technical discussion is supplemented with numerical examples that consider collective quadrupole vibrations in $^{40}$Ca, particularly focusing on the issues of convergence, treatment of linearly dependent bases, energy conservation, and prescriptions for the density-dependent part of an interaction.

Appendices

Appendix A: Local transition densities

The spin expansion of the non-local transition density [Eq. (42)] for isospin $\tau$ reads

$$\begin{aligned} \begin{aligned} \rho _{\varvec{q}\varvec{q'}}^{(\tau )}(\varvec{r}\sigma ,\varvec{r'}\sigma ';t)&= \frac{1}{2}\rho _{\varvec{q}\varvec{q'}}^{(\tau )}(\varvec{r},\varvec{r'};t) \delta _{\sigma \sigma '} \\&+ \frac{1}{2} \sum _\mu {\langle {\sigma |{\hat{\sigma }}_\mu |\sigma '}\rangle }s_{\varvec{q}\varvec{q'}, \mu }^{(\tau )} (\varvec{r},\varvec{r'};t), \end{aligned} \end{aligned}$$

(A.1)

where $\rho _{\varvec{q}\varvec{q'}}^{(\tau )}(\varvec{r},\varvec{r'};t)$ is the non-local one-body transition particle density,

$$\begin{aligned} \rho _{\varvec{q}\varvec{q'}}^{(\tau )}(\varvec{r},\varvec{r'};t) = \sum _\sigma \rho _{\varvec{q}\varvec{q'}}^{(\tau )}(\varvec{r}\sigma ,\varvec{r'}\sigma ;t), \end{aligned}$$

(A.2)

$s_{\varvec{q}\varvec{q'}, \mu }^{(\tau )} (\varvec{r},\varvec{r'};t)$ is the $\mu$-th component of the non-local one-body transition spin density,

$$\begin{aligned} s_{\varvec{q}\varvec{q'}, \mu }^{(\tau )} (\varvec{r},\varvec{r'};t) = \sum _{\sigma \sigma '} \rho _{\varvec{q}\varvec{q'}}^{(\tau )}(\varvec{r}\sigma ,\varvec{r'}\sigma ';t) {\langle {\sigma '|{\hat{\sigma }}_\mu | \sigma }\rangle }, \end{aligned}$$

(A.3)

and ${\hat{\sigma }}_\mu$ are the Pauli operators. The local variants of the particle density $\rho _{\varvec{q} \varvec{q'}}^{(\tau )}(\varvec{r};t)$, spin density $\varvec{s}_{\varvec{q} \varvec{q'}}^{(\tau )}(\varvec{r};t)$, kinetic density $\tau _{\varvec{q} \varvec{q'}}^{(\tau )}(\varvec{r};t)$, current density $\varvec{j}_{\varvec{q} \varvec{q'}}^{(\tau )}(\varvec{r};t)$, spin-current pseudotensor density $J_{\varvec{q} \varvec{q'}, \mu \nu }^{(\tau )}(\varvec{r};t)$, and spin-orbit current vector density $\mathrm{{\varvec{J}}}_{\varvec{q} \varvec{q'}}^{(\tau )}(\varvec{r};t)$ read

$$\begin{aligned} \rho _{\varvec{q} \varvec{q'}}^{(\tau )}(\varvec{r};t)&= \rho _{\varvec{q} \varvec{q'}}^{(\tau )}(\varvec{r}, \varvec{r};t), \end{aligned}$$

(A.4a)

$$\begin{aligned} s_{\varvec{q} \varvec{q'},\mu }^{(\tau )}(\varvec{r};t)&= s_{\varvec{q} \varvec{q'},\mu }^{(\tau )}(\varvec{r}, \varvec{r};t), \end{aligned}$$

(A.4b)

$$\begin{aligned} \tau _{\varvec{q} \varvec{q'}}^{(\tau )}(\varvec{r};t)&= \nabla \cdot \nabla ' \rho _{\varvec{q} \varvec{q'}}^{(\tau )} (\varvec{r},\varvec{r'};t)\vert _{\varvec{r'}=\varvec{r}}, \end{aligned}$$

(A.4c)

$$\begin{aligned} j_{\varvec{q} \varvec{q'}, \mu }^{(\tau )}(\varvec{r};t)&= \frac{1}{2i}(\nabla _\mu - \nabla '_\mu )\rho _{\varvec{q} \varvec{q'}}^{(\tau )} (\varvec{r},\varvec{r'};t)\vert _{\varvec{r'}=\varvec{r}}, \end{aligned}$$

(A.4d)

$$\begin{aligned} J_{\varvec{q} \varvec{q'},\mu \nu }^{(\tau )}(\varvec{r};t)&= \frac{1}{2i}(\nabla _\mu - \nabla '_\mu ) s_{\varvec{q} \varvec{q'},\nu }^{(\tau )}(\varvec{r},\varvec{r'};t)\vert _{\varvec{r'}=\varvec{r}}, \end{aligned}$$

(A.4e)

$$\begin{aligned} \mathrm{{J}}_{\varvec{q} \varvec{q'}, \lambda }^{(\tau )}(\varvec{r};t)&= \sum _{\mu \nu } \epsilon _{\lambda \mu \nu } J_{\varvec{q} \varvec{q'},\mu \nu }^{(\tau )}(\varvec{r};t). \end{aligned}$$

(A.4f)

In the following paragraph, the explicit dependence on time and isospin is omitted for compactness.

For Slater generating states, the coordinate space representation of the non-local transition density (for either neutrons or protons) can be written as

$$\begin{aligned} \rho _{\varvec{q} \varvec{q'}}(\varvec{r}\sigma ,\varvec{r'}\sigma ') = \sum _{kl}\varphi _k^{\varvec{q'}}(\varvec{r}\sigma ) \Big [M_{\varvec{q} \varvec{q'}}^{-1}\Big ]_{kl}\varphi _l^{\varvec{q}*} (\varvec{r'}\sigma '), \end{aligned}$$

(A.5)

where $\Big [M_{\varvec{q} \varvec{q'}}^{-1}(t)\Big ]_{kl}$ are (generally complex) elements of the inverted matrix of single-particle overlaps [Eq. (37)]. Given the decomposition (33), the local transition particle density reads

$$\begin{aligned} \rho _{\varvec{q}\varvec{q'}}(\varvec{r}) = \sum _{kl} \Big [M_{\varvec{q}\varvec{q'}}^{-1} \Big ]_{kl} \Big [ \rho _{\varvec{q}\varvec{q'}}^{R}(\varvec{r})+i\rho _{\varvec{q}\varvec{q'}}^{I}(\varvec{r}) \Big ]_{kl} \end{aligned}$$

(A.6)

with

$$\begin{aligned} \Big [\rho _{\varvec{q}\varvec{q'}}^{R}(\varvec{r})\Big ]_{kl}&= \sum _{\alpha } \varphi _{k,\alpha }^{\varvec{q'}} (\varvec{r}) \varphi _{l,\alpha }^{\varvec{q}}(\varvec{r}), \end{aligned}$$

(A.7a)

$$\begin{aligned} \Big [\rho _{\varvec{q}\varvec{q'}}^{I}(\varvec{r})\Big ]_{kl}&= \varphi _{k,1}^{\varvec{q'}}(\varvec{r}) \varphi _{l,0}^{\varvec{q}}(\varvec{r}) -\varphi _{k,0}^{\varvec{q'}}(\varvec{r}) \varphi _{l,1}^{\varvec{q}}(\varvec{r}) \nonumber \\ {}&\quad +\varphi _{k,3}^{\varvec{q'}}(\varvec{r}) \varphi _{l,2}^{\varvec{q}}(\varvec{r}) -\varphi _{k,2}^{\varvec{q'}} (\varvec{r}) \varphi _{l,3}^{\varvec{q}}(\varvec{r}), \end{aligned}$$

(A.7b)

for $\alpha = 0, 1, 2, 3$. Similarly, the local transition kinetic density reads

$$\begin{aligned} \tau _{\varvec{q}\varvec{q'}}(\varvec{r}) = \sum _{kl} \Big [M_{\varvec{q}\varvec{q'}}^{-1}\Big ]_{kl} \Big [\tau _{\varvec{q}\varvec{q'}}^{R}(\varvec{r})+i\tau _{\varvec{q}\varvec{q'}}^{I}(\varvec{r})\Big ]_{kl}, \end{aligned}$$

(A.8)

with

$$\begin{aligned} \Big [\tau _{\varvec{q}\varvec{q'}}^{R}(\varvec{r})\Big ]_{kl}&= \sum _{\alpha } \big (\nabla \varphi _{k,\alpha }^{\varvec{q'}}(\varvec{r})\big )\big (\nabla \varphi _{l,\alpha }^{\varvec{q}}(\varvec{r})\big ), \end{aligned}$$

(A.9a)

$$\begin{aligned} \Big [\tau _{\varvec{q}\varvec{q'}}^{I}(\varvec{r})\Big ]_{kl}&= \big (\nabla \varphi _{k,1}^{\varvec{q'}}(\varvec{r})\big ) \big (\nabla \varphi _{l,0}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\ {}&\quad - \big (\nabla \varphi _{k,0}^{\varvec{q'}}(\varvec{r})\big )\big (\nabla \varphi _{l,1}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\&\quad + \big (\nabla \varphi _{k,3}^{\varvec{q'}}(\varvec{r})\big ) \big (\nabla \varphi _{l,2}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\&\quad - \big (\nabla \varphi _{k,2}^{\varvec{q'}}(\varvec{r})\big )\big (\nabla \varphi _{l,3}^{\varvec{q}}(\varvec{r})\big ), \end{aligned}$$

(A.9b)

with

$$\begin{aligned} \big (\nabla \varphi _{k,\alpha }^{\varvec{q'}}(\varvec{r})\big )\big (\nabla \varphi _{l,\beta }^{\varvec{q}}(\varvec{r})\big ) = \sum _{\mu }\big (\partial _\mu \varphi _{k,\alpha }^{\varvec{q'}}(\varvec{r})\big ) \big (\partial _\mu \varphi _{l,\beta }^{\varvec{q}}(\varvec{r})\big ) \nonumber \\ \end{aligned}$$

(A.10)

and $\mu = x, y, z$. Furthermore, the $\mu$-th component of the local transition current density reads

$$\begin{aligned} j_{\varvec{q}\varvec{q'}}^{\mu }(\varvec{r}) = \frac{1}{2}\sum _{kl}\Big [M_{\varvec{q}\varvec{q'}}^{-1}\Big ]_{kl} \Big [j_{\varvec{q}\varvec{q'}}^{\mu ,R}(\varvec{r})+ij_{\varvec{q} \varvec{q'}}^{\mu ,I}(\varvec{r})\Big ]_{kl}, \end{aligned}$$

(A.11)

with

$$\begin{aligned} \Big [j_{\varvec{q}\varvec{q'}}^{\mu ,R}(\varvec{r})\Big ]_{kl}&= \big (\partial _\mu \varphi _{k,1}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,0}^{\varvec{q}}(\varvec{r}) -\big (\partial _\mu \varphi _{k,0}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,1}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad + \big (\partial _\mu \varphi _{k,3}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,2}^{\varvec{q}}(\varvec{r}) - \big (\partial _\mu \varphi _{k,2}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,3}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad - \varphi _{k,1}^{\varvec{q'}}(\varvec{r})\big (\partial _\mu \varphi _{l,0}^{\varvec{q}}(\varvec{r})\big ) + \varphi _{k,0}^{\varvec{q'}}(\varvec{r})\big (\partial _\mu \varphi _{l,1}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\&\quad - \varphi _{k,3}^{\varvec{q'}}(\varvec{r})\big (\partial _\mu \varphi _{l,2}^{\varvec{q}}(\varvec{r})\big ) + \varphi _{k,2}^{\varvec{q'}}(\varvec{r})\big (\partial _\mu \varphi _{l,3}^{\varvec{q}}(\varvec{r})\big ), \end{aligned}$$

(A.12a)

$$\begin{aligned} \Big [j_{\varvec{q}\varvec{q'}}^{\mu ,I}(\varvec{r})\Big ]_{kl}&= \sum _{\alpha } \varphi _{k,\alpha }^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,\alpha }^{\varvec{q}} (\varvec{r})\big ) \nonumber \\&\quad - \sum _{\alpha }\big (\partial _\mu \varphi _{k,\alpha }^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,\alpha }^{\varvec{q}}(\varvec{r}), \end{aligned}$$

(A.12b)

The components of the local transition spin density then read

$$\begin{aligned} s_{\varvec{q}\varvec{q'}}^{\mu }(\varvec{r}) = \sum _{kl} \Big [M_{\varvec{q}\varvec{q'}}^{-1}\Big ]_{kl} \Big [s_{\varvec{q}\varvec{q'}}^{\mu ,R}(\varvec{r}) +is_{\varvec{q}\varvec{q'}}^{\mu ,I}(\varvec{r})\Big ]_{kl}, \end{aligned}$$

(A.13)

with

$$\begin{aligned} \Big [s_{\varvec{q}\varvec{q'}}^{x,R}(\varvec{r})\Big ]_{kl}&= \varphi _{k,0}^{\varvec{q'}}(\varvec{r})\varphi _{l,2}^{\varvec{q}}(\varvec{r}) + \varphi _{k,1}^{\varvec{q'}}(\varvec{r}) \varphi _{l,3}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad + \varphi _{k,2}^{\varvec{q'}}(\varvec{r})\varphi _{l,0}^{\varvec{q}}(\varvec{r}) + \varphi _{k,3}^{\varvec{q'}}(\varvec{r}) \varphi _{l,1}^{\varvec{q}}(\varvec{r}), \end{aligned}$$

(A.14a)

$$\begin{aligned} \Big [s_{\varvec{q}\varvec{q'}}^{x,I}(\varvec{r})\Big ]_{kl}&= \varphi _{k,1}^{\varvec{q'}}(\varvec{r}) \varphi _{l,2}^{\varvec{q}}(\varvec{r}) - \varphi _{k,2}^{\varvec{q'}}(\varvec{r}) \varphi _{l,1}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad + \varphi _{k,3}^{\varvec{q'}}(\varvec{r})\varphi _{l,0}^{\varvec{q}}(\varvec{r}) - \varphi _{k,0}^{\varvec{q'}}(\varvec{r}) \varphi _{l,3}^{\varvec{q}}(\varvec{r}), \end{aligned}$$

(A.14b)

$$\begin{aligned} \Big [s_{\varvec{q}\varvec{q'}}^{y,R}(\varvec{r})\Big ]_{kl}&= \varphi _{k,0}^{\varvec{q'}}(\varvec{r}) \varphi _{l,3}^{\varvec{q}}(\varvec{r}) + \varphi _{k,3}^{\varvec{q'}}(\varvec{r}) \varphi _{l,0}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad - \varphi _{k,1}^{\varvec{q'}}(\varvec{r}) \varphi _{l,2}^{\varvec{q}}(\varvec{r}) - \varphi _{k,2}^{\varvec{q'}}(\varvec{r}) \varphi _{l,1}^{\varvec{q}}(\varvec{r}), \end{aligned}$$

(A.14c)

$$\begin{aligned} \Big [s_{\varvec{q}\varvec{q'}}^{y,I}(\varvec{r})\Big ]_{kl}&= \varphi _{k,0}^{\varvec{q'}}(\varvec{r}) \varphi _{l,2}^{\varvec{q}}(\varvec{r}) - \varphi _{k,2}^{\varvec{q'}}(\varvec{r}) \varphi _{l,0}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad + \varphi _{k,1}^{\varvec{q'}}(\varvec{r}) \varphi _{l,3}^{\varvec{q}}(\varvec{r}) - \varphi _{k,3}^{\varvec{q'}}(\varvec{r}) \varphi _{l,1}^{\varvec{q}}(\varvec{r}), \end{aligned}$$

(A.14d)

$$\begin{aligned} \Big [s_{\varvec{q}\varvec{q'}}^{z,R}(\varvec{r})\Big ]_{kl}&= \varphi _{k,0}^{\varvec{q'}}(\varvec{r}) \varphi _{l,0}^{\varvec{q}}(\varvec{r}) + \varphi _{k,1}^{\varvec{q'}}(\varvec{r}) \varphi _{l,1}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad - \varphi _{k,2}^{\varvec{q'}}(\varvec{r})\varphi _{l,2}^{\varvec{q}}(\varvec{r}) - \varphi _{k,3}^{\varvec{q'}}(\varvec{r}) \varphi _{l,3}^{\varvec{q}}(\varvec{r}), \end{aligned}$$

(A.14e)

$$\begin{aligned} \Big [s_{\varvec{q}\varvec{q'}}^{z,I}(\varvec{r})\Big ]_{kl}&= \varphi _{k,1}^{\varvec{q'}}(\varvec{r}) \varphi _{l,0}^{\varvec{q}}(\varvec{r}) - \varphi _{k,0}^{\varvec{q'}}(\varvec{r}) \varphi _{l,1}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad + \varphi _{k,2}^{\varvec{q'}}(\varvec{r})\varphi _{l,3}^{\varvec{q}}(\varvec{r}) - \varphi _{k,3}^{\varvec{q'}}(\varvec{r}) \varphi _{l,2}^{\varvec{q}}(\varvec{r}). \end{aligned}$$

(A.14f)

Finally, the components of the spin-current pseudotensor density read

$$\begin{aligned} J_{\varvec{q}\varvec{q'}}^{\mu \nu }(\varvec{r}) = \frac{1}{2} \sum _{kl}\Big [M_{\varvec{q}\varvec{q'}}^{-1}\Big ]_{kl} \Big [J_{\varvec{q}\varvec{q'}}^{\mu \nu , R}(\varvec{r}) +iJ_{\varvec{q}\varvec{q'}}^{\mu \nu , I}(\varvec{r})\Big ]_{kl} \nonumber \\ \end{aligned}$$

(A.15)

with

$$\begin{aligned} \Big [J_{\varvec{q}\varvec{q'}}^{\mu x, R}(\varvec{r})\Big ]_{kl}&= \big (\partial _\mu \varphi _{k,1}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,2}^{\varvec{q}}(\varvec{r}) - \varphi _{k,1}^{\varvec{q'}}(\varvec{r})\big (\partial _\mu \varphi _{l,2}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\&\quad - \big (\partial _\mu \varphi _{k,0}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,3}^{\varvec{q}}(\varvec{r}) + \varphi _{k,0}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,3}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\&\quad + \big (\partial _\mu \varphi _{k,3}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,0}^{\varvec{q}}(\varvec{r}) - \varphi _{k,3}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,0}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\&\quad - \big (\partial _\mu \varphi _{k,2}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,1}^{\varvec{q}}(\varvec{r}) + \varphi _{k,2}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,1}^{\varvec{q}}(\varvec{r})\big ), \end{aligned}$$

(A.16a)

$$\begin{aligned} \Big [J_{\varvec{q}\varvec{q'}}^{\mu x, I}(\varvec{r})\Big ]_{kl}&= \varphi _{k,0}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,2}^{\varvec{q}}(\varvec{r})\big ) - \big (\partial _\mu \varphi _{k,0}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,2}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad + \varphi _{k,1}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,3}^{\varvec{q}}(\varvec{r})\big ) - \big (\partial _\mu \varphi _{k,1}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,3}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad + \varphi _{k,2}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,0}^{\varvec{q}}(\varvec{r})\big ) - \big (\partial _\mu \varphi _{k,2}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,0}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad + \varphi _{k,3}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,1}^{\varvec{q}}(\varvec{r})\big ) - \big (\partial _\mu \varphi _{k,3}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,1}^{\varvec{q}}(\varvec{r}), \end{aligned}$$

(A.16b)

$$\begin{aligned} \Big [J_{\varvec{q}\varvec{q'}}^{\mu y,R}(\varvec{r})\Big ]_{kl}&= \big (\partial _\mu \varphi _{k,0}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,2}^{\varvec{q}}(\varvec{r}) - \varphi _{k,0}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,2}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\&\quad + \big (\partial _\mu \varphi _{k,1}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,3}^{\varvec{q}}(\varvec{r}) - \varphi _{k,1}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,3}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\&\quad - \big (\partial _\mu \varphi _{k,2}^{\varvec{q'}}(\varvec{r}) \big ) \varphi _{l,0}^{\varvec{q}}(\varvec{r}) + \varphi _{k,2}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,0}^{\varvec{q}}(\varvec{r}) \big ) \nonumber \\&\quad - \big (\partial _\mu \varphi _{k,3}^{\varvec{q'}} (\varvec{r})\big ) \varphi _{l,1}^{\varvec{q}}(\varvec{r}) + \varphi _{k,3}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,1}^{\varvec{q}} (\varvec{r})\big ), \end{aligned}$$

(A.16c)

$$\begin{aligned} \Big [J_{\varvec{q}\varvec{q'}}^{\mu y,I}(\varvec{r})\Big ]_{kl}&= \big (\partial _\mu \varphi _{k,1}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,2}^{\varvec{q}}(\varvec{r}) - \varphi _{k,1}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,2}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\&\quad - \big (\partial _\mu \varphi _{k,0}^{\varvec{q'}} (\varvec{r})\big ) \varphi _{l,3}^{\varvec{q}}(\varvec{r}) + \varphi _{k,0}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,3}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\&\quad + \big (\partial _\mu \varphi _{k,2}^{\varvec{q'}} (\varvec{r})\big ) \varphi _{l,1}^{\varvec{q}}(\varvec{r}) - \varphi _{k,2}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,1}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\&\quad - \big (\partial _\mu \varphi _{k,3}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,0}^{\varvec{q}}(\varvec{r}) + \varphi _{k,3}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,0}^{\varvec{q}}(\varvec{r})\big ), \end{aligned}$$

(A.16d)

$$\begin{aligned} \Big [J_{\varvec{q}\varvec{q'}}^{\mu z, R}(\varvec{r})\Big ]_{kl}&= \big (\partial _\mu \varphi _{k,1}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,0}^{\varvec{q}}(\varvec{r}) - \varphi _{k,1}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,0}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\&\quad - \big (\partial _\mu \varphi _{k,0}^{\varvec{q'}} (\varvec{r})\big ) \varphi _{l,1}^{\varvec{q}}(\varvec{r}) + \varphi _{k,0}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,1}^{\varvec{q}}(\varvec{r})\big ) \nonumber \\&\quad + \big (\partial _\mu \varphi _{k,2}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,3}^{\varvec{q}}(\varvec{r}) - \varphi _{k,2}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,3}^{\varvec{q}}(\varvec{r}) \big ) \nonumber \\&\quad - \big (\partial _\mu \varphi _{k,3}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,2}^{\varvec{q}}(\varvec{r}) + \varphi _{k,3}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,2}^{\varvec{q}}(\varvec{r})\big ), \end{aligned}$$

(A.16e)

$$\begin{aligned} \Big [J_{\varvec{q}\varvec{q'}}^{\mu z, I}(\varvec{r})\Big ]_{kl}&= \varphi _{k,0}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,0}^{\varvec{q}}(\varvec{r})\big ) - \big (\partial _\mu \varphi _{k,0}^{\varvec{q'}}(\varvec{r}) \big ) \varphi _{l,0}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad + \varphi _{k,1}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,1}^{\varvec{q}}(\varvec{r}) \big ) - \big (\partial _\mu \varphi _{k,1}^{\varvec{q'}}(\varvec{r}) \big ) \varphi _{l,1}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad - \varphi _{k,2}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,2}^{\varvec{q}}(\varvec{r})\big ) + \big (\partial _\mu \varphi _{k,2}^{\varvec{q'}}(\varvec{r}) \big ) \varphi _{l,2}^{\varvec{q}}(\varvec{r}) \nonumber \\&\quad - \varphi _{k,3}^{\varvec{q'}}(\varvec{r}) \big (\partial _\mu \varphi _{l,3}^{\varvec{q}}(\varvec{r})\big ) + \big (\partial _\mu \varphi _{k,3}^{\varvec{q'}}(\varvec{r})\big ) \varphi _{l,3}^{\varvec{q}}(\varvec{r}). \end{aligned}$$

(A.16f)

Appendix B: Coupling constants

The coupling constants appearing in the Skyrme energy density (44) read

$$\begin{aligned} B_1&= \frac{1}{2}t_0\left( 1+\frac{1}{2}x_0\right) , \end{aligned}$$

(B.17a)

$$\begin{aligned} B_2&= - \frac{1}{2}t_0\left( \frac{1}{2} + x_0\right) , \end{aligned}$$

(B.17b)

$$\begin{aligned} B_3&= \frac{1}{4} \bigg (t_1 \left( 1 + \frac{1}{2}x_1\right) + t_2 \left( 1 + \frac{1}{2}x_2\right) \bigg ), \end{aligned}$$

(B.17c)

$$\begin{aligned} B_4&= -\frac{1}{4} \bigg (t_1\left( \frac{1}{2} + x_1\right) - t_2\left( \frac{1}{2} + x_2\right) \bigg ), \end{aligned}$$

(B.17d)

$$\begin{aligned} B_5&= -\frac{1}{16} \bigg (3t_1\left( 1+\frac{1}{2}x_1\right) - t_2\left( 1+\frac{1}{2}x_2\right) \bigg ), \end{aligned}$$

(B.17e)

$$\begin{aligned} B_6&= \frac{1}{16} \bigg (3t_1 \left( \frac{1}{2}+ x_1\right) + t_2(\frac{1}{2} + x_2)\bigg ), \end{aligned}$$

(B.17f)

$$\begin{aligned} B_7&= \frac{1}{12}t_3\left( 1+\frac{1}{2}x_3\right) , \end{aligned}$$

(B.17g)

$$\begin{aligned} B_8&= - \frac{1}{12}t_3\left( \frac{1}{2} + x_3\right) , \end{aligned}$$

(B.17h)

$$\begin{aligned} B_9&= -\frac{1}{2}W, \end{aligned}$$

(B.17i)

$$\begin{aligned} B_{10}&= \frac{1}{4}t_0x_0, \end{aligned}$$

(B.17j)

$$\begin{aligned} B_{11}&= -\frac{1}{4}t_0 , \end{aligned}$$

(B.17k)

$$\begin{aligned} B_{12}&= \frac{1}{24}t_3x_3, \end{aligned}$$

(B.17l)

$$\begin{aligned} B_{13}&= -\frac{1}{24}t_3. \end{aligned}$$

(B.17m)

Parameters $t_i$, $x_i$ ($i = 0, 1, 2, 3$), W, and $\alpha$ are the standard parameters of the Skyrme pseudopotential [7, 73].

Appendix C: Variance of a one-body operator in a normalized MC-TDDFT state

To evaluate the variance of Eq. (65), we need an expectation value of the ${\hat{O}}^2$ operator in a normalized MC-TDDFT state. We start from

$$\begin{aligned} {\langle {\varPsi (t)|{\hat{O}}^2|\varPsi (t)}\rangle } = \int _{\varvec{q} \varvec{q'}} \! \! \,d\varvec{q} \,d\varvec{q'} g^*_{\varvec{q}}(t) {{\mathcal {O}}^2}^c_{\varvec{q}\varvec{q'}}(t) g_{\varvec{q'}}(t). \end{aligned}$$

(C.18)

Again, the collective kernel of the ${\hat{O}}^2$ operator is calculated from (24) and the corresponding usual kernel follows from the generalized Wick theorem,

$$\begin{aligned} \begin{aligned} {\mathcal {O}}^2_{\varvec{q} \varvec{q'}}(t) = {\mathcal {N}}_{\varvec{q} \varvec{q'}}(t) \Big [&{{\textrm{Tr}}}^2\big (O\rho ^{\varvec{q}\varvec{q'}}(t)\big ) \\ +&{{\textrm{Tr}}}\big (O\rho ^{\varvec{q}\varvec{q'}}(t)O(1-\rho ^{\varvec{q}\varvec{q'}}(t)\big )\Big ]. \end{aligned} \end{aligned}$$

(C.19)

Here,

$$\begin{aligned} {{\textrm{Tr}}}\big (O\rho ^{\varvec{q}\varvec{q'}}(t)\big ) = \int \,d^3 \varvec{r} O(\varvec{r}) \rho _{\varvec{q} \varvec{q'}}(\varvec{r}; t). \end{aligned}$$

(C.20)

Furthermore, the second trace corresponds to the sum of two terms,

$$\begin{aligned} {{\textrm{Tr}}}\big (O\rho ^{\varvec{q}\varvec{q'}}(t)O(1-\rho ^{\varvec{q}\varvec{q'}}(t)\big ) = C_1^{\varvec{q} \varvec{q'}}(t) + C_2^{\varvec{q} \varvec{q'}}(t). \nonumber \\ \end{aligned}$$

(C.21)

The first term reads

$$\begin{aligned} \begin{aligned} C_1^{\varvec{q} \varvec{q'}}(t)&= \int \,d^3 \varvec{r} O^2(\varvec{r}) \sum _{kl} \Big [M_{\varvec{q} \varvec{q'}}^{-1}(t)\Big ]_{kl} \\ {}&\quad \times \biggl \{ A_{kl}^{\varvec{q}\varvec{q'}}(\varvec{r};t) + \quad iB_{kl}^{\varvec{q}\varvec{q'}}(\varvec{r};t) \biggl \}, \end{aligned} \end{aligned}$$

(C.22)

with

$$\begin{aligned} A_{kl}^{\varvec{q}\varvec{q'}}(\varvec{r};t) = \sum _{\alpha } \varphi _{k,\alpha }^{\varvec{q'}}(\varvec{r};t) \varphi _{l,\alpha }^{\varvec{q}}(\varvec{r};t) \end{aligned}$$

(C.23)

and

$$\begin{aligned} \begin{aligned} B_{kl}^{\varvec{q}\varvec{q'}}(\varvec{r};t)&= \varphi _{k,1}^{\varvec{q'}}(\varvec{r};t) \varphi _{l,0}^{\varvec{q}}(\varvec{r};t) \\ {}&\quad - \varphi _{k,0}^{\varvec{q'}}(\varvec{r};t) \varphi _{l,1}^{\varvec{q}}(\varvec{r};t) \\ {}&\quad + \varphi _{k,3}^{\varvec{q'}}(\varvec{r};t) \varphi _{l,2}^{\varvec{q}}(\varvec{r};t) \\ {}&\quad - \varphi _{k,2}^{\varvec{q'}}(\varvec{r};t) \varphi _{l,3}^{\varvec{q}}(\varvec{r};t). \end{aligned} \end{aligned}$$

(C.24)

The second term reads

$$\begin{aligned} \begin{aligned} C_2^{\varvec{q} \varvec{q'}}(t)&= \int \,d^3 \varvec{r} \int \,d^3 \varvec{r'} O(\varvec{r}) O(\varvec{r'}) \\ {}&\quad \times \sum _{klmn} \Big [M_{\varvec{q} \varvec{q'}}^{-1}(t)\Big ]_{kl} \Big [M_{\varvec{q} \varvec{q'}}^{-1}(t)\Big ]_{mn} \\&\quad \times \biggl \{ A_{kn}^{\varvec{q}\varvec{q'}}(\varvec{r};t) + iB_{kn}^{\varvec{q}\varvec{q'}}(\varvec{r};t) \biggl \} \\ {}&\quad \times \biggl \{ A_{ml}^{\varvec{q}\varvec{q'}}(\varvec{r'};t) + iB_{ml}^{\varvec{q}\varvec{q'}}(\varvec{r'};t) \biggl \}. \end{aligned} \end{aligned}$$

(C.25)