Time-dependent Covariant Density Functional Theory

• TD-CDFT is a relativistic extension of density functional theory where many-body states evolve as Dirac spinors in self-consistent Lorentz scalar and vector fields.
• It integrates both time-even and time-odd mean fields, ensuring nuclear magnetism and rotational effects such as alignment and moments of inertia are treated consistently.
• The framework is applied in 3D coordinate-space implementations to accurately simulate reaction dynamics including fusion, dissipation, quasi-fission, and multinucleon transfer.

Time-dependent covariant density functional theory (TD-CDFT) is the time-dependent extension of covariant density functional theory in nuclear physics. It treats the many-body state as a Slater determinant of occupied single-particle Dirac spinors evolving self-consistently in Lorentz scalar and vector mean fields that are functionals of local densities and currents. In practical nuclear applications, TD-CDFT is used for large-amplitude collective motion and reaction dynamics, including fusion, dissipation, quasi-fission, and multinucleon transfer, while its rotating stationary limit clarifies how time-odd mean fields govern currents, angular-momentum alignment, and inertial response [(Ren et al., 2020); (Zhang et al., 2024); (Afanasjev et al., 2010)].

1. Concept and domain of the theory

Covariant density functional theory is a relativistic Kohn–Sham framework in which the nucleon is described by a Dirac spinor and the energy density functional is built from Lorentz scalars and four-vectors. TD-CDFT inherits that structure at the time-dependent level: the densities and currents become functions of $t$, and the occupied Dirac orbitals evolve in self-consistent scalar and vector fields. The resulting formalism is fully microscopic at the mean-field level and can be implemented in three-dimensional coordinate space without symmetry restrictions (Ren et al., 2020).

A central feature of the covariant formulation is that time-even and time-odd sectors are not independently parameterized. Because the fields are assembled from Lorentz-covariant bilinears such as $j^\mu j_\mu$, the same couplings control the time-like and space-like parts of the vector field. In nuclear dynamics this is consequential: currents, spin-orbit currents, and other time-odd contributions are fixed by the same functional that reproduces ground-state properties, rather than introduced as separate empirical terms [(Ren et al., 2020); (Afanasjev et al., 2010)].

The theory has been formulated with both point-coupling and meson-exchange covariant energy density functionals. In recent reaction applications, the point-coupling functional PC-PK1 and the finite-range meson-exchange functional DD-ME2 have both been used, and both generate very similar multinucleon-transfer cross sections for the systems studied (Zhang et al., 2024).

2. Covariant time-dependent equations and self-consistent fields

In the time-dependent formulation, the occupied single-particle orbitals satisfy a time-dependent Kohn–Sham-like Dirac equation,

$$i\hbar\frac{\partial}{\partial t}\psi_k(\bm{r},t) = \hat{h}(\bm{r},t)\psi_k(\bm{r},t),$$

with single-particle Hamiltonian

$$\hat{h}(\bm{r},t) = \bm{\alpha}\cdot(\hat{\bm{p}}-\bm{V}) + V^0 + \beta(m+S).$$

Here $S(\mathbf r,t)$ is the Lorentz scalar potential and $V^\mu=(V^0,\mathbf V)$ is the four-vector potential. Both are functionals of the time-dependent local densities and currents (Zhang et al., 2024).

The basic local covariant densities used in practical implementations are

$$\rho_S(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\psi_k(\mathbf r,t),$$

$$j^\mu(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\psi_k(\mathbf r,t),$$

$$j_{TV}^{\mu}(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\tau_3\psi_k(\mathbf r,t).$$

For the point-coupling realization used in the three-dimensional lattice implementation, the static covariant functional contains four-fermion point couplings, higher-order self-interactions, derivative terms simulating finite range, and the electromagnetic field. Functional variation yields the Dirac Hamiltonian with scalar and vector self-consistent fields $S(\mathbf r,t)$ and $j^\mu j_\mu$0 evaluated from the instantaneous densities and currents (Ren et al., 2020).

Practical nuclear TD-CDFT calculations commonly adopt the adiabatic approximation: the instantaneous Hamiltonian is taken to be exactly the static CDFT Hamiltonian functional evaluated at the instantaneous densities and currents. In the reaction calculations reported for multinucleon transfer, the approximations are explicitly the mean-field approximation, the no-sea approximation, pairing neglected, and local instantaneous mean fields. Within that framework the time evolution describes the most probable path for a chosen set of initial conditions (Zhang et al., 2024).

The formalism accommodates distinct classes of covariant energy density functionals. PC-PK1 is a zero-range point-coupling functional with contact interactions in isoscalar-scalar, isoscalar-vector, and isovector-vector channels and non-linear terms in the isoscalar couplings; DD-ME2 is a finite-range meson-exchange functional with density-dependent couplings of $j^\mu j_\mu$1, $j^\mu j_\mu$2, and $j^\mu j_\mu$3 mesons to nucleons, plus the Coulomb field (Zhang et al., 2024).

3. Time-odd mean fields, nuclear magnetism, and the rotating limit

The stationary rotating limit of TD-CDFT is obtained in the cranking approximation, where the system is described in a frame rotating with constant angular velocity $j^\mu j_\mu$4 about the $j^\mu j_\mu$5-axis. The single-particle equation becomes

$j^\mu j_\mu$6

with

$j^\mu j_\mu$7

This stationary cranking equation is the rotating-frame static realization of the same covariant functional structure that appears in full TD-CDFT (Afanasjev et al., 2010).

The decisive time-odd ingredients are the currents $j^\mu j_\mu$8 and the space-like components of the vector fields. In the meson-exchange formulation, the magnetic potential is

$j^\mu j_\mu$9

and the space-like $$i\hbar\frac{\partial}{\partial t}\psi_k(\bm{r},t) = \hat{h}(\bm{r},t)\psi_k(\bm{r},t),$$0 and $$i\hbar\frac{\partial}{\partial t}\psi_k(\bm{r},t) = \hat{h}(\bm{r},t)\psi_k(\bm{r},t),$$1 fields are sourced by the nuclear currents. By construction, $$i\hbar\frac{\partial}{\partial t}\psi_k(\bm{r},t) = \hat{h}(\bm{r},t)\psi_k(\bm{r},t),$$2 and $$i\hbar\frac{\partial}{\partial t}\psi_k(\bm{r},t) = \hat{h}(\bm{r},t)\psi_k(\bm{r},t),$$3 are time-even, whereas $$i\hbar\frac{\partial}{\partial t}\psi_k(\bm{r},t) = \hat{h}(\bm{r},t)\psi_k(\bm{r},t),$$4 is time-odd. These space-like vector fields are the nuclear-magnetism sector of CDFT (Afanasjev et al., 2010).

A distinctive result of the covariant formulation is that time-odd mean fields related to nuclear magnetism are fixed by Lorentz invariance and do not require additional coupling constants: the coupling constants of time-even mean fields are used also for time-odd mean fields. This makes the time-odd sector comparatively constrained. In rotating nuclei, these fields have a profound effect on the dynamic and kinematic moments of inertia, alter single-particle alignments, shift band crossings, and produce phenomena such as signature separation in odd-odd nuclei (Afanasjev et al., 2010).

Quantitatively, nuclear magnetism typically increases $$i\hbar\frac{\partial}{\partial t}\psi_k(\bm{r},t) = \hat{h}(\bm{r},t)\psi_k(\bm{r},t),$$5 by $$i\hbar\frac{\partial}{\partial t}\psi_k(\bm{r},t) = \hat{h}(\bm{r},t)\psi_k(\bm{r},t),$$6–$$i\hbar\frac{\partial}{\partial t}\psi_k(\bm{r},t) = \hat{h}(\bm{r},t)\psi_k(\bm{r},t),$$7 at normal deformation and by $$i\hbar\frac{\partial}{\partial t}\psi_k(\bm{r},t) = \hat{h}(\bm{r},t)\psi_k(\bm{r},t),$$8–$$i\hbar\frac{\partial}{\partial t}\psi_k(\bm{r},t) = \hat{h}(\bm{r},t)\psi_k(\bm{r},t),$$9 at superdeformation and hyperdeformation for unpaired systems. In $$\hat{h}(\bm{r},t) = \bm{\alpha}\cdot(\hat{\bm{p}}-\bm{V}) + V^0 + \beta(m+S).$$0Pb superdeformed protons, the unpaired crossing between $$\hat{h}(\bm{r},t) = \bm{\alpha}\cdot(\hat{\bm{p}}-\bm{V}) + V^0 + \beta(m+S).$$1 and $$\hat{h}(\bm{r},t) = \bm{\alpha}\cdot(\hat{\bm{p}}-\bm{V}) + V^0 + \beta(m+S).$$2 is shifted from $$\hat{h}(\bm{r},t) = \bm{\alpha}\cdot(\hat{\bm{p}}-\bm{V}) + V^0 + \beta(m+S).$$3 MeV in the WNM calculation to $$\hat{h}(\bm{r},t) = \bm{\alpha}\cdot(\hat{\bm{p}}-\bm{V}) + V^0 + \beta(m+S).$$4 MeV in the NM calculation. The same study also finds that the current patterns in rotating superdeformed and hyperdeformed bands are far from rigid rotation even when the moments of inertia are close to the rigid-body value, indicating vortex-dominated quantal flow rather than hydrodynamically rigid flow (Afanasjev et al., 2010).

These results are directly relevant for TD-CDFT beyond stationary rotation. They indicate that inertial parameters, current patterns, and level-crossing dynamics depend sensitively on the self-consistent time-odd vector sector, not merely on the time-even density dependence. A plausible implication is that TD-CDFT calculations that suppress or incompletely treat $$\hat{h}(\bm{r},t) = \bm{\alpha}\cdot(\hat{\bm{p}}-\bm{V}) + V^0 + \beta(m+S).$$5 will misrepresent rotational inertia, alignment dynamics, and other current-dominated observables (Afanasjev et al., 2010).

4. Three-dimensional coordinate-space implementation and benchmark properties

A full three-dimensional coordinate-space realization of TD-CDFT with PC-PK1 has been developed without any symmetry restrictions. In that implementation, the static $$\hat{h}(\bm{r},t) = \bm{\alpha}\cdot(\hat{\bm{p}}-\bm{V}) + V^0 + \beta(m+S).$$6O ground state is obtained on a $$\hat{h}(\bm{r},t) = \bm{\alpha}\cdot(\hat{\bm{p}}-\bm{V}) + V^0 + \beta(m+S).$$7 box with mesh spacing $$\hat{h}(\bm{r},t) = \bm{\alpha}\cdot(\hat{\bm{p}}-\bm{V}) + V^0 + \beta(m+S).$$8, and the time-dependent reaction calculations use a $$\hat{h}(\bm{r},t) = \bm{\alpha}\cdot(\hat{\bm{p}}-\bm{V}) + V^0 + \beta(m+S).$$9 box. The Coulomb potential is solved by Hockney’s method for the Poisson equation with isolated boundary conditions, and spatial derivatives are handled using Fourier spectral methods (Ren et al., 2020).

Time propagation is performed with a predictor–corrector scheme and a Taylor expansion of the evolution operator. The benchmark calculations use $S(\mathbf r,t)$0 and Taylor-expansion order $S(\mathbf r,t)$1. The implementation was explicitly tested against relativistic kinematics, conservation laws, and time-reversal invariance. For a boosted $S(\mathbf r,t)$2O nucleus evolved up to $S(\mathbf r,t)$3 fm/$S(\mathbf r,t)$4, momentum deviations are of order $S(\mathbf r,t)$5; for a head-on $S(\mathbf r,t)$6O+$S(\mathbf r,t)$7O collision at $S(\mathbf r,t)$8 MeV, the total-energy deviation is $S(\mathbf r,t)$9 over $V^\mu=(V^0,\mathbf V)$0 fm/$V^\mu=(V^0,\mathbf V)$1 and the particle-number deviation is $V^\mu=(V^0,\mathbf V)$2 (Ren et al., 2020).

The benchmark also shows that the excitation energy of a boosted $V^\mu=(V^0,\mathbf V)$3O follows the relativistic kinetic-energy expression $V^\mu=(V^0,\mathbf V)$4 rather than the nonrelativistic $V^\mu=(V^0,\mathbf V)$5 once $V^\mu=(V^0,\mathbf V)$6. In head-on $V^\mu=(V^0,\mathbf V)$7O+$V^\mu=(V^0,\mathbf V)$8O collisions, the dissipation fraction

$V^\mu=(V^0,\mathbf V)$9

decreases with increasing collision energy: $$\rho_S(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\psi_k(\mathbf r,t),$$0 at $$\rho_S(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\psi_k(\mathbf r,t),$$1 MeV, $$\rho_S(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\psi_k(\mathbf r,t),$$2 at $$\rho_S(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\psi_k(\mathbf r,t),$$3 MeV, and $$\rho_S(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\psi_k(\mathbf r,t),$$4 at $$\rho_S(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\psi_k(\mathbf r,t),$$5 MeV. The corresponding density profiles become progressively less diffused as the incident energy increases, consistent with shorter contact times and reduced mean-field rearrangement (Ren et al., 2020).

Above-barrier fusion cross sections can be extracted through a partial-wave decomposition,

$$\rho_S(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\psi_k(\mathbf r,t),$$6

with fusion probabilities smoothed by a Hill–Wheeler Fermi-type expression. For $$\rho_S(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\psi_k(\mathbf r,t),$$7O+$$\rho_S(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\psi_k(\mathbf r,t),$$8O, the resulting cross sections oscillate with $$\rho_S(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\psi_k(\mathbf r,t),$$9 because of discrete $$j^\mu(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\psi_k(\mathbf r,t),$$0 quantization; the oscillatory behavior is in reasonable agreement with experimental data and close to nonrelativistic Skyrme-TDDFT results that include time-odd spin-orbit terms (Ren et al., 2020).

5. Multinucleon transfer and reaction dynamics

The extension of TD-CDFT to multinucleon transfer combines real-time covariant dynamics with particle-number projection. In the reaction calculations, projectile and target ground states are prepared by static self-consistent CDFT on a $$j^\mu(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\psi_k(\mathbf r,t),$$1 grid with $$j^\mu(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\psi_k(\mathbf r,t),$$2 fm, then placed at an initial center-to-center distance of $$j^\mu(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\psi_k(\mathbf r,t),$$3 fm and boosted along a Rutherford trajectory for a chosen $$j^\mu(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\psi_k(\mathbf r,t),$$4 and impact parameter $$j^\mu(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\psi_k(\mathbf r,t),$$5. Time evolution is carried out in a fully three-dimensional Cartesian box, usually $$j^\mu(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\psi_k(\mathbf r,t),$$6 with the same mesh spacing, by a predictor–corrector scheme with fourth-order Taylor expansion and $$j^\mu(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\psi_k(\mathbf r,t),$$7 fm/$$j^\mu(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\psi_k(\mathbf r,t),$$8 (Zhang et al., 2024).

After the fragments separate, transfer probabilities are extracted by projecting onto definite proton and neutron numbers in a spatial region $$j^\mu(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\psi_k(\mathbf r,t),$$9. For neutrons or protons,

$$j_{TV}^{\mu}(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\tau_3\psi_k(\mathbf r,t).$$0

and the probability for a fragment in region $$j_{TV}^{\mu}(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\tau_3\psi_k(\mathbf r,t).$$1 to have $$j_{TV}^{\mu}(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\tau_3\psi_k(\mathbf r,t).$$2 neutrons and $$j_{TV}^{\mu}(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\tau_3\psi_k(\mathbf r,t).$$3 protons is

$$j_{TV}^{\mu}(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\tau_3\psi_k(\mathbf r,t).$$4

Channel cross sections are then obtained from

$$j_{TV}^{\mu}(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\tau_3\psi_k(\mathbf r,t).$$5

Within the mean-field framework this factorization implies no correlations between neutron and proton number distributions at the level of the projection formula (Zhang et al., 2024).

The method has been applied to $$j_{TV}^{\mu}(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\tau_3\psi_k(\mathbf r,t).$$6 at $$j_{TV}^{\mu}(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\tau_3\psi_k(\mathbf r,t).$$7 MeV, $$j_{TV}^{\mu}(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\tau_3\psi_k(\mathbf r,t).$$8 at $$j_{TV}^{\mu}(\bm{r},t) = \sum_k \bar{\psi}_k(\mathbf r,t)\gamma^{\mu}\tau_3\psi_k(\mathbf r,t).$$9 MeV, $S(\mathbf r,t)$0 at $S(\mathbf r,t)$1 MeV, $S(\mathbf r,t)$2 at $S(\mathbf r,t)$3 MeV, and $S(\mathbf r,t)$4 at $S(\mathbf r,t)$5 MeV. The reactions display the expected charge-equilibration tendency: neutron transfer proceeds from the more neutron-rich partner to the less neutron-rich one, while proton transfer tends to compensate the initial $S(\mathbf r,t)$6 asymmetry. For systems with large $S(\mathbf r,t)$7, small-impact-parameter dynamics produces elongated necks and quasi-fission-like binary exit channels with substantial mass and charge redistribution (Zhang et al., 2024).

For the three benchmark systems previously studied with Skyrme SLy5 TDHF, the TD-CDFT results with PC-PK1 and DD-ME2 are very similar to each other and reproduce the qualitative pattern of the SLy5 calculations. For $S(\mathbf r,t)$8 and $S(\mathbf r,t)$9, the calculations constitute the first microscopic total-cross-section predictions reported in that work. Compared with the GRAZING model, TD-CDFT is in much better agreement with data, especially in proton pick-up channels where neck dynamics and quasi-fission-like behavior are important. The $j^\mu j_\mu$00 channel in $j^\mu j_\mu$01 is reproduced very well on both the neutron-pickup and neutron-stripping sides (Zhang et al., 2024).

The systematic discrepancies are also instructive. As in earlier TDHF studies, neutron pickup is often overestimated and neutron stripping underestimated in higher-multiplicity channels, and the primary-fragment peaks are shifted toward higher neutron number. The reported explanation is that neutron evaporation from primary fragments is not included in the mean-field dynamics, so the calculated distributions remain too neutron-rich relative to experiment (Zhang et al., 2024).

6. Approximation structure, nomenclature, and research context

The present nuclear form of TD-CDFT is a mean-field theory. In the three-dimensional lattice benchmark and in the multinucleon-transfer applications, the many-body wave function is approximated by a single Slater determinant, pairing is neglected, and explicit two-body correlations beyond the energy density functional are absent. In the benchmark reaction study, the adiabatic approximation is used and no many-body tunneling is available, so sub-barrier fusion is outside the method’s scope. In the multinucleon-transfer calculations, fluctuations around the most probable mean-field path, post-collision evaporation, and secondary fission are not treated explicitly (Ren et al., 2020, Zhang et al., 2024).

These limitations define the main directions of extension. The multinucleon-transfer study explicitly identifies dynamical pairing correlations as a target for future work, and also points to systematic treatment of initial deformation and orientation degrees of freedom, coupling to statistical decay models, and applications to transuranium and superheavy-nucleus production. The benchmark work likewise points toward pairing, beyond-mean-field or stochastic extensions, and broader applications to heavy systems, collective excitations, and fission dynamics (Zhang et al., 2024, Ren et al., 2020).

A frequent source of confusion is the acronym itself. In electronic-structure and condensed-matter theory, “TDCDFT” commonly denotes time-dependent current-density-functional theory rather than time-dependent covariant density functional theory. In that literature, the basic objects are the current density and the vector potential, the exchange-correlation kernel is a tensor $j^\mu j_\mu$02, and rigorous lattice formulations reduce representability to existence and uniqueness of a nonlinear lattice Schrödinger equation [(Gatti, 2010); (Tokatly, 2010)]. By contrast, nuclear TD-CDFT is a relativistic mean-field framework based on Kohn–Sham Dirac dynamics and covariant energy density functionals (Ren et al., 2020, Zhang et al., 2024).

Despite that terminological overlap, the nuclear literature gives the covariant label a specific technical meaning: Lorentz covariance constrains the relation between scalar and vector fields and between time-even and time-odd channels. The detailed rotating-frame analysis of nuclear magnetism shows why this matters dynamically. Time-odd currents, spin densities, and inertial properties are not peripheral corrections but structural components of the covariant theory, and their self-consistent treatment appears essential for moments of inertia, band crossings, dissipation, fusion, quasi-fission, and multinucleon transfer [(Afanasjev et al., 2010); (Ren et al., 2020); (Zhang et al., 2024)].