Extended Quantum Molecular Dynamics (EQMD)

• Extended Quantum Molecular Dynamics (EQMD) is a transport model where each nucleon is represented by a dynamical Gaussian wave packet that adapts its width based on the nuclear environment.
• The model utilizes frictional cooling, a phenomenological Pauli potential, and flexible Skyrme-based interactions to produce stable ground states and accurately simulate nuclear resonances and cluster formations.
• Recent improvements include Monte Carlo integration for non-integer EOS exponents and stochastic collision treatments that better account for intrinsic momentum spreads in reactions.

Searching arXiv for recent and foundational EQMD papers relevant to model formulation, applications, and methodological developments. Extended Quantum Molecular Dynamics (EQMD) is a QMD-like transport framework in which each nucleon is represented by a Gaussian wave packet whose width is itself a dynamical variable rather than a fixed parameter. Across the literature, EQMD is used for stable ground-state nucleus modeling, cluster structure, heavily deformed nuclei, and low- to intermediate-energy nuclear reactions, with particular prominence in studies of giant resonances, clustering phenomena, and structure-sensitive reaction observables (Wang et al., 2017, Shi et al., 2024). Its defining refinements over standard QMD are the use of dynamical wave-packet widths, a phenomenological Pauli potential, frictional cooling for initialization, and subtraction of spurious center-of-mass zero-point motion, which together improve ground-state stability and extend the model’s applicability to collective dynamics and nontrivial nuclear morphologies (Shi et al., 2024, Shi et al., 2024).

1. Formal structure of the model

In EQMD, the total many-body wave function is taken as a direct product of single-particle Gaussian packets,

$\Psi=\prod_i \varphi(\boldsymbol r_i),$

with

$$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$

or, in equivalent notation used in other papers,

$v_i=\frac{1}{\lambda_i}+i\delta_i.$

Here $\boldsymbol R_i$ and $\boldsymbol P_i$ are the coordinate- and momentum-space centroids, while the complex width parameter is dynamical (Cao et al., 2022, Wang et al., 2017).

This dynamical width is the main formal distinction from standard QMD emphasized throughout the EQMD literature. Because $\lambda_i$ and $\delta_i$ evolve in time, the model can adapt packet localization to the nuclear environment rather than imposing a fixed packet scale. The papers repeatedly connect this feature to improved ground-state stability, better reproduction of static properties, and more realistic collective motion, especially in deformed nuclei and clusterized systems (Wang et al., 2017, Shi et al., 2024).

The Hamiltonian expectation value is written as kinetic terms plus an interaction part,

$$H_{\rm int}=H_{\rm Skyrme}+H_{\rm Coulomb}+H_{\rm Symmetry}+H_{\rm Pauli},$$

with additional subtraction of spurious center-of-mass zero-point motion. In several implementations, the kinetic energy explicitly contains both the centroid contribution $\mathbf P_i^2/2m$ and a width-dependent quantum term. This distinction becomes important in applications where internal packet momentum spread affects observables, such as direct hard-photon production (Shi et al., 2024, Shi et al., 2020).

2. Initialization, frictional cooling, and effective interactions

EQMD ground states are prepared by evolving the variational parameters with damped equations of motion derived from the time-dependent variational principle,

$$\dot{\mathbf R}_i=\frac{\partial H}{\partial \mathbf P_i}+\mu_{\mathbf R}\frac{\partial H}{\partial \mathbf R_i},\qquad \dot{\mathbf P}_i=-\frac{\partial H}{\partial \mathbf R_i}+\mu_{\mathbf P}\frac{\partial H}{\partial \mathbf P_i},$$

$$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$0

During cooling, the friction coefficients are negative and drive the system toward a local minimum; during real-time dynamics they are set to zero so that energy is conserved (Wang et al., 2017, Cao et al., 2022).

This initialization strategy is central to essentially all EQMD applications. In studies of deformed giant dipole resonances, the cooled states are selected so that the resulting deformation agrees with experiment, enabling axis-resolved resonance analysis in Nd and Sm isotopes (Wang et al., 2017). In broad-nuclide surveys of bubble-like morphologies, frictional cooling is used to obtain relaxed low-energy cluster configurations over the AME2020 chart, with the stopping condition defined by vanishing time derivatives of $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$1, $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$2, $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$3, and $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$4 (Ren et al., 23 May 2026).

The effective interaction is usually decomposed into Skyrme, Coulomb, symmetry, and Pauli terms. In the older EQMD setup used for many GDR calculations, the Skyrme term employs a soft equation of state with

$$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$5

and the symmetry energy coefficient is often taken as $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$6 MeV or varied to test sensitivity (Cao et al., 2022, Wang et al., 2017). The Pauli potential is introduced phenomenologically to suppress close approach of identical nucleons in phase space; in the deformed-nucleus GDR study it is written as

$$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$7

with $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$8 MeV, $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$9, and $v_i=\frac{1}{\lambda_i}+i\delta_i.$0 (Wang et al., 2017).

A recurring practical point is that EQMD’s stability depends not only on the interaction choice but also on how density-dependent terms are propagated. This becomes especially significant once one departs from the original $v_i=\frac{1}{\lambda_i}+i\delta_i.$1 setup and attempts to use softer, non-integer-exponent equations of state (Shi et al., 2024).

3. Collective dipole and monopole dynamics

Giant resonances are among the most developed EQMD application domains. In GDR studies, the standard isovector dipole observables are

$v_i=\frac{1}{\lambda_i}+i\delta_i.$2

and the $v_i=\frac{1}{\lambda_i}+i\delta_i.$3-ray spectrum is extracted from the Fourier transform of the second derivative of the dipole moment (Wang et al., 2017, Cao et al., 2022). In proton-capture calculations, the compound nucleus is formed dynamically and evolved for about $v_i=\frac{1}{\lambda_i}+i\delta_i.$4, with $v_i=\frac{1}{\lambda_i}+i\delta_i.$5 events per energy point and $v_i=\frac{1}{\lambda_i}+i\delta_i.$6 fm to avoid angular-momentum effects (Wang et al., 2017). In deformed heavy nuclei, the GDR is decomposed into major-axis and minor-axis components, and the splitting satisfies

$v_i=\frac{1}{\lambda_i}+i\delta_i.$7

with fitted relations given separately for Sm and Nd chains (Wang et al., 2017).

The same formalism has been used to address deformation, temperature, symmetry energy, angular momentum, and external electromagnetic fields. In deformed Nd and Sm isotopes, EQMD reproduces the transition from a single-hump spectrum in nearly spherical nuclei to double-hump spectra in strongly prolate systems, and for $v_i=\frac{1}{\lambda_i}+i\delta_i.$8Nd the calculation is reported to be perfectly consistent with experiment when $v_i=\frac{1}{\lambda_i}+i\delta_i.$9 MeV (Wang et al., 2017). In proton capture reactions such as $\boldsymbol R_i$0B$\boldsymbol R_i$1C, $\boldsymbol R_i$2Al$\boldsymbol R_i$3Si, $\boldsymbol R_i$4K$\boldsymbol R_i$5Ca, and $\boldsymbol R_i$6Co$\boldsymbol R_i$7Ni, the model yields systematic trends in peak energy, strength, and FWHM, including a temperature-dependent width that is almost constant for $\boldsymbol R_i$8 MeV, rises sharply for $\boldsymbol R_i$9 MeV, and supports saturation above $\boldsymbol P_i$0 MeV (Wang et al., 2017).

EQMD has also been used for Coulomb-excited $\boldsymbol P_i$1Ca in ultra-peripheral $\boldsymbol P_i$2O+$\boldsymbol P_i$3Ca collisions with impact parameter $\boldsymbol P_i$4 fm and beam energies from 50 to 500 MeV/nucleon. In that study, inclusion of the collision-generated magnetic field enhances the GDR peak energy, strength, and width. The magnetic field peaks at about $\boldsymbol P_i$5 G around $\boldsymbol P_i$6 for 100 MeV/nucleon and remains significant up to roughly $\boldsymbol P_i$7; its effect on GDR broadening is interpreted as a combined consequence of temperature and magnetic-field-enhanced angular momentum rather than heating alone (Cao et al., 2022). The paper further notes that in the low-energy region the observed “GDR” likely contains mixed vibrational and rotational character rather than a pure vibrational dipole mode (Cao et al., 2022).

Monopole dynamics provide a complementary benchmark. In the 2024 soft-EOS study, the isoscalar giant monopole resonance of $\boldsymbol P_i$8Pb is used to test newly introduced Skyrme-based interactions. The original EQMD gives a peak energy around $\boldsymbol P_i$9 MeV, far above the RCNP experimental value, which is interpreted as evidence that the original mean field is too stiff. Among the new parameterizations, SkP with $\lambda_i$0 MeV gives the best agreement for $\lambda_i$1Pb, while heavier nuclei are described well overall and lighter nuclei are overestimated by about $\lambda_i$2 MeV (Shi et al., 2024).

4. Equation of state, collision terms, and propagation improvements

A major development in recent EQMD work is the relaxation of the model’s original EOS restrictions. One 2024 study replaces the schematic Skyrme-like interaction by a standard Skyrme energy density functional including bulk, gradient, symmetry, and Coulomb terms, and introduces SkP, SkT1, and SKXce parameter sets with incompressibilities $\lambda_i$3, 236, and 268 MeV, respectively (Shi et al., 2024). Another 2024 study addresses the numerical instability that arises when the density exponent $\lambda_i$4 is non-integer. It replaces the standard approximate treatment of $\lambda_i$5 by a Monte Carlo integral method and shows that, for $\lambda_i$6, the new method reproduces the original analytical EQMD result, while for non-integer $\lambda_i$7 it remains stable where the old approximation diverges or leads to non-positive width parameters (Shi et al., 2024).

These two developments are complementary. The Skyrme-EDF reformulation provides softer incompressibility with long-time ground-state stability over $\lambda_i$8 for $\lambda_i$9Pb (Shi et al., 2024), while the Monte Carlo propagation method makes it practical to use softer EOS parameterizations in IVGDR calculations without the initialization failures associated with the older approximation (Shi et al., 2024). This suggests a broader transition of EQMD from a historically stiff and limited interaction model toward a more flexible transport framework.

Binary-collision modeling has also been revised. In the original EQMD collision prescription, collisions are handled geometrically with $\delta_i$0 fm and

$\delta_i$1

A 2025 study of $\delta_i$2Pb GDR replaces this by a stochastic collision treatment based on Gaussian-overlap probabilities and free elastic cross sections from Cugnon et al. (Shi et al., 24 Aug 2025). With the stochastic term, the GDR width becomes strongly dependent on the $\delta_i$3 cross section, and the best agreement with evaluated data is obtained with

$\delta_i$4

reproducing

$\delta_i$5

The same work concludes that a significant in-medium reduction of free $\delta_i$6 elastic cross sections is needed to reproduce the width (Shi et al., 24 Aug 2025).

A different transport refinement appears in direct hard-photon calculations. Because EQMD packets carry intrinsic momentum spread, using only centroid momenta in $\delta_i$7 collisions misses a substantial fraction of the available kinetic energy. The 2020 study remedies this by sampling the internal momentum distribution of the wave packet for the first photon-producing $\delta_i$8 collision and modifying Pauli blocking accordingly, improving the yield, inverse slope, and angular distribution for $\delta_i$9N+$$H_{\rm int}=H_{\rm Skyrme}+H_{\rm Coulomb}+H_{\rm Symmetry}+H_{\rm Pauli},$$0C at $$H_{\rm int}=H_{\rm Skyrme}+H_{\rm Coulomb}+H_{\rm Symmetry}+H_{\rm Pauli},$$1, 30, and 40 MeV (Shi et al., 2020). The paper’s central point is that the width-dependent quantum kinetic term is not a formal detail but directly affects inelastic observables (Shi et al., 2020).

5. Clustering, short-range correlations, and nuclear morphology

Beyond giant resonances, EQMD is widely used because it can generate and preserve cluster-sensitive configurations. The 2021 short-range-correlation study modifies EQMD by adding a repulsive short-range term

$$H_{\rm int}=H_{\rm Skyrme}+H_{\rm Coulomb}+H_{\rm Symmetry}+H_{\rm Pauli},$$2

motivated by the absence of a sufficiently strong repulsive core in the original interaction (Shen et al., 2021). In $$H_{\rm int}=H_{\rm Skyrme}+H_{\rm Coulomb}+H_{\rm Symmetry}+H_{\rm Pauli},$$3C, strengthening the short-range repulsion enhances the high-momentum tail of the nucleon momentum distribution, shifts RMS radii and binding energies, and weakens the emitted proton-pair momentum correlation function calculated with the Lednicky–Lyuboshitz method (Shen et al., 2021). The same work stresses that this remains a simplified SRC picture because explicit tensor forces, spin, and isospin dynamics are absent (Shen et al., 2021).

In proton-induced reactions on $$H_{\rm int}=H_{\rm Skyrme}+H_{\rm Coulomb}+H_{\rm Symmetry}+H_{\rm Pauli},$$4C, EQMD has been coupled to GEMINI to examine how initial structure affects fragment production and event information entropy. For $$H_{\rm int}=H_{\rm Skyrme}+H_{\rm Coulomb}+H_{\rm Symmetry}+H_{\rm Pauli},$$5C between 5 and 200 MeV/nucleon, the triangular $$H_{\rm int}=H_{\rm Skyrme}+H_{\rm Coulomb}+H_{\rm Symmetry}+H_{\rm Pauli},$$6 structure with binding energy $$H_{\rm int}=H_{\rm Skyrme}+H_{\rm Coulomb}+H_{\rm Symmetry}+H_{\rm Pauli},$$7 MeV exhibits an extra quasi-elastic $$H_{\rm int}=H_{\rm Skyrme}+H_{\rm Coulomb}+H_{\rm Symmetry}+H_{\rm Pauli},$$8C branch relative to the spherical structure with binding energy $$H_{\rm int}=H_{\rm Skyrme}+H_{\rm Coulomb}+H_{\rm Symmetry}+H_{\rm Pauli},$$9 MeV. This excess is most visible around $\mathbf P_i^2/2m$0–$\mathbf P_i^2/2m$1 MeV and peaks near $\mathbf P_i^2/2m$2 MeV; it produces a small dent in both fragment information entropy and multiplicity information entropy, leading the authors to propose event information entropy as a probe for $\mathbf P_i^2/2m$3-cluster structure (Shen et al., 30 Jan 2025).

A more global structural application appears in the 2026 survey of bubble-like nuclei across the AME2020 database. There, EQMD relaxed low-energy cluster configurations are classified by the dimensionless $\mathbf P_i^2/2m$4 scheme: $\mathbf P_i^2/2m$5 counts radial-density inflection points and distinguishes droplet ($\mathbf P_i^2/2m$6), bubble ($\mathbf P_i^2/2m$7), and toroidal bubble ($\mathbf P_i^2/2m$8) nuclei; $\mathbf P_i^2/2m$9 measures central density depletion; $$\dot{\mathbf R}_i=\frac{\partial H}{\partial \mathbf P_i}+\mu_{\mathbf R}\frac{\partial H}{\partial \mathbf R_i},\qquad \dot{\mathbf P}_i=-\frac{\partial H}{\partial \mathbf R_i}+\mu_{\mathbf P}\frac{\partial H}{\partial \mathbf P_i},$$0 measures relative surface thickness; and $$\dot{\mathbf R}_i=\frac{\partial H}{\partial \mathbf P_i}+\mu_{\mathbf R}\frac{\partial H}{\partial \mathbf R_i},\qquad \dot{\mathbf P}_i=-\frac{\partial H}{\partial \mathbf R_i}+\mu_{\mathbf P}\frac{\partial H}{\partial \mathbf P_i},$$1 measures the relative size of the internal low-density region (Ren et al., 23 May 2026). Light nuclei are predominantly droplet-like with $$\dot{\mathbf R}_i=\frac{\partial H}{\partial \mathbf P_i}+\mu_{\mathbf R}\frac{\partial H}{\partial \mathbf R_i},\qquad \dot{\mathbf P}_i=-\frac{\partial H}{\partial \mathbf R_i}+\mu_{\mathbf P}\frac{\partial H}{\partial \mathbf P_i},$$2, $$\dot{\mathbf R}_i=\frac{\partial H}{\partial \mathbf P_i}+\mu_{\mathbf R}\frac{\partial H}{\partial \mathbf R_i},\qquad \dot{\mathbf P}_i=-\frac{\partial H}{\partial \mathbf R_i}+\mu_{\mathbf P}\frac{\partial H}{\partial \mathbf P_i},$$3, $$\dot{\mathbf R}_i=\frac{\partial H}{\partial \mathbf P_i}+\mu_{\mathbf R}\frac{\partial H}{\partial \mathbf R_i},\qquad \dot{\mathbf P}_i=-\frac{\partial H}{\partial \mathbf R_i}+\mu_{\mathbf P}\frac{\partial H}{\partial \mathbf P_i},$$4, $$\dot{\mathbf R}_i=\frac{\partial H}{\partial \mathbf P_i}+\mu_{\mathbf R}\frac{\partial H}{\partial \mathbf R_i},\qquad \dot{\mathbf P}_i=-\frac{\partial H}{\partial \mathbf R_i}+\mu_{\mathbf P}\frac{\partial H}{\partial \mathbf P_i},$$5, while most medium-mass nuclei have $$\dot{\mathbf R}_i=\frac{\partial H}{\partial \mathbf P_i}+\mu_{\mathbf R}\frac{\partial H}{\partial \mathbf R_i},\qquad \dot{\mathbf P}_i=-\frac{\partial H}{\partial \mathbf R_i}+\mu_{\mathbf P}\frac{\partial H}{\partial \mathbf P_i},$$6, especially near $$\dot{\mathbf R}_i=\frac{\partial H}{\partial \mathbf P_i}+\mu_{\mathbf R}\frac{\partial H}{\partial \mathbf R_i},\qquad \dot{\mathbf P}_i=-\frac{\partial H}{\partial \mathbf R_i}+\mu_{\mathbf P}\frac{\partial H}{\partial \mathbf P_i},$$7Ca and in neutron-rich regions. Toroidal bubble nuclei emerge for $$\dot{\mathbf R}_i=\frac{\partial H}{\partial \mathbf P_i}+\mu_{\mathbf R}\frac{\partial H}{\partial \mathbf R_i},\qquad \dot{\mathbf P}_i=-\frac{\partial H}{\partial \mathbf R_i}+\mu_{\mathbf P}\frac{\partial H}{\partial \mathbf P_i},$$8 and become prevalent in heavy systems, and bubble structures are reported to be widespread in the superheavy region (Ren et al., 23 May 2026).

These applications indicate the niche EQMD occupies within transport modeling: it is not limited to reaction kinematics, but is repeatedly used as a structure-sensitive dynamical generator for deformed, clustered, and hollow configurations. A plausible implication is that the model’s phenomenological ingredients are being used not only to propagate reactions but also to define effective nuclear morphology classes.

6. Limitations, interpretive issues, and related generalizations

Several limitations recur across the EQMD literature. The Pauli principle is enforced through a phenomenological Pauli potential rather than exact antisymmetrization, which is repeatedly presented as a practical approximation rather than a first-principles treatment (Wang et al., 2017, Shi et al., 2024). The original interaction set was effectively limited to hard incompressibility, motivating the introduction of softer Skyrme-based parameterizations (Shi et al., 2024). Momentum-dependent interactions are still absent in the 2024 soft-EOS implementation, and the authors explicitly identify that as a remaining limitation (Shi et al., 2024).

Collision modeling is another area of active revision. The original geometric collision term is criticized for possible spurious repeated collisions and for the arbitrariness of $$\dot{\mathbf R}_i=\frac{\partial H}{\partial \mathbf P_i}+\mu_{\mathbf R}\frac{\partial H}{\partial \mathbf R_i},\qquad \dot{\mathbf P}_i=-\frac{\partial H}{\partial \mathbf R_i}+\mu_{\mathbf P}\frac{\partial H}{\partial \mathbf P_i},$$9 and $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$00; the stochastic reformulation in the 2025 $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$01Pb study is presented as more suitable when the cross section is large (Shi et al., 24 Aug 2025). Likewise, the hard-photon work shows that centroid-only collision kinematics can be inadequate when intrinsic wave-packet momentum spread materially contributes to the available collision energy (Shi et al., 2020).

Interpretive caution also appears in resonance applications. In the magnetic-field study of $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$02Ca, the broadening of the fitted GDR spectrum cannot be understood from temperature alone because the field also changes angular momentum, and at low beam energy the resonance likely mixes vibrational and rotational character (Cao et al., 2022). In proton-capture studies, the real process includes strong $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$03 transitions that are difficult to reproduce in full detail, and for $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$04Co$$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$05Ni the temperature changes very little, suggesting proton capture may not be an ideal way to excite GDR in nuclei with $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$06 (Wang et al., 2017).

A related methodological direction is represented by a 2022 wave-packet molecular dynamics framework described as very closely related to EQMD in spirit and in mathematical structure, but generalized to anisotropic packets with full tensorial shape dynamics (Svensson et al., 2022). That model allows elongation and rotation in arbitrary directions, uses generalized Ewald summation, and reports a $$\varphi(\boldsymbol r_i)=\left(\frac{\nu_i+\nu_i^\ast}{2\pi}\right)^{3/4} \exp\!\left[-\frac{\nu_i}{2}(\boldsymbol r_i-\boldsymbol R_i)^2+\frac{i}{\hbar}\boldsymbol P_i\cdot \boldsymbol r_i\right],$$07 increase in DC conductivity in dense hydrogen relative to isotropic packets (Svensson et al., 2022). Although it is not an EQMD paper in the strict nuclear-transport sense, it indicates how the EQMD philosophy of dynamical packet-shape evolution can be extended when anisotropy itself becomes a leading-order physical degree of freedom.

Taken together, these developments define EQMD as a variational Gaussian-packet transport model whose distinctive contribution lies in the simultaneous treatment of dynamical widths, stabilized initialization, phenomenological fermionic repulsion, and structure-sensitive real-time evolution. The model’s most established achievements are in deformation-resolved GDR systematics, soft-EOS extensions, clustering-sensitive reaction studies, and the identification of exotic density morphologies, while its continuing development is concentrated on collision kernels, EOS flexibility, and more complete microscopic content (Wang et al., 2017, Shi et al., 2024, Ren et al., 23 May 2026).