---
title: Gogny D1M Effective Interaction
url: https://www.emergentmind.com/topics/gogny-d1m-effective-interaction
type: topic
---

# Gogny D1M Effective Interaction

to=arxiv_search.search  银航json
{"query":"Gogny D1M effective interaction D1M Gogny arXiv", "max_results": 10}
to=arxiv_search.search ＿国产json
{"query":"1408.6941 or 1001.4893 or 1806.02080 Gogny D1M", "max_results": 10}
to=arxiv_search.search  天天彩票中大奖json
{"query":"\"Ground state octupole correlation energies with effective forces\" OR \"Mean field study of structural changes in Pt isotopes with the Gogny interaction\" OR \"Spurious finite-size instabilities with Gogny-type interactions\"", "max_results": 10}
to=arxiv_search.search  全民彩票天天送钱json
{"query":"site:arxiv.org Gogny D1M", "max_results": 5}
The Gogny D1M effective interaction is a parametrization of the finite-range, density-dependent Gogny force used in self-consistent Hartree–Fock–Bogoliubov calculations of nuclear structure. Within the D1 family, D1M was adjusted to reproduce all known nuclear binding energies using HFB, supplemented by an approximate rotational energy correction and an approximate zero-point energy correction associated with quadrupole motion, while also constraining charge radii and requiring a realistic symmetric and neutron-matter equation of state; the resulting global mass fit achieved an rms deviation of 0.79 MeV [1408.6941]. In subsequent applications, D1M has been used as a global Gogny energy density functional for mean-field, beyond-mean-field, fission, spectroscopic, and astrophysical studies, with the same finite-range interaction generating both the particle–hole and pairing fields [1312.7229].

## 1. Historical placement and fitting rationale

D1M was introduced as a successor to earlier Gogny parametrizations such as D1S and D1N, with an explicit emphasis on global mass reproduction and realistic matter properties. In the fission literature, it is described as a parametrization designed to improve global nuclear mass predictions relative to D1S while maintaining good spectroscopic performance, and as curing the “mass drift” of D1S in heavy nuclei and improving \(Q_\alpha\) values [1405.6784]. In large-scale fission benchmarks, D1M is further characterized as fitted to realistic neutron matter and to the binding energies of all known nuclei, achieving a global rms of 0.798 MeV [1312.7229].

The fit strategy is central to the identity of D1M. One strand of the literature emphasizes that the parameters were adjusted to reproduce all known nuclear binding energies with HFB, supplemented by approximate rotational and quadrupole zero-point corrections, in the spirit of a 5D Bohr Hamiltonian [1408.6941]. Another strand stresses that the fit also incorporated a realistic neutron-matter equation of state and charge radii, making D1M simultaneously a mass model and a constrained effective interaction for bulk matter [1405.6784]. This construction explains why D1M is often chosen when binding-energy systematics, neutron-rich nuclei, or heavy-nucleus fission are the primary targets.

A recurring conclusion across applications is that D1M preserves the characteristic robustness of the Gogny framework. In Pt isotopes, D1M yields results extremely similar to those of D1S and D1N, with only modest quantitative differences traceable to pairing strength and barrier heights [1001.4893]. In octupole-correlation studies, the qualitative and quantitative conclusions obtained with D1M are reported to be similar to those found with D1S and D1N [1408.6941]. This suggests that D1M is best understood not as a radical departure from the Gogny tradition, but as a refit that shifts the balance toward global masses and matter constraints.

## 2. Formal structure of the interaction

D1M retains the standard Gogny operator structure: two finite-range Gaussian central terms with spin- and isospin-exchange operators, a zero-range density-dependent term, and a zero-range spin–orbit term; in finite nuclei, the Coulomb interaction is included for protons [1408.6941]. In conventional notation, the two-body interaction is written as
\[
V(\mathbf{r}_1,\mathbf{r}_2)=\sum_{i=1}^{2}\Big(W_i+B_iP_\sigma-H_iP_\tau-M_iP_\sigma P_\tau\Big)e^{-|\mathbf{r}_1-\mathbf{r}_2|^2/\mu_i^2}
+t_3(1+x_0P_\sigma)\delta(\mathbf{r}_1-\mathbf{r}_2)\rho^\alpha\!\left(\frac{\mathbf{r}_1+\mathbf{r}_2}{2}\right)
+iW_0(\boldsymbol{\sigma}_1+\boldsymbol{\sigma}_2)\cdot\big(\mathbf{k}'\times\delta(\mathbf{r}_1-\mathbf{r}_2)\mathbf{k}\big),
\]
with \(P_\sigma\) and \(P_\tau\) the spin and isospin exchange operators, \(\mu_i\) the Gaussian ranges, \(\rho\) the local density, and \(\mathbf{k},\mathbf{k}'\) the relative-momentum operators acting to the right and left [1408.6941].

The D1M parameter set reported in later Gogny neutron-star work is
\[
\mu_1=0.50\ \mathrm{fm},\qquad \mu_2=1.00\ \mathrm{fm},
\]
\[
W_1=-12797.57,\quad B_1=14048.85,\quad H_1=-15144.43,\quad M_1=11963.81,
\]
\[
W_2=490.95,\quad B_2=-752.27,\quad H_2=675.12,\quad M_2=-693.57,
\]
\[
t_3=1562.22\ \mathrm{MeV\,fm}^4,\qquad x_3=1,\qquad \alpha=\frac13,\qquad W_0=115.36\ \mathrm{MeV\,fm}^5
\]
[2109.02520]. The literature also writes the density-dependent exchange coefficient as \(x_0\) rather than \(x_3\); both notations appear in Gogny papers collected here.

A defining technical feature of the Gogny architecture is that the same finite-range interaction generates both the mean field and the pairing field. In HFB applications with D1M, no separate pairing force is introduced, and the finite range makes the pairing tensor finite and naturally regulates ultraviolet behavior [2104.08063]. This aspect distinguishes D1M from functionals in which pairing is appended phenomenologically.

## 3. Nuclear-matter properties and global EDF character

The bulk properties of D1M at saturation are explicitly tabulated in unified neutron-star calculations: saturation density \(\rho_0=0.1647\ \mathrm{fm}^{-3}\), energy per particle \(E_0=-16.02\ \mathrm{MeV}\), incompressibility \(K_\infty=224.98\ \mathrm{MeV}\), isoscalar effective mass \(m^*/m=0.746\), symmetry energy \(J=28.55\ \mathrm{MeV}\), symmetry energy at \(0.1\ \mathrm{fm}^{-3}\) of \(23.80\ \mathrm{MeV}\), and slope parameter \(L=24.83\ \mathrm{MeV}\) [2109.02520]. These numbers are consistent with the broader characterization of D1M as a Gogny functional that combines good finite-nucleus performance with a realistic symmetric and neutron-matter equation of state [1408.6941].

The same body of work also identifies a limitation: D1M produces a rather soft equation of state in neutron matter, and in neutron-star applications this softness leads to a maximum mass of about \(1.74\,M_\odot\), below the observational value of two solar masses [2109.02520]. Earlier astrophysical discussions make the same point more generally for D1S, D1N, and D1M, noting that the common problem is a too soft neutron-matter equation of state at high density [1712.06735]. The issue is therefore not a defect of D1M in finite nuclei, but a tension between finite-nucleus calibration and the high-density isovector sector.

This limitation motivated reparametrizations derived from D1M. D1M* and D1M** were constructed by modifying the finite-range strengths so as to stiffen the symmetry energy density dependence while preserving the good properties of D1M in finite nuclei [1712.06735; 1810.07469; 2109.02520]. The reparametrization strategy kept \(\mu_i\), \(\alpha\), and the spin–orbit strength fixed, constrained symmetric-matter properties and pairing combinations to remain equal to D1M, and used the remaining freedom to increase \(L\) [1712.06735]. The relation between D1M and D1M* is therefore structural: D1M* is best viewed as a targeted isovector modification of D1M rather than an unrelated force.

## 4. Mean-field and beyond-mean-field implementation

D1M is used within the constrained HFB framework in both axial and triaxial spaces. In representative calculations, the HFB equations are solved in a harmonic-oscillator basis with a gradient method to locate minima, with constraints imposed on multipole operators such as \(Q_{20}\), \(Q_{22}\), and \(Q_{30}\) [1001.4893]. In octupole applications, a family of axially symmetric HFB states \(|\Phi(Q_{30})\rangle\) is generated by varying the axial octupole moment, and the constrained energy
\[
E(Q_{30})=\langle \Phi(Q_{30})|\hat H|\Phi(Q_{30})\rangle
\]
is used to determine whether the ground state spontaneously breaks reflection symmetry [1408.6941].

Beyond mean field, D1M has been employed in parity restoration and in Generator Coordinate Method calculations. For parity projection one evaluates
\[
E_{\pi}(Q_{30})=\frac{\langle \Phi(Q_{30})|\hat H\,\hat P_\pi|\Phi(Q_{30})\rangle}{\langle \Phi(Q_{30})|\hat P_\pi|\Phi(Q_{30})\rangle},
\qquad
\hat P_\pm=\frac12(1\pm\hat\Pi),
\]
while in the GCM the collective state is written as
\[
|\Psi\rangle=\int dq\,f(q)\,|\Phi(q)\rangle,
\]
with the weights determined by the Hill–Wheeler equation [1408.6941]. In more elaborate quadrupole–octupole studies, D1M is combined with parity restoration and symmetry-conserving GCM in two collective coordinates, using mixed-density or projected-density prescriptions for the density-dependent term [2104.08063].

This methodology has two implications for the interpretation of D1M. First, the parametrization is routinely deployed not only as a static mean-field interaction but as a building block for correlation energies, negative-parity excitations, and collective amplitudes. Second, the density dependence of Gogny forces requires explicit care in symmetry restoration and configuration mixing, a point repeatedly emphasized in the D1M literature [1408.6941; 2104.08063].

## 5. Finite-nucleus applications and empirical performance

In large-scale octupole studies over 818 even–even nuclei from oxygen to copernicium, D1M yields modest but non-negligible octupole correlation energies: the mean-field octupole energy gain does not exceed about \(1.2\) MeV, parity-restored correlation energies reach up to about \(1.5\) MeV, and full octupole GCM correlation energies reach up to about \(2.5\) MeV [1408.6941]. When these correlations are added to binding energies, they tend to shift theoretical curves upward almost uniformly, with only minor local improvements; the too-large shell gaps predicted by self-consistent mean-field models are not quenched [1408.6941]. The specific conclusion is that octupole correlations do not affect in a significant way the trend and systematic of binding energies.

In Pt isotopes, D1M reproduces the same qualitative and nearly the same quantitative mean-field structural systematics as D1S and D1N. The robust picture reported there is: \(^{166\text{–}182}\)Pt prolate ground states, \(^{184\text{–}196}\)Pt triaxial \(\gamma\)-soft ground states, \(^{198\text{–}202}\)Pt oblate ground states, and \(^{204}\)Pt spherical [1001.4893]. The differences relative to D1S are limited to slightly lower spherical barrier heights and stronger pairing, the latter leading to somewhat smaller Thouless–Valatin moments of inertia [1001.4893]. This close agreement is one of the clearest demonstrations that D1M preserves Gogny spectroscopic systematics outside the mass-fit dataset.

Fission applications extend the range of validation. For uranium isotopes \(^{232\text{–}280}\)U, D1M is benchmarked against available experimental data on inner and second barrier heights, excitation energies of the fission isomers, and half-lives, and is concluded to represent a reasonable starting point to describe fission in heavy and superheavy nuclei [1312.7229]. For plutonium isotopes \(^{232\text{–}280}\)Pu, D1M is again described as a reasonable starting point for microscopic fission, with the important caveat that absolute spontaneous-fission half-lives are extremely sensitive to pairing strength and collective inertia [1405.6784]. In both uranium and plutonium studies, the trends with neutron number are reproduced more reliably than the absolute values.

D1M has also served as the microscopic input to mapped spectroscopic models. In odd-odd \(^{124\text{–}132}\)Cs, constrained Gogny-D1M HFB calculations provide the \((\beta,\gamma)\)-deformation energy surfaces of the even-even Xe cores, as well as single-particle energies and occupation probabilities, which are then mapped to an interacting boson–fermion–fermion Hamiltonian [1908.03322]. This use underscores a broader role of D1M: it is not only a direct HFB interaction, but also a generator of microscopic structure inputs for algebraic or collective Hamiltonians.

## 6. Stability, misconceptions, and descendant parametrizations

A recurrent misconception is to transfer the finite-size-instability discussion of D1M* to D1M itself. The linear-response study of Gogny-type interactions reports that D1M performs better than D1M* and D1N in the scalar–isovector channel, and finite-nucleus tests confirm the absence of spurious behavior for D1M [1806.02080]. In coordinate-space calculations for \(^{208}\)Pb, \(^{120}\)Sn, and \(^{4}\)He, D1M converges and yields physically reasonable proton and neutron densities [1806.02080]. The instability problem identified in 2018 is therefore specific to D1M* and, to a lesser extent, D1N, not to D1M.

The subsequent comment on D1M* sharpened that distinction. It independently confirmed the existence of a finite-size instability for D1M* in coordinate-space calculations, but also showed that harmonic-oscillator-basis calculations—the standard framework for Gogny forces—are robust, and that beyond-mean-field GCM calculations with D1M* remain stable and yield results very close to D1M [1807.10159]. For D1M itself, the same comment presents consistent HO-basis and mesh binding energies for doubly magic nuclei, reinforcing the view that D1M is a stable reference within the Gogny family.

In astrophysical work, the D1M descendants D1M* and D1M** were introduced precisely because D1M is too soft in neutron matter, not because it is unreliable in finite nuclei [1712.06735; 2109.02520]. D1M* raises the symmetry-energy slope to \(L=43.18\) MeV and D1M** to \(L=33.91\) MeV, producing maximum neutron-star masses around two solar masses while preserving D1M-like finite-nucleus performance [2109.02520]. A plausible implication is that D1M remains the reference Gogny parametrization when global finite-nucleus structure is the priority, whereas D1M* and D1M** are specialized extensions for neutron-star applications.

Taken together, the literature presents D1M as a global Gogny effective interaction with a canonical two-Gaussian finite-range structure, a density-dependent zero-range term, and a zero-range spin–orbit term; a fit protocol centered on masses, radii, and matter constraints; strong performance in mean-field and beyond-mean-field finite-nucleus calculations; and a well-defined limitation in the high-density isovector sector that later reparametrizations sought to remedy [1408.6941; 1312.7229; 2109.02520].

Source: https://www.emergentmind.com/topics/gogny-d1m-effective-interaction