## This content will become publicly available on January 29, 2021

# Exact restoration of Galilei invariance in density functional calculations with quantum Monte Carlo

## Abstract

We investigate some aspects of the connection between mean-field and ab-initio calculations for any-nucleon systems employing the Gogny effective interaction. In particular, we present variational Monte Carlo calculations of ^{4}He and ^{16}O, for which we extended the formalism of the linear method to the case of a density-dependent Hamiltonian. Monte Carlo calculations using an uncorrelated basis reduce to Hartree-Fock ones, but Galilean invariance can be preserved, contrary to the usual treatment. An analysis of the results obtained with the Gogny force when used in conjunction with correlated wave functions has also been performed. Here, the calculations show the collapse of the deuteron, leading to a set of unphysical predictions for the structure of heavier nuclei. This behavior remarks the different role of the Gogny effective interaction as compared to bare interactions typically used in fully microscopic calculations.

- Authors:

- Univ. of Trento (Italy)
- Univ. de Sevilla (Spain)
- Michigan State Univ., East Lansing, MI (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- INFN-TIFPA Trento Institute of Fundamental Physics and Applications, Trento (Italy); Argonne National Lab. (ANL), Argonne, IL (United States)
- Univ. of Trento (Italy); INFN-TIFPA Trento Institute of Fundamental Physics and Applications, Trento (Italy)
- INFN Sezione di Milano (Italy)

- Publication Date:

- Research Org.:
- Argonne National Lab. (ANL), Argonne, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Nuclear Physics (NP) (SC-26); European Union, Horizon 2020 Research and Innovation Programme; Spanish Ministerio de Economia y Competitividad (MINECO); USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Scientific User Facilities Division

- OSTI Identifier:
- 1597272

- Grant/Contract Number:
- AC02-06CH11357; 654002; SC0013617

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Physics. G, Nuclear and Particle Physics

- Additional Journal Information:
- Journal Volume: 47; Journal Issue: 3; Journal ID: ISSN 0954-3899

- Publisher:
- IOP Publishing

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; quantum Monte Carlo; density functional theory; Galilean invariance; density-dependent nuclear interactions; mean-field calculations

### Citation Formats

```
Massella, P., Barranco, F., Lonardoni, D., Lovato, A., Pederiva, F., and Vigezzi, Enrico. Exact restoration of Galilei invariance in density functional calculations with quantum Monte Carlo. United States: N. p., 2020.
Web. doi:10.1088/1361-6471/ab588c.
```

```
Massella, P., Barranco, F., Lonardoni, D., Lovato, A., Pederiva, F., & Vigezzi, Enrico. Exact restoration of Galilei invariance in density functional calculations with quantum Monte Carlo. United States. doi:10.1088/1361-6471/ab588c.
```

```
Massella, P., Barranco, F., Lonardoni, D., Lovato, A., Pederiva, F., and Vigezzi, Enrico. Wed .
"Exact restoration of Galilei invariance in density functional calculations with quantum Monte Carlo". United States. doi:10.1088/1361-6471/ab588c.
```

```
@article{osti_1597272,
```

title = {Exact restoration of Galilei invariance in density functional calculations with quantum Monte Carlo},

author = {Massella, P. and Barranco, F. and Lonardoni, D. and Lovato, A. and Pederiva, F. and Vigezzi, Enrico},

abstractNote = {We investigate some aspects of the connection between mean-field and ab-initio calculations for any-nucleon systems employing the Gogny effective interaction. In particular, we present variational Monte Carlo calculations of 4He and 16O, for which we extended the formalism of the linear method to the case of a density-dependent Hamiltonian. Monte Carlo calculations using an uncorrelated basis reduce to Hartree-Fock ones, but Galilean invariance can be preserved, contrary to the usual treatment. An analysis of the results obtained with the Gogny force when used in conjunction with correlated wave functions has also been performed. Here, the calculations show the collapse of the deuteron, leading to a set of unphysical predictions for the structure of heavier nuclei. This behavior remarks the different role of the Gogny effective interaction as compared to bare interactions typically used in fully microscopic calculations.},

doi = {10.1088/1361-6471/ab588c},

journal = {Journal of Physics. G, Nuclear and Particle Physics},

number = 3,

volume = 47,

place = {United States},

year = {2020},

month = {1}

}

Works referenced in this record:

##
Microscopic approach of fission dynamics applied to fragment kinetic energy and mass distributions in ${}^{238}\mathrm{U}$

journal, February 2005

- Goutte, H.; Berger, J. F.; Casoli, P.
- Physical Review C, Vol. 71, Issue 2

##
Mean field and beyond description of nuclear structure with the Gogny force: a review

journal, December 2018

- Robledo, L. M.; Rodríguez, T. R.; Rodríguez-Guzmán, R. R.
- Journal of Physics G: Nuclear and Particle Physics, Vol. 46, Issue 1

##
Density functionals in the laboratory frame

journal, January 2008

- Giraud, B. G.
- Physical Review C, Vol. 77, Issue 1

##
Ab initio no core shell model

journal, March 2013

- Barrett, Bruce R.; Navrátil, Petr; Vary, James P.
- Progress in Particle and Nuclear Physics, Vol. 69

##
Accurate nucleon-nucleon potential with charge-independence breaking

journal, January 1995

- Wiringa, R. B.; Stoks, V. G. J.; Schiavilla, R.
- Physical Review C, Vol. 51, Issue 1

##
*Ab Initio* Description of the Exotic Unbound ${}^{7}\mathrm{He}$ Nucleus

journal, January 2013

- Baroni, Simone; Navrátil, Petr; Quaglioni, Sofia
- Physical Review Letters, Vol. 110, Issue 2

##
Isovector properties of the Gogny interaction

journal, November 2014

- Sellahewa, Roshan; Rios, Arnau
- Physical Review C, Vol. 90, Issue 5

##
Electromagnetic structure of the helium isotopes

journal, April 1977

- McCarthy, J. S.; Sick, I.; Whitney, R. R.
- Physical Review C, Vol. 15, Issue 4

##
Testing nonlocal nucleon-nucleon interactions in four-nucleon systems

journal, October 2004

- Lazauskas, Rimantas; Carbonell, Jaume
- Physical Review C, Vol. 70, Issue 4

##
Elastic proton scattering on tritium below the $n\text{\u2212}{}^{3}\mathrm{He}$ threshold

journal, May 2009

- Lazauskas, Rimantas
- Physical Review C, Vol. 79, Issue 5

##
Elastische Elektronenstreuung an14N,15N,16O und18O bei kleiner Impuls�bertragung

journal, March 1975

- Sch�tz, Wolfgang
- Zeitschrift f�r Physik A Atoms and Nuclei, Vol. 273, Issue 1

##
Solution of the Center-Of-Mass Problem in Nuclear Structure Calculations

journal, August 2009

- Hagen, G.; Papenbrock, T.; Dean, D. J.
- Physical Review Letters, Vol. 103, Issue 6

##
Light-Nuclei Spectra from Chiral Dynamics

journal, February 2018

- Piarulli, M.; Baroni, A.; Girlanda, L.
- Physical Review Letters, Vol. 120, Issue 5

##
Some considerations on the restoration of Galilei invariance in the nuclear many-body problem: Part I: Mathematical tools, spectral functions and spectroscopic factors of simple bound states

journal, September 2001

- Schmid, K. W.
- The European Physical Journal A, Vol. 12, Issue 1

##
Density functional theory with spatial-symmetry breaking and configuration mixing

journal, April 2014

- Lesinski, Thomas
- Physical Review C, Vol. 89, Issue 4

##
Hartree-Fock-Bogolyubov calculations with the $D1$ effective interaction on spherical nuclei

journal, April 1980

- Dechargé, J.; Gogny, D.
- Physical Review C, Vol. 21, Issue 4

##
Time-dependent quantum collective dynamics applied to nuclear fission

journal, February 1991

- Berger, J. F.; Girod, M.; Gogny, D.
- Computer Physics Communications, Vol. 63, Issue 1-3

##
Nuclear Charge Distributions of Isotone Pairs. II. ${}_{}{}^{39}{}_{}{}^{}\mathrm{K}$ and ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$

journal, May 1973

- Sinha, B. B. P.; Peterson, G. A.; Whitney, R. R.
- Physical Review C, Vol. 7, Issue 5

##
Properties of ${}^{12}C$ in the *Ab Initio* Nuclear Shell Model

journal, June 2000

- Navrátil, P.; Vary, J. P.; Barrett, B. R.
- Physical Review Letters, Vol. 84, Issue 25

##
Spherical Hartree-Fock calculations with linear-momentum projection before the variation: Part II: Spectral functions and spectroscopic factors

journal, January 2004

- Rodríguez-Guzmán, R. R.; Schmid, K. W.
- The European Physical Journal A, Vol. 19, Issue 1

##
Four-nucleon scattering: *Ab initio* calculations in momentum space

journal, January 2007

- Deltuva, A.; Fonseca, A. C.
- Physical Review C, Vol. 75, Issue 1

##
Optimization of Ground- and Excited-State Wave Functions and van der Waals Clusters

journal, July 2001

- Nightingale, M. P.; Melik-Alaverdian, Vilen
- Physical Review Letters, Vol. 87, Issue 4

##
Quantum Monte Carlo calculations of light nuclei with local chiral two- and three-nucleon interactions

journal, November 2017

- Lynn, J. E.; Tews, I.; Carlson, J.
- Physical Review C, Vol. 96, Issue 5

##
Some considerations on the restoration of Galilei invariance in the nuclear many-body problem: Part II: Electromagnetic properties of simple bound states

journal, March 2002

- Schmid, K. W.
- The European Physical Journal A, Vol. 13, Issue 3

##
Analyzing the Fierz rearrangement freedom for local chiral two-nucleon potentials

journal, November 2017

- Huth, L.; Tews, I.; Lynn, J. E.
- Physical Review C, Vol. 96, Issue 5

##
Simple parametrization of nucleon form factors

journal, December 2004

- Kelly, J. J.
- Physical Review C, Vol. 70, Issue 6

##
A high-precision variational approach to three- and four-nucleon bound and zero-energy scattering states

journal, May 2008

- Kievsky, A.; Rosati, S.; Viviani, M.
- Journal of Physics G: Nuclear and Particle Physics, Vol. 35, Issue 6

##
Elastische Elektronenstreuung an4He zwischen 30 und 59 MeV

journal, April 1968

- Erich, U.; Frank, H.; Haas, D.
- Zeitschrift für Physik A Hadrons and nuclei, Vol. 209, Issue 2

##
$N\text{\u2212}d$ elastic scattering using the hyperspherical harmonics approach with realistic local and nonlocal interactions

journal, September 2009

- Marcucci, L. E.; Kievsky, A.; Girlanda, L.
- Physical Review C, Vol. 80, Issue 3

##
Alternate, well-founded way to treat center-of-mass correlations: Proposal of a local center-of-mass correlations potential

journal, February 2013

- Messud, Jérémie
- Physical Review C, Vol. 87, Issue 2

##
Nuclear charge distributions in $1{f}_{\frac{7}{2}}$ -shell nuclei from muonic x-ray measurements

journal, January 1981

- Wohlfahrt, H. D.; Shera, E. B.; Hoehn, M. V.
- Physical Review C, Vol. 23, Issue 1

##
Some considerations on the restoration of Galilei invariance in the nuclear many-body problem: Part III: Energies of simple bound states

journal, August 2002

- Schmid, K. W.
- The European Physical Journal A, Vol. 14, Issue 4

##
Equation of State Calculations by Fast Computing Machines

journal, June 1953

- Metropolis, Nicholas; Rosenbluth, Arianna W.; Rosenbluth, Marshall N.
- The Journal of Chemical Physics, Vol. 21, Issue 6

##
Some considerations on the restoration of Galilei invariance in the nuclear many-body problem: Part IV: Simple continuum states

journal, April 2003

- Schmid, K. W.
- The European Physical Journal A, Vol. 16, Issue 4

##
Generalization of internal density-functional theory and Kohn-Sham scheme to multicomponent self-bound systems, and link with traditional density-functional theory

journal, November 2011

- Messud, Jérémie
- Physical Review A, Vol. 84, Issue 5

##
Structure of the Ground State of a Fermion Fluid

journal, September 1981

- Schmidt, K. E.; Lee, Michael A.; Kalos, M. H.
- Physical Review Letters, Vol. 47, Issue 11

##
Neutron Skin of ${}^{208}\mathrm{Pb}$ , Nuclear Symmetry Energy, and the Parity Radius Experiment

journal, June 2011

- Roca-Maza, X.; Centelles, M.; Viñas, X.
- Physical Review Letters, Vol. 106, Issue 25

##
Minimally nonlocal nucleon-nucleon potentials with chiral two-pion exchange including $\Delta $ resonances

journal, February 2015

- Piarulli, M.; Girlanda, L.; Schiavilla, R.
- Physical Review C, Vol. 91, Issue 2

##
Benchmark test calculation of a four-nucleon bound state

journal, August 2001

- Kamada, H.; Nogga, A.; Glöckle, W.
- Physical Review C, Vol. 64, Issue 4

##
Structure of the ${\mathrm{He}}^{4}$ Nucleus from Elastic Electron Scattering

journal, August 1967

- Frosch, R. F.; McCarthy, J. S.; Rand, R. E.
- Physical Review, Vol. 160, Issue 4

##
Implementation of the linear method for the optimization of Jastrow–Feenberg and backflow correlations

journal, May 2015

- Motta, M.; Bertaina, G.; Galli, D. E.
- Computer Physics Communications, Vol. 190

##
Elastic Electron Scattering from ${}_{}{}^{3}{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^{4}{}_{}{}^{}\mathrm{He}$ at High Momentum Transfer

journal, May 1978

- Arnold, R. G.; Chertok, B. T.; Rock, S.
- Physical Review Letters, Vol. 40, Issue 22

##
Systematic study of infrared energy corrections in truncated oscillator spaces with Gogny energy density functionals

journal, November 2016

- Arzhanov, Alexander; Rodríguez, Tomás R.; Martínez-Pinedo, Gabriel
- Physical Review C, Vol. 94, Issue 5

##
Elastic electron scattering from 3He and 4He

journal, April 1985

- Ottermann, C. R.; Köbschall, G.; Maurer, K.
- Nuclear Physics A, Vol. 436, Issue 4

##
Ground-state properties of 4 He and 16 O extrapolated from lattice QCD with pionless EFT

journal, September 2017

- Contessi, L.; Lovato, A.; Pederiva, F.
- Physics Letters B, Vol. 772

##
Charge density of 40Ca

journal, December 1979

- Sick, I.; Bellicard, J. B.; Cavedon, J. M.
- Physics Letters B, Vol. 88, Issue 3-4

##
Electromagnetic structure of few-nucleon ground states

journal, January 2016

- Marcucci, L. E.; Gross, F.; Peña, M. T.
- Journal of Physics G: Nuclear and Particle Physics, Vol. 43, Issue 2

##
State dependent effective interaction for the hyperspherical formalism

journal, March 2000

- Barnea, Nir; Leidemann, Winfried; Orlandini, Giuseppina
- Physical Review C, Vol. 61, Issue 5

##
Variational Calculation on A = 3 and 4 Nuclei with Non-Local Potentials

journal, July 2006

- Viviani, M.; Marcucci, L. E.; Rosati, S.
- Few-Body Systems, Vol. 39, Issue 3-4

##
Inclusive electron-nucleus cross section within the self-consistent Green's function approach

journal, August 2018

- Rocco, N.; Barbieri, C.
- Physical Review C, Vol. 98, Issue 2

##
Optimization of quantum Monte Carlo wave functions by energy minimization

journal, February 2007

- Toulouse, Julien; Umrigar, C. J.
- The Journal of Chemical Physics, Vol. 126, Issue 8

##
Self-consistent mean-field models for nuclear structure

journal, January 2003

- Bender, Michael; Heenen, Paul-Henri; Reinhard, Paul-Gerhard
- Reviews of Modern Physics, Vol. 75, Issue 1

##
Spherical Hartree-Fock calculations with linear-momentum projection before the variation: Part I: Energies, form factors, charge densities and mathematical sum rules

journal, January 2004

- Rodríguez-Guzmán, R. R.; Schmid, K. W.
- The European Physical Journal A, Vol. 19, Issue 1

##
$\mathrm{Nd}$ elastic scattering as a tool to probe properties of $3N$ forces

journal, January 2001

- Witała, H.; Glöckle, W.; Golak, J.
- Physical Review C, Vol. 63, Issue 2

##
Elastic electron scattering from 12C and 16O

journal, July 1970

- Sick, I.; McCarthy, J. S.
- Nuclear Physics A, Vol. 150, Issue 3

##
Inelastic Electron-Nucleus Scattering and Nucleon-Nucleon Correlations

journal, February 1962

- McVoy, K. W.; Van Hove, L.
- Physical Review, Vol. 125, Issue 3

##
Energy and Variance Optimization of Many-Body Wave Functions

journal, April 2005

- Umrigar, C. J.; Filippi, Claudia
- Physical Review Letters, Vol. 94, Issue 15

##
Path integrals in the theory of condensed helium

journal, April 1995

- Ceperley, D. M.
- Reviews of Modern Physics, Vol. 67, Issue 2

##
Unified *ab initio* approaches to nuclear structure and reactions

journal, April 2016

- Navrátil, Petr; Quaglioni, Sofia; Hupin, Guillaume
- Physica Scripta, Vol. 91, Issue 5

##
Comparative study of three-nucleon potentials in nuclear matter

journal, February 2012

- Lovato, Alessandro; Benhar, Omar; Fantoni, Stefano
- Physical Review C, Vol. 85, Issue 2

##
Studies of pear-shaped nuclei using accelerated radioactive beams

journal, May 2013

- Gaffney, L. P.; Butler, P. A.; Scheck, M.
- Nature, Vol. 497, Issue 7448

##
Quantum Monte Carlo methods for nuclear physics

journal, September 2015

- Carlson, J.; Gandolfi, S.; Pederiva, F.
- Reviews of Modern Physics, Vol. 87, Issue 3