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## Orbital-free approach for high energy density physics

#### La Maison des Mines et des Ponts et ChaussĂ©es, Paris

#### Organisers

The challenge of high energy density physics

The start of the ignition campaign on the National Ignition Facility in Livermore (USA) in 2010 brings new opportunities to study matter in extreme regimes of density and temperature. The design of this type of experiment requires the precise knowledge of the behavior of matter in these regimes, which is still lacking. As an example, the most recent shots show an unpredicted amount of heavy elements, namely Germanium, in the central hot region composed of deuterium and tritium, one plausible reason for the low fusion rate reached up to now. The understanding of the radiative and hydrodynamic evolution of this mixture is therefore of high priority and relies on the evaluation of microscopic coefficients like opacities, equations of state or thermal conductivities. These two latter quantities are also of considerable interest for astrophysical problems like the the nature of exoplanets which is driven by hydrostatic equilibrium and thermal diffusion. An urgent need exists for these properties in order to guide current experiments and simulations across a broad span of disciplines.

In recent years, numerical simulations based on the Kohn-Sham or orbital formulation of Density Functional Theory (KS-DFT) have proven to be extremely powerful in computing such coefficients. Unfortunately, and despite the rapid growth of massively parallel machines, quantum molecular dynamics (QMD) - based on the orbital scheme - remains limited in terms of temperature because of the thermal Fermi-Dirac distribution. This latter leads indeed to the involvement of a huge number of one-body quantum states as soon as the temperature becomes comparable or higher to the Fermi energy. This limitation is even more severe for mixtures of low and high Z elements since the molecular dynamics requires a substantial number of atoms to be relevant.

An alternative approach to study mixed systems is classical molecular dynamics which has been successfully applied in various domains from biological systems to the behavior of explosive materials. Nevertheless, these methods are also limited to thermodynamic conditions and environments for which the many-body interaction potentials have been constructed. These potentials must be determined by more fundamental approaches like QMD or fit to experimental data which is unavailable in many regimes, for example ICF.

In the last few years, a semi-classical technique based on an orbital-free treatment of the electronic structure has been used to study this new field of physics. The only input variables are the nature of the system and the thermodynamical conditions like in KS-DFT but the approach describes the electronic free-energy solely in terms of the local electronic density in the true spirit of the Hohenberg and Kohn theorem, avoiding the time consuming computations of orbitals. The calculations based on these methods are on average several orders of magnitude faster than their orbital-based counterpart, depending on the physical system.

Following a successful CFCAM discussion meeting held in June 2011 that drew the preceding picture, we propose to organize a workshop in autumn 2012 on the development of orbital-free methods at finite temperature with applications from warm dense matter (temperatures of a few eV and densities of a few g/cc) up to the ignition region (temperatures of a few keV and densities of a few kg/cc). Even with the predicted advances in algorithms and hardware systems, orbital-free approaches represent the only viable means of providing microscopic properties in the warm, dense matter regime for the foreseeable future given the huge computational investment in generating

orbitals for many finite-temperature systems.

The challenge of high energy density physics

The start of the ignition campaign on the National Ignition Facility in Livermore (USA) in 2010 brings new opportunities to study matter in extreme regimes of density and temperature. The design of this type of experiment requires the precise knowledge of the behavior of matter in these regimes, which is still lacking. As an example, the most recent shots show an unpredicted amount of heavy elements, namely Germanium, in the central hot region composed of deuterium and tritium, one plausible reason for the low fusion rate reached up to now. The understanding of the radiative and hydrodynamic evolution of this mixture is therefore of high priority and relies on the evaluation of microscopic coefficients like opacities, equations of state or thermal conductivities. These two latter quantities are also of considerable interest for astrophysical problems like the the nature of exoplanets which is driven by hydrostatic equilibrium and thermal diffusion. An urgent need exists for these properties in order to guide current experiments and simulations across a broad span of disciplines.

In recent years, numerical simulations based on the Kohn-Sham or orbital formulation of Density Functional Theory (KS-DFT) have proven to be extremely powerful in computing such coefficients. Unfortunately, and despite the rapid growth of massively parallel machines, quantum molecular dynamics (QMD) - based on the orbital scheme - remains limited in terms of temperature because of the thermal Fermi-Dirac distribution. This latter leads indeed to the involvement of a huge number of one-body quantum states as soon as the temperature becomes comparable or higher to the Fermi energy. This limitation is even more severe for mixtures of low and high Z elements since the molecular dynamics requires a substantial number of atoms to be relevant.

An alternative approach to study mixed systems is classical molecular dynamics which has been successfully applied in various domains from biological systems to the behavior of explosive materials. Nevertheless, these methods are also limited to thermodynamic conditions and environments for which the many-body interaction potentials have been constructed. These potentials must be determined by more fundamental approaches like QMD or fit to experimental data which is unavailable in many regimes, for example ICF.

In the last few years, a semi-classical technique based on an orbital-free treatment of the electronic structure has been used to study this new field of physics. The only input variables are the nature of the system and the thermodynamical conditions like in KS-DFT but the approach describes the electronic free-energy solely in terms of the local electronic density in the true spirit of the Hohenberg and Kohn theorem, avoiding the time consuming computations of orbitals. The calculations based on these methods are on average several orders of magnitude faster than their orbital-based counterpart, depending on the physical system.

Following a successful CFCAM discussion meeting held in June 2011 that drew the preceding picture, we propose to organize a workshop in autumn 2012 on the development of orbital-free methods at finite temperature with applications from warm dense matter (temperatures of a few eV and densities of a few g/cc) up to the ignition region (temperatures of a few keV and densities of a few kg/cc). Even with the predicted advances in algorithms and hardware systems, orbital-free approaches represent the only viable means of providing microscopic properties in the warm, dense matter regime for the foreseeable future given the huge computational investment in generatingorbitals for many finite-temperature systems.

Orbital-free description of hot and dense matter: an active field of research

Over the last few years, a new effort has been brought to the development of orbital-free techniques to study hot and dense matter. Following the pioneering work initiated after a series of CECAM workshops from 1988 to 1990 (P. Madden, B. Alder, J.-P. Hansen, E. Smargiassi, I. Stich, G. Zérah and J. Clérouin) [1], the coupling of classical molecular dynamics for the nuclei with orbital-free description for the electrons has been updated to a powerful tool for atomic simulations by F. Lambert and co-workers with new insights into matter in extreme conditions:

- the equation of state of light element as well as heavy ones and the verification of mixing rules for the equation of state of mixtures [2,3];

- the ionic transport coefficients with particular emphasis on comparison with classical systems like the One Component Plasma allowing prescriptions for the hydrodynamics codes [4,5,6];

- the electronic transport coefficients with a particular interest in checking widespread models used in inertial confinement fusion designs [7].

In terms of numerical features, orbital-free codes have taken advantage of the advances in KS-DFT in terms of the algorithms used either for the electronic part - direct minimization of the free energy by conjugate gradient techniques - or for the ionic molecular dynamics [4,8,9]. Dealing only with the local electronic density, the orbital-free free energy minimization is consequently much faster than its Kohn-Sham counterpart. At the same time, alternative schemes based on a complete description of the system in real space with adaptive mesh refinement [10] allow for studying systems up to a million of particles with particularly low symmetries like plasmas, opening the road to the treatment of very asymmetric mixtures. Completing this opportunity, quantum and orbital-free approaches can be coupled through a subsytem point of view [11,12] with the environment of a quantum subsystem being described by an orbital-free model. These numerical features could perfectly be adapted to finite temperature systems.

When the quantum calculations are tractable, extensive comparisons have been made to the Kohn-Sham approach [13,5,14] demonstrating that orbital-free calculations constitute an effective technique for expanding atomistic simulations into thermodynamical regimes in which KS-DFT ultimately fails due to the prohibitive computational time. New results on the development of free energy functionals are starting to emerge in order to improve the transition between orbital-based and orbital-free domains [15]. Moreover, orbital-free techniques are now used to derive classical many-body potentials [16] bridging a gap towards multi-scale and multi-physics simulations.

The orbital free approach has also been proposed in the framework of a global microscopic time-dependent description of fusion plasmas in the inertial fusion context. In this perspective, the orbital free approach seems a promising development platform by explicitly eliminating the classical treatment of the electrons, which forms a particularly time-consuming step. As an example, OF-DFT is part of the development roadmap of the CIMARRON project [17] which is dedicated to the implementation of nuclear reactions and radiative transport to a molecular dynamics code. At the same time, it opens new fields of research, both in theoretical and computational terms, on time dependent properties in an orbital-free point of view.

## References

**France**

jean ClĂ©rouin (CEA/DAM/DIF) - Organiser

Flavien Lambert (Alternative Energies and Atomic Energy Commission (CEA), Arpajon) - Organiser

**United States**

Lee Collins (Los Alamos National Laboratory) - Organiser & speaker