Lawrence Livermore National Laboratory



Aurora Pribram-Jones

Overview

Warm dense matter (WDM) is a high-energy phase between solids and plasmas that is present in the centers of planets and on the path to ignition of inertial confinement fusion (ICF). The extreme conditions under which one finds WDM lead to complications in its simulation, as both classical and quantum effects must be included. One of the most successful simulation methods is density functional theory molecular dynamics (DFT MD). Despite successes, DFT MD remains computationally costly and suffers from poor handling of strong electronic correlation present in stretched and breaking molecular bonds.

This project aimed to improve WDM simulation capabilities at Lawrence Livermore National Laboratory (LLNL) via two approaches: (1) the formal development of temperature-dependent DFT approximations and (2) semiclassical methods. The first approach was chosen because it establishes the strictly correlated electron reference for thermal DFT. This formalism carefully defines the relationship between interaction strength, density, and electronic temperature and can provide approximations that work better than traditional DFT for highly correlated systems. This type of accuracy is important in systems where bonds between atoms in materials are stretched or broken. The second tactic within this project (semiclassical) was based on the previous work of the principal investigator to move to orbital-free methods capable of capturing quantum density oscillations present in warm dense materials at much lower computational cost. For this project, the researcher used this semiclassical approach to derive new methods for the calculation of electronic partition functions and dipole matrix elements for atomic systems.

The progress made within the brief funding period of this project provided first steps toward two separate new methods for WDM simulation and has the potential to impact LLNL's ICF efforts and stockpile stewardship mission by providing the Laboratory with new simulation and analytical capabilities for materials under extreme conditions.

Background and Research Objectives

WDM, a high-energy phase between solids and plasmas, has been characterized as the "malfunction junction" because of the inadequacy of traditional condensed matter and classical plasma descriptions of its complicated state (Department of Energy 2009). At temperatures over a thousand kelvin and with condensed matter densities, these materials demand theoretical treatments that incorporate quantum effects, strong correlation, and partial ionization (Department of Energy 2009, Graziani et al. 2014). Core structure modeling of giant planets and Earth-like planets relies on accurate WDM phase diagrams. Good predictions of WDM melt properties and thermal conductivity also contribute to the drive toward ICF (Atzeni and Meyer-Ter-Vehn 2004), as the fuel and capsule pass through WDM conditions toward those of even higher temperatures and pressures during ignition. This transition must be controlled through the aid of accurate theoretical models for both fuel and capsule materials, as inhomogeneity developed at any point during ignition can potentially quench the budding reaction.

Since not all WDM experiments are conducive to taking reliable or isolated measurements at such extreme conditions, DFT MD (Car and Parrinello 1985; Kresse and Hafner 1993 and 1994; Kresse and Furthmüller 1996; Iftimie, Minary, and Tuckerman, 2005) is often used to predict properties of interest. DFT MD, one of the most successful WDM simulation methods, uses DFT, a quantum mechanical method (Hohenberg and Kohn 1964, Burke 2007), to calculate the forces used in the classical simulations that generate ion distributions in the simulated material. This mixed quantum-classical method has been very successful (Mattsson and Desjarlais 2006, Knudsen and Desjarlais 2009), but it is hugely expensive, misses some temperature effects, and has an accuracy level that cannot be systematically improved using standard approaches. This is largely due to its reliance on Kohn-Sham DFT.

Kohn-Sham DFT uses a non-interacting system to calculate the energies of an interacting electronic system with the same electronic probability density (Kohn and Sham 1965). Exact expressions are known for all but a small (but crucial) piece of the energy, called the exchange-correlation (XC) energy. Many approximations for this piece exist at zero temperature (Burke 2012, Burke and Wagner 2013), which are often used for finite-temperature DFT (FT DFT) at this time. FT DFT combines statistical mechanical distributions of energy states with the energy minimization methods of DFT (Mermin 1965). Including statistical equilibrium in this way demands use of a temperature-dependent XC functional in the exact theory (Pribram-Jones et al. 2014) though this is rarely done in practice. The high cost of Kohn-Sham DFT at WDM conditions stems from the very large numbers of fractionally occupied, high-energy states (Karasiev et al. 2014). In order to use Kohn-Sham methods, massive eigensystems must be solved at every time step, leading to great computational expense.

The goals of this project were twofold: (1) derive and analyze the strictly correlated electron approach to finite-temperature density functional theory, and (2) speed up calculations of free energies and other observables for warm dense electronic systems using semiclassical methods. Toward that end, this six-month project yielded the following deliverables:

  • a framework for balanced treatment of strong correlation and temperature effects within the DFT framework currently in use across LLNL's simulation of materials under extreme conditions
  • derivation of mathematical conditions (Liu and Burke 2009, Burke et al. 2016, Pribram-Jones and Burke, 2016) on the new formalism for strictly correlated electrons DFT at finite temperature
  • derivation of a framework for calculating approximate free energies based on approximate partition functions

Impact on Mission

These fundamental results support the development of LLNL's core competency in high-energy-density (HED) science. The project is aligned with the HED science aspects of the NNSA and LLNL stockpile stewardship mission (properties of materials at extreme conditions are directly related to modeling and simulation) and could impact both experimental design and results analysis for projects on the National Ignition Facility and other HED facilities.

Conclusion

The principal investigator has accepted a position as Assistant Professor at University of California, Merced, where she will continue this research and train her students in these areas of interest. She is also part of the DOE-funded Consortium for High Energy Density Science, which supports her research group's work and its continued collaborations with LLNL. Next steps for research on these topics includes connection of the finite temperature; strong interaction reference with the standard Kohn-Sham formalism; demonstration of this strictly correlated approach to finite temperature in the asymmetric Hubbard dimer and uniform electron gas; creation of new temperature-dependent XC approximations using the strictly correlated formalism; continued development of semiclassical partition function approximation and application of these approximations to more model systems; development of the semiclassical partition function approximations into a practical method for WDM computation and connection to experimental techniques, such as spectroscopy.

References

Atzeni, S. and J. Meyer-Ter-Vehn. 2004. The Physics of Inertial Fusion: Beam-Plasma Interaction, Hydrodynamics, Hot Dense Matter.

Burke, K. 2007. "The ABC of DFT." available online.

Burke, K. 2012. "Perspective on Density Functional Theory." J. Chem. Phys. 136.

Burke, K. and Lucas O. Wagner. 2013. "DFT in a Nutshell." Int. J. Quant. Chem. 113, 96–101.

Burke, K., et al. 2016. "Exact Conditions on the Temperature Dependence of Density Functionals." Phys. Rev. B 93, 195132.

Car, R. and M. Parrinello. 1985. "United Approach for Molecular Dynamics and Density-Functional Theory." Phys. Rev. Lett. 55, 2471–2474.

Department of Energy. 2009. U.S. Department of Energy, Basic Research Needs for High Energy Density Laboratory Physics: Report of the Workshop on High Energy Density Laboratory Physics Research Needs. Tech.Rep., Office of Science and National Nuclear Security Administration.

Graziani, F., et al., eds. 2014. "Frontiers and Challenges in Warm Dense Matter." Lecture Notes in Computational Science and Engineering, Vol. 96 (Springer International Publishing.

Hohenberg, P. and W. Kohn. 1964. "Inhomogeneous Electron Gas." Phys. Rev. 136, B864–B871.

Iftimie, R., P. Minary, and M. E. Tuckerman. 2005. "Ab Initio Molecular Dynamics: Concepts, Recent Developments, and Future Trends." Proceedings of the National Academy of Sciences of the United States of America 102, 6654–6659.

Karasiev, V. V., et al. 2014. "Innovations in finite-temperature density functionals." Frontiers and Challenges in Warm Dense Matter, Lecture Notes in Computational Science and Engineering, Vol. 96, edited by F. Graziani, et al. Springer International Publishing. pp. 61–85.

Kohn, W. and L. J. Sham. 1965. "Self-Consistent Equations Including Exchange and Correlation Effects." Phys. Rev. 140, A1133–A1138.

Kresse, G. and J. Hafner. 1993. "Ab Initio Molecular Dynamics for Liquid Metals." Phys. Rev. B 47, 558–561.

Kresse, G. and J. Hafner. 1994. "Ab Initio Molecular-Dynamics Simulation of the Liquid-Metal Amorphous-Semiconductor Transition in Germanium." Phys. Rev. B 49, 14251–14269.

Kresse, G. and J. Furthmüller. 1996. "Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set." Phys. Rev. B 54, 11169-11186.

Liu, Z.-F. and K. Burke. 2009. "Adiabatic Connection for Strictly Correlated Electrons." J. Chem. Phys. 131, 124124.

Mattsson, T. R. and M. P. Desjarlais. 2006. "Phase Diagram and Electrical Conductivity of High Energy-Density Water from Density Functional Theory." Phys. Rev. Lett. 97, 017801.

Mermin, N. D. 1965. Thermal Properties of the Inhomogenous Electron Gas." Phys. Rev. 137, A: 1441.

Pribram-Jones, A. and K. Burke. 2016. "Connection Formulas for Thermal Density Functional Theory." Phys. Rev. B 93, 205140.

Pribram-Jones, A., et al. 2014. "Thermal Density Functional Theory in Context." Frontiers and Challenges in Warm Dense Matter, Lecture Notes in Computational Science and Engineering, Vol. 96, edited by F. Graziani, et al. Springer International Publishing. pp. 25–60.