Innovations in Finite-Temperature Density Functionals

Reliable, tractable computational characterization of warm dense matter is a challenging task because of the wide range of important aggregation states and effective interactions involved. Contemporary best practice is to do ab initio molecular dynamics on the ion constituents with the forces from the electronic population provided by density functional calculations. Issues with that approach include the lack of reliable approximate density functionals and the computational bottleneck intrinsic to Kohn-Sham calculations. Our research is aimed at both problems, via the so-called orbital-free approach to density functional theory. After a sketch of the relevant properties of warm dense matter to motivate our research, we give a survey of our results for constraint-based non-interacting free energy functionals and exchange-correlation free-energy functionals. That survey includes comparisons with novel finite-temperature Hartree-Fock calculations and also presents progress on both pertinent exact results and matters of computational technique. V.V. Karasiev • T. Sjostrom • D. Chakraborty • K. Runge • F.E. Harris • S.B. Trickey ( ) Quantum Theory Project, Departments of Physics and of Chemistry, University of Florida, P.O. Box 118435, Gainesville, FL 32611-8435, USA e-mail: [email protected]; [email protected] http://www.qtp.ufl.edu/ofdft J.W. Dufty Department of Physics, University of Florida, P.O. Box 118435, Gainesville, FL 32611-8435, USA F. Graziani et al. (eds.), Frontiers and Challenges in Warm Dense Matter, Lecture Notes in Computational Science and Engineering 96, DOI 10.1007/978-3-319-04912-0__3, © Springer International Publishing Switzerland 2014 61 62 V.V. Karasiev et al. 1 Setting and Perspective Materials under extreme conditions have been a major and rewarding focus of condensed matter physics for well over a century. Low-temperature physics is a familiar example, with its modern form dating to the liquefaction of Hydrogen by Dewar in 1898 and of Helium by Kamerlingh Onnes in 1908 [1]. High energy-density physics (HEDP), the focus of the 2012 IPAM Long Program “Computational Methods in High Energy Density Plasmas”, is less well-known, even in the scientific community. But from the perspective of shock and detonation phenomena at least, HEDP has as long and deep a scientific heritage as does low temperature physics. A history of early work on detonation waves [2], for example, notes the first detonation velocity measurements in condensed explosives by Abel in 1869, the beginnings of detonation theory (antecedent to the ChapmanJouguet equation) by Michelson in 1890, and the Chapman-Jouguet relation itself by Chapman in 1899 and Jouguet in 1905 and 1906. Other examples could be given from plasma physics. A sign of the richness of HEDP is that only much more recently has it been recognized that HEDP includes a complicated condensed matter regime now called warm dense matter (WDM). We discuss WDM traits briefly below. The challenge and opportunity to develop computationally tractable, predictive methods suitable for a comparatively unexplored condensed matter regime drew us together as a team. That same combination of challenge and opportunity led to Workshop IV “Computational Challenges in Warm Dense Matter” of the IPAM HEDP Long Program. The Workshop was a de facto review of theory, modeling, and simulation for WDM, with conversations among those already involved as well as with workers whose research clearly could be of relevance. In that context, we offer here a survey of our contributions, confident that we do not have to survey the entire area and, hence, the entire Workshop! It is fitting to begin with a system perspective. WDM comprises a condensed matter regime characterized roughly by electron temperatures T 1–15 eV and pressures to 1 Mbar or greater. (Aside: Theoretical and computational treatment of WDM involves diverse topics with many abbreviations. Most of the more common ones are listed in the Glossary; see the Appendix.) Recently WDM has attracted attention because of its importance in diverse physical systems, including exo-planet interiors, the path to inertial confinement fusion, and neutron stellar atmospheres [3–13]. The relationship between the WDM regime and other states of matter is illustrated schematically in Fig. 1. The regime is inherently challenging because its thermodynamics cannot be framed in terms of small perturbations from ideal, solvable models. WDM temperature and pressure ranges correspond to values of order unity in the two relevant expansion parameters: the Coulomb coupling parameter D Q=rskBT and the electron degeneracy parameter D kBT= F. (Here Q D the relevant charge, rs D Wigner radius, F D electron Fermi energy, T D temperature, kB D Boltzmann constant.) WDM thus does not fall neatly within the parameter space typical of either ordinary condensed-matter physics or plasma physics. One result is that plasma physics methods which originate in the classical Innovations in Finite-Temperature Density Functionals 63 Fig. 1 Qualitative positioning of WDM relative to other physical states and systems limit do not extend well into the WDM regime. At a fundamental level, such approaches cannot be entirely successful; the quantum limit cannot be recovered from classical physics. Conversely, condensed-matter physics methods must be extended well beyond their normal realms of application. The logic is inexorable. A foundational element for understanding and manipulating material behavior is the equation of state (EOS). For materials under near-ambient conditions, best practice has evolved to be a combination of electronic structure calculations using density functional theory (DFT) and molecular dynamics (MD). In many cases, the zero-temperature EOS (or cold curve), including crystalline phase transitions, can be predicted quite accurately with DFT alone [14,15]. For the nuclear (or ionic) contribution, so-called ab initio MD (AIMD; see Refs. [16–20]) is quite successful. In its simplest form (Born-Oppenheimer MD), AIMD gets the electronic forces on the nuclei (or ions) from a DFT calculation at each nuclear step. AIMD thereby combines chemical realism from explicit quantum mechanical treatment of the electrons with the moreor-less classical contributions of the nuclear species. There are both conceptual and technical problems associated with this rosy picture, however, and they are worsened by extending into the WDM regime. A brief sketch of the customary DFT approach is needed for context. The variational minimization of the density functional customarily is via the well-known KohnSham (KS) procedure [21]. It introduces a model (sometimes called “fictitious”) non-interacting many-fermion system of the same density, n, as the physical system of interest. The density is expanded in the one-body states (orbitals) of the KS Hamiltonian hKSŒn 'i D "i'i ; (1)