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The liquid-liquid phase transition in high-pressure Hydrogen is a problem of longstanding and controversy. The recent Nature paper by Cheng et al. [vol. 585, p. 217] makes a set of strong claims to the effect that all the previous density functional theory molecular dynamics (MD-DFT) and quantum Monte Carlo calculations of that transition are incorrect because of finite size effects and, in the MD-DFT case, short run times. The basis of those claims is their use of large systems and long durations for classical MD driven by a machine-learnt potential (MLP) which they developed. The straightforward test of their claims is to do MD-DFT on systems as large or larger than Cheng et al. used and for significantly longer durations than in the previous MD-DFT simulations. We have done so and find that neither diagnosis of theirs (size effects, duration limits) is correct. Instead, we find that the MLP does not drive MD in fidelity with the underlying DFT electronic structure that it is supposed to replicate. The result is that the MLP-driven MD results are artifactual, not systematically connected to the theoretical underpinning on which the MLP was trained.

Using conceptually and procedurally consistent density functional theory (DFT) calculations with an advanced meta-GGA exchange-correlation functional in ab initio molecular dynamics simulations, we determine the insulator-metal transition (IMT) of warm dense fluid hydrogen over the pressure range 50 to 300 GPa. Inclusion of nuclear quantum effects via path-integral molecular dynamics (PIMD) sharpens the metallic transition and lowers the transition temperature relative to results from Born-Oppenheimer (BO) MD. BOMD itself gives improved agreement with experimental results compared to previous DFT predictions. Examination of the ionic pair correlation function in the context of the abrupt conductivity increase at the transition confirms a metallic transition due to the dissociation of molecular hydrogen that coincides with an abrupt band gap closure. Direct comparison of the PIMD and BOMD results clearly demonstrates an isotope effect on the IMT. Distinct from stochastic simulations, these results do not depend upon any ad hoc combination of ground-state and finite-T methodologies.

For orbital-free {\it ab initio} molecular dynamics, especially on systems in extreme thermodynamic conditions, we provide the first pseudo-potential-adapted generalized gradient approximation (GGA) functional for the non-interacting free energy. This is achieved by systematic finite-temperature extension of our recent LKT ground state non-interacting kinetic energy GGA functional (Phys. Rev. B \textbf{98}, 041111(R) (2018)). We test the performance of the new functional first via static lattice calculations on crystalline aluminum and silicon. Then we compare deuterium equation of state results against both path-integral Monte Carlo and conventional (orbital-dependent) Kohn-Sham results. The new functional, denoted LKTF, outperforms the previous best semi-local free energy functional, VT84F (Phys.\ Rev.\ B \textbf{88}, 161108(R) (2013)), and provides modestly faster simulations. We also discuss subtleties of identification of kinetic and entropic contributions to non-interacting free-energy functionals obtained by extension from ground state orbital-free kinetic energy functionals.

Realizing the potential for predictive density functional calculations of matter under extreme conditions depends crucially upon having an exchange-correlation (XC) free energy functional accurate over a wide range of state conditions. Unlike the ground-state case, no such functional exists. We remedy that with systematic construction of a generalized gradient approximation XC free-energy functional based on rigorous constraints, including the free energy gradient expansion. The new functional provides the correct temperature dependence in the slowly varying regime and the correct zero-T, high-T, and homogeneous electron gas limits. Its accuracy in the warm dense matter regime is attested by excellent agreement of the calculated deuterium equation of state with reference path integral Monte Carlo results at intermediate and elevated T. Pressure shifts for hot electrons in compressed static fcc Al and for low density Al demonstrate the combined magnitude of thermal and gradient effects handled well by this functional over a wide T range.

A simple expression for the uniform electron gas (UEG) correlation energy, recently presented in Ref. [J. Chem. Phys. 145, 021101 (2016)], deviates from the reference quantum Monte-Carlo (QMC) data at large r_s. We propose to define one of the parameters from a requirement to match the large-rs QMC data. Functional with the new parameter provides much better agreement with the QMC data at large r_s without deterioration of the functional quality at small and intermediate r_s.

Effects of explicit temperature dependence in the exchange-correlation (XC) free-energy functional upon calculated properties of matter in the warm dense regime are investigated. The comparison is between the KSDT finite-temperature local density approximation (TLDA) XC functional [Phys.\ Rev.\ Lett.\ \textbf{112}, 076403 (2014)] parametrized from restricted path integral Monte Carlo data on the homogeneous electron gas (HEG) and the conventional Monte Carlo parametrization ground-state LDA XC functional (Perdew-Zunger, "PZ") evaluated with $T$-dependent densities. Both Kohn-Sham (KS) and orbital-free density functional theory (OFDFT) are used, depending upon computational resource demands. Compared to the PZ functional, the KSDT functional generally lowers the direct-current (DC) electrical conductivity of low density Al, yielding improved agreement with experiment. The greatest lowering is about 15\% for T= 15 kK. Correspondingly, the KS band structure of low-density fcc Al from KSDT exhibits a clear increase in inter-band separation above the Fermi level compared to the PZ bands. In some density-temperature regimes, the Deuterium equations of state obtained from the two XC functionals exhibit pressure differences as large as 4\% and a 6\% range of differences. However, the Hydrogen principal Hugoniot is insensitive to explicit XC $T$-dependence because of cancellation between the energy and pressure-volume work difference terms in the Rankine-Hugoniot equation. Finally, the temperature at which the HEG becomes unstable is $T\geq$ 7200 K for $T$-dependent XC, a result that the ground-state XC underestimates by about 1000 K.

Within the framework of density functional theory, we present a study of approximations to the enhancement factor of the non-interacting kinetic energy functional $T_s[\rho]$. For this purpose, we employ the model of Liu and Parr [S. Liu and R.G. Parr, Phys. Rev. A {\bf 55}, 1792 (1997)] based on a series expansion of $T_s[\rho]$ involving powers of the density. Applications to 34 atoms, at the Hartree-Fock level showed that the enhancement factors present peaks that are in excellent agreement with those of the exact ones and give an accurate description of the shell structure of these atoms. The application of Z-dependent expansions to represent some of the terms of these approximation for neutral atoms and for positive and negative ions, which allows $T_s[\rho]$ to be cast in a very simple form, is also explored. Indications are given as to how these functionals may be applied to molecules and clusters

F.E. Harris has been a significant partner in our work on orbital-free density functional approximations for use in ab initio molecular dynamics. Here we mention briefly the essential progress on single-point functionals since our original paper (2006). Then we focus on the advantages and limitations of generalized gradient approximation (GGA) non-interacting kinetic-energy functionals. We reconsider the constraints provided by near-origin conditions in atomic-like systems and their relationship to regularized versus physical external potentials. Then we seek the best empirical GGA for the non-interacting KE for a modest-sized set of molecules with a well-defined near-origin behavior of their densities. The search is motivated by a desire for insight into GGA limitations and for a target for constraint-based development.

The thermodynamics for a system with given temperature, density, and volume is described by the Canonical ensemble. The thermodynamics for a corresponding system with the same temperature, volume, and average density is described by the Grand Canonical ensemble. In general a chosen thermodynamic potential (e.g., free energy) is different in the two cases. Their relationship is considered here as a function of the system size. Exact expressions relating the fundamental potential for each (free energy and pressure, respectively) are identified for arbitrary system size. A formal asymptotic analysis for large system size gives the expected equivalence, but without any characterization of the intermediate size dependence. More detailed evaluation is provided for the simple case of a homogeneous, non-interacting Fermi gas. In this case, the origin of size dependence arises from only two length scales, the average inter-particle distance and quantum length scale (thermal deBroglie or Fermi length). The free energies per particle calculated from each ensemble are compared for particle numbers $2\le N\le 64$ for a range of temperatures above and below the Fermi temperature. The relevance of these results for applications of density functional theory is discussed briefly., Comment: 13 pages, 8 figures

Smooth, highly accurate analytical representations of Fermi-Dirac (FD) integral combinations important in free-energy density functional calculations are presented. Specific forms include those that occur in the local density approximation (LDA), generalized gradient approximation (GGA), and fourth-order gradient expansion of the non-interacting free energy as well as in the LDA and second-order gradient expansion for exchange. By construction, all the representations and their derivatives of any order are continuous on the full domains of their independent variables. The same type of technique provides an analytical representation of the function inverse to the FD integral of order $1/2$. It plays an important role in physical problems related to the electron gas at finite temperature. From direct evaluation, the quality of these improved representations is shown to be substantially superior to existing ones, many of which were developed before the era of large-scale computation or early in the era.

Implementation of orbital-free free-energy functionals in the Profess code and the coupling of Profess with the Quantum Espresso code are described. The combination enables orbital-free DFT to drive ab initio molecular dynamics simulations on the same footing (algorithms, thermostats, convergence parameters, etc.) as for Kohn-Sham (KS) DFT. All the non-interacting free-energy functionals implemented are single-point: the local density approximation (LDA; also known as finite-T Thomas-Fermi, ftTF), the second-order gradient approximation (SGA or finite-T gradient-corrected TF), and our recently introduced finite-T generalized gradient approximations (ftGGA). Elimination of the KS orbital bottleneck via orbital-free methodology enables high-T simulations on ordinary computers, whereas those simulations would be costly or even prohibitively time-consuming for KS molecular dynamics (MD) on very high-performance computer systems. Example MD simulations on H over a temperature range 2,000 K <= T <=4,000,000 K are reported, with timings on small clusters (16-128 cores) and even laptops. With respect to KS-driven calculations, the orbital-free calculations are between a few times through a few hundreds of times faster.

An accurate analytical parametrization for the exchange-correlation free energy of the homogeneous electron gas, including interpolation for partial spin-polarization, is derived via thermodynamic analysis of recent restricted path integral Monte-Carlo (RPIMC) data. This parametrization constitutes the local spin density approximation (LSDA) for the exchange-correlation functional in density functional theory. The new finite-temperature LSDA reproduces the RPIMC data well, satisfies the correct high-density and low- and high-$T$ asymptotic limits, and is well-behaved beyond the range of the RPIMC data, suggestive of broad utility.

We report the first wholly non-empirical generalized gradient approximation, non-interacting free energy functional for orbital-free density functional theory and use that new functional to provide forces for finite-temperature molecular dynamics simulations in the warm dense matter (WDM) regime The new functional provides good-to-excellent agreement with reference Kohn-Sham calculations under WDM conditions at a minuscule fraction of the computational cost of corresponding orbital-based simulations.

New local "hybrid" functionals proposed by V. V. Karasiev in [J. Chem. Phys. {\bf 118}, 8567 (2003)] are benchmarked against nonlocal hybrid functionals. Their performance is tested on the total and high occupied orbital energies, as well as the electric moments of selected diatomic molecules. The new functionals, along with the Hartree-Fock and non-hybrid functionals, are employed for finite-difference calculations, which are basis-independent. Basis set errors in the total energy and electric moments are calculated for the 6-311G, 6-311G++G(3df,3pd) and AUG-cc-pVnZ (n=3,4,6) basis sets used in conjunction with the Hartree-Fock and conventional density functional methods. A comparison between the results of the finite-difference local "hybrid" and basis set nonlocal hybrid functional shows that total energies of local and nonlocal hybrid functionals agree to within the basis set error. Discrepancies for multipole moments are larger in magnitude when compared to the basis set errors, but still reasonably small (smaller than errors produced by the 6-311G basis set). Thus, we recommend using the new local "hybrid" functionals whenever the accuracy is expected to be sufficient, because they require a solution of just differential Kohn-Sham equations, instead of integro-differential ones in the case of hybrid functionals.

We compare the behavior of the finite-temperature Hartree-Fock model with that of thermal density functional theory using both ground-state and temperature-dependent approximate exchange functionals. The test system is bcc Li in the temperature-density regime of warm dense matter (WDM). In this exchange-only case, there are significant qualitative differences in results from the three approaches. Those differences may be important for Born-Oppenheimer molecular dynamics studies of WDM with ground-state approximate density functionals and thermal occupancies. Such calculations require reliable regularized potentials over a demanding range of temperatures and densities. By comparison of pseudopotential and all-electron results at ${\mathrm T} = 0$K for small Li clusters of local bcc symmetry and bond-lengths equivalent to high density bulk Li, we determine the density ranges for which standard projector augmented wave (PAW) and norm-conserving pseudopotentials are reliable. Then we construct and use all-electron PAW data sets with a small cutoff radius which are valid for lithium densities up to at least 80 g/cm$^3$.

Self interaction error remains an impotrant problem in density functional theory. A number of approximations to exact exchange aimed to correct for this error while retainining computational efficiency had been suggested recently. We present a critical comparison between model exchange potentials generated through the application of the asymptotically-adjusted self-consistent $\alpha$, AASC$\alpha$, method and BJ effective exchange potential advanced in [A.D. Becke and E.R. Johnson, J. Chem. Phys. 124, 221101 (2006)] and [V.N. Staroverov, J. Chem. Phys. 129, 134103 (2008)]. In particular we discuss their compliance with coordinate-scaling, virial and functional derivative conditions. We discuss the application of the AASC$\alpha$ method to generate the AA-BJ potential. A numerical comparison is carried out through the implementation of a fully-numerical diatomic molecule code yielding molecular virial energies and ionization potentials approximated by the energies of the HOMO orbitals. It is shown that some of the shortcomings of these model potentials, such as the non-compliance with the Levy-Perdew virial relation, may be eliminated by multiplying the response term by an orbital-dependent functional $\alpha$, which can be simplified to a constant determined during the self-consistent procedure (self-consistent $\alpha$).