Temperature-dependent behavior of confined many-electron systems in the Hartree-Fock approximation

Many-electron systems confined at substantial finite temperatures and densities present a major challenge to density functional theory. In particular, there is comparatively little systematic knowledge about the behavior of free-energy density functionals for temperatures and pressures of interest, for example, in the study of warm dense matter (WDM). As with ground-state functionals, development of approximate free-energy functionals is faced with significant needs for reliable assessment and calibration data. Here we address, in part, this need for detailed results on well-characterized systems. We present results on a comparatively simple, well-defined, computationally feasible but previously unexplored model, the thermal Hartree-Fock approximation. We discuss the main technical tasks (defining a suitable basis and evaluation of the required matrix elements) and give an illustrative initial application that probes both the content of the model and the solution techniques: a system of eight one-electron atoms with nuclei at fixed, arbitrary positions in a hard-walled box. Even this simple system produces physical behavior different from that produced by simple ground-state density functionals used at finite temperature (a common approximation in the study of WDM).

Figure 1

(a) Ground-state energy of an H atom in the center of a cube with edge length $L$. Fit is for function $f\left(L\right)=a/L^2+b/L+c$. (b) Pressure calculated from the energy fit.

Figure 2

Comparison of ground-state energy of an H atom in a cube with its energy within two bounding spheres.

Figure 3

(a) Comparison of energy of an H atom in a cube with that in a sphere of equal volume. (b) The difference of the two ground-state energies.

Figure 4

Energy vs atomic separation for the H${}_2$ molecule centered in a 30-bohr cube. The ground state agrees exactly with the free GTO calculation using the same primitives as a basis.

Figure 5

Ground-state energy for H${}_2$ with a bond length of $1.4$ bohrs confined in a cube of edge length $L$.

Figure 6

Energy vs bond length for the H${}_2$ molecule centered in cubes $L=30$ and $L=5$ bohrs. The minima are at $1.383$ and $1.178$ bohrs, respectively.

Figure 7

Optimized bond length $R$ for the H${}_2$ molecule vs pressure.

Figure 8

Lowest five eigenstate energies of H as a function of cube edge $L$. Energy values are from Table .

Figure 9

(a) Single H atom total energy, $E={\mathcal{F}}_{\mathrm{FTHF}}+T{\mathcal{S}}_{\mathrm{FTHF}}$, and (b) free energy ${\mathcal{F}}_{\mathrm{FTHF}}$ as a function of cube edge $L$, with the one-electron levels populated according to the Fermi distribution.

Figure 10

Energy components of a cubically confined single H atom vs temperature: (a) kinetic energy, (b) electron-nuclear energy, (c) total energy, (d) free energy, and (e) entropic energy [(d) minus (c)]. The key for all plots is shown in (e).

Figure 11

Zero- and finite-temperature total energy and free energy for eight H atoms in different sized cubes: (a) $E={\mathcal{F}}_{\mathrm{FTHF}}+T{\mathcal{S}}_{\mathrm{FTHF}}$ and (b) ${\mathcal{F}}_{\mathrm{FTHF}}$.

Figure 12

Self-consistent FTHF free-energy contributions for eight confined H atoms in a cubic array as a function of $T$ for four different cube edge lengths $L$. The key for all plots is given in (d).

Figure 13

FTHF kinetic energy compared with finite-temperature Thomas-Fermi KE and two forms of von Weizsäcker augmentation of FTTF for the eight-H cubic system at $L=6$ bohrs. See text for details.

Figure 14

FTHF exchange free-energy ground-state LDA and Perrot and Dharma-wardana temperature-dependent LDA (eight-H cubic system, $L=6$ bohrs). See text for details.

Figure 15

FTHF internal energy per atom for the eight-H cubic-symmetry system at $L=7$ bohrs compared with $r_s=2$ hydrogen plasma average atom DFT calculation.

Figure 16

Fermi distribution for eight H atoms in a box, $L=6$ bohrs. Points may represent more than one state due to energy degeneracies.

Figure 17

Energy difference between the fourth and fifth energy levels along with the occupation number of the fifth energy level. Eight H atoms, cubic box, $L=6$ bohrs.

Figure 18

Chemical potential $\mu$ and the fourth and fifth energy levels for eight H atoms in a cubic box, $L=6$ bohrs. The curve labeled “middle” is half way between ${\varepsilon }_4$ and ${\varepsilon }_5$.