Lattice structure of mercury: Influence of electronic correlation

Mercury condenses at $233\mathrm{K}$ into the rhombohedral structure with an angle of 70.53°. Theoretical predictions of this structure are difficult. While a Hartree-Fock treatment yields no binding at all, density-functional theory (DFT) approaches with gradient-corrected functionals predict a structure with a significantly too large lattice constant and an orthorhombic angle of about 60°, which corresponds to an fcc structure. Surprisingly, the use of the simple local density approximation (LDA) functional yields the correct structure and lattice constants in very good agreement with experiment; relativistic effects are shown to be essential for reaching this agreement. In addition to DFT results, we present a wave-function-based correlation treatment of mercury and discuss in detail the effects of electron correlation on the lattice parameters of mercury including $d$-shell correlation and the influence of three-body terms in the many-body decomposition of the interatomic correlation energy. The lattice parameters obtained with this scheme at the coupled cluster level of theory agree within 1.5% with the experimental values. We further present the bulk modulus calculated within the wave-function approach, and compare to LDA and experimental values.

Figure 1

The rhombohedral primitive cell is shown with reference to the face-centered cubic (fcc) structure. If the angle $\alpha$ is 60°, it is an ideal fcc close packed structure. This angle is 70° in Hg. If this angle were 90°, then it would be a simple cubic lattice. An angle of 109.28° corresponds to a body-centered cubic (bcc) lattice.

Figure 2

The rhombohedral primitive cell is shown with reference to the conventional hexagonal unit cell described by two lattice parameters (here labeled $a_{\mathrm{hex}}$ and $c_{\mathrm{hex}}$). The primitive cell in the hexagonal description is also shown (gray lines). The conventional cell described by hexagonal lattice parameters contains three atoms.

Figure 3

(Color online) The relationship of the hcp (left), fcc (middle), and rhombohedral (right) lattices is shown in terms of the arrangement of the 12 nearest neighbors in the solid. For the rhombohedral case, the rhombohedral lattice parameter, $a=a\left(\mathrm{nn}\right)$, is shown with solid bonds while the second nearest-neighbor $a\left(\mathrm{n}2\right)$ is shown with dotted bonds. It should be possible to see that the stacking order in the rhombohedral lattice is ABC as in fcc, as opposed to ABA in hcp.

Figure 4

The effect of relativity on the stability of the rhombohedral lattice in mercury. At each bond length, the optimized energy and corresponding angle is plotted. The relativistic (R) and nonrelativistic (NR) values are given, with the experimental rhombohedral angle marked by a dotted line. Energy is in eV and nearest-neighbor distance is in Å.

Figure 5

The effect of relativity on the stability of the hexagonal lattice in mercury. At each value of $a$, the optimized energy and corresponding ratio of $c∕a$ is plotted. For mercury, the lattice parameter $a$ is only the nearest-neighbor distance for values of $c∕a>1.63$. The relativistic (R) and nonrelativistic (NR) values are given, with the ideal $c∕a$ ratio marked by a dotted line. Energy is in eV and lattice parameter $a$ is in Å.

Figure 6

(Color online) The cohesive energy of Hg with respect to the nearest-neighbor-distance $a$.

Figure 7

(Color online) The cohesive energy of Hg with respect to the angle in the rhombohedral structure.

Figure 8

(Color online) Three-body clusters considered in the incremental scheme, drawn on a background of the 12 nearest neighbors with all bonds of length $a_0$ drawn.

Figure 9

Three-body results for the nearest-neighbor distance of solid mercury. Left: results of the $\left(5d6s\right)$ correlation treatment; Right: $6s$ correlation only. The energies are weighted for the total contribution to the lattice of each increment. The scale is the same in each subplot.

Figure 10

Three-body results for the optimized angle of solid mercury. Left: results of the $\left(5d6s\right)$ correlation treatment; Right: $6s$ correlation only. The energies are weighted for the total contribution to the lattice of each increment. The scale is the same in each subplot.