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FIG. 12. Distribution of cavity volumes in Bi at (a) 573 K and (b) 1023 K. ] The panels have the same units. Inset: Large cavities at 1023 K. ( The diffusion constants for the four temperatures (Eq. (4)) are given in Table III. A linear relationship between log D and 1/T gives an activation barrier of 0.126 eV for self- diffusion in liquid Bi. The value of D at 573 K is consis- tent with earlier DF/MD calculations at 600 K (1.91 4 x 107>cm?/s)”® and the interpolation formula based on E 0.05 tracer diffusion experiments (1.74 x 10~>cm?/s),°” although the er- ror bars in the latter are large (~30%).

Figure 12 Distribution of cavity volumes in Bi at (a) 573 K and (b) 1023 K. ] The panels have the same units. Inset: Large cavities at 1023 K. ( The diffusion constants for the four temperatures (Eq. (4)) are given in Table III. A linear relationship between log D and 1/T gives an activation barrier of 0.126 eV for self- diffusion in liquid Bi. The value of D at 573 K is consis- tent with earlier DF/MD calculations at 600 K (1.91 4 x 107>cm?/s)”® and the interpolation formula based on E 0.05 tracer diffusion experiments (1.74 x 10~>cm?/s),°” although the er- ror bars in the latter are large (~30%).

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TABLE I. Simulation parameters.
TABLE I. Simulation parameters.
FIG. 2. Isomers of Big, (a)-(e) in order of decreasing stability. The weak “bonds” between the tetrahedra in Big(b) are 4.30 A long. FIG. 1. Isomers of Bi., Big, and Bi. (a) denotes the most stable isomer, (b) the next highest in energy.
FIG. 2. Isomers of Big, (a)-(e) in order of decreasing stability. The weak “bonds” between the tetrahedra in Big(b) are 4.30 A long. FIG. 1. Isomers of Bi., Big, and Bi. (a) denotes the most stable isomer, (b) the next highest in energy.
TABLE II. Spectroscopic constants of Bi,. R-CI: relativistic configuration interaction, DF: density functional.
TABLE II. Spectroscopic constants of Bi,. R-CI: relativistic configuration interaction, DF: density functional.
FIG. 4. Isomers of Bi,,, Bi,;, and Bi,,. (a) denotes the most stable isomer, (b) the next highest in energy. Low-lying isomers of Big are shown in Fig. 2. Of partic- ular interest is the relative stability of a pair of Bi, tetrahedra, which was also found by Hirschfeld and Lustfeld.°’ There are several local minima in the energy surface of the Bi,(7,,)- dimer, but the lowest energy [0.14 eV above (a)] is found when faces of the two tetrahedra (bond lengths 3.04 A) are
FIG. 4. Isomers of Bi,,, Bi,;, and Bi,,. (a) denotes the most stable isomer, (b) the next highest in energy. Low-lying isomers of Big are shown in Fig. 2. Of partic- ular interest is the relative stability of a pair of Bi, tetrahedra, which was also found by Hirschfeld and Lustfeld.°’ There are several local minima in the energy surface of the Bi,(7,,)- dimer, but the lowest energy [0.14 eV above (a)] is found when faces of the two tetrahedra (bond lengths 3.04 A) are
FIG. 3. Isomers of Big, Big, and Bi,,- (a) denotes the most stable isomer (b) the next highest in energy. twofold coordinated dimer atoms (2.83 A), the bond lengths in these structures lie in a narrow range (3.02-3.10 A). The longest bonds occur in the C,,, isomer of Bi; (3.09 A, 3.15 A), where the out-of-plane atom is fourfold coordinated. The en- ergies of 7(a) and 7(b) are almost degenerate.
FIG. 3. Isomers of Big, Big, and Bi,,- (a) denotes the most stable isomer (b) the next highest in energy. twofold coordinated dimer atoms (2.83 A), the bond lengths in these structures lie in a narrow range (3.02-3.10 A). The longest bonds occur in the C,,, isomer of Bi; (3.09 A, 3.15 A), where the out-of-plane atom is fourfold coordinated. The en- ergies of 7(a) and 7(b) are almost degenerate.
HIG. 6. Vibration densities of states for isomers of Bi,. Numbers withou subscripts denote the most stable isomer. With the exception of Bi,, there are few measurements of vibration frequencies in Bi, clusters. Neon matrix stud- ies of Bi,”! show vibration frequencies of 89.8, 120.4, and 149.7 cm7!. Our calculated values (T,: 93, 120, 122, and
HIG. 6. Vibration densities of states for isomers of Bi,. Numbers withou subscripts denote the most stable isomer. With the exception of Bi,, there are few measurements of vibration frequencies in Bi, clusters. Neon matrix stud- ies of Bi,”! show vibration frequencies of 89.8, 120.4, and 149.7 cm7!. Our calculated values (T,: 93, 120, 122, and
FIG. 5. Cohesive energies of Bi, clusters for CPMD-optimized geometries. Thick red: CPMD calculations (no s-o coupling), thin red: VASP calculations (no s-o coupling), blue: VASP calculations (s-o coupling), green: experimen- tal cohesive energy of Bi (Ref. 58).
FIG. 5. Cohesive energies of Bi, clusters for CPMD-optimized geometries. Thick red: CPMD calculations (no s-o coupling), thin red: VASP calculations (no s-o coupling), blue: VASP calculations (s-o coupling), green: experimen- tal cohesive energy of Bi (Ref. 58).
FIG. 7. Snapshots of liquid Bi at (a) 573 K and (b) 1023 K. The cavities are shown in red. The bond angle distribution at 573 K is shown in tig. 10. The dependence of the distribution depends on the
FIG. 7. Snapshots of liquid Bi at (a) 573 K and (b) 1023 K. The cavities are shown in red. The bond angle distribution at 573 K is shown in tig. 10. The dependence of the distribution depends on the
Rings and cavities are analyzed as described above Sec. II B). Simulations of hundreds of atoms are needed to FIG. 9. Near-neighbor distributions in liquid Bi at 573 K. Below are shown the average distances of each neighbor at all four simulation temperatures.
Rings and cavities are analyzed as described above Sec. II B). Simulations of hundreds of atoms are needed to FIG. 9. Near-neighbor distributions in liquid Bi at 573 K. Below are shown the average distances of each neighbor at all four simulation temperatures.
FIG. 8. (a) Pair distribution functions for liquid Bi at the temperatures shown. Black: present calculations, red: neutron scattering data (Ref. 15). (b) Corre- sponding structure factors.
FIG. 8. (a) Pair distribution functions for liquid Bi at the temperatures shown. Black: present calculations, red: neutron scattering data (Ref. 15). (b) Corre- sponding structure factors.
FIG. 10. Bond angle distribution at 573 K as a function of cutoff distance.
FIG. 10. Bond angle distribution at 573 K as a function of cutoff distance.
TABLE III. Self-diffusion constant D (1075 cm2/s) of Bi at temperatures T (K).
TABLE III. Self-diffusion constant D (1075 cm2/s) of Bi at temperatures T (K).
FIG. 11. Ring distributions in liquid Bi at 573 and 1023 K.
FIG. 11. Ring distributions in liquid Bi at 573 and 1023 K.
FIG. 13. Calculated power spectra for liquid Bi at 573 K (blue) and 1023 K (red). Curves are normalized to unity at zero frequency. The general features of the electronic band structure of crystalline Bi are both well known® and consistent with x-ray photoemission spectroscopy on amorphous and crys- talline Bi.°? The three bands near the Fermi energy arising from 6p-electrons have antibonding, lone-pair, and bonding components and are separated by ~4 eV from two bands at lower energies arising from the 6s-electrons. The Kohn-Sham eigenvalue spectra (electronic density of states, DOS) for 573 K and 1023 K are shown in Fig. 14. The overall picture is similar to that in amorphous and crystalline Bi, with two broad bands from the Fermi energy to —4 eV, and between —8 and —12 eV. The non-zero density of states at the Fermi energy is consistent with the metallic behavior of Bi at these temperatures.
FIG. 13. Calculated power spectra for liquid Bi at 573 K (blue) and 1023 K (red). Curves are normalized to unity at zero frequency. The general features of the electronic band structure of crystalline Bi are both well known® and consistent with x-ray photoemission spectroscopy on amorphous and crys- talline Bi.°? The three bands near the Fermi energy arising from 6p-electrons have antibonding, lone-pair, and bonding components and are separated by ~4 eV from two bands at lower energies arising from the 6s-electrons. The Kohn-Sham eigenvalue spectra (electronic density of states, DOS) for 573 K and 1023 K are shown in Fig. 14. The overall picture is similar to that in amorphous and crystalline Bi, with two broad bands from the Fermi energy to —4 eV, and between —8 and —12 eV. The non-zero density of states at the Fermi energy is consistent with the metallic behavior of Bi at these temperatures.
FIG. 14. Electronic density of states for liquid Bi at T = 573 K (blue) and 1023 K (red). The Fermi energy (dashed) is zero in each case.
FIG. 14. Electronic density of states for liquid Bi at T = 573 K (blue) and 1023 K (red). The Fermi energy (dashed) is zero in each case.
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