Table 1 Lattice properties obtained from the Birch–Murnaghan equation of state for the diamond and β-Sn phases of silicon

From: A shortcut to the thermodynamic limit for quantum many-body calculations of metals

Structure

Property

TA-HF

sfTA-CCSD-FS

sfTA-DCSD-FS

Experiments

DMCa

DMCb

DMC + EMP-ppc

AFQMC

D-Si

Vt (Å3 per atom)

15.52

17.62

–

18.15

–

18.14

17.83

18.15

 

B0 (GPa)

104

105.1

–

99.2

103.0

98.0

96.2

–

 

\({B}_{0}^{\prime}\)

3.82

3.83

–

4.11

–

4.6

4.19

–

 

V0 (Å3 per atom)

20.78

20.04

–

20.0

20.11

19.98

19.75

–

β-Sn Si

Vt (Å3 per atom)

12.16

13.5

–

13.96

–

13.9

13.81

13.955

 

B0 (GPa)

112

118.3

–

–

114

107

104.2

–

 

\({B}_{0}^{\prime}\)

4.05

4.6

–

–

–

4.6

4.7

–

 

V0 (Å3 per atom)

15.96

14.95

–

–

15.26

15.2

15.17

–

 

ΔE (eV per atom)

1.302

0.562

0.494

–

0.505

0.424

0.329

0.365d

 

Pt (GPa)

52.96

17.37

15.26

–

17.8

15.3

13.16

13.9

 

Pt vib. (GPa)

51.66

16.07

13.96

11.3–12.5

16.5

14.0

12.2

12.6

  1. Lattice properties for the two silicon phases, including the transition volume (Vt), bulk modulus (B0), pressure derivative of the bulk modulus (\(B_0\)), volume at equilibrium (V0), energy difference between the two minima (ΔE), transition pressure (Pt) and fully corrected transition pressures (Pt vib.) (see Methods for details). Our TA-HF and sfTA-CCSD energies are compared with those from DMC, AFQMC and experiments. The following superscripts distinguish between different DMC studies: aAlfè et al.19, bHennig et al.18 and cMaezono et al.21. AFQMC numbers come from Purwanto et al.20 and to obtain d we applied a core-polarization correction ourselves. The column marked DMC + EMP-pp used an empirical pseudopotential correction (EMP-pp). The energies and transition pressures consistently include core-polarization contributions (see Methods for details). Experimental numbers come from various sources22,23,24 as organized by Hennig and colleagues18.
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