Supporting information.
Benson et al.
Details of theoretical calculations:
The enthalpy vs. pressure phase diagrams of CaC2 and Li2C2 and the ab initio
calculations during structure prediction were performed using first principles all-electron
projector augmented waves (PAW)[1] method as implemented by the Vienna Ab Initio
Simulation Package (VASP)[2]. Exchange-correlation effects were treated within the generalized
gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) parameterization[3].
The structures were relaxed with respect to pressure, lattice parameters, and atomic positions.
Forces were converged to better than 10-3 eV/Å. Integration over the Brillouin Zone (BZ) was
done on a grid of special k-points of size 11 × 11 × 11 (for enthalpy vs. pressure phase diagrams)
and resolutions better than 2π × 0.06 Å-1 (during structure prediction, see reference 4) determined
according to the Monkhorst-Pack scheme[5]. For all calculations using VASP the plane-wave
energy cutoff was set to 675 eV (Li2C2) and 550 eV (CaC2) with 3p64s2, 1s22s1 and 2s22p2 treated
as valence electrons for Ca, Li and C, respectively. To obtain the band structure and enthalpies
VASP calculations were performed using the tetrahedron method with Blöchl correction for BZ
integration.[6] The equations of state were determined using the third-order Birch-Murnaghan
method. [7] Bader analysis[8] of charge densities was performed according to Ref. 9. To achieve a
high accuracy the mesh for the augmentation charges was substantially increased. The error of
calculated Bader charges is smaller than 0.01 e/atom.
The evolutionary algorithm USPEX[4,10,11] was applied to the CaC2 and Li2C2 systems for
structure prediction. Structure searches were performed at the target pressures 5.5, 8, 10, and 20
GPa with 1, 2, 3, 4 and 6 formula units per simulation cell. VASP was used for all ab initio
structural relaxation and enthalpy calculations. The first generation of structures was generated
randomly and each subsequent generation was produced from 60% of the lowest-enthalpy
structures in the previous generation. The lowest-enthalpy structures of every generation
survived into the next generation. For producing the next generations’ structures the operators
used were heredity (60% structures), atomic permutation (10%), lattice mutation (20%), and soft
mutation (10%). During structure prediction each individual experienced four ab initio structural
relaxations per generation to target pressure and relaxed atomic forces. The series of relaxations
were designed to quickly converge the individual to its local minima but maintain high
convergence standards. The number of generations for each complete structure prediction
calculation varied (this depends heavily on the number of atoms in simulation cell), but the
global minimum was usually reached within 15-20 generations, which is the typical range for
USPEX calculations. Each individual generation contained a population of 1.5-2 times the
number of atoms in the simulation cell (with a minimum of 15), except the first generation which
contained a population of ~20 times the the number of atoms in the cell. The large population in
the first generation was designed to sample a large piece of the energy landscape, which
subsequent generations could then use.
Structure prediction using random structure searching methodology used target pressures 5,
10 and 20 GPa. For sake of computational cost, we utilized space group symmetry constraints
when generating the initial structures (at least 1000 configurations were generated and then
minimized without symmetry constraints using the VASP code), the computational details could
be found in Ref. [12].
1
Phonon and Electron-phonon coupling calculations were performed using the QUANTUMESPRESSO package[13] using the PBE exchange-correlation. The Monkhorst-Pack grid for
calculating the electronic density of states for electron-phonon coupling was set at 40 × 40 × 40.
The plane-wave cutoff was set at 60 Ry using Gaussian smearing with a smearing parameter of
0.05 Ry. For calculations of phonons an 8 × 8 × 8 k-point mesh was using for electronic
integration over the BZ with a 4 × 4 × 4 q grid used for calculating the phonon dynamical matrix
elements. Ultrasoft pseudopotentials used were generated by the Vanderbilt method[14] with
3s23p64s2, 1s22s1 and 2s22p2 as valence electrons for Ca, Li and C, respectively.
To explore the superconductivity of the new materials electron-phonon coupling was
performed on the metallic structures. The superconductivity critical temperature TC was
calculated via the Allen-Dynes modified McMillan equation[15]:


1.04 1    
TC  log exp  
(1)
,
*



1.2
1
0.62






where μ* is the Coulomb pseudopotential, and the electron-phonon coupling constant λ and the
phonon frequencies logarithmic average ωlog are calculated directly from the eliashberg function
α2F(ω) via:
 2   2 F ( )

log  exp  
(2)
ln( )d  ,

 0

and

 2 F ( )
(3)
d .
 2

0
References for Supporting Information:
[1] Blöchl, P. E. Phys. Rev. B 1994, 50, 17953; Kresse, F.; Joubert, J. Phys. Rev. B 1999, 59,
1758.
[2] Kresse, G.; Hafner, J. Phys. Rev. B 1993, 48, 13115; Kresse, G.; Furthmüller, J. Comput. Mat.
Sci. 1996, 6, 15
[3] Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865.
[4] Glass, C. W.; Oganov, A. R.; Hansen, N. Comp. Phys. Comm. 2006, 175, 713.
[5] Monkhorst, H. J.; Pack, J. D. Phys. Rev. B 1976, 13, 5188.
[6] Blöchl, P. E.; Jepsen, O.; Andersen, O. K. Phys. Rev. B 1994, 49, 16223.
[7] Birch, F. Phys. Rev. 1947, 71, 809.
2
[8] Bader, R. F. W. Atoms in Molecules: A Quantum Theory (Oxford University Press: Oxford,
1990).
[9] Armaldsen, A.; Tang, W.; Henkelman, G. http://theory.cm.utexas.edu/bader/; Tang, W.;
Sanville, E.; Henkelman, G. J. Phys.: Condens. Matter 2009, 21, 084204.
[10] Oganov, A. R.; Glass, C. W. J. Chem. Phys. 2006, 124, 244704.
[11] Lyakhov, A. O.; Oganov, A. R.; Valle, M. Comp. Phys. Comm. 2010, 181, 1623.
[12] [11] Srepusharawoot, P.; Blomqvist, A.; Araújo, M.; Scheicher, R. H.; Ahuja, R. Phys. Rev.
B 2010, 25, 125439.
[13] Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.;
Chiarotti, G. L.; Cococcioni, M.; Dabo, I.; Dal Corso, A.; de Gironcoli, S.; Fabris, S.; Fratesi, G.;
Gebauer, R.; Gerstmann, U.; Gougoussis, C.; Kokalj, A.; Lazzeri, M.; Martin-Samos, L.; Marzari,
N.; Mauri, F.; Mazzarello, R.; Paolini, S.; Pasquarello, A.; Paulatto, L.; Sbraccia, C.; Scandolo,
S.; Sclauzero, G.; Seitsonen, A. P.; Smogunov, A.; Umari, P.; Wentzcovitch, R. M. J. Phys.
Condens. Matter 2009, 21, 395502.
[14] Vanderbilt, D. Phys. Rev. B 1990, 41, 7892.
[15] Allen, P. B.; Dynes, R. C. Phys. Rev. B 1975, 12, 905.
3
Table S1: Compilation of distances at the equilibrium volume
Structure
GS-Li2C2
Cmcm-Li2C2
P-3m1-Li2C2
GS-CaC2
Cmcm-CaC2
Immm-CaC2
Nearest
C-C
(Å)
1.26
1.45×2
1.51×3
1.26
1.40, 1.46
1.48
C1-C1
1.55×2 C1-C2
Nearest M-C (Å)
2.19×2, 2.37×4
2.25×2, 2.26×4, 2.39
2.02, 2.29×3
2.57×2, 2.64×2, 2.79×2, 2.82×2
2.62×4, 2.65×2, 2.71 × 4
2.65×4, 2.93×2 C1-M
2.64×2, 2.66×4 C2-M
Table S2: Compilation of structural parameters for predicted structures at their equilibrium
volume
P-3m1-Li2C2
a = 2.5708 Å, c = 6.1940 Å
Li 2d 1/3, 2/3, 0.3043
C 2d 1/3, 2/3, 0.9782
Cmcm-Li2C2
a = 3.3350 Å, b = 7.8128 Å, c = 2.5580 Å
Li 4c 0.5000 0.1499 0.7500
C 4c 0.5000 0.4563 0.7500
Cmcm-CaC2
a = 3.7459 Å, b = 8.7559 Å, c = 4.7587 Å
Ca 4c 0.5000 0.8535 0.7500
C 8f 0.5000 0.4383 0.1029
Immm-CaC2
a = 2.6818 Å, b = 7.3506 Å, c = 6.4867 Å
Ca 4g 0.0000 0.2059 0.0000
C1 4i 0.0000 0.0000 0.3860
C2 4j 0.0000 0.5000 0.2325
4
Table S3: Equation of states parameters
Li2C2
GS
P-3m1
Cmcm
E0 (eV/fu)
-22.255
-22.024
-21.863
V0 (Å3/fu)
47.106
35.632
33.579
B0 (GPa)
39.782
94.778
74.925
B0’
3.921
3.472
4.892
CaC2
GS
Cmcm
Immm
E0 (eV/fu)
-20.412
-20.411
-19.826
V0 (Å3/fu)
48.898
39.072
31.994
B0 (GPa)
50.381
90.179
135.047
B0’
3.649
4.189
4.215
5
Figure S1: Enthalpy-pressure relations (per formula unit) for CaC2 with respect to the tetragonal
CaC2-I structure (space group 139) and referring to zero Kelvin. Results were obtained from
VASP calculations. CaC2-I, -II, -III, -V, -VI, -VII denote polymorphs consisting of acetylide C22units. Their labeling is according to A. Kulkarni, K. Doll, J. C. Schön, and M. Jansen, J. Phys.
Chem. B 114, 15573 (2010). It is seen that the polymeric forms Cmcm-CaC2 (Sg 63) and ImmmCaC2 (Sg 71) stabilize rapidly with increasing pressure with respect to the modifications based
on dumbbell units.
6