Supplementary material for “Lattice energies of molecular solids from the random
phase approximation with singles corrections”
Jiřı́ Klimeš1, 2
1
I.
J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic,
Dolejškova 3, CZ-18223 Prague 8, Czech Republic
2
Department of Chemical Physics and Optics, Faculty of Mathematics and Physics,
Charles University, Ke Karlovu 3, CZ-12116 Prague 2, Czech Republic
(Dated: August 19, 2016)
PAW POTENTIALS USED
The details describing the standard and hard PAW potentials are listed in Table S1.
II.
LATTICE ENERGIES OF DISPERSION
CORRECTED DFT FUNCTIONALS
The lattice energies obtained for density functional
theory functionals without and with dispersion corrections are listed in Table S2 and Table S3, respectively.
The isolated molecule was simulated by a cubic box with
a side of 18 Å, the solid with structure and k-points as
discussed in Section III of this SI. Values obtained at the
experimental volume (V0 ) and using a Murnaghan fit to
seven points around the experimental volume (Vfit ) are
given. Numbers in italics show results which are outside
of the fitted region, this occurs for systems dominantly
bound by dispersion when treated by DFT functionals
without dispersion corrections.
III.
INPUTS
The structures used for the calculations are stored in a
separate file structures.tar.gz in the root directory of
SI. They contain the optB88-vdW optimised structures
of the isolated molecules and of the solid at the experimental volume (POSCAR.1.000) and at six other volumes
around the experimental volume. The k-point set used
for the DFT calculation is given in the KPOINTS file in
the VASP format.
The input files are stored in file incars.tar.gz,
one directory with inputs for DFT calculations, second for RPA calculations.
In the first case the
final input file (INCAR) was made by performing
cat ${xc} INCAR.def > INCAR , with xc denoting the
chosen DFT method. For the RPA-based methods the
calculations proceeded by performing the following sequence of calculations:
cp INCAR.dft INCAR
echo "ENCUT=$encut" >> INCAR
$VASP
cp OUTCAR OUTCAR.dft
cp INCAR.exx INCAR
echo "ENCUT=$encut" >> INCAR
$VASP
cp OUTCAR OUTCAR.EXX.$encut
NBANDS_=‘grep -m 1 "maximum number" OUTCAR
| awk ’{print $5}’ ‘
NBANDS=‘echo "$NBANDS_ * $mul" | bc -l‘
cp INCAR.exact INCAR
echo "ENCUT=$encut" >> INCAR
echo "NBANDS=$NBANDS" >> INCAR
$VASP
cp OUTCAR OUTCAR.diag.$encut
cp INCAR.rpar INCAR
echo "ENCUT=$encut" >> INCAR
echo "ENCUTGW=$encutgw" >> INCAR
echo "NBANDS=$NBANDS" >> INCAR
$VASP > out.$encut.$encutgw
cp OUTCAR OUTCAR.$encut.$encutgw
cp INCAR.GW INCAR
echo "ENCUT=$encut" >> INCAR
echo "ENCUTGW=$encutgw" >> INCAR
echo "NBANDS=$NBANDS" >> INCAR
$VASP > out.RGW.$encut.$encutgw
cp OUTCAR OUTCAR.RGW.$encut.$encutgw
Here VASP is the VASP binary, encut is the basis-set
cut-off, encutgw is the basis-set cut-off for response related properties, mul = 1 for complex version of VASP
and mul = 2 for gamma-only version of VASP.
IV.
COMPARISON TO ALL-ELECTRON
CALCULATIONS
To assess the accuracy of our DFT calculations we
compared the PBE and PBE0 results obtained using
VASP to those of the Gaussian09 code. We used an oxalic
acid dimer taken from the structure of the β polymorph
of oxalic acid at experimental volume. The structure
used is also available in the structures.tar.gz file. The
VASP calculations used our standard set-up including a
cut-off of 1000 eV. The data are summarised in Table S4.
2
TABLE S1. The core radii in a.u. and the number of projectors for each angular momentum of the PAW potentials used in
this study. We also give the core radius of the local potential or the potential used as a local potential and the default basis-set
pw
cut-off energy Ecut
.
System
rs
Ns
rp
Np
rd
Nd
rlocal
Ecut
1.1
1.5
1.5
1.5
1
1
1
1
0.7
d
d
d
300
414
421
415
0.8
1.1
1.1
1.1
1
2
2
2
0.7
0.8
0.9
0.9
700
742
756
766
Standard
H
C
N
O
0.95
1.2
1.3
1.2
2
2
2
2
1.1
1.5
1.5
1.52
2
2
2
2
H
C
N
O
0.8
1.0
0.9
1.0
2
3
3
3
0.8
1.1
1.1
1.1
2
3
3
3
Hard
TABLE S2. Lattice energies of the studied molecular solids
in kJ/mol as calculated with PBE and PBE0 compared to
the reference data of Yang et al. for benzene1 and of Reilly
and Tkatchenko (Ref. 2) for the other systems. The columns
E(V0 ) give the energy obtained using the structure at the experimental volume while E(Vfit ) is the value calculated using
a fit to energies at different volumes. Data in italics denote
situation, where the optimal volume was outside of the fitting
region.
System
Ref.
Adamantane
Anthracene
Naphthalene
Benzene
CO2
Urea
Ammonia
Cyanamide
Oxalic acid α
Oxalic acid β
V.
−69.4
−112.7
−81.7
−55.3
−28.4
−102.5
−37.2
−79.7
−96.3
−96.1
PBE
E(V0 )
4.7
5.0
4.4
−2.8
−6.5
−72.9
−28.2
−62.5
−49.2
−53.2
PBE
E(Vfit )
− 7 .2
−12 .9
−10 .7
−10 .9
−9 .8
−74.4
−28.4
−63.5
−51.6
−54.0
PBE0
E(V0 )
5.9
1.0
1.2
−3.9
−8.1
−74.9
−26.8
−61.4
−53.3
−54.8
PBE0
E(Vfit )
−2 .7
−11 .6
−9 .4
−9 .4
−9 .8
−75.5
−26.9
−61.9
−54.0
−55.0
CONVERGED DATA WITH STANDARD
PAW POTENTIALS
The data that were used to make a Figure 3 in the
main text are presented in Tables S5 and S6.
VI.
CONVERGENCE OF EXX, RSE, AND
GWSE ENERGIES
Here we show an example of the convergence of the
extrapolated EXX, rSE, and GWSE energies, in Table S7
for solid oxalic acid β and in Table S8 for the oxalic acid
molecule. The bare data are shown in Figs. S1, S2, S3,
FIG. S1. Convergence of the EXX energy with respect to the
inverse of the total number of k-points for oxalic acid β. A
plane-wave basis-set cut-off of 500 eV was used.
and S4
VII.
TIME AND MEMORY REQUIREMENTS
OF THE CALCULATIONS
Time and memory requirements for oxalic acid using
a plane-wave basis-set cut-off of 600 eV are given in Table S9 for the α solid phase and in Table S10 for the
molecule. This set-up is sufficient to converge the RPA
energy to about 0.5 kJ/mol, and the GWSE energy to
about 2 kJ/mol. Larger basis-set cut-offs need to be used
to further converge the values. Moreover, we used hard
PAW potentials to improve the accuracy of the values.
The time required by all these calculations is summarised
in Table S11. Note that for dense k-point grids or large
cell sizes a the time is smaller than for less dense grids
or smaller cell sizes a as a smaller set of basis sets was
3
TABLE S3. Lattice energies of the studied molecular solids in kJ/mol as calculated with PBE0-rsMBD, PBE0-D3BJ , and PBE01
D3BJ
ATM of Yang et al. for benzene and of Reilly and Tkatchenko (Ref. 2) for the other systems. Two datasets correspond to
the energy of solid taken from a structure at the experimental volume (E(V0 )) and energy obtained by fitting to energies of
seven structures around the experimental volume (E(Vfit )).
System
Adamantane
Anthracene
Naphthalene
Benzene
CO2
Urea
Ammonia
Cyanamide
Oxalic acid α
Oxalic acid β
Ref.
−69.4
−112.7
−81.7
−55.3
−28.4
−102.5
−37.2
−79.7
−96.3
−96.1
PBE0
rsMBD
E(V0 )
−74.3
−107.8
−80.7
−53.7
−23.5
−109.2
−39.8
−88.6
−95.3
−96.3
PBE0
rsMBD
E(Vfit )
−79.3
−108.9
−81.3
−54.7
−23.6
−109.5
−40.5
−88.8
−95.9
−97.2
PBE0
D3BJ
E(V0 )
−70.5
−112.4
−84.2
−56.3
−25.5
−108.7
−40.4
−90.5
−94.3
−95.6
PBE0
D3BJ
E(Vfit )
−72.2
−113.4
−84.7
−57.2
−25.5
−108.9
−40.9
−90.9
−94.9
−96.4
PBE0
D3BJ
ATM
E(V0 )
−63.3
−102.6
−77.3
−52.5
−24.7
−106.2
−39.6
−88.8
−91.3
−92.6
PBE0
D3BJ
ATM
E(Vfit )
−64.0
−103.0
−77.4
−53.0
−24.7
−106.3
−40.0
−89.0
−91.6
−93.2
TABLE S4. Interaction energy of oxalic acid dimer taken from the β polymorph calculated for the PBE and PBE0 functionals
with VASP and with Gaussian09. The basis sets used in the Gaussian calculations were from the aug-cc-pVN Z set and different
cell sizes and PAW potentials were used in VASP calculations.
G09
PBE
PBE0
aVTZ
−85.86
−87.01
aVQZ
−85.61
−86.63
VASP
Hard
aV5Z
−85.04
−86.15
20 Å
−84.83
−85.99
VASP
Standard
22 Å
−84.79
−85.95
20 Å
−86.39
−87.46
22 Å
−86.36
−87.44
FIG. S2. Convergence of the rSE and GWSE energies with
respect to the inverse of the total number of k-points for oxalic
acid β. A plane-wave basis-set cut-off of 500 eV was used.
FIG. S3. Convergence of the EXX energy with respect to the
inverse volume of the simulation cell for oxalic acid molecule.
A plane-wave basis-set cut-off of 600 eV was used.
used.
For comparison, a reference quality PBE0 calculation
with basis-set cut-off of 1000 eV, hard PAW potentials
requires 315 CPU hours when a 4×3×4 k-point grid is
used and 8260 CPU hours when 5×4×5 grid is used. If
the Coulomb cut-off technique is not used, the energy
4
TABLE S5. The EXX and RPA correlation contributions to the lattice energy obtained using standard potentials, the ”hardnormal” correction ∆ obtained at a finite k-point set and at a finite volume of the simulation cell used for the isolated molecule,
and the final estimates of the contributions of the interaction energy for hard potentials.
System
Adamantane
Anthracene
Naphthalene
Benzene
CO2
Urea
Ammonia
Cyanamide
Oxalic acid α
Oxalic acid β
E(norm)
56.5
64.3
51.5
24.3
0.2
−36.3
−0.1
−17.4
−22.1
−12.5
EXX
∆
−0.4
0.2
−0.3
−0.3
0.7
3.7
0.2
1.2
4.6
4.1
E(hard)
56.1
64.5
51.2
24.1
0.9
−32.6
0.1
−16.2
−17.5
−8.4
E(norm)
−113.2
−157.0
−119.3
−69.1
−23.6
−59.4
−31.4
−53.2
−64.5
−74.2
RPA
∆
0.5
−0.2
−0.3
−0.1
−1.4
−4.0
−0.2
−2.5
−4.8
−4.6
E(hard)
−112.7
−157.1
−119.6
−69.2
−25.0
−63.4
−31.6
−55.7
−69.3
−78.7
TABLE S6. The rSE and GWSE contributions to the lattice energy obtained using standard potentials, the ”hard-normal”
correction ∆ obtained at a finite k-point set and at a finite volume of the simulation cell used for the isolated molecule, and
the final estimates of the contributions of the interaction energy for hard potentials.
System
Adamantane
Anthracene
Naphthalene
Benzene
CO2
Urea
Ammonia
Cyanamide
Oxalic acid α
Oxalic acid β
E(norm)
−10.4
−6.2
−5.2
−3.8
−2.8
−8.5
−6.4
−9.4
−11.7
−12.9
rSE
∆
−0.1
0.0
−0.1
−0.1
0.0
−0.2
0.0
−0.1
−0.1
−0.1
obtained with the first grid deviates by 8 meV from the
converged energy and the latter by 4 meV.
REFERENCES
1
2
J. Yang, W. Hu, D. Usvyat, D. Matthews, M. Schütz, and
G. K. Chan, Science 345, 640 (2014).
A. M. Reilly and A. Tkatchenko, J. Phys. Chem. 139,
024705 (2013).
E(hard)
−10.5
−6.3
−5.3
−4.0
−2.8
−8.7
−6.4
−9.5
−11.8
−13.0
E(norm)
−10.7
−10.6
−8.5
−6.1
−2.1
−7.1
−5.8
−9.1
−8.4
−9.2
GWSE
∆
−0.5
−0.3
−0.7
−0.3
−1.2
−1.5
−0.3
−1.0
−2.9
−2.7
E(hard)
−11.2
−10.9
−9.2
−6.4
−3.2
−8.6
−6.1
−10.1
−11.2
−11.9
5
TABLE S7. Results of extrapolations of the EXX, rSE, and GWSE energies using different k-point sets for oxalic acid β. The
numbers denote the range of k-point sets that were used for extrapolation. For example, the “2–4” column gives results of
extrapolation that used data from 2×2×2, 3×3×3, and 4×4×4 k-point sets. The data are in eV and for a plane-wave basis set
cut-off of 500 eV.
E(EXX)
E(rSE)
E(GWSE)
2–3
−108.0984
−1.8428
−1.3639
2–4
−108.0978
−1.8427
−1.3639
2–5
−108.0975
3–4
−108.0970
−1.8424
−1.3639
3–5
−108.0969
TABLE S8. Results of extrapolations of the EXX, rSE, and GWSE energies using different ranges of cell sizes for oxalic acid
molecule. Individual energies were calculated for molecules placed in cells a × a + 1 × a + 2. Columns amin and amax then denote
respectively the smallest and largest cell side of the cells that were used for the calculations. Linear regression was performed
on the data with the x axis set to the inverse of the cell volume. The last line shows the bare data for a cell with a = 10 Å.
The data are in eV for a plane-wave basis set cut-off of 600 eV.
amin
6
6
6
6
7
7
7
8
8
9
10
10
amax
7
8
9
10
8
9
10
9
10
10
20
–
E(EXX)
−107.9130
−107.9278
−107.9377
−107.9441
−107.9452
−107.9519
−107.9558
−107.9597
−107.9616
−107.9638
−107.9667
−108.3565
E(rSE)
−1.7118
−1.7135
−1.7146
−1.7153
−1.7156
−1.7161
−1.7166
−1.7168
−1.7173
−1.7179
–
−1.6921
E(GWSE)
−1.3214
−1.3217
−1.3220
−1.3222
−1.3221
−1.3224
−1.3225
−1.3226
−1.3227
−1.3227
–
−1.3054
6
TABLE S11. Time requirements in CPUhours for the calculation of the oxalic acid lattice energy. The line Σ gives the
total time requirements for the solid phase calculations and
for the molecular calculations, Σall gives the complete sum,
including the time requirements for the calculations with hard
PAW potentials. Salomon supercomputer equipped with Intel
Xeon E5-2680v3 processors and Infiniband FDR network was
used.
Solid
Σ
hard PAW
Molecule
FIG. S4. Convergence of the singles corrections with respect
to the inverse volume of the simulation cell for oxalic acid
molecule. A plane-wave basis-set cut-off of 600 eV was used.
Σ
hard PAW
Σall
TABLE S9. Time and memory requirements for the oxalic
acid solid in the α polymorph. A plane-wave cut-off of 600 eV
was used, required time is given in CPUhours and the required
memory in GB. Total time requirements are given on the line
Σ. Salomon supercomputer equipped with Intel Xeon E52680v3 processors and Infiniband FDR network was used.
k-points
2×2×2
3×3×3
Σ
time
34
192
226
RPA
memory
288
883
–
time
272
920
1192
Singles
memory
480
1024
–
TABLE S10. Time and memory requirements for the oxalic
acid molecule. A plane-wave cut-off of 600 eV was used, the
simulation cell dimensions were a × a + 1 × a + 2, required
time is given in CPUhours and the required memory in GB.
Total time requirements are given on the line Σ. Salomon supercomputer equipped with Intel Xeon E5-2680v3 processors
and Infiniband FDR network was used.
a
9
10
11
Σ
time
26
63
157
246
RPA
memory
192
365
653
–
time
288
775
–
1063
Singles
memory
384
768
–
–
k-pts/a
2×2×2
3×3×3
4×4×4
2×2×2
8
9
10
11
8
RPA
400
1000
100
1500
700
140
180
380
240
940
200
3340
Singles
4040
6300
800
11140
7100
1450
1970
4630
1000
9050
2300
29590