94
S. Yanagisawa and I. Hamada
vdW-DF. As the initial structures for the structural optimization, we used the Xray diffraction data of naphthalene [43] and anthracene [44] with the space group
of P 2 1 /a measured at 295 K and those of tetracene with the P 1 symmetry reported
by Robertson [45, 46]. For additional calculations in this work, we used the Xray diffraction data of the pentacene crystalline phase obtained by vapor deposition
[47]. The crystalline phase has spacing for the ac planes [(001)d spacing] of
1.45 nm [47], and similar spacing has been reported in other experiments of the
single crystals [48–50]. Recent theoretical studies [51, 52] showed that with the
recent vdW-inclusive method, similar polymorph is more stable than the other
experimentally reported polymorphs such as the bulk phase reported by Campbell
et al. [53] and the thin film phases [54, 55]. The initial crystal structure of hexacene
came from the diffraction data at 123 K of the crystalline phase fabricated with
physical vapor transport method [56]. For the total energy calculation, we used
the projector augmented-wave (PAW) method [57] as implemented in the Vienna
ab initio simulation package (VASP) [58, 59]. The vdW-DF calculations were
performed using the Román-Pérez-Soler algorithm [40] implemented by Klimeš
et al. [39]. We used the revised vdW-DF2 (rev-vdW-DF2) [10], which uses the
revised Becke’s exchange functional [60] and the nonlocal correlation for the second
version of vdW-DF [37].
We used the kinetic energy cutoff of 1000 eV to expand the wave functions in
terms of a plane-wave basis set, along with hardest PAW potentials supplied with
the VASP code [61]. The Brillouin zone integration was performed using a 4 × 4 × 4
Monkhorst-Pack (MP) [62] k-point set for naphthalene and anthracene, a 4 × 4 × 2
MP k-point set for tetracene. In the calculations of pentacene and hexacene in this
work, we used a 4 × 3 × 2 and 4 × 4 × 2 k-point set, respectively. Cell parameters
and internal degrees of freedom were optimized until the forces acting on atoms
became smaller than the threshold value of 1.0 × 10 −3 eV Å −1 . With this setting
the cell parameters and lattice energy are estimated to converge within 0.039 Å,
0.49 ◦ , and 2.37 Å 3 and 1 meV, respectively. The lattice energy was calculated by
subtracting the sum of the total energies of the constituent molecules from that of
the molecular crystal.
We also investigated the effect of the zero-point vibrational energy (ZPE)
to the equilibrium volume for selected crystals as follows. Starting from the
equilibrium crystal structure, the cell volume was varied by ±2% up to ±10%, and
at each volume, we performed the fixed volume structural optimization followed
by the normal mode analysis at the point to calculate the ZPE contribution
to the total energy. We then calculated the total energy with ZPE and fitted it
to the Murnaghan [63] equation of state to get the ZPE corrected equilibrium
volume.
Table 4.1 displays the resulting lattice constants. Overall, the cell parameters
and equilibrium volumes are in reasonable agreement with experimental values
in the literature, which were measured at 5−296 K [47, 48, 50, 56, 66, 67]. They
are also comparable to other vdW-inclusive methods such as vdW-DF-cx [35],
S. Yanagisawa and I. Hamada
vdW-DF. As the initial structures for the structural optimization, we used the Xray diffraction data of naphthalene [43] and anthracene [44] with the space group
of P 2 1 /a measured at 295 K and those of tetracene with the P 1 symmetry reported
by Robertson [45, 46]. For additional calculations in this work, we used the Xray diffraction data of the pentacene crystalline phase obtained by vapor deposition
[47]. The crystalline phase has spacing for the ac planes [(001)d spacing] of
1.45 nm [47], and similar spacing has been reported in other experiments of the
single crystals [48–50]. Recent theoretical studies [51, 52] showed that with the
recent vdW-inclusive method, similar polymorph is more stable than the other
experimentally reported polymorphs such as the bulk phase reported by Campbell
et al. [53] and the thin film phases [54, 55]. The initial crystal structure of hexacene
came from the diffraction data at 123 K of the crystalline phase fabricated with
physical vapor transport method [56]. For the total energy calculation, we used
the projector augmented-wave (PAW) method [57] as implemented in the Vienna
ab initio simulation package (VASP) [58, 59]. The vdW-DF calculations were
performed using the Román-Pérez-Soler algorithm [40] implemented by Klimeš
et al. [39]. We used the revised vdW-DF2 (rev-vdW-DF2) [10], which uses the
revised Becke’s exchange functional [60] and the nonlocal correlation for the second
version of vdW-DF [37].
We used the kinetic energy cutoff of 1000 eV to expand the wave functions in
terms of a plane-wave basis set, along with hardest PAW potentials supplied with
the VASP code [61]. The Brillouin zone integration was performed using a 4 × 4 × 4
Monkhorst-Pack (MP) [62] k-point set for naphthalene and anthracene, a 4 × 4 × 2
MP k-point set for tetracene. In the calculations of pentacene and hexacene in this
work, we used a 4 × 3 × 2 and 4 × 4 × 2 k-point set, respectively. Cell parameters
and internal degrees of freedom were optimized until the forces acting on atoms
became smaller than the threshold value of 1.0 × 10 −3 eV Å −1 . With this setting
the cell parameters and lattice energy are estimated to converge within 0.039 Å,
0.49 ◦ , and 2.37 Å 3 and 1 meV, respectively. The lattice energy was calculated by
subtracting the sum of the total energies of the constituent molecules from that of
the molecular crystal.
We also investigated the effect of the zero-point vibrational energy (ZPE)
to the equilibrium volume for selected crystals as follows. Starting from the
equilibrium crystal structure, the cell volume was varied by ±2% up to ±10%, and
at each volume, we performed the fixed volume structural optimization followed
by the normal mode analysis at the point to calculate the ZPE contribution
to the total energy. We then calculated the total energy with ZPE and fitted it
to the Murnaghan [63] equation of state to get the ZPE corrected equilibrium
volume.
Table 4.1 displays the resulting lattice constants. Overall, the cell parameters
and equilibrium volumes are in reasonable agreement with experimental values
in the literature, which were measured at 5−296 K [47, 48, 50, 56, 66, 67]. They
are also comparable to other vdW-inclusive methods such as vdW-DF-cx [35],