4 Nanoscale First-Principles Electronic Structure Simulations of Materials. . .
95
more elaborated ones such as Tkatchenko-Scheffler vdW correction (TS-vdW)
[5], that with the many-body dispersion (MBD) [8], and the exchange-hole dipole
moment model with B86b exchange (B86b-XDM) [69, 70]. As for comparison to
experiments, the average absolute relative deviations from the experimental values
[50, 66, 67] are 0.6% (length) and 0.7% (angle) for naphthalene, 0.5% (length) and
0.02% (angle) for anthracene, and 1.0% (length) and 0.3% (angle) for tetracene.
Among the three oligoacene crystals, the maximum absolute relative deviation of
the lattice constants of 1.93% was found for b of tetracene crystal, the experimental
value of which was measured at 175 K [50].
For pentacene, the absolute relative error from the experimental value measured
at 295.5 K [47] is 1.5% (length) and 0.9% (angle) on average, and the maximum
absolute relative deviation from experiments is found for b (2.98%). However, if
the calculated values are compared to the measurement of the same polymorph at
90 K [48], the maximum absolute relative deviation decreases to 1.07% (b), and the
average absolute relative deviations are 0.6% and 0.5% for the cell length and cell
angle, respectively. The calculated lattice constants and equilibrium volume are also
in good agreement with those predicted with vdW-DF-cx for the same polymorph
[52]. In the case of hexacene, the absolute relative deviation from experiments is
found to be largest of all the oligoacene crystals, i.e., the averaged deviation of 3.8%
and 1.0% for length and angle, respectively, and the maximum deviation of 6.15%
is found for length b, in comparison to the experimental values measured at 123 K
[56]. The deviation might be diminished if the theoretical result was compared
to experiments measured at lower temperature, as demonstrated for pentacene. In
addition to that, one should take into account the thermal expansion of the volume
at finite temperature, i.e., the vibrational effect of the crystal.
The effect of ZPE to the equilibrium volume was investigated by the fixed volume
structural optimization, followed by the normal mode analysis at the point. The
ZPE effect was taken into account, while the volume was incrementally varied. The
resulting volume of naphthalene and anthracene was reasonably larger than that
determined without including ZPE (see Table 4.1). Here, the phonon frequency was
calculated by sampling only the -point. The deviation from experiments might
decrease if the phonon frequencies in the entire Brillouin zone were taken into
account.
The theoretical lattice energies per molecule of the oligoacene crystals, slightly
underestimated, are in reasonable agreement with experiments, with the absolute
relative deviations (deviations) being 5.2% (0.044 eV) and 8.6% (0.087 eV) for
naphthalene and anthracene, respectively. Notice that they are in good agreement
with those at the level of RPA with single excitations [29]. Taking into account
the calculated lattice energy of the tetracene crystal (−1.383 eV), along with the
theoretical vibrational contribution to the sublimation enthalpy of organic crystals
(0.07−0.11 eV) [64], we estimate the theoretical sublimation enthalpy would be in
agreement with the experimental value (−1.409 eV) [68] within 0.14 eV.
95
more elaborated ones such as Tkatchenko-Scheffler vdW correction (TS-vdW)
[5], that with the many-body dispersion (MBD) [8], and the exchange-hole dipole
moment model with B86b exchange (B86b-XDM) [69, 70]. As for comparison to
experiments, the average absolute relative deviations from the experimental values
[50, 66, 67] are 0.6% (length) and 0.7% (angle) for naphthalene, 0.5% (length) and
0.02% (angle) for anthracene, and 1.0% (length) and 0.3% (angle) for tetracene.
Among the three oligoacene crystals, the maximum absolute relative deviation of
the lattice constants of 1.93% was found for b of tetracene crystal, the experimental
value of which was measured at 175 K [50].
For pentacene, the absolute relative error from the experimental value measured
at 295.5 K [47] is 1.5% (length) and 0.9% (angle) on average, and the maximum
absolute relative deviation from experiments is found for b (2.98%). However, if
the calculated values are compared to the measurement of the same polymorph at
90 K [48], the maximum absolute relative deviation decreases to 1.07% (b), and the
average absolute relative deviations are 0.6% and 0.5% for the cell length and cell
angle, respectively. The calculated lattice constants and equilibrium volume are also
in good agreement with those predicted with vdW-DF-cx for the same polymorph
[52]. In the case of hexacene, the absolute relative deviation from experiments is
found to be largest of all the oligoacene crystals, i.e., the averaged deviation of 3.8%
and 1.0% for length and angle, respectively, and the maximum deviation of 6.15%
is found for length b, in comparison to the experimental values measured at 123 K
[56]. The deviation might be diminished if the theoretical result was compared
to experiments measured at lower temperature, as demonstrated for pentacene. In
addition to that, one should take into account the thermal expansion of the volume
at finite temperature, i.e., the vibrational effect of the crystal.
The effect of ZPE to the equilibrium volume was investigated by the fixed volume
structural optimization, followed by the normal mode analysis at the point. The
ZPE effect was taken into account, while the volume was incrementally varied. The
resulting volume of naphthalene and anthracene was reasonably larger than that
determined without including ZPE (see Table 4.1). Here, the phonon frequency was
calculated by sampling only the -point. The deviation from experiments might
decrease if the phonon frequencies in the entire Brillouin zone were taken into
account.
The theoretical lattice energies per molecule of the oligoacene crystals, slightly
underestimated, are in reasonable agreement with experiments, with the absolute
relative deviations (deviations) being 5.2% (0.044 eV) and 8.6% (0.087 eV) for
naphthalene and anthracene, respectively. Notice that they are in good agreement
with those at the level of RPA with single excitations [29]. Taking into account
the calculated lattice energy of the tetracene crystal (−1.383 eV), along with the
theoretical vibrational contribution to the sublimation enthalpy of organic crystals
(0.07−0.11 eV) [64], we estimate the theoretical sublimation enthalpy would be in
agreement with the experimental value (−1.409 eV) [68] within 0.14 eV.