Force-field parameters for molecular building blocks of fluorene and thiophene are generated using pre-existing bond-stretching and angle-bending terms for monomer
Table 5.6: Comparison of the un-modified and modified OPLS force-fields (MD0
and MD1 respectively) with DFT calculations for 2mers of fluorene (F0), methyl-
fluorene (F1), thiophene (T0), and ethyl-thiophene (T2). The properties compared are: the difference in energy between the cis and trans minima (∆E); the dihedral angle of each minimum (φ); and the end-end length associated with each minimum (lEE).
∆E (kJ/mol) φtrans (°) φcis (°) lEE,trans (Å) lEE,cis (Å)
DFT MD0 MD1 DFT MD0 MD1 DFT MD0 MD1 DFT MD0 MD1 DFT MD0 MD1
F0 0.04 0.00 0.08 39 39 40 141 135 140 15.3 15.2 15.4 15.1 15.0 15.2 F1 -0.05 0.03 0.05 39 41 38 141 139 142 15.3 15.2 15.4 15.1 15.0 15.2 T0 2.23 2.23 2.09 26 31 28 147 145 145 6.4 6.4 6.4 6.3 6.3 6.3 T2 0.84 1.43 0.86 45 20 50 129 129 135 6.4 6.5 6.4 6.3 6.5 6.3 units from the OPLS force-field. In the case of fluorene also parameters for the bonds and angles around the inter-monomer junction are available. However, since inter-monomer parameters are lacking for oligo-thiophenes, the fluorene inter-monomer parameters are used.
Partial charges which were generated following the procedure described in section 5.5 are directly implemented in the force-field parameter set. However, in the case of dihedral potentials these require the extra ’substraction’ step, as already described in section 5.2, before implementation in the force-field (FF). This is to avoid a double-counting of interactions, such as dispersive and electrostatic interaction, that are already described by the existing FF terms.
It was found that in order to achieve better agreement between the DFT and MD results the equilibrium bond lengths and angles of the parameters taken from the OPLS force-field need to modified while keeping the original force-constants. Equi- librium values were taken from the global minimal geometries obtained from the CAM-B3LYP/6-31G* optimisation. Table 5.6 contains a breakdown of these changes on: the resulting minimal energy difference, ∆E; the values of each dihedral min- imum, φtrans/cis; and the end-to-end length of each minimal geometry, lEE,trans/cis.
In all cases, it was found that there is either improvement in particular values or changes which are negligible (∼ 0.1 kJ/mol, 1°, 0.1 Å in each value).
In the case of fluorene with and without methyl-side chains (F0 and F1, respec- tively), it was found that the only slight improvement is in the location of φcis of
F0. While there are other values which seem to agree less well after modification, these deviations are generally negligible as mentioned above. Overall, there is good correspondence between F0 and F1 values.
In the case of thiophene with and without ethyl-side chains (T0, T2), there are some substantial improvements introduced by this modification as the changes are
made to the inter-monomer junction parameters. While there are significant im- provements in ∆E and φtrans for T2, the deviation of φcis slightly increased by ≈ 6°.
For the T0 there is a small improvement in the φtrans, but on the other hand the ∆E
value is worsened. This may play a role when considering properties of more mi- croscopic interest (e.g. if utilising geometries for optical calculations). Overall, the improvements made in T2 are substantial and the energetic improvement strongly impacts the resulting dynamics.
The force-field contribution to the dihedral potential is isolated by performing a dihedral scan using the FF parameters (without the required dihedral) over intervals of 10° from 0° to 180° in a manner analogous to that of the scans performed using DFT. In order to isolate all interactions relevant in the dihedral rotation which are not the covalent interaction and also to restrain the dihedral at each value in the scan, the four covalent energetic functions at each inter-monomer juncture are used to impose restraints.
The effective restraints at a given angle, φ0, are generated by placing each of the
four dihedral terms under the influence of a periodic potential, VR, given by:
VR(φ) = kc[1 − cos(φ − φ0)]. (5.1)
When it comes to the choice value kc care must be taken so as to find a balance
between forming an effective restraint without inducing any unwanted distortion to the molecule. For example, for molecules with methyl or no side-chains, the choice of kc= 5 × 104 kJ/mol is suitable. However, in the case of ethyl-thiophene, a large
reduction is necessary kc = 103 kJ/mol which may be attributed to the prevalence
of large forces in the side-chain - dihedral area. For each point along the scan the geometry is then optimised in vacuum using the conjugate-gradients minimisation algorithm within Gromacs 4.6.535,36 and the total energy of each point is used to
generate the corresponding profile.
With the FF contribution isolated, the required dihedral profile is obtained by sub- tracting the FF contribution from the DFT scan. The resulting ’subtracted’ profile is then fit to a 5th order Ryckaert-Bellmans function:
VRB(φ) = 4 5
X
n=0
cn[± cos(φ)]n (5.2)
from the difference of 180° between one pair of dihedral angles and another of the four used. For example, for a dihedral angle of 0° in the polymer convention, the dihedral angle of the two pairs of four atoms in the trans position is φ° while the two pairs in the cis position have a (φ + 180)° dihedral angle. As such, the function cosine terms in Equation 5.2 must be modified to cos(φ + 180) = − cos(φ) in order to yield the appropriate energy.
0 30 60 90 120 150 180 Dihedral Angle (º) -6 -4 -2 0 2 4 6 8 10 Energy (kJ/mol)
(b)
0 30 60 90 120 150 180 Dihedral Angle (º) -15 -10 -5 0 5 10 15 20 25 Energy (kJ/mol) -15 -10 -5 0 5 10 15 20 25 30 Energy (kJ/mol) DFT Profile FF Contribution Subtraction Fitted Profile Fit + FF Contribution(a)
(
c
)
(b)
(c)
Figure 5.25: Subtraction profiles for (a) fluorene, (b) thiophene and (c) ethyl- thiophene. Each figure displays the calculated DFT profile; the profile obtained from the MD ’scan’; the resulting subtracted profile; the fit of the subtracted profile to a 5th order Ryckaert-Bellmans function; and the resulting ’effective’ profile given by the addition of the MD scan profile and the fitted profile. (Legend applies to all graphs.)
Figure 5.25 provides examples of the curves obtained in the subtraction process for 2mers of fluorene, thiophene, and ethyl-thiophene. In the cases with no side-chains,
Table 5.7: Height of the energetic barriers at 0° (E0) and 180° (E180) for 2mers
ethyl-thiophene, methyl-thiophene, and thiophene using fitted profiles obtained from scans using different side-chain lengths. Each barrier is calculated relative to the closest local minimum (i.e. the trans minimum for ∆E0 and the cis minimum for
∆E180). The labels Tx (Ty) denote the energies of a 2mer with an x-yl side-chain
with energetic profile taken from a y-yl side-chain scan. The DFT values shown are those from the dihedral scan of the Tx molecule.
∆E0 (kJ/mol) ∆E180 (kJ/mol)
DFT MD DFT MD T0 (T0) 0.59 0.33 2.02 1.75 T1 (T0) 1.20 5.35 7.21 T1 (T1) 1.20 1.33 4.65 5.00 T2 (T0) 4.02 11.07 6.71 12.00 T2 (T1) 4.02 6.67 6.71 9.18 T2 (T2) 4.02 3.90 6.71 7.66
Ryckaert-Bellemans function result in a force-field which quantitatively mimics the DFT dihedral potential to a very high degree of accuracy.
It is noted that this procedure results in a far less smooth fit for ethyl-thiophene, Fig- ure 5.25(c). As discussed with respect to the DFT scans, the introduction of side- chains is a potential source of inconsistency in the calculations due to the relative freedom of side-chains. To minimise this inconsistency, the starting geometries are taken from the DFT calculations so as to reproduce the side-chain conformations as well as possible. However, this does not, in all cases, lead to a perfect agreement in the side-chain conformations between the DFT and FF scans.
As seen from the DFT results for alkyl-fluorenes, Figure 5.18(a), addition of methyl side-chains to fluorene has no effect on the dihedral profile which means no fur- ther modification are needed. In the case of alkyl-thiophenes, section 5.4, it was argued that the steric interactions responsible for large changes in dihedral po- tential should be already accounted for by the force-field. However, as is shown in Table 5.7, utilising the dihedral potential fitted from a thiophene with a shorter side-chain leads to drastically overestimated barriers at the planar positions. As such, reparameterisation of the dihedral term must be performed to accomodate for this. Therefore, given the tendency for inconsistency observed in ethyl-thiophene due to the side-chain degrees of freedom and that the difference in DFT dihedral potential between ethyl and propyl-thiophene is small (≈ 0.5 kJ/mol at the pla- nar barriers, Figure 5.18(b)), the ethyl-thiophene potential is used for thiophene molecules with longer side-chains.