Following the scheme described in Section 2.1.4, the required force-field contribution to the dihedral potential is isolated by performing force-field scans over intervals of 10° from 0° to 180° in a manner analogous to that of the DFT scans performed in Chapter 3. Each point in the scan is initiated using the corresponding geometry
from the DFT scan calculation so as to ensure optimal correspondence between each geometry and, thus, comparability of the resulting energies. As the force- field minimisation is more sensitive to distortions in initial geometries than the DFT procedure, this measure is taken to provide greater consistency between the two procedures while also aiding the force-field minimisation in finding the correct minimum.
As we wish to isolate all interactions relevant in the dihedral rotation which are not the covalent interaction and also wish to restrain the dihedral at each value in the scan, the four covalent energetic functions at each inter-monomer juncture are free to be utilised as restraints. In order to generate an effective restraint at a given angle, φ0, each of the four dihedral terms are placed under the influence of a periodic
potential, VR, given by:
VR(φ) = kc[1− cos(φ − φ0)]. (4.1)
While we have found it to be sufficient to impose only two restraints on the corre- sponding DFT scans, we find that it is necessary to impose restraints on all four dihedral terms in the FF scan. Failure to do so results in significant deviations in the free dihedral terms (± 5°). Deviations of this scale result in poor comparability between the force-field and DFT scans and lead to ill-fitted potentials. In turn, this leads to inconsistency between the subtraction profiles and resulting unrestrained optimisations and simulations.
In choosing the value of kc for the periodic restraint, care must be taken so as to
find a balance between forming an effective restraint without inducing any unwanted distortion in the molecule. We obtained values of kcby a trial and error procedure of
maximising kc, for optimal correspondence of the final dihedral angle with the target
dihedral angle, without risking distortions of the given molecule. For molecules with methyl or no side-chains, the choice of a large value, kc= 5× 104 kJ/mol, provides a
suitable restraint with very low error in the resulting dihedral values (± ∼ 0.25°). In the case of ethyl-thiophene, a significant reduction to kc = 103kJ/mol is necessary
which can be attributed to the prevalence of large forces in the side-chain - dihedral area. This reduction results in an increased error in the dihedral values (± ∼ 1°) though this error is still within an acceptable tolerance. From this, the geometry is then optimised in vacuum using the conjugate-gradients minimisation algorithm within Gromacs and the total energy of each point along the scan is calculated to form the corresponding profile.
With the force-field (FF) contribution to the dihedral potential isolated, the required dihedral profile is obtained by subtracting the FF contribution from the DFT potential. The resulting ‘subtracted’ potential is then fitted to a 5th order
0 30 60 90 120 150 180 Dihedral Angle (º) -15 -10 -5 0 5 10 15 20 25 30 35 Energy (kJ/mol) DFT Profile FF Contribution Subtraction Fitted Profile Fit + FF Contribution 0 30 60 90 120 150 180 Dihedral Angle (º) -6 -4 -2 0 2 4 6 8 10 12
(a) Fluorene
(b) Thiophene
Figure 4.1: Subtraction potentials for (a) fluorene and (b) thiophene. Each figure displays the calculated DFT potential; the FF contribution; the resulting subtracted potential; the fit of the subtracted potential to a 5th order Ryckaert-Bellmans func- tion; and the resulting effective potential given by the addition of the FF scan potential and the fitted potential. (Legend in (a) applies to both graphs.)
Ryckaert-Bellmans (RB) function: VRB(φ) = 5 X n=0 cn[cos(φ)]n. (4.2)
In Figure 4.1, the subtraction curves and fits are shown for 2mers of fluorene and thiophene. In both cases, the fit from the RB function is accurate and the combined fit and FF contribution of each closely resembles the corresponding DFT potential. To implement the fitted potential, we divide the potential across each of the four available four-atom dihedrals at the inter-monomer junction. This is shown schematically in Figure 4.2. As can be seen, two of the four dihedrals are in the same convention as the fitted potential (the polymer convention) with the trans angle set to 0° while the other two are in the trans = 180° convention. Given that cos(φ + π) =− cos(φ), the potential for the latter two is expressed with alternating positive and negative coefficients as is shown.
With the potential in place in the 2mer FF, a first test is to determine the correspondence between the minima obtained from free optimisation in the FF and using DFT. For each 2mer, we have calculated the difference in energy, ∆Em =
Ecis−Etrans, between the cis and trans minima of each molecule using both methods.
As is shown in Table 4.1, in both cases without side-chains, the FF and DFT values are comparable to within ' 0.1 kJ/mol (' 0.04 kBT ). This test demonstrates both
the accuracy of the correspondence found by the fitting procedure as well as the ability of the fitted force-field to reproduce the quantum chemical minima in these examples. This point is discussed further in Section 4.1.3.
For both fluorene and thiophene, we see that this procedure is fairly straight- forward and yields both accurate fits to the subtraction potentials and provides an accurate representation of the cis and trans optimal geometries. This correspon-
VRB( ) = 1 4 5 X n=0 cn[cos( )]n VRB( ) = 1 4 5 X n=0 cn[ cos( )]n
}
}
Figure 4.2: Schematic depicting the four dihedral terms at a thiophene junction and the sign convention of the RB function (Eq. 4.2) associated with it. For fluorene, this convention is followed analogously.
Table 4.1: Comparison of the difference in energy between the cis and trans minima, ∆Em, of 2mers of fluorene, C1-fluorene, thiophene, and C2-thiophene calculated from
DFT and from the force-field (FF).
∆Em (kJ/mol) DFT FF fluorene 0.04 0.08 C1-fluorene -0.05 0.05 thiophene 2.23 2.09 C2-thiophene 0.84 0.86
dence was enhanced in both cases by modifying the equilibrium bond lengths and angles of the OPLS force-field which we shall discuss in greater detail in Section 4.1.3. For now, we proceed to apply the same procedure to fluorenes and thiophenes with side-chains.