ACTIVIDADES Procariota

An example of the parameter fitting process for the cp..cp..c=1 angle potential was shown in Figure 3.5a; this corresponds to the angle between two aromatic carbon atoms and the sp2 _{hybridised carbon of the imine group. This imine}

functionality directly attached to an aromatic ring was not specifically parameterised, and therefore did not accurately reproduce the angle upon minimisation with PCFF.

Figure 3.5 – Example of the fitting approach for intra-molecular potentials. a) The cp..cp..c=1 angle was shown as a shaded yellow arc, this angle was scanned at the B3LYP/6-31G(d,p) level to generate reference data. b) Calculated energy as a function of angle for B3LYP/6-31G(d,p) (red crosses), unmodified PCFF (dotted line), and CSFF (solid line).

Using the fragment geometry shown in Figure 3.4, a scan from 100° to 136° was carried out in 1° intervals; at each step the structure was geometry optimised with a B3LYP14_{/6-31G(d,p) level of theory, under the constraint of the fixed}

angle. This covered both sides of the expected minima for this angle, around 120°, therefore a PES representation could be generated. This enabled a FF parameter to be fitted to this data, with the view to successfully describe this specific angle.

A scan of this type will not only alter the angle for which parameters were required, but will also affect other geometric factors that influence the energy calculated by the FF. There will also be a knock-on effect other parameters – this was dependent on which dependency tree the cp..cp..c=1 angle was in. To quantify this, FF atom types were assigned to the B3LYP/6-31G(d,p) optimised structures using the Discover module in Materials Studio.17 _{The difference in}

energy from this reference for each scan point then gave the “B3LYP” and “FF” energies for comparison in the fitting procedure. The results were compared for the cp..cp..c=1 scan in Figure 3.5b, in this case the FF underestimates the energy seen in the B3LYP data although the positions of the minimum were in good agreement. The difference between the B3LYP energy and the PCFF calculations for each scan point allowed the variation in the energy due to the missing potentials to be estimated.

The required minimum angle, θ0, and the quadratic term for the angle potential

were initially set from a quadratic fit to this difference data in the vicinity of the minimum. The full set of force constant parameters were then obtained from the full data set using linear regression with the deviation of the angle from θ0 as the

0 4 8 12 16 20 100 120 140 E n er g y / k ca l m o l -1 cp..cp..c=1 angle / °

a)

b)

independent variable. This example was complicated by the fact that two cp..cp..c=1 angles were varied simultaneously in the scan, since the second exo- ring angle at the substituted cp atom has the same atom types. As a first approximation, the fitted force constants were simply halved.

This process was repeated for the entire set of intramolecular bonds, angles and dihedral potentials for the fragment. The parameters were then added to the FF and this was used to recalculate the FF estimate for the conformational energies for each of the corresponding scans. Since the energy calculated for each geometry was dependent on all the FF parameters, plotting of curves to compare B3LYP/6-31G(d,p) and FF energies for each scan showed some disagreement. Further refinement of the parameters was undertaken by cycling between fitting for individual curves, and regenerating the entire set of scan data. To compare parameter sets, we used the sum of squares:



S

_

1

N

E

_E







where i was an index for the N scan conformations, and the energies, Ei, were

taken from the methods indicated by the superscript labels. The process of refining the potentials was carried out with the aim of minimising this sum of squares and continued until a self-consistent parameter set was obtained. A plot including the resulting data for the example cp..cp..c=1 angle potential was included in Figure 3.5b and it is clear that the agreement with the B3LYP reference data was very good. A full list of the parameters can be found in Appendix A.2, A.3, and A.4.

Some of the potentials fitted in this way were already present in the PCFF parameter set, most notably the aromatic ring carbons. Figure 3.6 gives a comparison of the new CSFF cp..cp bond, cp..cp..cp..cp dihedral potentials, and the original PCFF forms. The plot in Figure 3.6a shows that the cp..cp bond distance was shorter in CSFF than in the PCFF version by 0.033 Å (1.384 cf. 1.417 Å). Earlier DFT calculations using a B3LYP density functional and the TZ2P basis set on benzene18 _{gave 1.403 Å and 1.391 Å, while our method gives a C-C}

distance in benzene of 1.397 Å. The slight shortening of the aromatic C-C distance is also consistent with the available X-ray diffraction data on the cage structures, for example the average aromatic C-C distance in CC1α is 1.400 Å.1

Near to their respective minima, the PES of the cp..cp bond for PCFF and CSFF potentials was similar, the width of the potential well at 2 kcal mol-1 _{being 0.133}

Å (CSFF) compared to 0.130 Å (PCFF), so that near the optimal bond lengths the two FF have a similar bond stiffness.

Figure 3.6 – Comparison of CSFF (solid lines) and PCFF (dashed lines) terms for aromatic carbon atoms. a) cp..cp bond stretch and b) cp..cp..cp..cp dihedral potential.

The cp..cp..cp..cp dihedral potential shown in Figure 3.6b shows a stronger dependence on the dihedral angle for the CSFF parameterisation than for the PCFF case. This was one of the contributions that will take part in maintaining the planarity of the conjugated aromatic and imine part of the cage structures and this difference suggested that the CSFF potential would hold in plane more closely than PCFF.

Figure 3.7 shows the dihedral potentials produced for the torsion between the imine group, the phenyl ring, and for the imine bond itself. Figure 3.7a and 3.7b demonstrate that the torsion potentials responsible for the planarity of the phenyl-imine system were basically absent from the PCFF parameterisation. For the B3LYP/6-31G(d,p) and CSFF representations, the rotation by 100° around the cp..cp..c=1..n= torsion has an associated barrier of around 8 kcal mol-1_{, this}

lends some flexibility to the phenyl/imine system, but was still twice as large as the barrier for the unhindered rotation of the sp3 _{carbon bond, exemplified by}

n=..c..c..n= (refer to Appendix A.4.). Comparison of Figures 3.7a and 3.7b shows that the imine bond was parameterised to give a greater energetic penalty for a given dihedral rotation than were the bonds in the aromatic ring. This was consistent with the imine having a greater double bond character than the aromatic cp..cp bonds. 0 2 4 6 8 10 1.2 1.3 1.4 1.5 1.6 E n er g y / k ca l m o l -1 r (cp..cp) / Å 0 2 4 6 8 10 -50 -25 0 25 50 E n er g y / k ca l m o l -1 cp..cp..cp..cp dihedral / °

b)

a)

Figure 3.7 – Comparison of CSFF potential (solid lines), the PCFF potential (dotted lines) and the B3LYP/6-31G(d,p) fragment reference data (crosses) for the dihedral potentials around the imine bond. a) cp..cp..c=1..n= dihedral between phenyl ring and imine and b) c..n=..c=1..cp dihedral around the imine bond itself. c) Comparison of the CC1 cage structure as optimised using PCFF and CSFF potentials and the reference crystal structure. In each case an example cp..cp..c=1..n= dihedral was highlighted in orange. Hydrogen atoms have been omitted for clarity.