Quantum Mechanics for the Dihedral Angles in the [aam-aam] Dimer 29

3.3 FENE Potential

3.3.1 Dihedral Conformation and the FENE Potential

3.3.1.1 Quantum Mechanics for the Dihedral Angles in the [aam-aam] Dimer 29

O O

NH

₂

H

₂

N

(a) aam-aam dihedral (b) aam-aam dihedral

-495.66 -495.658 -495.656 -495.654 -495.652 -495.65 -495.648 -495.646 -495.644 -495.642 -495.64

0 50 100 150 200 250 300 350

E (/au)

Dihedral angle (degrees) DFT/MO6-2X

dihedral-aam-aam dihedral-aam-aam

Figure 3.3: The dihedral used for the aam-aam dimer is shown in (a) and (b) with magenta and green colors, respectively. The 360^◦ point should be equivalent to the O^◦for a constrained dihedral scan, however we executed a relaxed scan.

The value for the dihedral of the starting optimized structure shown in Figure 3.3b is ⁶ dihedral

= 204.1^◦. However, the scan showed there are other conformers with lower energies: ⁶ dihe = 170^◦,

6 dihe= 180^◦and⁶ dihe= 187^◦. Thus, optimizations with these starting geometries were performed and the results are depicted by the blue squares in Figure 3.3c.

After further optimization we found that: ⁶ dihe = 170.0^◦ goes to⁶ dihe= 173.9^◦,⁶ dihe = 180.0^◦ goes to ⁶ dihe = 174.0^◦, ⁶ dihe = 187.0^◦ goes to ⁶ dihe = 179.4^◦. We observed that two structures converge to almost the same dihedral angle and almost the same energy, see Table 3.1. We pick the geometry with lowest energy, i.e. dihedral 173.9^◦.

Table 3.1: Energies for dihedral in the aam-aam Angle (/degrees) Energy (/au)

6 dihe = 173.9^◦ −495.657144

6 dihe = 174.0^◦ −495.657138

We then explored how the different most stable dihedral correlates to the FENE potential terms.

The QM curve for investigating the bond strength is shown in Figure 3.4. We can see that the dihedrals have a different well depth, however both have the same equilibrium value. This is expected considering it represents the same C-C bond as shown in Figure 3.7.

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

1 1.2 1.4 1.6 1.8 2 2.2 2.4

E (/au)

Coordinate distance (A) DFT/MO6-2X

aam-aam-dihedral=173.9 aam-aam-dihedral=179.4

(a) aam-aam dihedrals

-0.192 -0.19 -0.188 -0.186 -0.184 -0.182 -0.18 -0.178 -0.176 -0.174

1.4 1.45 1.5 1.55 1.6 1.65 1.7

E (/au)

Coordinate distance (A) DFT/MO6-2X

aam-aam-dihedral=173.9 aam-aam-dihedral=179.4

(b) aam-aam dihedrals

Figure 3.4: Bond energies for aam-aam with dihedrals 173.9^◦and 179.4^◦

Table 3.2 shows the FENE parameters obtained by fitting to the dihedral terms described. For comparison we also include the FENE parameters obtained from another optimized structure with less stable dihedral (204.1^◦).

As we discussed earlier, the QM results shows that the same distance for the C-C bond should be expected independently of the configuration used. This is captured by the FENEσparameters, i.e.

it is the same for all configurations. On the other hand, our QM results suggested that the different configurations would result in a different well depth, which also captured by the FENE potential and the different values for theparameter.

From this we can conclude that the FENE potential capture the qualitative parts of QM. From this we can conclude that the dihedral configuration used does not influe significatively the result

Table 3.2: FENE parameters obtained for different dihedral for the [aam-aam] dimer Combination Bond σ(˚A) (au) R0 (˚A) K(au/˚A²)

[aam-aam] (204.1^◦) (C-C) 1.55 0.188 2.33 2.35 [aam-aam] (173.9^◦) (C-C) 1.55 0.189 2.33 2.36 [aam-aam] (179.4^◦) (C-C) 1.55 0.191 2.33 2.38

obtained for the FENE terms. The absolute numbers of the FENE potential does not have a physical meaning but the relative quantities does.

3.3.1.2 Quantum Mechanics for the Dihedral Angles in the[xlinker] Monomer

We repeat the process for the different units involved in the polymerization, this section corresponds to the xlinker. The dihedral is shown in Figure 3.5a.

O N H N H O

(a) xlinker dihedral (b) xlinker dihedral

-456.346 -456.344 -456.342 -456.34 -456.338 -456.336 -456.334 -456.332

0 50 100 150 200 250 300 350

E (/au)

Dihedral angle (degrees) DFT/MO6-2X

dihedral-xlinker dihedral-xlinker

Figure 3.5: xlinker dihedral used. (a) The structure used is shown in red and the dihedral angle explored is magenta. (b) 3D representation of the xlinker with the dihedral used shown in green.

The blue dots in (c) are reoptimized structures with the dihedrals shown in green.

We started with the optimized geometry shown in Figure 3.5b. The value for the dihedral of this structure is ⁶ dihedral = 103.4^◦. The results of a full dihedral scan are shown in Figure 3.5b.

The lowest laying conformers have a dihedral value of, ⁶ dihe = 103.4^◦, ⁶ dihe = 90.0^◦, ⁶ dihe = 270.0^◦. Dihedral angles of 90.0^◦and 270.0^◦ should be similar in energy, however we re-optimize the geometries. The results for this second optimization is shown in Figure 3.6c as blue dots. Some of the dihedral angles change slightly while one remains the same. This is, the starting⁶ dihe = 90.0^◦ reconverged to ⁶ dihe = 90.0^◦, while the starting ⁶ dihe = 103.4^◦ reconverged to⁶ dihe = 100.8^◦ and the starting⁶ dihe = 270.0^◦ reconverged to⁶ dihe = 281.3^◦.

Then we scan the bond strength for the most stable dihedral angles with our QM procedure.

The scanned bond is shown in Figure 3.13. The results for the three dihedral angles are show in Figure 3.6. We observe that all the energies, minima and distance to the minima are the same. This is different to the dimer [aam-aam] case where there is a small difference in the depth well.

Therefore, the corresponding xlinker dihedral does not affect the FENE potential parameters.

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15

1 1.2 1.4 1.6 1.8 2 2.2 2.4

E (/au)

Coordinate distance (A) DFT/MO6-2X

dihe=90 dihe=100.8 dihe=281.3

(a) xlinker dihedrals

-0.195 -0.19 -0.185 -0.18 -0.175

1.2 1.3 1.4 1.5 1.6 1.7 1.8

E (/au)

Coordinate distance (A) DFT/MO6-2X

dihe=90 dihe=100.8 dihe=281.3

(b) xlinker dihedrals

Figure 3.6: Bond scan for most stable xlinker dihedrals

Table 3.3: [xlinker] FENE parameters for the FENE potential from QM Combination bond σ(˚A) (au) R0(˚A) K (au/˚A²) [xlinker] (90.0^◦) (N-C) 1.45 0.192 2.175 2.74 [xlinker] (100.8^◦) (N-C) 1.45 0.192 2.175 2.74 [xlinker] (281.3^◦) (N-C) 1.45 0.192 2.175 2.74

3.3.2 Bond Strength and the FENE Potential

This section describes in more detail the QM results for the monomer and dimer bonds involved in the hydrogel polymerization. We will use only one dihedral angle of one optimized geometry since we demonstrated in the previous section that any optimized structure with a given dihedral angle will give FENE terms that do not vary considerably from the the dihedral global minima. The FENE potential is intended for coarse grained systems, which means that many vibrational modes are smeared out, in order to make the calculation faster. Therefore small variation in the for a given interaction will not have a big effect when the full simulation is considered.

Dalam dokumen Design of molecules and materials for applications in clean energy, catalysis and molecular machines through quantum mechanics, molecular dynamics and Monte Carlo simulations (Halaman 50-53)