Angle torsion

Atomistically detailed models account for all atoms. The force field contains additive contributions specified in tenns of bond lengtlis, bond angles, torsional angles and possible crosstenns. It also includes non-bonded contributions as tire sum of van der Waals interactions, often described by Lennard-Jones potentials, and Coulomb interactions. Atomistic simulations are successfully used to predict tire transport properties of small molecules in glassy polymers, to calculate elastic moduli and to study plastic defonnation and local motion in quasi-static simulations [fy7, ( ]. The atomistic models are also useful to interiDret scattering data [fyl] and NMR measurements [70] in tenns of local order. 

The first study in which a full CASSCE treatment was used for the non-adiabatic dynamics of a polyatomic system was a study on a model of the retinal chromophore [86]. The cis-trans photoisomerization of retinal is the primary event in vision, but despite much study the mechanism for this process is still unclear. The minimal model for retinal is l-cis-CjH NHj, which had been studied in an earlier quantum chemisti7 study [230]. There, it had been established that a conical intersection exists between the Si and So states with the cis-trans defining torsion angle at approximately a = 80° (cis is at 0°). Two... 

The second method for representing a molecule in 3D space is to use internal coordinates such as bond lengths, bond angles, and torsion angles. Internal coordinates describe the spatial arrangement of the atoms relative to each other. Figure 2-91 illustratc.s thi.s for 1,2-dichlorocthanc. 

A set of rules determines how to set up a Z-matrix properly, Each line in the Z-matiix represents one atom of the molecule. In the first line, atom 1 is defined as Cl, which is a carbon atom and lies at the origin of the coordinate system. The second atom, C2, is at a distance of 1.5 A (second column) from atom 1 (third column) and should always be placed on one of the main axes (the x-axis in Figure 2-92). The third atom, the chlorine atom C13, has to lie in the xy-planc it is at a distanc e of 1.7 A from atom 1, and the angle a between the atoms 3-1-2 is 109 (fourth and fifth columns). The third type of internal coordinate, the torsion angle or dihedral r, is introduced in the fourth line of the Z-matiix in the sixth and seventh column. It is the angle between the planes which arc... 

Figure 2-91. Internal coordinates of 1,2-dichloroethane bond lengths and r2, bond angle a, and torsion angle r.

Spanned by tbc atoms 4, 2, and 1, and 2, 1, and 3 (tlic ry-planc), Except of the first three atoms, each atom is described by a set of three internal coordinates a distance from a previously defined atom, the bond angle formed by the atom with two previous atoms, and the torsion angle of the atom with three previous atoms. A total of 3/V - 6 internal coordinates, where N is the number of atoms in the molecule, is required to represent a chemical structure properly in 3D space. The number (,3N - 6) of internal coordinates also corresponds to the number of degrees of freedom of the molecule. 

Figure 2-102. Dependence of the potential energy ctirve of n-butane on the torsion angle r between carbon atoins C2 and C3.

Usually, two conformations are regarded as geometrically different if their minimized RMS deviation is equal to or larger than 0.3 A in Cartesian space RMSx z)> or 30 " in torsion angle space (RMSja). respectively. 

Figure 2-107. Derivation of the torsion angle library (TA Library).

Figure 2-108. Derivation of a syrMbolic potential energy function from the torsion angle distribution of a torsion fragment.

Figure 2-108 shows the correspondence between a histogram and the derived empirical energy function for the torsion angle fragment C-N H)-C(H)(H -C. 

Another way is to define an improper torsion angle e- (for atoms 1-2-3-4 in Figure 7-11 in combination with a potential lihe V((r- = fc l-cos2fi.-), which has its minima at <> = 0 and 7t. This of course implies the risk that, if the starting geometry is far from reality, the optimi2 ation will perhaps lead to the wrong local minimum. 

You can include geometric restraints—for interatomic distances, bond angles, and torsion angles—in any molecular dynamics calculation or geometry optim i/.ation. Here are some applications of restrain ts ... 

Evalii atm g average distances, angles, and torsion angles, pins their deviations, can facilitate understanding of detailed molecular properties and functional characteristics. 

Example Yon can monitor improper torsion angles to determine wh ich side of a substrate m olecn le faces the active site of a protein. Select three atoms on the substrate molecule and a fourth in the active site. These atom s define an improper torsion angle. Save th is selection as a named selection. Then observe a plot of this improper torsion angle (in the Molecular Dynam ics Results dialog... 

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