Internal torsion angle

Methods using internal (torsion angles) coordinate frames, which are very common in organic chemistry, are not always applicable to inorganic systems. 

The internal torsion angles, which are characteristic for the furanose ring of deoxyribose (noted by T0-T4) determine the appearance of endo- and exo-conformations of deoxyribose. The Watson-Crick model corresponds to B-DNA type, having the deoxyribose in the C-2 -endo conformational form. The secondary structure of B-DNA... 

There will be relaxation processes on many time scales. The most rapid changes will occur between chemically bonded atoms. Chemical bonds lhat have been stretched or compressed will relax to some excited vibrational level and eventually will therm ize and return to the ground vibrational state. Bond angles will also thermalize and return to the ground vibrational state. Internal torsional angles will initially thermalize within the rotational isomeric state that has been achieved after the deformation. The internal relax-... 

The chain molecule solution will be characterized by a set of rotational isomeric states. After the initial thermalization, this set will still be in a nonequilibrium state due to the overall shear deformation. For the internal torsional angle distribution to relax to equilibrium, it is necessary for the solution liquid structure to relax and the solvent molecule distribution to reach equilibrium. In a concentrated polymer solution, these processes are highly coupled. Rotational isomeric state changes depend on both internal potentials and the local viscosity and are often in the nanosecond range, well above the glass transition for the solution. Total stress relaxation cannot occur any faster than the chains can change their local rotational isomeric states. 

The basis set for the spin coordinates, cr)) is the space of spin transitions and is defined elsewhere [2, 9, 82]. For the stochastic part we make the standard choice of employing Wigner rotation matrices for the global rotation and complex exponentials for the internal torsional angle, that is, A) = LMK) ) with,... 

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. 

Internal rotation in isooctane (2,2,4-trimethylpentane) creates a large number of staggered conformations. However, only rotation about the C3-C4 bond produces conformations with different structures. Plot the energy of isooctane (vertical axis) vs. HCCCtBu torsion angle, i.e., about the C3-C4 bond (horizontal axis). How many minimum energy structures are there Are they all fully staggered Draw Newman projections that show the conformation of these structures. How does steric repulsion affect isooctane conformation ... 

Another way of removing the six translational and rotational degrees of freedom is to use a set of internal coordinates. For a simple acyclic system these may be chosen as 3N — I distances, 3N — 2 angles and 3N -3 torsional angles, as illustrated in the construction of Z-matrices in Appendix E. In internal coordinates the six translational and rotational modes are automatically removed (since only 3N — 6 coordinates are defined), and the NR step can be formed straightforwardly. For cyclic systems a choice of 3A — 6 internal variables which span the whole optimization space may be somewhat more problematic, especially if symmetry is present. 

The transformation from a set of Cartesian coordinates to a set of internal coordinates, wluch may for example be distances, angles and torsional angles, is an example of a non-linear transformation. The internal coordinates are connected with the Cartesian coordinates by means of square root and trigonometric functions, not simple linear combinations. A non-linear transformation will affect the convergence properties. This may be illustrate by considering a minimization of a Morse type function (eq. (2.5)) with D = a = ] and x = AR. 

The trisulfane molecule exists as two conformers which have been termed as cis- and trans-HzSi. While the trans-form is a helical molecule of C2 symmetry with the motif ++ (or — for the enantiomer), the cfs-form is of Q symmetry with the motif +- (identical to -+). Both forms have been detected by rotational spectroscopy [17, 45, 46]. The motif gives the order of the signs of the torsion angles at the SS bonds. The geometrical parameters [17] are presented in Table 4. The trans-isomer is by only 1 kj mol more stable than the cfs-form but the barrier to internal rotation from tmns to cis is 35 kJ mor [46]. The dipole moments were calculated by ab initio MO theory at the QCISD/TZ+P level as 0.68 D (trans) and 2.02 D (cis) [46]. For geometrical parameters of cis- and trans-trisulfane calculated at the MP2/6-311++G> > level, see [34]. 

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