Internal coordinates symmetry preservation

Other than in the preceding section, the matrix D must now relate the increments of symmetrically equivalent internal (or bond) coordinates with the increment of the respective base internal coordinate. Let us return to the example, Eq. 56, and let the matrix D, as depicted there, be a portion of the whole matrix D. The row and column headings of each 3x3 block of D for a particular atom now specify the increments of bond length Ar, bond angle Atp, and dihedral (or torsional) angle AO used for the attachment of this atom to the chain, instead of the increments of Cartesian coordinates of the atom. It is assumed that the description by internal coordinates has been chosen for the best possible preservation of molecular symmetry. The three symmetrically equivalent atoms a, a, and a" will then be attached to an interatomic bond along thejr-axis (or, by convention, to an equivalent... 

Fig. 2. Symmetry-unique portion of the pseudorotation path for H3, outer plates, around the Dih conical intersection point, center plates, as originally discussed by Longuet-Higgins [4]. Each point on the path is the constructed from (cos 0 Qy -h sinO Qx). Present drawing from Ref. [15]. The third internal coordinate, not shown, is the -preserving ring breathing mode, Qs...

Here the nonrelativistic (Coulomb, r] = 2) and relativistic (Coulomb - -spin-orbit, = 3) seams of conical intersection are compared. In Cg symmetry, there are 5 symmetry preserving internal coordinates. The seam conditions, the conditions that require X to be a point of conical intersection, define r] internal coordinates. Thus there is a many-to-one association of nonrelativistic and relativistic seam points. This many-to-one association can be reduced to one-to-one association by comparing the lowest energy point of conical intersection for a fixed-value of the H -H distance, — H ), see Fig. 8(a) for atom labelling. Below we show that this approach results in meaningful comparisons. 

An atom (or a molecule) is said to be polarized with respect to a particular coordinate system (e.g., laboratory or parent molecule body fixed system) if any of its internal angular momenta are either aligned or oriented. The limit of a non-polarized fragment corresponds to equal populations in all the Mj sublevels of a given J. The system exhibits alignment when the populations are unequal in the Mj levels for a given J but cylindrical symmetry is preserved by equal populations in +Mj and —Mj levels. Orientation corresponds to the case where the +Mj and —Mj populations are not equal for any one of the Mj values (see Fig. 7.11). 

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