Horizontal symmetry plane

It is useful to note that the same issue does not arise in complexes that have a horizontal symmetry plane. There, a wedge geometry allows L and hj to attain equivalent positions above and below that plane. These conceptual issues have been addressed in the case of oxorhenium(V) complexes by two experimental studies, each of which supports intervention of intermediate(s) that undergo turnstile or trigonal twist mechanisms. In so doing, L and L7 attain equivalent or at least interchangeable positions. These studies are the subjects of the next two sections. 

We now turn back to the character table for D3h and note that an A2 orbital must go into itself on reflection through the horizontal symmetry plane. The effect of this symmetry operation on the individual atomic orbitals is as follows ... 

Group Cnh This group is obtained from Cn by adding a horizontal symmetry plane perpendicular to the axis of rotation. Apart from the n rotations of Cn/ this group contains the n improper rotations including the pure reflection ohCAll these... 

Typical electron momentum densities with (3, — 1) and (3, +1) saddle points at zero momentum are found in MgO and acetylene (HCCH), respectively. A momentum density with a zero momentum (3, — 1) critical point is shown for MgO in Fig. 19.6. In the vertical plane of symmetry /T(p, 0,pj has the structure of two hills separated by a ridge or col, and one sees two local (and global) maxima located symmetrically along the p axis. The plot in the horizontal symmetry plane has the structure of a hill. 

When the rotation around the Si—Si bond occurs, the horizontal symmetry plane is lost, and the molecule belongs to 03 point group. 

Another simple amine with a planar (or nearly planar) N atom is tri(/so-propyl)amine. See Fig. 15.4. The molecule has a threefold symmetry axis through the N atom. In addition there is a symmetry plane through the N atom and the three ligating C atoms. Since we use the symmetry axis to define the vertical direction, the symmetry plane is horizontal. Note that there is no symmetry axis along the N-C bonds. Molecules with a threefold axis and a horizontal symmetry plane (and no more symmetry) are said to have Csf, symmetry. 

In both molecules the coordination polyhedron of the P atom is a trigonal bipyramid or tbp. The phosphorus and the two axial halogen atoms define a vertical threefold symmetry axis, the three equatorial halogen atoms a horizontal symmetry plane. In addition to the threefold symmetry axis and the horizontal symmetry plane, each molecule has a twofold symmetry axes through each of the three equatorial P-X bonds. The symmetry of a molecules with one n-fold symmetry axis and n twofold symmetry axes perpendicular to it are denoted by D . The presence of a horizontal symmetry plane is indicated by the additional subscript h. The symmetry of PF5 or PCI5 then, is 1)3. ... 

This is certainly the case when a group is centrosymmetric, i.e., when it contains an inversion centre. Since the inversion operation commutes with all operations, a centrosymmetric group can be written as the direct product C x Hrot. where Q = E,i. However, direct product groups are not limited to centrosymmetry. In the group Dsh, for example, the horizontal symmetry plane forms a separate conjugacy class, which means that it commutes with all the operations of the group. It thus... 

As a final application, we discuss an example of a molecular radical, where more symmetry is present than the eye meets. The triphenylmethyl radical, C19H15, is a planar, conjugated, hydrocarbon-radical, with 19 tt-electrons. The molecular point group for the planar configuration is >3a, but, since all valence -orbitals are antisymmetric with respect to the horizontal symmetry plane, the relevant symmetry of the valence shell is only as seen from Fig. 4.10. The molecular symmetry group distributes the 19 atoms over five trigonal orbits of atoms that, under C3 , can solely be permuted with partners in the same orbit. 

The zx and xy planes correspond to the vertical and horizontal symmetry planes of the oxirane molecule, respectively. The broken line is the projection of the Ar-cm line onto the x-z plane. 6 angle between the normal (z) to the ring-plane and Rem ... 

The surface plot of n( p) in the p Pz plane is simply a hill. Symmetry ensures that an identical plot would be obtained for all other vertical planes of symmetry. A similar plot is seen for the horizontal pxPy plane of symmetry. 

Dyvy, = in addition to an TV-fold (vertical) rotation axis there are TV horizontal twofold axes, TV vertical reflection planes and one horizontal reflection plane [TV2/m if TV is odd Nlml/mllm, for short N/mmm, if TV is even]. = symmetry of a cylinder [oo/m2/ffl, for short oo/m m or m]. 

We always choose the z axis of the coordinate system as coinciding with the highest-order (proper or improper) rotation axis of the molecule. A symmetry plane that contains this axis is called a av plane a symmetry plane perpendicular to this axis is called a ah plane (where v and h stand for vertical and horizontal). 

We have now reached a point of departure in the process of adding further symmetry elements to a C axis. We shall consider (1) the addition of different kinds of symmetry planes to the C axis only, and (2) the addition of symmetry planes to a set of elements consisting of the C axis and the n C2 axes perpendicular to it. In the course of this development it will be useful to have some symbols for several kinds of symmetry planes. In defining such symbols we shall consider the direction of the C axis, which we call the principal axis or reference axis, to be vertical. Hence, a symmetry plane perpendicular to this axis will be called,a horizontal plane and denoted ah. Planes that include the C axis are generally called vertical planes, but there are actually two different types. In some molecules all vertical planes are equivalent and are symbolized av. In others there may be two different sets of vertical planes (as in PtClJ" cf. page 32), in which case those of one set will be called ov and those of the other set crrf, the d standing for dihedral. It will be best to discuss these differences more fully as we meet them. 

Linear molecules with a center of symmetry, such as BeF, and all of the X, molecules (Fig. 3.I4e-g), possess a horizontal mirror plane and an infinite number of C2 axes perpendicular to the principal axis and thus have Oxh symmetry. 

FIGURE 4. (a) Coordinate system for C=0 group, (b) Relevant n-orbital of ketone carbonyl n - tt transition. The vertical XZ plane is a nodal plane for the n-orbital that bisects the R—C—R angle and lies perpendicular to the R(R )C=0 local symmetry plane, (c) Relevant w -orbital of ketone n - tt transition. The horizontal YZ plane is a nodal plane for the tt orbital and lies on the R(R )C=0 local symmetry plane, (d) Carbonyl oxygen, as viewed looking down the Z-axis from O to C, showing circular movement of electron from the n to the tt orbital... 

The number of classes, equal to 5, is derived considering all the possible conditions for chirotopicity of the catalytic sites corresponding to L. If they are not chirotopic, i.e. if they are achirotopic (e.g. they are bisected by a horizontal mirror plane), there are two possibilities only the two sites are equal (class I catalysts) or different from each other (class II catalysts). If, on the contrary, they are chirotopic, three possibilities exist the two catalytic sites are homotopic (equal) - related by a twofold symmetry axis (class III catalysts), enan-tiotopic - related by a vertical mirror plane (class IV catalysts) or diastereotopic (different from each other) - no symmetry element is present (class V catalysts). As a consequence, only five classes of metallocene catalysts may exist if interconversion among stereoisomers is not taken into account [122]. 

Symmetry operations in the point group D3h (6m2), applicable to a triangular bipyramid of that symmetry. The corners of the bipyramid are labeled A, B, C, D, E. The symmetry operations are two vertical threefold axes C3 (clockwise rotation by 120°) and C3 (counterclockwise rotation by 120°, or counterclockwise by 240°), three horizontal twofold axes C2, C2, and C2", and three vertical symmetry planes av, [Pg.395]

The Re2Cl ion has three different kinds of symmetry plane [see Fig. A5-4(b)]. There is a unique one that bisects the Re—Re bond and all the vertical edges of the prism. Since it is customary to define the direction of the highest proper axis of symmetry, C4 in this case, as the vertical direction, this symmetry plane is horizontal and the subscript h is used to identify it, ah. There are then two types of vertical symmetry plane, namely, the two that contain opposite vertical edges, and two others that cut the centers of the opposite vertical faces. One of these two sets may be designated cr and [Pg.1309]

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