Symmetry plane reflection through

We consider four kinds of symmetry elements. For an n fold proper rotation axis of symmetry Cn, rotation by 2n f n radians about the axis is a symmetry operation. For a plane of symmetry a, reflection through the plane is a symmetry operation. For a center of symmetry /, inversion through this center point is a symmetry operation. For an n-fold improper rotation axis Sn, rotation by lir/n radians about the axis followed by reflection in a plane perpendicular to the axis is a symmetry operation. To denote symmetry operations, we add a circumflex to the symbol for the corresponding symmetry element. Thus Cn is a rotation by lit/n radians. Note that since = o, a plane of symmetry is equivalent to an S, axis. It is easy to see that a 180° rotation about an axis followed by reflection in a plane perpendicular to the axis is equivalent to inversion hence S2 = i, and a center of symmetry is equivalent to an S2 axis. 

There are three symmetry operations each involving a symmetry element rotation about a simple axis of symmetry (C ), reflection through a plane of symmetry (a), and inversion through a center of symmetry (i). More rigorously, symmetry operations may be described under two headings Cn and Sn. The latter is rotation... 

Quantity Relative symmetry of A and A Reflection through xz or yz plane Reflection through jcy plane Time reversal... 

SOLUTION idxy is antisymmetric for reflection only throngh the x, z and y, z planes. 3dyz is antisymmetric for reflection only through the x, z and x, y planes. Hence, these orbitals differ in their symmetries for reflection through the y, z and x, y planes. 

EXAMPLE 13-1 BH3, like ammonia, has a three-fold symmetry axis. However, BH3 is planar. As a result, there is an extra symmetry operation—reflection through the molecular plane—that does not move any nuclei. Thus, both the identity operation and this reflection leave all nuclei unmoved. Is this an example of redundant operations ... 

Symmetry for reflection through a plane perpendicular to the polymer axis is not useful for this analysis. Symmetry for reflection through a plane containing the atoms divides the COs into a and tt types. Only a-it crossings are allowed. 

A model of F is identical with its mirror image. It is achiral, although it does not have a plane of symmetry, due to the presence of a center of symmetry that is located between C2 and C3. A center of symmetry, like a plane of symmetry, is a reflection symmetry element. A center of symmetry involves reflection through a point a plane of symmetry requires reflection about a plane. A model of the mirror image of S (structure T) is not identical to S, but is a conformational enantiomer of S. They can be made identical 1 rotation about the C2, C3 bond. Since S and T are conformational enantiomers, the two will be present in equal amounts in a solution of this configurational stereoisomer. Both conformation S and conformation T are chiral and therefore should rotate the plane of plane polarized light. 

In the case of a planar linear polyene the molecular orbitals may be classified according to their symmetry under reflection through the molecular plane, a orbitals are symmetric under reflection, n orbitals are antisymmetric. In discussing the low energy excited singlet states of linear polyenes it is appropriate to limit attention to many electron states built from orbitals with n symmetry (the 7C-electron approximation). In the simplest treatments these 7C-electron orbitds may be written as linear combinations of 2p-7c atomic orbitals, one per sp2 hybridized carbon atom. In the case of an all-trans planar polyene which has 2/m symmetry, these 7C-electron orbitals will transform as Ay or Bg. Since the number of electrons is even, the many electron states will have either Ag or Bu symmetry. 

Planes of symmetry. Planes through which there is reflection to an identical point in the pattern. In a lattice there may be a lateral movement parallel to one or more axes (glide plane). 

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of ineitia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes a,b,c). In order to detemiine the parity of the molecule through inversions in SF, we first rotate all the electrons and nuclei by 180° about the c axis (which is peipendicular to the molecular plane) and then reflect all the electrons in the molecular ab plane. The net effect is the inversion of all particles in SF. The first step has no effect on both the electronic and nuclear molecule-fixed coordinates, and has no effect on the electronic wave functions. The second step is a reflection of electronic spatial coordinates in the molecular plane. Note that such a plane is a symmetry plane and the eigenvalues of the corresponding operator then detemiine the parity of the electronic wave function. 

Finally, for linear molecules in Z states, the wavefunctions can be labeled by one additional quantum number that relates to their symmetry under reflection of all electrons through a ay plane passing through the molecule s Coo axis. If F is even, a + sign is appended as a superscript to the term symbol if F is odd, a - sign is added. 

Recall that the symmetry labels e and o refer to the symmetries of the orbitals under reflection through the one Cy plane that is preserved throughout the proposed disrotatory closing. Low-energy configurations (assuming one is interested in the thermal or low-lying photochemically excited-state reactivity of this system) for the reactant molecule and their overall space and spin symmetry are as follows ... 

The hi orbitals would maintain their identity going to a" symmetry. Thus Bi and A2 (both A" in Cs symmetry and odd under reflection through the molecular plane) can mix. The system could thus follow the A2 component of the C( P) + H2 surface to the place... 

Although the ultimate criterion is, of course, nonsuperimposability on the mirror image (chirality), other tests may be used that are simpler to apply but not always accurate. One such test is the presence of a plane of symmetry A plane of symmetry (also called a mirror plane) is a plane passing through an object such that the part on one side of the plane is the exact reflection of the part on the other side (the plane acting as a mirror). Compounds possessing such a plane are always optically inactive, but there are a few cases known in which compounds lack a plane of symmetry and are nevertheless inactive. Such compounds possess a center of symmetry, such as in a-truxillic acid, or an alternating axis of symmetry as in 1. A... 

Figure 9.9. The -n bases orbitals for a 2, + 2, cycloaddition S, A indicate symmetric or antisymmetric reflection through the symmetry plane

First, it is apparent that reflection through the xz plane, indicated by transforms H into H". More precisely, we could say that H and H" are interchanged by reflection. Because the z-axis contains a C2 rotation axis, rotation about the z-axis of the molecule by 180° will take H into H" and H" into H, but with the "halves" of each interchanged with respect to the yz plane. The same result would follow from reflection through the xz plane followed by reflection through the yz plane. Therefore, we can represent this series of symmetry operations in the following way ... 

Figure 3.50 Carbon-carbon NBOs of CH2=CHNH2 at

For information about point groups and symmetry elements, see Jaffd, H. H. Orchin, M. Symmetry in Chemistry Wiley New York, 1965 pp. 8-56. The following symmetry elements and their standard symbols will be used in this chapter An object has a twofold or threefold axis of symmetry (C2 or C3) if it can be superposed upon itself by a rotation through 180° or 120° it has a fourfold or sixfold alternating axis (S4 or Sh) if the superposition is achieved by a rotation through 90° or 60° followed by a reflection in a plane that is perpendicular to the axis of the rotation a point (center) of symmetry (i) is present if every line from a point of the object to the center when prolonged for an equal distance reaches an equivalent point the familiar symmetry plane is indicated by the symbol a. 

An example is the (110) plane of III-V semiconductors, such as GaAs(llO). The only nontrivial symmetry operation is a mirror reflection through a line connecting two Ga (or As) nuclei in the COOl] direction, which we labeled as the X axis. The Bravais lattice is orthorhombic primitive (op). In terms of real Fourier components, the possible corrugation functions are... 

Symmetry operators leave the electronic Hamiltonian H invariant because the potential and kinetic energies are not changed if one applies such an operator R to the coordinates and momenta of all the electrons in the system. Because symmetry operations involve reflections through planes, rotations about axes, or inversions through points, the application of such an operation to a product such as H / gives the product of the operation applied to each term in the original product. Hence, one can write ... 

The equivalent symmetry element in the Schoenflies notation is the improper axis of symmetry, S which is a combination of rotation and reflection. The symmetry element consists of a rotation by n of a revolution about the axis, followed by reflection through a plane at right angles to the axis. Figure 1.14 thus presents an S4 axis, where the Fi rotates to the dotted position and then reflects to F2. The equivalent inversion axes and improper symmetry axes for the two systems are shown in Table 1.1. 

Improper rotation axis. Rotation about an improper axis is analogous to rotation about a proper synunetry axis, except that upon completion of the rotation operation, the molecule is mirror reflected through a symmetry plane perpendicular to the improper rotation axis. These axes and their associated rotation/reflection operations are usually abbreviated X , where n is the order of the axis as defined above for proper rotational axes. Note that an axis is equivalent to a a plane of symmetry, since the initial rotation operation simply returns every atom to its original location. Note also that the presence of an X2 axis (or indeed any X axis of even order n) implies that for every atom at a position (x,y,z) that is not the origin, there will be an identical atom at position (—x,—y,—z) the origin in such a system is called a point of inversion , since one may regard every atom as having an identical... 

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