Basic Symmetry Operations

A symmetry operation is a transformation of a system that leaves an object in an indistinguishable position. For molecular systems, we need be concerned with only two types of symmetry operations proper rotations (C ) and improper rotations (S ). A C is a rotation around an axis by (360/ )° that has the net effect of leaving the position of the object unchanged. Thus, a C2 is a 180° rotation, a C3 a 120° rotation, and so on. These are termed proper rotations, because it is actually physically possible to rotate an object by 180° or 120°. Some examples are shown below, with the atoms labeled only to highlight the operation. 

In contrast, improper rotations are not physically possible. An S involves a rotation of (360/ )°, combined with a reflection across a mirror plane that is perpendicular to the rotation axis (see examples on the next page). Note that Si is equivalent to just a mirror reflection (denoted with a a), while Sz is equivalent to a center of inversion (denoted with an i). The Cj operation also exists. It leaves an object completely unmoved and is also termed the identity operation, sometimes symbolized as . An internal cr plane that includes a C2 axis is designated a (Ty, while a a plane perpendicular to a C2 axis is designated Oh- 

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Fig. 1. The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. The chiral vector OA or Cf, = nOf + tnoi defined on the honeycomb lattice by unit vectors a, and 02 and the chiral angle 6 is defined with respect to the zigzag axis. Along the zigzag axis 6 = 0°. Also shown are the lattice vector OB = T of the ID tubule unit cell, and the rotation angle 4/ and the translation r which constitute the basic symmetry operation R = (i/ r). The diagram is constructed for n,m) = (4,2).

The symmetry groups for the chiral tubules are Abelian groups. The corresponding space groups are non-symmorphic and the basic symmetry operations... 

CHECKPOINT The five basic symmetry operations we ve just seen account for every possibie way thait equivaient attorns in a moiecuie can be exchanged, without aitering which atoms are connected to which. We need these symmetry operations in order to determine how the shape of the moiecuie may iimit the possibie moiecuiar orbitais and vibraitionai motions, and aiso how the moiecuie interacts with radiaition. 

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

SymApps converts 2D structures From the ChemWindow drawing program into 3D representations with the help of a modified MM2 force field (see Section 7.2). Besides basic visualization tools such as display styles, perspective views, and light source adjustments, the module additionally provides calculations of bond lengths, angles, etc, Moreover, point groups and character tables can be determined. Animations of spinning movements and symmetry operations can also he created and saved as movie files (. avi). 

The basic idea of symmetry analysis is that any basis of orbitals, displacements, rotations, etc. transforms either as one of the irreducible representations or as a direct sum (reducible) representation. Symmetry tools are used to first determine how the basis transforms under action of the symmetry operations. They are then used to decompose the resultant representations into their irreducible components. 

It is then possible to represent the above-mentioned symmetry operation by the 3x3 matrix of Equation (7.1). In a more general way, we can associate a matrix M with each specific symmetry operation R, acting over the basic functions x, y, and z of the vector (x,y, z). Thus, we can represent the effect of the 48 symmetry operations of group Oh (ABe center) over the functions (x, y, z) by 48 matrices. This set of 48 matrices constitutes a representation, and the basic functions x, y, and z are called basis functions. 

Such a set of eigenfunctions must form the basis for a representation of the symmetry group of the Hamiltonian, because for every symmetry operation S, Tipi = pi implies that H Spi) = Sp>i) and hence that the transformed wave function Spi must be a linear combination of the basic set of eigenfunctions (/ ,... Pn-... 

A group is a collection of elements that are interrelated according to certain rules. We need not specify what the elements are or attribute any physical significance to them in order to discuss the group which they constitute. In this book, of course, we shall be concerned with the groups formed by the sets of symmetry operations that may be carried out on molecules or crystals, but the basic definitions and theorems of group theory are far more general. 

The procedure just used, although routine and reliable, is lengthy, particularly for the two-dimensional representation. The results could have been obtained with less labor by recognizing that the rotational symmetry, the behavior of the SALCs upon rotation about a principal axis, of order 3 or more, alone fixes their basic form. Their behavior under the other symmetry operations is a direct consequence of the inherent symmetry of an individual orbital toward these operations, ah or a C2 passing through it, being added to the symmetry properties under the pure rotations about the principal axis. Let... 

A frequently asked question is What are the differences between nuclear physics and nuclear chemistry Clearly, the two endeavors overlap to a large extent, and in recognition of this overlap, they are collectively referred to by the catchall phrase nuclear science. But we believe that there are fundamental, important distinctions between these two fields. Besides the continuing close ties to traditional chemistry cited above, nuclear chemists tend to study nuclear problems in different ways than nuclear physicists. Much of nuclear physics is focused on detailed studies of the fundamental interactions operating between subatomic particles and the basic symmetries governing their behavior. Nuclear chemists, by contrast, have tended to focus on studies of more complex phenomena where statistical behavior is important. Nuclear chemists are more likely to be involved in applications of nuclear phenomena than nuclear physicists, although there is clearly a considerable overlap in their efforts. Some problems, such as the study of the nuclear fuel cycle in reactors or the migration of nuclides in the environment, are so inherently chemical that they involve chemists almost exclusively. 

It is equivalent to describe the symmetry class of the tetrahedron as 3/2-m or 3/4. The skew line relating two axes means that they are not orthogonal. The symbol 3/2-m denotes a threefold axis, and a twofold axis which are not perpendicular and a symmetry plane which includes these axes. These three symmetry elements are indicated in Figure 2-50. The symmetry class 3/2-m is equivalent to a combination of a threefold axis and a fourfold mirror-rotation axis. In both cases the threefold axes connect one of the vertices of the tetrahedron with the midpoint of the opposite face. The fourfold mirror-rotation axes coincide with the twofold axes. The presence of the fourfold mirror-rotation axis is easily seen if the tetrahedron is rotated by a quarter of rotation about a twofold axis and is then reflected by a symmetry plane perpendicular to this axis. The symmetry operations chosen as basic will then generate the remaining symmetry elements. Thus, the two descriptions are equivalent. 

There are analogous phenomena for all kinds of molecular motion which may be symmetric and antisymmetric with respect to the various symmetry operations of the molecular point group. The two main kinds of motion in a molecule are nuclear and electronic. The nuclear motion may be translational, rotational and vibrational (Chapter 5). The electronic motion is basically the changes in the electron density distribution (Chapter 6). 

The Pentagonal Dodecahedron and the Icosahedron. These bodies (Fig. A5-8) are related to each other in the same way as are the octahedron and the cube, the vertices of one defining the face centers of the other, and vice versa. Both have the same symmetry operations, a total of 120 We shall not list them in detail but merely mention the basic symmetry elements six C5 axes ten C3 axes, fifteen C2 axes, and fifteen planes of symmetry. The group of 120 operations is designated Ih and is often called the icosahedral group. 

The commutation relation between two arbitrary operators is not conserved upon transformation to effective operators by any of the definitions. Many state-independent effective operator definitions preserve the commutation relations involving // or a constant of the motion, as well as those involving operators which are related to P in a special way, for example, A with [P, 4] = 0. Many state-dependent definitions also conserve these special commutation relations. However, state-dependent definitions are not as convenient for formal and possibly computational reasons. The most important preserved commutation relations are those involving observables, since, as discussed in Section VII, they ensure that the basic symmetries of the system are conserved in effective Hamiltonian calculations. 

A word about notation. For the points groups we shall generally use the Schonflies notation. In this notation basically the same symbols are used for the point groups as for the symmetry operations. Whenever there is any doubt as to the meaning we shall use a caret (example Cn) to indicate a symmetry operation. 

The ability of a body to coincide with itself in its different positions regarding a coordinate system is called its symmetry. This property reveals itself in iteration of the parts of the body in space. The iteration may be done by reflection in mirror planes, rotation about certain axes, inversions and translations. These actions are called the symmetry operations. The planes, axes, points, etc., are known as symmetry elements. Essentially, mirror reflection is the only truly primitive symmetry operation. All other operations may be done by a sequence of reflections in certain mirror planes. Hence, the mirror plane is the only true basic symmetry element. But for clarity, it is convenient to use the other symmetry operations, and accordingly, the other aforementioned symmetry elements. The symmetry elements and operations are presented in Table 1. 

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