Symmetry and Point Groups
In this chapter on "Symmetry and point groups," elements of molecular symmetry and point group are introduced. A transformation matrix for each symmetry operation is added to provide a computational know-how. An introduction to the group representation of
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Symmetry and Point Groups
2.1 Introduction Symmetry plays a vital role in the analysis of the structure, bonding, and spectroscopy of molecules. We will explore the basic symmetry elements and operations and their use in determining the symmetry classification (point group) of different molecules. The symmetry of objects (and molecules) may be evaluated through certain tools known as the elements of symmetry.
2.2 Symmetry Operations and Symmetry Elements A symmetry operation is defined as an operation performed on a molecule that leaves it apparently unchanged. For example, if a water molecule is rotated by 180◦ around a line perpendicular to the molecular plane and passing through the central oxygen atom, the resulting structure is indistinguishable from the original one (Fig. 2.1). A symmetry element can be defined as the point, line or plane with respect to which a symmetry operation is performed. The symmetry element associated with the rotation drawn above is the line, or rotation axis, around which the molecule was rotated. The water molecule is said to possess this symmetry element. Table 2.1 includes the types of symmetry elements, operations and their symbols [2].
Fig. 2.1 Water molecule undergoing rotation by 180◦
K. I. Ramachandran et al., Computational Chemistry and Molecular Modeling DOI: 10.1007/978-3-540-77304-7, ©Springer 2008
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2 Symmetry and Point Groups
Table 2.1 Types of symmetry elements, operations, and their symbols Element
Operation
Symbol
Symmetry plane Inversion center Proper axis Improper axis
Reflection through the plane Inversion: Every point x, y, z translated into −x, −y, −z Rotation about the axis by 360/n 1. Rotation by 360/n degrees 2. Reflection through the plane perpendicular to the rotation axis
σ i Cn Sn
2.3 Symmetry Operations and Elements of Symmetry 2.3.1 The Identity Operation Every molecule possesses at least one symmetry element, the identity. The identity operation amounts to doing nothing to a molecule or a rotation of the molecule by 360◦ and so leaving the molecule completely unchanged. The symbol of the ˆ Let us identity element is E and the corresponding operation is designated as E. assign the coordinates (x1 , y1 , z1 ) to any atom of the molecule. The identity operation does not alter these coordinates. If the coordinates after the operation are designated as(x2 , y2 , z2 ), then we get the following equations: x2 = 1x1 + 0y1 + 0z1 y2 = 0x1 + 1y1 + 0z1
(2.1) (2.2)
z2 = 0x1 + 0y1 + 1z1 .
(2.3)
Or, the identity operation matrix can be represented as: ⎡
⎤ ⎡ ⎤⎡ ⎤ x2 1 0 0 x1 ⎣ y2 ⎦ = ⎣ 0 1 0 ⎦ ⎣ y1 ⎦ z2 z1 0 0 1
(2.4)
Or, the transformation matrix (T ) corresponding to E becomes: ⎡
⎤ 1 0 0 T = ⎣0 1 0⎦ 0 0 1
(2.5)
The identity operation will get the same representation as Eq. 2.4 for a molecule belonging to any point group. We can take internal coordinates of all the atoms (e.g. water) of the molecule for determining the transformation matrix corresponding to the identity operation as shown in Fig. 2.2.
2.3 Symmetry Operations and Elemen
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