Symmetry Elements

10 Key excerpts on "Symmetry Elements"

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  Molecular Symmetry

  • David Willock, David J. Willock(Authors)
  • 2009(Publication Date)
  • Wiley

    (Publisher)

  To achieve this we look for features in the geometry of a molecule that give rise to its symmetry. The most easily recognized of these features, or Symmetry Elements , are rotational axes (lines of symmetry) and mirror planes (planes of symmetry). These will be discussed in the remaining sections of this chapter, along with the inversion centre, which is a point of symmetry. There are other Symmetry Elements and operations that are possible, and we will meet these in Chapter 2. The Symmetry Elements imply that 6 Molecular Symmetry there are symmetry operations: actions that can be carried out which appear to leave the molecule unchanged. If a molecule has multiple Symmetry Elements then there will be at least one point in space which lies within them all. For example, Figure 1.8 shows that all the rotation axes of ferrocene meet at the central point where the Fe atom is located. For this reason, the symmetry of molecules is often referred to as point group symmetry . The idea of this book is to introduce the ideas of point group symmetry and its application in vibrational spectroscopy and the molecular orbital (MO) description of chemical bonding. In periodic systems (such as crystal structures), other symmetries exist to do with trans-lation between equivalent molecules. See the Further Reading section at the end of this chapter for a book on this topic. 1.2 Symmetry Elements and Operations 1.2.1 Proper Rotations: C n The geometric properties of shapes that make them symmetric can be classified by their Symmetry Elements. The validity of a symmetry element can be checked by carrying out the corresponding operation and then comparing the object with the starting point. For example, imagine constructing an axis for a water molecule which runs through the oxygen atom, bisecting the H O H angle, with the axis in the plane of the molecule. This con-struction is shown in Figure 1.6, which also illustrates the result of rotating the molecule by 180 ◦ around the axis.

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  Condensed Matter Optical Spectroscopy

  An Illustrated Introduction

  • Iulian Ionita(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  Symmetry Elements are geometric entities such as plane, line, and point (Figure 1.2). Every symmetry operation is defined with respect to an element, but at the same time, every symmetry element exists only if the appropriate operation (which leaves the object unchanged) exists. Thus, we will discuss operations and elements together. Symmetry operations are also geometric transformations, which, applied to a molecule, change only the positions of some atoms (move an atom in place of another) without changing the size, shape, and position of the molecule . In other words, the final state of the molecule is the same as the initial state except the labeled numbering of atoms. There are four types of Symmetry Elements and related operations as shown in Table 1.1. 1.1.2 SYMMETRY PLANES AND REFLECTIONS As we have already mentioned, the symmetry plane (mirror plane) should be chosen for the purpose of passing through the molecule, or through the box containing the molecule. Thus, the reflection in the mirror plane must transform the molecule in itself, that is, the image will be located in the same position as the original molecule. We use the symbol σ for the FIGURE 1.1 Water molecule “looking” in the mirror. Vertical plane Line Line Horizontal plane Point FIGURE 1.2 Symmetry Elements are geometrical entities. TABLE 1.1 Symmetry Elements and Appropriate Symmetry Operations in Molecular Symmetry Symmetry Element Element Symbol Symmetry Operation Operation Symbol 1 Plane, can be horizontal, vertical, dihedral σ Reflection Σ 2 Center of inversion i Inversion of all atom coordinates through the center I 3 Proper axis c Rotation around the axis C n k 4 Improper axis s Rotation around the axis followed by reflection in the plane perpendicular to the axis S n k 5 Identity E Source: Adapted from Cotton, A. F. 1990. Chemical Applications of Group Theory , 3rd edition. New York: John Wiley & Sons. Inc.

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  Molecular Symmetry and Group Theory

  Approaches in Spectroscopy and Chemical Reactions

  • R. C. Maurya, J.M. Mir(Authors)
  • 2019(Publication Date)
  • De Gruyter

    (Publisher)

  1 Symmetry Elements and symmetry operations: molecular symmetry 1.1 Introduction Symmetry is a very important and fascinating property of molecules. The concept of molecular symmetry is important from the view point of their applications to chemical problems, viz., molecular orbitals in polyatomic molecules, and thus their structure and bonding, crystal field and molecular orbital theory of complex compounds, electronic and vibrational spectra of simple and complex compounds and so on. In order to use the symmetry properties of molecules in solving chemical problems as mentioned above, it is necessary to have some acquaintance with the branch of mathematics known as the Group Theory. It can be applied to any set of elements, which obey the necessary conditions to be called a Group. A Group in the mathematical sense is a collection of elements having certain properties in common, which enable a wide variety of algebraic manipulations to be carried out. The elements could be numbers, matrices, symmetry operations or Symmetry Elements. All the symmetry operations present in a molecule forms a Group. Moreover, each symmetry operation in a molecule can be represented by a matrix. Hence, the symmetry operations/Symmetry Elements of a molecule constitute what is known as a mathematical group. Hence, the formal treatment of the concept of molecular symmetry is the subject matter of this chapter. 1.2 Molecular symmetry: in non-mathematical and geometrical sense The word Symmetry is derived from the Greek word Summetria meaning “similar measure.” It implies that each part of an object is in harmony with each other and is well balanced. Hence, the term Symmetry is almost synonymous to beauty because the nature has made most of its creation symmetrical. The Sun, the planets, the human beings, animals and plants are all symmetrical

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  Symmetry and Group theory in Chemistry

  • M Ladd(Author)
  • 1998(Publication Date)
  • Woodhead Publishing

    (Publisher)

  2 Symmetry operations and Symmetry Elements Our torments also may in length of time Become our elements. John Milton (1608-1674): Paradise Lost 2.1 INTRODUCTION: THE TOOLS OF SYMMETRY In order that the concept of symmetry shall be generally useful, it is necessary to develop precisely the tools of symmetry, the symmetry operations and Symmetry Elements appropriate to finite bodies which, for our purposes, are mainly chemical molecules. Then, as a prerequisite to group theory and its applications to chemistry, we shall consider some of the basic manipulations of vectors and matrices that can be used to simpllfy the discussion of symmetry operations and their combinations. There exist two important notations for symmetry, and both of them are in general use. In studying the symmetry of molecules and the applications of group theory in chemistry, we shall make use of the Schonilies notation, as is customary. When we come to consider the symmetry of the extended patterns of atomic arrangements in crystals, the Hermann-Mauguin notation is always to be preferred. Once we have become familiar with symmetry concepts in the first of these notations, the Hermann-Mauguin notation will produce little difficulty. 2.2 DEFINING SYMMETRY OPERATIONS, ELEMENTS AND We follow our statement of symmetry in Section 1.2, and define a symmetry operation as an action that moves a body into a position that is indistinguishable fiom its initial position: it is the action of a symmetry operation that reveals the symmetry inherent in a body. A symmetry operation may be considered to take place with respect to a symmetry element. A symmetry element is a geometrical entity, a plane, a line or a point, which is associated with its corresponding symmetry operation.

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  Gigantic Challenges, Nano Solutions

  The Science and Engineering of Nanoscale Systems

  • Maher S. Amer(Author)
  • 2021(Publication Date)
  • Jenny Stanford Publishing

    (Publisher)

  “somehow” turns out to be simply the operation of any of the following five Symmetry Elements.

  6.2 Symmetry Elements and Their Operations

  For symmetry, we need to have Symmetry Elements and symmetry operations. A symmetry element is a geometrical entity around which a symmetry operation is performed. A symmetry element can be a point, axis, or plane. A symmetry operation is the movement of a body (molecule, crystal, shape, etc.) around such symmetry element. If after the movement the shape appears the same as before, then, the geometry possesses that symmetry element. We will start with describing the five different Symmetry Elements then we will distinguish, further between the Symmetry Elements and operations.

  6.2.1 Identity (E)

  The identity symmetry element exists in everything in the universe. It is usually given the symbol (E) for the German word ‘Einheit”1 meaning unity. Loosely, the word can be translated as “the same” or “identical.”

  1 In crystallography and spectroscopy, the reader will come across many German nomenclature due to the ground-breaking work done by German scientists in these fields.

  6.2.2 Center of Symmetry (i)

  A center of symmetry is a point in space that occupies a midpoint on a line connecting two indistinguishable positions. The center of symmetry is also known as inversion center, hence, the designation “i.” If one connects a line from an atom in a molecule, or generally speaking, a site or position in space, through a center of symmetry, extending the line for the same distance should lead to an equivalent indistinguishable atom, or position. For example, the carbon atom in a CO2 molecule (see Figure 6.1 for example) occupies a center of symmetry. If we consider any of the oxygen atoms, connecting a line from that oxygen to the carbon and extending the line to an equal distance will lead us to the second oxygen atom in the molecule that is indistinguishable from the one we started with. Figure 6.1 illustrates the concept of a center of symmetry. It is important to note that a center of symmetry of a molecule may, or may not be occupied by an atom. For example, both CO2 and C2 H2 molecules possess a center of symmetry. While that center of symmetry is occupied by a carbon atom in the case of CO2 , it is unoccupied in the case of ethene, and actually lies on the midpoint between the two carbon atoms. As we mentioned above, if we operate the center of symmetry operation on one of the oxygen atoms in the CO2 molecules, we will move that atom to the second oxygen atom position. Now what if we operate the center of symmetry on the same oxygen atom twice? This will bring the oxygen atom back to its original (i.e., identical) position. It is clear, then, that operating a center of symmetry element twice (this is mathematically expressed as i2 ) is equivalent to the identity element of symmetry (E). Mathematically this can be expressed as i2 ≡ E.

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  Principles of Inorganic Chemistry

  • Brian W. Pfennig(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  As we shall see, it is the three-dimensional shape of a molecule that dictates its molecular symmetry and we can use a mathematical description of symmetry properties, known as group theory, to describe the structure, bonding, and spectroscopy of molecules. The more symmetric the molecule, the greater its degeneracy, and the simpler the construction of its energy levels. In large mole- cules having a high degree of symmetry, we can therefore take advantage of the symmetrical relationships of the different atoms in molecules to simplify the mathematical solutions to the wave equation. The first step in the application of symmetry to molecular properties is there- fore to recognize and organize all the Symmetry Elements that the molecule pos- sesses. A symmetry element is an imaginary point, line, or plane in the molecule about which a symmetry operation is performed. An operator is a symbol that tells you to do something to whatever follows it. Thus, for example, the Hamiltonian operator is the sum of the partial differential equations relating to the kinetic and potential energy of a system. When we apply the Hamiltonian operator to the wave function of an atom, the solutions correspond to the total energy of the sys- tem. A symmetry operation is a geometrical operation that moves an object about some symmetry element in a way that brings the object into an arrangement that is indistinguishable from the original. By indistinguishable, we mean that the object was returned to an equivalent position, one in which every equivalent part of the object has the same orientation and the same relative position in space as it did in the original. One way of determining whether the object was brought into an indistinguishable arrangement is to close your eyes while somebody else per- forms a symmetry operation on the object.

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  Introduction to Crystallography

  • Donald E. Sands(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  All points on a proper rotation axis or on a mirror plane are stationary, and no new points are generated from these by the action of the operator. In the cases of the center of inversion and the improper rotations, there is a unique point that is left fixed. It is possible to conceive of symmetry operations that leave no point unchanged. If a set of points (or atoms) has such a symmetry element, and the operation is applied to produce a new set of points, then the new set must also contain the symmetry element (otherwise, the new set would be distinguishable from the first, in contradiction to our definition of symmetry). Operation by the new symmetry element will generate still another set of equivalent points, which will also contain the symmetry element. The result of the repeated application of newly generated Symmetry Elements is that an infinite number of equivalent points will be generated for each of our original points. It, therefore, follows that a finite molecule cannot have Symmetry Elements that do not leave at least one point fixed. For this reason, if we want to discuss the symmetry of molecules we need only the elements summarized in Table 2-1, which are referred to as elements of point symmetry. Crystals, however, can have Symmetry Elements that leave no point fixed (since we have agreed to regard our crystalline arrays as infinite), and translational symmetry will be dealt with at length in the following chapter. 2-10 Combinations of Symmetry Elements We have already observed that molecules sometimes possess more than one symmetry element. Every molecule has a C 1 axis, and even the very simple water molecule has, in addition, a C 2 axis and two mirror planes (one of them is the plane of the molecule). We could describe the symmetry of a molecule by listing all of its Symmetry Elements. For CH 4 such a list would contain 24 entries, and SF 6 would require 48 entries

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  Principles of Quantum Chemistry

  • David V. George(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  8-3 MOLECULAR SYMMETRY OPERATIONS Many molecules have a certain amount of symmetry that can be associated with Symmetry Elements. In order to indicate how we can classify these Symmetry Elements we consider, as an example, the water molecule. We shall examine its symmetry properties and show that these lend themselves to the use of group theory. The molecule is shown in Fig. 8-1. If we rotate the molecule by 180° about the axis labeled C 2 , the new configuration is equivalent to the original one; we say that we have performed a symmetry operation on the molecule. Another symmetry operation for the water molecule is reflection in the plane labeled cr v ; this operation also gives an equivalent configuration to the original one. Similarly, we can reflect the molecule through the σ' ν plane. Since the molecule is totally in this plane, this operation must be a symmetry opera-tion also. There is yet another symmetry operation for the water molecule —leave the molecule as it is. This last operation may seem quite trivial, but it is an important one from the point of view of application of group theory to the molecule; it corresponds to the identity element, which is necessary for a set of elements to constitute a group. These four symmetry operations for the water molecule correspond to Molecular Symmetry Operations 117 > -Y Fig. 8-1 The symmetry operations for the water molecule (point group C 2i ). Symmetry Elements that form a group. The law for combination of the elements is to perform the symmetry operations successively in the order given. Let us, by way of some examples, show that these symmetry ele-ments do in fact form a group. The combination requirement for the elements to form a group can be illustrated as follows: Reflect the molecule through the cr r plane; then rotate it through 180° about the C> axis. This is equivalent to leaving it as it is, i.e.. the operation corresponding to the identity element.

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  Modern Spectroscopy

  • J. Michael Hollas(Author)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  1 symbol is not used.

  4.1.6 Generation of elements

  Equation (4.1) illustrates how the elements

  Cn

  and

  σh

  generate

  Sn .

  Figure 4.5 shows how C 2 and

  σv

  in difluoromethane CH2 F2 generate ; that is

  (4.2)

  where

  σv

  is taken to be the plane containing CH2 and that containing CF2 . Similarly,

  (4.3)

  Figure 4.5

  Illustration that, in CH2 F2 , C2 and

  σv

  generate

  From a

  Cn

  element we can generate other elements by raising it to the powers 1, 2, 3, …, (n − 1). For example, if there is a C 3 element there must also be , where

  (4.4)

  and the operation is a rotation, which we take to be clockwise, by 2 × (2π /3) radians. Similarly, if there is a C 6 element there must necessarily be , and . The operation is equivalent to an anticlockwise rotation by 2π /6, an operation which is given the symbol . Similarly, is equivalent to and, in general,

  (4.5)

  From an

  Sn

  element, also, we can generate

  Sn

  to the powers 1, 2, 3,…, (n − 1). Figure 4.6 , for example, illustrates the and operations in allene and shows that

  (4.6)

  where implies an anticlockwise rotation by 2π /4 followed by a reflection (note that a reflection is the same as its inverse, σ −1 ).

  4.1.7 Symmetry conditions for molecular chirality

  A chiral molecule is one which exists in two forms, known as enantiomers. Each of the enantiomers is optically active, which means that they can rotate the plane of plane-polarized light. The enantiomer that rotates the plane to the right (clockwise) has been called the d (or dextro) form and the one that rotates it to the left (anticlockwise) the l (or laevo) form. Nowadays, it is more usual to refer to the d and l

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  Symmetries of Nature

  A Handbook for Philosophy of Nature and Science

  • Klaus Mainzer(Author)
  • 2013(Publication Date)
  • De Gruyter

    (Publisher)

  On account of the finite number of combinations of the symmetry ele-ments, it is clear that there can only be a finite number of groups of points. 160 160 See also J. M. Hollas, see Note 158, Chapter 3; R. L.Flurry, see Note 158, Chapter 3. 4.4 Symmetries in Chemistry, Biology and the Theory of Evolution 4 9 3 •CI •CI Fig. 7 Thereby many different molecules can belong to the same point group, i.e. they can have the same symmetry structure. For example, H2O and CH2F2 belong to the same point group C2 V , since both molecules have only the sym-metry elements I, C2, σ ν and σ ν ' (Figure 9). The classification of the point groups also makes it possible to explain the relationship of optical activity and molecular structure in terms of group the-ory. According to Pasteur, a compound had optical activity if the molecules in question could not be made to coincide with their reflection. In that case, Pasteur spoke of dissymmetry. 161 Other terms are enantiomery, which in the Greek translation means opposite shape, or chirality, which alludes to the left and right-handedness of the reflective orientation. In terms of group theory, it is a matter of determining the elements of symmetry which lead to optical activity. In general, 1) a molecule with an arbitrary axis of reflection 161 L. Pasteur, Leçons sur la dissymétrie moléculaire, Paris 1861; P. Curie also speaks of dissymétrie in: Sur la symétrie dans les phénomènes physiques, symétrie d'un champ électrique et d'un champ magnétique, in: Journal de Physique 3 1894, 393. Chirality goes back to Lord Kelvin, Baltimore Lectures 1884/1893, London 1904. 494 4. Symmetries in Modern Physics and Natural Sciences y S n cannot be optically active, and 2) a molecule without an axis of reflection is optically active. 162 For a long time, opinion held that the lack of planes of symmetry or cen-ters of inversion caused dissymmetry.

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