INTRODUCTION. Fig. 1.1

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1 1 INTRODUCTION 1.1 SYMMETRY: AN INTRODUCTION In nature, when we see the fascinating world of plants, flowers, birds, architectural buildings (Lotus Temple of Delhi, Taj Mahal, Ashoka Pillar, Rastrapati hawan, etc.) they have an aesthetic appeal. eauty is a sense and it can t be defined but everybody feels it and derive pleasure out of it. The famous poet John Keats said A thing of beauty is a joy for ever or beauty is truth and truth is beauty. eauty of an object is also very much related to its geometry, for example, a square is more beautiful than a triangle and a circle looks more beautiful than a square (see Fig. 1.1). We have also heard of a famous quotation God geometrises. Every object in the natural physical world has a specific geometry and the idea of beauty and harmony (rhythm) was long ago (Greek sculptors and architects) associated with a term Symmetry. ut what is symmetry? In Greek, it means syn-metron that is to measure together. When we see any object, the human brain unknowingly does the work of rotation, or reflections, through imaginary axes and planes to feel the sense of symmetry. Symmetry here means that certain part of an object looks exactly like another part (or mirror images of each other). It means that it can be described as a regular arrangements of objects with pleasing proportions and periodicity. Therefore, symmetry is a balancing phenomenon and expressed in harmony of proportions. For example, a butterfly or a flower or a human body looks symmetrical around a point or an imaginary axis or a plane and that particular symmetry gives us a sense of beauty. A triangle A square A circle A sphere Fig. 1.1 In Fig. 1.2, the butterfly is symmetrical around a line passing just half of the body of the butterfly, in Fig. 1.2 the human body is symmetrical about the dotted line and the flower looks 1

2 2 SYMMETRY AND GROUP THEORY IN CHEMISTRY symmetrical around an imaginary point and planes. The concept of symmetry was popular at the time of Pythagoras (a Greek mathematician). Man has learned the art of thinking, building and living from nature, therefore, symmetry has become primarily important in human creations. So now for chemists the question arises that what we actually mean by the term symmetry in scientific terms? For this, first of all geometry of the objects (atoms, molecules, ions, etc.) is known by the help of valence bond theory (Hybridization) or VSEPR (valence shell electron pair repulsion) theory and then the object is analysed microscopically in terms of elements of symmetry present in the molecule. Therefore, symmetry is a microscopic analysis of geometry. In scientific terminology, symmetry of a molecule or any object is determined on the basis of the symmetry elements present. Molecule containing more number of symmetry elements are more symmetrical. Why the symmetry is so important to learn? It is because of the fact that Fundamental laws of nature are related to symmetry and therefore it is the basic concept in the field of scientific world. The concept of symmetry is applied through abstract mathematical group theory (which we deal later) to predict the properties like molecular vibrations, dipole moments, energy states, electronic transitions, optical activity, crystal structure, degeneracies of energy states, etc. of compounds (molecules), thus making symmetry quantitative. It will solve many problems associated with structure, bonding, spectra, etc. of the molecules. Therefore, symmetry and group theory are important aspects of theoretical chemistry in understanding the fine structure and bonding in molecule and their behaviour. That is why group theory may be called as Algebra of geometry and symmetry is called a Science of Aesthetics. utterfly Human body (c) Flower Fig. 1.2 Theoretical chemists generally deal bonding, structure or spectra of a molecule by the methods of Quantum mechanics in which a Schrödinger equation is framed and then it is solved. This needs a knowledge of high level mathematics and physics. Instead of complicated mathematical calculations, symmetry and group theory provides an alternate path to understand the chemical bonding and electronic structure of molecules by using the symmetry properties of the molecule. For molecules of simple geometry, one can solve the Schrödinger equation but as the geometry of the molecules become more complex, it becomes very difficult to obtain even approximate solutions. In such cases, symmetry properties are very much helpful to predict physical, chemical and spectral properties of the compounds. In the Chapter, 3 it will be explained in detail that how on the basis of symmetry of the molecule, useful informations about eigen functions and eigen values can be obtained without solving the Schrödinger equation for the molecule under consideration. Here, we will confine our discussions to physical/or chemical problems of isolated molecules or complex ions only, i.e., we consider the symmetry of the molecule itself and not to any symmetry due to association of molecules.

3 INTRODUCTION SYMMETRY ELEMENTS AND SYMMETRY OPERATIONS As discussed above, symmetry of a molecule can be defined in terms of symmetry elements and symmetry operations. ut, before these are to be defined, some fundamental aspects of coordinate system and mathematical operators are to be understood to make the idea of molecular symmetry as useful as possible. The following three points criterion will be adopted for further discussions throughout the book: (i) The centre of the molecule (often termed as centre of gravity of the molecule) is considered to be coincident with the centre (origin) of the Cartesian coordinate system [Fig. 1.3 and ], which will follow the Right Hand Rule. In this convention the positive directions of the three Cartesian axis are defined in the same way as the thumb, index finger and middle fingers of the right hand. It can be understood in this way: If we hold our right hand in such a manner that the thumb is pointing up, index finger pointing like a gun and middle finger points in such a way that it is perpendicular to the other two (index finger and thumb), then the thumb, index finger and middle finger now will represent the z, x and y directions of the Cartesian coordinate system respectively, as shown in Fig. 1.4 or alternatively, if palm is rotated such that it faces up, to present the z-axis in its usual perpendicular direction [Figs. 1.3 and 1.4]. z z y y or x x Fig. 1.3 Index finger x z z y y Middle finger Thumb or x Fig. 1.4 Right Hand Rule

4 4 SYMMETRY AND GROUP THEORY IN CHEMISTRY (ii) Mathematical operators: An operator gives a rule by which a function is transformed into another function, for example, when we write d d dx x 2 i, here, d is a mathematical dx operator, which carry out differentiation of x 2, and when differentiated, it is transformed to another function 2x. Similarly, +,,, are operators which carries out addition, subtraction, multiplication and division, respectively. When more than one operators are there, we should follow the conventions of operator algebra, according to which operations are to be carried out from right to left (this is discussed in combined operations). (iii) To distinguish a symmetry operation from symmetry element, a circumflex (cap) is written over the symbol of the symmetry element. The existence of a symmetry element is indicated by the effect of an operator on the molecular wave function or on geometrical figure to result into an another function or figure equivalent or identical with the original one. Now, we will discuss what are symmetry elements and symmetry operations? A symmetry element is an imaginary line, plane or point in a molecule (or in any object) about which one or more symmetry operations are carried out. Here, symmetry operation means that the configuration (geometrical form) obtained after the operation is indistinguishable from the original one. Operations here mean, movement (rotation, reflection, inversion, etc.) about a symmetry element. Symmetry element and symmetry operations are closely interrelated, but they are quite different. For the beginners, it creates confusion. A symmetry operation may be defined as a movement of an object such that it brings the object into an equivalent configuration (indistinguishable configuration). It may not necessarily be identical with the original one, because some equivalent parts may have been interchanged. For example, take H 2 molecule. If it is rotated about an imaginary line passing through the centre of the covalent bond (as shown by dotted line in Fig. 1.5) by an angle of 180, we get equivalent configuration, but if this is further rotated about the same imaginary line by 180, we get identical configuration. Here and are Hydrogen atoms and are indistinguishable, for convenience, we have written them as and (to understand the movement of atoms on rotation) Original configuration Equivalent configuration Fig. 1.5 Identical configuration Therefore, a symmetry element is a line, point or plane about which one or more symmetry operations can be carried out whereas symmetry operations are the movements of the different parts of a molecule about the symmetry element to give indistinguishable configurations (equivalent) from original one. Now, we will describe the kinds of symmetry elements that a molecule may possess and the symmetry operations generated by the symmetry elements. Here, students are advised to visualize three dimensional model of molecule to understand the operations. Further, the symmetry operations

5 INTRODUCTION 5 should be carried out within the molecule and not outside the molecule, because operation outside the molecule will lead to the translation motion of the molecule, which is not the part of our discussion. In symmetry operations, at least one point in the molecule remain unaffected by all the symmetry operations. All the symmetry elements intersect at this point. This point may be the centre of mass of the object and due to this the description of symmetry of molecule is also called Point-group symmetry. In all, there are five types of basic symmetry elements and they are summarized in the Table 1.1. Table 1.1 Symmetry Element Symmetry Operation Symbol 1. Axis of symmetry by an angle C n (proper or principal) θ = 360/n about the axis 2. Plane of symmetry Reflection in the plane σ 3. Improper axis of symmetry about the axis followed S n by reflection in a plane perpendicular to rotation axis 4. Centre of symmetry Inversion through the centre i of symmetry 5. Identity No operation E Proper or Principal Axis of Symmetry (C n ) Axis of symmetry is an imaginary line in a molecule about which if the molecule is rotated by such a minimum angle (θ) that after rotation, indistinguishable configuration of the molecule is obtained. Then, those imaginary lines are called al Axis of symmetry and the rotation by an angle θ about this line is called symmetry operation. Here we choose clockwise rotation as positive and anticlockwise rotation as negative. The symbol for the axis of rotation is C n, where n is the order of the axis of symmetry and it can be expressed as, n = 360/θ or 2π/θ. When n = 2, 3, 4, then it is called two-fold, three-fold, tetra-fold, axis of symmetry, respectively. It means that n is the number of times the molecule can be rotated to reach at identical configuration. For example, in H 2 molecule we have (Fig. 1.6) Fig. 1.6 ( a and b subscripts are written for convenience only otherwise they are indistinguishable) Here, the dotted line is the Axis of symmetry whose order (n) is 2 because by rotating the molecule about this line twice by 180, identical configuration is achieved. Therefore, this line is twofold axis of symmetry and designated as. Similarly, take the case of F 3, a triangular planar molecule. (Fig. 1.7) We can have the following symmetry operations:

6 6 SYMMETRY AND GROUP THEORY IN CHEMISTRY Here, we see that if the molecule is rotated by an angle of 120 about a line passing through the centre of the oron atom and perpendicular to the molecular plane (i.e., plane of the paper), it gives an indistinguishable configuration (P), therefore, this line is a three-fold axis of symmetry (C 3 ) and another indistinguishable configuration (Q) is obtained if the molecule is rotated by 180 about a line passing along ond, therefore the line is a two-fold axis of symmetry ( ). Thus, this molecule, (F 3 ) have one C 3 and three axes of symmetry. Here, C 3 is perpendicular to axis. Now question arises that if there are more number of axes of symmetry present, which one is the proper or principal axis of symmetry? For this, the convention is this that highest order axis of symmetry is called Principal axis of symmetry. Therefore, in case of F 3 molecule, the C 3 axis of symmetry is the Principal Axis of symmetry. If there are C n axes of different orders, then highest order axis will be the Principal or Proper axis of symmetry. 120 (P) 180 C3 Fig. 1.7 Problem: Find out the principal axis of symmetry in enzene, 2 4 and NH 3. Solution: enzene, has a hexagonal planar structure because all the six carbons atoms of the benzene are sp 2 hybridized. It has six axes (as shown by dotted lines Fig. 1.8) and one C 6 axis, which passes through the centre of the molecule and perpendicular to the plane of the molecule. Therefore, principal axis of symmetry is C 6. Similarly, 2 4 and NH 3 have C 4 and C 3 as principal axis of symmetry, respectively. Now, if we try to find axis of symmetry in ethylene molecule, again it is a planar molecule (because both carbon atoms are sp 2 hybridized), and it has three axes of symmetry as shown in Fig. 1.9 by dotted lines. Now, question arises that which line is the proper axis of symmetry because order of all the three axes of symmetry is same. Here, in such cases, the convention is this that when there are more number of axes of symmetry the same order, then that axis which passes through maximum number (Q) Fig. 1.8 H H C C Fig. 1.9 H H

7 INTRODUCTION 7 of atoms of the molecule is principal axis of symmetry. So, in this case, the dotted line which is passing through the two carbons of the molecule is the principal axis of symmetry. Further, in all molecules, the principal axis of symmetry is taken as z-axis in the Cartesian coordinate system. This element of symmetry is also commonly called axis of symmetry. For linear molecules like H 2, CO 2, H, COS this axis of symmetry can be C, here the order (n) of the axis is infinite. It means that rotation about this axis by the smallest possible fraction of angle (tending to 0 ) will give equivalent orientation of the molecule and we recall that when θ tends to zero, n tends to infinity. (when θ 0, n ). For example, in case of H, OCS, HCN, H 2, CO 2 etc. molecules, the line along the bond is C as shown in Fig C H H H H Fig Now, as we know that each symmetry element generates some symmetry operations, i.e., symmetry operations can be carried out on each symmetry element. Obviously, the question arises that how many symmetry operations are generated by the axis of symmetry (C n ) or how many symmetry operations can be carried out on this element of symmetry. The answer lies in the fact that the nth rotation of a molecule will bring back the molecule to the original orientation. For example, take a simple molecule like H 2 O, which is V-shape and has a axis of symmetry. The first rotation by 180 once (C 1 2 ) gives Fig and second rotation by 180 about the axis C 1 2 gives Fig. 1.11(c) which is the original orientation of the water molecule. That is, by rotating twice by 180 ( 2 by 360/n carried out in succession of m times is represented by a symbol C m n where n and m are integers. ), the identical configuration of the molecule is generated. Therefore, here we have two operations on axis of symmetry and these are C 1 2 and C2 2 but C2 2 operation produces identical configuration therefore it is not a separate operation (as we see later that it is included in Identity operation). Therefore, the distinct operation possible on axis of symmetry is one only and that is C 1 2. Similarly on C 3 axis of symmetry, two distinct operations are possible and they are C1 3 and C2 3 because again C 3 3 operation will produce the original orientation of the molecule (which is included in Identity symmetry element). On C 4, three distinct symmetry operations can be carried out to get C n axis of symmetry generates (n 1) distinct equivalent configurations of the molecule. Therefore, symmetry operations. it can be understood that axis of symmetry of order n (C n ) has n 1 symmetry operations. O ^ O ^ O Fig (c)

8 8 SYMMETRY AND GROUP THEORY IN CHEMISTRY Now it can be easily seen that there are many equivalent operations possible, for example, C 1 3 C2 6, C2 3 C4 6, C1 2 C2 4 C3 6, etc. and Cn n E where E is the Identity element of symmetry. As mentioned earlier, here the rotations are carried out in clockwise direction, one can argue that anticlockwise rotation may also give equivalent configuration, then they are also symmetry operation. Yes, they are also symmetry operations. For example, in F 3 molecule, we can easily recognise that anticlockwise rotation by 120 gives Fig. 1.12, whereas, same configuration is obtained by successively twice rotating the molecule by 120 about C 3 proper axis of symmetry. Hence, we can write C 1 3 C2 3 and therefore, C 1 3 is not to be considered as a separate operation. 1 (C 3 ), 120 Anticlockwise (C 3) ockwise C ockwise Fig (c) Problem: How many rotational axis of symmetry are present in 2 4 and C 5 H 5 and which is the proper axis of symmetry in these molecules? Solution: 2 4 has square planar geometry and cyclopentadienyl anion (C 5 H 5 ) is also planar. As we see here, one four-fold axis (C 4 ) which is passing through the centre of Platinum atom and perpendicular to the plane of the molecule and four two-fold axes of symmetry, which are perpendicular to C 4. So 2 4 has 1C C 4. The highest order axis is C 4, therefore, C 4 is the principal axis of symmetry. Similarly, the geometry of C 5 H 5 is shown here. C 4 Fig C 5 Fig. 1.14

9 INTRODUCTION 9 It can be easily known that this species contain 1C 5 + 5, C 5. The principal axis of symmetry is C 5 perpendicular to the plane of C 5 H 5 anion, passing through the centre of the molecule. Plane of Symmetry (σ) If there exists an imaginary plane in the molecule about which the reflection of the molecule is carried out and after reflection if configuration is indistinguishable from the original one, then the imaginary plane (or mirror plane) is called a Plane of symmetry. This element of symmetry is particularly important in the discussion on the optical activity of organic compounds. For example, in H 2 O we can have a plane perpendicular to molecular plane passing through oxygen atom and bisecting the angle between two O onds. Figure 1.15 and 1.15 are indistinguishable, therefore, the plane (σ xz ) about which reflection operation was carried out is a plane of symmetry. z O y Molecular plane ( s xy ) x ^ s xz (Reflection) O Plane of symmetry Fig Here, H 2 O molecule is planar, therefore, molecular plane (σ xy ) which contains all the atoms of the molecule is also a plane of symmetry because reflection through this plane will leave all atoms unshifted and we get indistinguishable configuration. Whereas, reflection through σ xz leaves oxygen atom at the same place but the two H-atoms are exchanged. It is very easy to recognize that the operation of reflection produce equivalent configuration but if the operation is carried out twice, original configuration is obtained. Hence, a plane of symmetry (σ) generates only one distinct operation. Therefore, we can write, σ n = σ, if n is odd. σ σ = σ 2 E and σ n = E, if n is even. Here E stands for Identity element. There are three types of planes of symmetry: (i) Vertical plane of symmetry (σ v ): It is a plane of symmetry passing through (or coinciding with) the principal axis of symmetry and one of the subsidiary axis (if present). In other words it is a plane containing Principal Axis of Symmetry. (ii) Horizontal plane of symmetry (σ h ): It is a plane of symmetry which is perpendicular to the principal axis of symmetry.

10 10 SYMMETRY AND GROUP THEORY IN CHEMISTRY (iii) Dihedral plane of symmetry (σ d ): It is the plane of symmetry passing through the principal axis of symmetry and bisecting the angle between two subsidiary axes ( -axes). In other words, dihedral plane σ d, are those planes which bisect the angle between two successive -axis. Also, if the angle between a set of two planes is bisected by a axis, then this set of planes is also designated as σ d s or we can say that dihedral planes (σ d ) are those σ v s which bisect the angle between two successive c 2 axes. For example, in case of H 2 O molecule, σ xz and σ xy are vertical planes of symmetry (s v s), because both planes contains principal axis of symmetry ( ). Similarly in 2 4 we can see that it contains 1σ h, 2σ v and 2σ d (Fig. 1.16). s v s h s v s d s d Fig A good example to describe the dihedral plane of symmetry (σ d ) is Allene (C 3 H 4 ). The geometry of the molecule is shown below in which the central carbon is sp hybridized, whereas the terminal carbons are sp 2 hybridized. The H-atoms of terminal carbons lie in perpendicular planes (Fig. 1.17). C 2 H 3 H 1 s d s d C C C C() 2 z H 4 H 2 Fig. 1.17