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Symmetry Operations and Symmetry Oparators

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Symmetry Operations and Symmetry Oparators Dr. Lokesh Chandra Pati Dept. of Chemistry, J. K. College, Purulia Symmetry Operations and Symmetry Oparators: On order to study the symmetry of a molecule, certain operations such as rotation and reflection are performed and if by so doing, an arrangement is obtained which is indistinguishable from (superposable on) the original one, the operations is called a symmetry operation and the molecule is said to possess an element of symmetry defined by the operation performed. The symmetry operations are the ways of interchanging parts of a molecule. The symmetry operation and symmetry element are thus inseparably linked and often represented by the same symbols. Symmetry based solely on simple rotation is called symmetry of the first kind whereas symmetry on reflection or rotation- reflection is known as symmetry of second kind. Four fundamental elements of symmetry are present in the organic molecules. They are (i) Rotational axis of symmetry (Cn) (ii) Plane of symmetry (v) (iii) Centre of symmetry (i) (iv) Alternating of symmetry (Sn). Different manipulations of elements of symmetry that transform molecules into indistinguishable and identical structures are called symmetry operations and operation of identity respectively. The element of identity is designated as E or I. (i) Rotational axis of symmetry or proper axis of symmetry (Cn, where C stands for circulate and n is the fold):- If a molecule is rotated around an appropriate imaginary axis by an angle of 360/n and arrives at an arrangement indistinguishable from the original, the axis is called an n-fold simple or proper axis of symmetry or a rotational axis of symmetry of order n. The axis is designated as Cn and the operation is called a Cn operation. If the operation is repeated n times, it leads to an orientation identical with the original one. The value of n can never be a fraction, because in that case every Cn operation will not give superposable structure i.e. equivalent structure. If a molecule possess Cn axes with different values of n then the Cn axis having maximum value of n (fold) is called the principal axis. In the case where a molecule has several symmetry axes of the same order, the one passing through the greatest number of atoms is taken as the principal axis (e.g. as in naphthalene). Examples: (a) H2O molecule is angular in shape and it has one C2 rotational axis, passing through the O atom i.e. H2O has two-fold axis of symmetry (C2) bisecting the H-O-H angle, A and B are indistinguishable. O O 0 a rotation of 180 about b bH H aH H the C2 axis (B) C2 axis (A) (b) Chloroform (as also as NH3 and NR3) has one C3 axis along the C-H bond. H N C H Cl H H Cl Cl C3 C3 (c) Benzene (X) (planar) has one perpendicular C6 axis in addition, six C2 axes lying in the molecular plane (three passing through opposite atoms and three bisecting the opposite C-C bonds). C6 axis passing vertically through the center of the molecule. Here C6 is the principal axis because of higher value of n. Benzene also has a C3 axis collinear with C6 and a C2 also collinear with C6. C2 C2 6 C2 1 C2 2 C2 C2 C6 (X) A molecule can have only one principal axis (unique). (d) Cis and trans-1,3-dimethlcyclobutanes (III) and (IV) respectively have one C2 axis (vertical in (III) and horizontal in (IV)) as in fig. bellow. C2 H Me H H Me H Me H H H H Me H H H H C2 (IV) trans-1,3-dimethylcyclobutane (III) cis-1,3-dimethylcyclobutane (e) Trans-dichloroethylene (C) has one C2 axix perpendicular to the molecular plane and cyclopropane (D) has one vertical C3 (passing through the center of the molecule) and three horizontal C2 axes (perpendicular C3 axis) like BF3. C2 lC H H Cl C3 Trans dichloroethylene (C) Cyclopropane (D) 0 A C1 axis is trivial since rotations of any molecule around any axis by 360 leads to the original arrangement (an identity operator E or I). All molecules and objects in the universe must possess C1 axis. Linear molecules like H—C≡C—H (F) and H—C≡N (E), O=C=O etc possess a C∞ axis coincident with the internuclear axis since rotation around it by any angle gives an equivalent structures. H—C≡C—H, O=C=O in addition to C∞ axis possess an infinite number of C2 axes perpendicular to the center of the C∞ axis. C2 H C N C H C C H C F C2 (f) In case of naphthalene (G), ethylene (H) and allene (I), all of them contains several C2 axes but those shown vertically pass through more than one atom is taken as principal axis. H H C2 H H C C C2 C C2 C C2 C2 C2 C G H H H H (H) C2 (I) C2 It should be noted that during the operations, one point in the molecule (the center of gravity) remains unchanged in space i.e. must be invariant. Symmetry of this kind is called point symmetry to distinguish it form translational symmetry which involves displacement in space. Plane of symmetry: A plane of symmetry is a plane, which divides the molecule (or an object) into two halves, which are mirror images of each other. The plane is called a -plane and the operation a -operation. Two -operations are equivalent to an identity operation (2 = I) since they turn the molecules into the original. It is important to note that the two halves themselves may not be superposable. plane H v plane Molecular plane O C H H Cl Cl lC C2 The molecule of water has two mutually perpendicular -planes, Chloroform has three, each containing a H—C—Cl grouping etc. It should be remembered that every planer molecule necessarily has a plane of symmetry, namely. -Planes and Cn axes often occur together. The convention has been set up by placing the principal axis vertically (along the Z-axis), the -planes may be designated in relation to it. Thus h refers to a horizontal plane perpendicular to the principal axis and is unique; v, stands for a vertical plane containing the principal axis, and d, represents a diagonal plane bisecting the angle between two C2 axes. The number of v and d planes may be and usually greater than one. Examples: Benzene (X) has one h-plane (the molecular plane) and six v(=d)- planes (a set of three passing through the opposite atoms and another set of three bisecting the opposite C-C bonds); Cis-1,3-dimethlcyclobutane (III) has two v- planes (one passing through the methyl bearing carbons and the other passing through the two methylene carbons); the trans-1,3-dimethlcyclobutane (IV) one v-plane (passing through the two methylene carbons); trans-dichloroethylene (C) has one h-plane, coincident with the molecular plane; cyclopropane (D) has one h-plane, coincident with the molecular plane and three v-planes (each passing through an apex of the ring and bisecting the opposite C-C bond); acetylene, H—C≡C—H (F) has one h-plane at the center and an infinite number of v-planes passing through the internuclear axis (C); naphthalene (G) has one h-plane and two v-planes; ethylene (H) has two v-planes and one h- plane; allene (I) has two v-planes mutually perpendicular. Center of symmetry or inversion center (i): It is a point within a molecule such that if an atom (or point) is joined to it and the line extrapolated to an equal distance beyond it encounters an equivalent atom (or point). Mathematically, for every atom with co-ordinates x,y,z there must be a similar atom with co-ordinates -x,-y,-z, the inversion center being the origin of the co-ordinates. There can be only one inversion center in a molecule. Examples: Trans-1,3-dimethylcyclobutane (IV) has ‘i’at the centre of ring. Trans- dichloroethylene (C) has one ‘i’ at the middle point of the C=C bond; α-truxillic acid and the anti conformation of meso tartaric acid have one each. Ph H H OH H CO2H CO2H CO2H CO2H H HO H Ph H Anti form of meso tartaric acid -truxilic acid (iv) Alternating axis of symmetry (Sn) or rotation-reflection symmetry or improper axis of rotation: An improper or an alternating or rotation-reflection axis of symmetry of order ‘n’ is an ‘n’-fold axis such that a rotation of 3600/n around it followed by reflection in a plane perpendicular to the axis generates a structure indistinguishable from the original. The order of the two operations may be reversed without change in the result. The fold of symmetry of alternating axis is only even (n= 2,4,6 etc). When ‘n’ is odd, then Sn is equivalent to other symmetry elements. Examples: (i) Structure (I) and (II) are indistinguishable. The vertical axis in -truxilic 0 acid (I) is an S2 axis since a rotation of 180 around it leads to the structure (Ia) which on being reflected across the plane of the ring, gives a structure (II), superposable on the original (I). It should be noted that S2 axis is equivalent to i and S1 axis is equivalent to - Plane. H CO2H H Ph S2 Ph H Ph H H CO2H H CO2H 0 (Ia) 180 rotation about S CO2H 2 mirror H Ph H H Ph -truxilic acid (I) CO2H H H CO2H H Ph (II) (ii) Anti conformation (III) of meso tartaric acid on being similarly reflected in a plane placed at the center of the C-C bond and at right angles to it given an orientation (IIIa) which on being rotation around the axis by 1800 becomes superposable with the original (III).
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