44 An introduction to molecular symmetry 4.2 (a) E is the identity operator. It effectively identifies the molecular configuration. The operator E leaves the molecule unchanged. All objects can be operated upon by the identity operator E. (b) A plane of symmetry (mirror plane) is denoted by G. (c) The symmetiy operation of rotation about anra-fold axis (the symmetry element) is denoted by C,,, in which the angle of rotation is 360y«, where H - 1,2,3,4 ... (d) An S^ axis is an n-fold improper rotation axis: rotation through-360°M about the axis is followed by reflection through a plane perpendicular to the same axis. If the mirror plane lies perpendicular to the principal axis, it is denoted a^. If the plane contains the principal axis, it is denoted o;. To see the difference between o^ and cr,', it is best to use an examplei'-'H^O is a simple example. Figure 4.1 shows the principal (Cj) axis in a molecule of HjO^.and the two mirror planes, both of which contain the principal rotation axis. The plane which bisects the molecule is labelled cr,, and the plane in which the molecule lies Figure 4.1 For answer 4.2: the is labelled o^'. principal axis of rotation, and the two mirror pianes in H^O. A <J^ plane contains the principal rotation axis, and also bisects the angle between two adjacent 2-fold axes. 4.3 (a) An 8"Vertex star. The symmetry of the star is such that rotation thi'ough (360/8)° - 43° gives another star superimposable on the first one; the * is used in the diagram to clarify the rotation that has occurred. This operation is repeated 7 more times to get back to the first orientation. The highest- order axis is an 8-fold axis (Cg) running through the centre of the star, perpendicular to the plane of the paper. (b) An ellipse. The symmetry is such that rotation thi'ough (360/2)° ^ 180" as shown Rotate in the diagram gives another ellipse through superimposable on the first one. This 180° operation is repeated once more to get back to the first orientation. The highest-order rotation axis of the ellipse is a 2-fold axis (C2). There are 2 other Cj axes: both lie in the plane of the paper, one horizontal, and one vertical with respect to the ellipse drawn. (c) A pentagon. The symmetry is such that Rotate rotation through (360/5)° = TZ'' gives through another pentagon superimposable on the first one. This operation is repeated 4 more times to get back to thefirst orientation. The principal axis is a 5-fold axis (Cg). (d) The symmetry of this shape is such that rotation through (360/3)° = 120° ^ves a shape superimposable on the first one. This operation is repeated twice more to get back to thefirst orientation. The principal axis is a 3-fold axis (C3). An introduction to molecular symmetry 45 4.4 SOj is a bent molecule (see answer 2.4b). It must possess an E operator - all molecules do. The other symmetry operators are: A Cj axis: A o" plane A a' plane: The symmetry operators are E, C_, &'• 'md a'. 4.5 Figure 2.1 in H&S shows one view of the Hp^ molecule. Another 2 views are shown here in structures 4.8 and 4.9. In 4.9, the molecule is viewed along the 0-0 bond. The symmetry operator that Kp^ possesses is a C^ axis running through the midpoint of the 0-0 bond in the direction shown in the left-hand diagram below. Its operation is shown in the right-hand diagram below; look carefully at the perspective in the diagrams. (4.8) (4.9) H(l) H(2) HC2) ^H(l) 4.6 The diagrams below summarize the answer. ci Each C7^ plane contains one B-F bond. (4.10) (4.11) 4.7 (a) On going from BF3 to BClFj (4.10 to 4.11): the C3 axis is lost, two C^ axes are Cl lost, and two o; planes are lost. (Colour the left-hand F atom in the diagrams above to signify a change to Cl, and confirm which of the symmetry operators are no longer valid). Br (b) On going from BCIF2 to BBrClF (4.11 to 4.12): C^ axis is lost, and the a^ plane (4.12) is lost. (Use the diagrams above again to help you confirm these). (c) Bach molecule has a c7^ plane ~ the plane containing the atoms. 46 An introduction to moleculac-^ymmetry 4.8 First, draw out the structm-es of the molecules and ions: H C! c) c) H A! See Chapters 15 and 16 for v:"o ^^ Br O'^ ^ O Cl "Cl 0 \ ^ 0 0 Br' representations of the Br bonding in SO3, [SO^]^- and [NO3]" so that each atom The structures of [SOJ^- and [NO3]- are drawn so as to emphasize the equivalence obeys the octet rule of the bonds (delocalized bonding), and therefore the symmetry; see margin-note. Note; (i) each possesses a C3 axis; . j,--^ _ (ii) only the planar molecules possess a a^, plane. The answers are therefore: (a) C3 axis but no a^^ plane: NH3, PBr3, [SO^F". (b) C3 axis and a cTj^ plane: SO3, AICI3, [NO3]-. 4.9 First, draw out the structures of the molecules and ions. Use the VSEPR model to show why [ICIJ- and XeF, are square planar (2 lone pairs and 4 bonding pairs): 2— Cl O '"'^Xe-'" As above, the structure drawn •Si);/,, C! F F' for [SOJ^" emphasizes its cr Cl symmetry and does not illustrate a bonding picture The only two species with a Q axis and a <J^^ plane are the square planar ones: [ICy-andXeF^. 4.10 For each, first draw out the structure (using VSEPR to help you) and then work out the number of mirror planes. (a) SF^: the structure is disphenoidal (4.13). SF^ contains 2 mirror planes as shown in 4.14: one plane contains the S and 2 F(axial) atoms, and the other contains the S __c.,o^>'F ^ • and 2 F(equatorial) atoms. (b) HjS: the bent structure is analogous to that of HjO (although the bond angle is different) and so the 2 mirror planes are as shown in Figure 4.1. (c) SFg: octahedral structure (6 bonding pairs, 0 lone pairs). There are 9 mirror planes. The 3 Oj, planes are shown in 4.15, and each contains the S and 4 F atoms. (4.13) (4.14) The 6 (7d planes can be considered in 3 sets of 2., One set is shown in 4.16. Each plane contains the S and two opposite F atoms (i.e. it contains a Q axis), and bisects the other two F-S-F axes. The other 2 sets of a^ planes are similarly constructed, starting with a different Q axis (i.e. F-S-F axis). (4.15) (4.16) (d) SOF^: the structure is disphenoidal (Figure 2.2, p. 21). SOF^ contains 2 mirror planes, analogous to those of SF^ (4.14). An introduction to molecular symmetry 47 (e) SO : this has a bent structure and so has the same symmetry as H,S in part (b) it has 2 mirror planes. ^ ^ (f) SO3: the stmcture is trigonal planar (see answer 2.17i, p. 20). There are 4 mirror planes: one <y, and 3 a^ planes. Their positions are analogous to those in BF answer 4,6. ^' Questions 4.11 -4.12: A centre ofinversion: general notes If reflection of all parts of a molecule through the centre of the rnolecule produces an mdistmguishable configuration, the centre is a centre of inversion (centre of symmetry), designated by the symboh-. , • j.;-, . (b)Astaggeredconformer(4.17) will be the most favoured in terms of steric energy s^uItuLT^^^^^^^^ ; ''f 'f ' ^^ ^" ^"^^"^°" ^^"^^- ™^ p-^ i^^ ^^o- ^n structu e 4 17. To confirm that this is acentre of inversion, reflect each point of the (4.17) moeculehrough cen^e.-andshowthatan identical pointis generated. For example, take H atom a ... reflect through i ... you end up at H atom a' which is indistinguishable from atom a. (d) The eclipsed confoiT^er 4.18 is the least favoured in terms of steric energy (e) This confonner does not contain a centre of inversion. To confinn this try a ing the miJoint of the Si-Si bond again as in part (c). Reflection of^l pSs of the molecule through this pomt does not lead to an indistinguishable configuration. (4.18) 4.12 l"-chpartofthisanswer,ifyouareunsureofthereasonwhyamoleculedpesor andTlr'.'° '""'''''f''^''^' ^^^^ '^' -«"tral atom as a trial inversion centre, and try reflecting parts of the molecule through this point. (a) EF3: B, group 13; 3 valence electrons; trigonal planar; structure 4 19 No inversion centre. (b) SiF,: Si, group 14; 4 valence electrons; tetrahedral; stmcture 4.20. No inversion centre. (c)_ XeF,: Xe, group 18; 8 valence electrons; square planar (4 bonding and 2 lone F ^ F pairs); stmcture 4.21. The Xe centre lies on an inversion centre. (4.19) inv5o n c f °"^ ^^' ^ "^^''''^ ^^e^trons; trigonal bipyramidal; structure 4.22. No (e) EXeF ]-: Xe, group 18; 8 valence electrons; pentagonal planar (5 bonding and F 2 lone pairs); stmcture 4.23. No inversion centre. (f) SFg: S, group 16; 6 valence electrons; octahedral; stmcture 4.24 The S centre lies on an inversion centre. F \ ' (g)C3F. planar smacture 4.25; this is related to ethene. There is an inversion centre (4.20) at the midpoint of the C=C bond.