Symmetry in Nature and in Molecules, Lecture notes of Chemistry

Download Symmetry in Nature and in Molecules and more Lecture notes Chemistry in PDF only on Docsity! Lecture 12 February 11, 2019  Symmetry in Nature and in Molecules   Symmetry Operations  Symmetry Elements  Point Groups and Assignments Symmetry Intuitively, we know symmetry when we see it.   But how do we put in quantitative terms that allows us to compare, assign, classify? Let’s look for these in molecules What is a point group?  A collection of symmetry elements for a specific symmetry, intersecting at a specific point for molecules, and displayed in a character table.   The C, point group: Molecules that have no symmetry elements at all except the trivial one where they are rotated through 360° and remain unchanged, belong to the C; point group. In other words, they have an axis of 360°/360° = 1-fold, so have a C; axis. Examples are: C; Bromo-chloro-fluoro-iodo- chloro-iodo-amine methane  The C,, point groups: These have a C, axis as their only symmetry element. They generally resemble propellers which have the frontand back different. Important examples are (hydrogens omitted for clarity): tripheny| phosphine viewed down C; axis Cobailt(l!I) tris-glycinate viewed down C; axis tripheny| phosphine viewed from the side C3 as ce Cobalt(I!1) tris-glycinate viewed from the side  Symbol for axes of symmetry Cn where rotation about axis gives indistinguishable configuration every (360/n)o (i.e. an n-fold axis) Thus H2O has a C2 (two-fold) axis, BF3 a C3 (three-fold) axis. One axis can give rise to >1 rotation, e.g. for BF3, what if we rotate by 240o? B (1)F F(2) F(3) B (3)F F(1) F(2) 240o Must differentiate between two operations. Rotation by 120o described as C3 1, rotation by 240o as C3 2. In general Cn axis (minimum angle of rotation (360/n)o) gives operations Cn m, where both m and n are integers. When m = n we have a special case, which introduces  a new type of symmetry operation..... IDENTITY OPERATION For H2O, C2 2 and for BF3 C3 3 both bring the molecule to an IDENTICAL arrangement to initial one. Rotation by 360o is exactly equivalent to rotation by 0o, i.e. the operation of doing NOTHING to the molecule. If a C2n axis (i.e. even order) present, then Cn must also be present: C4 Xe(4)F F(1) F(3) F(2) Xe(3)F F(4) F(2) Xe(2)F F(1) F(3) F(1) F(4) 90o i.e. C4 1 180oi.e. C4 2 ( C2 1) Therefore there must be a C2 axis coincident with C4, and the operations generated by C4 can be written: C4 1, C4 2 (C2 1), C4 3, C4 4 (E) Similarly, a C6 axis is accompanied by C3 and C2, and the operations generated by C6 are: C6 1, C6 2 (C3 1), C6 3 (C2 1), C6 4 (C3 2), C6 5, C6 6 (E) Molecules can possess several distinct axes, e.g. BF3: C3 F B F F C2 C2 C2 Three C2 axes, one along each B-F bond, perpendicular to C3 n C2 perpendicular to Cn puts molecule in D point group Inversion (i Each atom in the molecule is moved along a straight line through the inversion center to a point an equal distance from the inversion center. XYZ -X, -Y, -Z => Center of inversion Molecules can possess several distinct axes, e.g. BF3: C3 F B F F C2 C2 C2 Three C2 axes, one along each B-F bond, perpendicular to C3 n C2 perpendicular to Cn puts molecule in D point group So, What IS a group? And, What is a Character??? Symmetry elements/operations can be manipulated by  Group Theory, Representations and Character Tables A GROUP is a coHection of entities or elements which satisfy the following four conditions: 1) The product of any two elements (including the square of each element) must be an element of the group. For symmetry operations, the multiplication rule is to successively perform operations. 2) One element in the group must commute with all others and leave them unchanged. Therefore the “E”, EX = XE =X 3) The associative law of multiplication must hold A(BC) = (AB)C 4) Every element must have a reciprocal which is also an element of the group. i.e., ~ X(X-1) = (X-1)X = E Note: An element may be its own reciprocal. roups may be composed of anything: symmetry operations, nuclear particles, etc. Simplest is +1, -1. All the groups which follow the same multiplication table are called representations of the same group. > Character Tables  cubic functions z3, x(x2-3y), 2(x2+y2) Rz y(3x?-y7) (x, y) (Rx Ry) (xz2, yz2) [xyz, z(x?-y2)] [x(x2+y?), y(x2+y2)]  Ai | 1 1 1 1 1 1 + y, 2 An | 1 1 -1 1 1 -1/R, E'}2 -1 0 | 0 | (x,y) (x? — y*, xy) Ay | 1 1 1 -1 -1 -1 AZ | 1 1 =-L <1 <1 1 | z Ey |2 -1 0 —-2 1 O | (Ry, Ry) | (xz, yz) Character table for Day point group linears, E/2C, @)|C,/2C',/2C",| j |284|6,,|20,)20, rotations Wadratic Ail} i ojif tyr jijajafada x ay’, 27 Aggll] 1 P}-b] -r fifa pay -r}-i R, Bygfl) -1 Ji] i] -2 flj-bjadyay-l x?-y* Byy/l} -1 [tf -tf do fi}-b]ad}-t} i xy E, 2) oO |-2] 0 | Oo }2] 0 |-2] 0 | o | (R,, Ry) (xz, yz) Ayyll} 1 Pf} ob J a jet} -t |-t]-1 | -1 Agll) 1 Py} -b } -2 f-l}-1]-l} dj z Biull} -1 Ja} od fo -b f-l} a f-t}-1} 1 Boll} -1 Jaf} -t fod fet} a f-1] ot yl E,/2} 0 |-2) 0 | 0 J-2}0]/2}/0/]0] @y POINT GROUPS A collection of symmetry operations all of which pass through a single point  A point group for a molecule is a quantitative measure of the symmetry of that  molecule Assignment of Symmetry Elements to Point Group:  At first  Looks Daunting.   Inorganic Chemistry Chapter 1: Figure 6.9 © 2009 W.H. Freeman Daunting?  However almost all we will be concerned  with belong to just a few symmetry point groups A Simpler Approach Special Groups Start a (a) Linear? Cooy, Desh? ae (b) Multiple high-order axes? T,Th, Ta,O, Op, I, Ih? Step 2} > Low Symmetry (no axes): Cy, Cs, C; Step 3 > Only S,(n even) axis: S4, S¢, Sg,..-, Cy axis (not simple consequence of San) ep ie Sep 4 Steps—_ No Cy’s axes L to Cy nCy’s axes 1 to Cp On no o’s | | Chh Cy LINEAR MOLECULES Molecular axis is C - rotation by any arbitrary angle (360/)o, so infinite number of rotations. Also any plane containing axis is symmetry plane, so infinite number of planes of symmetry. Divide linear molecules into two groups: Do in fact fit into scheme - but they have an infinite number of symmetry operations. (i) No centre of symmetry, e.g.: H C N C No C2's perp. to main axis, but  v's containing main axis: point group Cv (ii) Centre of symmetry, e.g.: C2 O C O C2 C h i.e. C + C2's + h Point group Dh A few geometries have several, equivalent, highest order axes. Two geometries most important: Highly symmetrical molecules Regular tetrahedron e.g. Cl Si Cl Cl Cl 4 C3 axes (one along each bond) 3 C2 axes (bisecting pairs of bonds) 3 S4 axes (coincident with C2's) 6 d's (each containing Si and 2 Cl's) Point group: Td Regular octahedron e.g. S F F F F F F 3C4's (along F-S-F axes) also 4 C3's. 6 C2's, several planes, S4, S6 axes, and a centre of symmetry (at S atom) Point group Oh These molecules can be identified without going through the usual steps. Note: many of the more symmetrical molecules possess many more symmetry operations than are needed to assign the point group. 4. The C,, Groups C2 f & cz oixz) of(yz) At 1 1 [ 1 z x7, 7, z? Aa 1 1 -1 —1 R: xy By 1 —1 L —-1 x, Ry xE Bz 1 —I1 —1 1 », Re vz Ose EE 2C3 3e, AL 1 1 1j}ez x? yt, 2? Az 1 ! —1 Rs = 2 1 0 | Ge, yy R., Ry | Ce? — »?, xy)Cxz, yz) Cae | HE 204 Cz 204 2e0 A; U 1 1 1 riz x? + y?, 2? Aa 1 1 t _ —1 R B I —1 1 —1 x? — y? Bi T —!1 L —1 1 xy z£ 2 o -2 ° 0 | Ge, xR, RY | Cez, yz) Cs E 2Cs 2C37 Sov Ay 1 1 I 1 Zz x? + y?, 2? Ad 1 1 1 —1 Re Ey 2 2 cos 72° 2cos 144° O | &, »UR., RY) (xz, yz) Ez 2 2 cos 144° 2cos 72° o Gc? — »?, xy) Cee | EF 2C6 2€3 Cz 30, 34 ‘At 1 1 1 1 1 1[2z x? + y?, 27 Az L I I 1 -—1 —!1 R By L —Ft 1 —1 1 —1 Bz 1 — t —1 —-1 1 zy 2 —-1 —2 °o O | G, »ICRs, Re) | Cz, vz) Ex 2 _ —1 2 ° o (x? — y*, xy) 6. The D,, Groups Das £ CZ 6200 Cid 7 elev) ofez) ofpz) Ae 1 I 1 I 1 I 1 1 x2, pt, 22 Bie ! ! —1 —1 I I -1 —1 R. xy Bag 1-1 I —1 tot 1 —1 Ry xz 30 1 —1 —t 1 1-1 —1 1 Re yz Ay t 1 1 rt o—1 —1 —i —1 Bis 1 1 -1 —~1o o—t —1 I 1 z Baw I —1 I —1 —-1 t —t 1 ¥ Bau 1 1 —1 1 —1 1 r —1 x Dan E2Cs 3C2 on 253 3ey | Ay’ 1 1 t 1 1 1 xt y?, z? A; 1 1 —t 1 1-1 . = 2-1 ° 2-1! o cx. ») (x? — y?, xy) Ai” 1 1 [ —1 —1 —-1 Ay” 1 1 o—t ~—1 -1 1 z £* 2 —-1 o —2 1 ° (Rx, Ry) (xz, ¥z) Dar Eo 2Ca C2 202" 202" 5 254 on 200 Boe | Ais 1 1 1 1 1 1 I 1 1 t x7 + y?, 2? Ane 1 1 1 —1 —1 1 1 1 —i —1 Ry Bie 1 —1 1 1-1 1-1 1 Io —1 x? — y? Bag 1 —1 1 —1t 1 1-1 1-2 1 xy Ee 2 o —2 Q o 2 o —2 ° ° (Rx. Ry) (xz, yz) Aw 1 1 1 1 1 —t —! —1 ~—i ~—1 Aa 1 1 1 o—1 ~1 —t ~—1t —1 1 1 z Bus 1-1 1 I —1 ~1 1 —1 -1 1 Bu 1 ~t rt —r 1 —1 to—1 1 —1 Ee 2 o —2 ° o —2 o 2 oO o &, otpr) a otxy) ofar) Calx pty) & ©&;{z) =, y»?, 7 =F az RE R, = Wak ae bot | 35 = ee see Plt << =e See wt yF, rt tou et fe yt Cx? — »?, xy) Caz, yz) io, 254 30, mm 253 Jou €,; 2," 20," f E 205 £& 20.4 (xz, yz) ax? — pi =F (K,, yy) t=. ¥) & 4 ————— ssa Qe ==) ee anne gane=6 sss se0n seen geenad rl ot ase heos ef It eee Gee 6 II | ae ese > ee ee: ee et | 1 | It ee | Pin i. The 9,, Groups Ban Pas Inorganic Chemistry Chapter 1: Table 6.5 © 2009 W.H. Freeman Character table for C∞v point group E 2C∞ ... ∞ &sigmav linear, rotations quadratic A1=Σ+ 1 1 ... 1 z x2+y2, z2 A2=Σ‐ 1 1 ... ‐1 Rz E1=Π 2 2cos(Φ) ... 0 (x, y) (Rx,  Ry) (xz, yz) E2=Δ 2 2cos(2φ) ... 0 (x2‐y2, xy) E3=Φ 2 2cos(3φ) ... 0 ... ... ... ... ... E 2C∞ ... ∞σv i 2S∞ ... ∞C'2 linear  functions, rotations quadratic A1g=Σ+g 1 1 ... 1 1 1 ... 1 x2+y2, z2 A2g=Σ‐g 1 1 ... ‐1 1 1 ... ‐1 Rz E1g=Πg 2 2cos(φ) ... 0 2 ‐2cos(φ) ... 0 (Rx, Ry) (xz, yz) E2g=Δg 2 2cos(2φ) ... 0 2 2cos(2φ) ... 0 (x2‐y2, xy) E3g=Φg 2 2cos(3φ) ... 0 2 ‐2cos(3φ) ... 0 ... ... ... ... ... ... ... ... ... A1u=Σ+u 1 1 ... 1 ‐1 ‐1 ... ‐1 z A2u=Σ‐u 1 1 ... ‐1 ‐1 ‐1 ... 1 E1u=Πu 2 2cos(φ) ... 0 ‐2 2cos(φ) ... 0 (x, y) E2u=Δu 2 2cos(2φ) ... 0 ‐2 ‐2cos(2φ) ... 0 E3u=Φu 2 2cos(3φ) ... 0 ‐2 2cos(3φ) ... 0 ... ... ... ... ... ... ... ... ... Character table for D∞h point group