Molecular Symmetry

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(b) The groups Cn, Cnv, and Cnh Objects that in addition to the MOLECULAR SYMMETRY identity and an n-fold principal axis also have a horizontal mirror

plane σh belong to the groups Cnh. An example is trans-CHCl=CHCl

(6), which has the elements E, C2, • Bundet Boekfa and σh, so belongs to the group • Chem Div, Faculty Lib Arts & Sci C2h; the molecule B(OH)3 in the conformation shown in (7) belongs • Kasetsart University Kamphaeng Saen Campus to the group C3h. 1 2

Molecular Symmetry Molecular Symmetry

It process a horizontal mirror The C6H6 molecule has plane. The planar trigonal BF3 the elements E, C , 3C , molecule has the elements 6 2 3C2’, and σh together with E, C3, 3C2, and σh (with one C2 some others that these axis along each B-F bond), so elements imply, so it belongs to D3h . belongs to so belong to

D6h.

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Molecular Symmetry Molecular Symmetry

(c) The groups Dn, Dnh, and Dnd (c) The groups Dn, Dnh, and Dnd All homonuclear diatomic A molecule belongs to the group Dnd molecules, such as N2, belong to the if in addition to the elements of Dn it group D∞h because all rotations around the axis are symmetry possesses n dihedral mirror planes operations, as are end-to-end σd. The twisted, 90° allene (12) rotation and end-to-end reflection; D belongs to D2d, and the staggered conformation of ethane (13) belongs ∞h is also the group of the linear OCO and HCCH molecules and of a to D3d. uniform cylinder.Other examples of

Dnh molecules are shown in (9), (10), and (11). 5 6

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Molecular Symmetry Molecular Symmetry

(d) The groups Sn (e) The cubic groups

Molecules that have not been classified A number of very important molecules (e.g. into one of the groups mentioned so CH4 and SF6) possess more than one far, but that possess one Sn axis, principal axis. Most belong to the cubic belong to the group Sn. An example is groups, and in particular to the tetrahedral tetraphenylmethane, which belongs to groups T, Td and Th the point group S4 (14). Molecules belonging to Sn with n > 4 are rare. Note that the group S2 is the same as Ci, so such a molecule will already have been classified as Ci. 7 8

Molecular Symmetry Molecular Symmetry (e) The cubic groups (e) The cubic groups The groups Td and Oh are the groups of the tetrahedron (CH ) and the octahedron (SF ). If the object possesses A few icosahedral (20-faced) 4 6 the rotational symmetry of the tetrahedron or the molecules belonging to the octahedron, but none of their planes of reflection, then icosahedral group, the boranes and it belongs to the simpler groups T or O.

buckminsterfullerene, C60 (15).

The group This based on T but also contains a centre 9 of inversion. 10

Molecular Symmetry (f) The full rotation group

The full rotation group, R3 (the 3 refers to rotation in three dimensions), consists of an infinite number of rotation axes with all possible values of n. A

sphere and an atom belong to R3, but no molecule does. Exploring the D5h consequences of R3 is a very important way of applying symmetry arguments to at atoms, and is an alternative approach to the theory of 11 orbital angular momentum. 12

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Molecular Symmetry Molecular Symmetry Some immediate consequences of symmetry Some immediate consequences of symmetry Chirality A molecule may be chiral, and therefore Polarity only molecules belonging optically active, only if it does not to the groups Cn , Cnv, and possess an axis of improper rotation, Sn. Cs may have a permanent electric dipole moment

For Cn , Cnv, that dipole moment must lie along the symmetry axis. Thus ozone,O3, which is angular and belongs to the group C2V, may be polar (and is), but carbon dioxide, CO2, which is linear and belongs to the group D∞h, is not. 13 14

Determining the Point Group Molecular Symmetry • A flowchart for determining the point group of a molecule

Some immediate consequences of symmetry Some immediateมีแกน consequencesCn (n>2) ofมีแกน Csymmetry2 n แกนที่ตงั้ มากกวา่ 2 แกน ฉากกบแกนหลั กั 2 C s Molecules with the same list of elements are Molecule Linear n C ? n C ?  ? n>2 n 2 h D classified to the same group. nh The groups C , C and C (no rotational axis) 1 i s h ? i ? Cnh The groups C , C and C (n-fold axis) Dh Cv n nv nh  ? n  ? Cs d Dnd The groups Dn, Dnh, Dnd (n-fold axis and n i ? n  ? perpendicular C2s) Td Cnv v Dn The groups S (n-fold improper axis) i ? n Ci The cubic groups C ? S ? Oh 5 S2n 2n Tetrahedral groups (T, Td, Th) C1 Octahedral groups (O, O ) Yes h I C No Icosahedral groups (I) h n

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Molecular Symmetry Applications to molecular orbital theory and spectroscopy Character tables and symmetry labels

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Character Table Representations and Characters . Character Table is a table that characterizes the . All the operators can be written in the matrix form. different symmetry types possible in the point group. . The matrix is called a representation of an operator. • . The entries in a complete character table are derived C2v by using the formal techniques of group theory. • SO2 (px) C2V (E, C2, v v’) PS - + PA . The Matrix representative is called (n), where n is the - + dimention of the matrix + . The character of the representation matrix is the sum of - diagonal elements. PB

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Top view Side view S O S O O S O S O O

O O A S B A S B

p(S) p(O)

c2 v

 ’

v (A+B) (A-B)

B A p(O) S B S A

p(S) S

B A S

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Reduce- and Inreducible Representation

. Inspection of the representatives reveals that they are . According to the matrix representation, pA and pB are all of block-diagonal form. mixed together. . This shows that the ps is never mixed with the rest. . Using the LC, we can write the new basis as p1=pA+pB and p2=pA-pB

- + + - - + . The 3-D representative matrix ((3)) can be separated - + into (1)+ (2))

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Character Table . The characters of all representations are tabulated in a A S B character table.

C2v EC2 v v’ h=4 2 2 2 A1 1111Z z,y ,x A S B A S B A S B

A2 1 1 -1 -1 xy

B1 1-11-1X xz

B2 1-1-11y yx

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The Structure of Character Tables The Classification of LC of Orbitals Order Group Symmetry Operations . sA Class (# operations) NH3 • LCAO: s s C3v E2C3 3v h=6 B C 2 2 2 A1 111Z z, x + y this orbital is of symmetry species A1 and it contributes to A2 11-1 a1 MO in NH3. E 2 -1 0 (x,y) (xy, x2-y2),(xz,yz) . # of degerneracy of each representative is specified NO2 by the symmetry property of E operation or (E). Irreducible Symmetry • LCAO: Representations Properties () N Labels A, B: 1-D E: 2-D T: 3-D O O A (Cn) = 1 B (Cn) = -1

1 (v) = 1 2 (v) = -1

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Orbitals with nonzero overalp Vanishing Integrals & Orbital Overlap . Only orbitals of the same symmetry species may have . The value of integrals and orbital overlap is independent of nonzero overlap, so only orbitalsof the same symmetry the orientation of the molecule. species form bonding and antibonding combinations. • I is invariant under any symmetry operation of the molecule, otherwise it must be zero.

• For I not to be zero, the integrand f1f2 must have symmetry species A1. . Example: f1 = sB and f2 = sC of NH3 sB sC

not A1 . Problem: f1 = sN and f2 = sA +sB +sC of NH3

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Vanishing Integrals and Selectrion Rules

. Integrals of the form are common in quantum . In many cases, the product of functions f1 and f2 spans mechanics. a sum of irreducible representations. . For the integral to be nonzero, the product must span . In these cases, we have to decompose the reducible A1 or contain a component that span A1. representation into irreducible representations . The intensity of line spectra arises from a molecular transition between some initial state i and a final state f and depends on C2v EC2 v v’ the electric transition dipole moment fi. A2 1 1 -1 -1

B1 1-11-1 C2v EC2 v v’ B 1-11-1 1 A2+B1 200-2 z 1111

A1 1111 if fzi does not A1zB1 1-11 1 span species A1 31 32

SALC of H O AO of H O O-1s 2 2 HA-1s

a1 O-2s HA-1s + HB-1s b1 a1 a1 a 1 b1 b2 O-2py H -1s H -1s A B a b1 1 b1 a1 C2V EC2 (z) σv σ’v 1111 A1 O-2pz z

A2 1 1 -1 -1 a1

B1 1-11-1 O-2px y B2 1-1-11 b2 x

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MO of H2O

b1 a1

a1 b1 b2

a1 a1

a1 35 36

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Symmetry-adapted Linear Combinations Example of building SALC

. Symmetry-adapted linear combination (SALC) are the . s-orbitals of NH3 building blocks of LCAO-MO • Original basis are sN, sA, sB, sC . Original basis To construct the SALC from basis: NH 3 s s s s 1. Construct a table showing the effect of each operation N A B C EsN sA sB sC on each orbtial of the original basis. + N C3 sN sB sC sA C 2. To generate the combination of a specified symmetry - C3 sN sC sA sB A species, take each column in turn and:  s s s s v N A C B B ’ a) Multiply each member of the column by the character of v sN sB sA sC  “ s s s s the corresponding operation. v N C B A For A1 combination (1,1,1,1,1,1) b) Add together all the orbitals in each column with the factors as determined in a). c) Divide the sum by the order of the group.

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References 1. Peter Atkin et. al, Physical Chemistry, Oxford Univ press. 2. David O Hayward, Quantum Mechanics for Chemists, RSC, 184 p. 3. Jack Barrett, Structure and Bonding, RSC press, 178 p.