Tables for Group Theory

Tables for Group Theory By P. W. ATKINS, M. S. CHILD, and C. S. G. PHILLIPS This provides the essential tables (character tables, direct products, descent in symmetry and subgroups) required for those using group theory, together with general formulae, examples, and other relevant information. Character Tables: The Groups C, C s, C i The Groups C n (n =,,, 8) 4 The Groups D n (n =,, 4, 5, 6) 6 4 The Groups C nv (n =,, 4, 5, 6) 7 5 The Groups C nh (n =,, 4, 5, 6) 8 6 The Groups D nh (n =,, 4, 5, 6) 0 7 The Groups D nd (n =,, 4, 5, 6) 8 The Groups S n (n = 4, 6, 8) 4 9 The Cubic Groups: 5 T, T d, T h O, O h 0 The Groups I, I h 7 The Groups C v and D h 8 The Full Rotation Group (SU and R ) 9 Direct Products: General Rules 0 C, C, C 6, D, D 6, C v, C v, C 6v, C h, C h, C 6h, D h, D 6h, D d, S 6 0 D, D h 0 4 C 4, D 4, C 4v, C 4h, D 4h, D d, S 4 0 5 C 5, D 5, C 5v, C 5h, D 5h, D 5d 6 D 4d, S 8 7 T, O, T h, O h, T d 8 D 6d 9 I, I h 0 C v, D h The Full Rotation Group (SU and R ) The extended rotation groups (double groups): character tables and direct product table 4 Descent in symmetry and subgroups 6 Notes and Illustrations: General formulae 9 Worked examples Examples of bases for some representations 5 Illustrative examples of point groups: I Shapes 7 II Molecules 9 Oxford University Press, 006. All rights reserved.

Character Tables Notes: Atkins, Child, & Phillips: Tables for Group Theory () Schönflies symbols are given for all point groups. Hermann Maugin symbols are given for the crystaliographic point groups. () In the groups containing the operation C 5 the following relations are useful: η + = ( + 5 ) = 680L = cos44 o η = ( 5 ) = 0 680L = cos 7 o η + η + = + η + η η = + η η + η = η + + η = cos 7 o + cos44 o = Oxford University Press, 006. All rights reserved.

. The Groups C, C s, C i C E () A C s =C h (m) E σ h A x, y, R z x, y, z, xy A z, R x, R y yz, xz C i = S () E i A g R x, R y, R z x, y, z, xy, xz, yz A u x, y, z Oxford University Press, 006. All rights reserved.

. The Groups C n (n =,,,8) Atkins, Child, & Phillips: Tables for Group Theory C E C () A z, R z x, y, z, xy B x, y, R x, R y yz, xz C E C C ε = exp (πi/) () A z, R z x + y, z * ε ε E * ε ε (x, y)(r x, R y ) (x y, xy)(yz, xz) C 4 E C 4 C C 4 (4) A z, R z x + y, z B x y, xy E i i i i (x, y)(r x, R y ) (yz, xz) C 5 E C 5 C 5 C 5 4 C 5 ε = exp(πi/5) A z, R z x + y, z E * * ε ε ε ε * * ε ε ε ε (x,y)(r x, R y ) (yz, xz) E * * ε ε ε ε * * ε ε ε ε (x y, xy) C 6 E C 6 C C C 5 C 6 ε = exp(πi/6) (6) A z, R z x + y, z B E E * * ε ε ε ε * * ε ε ε ε * * ε ε ε ε * * ε ε ε ε (x, y) (R z, R y ) (xy, yz) (x y, xy) Oxford University Press, 006. All rights reserved. 4

. The Groups C n (n =,,,8) (cont..) C 7 E C 7 C 7 C 7 4 C 7 5 C 7 6 C 7 ε = exp (πi/7) A z, R z x + y, z * * * ε ε ε ε ε ε * E * * (x, y) ε ε ε ε ε ε (xz, yz) (R x, R y ) E * * * ε ε ε ε ε ε * * * ε ε ε ε ε ε (x y, xy) E * * * ε ε ε ε ε ε * * * ε ε ε ε ε ε C 8 E C 8 C 4 C 4 5 C C8 C 8 7 C ε = exp (πi/8) 8 A z, R z x + y, z B E * * ε i i ε ε ε * * ε i i ε ε ε E i i i i i i i i E * * ε i i ε ε ε * * ε i i ε ε ε (x, y) (R x, R y ) (xz, yz) (x y, xy) Oxford University Press, 006. All rights reserved. 5

. The Groups D n (n =,, 4, 5, 6) Atkins, Child, & Phillips: Tables for Group Theory D E C (z) C (y) C (x) () A x, y, z BB z, R z xy BB y, R y xz BB x, R x yz D E C C () A x + y, z A z, R z E 0 (x, y)(r x,, R y ) (x y, xy) (xz, yz) D 4 E C ' " 4 C( = C4) C C (4) A x + y, z A z, R z BB x y BB xy E 0 0 0 (x, y)(r x, R y ) (xz, yz) D 5 E C 5 C5 5C A x + y, z A z, R z E cos 7 º cos 44 0 (x, y)(r x, R y ) (xz, yz) E cos 44 º cos 7 0 (x y, xy) D 6 E (6) C 6 C C C C A x + y, z A z, R z BB BB E 0 0 (x, y)(r x, R y ) (xz, yz) E 0 0 (x y, xy) Oxford University Press, 006. All rights reserved. 6

4. The Groups C nν (n =,, 4, 5, 6) C ν E (mm) C σ ν (xz) σ v (yz) A z x, y, z A R z xy BB x, R y xz BB y, R x yz C ν E C σ ν (m) A z x + y, z A R z E 0 (x, y)(r x, R y ) (x y, xy)(xz, yz) C 4ν E C 4 C σ ν σ d (4mm) A z x + y, z A R z BB x y BB xy E 0 0 0 (x, y)(r x, R y ) (xz, yz) C 5ν E C 5 C 5 5σ ν A z x + y, z A R z E cos 7 cos 44 0 (x, y)(r x, R y ) (xz, yz) E cos 44 cos 7 0 (x y, xy) C 6ν E C 6 C C σ ν σ d (6mm) A z x + y, z A R z BB BB E 0 0 (x, y)(r x, R y ) (xz, yz) E 0 0 (x y, xy) Oxford University Press, 006. All rights reserved. 7

5. The Groups C nh (n =,, 4, 5, 6) C h E C I σ h (/m) A g R z x, y, z, xy BBg R x, R y xz, yz A u z BBu x, y C h ( 6) E C C σ h S 5 S ε = exp (πi/) A' R z x + y, z * * ε ε ε ε E' (x, y) (x y, xy) * * ε ε ε ε A'' z * * ε ε ε ε E'' (R * * x, R y ) (xz, yz) ε ε ε ε C 4h (4/m) E C 4 C C i 4 S σ 4 h S 4 A g R z x + y, z BBg (x y, xy) E g i i i i i i i i (R x, R y ) (xz, yz) A u z BBu E u i i i i i i i i (x, y) Oxford University Press, 006. All rights reserved. 8

BB Atkins, Child, & Phillips: Tables for Group Theory 5. The Groups C nh (n =,, 4, 5, 6) (cont ) C 5h E C 5 C 5 C 5 4 C σ 5 h S 7 5 S 5 S 5 9 S 5 ε = exp(πi/5) A R z x +y, z * * * * ε ε ε ε ε ε ε ε E * * * * ε ε ε ε ε ε ε ε (x, y) E * * * * ε ε ε ε ε ε ε ε * * * * ε ε ε ε ε ε ε ε z (x y, xy) A E * * * * ε ε ε ε ε ε ε ε * * * * ε ε ε ε ε ε ε ε (R x, R y ) (xz, yz) E * * * * ε ε ε ε ε ε ε ε * * * * ε ε ε ε ε ε ε ε C 6h (6/m) E C 6 C C C 5 C i 5 6 S 5 S σ h S 6 S ε = exp(πi/6) 6 A g x +y, z BBg (R x, R y ) (xz, yz) E g ε ε ε ε ε ε ε * * * * ε ε ε ε ε ε ε ε ε * * * * E g ε ε ε ε ε ε ε * * * * * * * * ε ε ε ε ε ε ε ε (x y, xy) ε A u Z u * * * * ε ε ε ε ε ε ε ε * * E * * u ε ε ε ε ε ε ε ε (x, y) E u * * * * ε ε ε ε ε ε ε ε * * * * ε ε ε ε ε ε ε ε Oxford University Press, 006. All rights reserved. 9

6. The Groups D nh (n =,, 4, 5, 6) D h E C (z) C (y) C (x) i σ(xy) σ(xz) σ(yz) (mmm) A g x, y, z BBg R z xy BBg R y xz BBg R x yz A u BBu z BBu y BBu x D h ( 6) m E C C σ h S σ v A x + y, z A R z E 0 0 (x, y) (x y, xy) A A z E 0 0 (R x, R y ) (xy, yz) D 4h E C 4 C C C i S 4 σ h σ v σ d (4/mmm) A g x + y, z A g R z x y BBg BBg xy E g 0 0 0 0 0 0 (R x, R y ) (xz, yz) A u A u Z BBu BBu E u 0 0 0 0 0 0 (x, y) Oxford University Press, 006. All rights reserved. 0

6. The Groups D nh (n =,, 4, 5, 6) (cont ) Atkins, Child, & Phillips: Tables for Group Theory D 5h E C 5 C 5C 5 σ h S 5 S 5σ v A x + y, z A R z E cos 7 cos 44 0 cos 7 cos 44 0 (x, y) E cos 44 cos 7 0 cos 44 cos 7 0 (x y, xy) A A z E cos 7 cos 44 0 cos 7 cos 44 0 (R x, R y ) (xy, yz) E cos 44 cos 7 0 cos 44 cos 7 0 5 D 6h (6/mmm) E C 6 C C C C i S S 6 σ h σ d σ v A g x + y, z A g R z BBg BBg E g 0 0 0 0 (R x R y ) (xz, yz) E g 0 0 0 0 (x y, xy) A u A u z BBu BBu E u 0 0 0 0 (x, y) E u 0 0 0 0 Oxford University Press, 006. All rights reserved.

7. The Groups D nd (n =,, 4, 5, 6) D d = V d ( 4) m E S 4 C C σ d A x + y, z A R z BB x y BB z xy E 0 0 0 (x, y) (xz, yz) (R x, R y ) D d ()m E C C i S 6 σ d A g x + y, z A g R z E g 0 0 (R x, R y ) (x y, xy) (xz, yz) A u A u z E u 0 0 (x, y) D 4d E S 8 C 4 S C 4C 4σ 8 d A x + y, z A R z BB BB z E 0 0 0 (x, y) E 0 0 0 0 (x y, xy) E 0 0 0 (R x, R y ) (xz, yz) Oxford University Press, 006. All rights reserved.

7. The Groups D nd (n =,, 4, 5, 6) (cont..) D 5d E C 5 C 5C 5 i S0 S 0 5σ d A g x + y, z A g R z E g cos 7 cos 44 0 cos 7 cos 44 0 (R x, R y ) (xy, yz) E g cos 44 cos 7 0 cos 44 cos 7 0 (x y, xy) A u A u z E u cos 7 cos 44 0 cos 7 cos 44 0 (x, y) E u cos 44 cos 7 0 cos 44 cos 7 0 D 6d E S C 6 S 4 C 5 S C 6C 6σ d A x + y, z A R z BB BB z E 0 0 0 (x, y) E 0 0 (x y, xy) E 0 0 0 0 0 E 4 0 0 E 5 0 0 0 (R x, R y ) (xy, yz) Oxford University Press, 006. All rights reserved.

8. The Groups S n (n = 4, 6, 8) Atkins, Child, & Phillips: Tables for Group Theory S 4 (4) E S 4 C S 4 A R z x + y, z B z (x y, xy) E i i i i (x, y) (R x, R y ) (xz, yz) S 6 () E C C i 5 S S 6 ε = exp (πi/) 6 A g R z x + y, z * * ε ε ε ε E g * * (R x, R y ) (x y, xy) (xy, yz) ε ε ε ε A u z * * ε ε ε ε E u * * (x, y) ε ε ε ε S 8 E S 8 C 4 S C 5 8 S 8 C 4 7 S ε = exp (πi/8) 8 A R z x + y, z B z * * ε i ε ε i ε E * * ε i ε ε i ε (x, y) i i i i E i i i i (x y, xy) E * * i i ε ε ε ε * * ε i ε ε i ε (R x, R y ) (xy, yz) Oxford University Press, 006. All rights reserved. 4

9. The Cubic Groups Atkins, Child, & Phillips: Tables for Group Theory T () E 4C 4C C ε = exp (πi/) A x + y + z * ε ε E * ( (x y )z x y ) ε ε T 0 0 (x, y, z) (R x, R y, R z ) (xy, xz, yz) T d E 8C C 6S 4 (4 m) A x + y + z A E 0 0 (z x y, (x y ) T 0 (R x, R y, R z ) T 0 (x, y, z) (xy, xz, yz) 6σ d T h E 4C 4C C i 4S 6 4S σ d ε = exp (πi/) 6 (m) A g x + y + z * ε ε E g ε ε (z x y, * * ε ε ε ε (x y ) T g 0 0 0 0 (R x, R y, R z ) (xy, yz, xz) A u * * * ε ε ε ε E u * * ε ε ε ε T u 0 0 0 0 (x, y, z) O E (4) 8C C 6C 4 6C A x + y + z A E 0 0 (z x y, (x y )) T 0 (x, y, z) (R x, R y, R z ) T 0 (xy, xz, yz) Oxford University Press, 006. All rights reserved. 5

9. The Cubic Groups (cont ) Atkins, Child, & Phillips: Tables for Group Theory O h (mm) E 8C 6C 6C 4 C ( = C4 ) i 6S 4 8S 6 σ h 6σ d A g x + y + z A g E g 0 0 0 0 (z x y, (x y )) T g 0 0 (R x, R y, R z ) T g 0 0 (xy, xz, yz) A u A u E u 0 0 0 0 T u 0 0 (x, y, z) T u 0 0 Oxford University Press, 006. All rights reserved. 6

0. The Groups I, I h I E C 5 C 0C 5 5C η ± 5 = ± A x +y +z T η + η 0 (x, y, z) (R x, R y, R z ) T η η + 0 G 4 0 H 5 0 0 (z x y, (x y ) xy, yz, zx) I h E C 5 C 0C 5 5C i S 0 S 0S 0 6 5 σ η ± 5 = ± A g x + y + z T g η η + 0 η + η 0 T g η + η 0 η η + 0 (R x,r y,r z ) G g 4 0 4 0 H g 5 0 0 5 0 0 (z x y, (x y )) (xy, yz, zx) A u T u η + η 0 η η + 0 (x, y, z) T u η η + 0 η + η 0 G u 4 0 4 0 H u 5 0 0 5 0 0 Oxford University Press, 006. All rights reserved. 7

. The Groups C v and D h C v E C C φ σ v A + z x + y, z A R z E Π cos φ 0 (x,y) (R x,r y ) (xz, yz) E Δ cos φ 0 (x y, xy) E Φ cos φ 0 D h E C φ σ v i S φ C + Σ x + y, z g Σ R g z g cos φ 0 cos φ 0 (R x, R y ) (xz, yz) Δ g cos φ 0 cos φ 0 (x y, xy) + Σ u z Σ u u cos φ 0 cos φ 0 (x,y) Δ u cos φ 0 cos φ 0 Oxford University Press, 006. All rights reserved. 8

. The Full Rotation Group (SU and R ) sin j + φ ( j) φ 0 χ ( φ ) = sin φ j + φ= 0 Notation : Representation labelled Γ (j) with j = 0,/,, /,, for R j is confined to integral values (and written l or L) and the labels S Γ (0), P Γ (), D Γ (), etc. are used. Oxford University Press, 006. All rights reserved. 9

Direct Products. General rules (a) For point groups in the lists below that have representations A, B, E, T without subscripts, read A = A = A, etc. (b) g u g g u u g (c) Square brackets [ ] are used to indicate the representation spanned by the antisymmetrized product of a degenerate representation with itself. Examples For D h E E A + A + E For D 6h E g E g = B g + E g.. For C, C, C 6, D, D 6,C v,c v, C 6v,C h, C h, C 6h, D h, D 6h, D d, S 6 A A BB BB E E A A A BB BB E E A A BB BB E E BB A A E E BB A E E E A + [A ]+ E BB + B + E E A + [A ] + E. For D, D h A B BB BB A A B BB BB BB A B BB BB A B BB A Oxford University Press, 006. All rights reserved. 0

4. For C 4, D 4, C 4v, C 4h, D 4h, D d, S 4 A A BB BB E A A A BB BB E A A BB BB E BB A A E BB A E E A + [A ] +B + B 5. For C 5, D 5, C 5v, C 5h, D 5h, D 5d A A E E A A A E E A A E E E A + [A ] + E E + E E A + [A ] + E 6. For D 4d, S 8 A A BB BB E E E A A A BB BB E E E A A BB BB E E E BB A A E E E BB A E E E E A + [A ] + E E + E BB + B + E E A + [A ] + E + E BB + B E A + [A ] + E 7. For T, O, T h, O h, T d A A E T T A A A E T T A A E T T E A + [A ] + E T + T T + T T A + E + [T ] + T A + E + T + T T A + E + [T ] + T Oxford University Press, 006. All rights reserved.

8. For D 6d A A BB BB E E E E 4 E 5 A A A BB BB E E E E 4 E 5 A A BB BB E E E E 4 E 5 BB A A E 5 E 4 E E E BB A E 5 E 4 E E E E A + [A ] + E E + E E + E 4 E + E 5 BB + B + E 4 E A + [A ] E + E 5 BB + B + E + E 5 + E 4 E E A + [A ] + BB + B E + E 5 E + E 4 E + E E 4 A + [A ] + E 4 E 5 A + [A ] + E 9. For I, I h A T T G H A A T T G H T A + [T ] + G + H T + G + H T +T + G + H H T A + [T ] + H T + G + H T +T + G + H G A + [T +T ] T +T + G + H + G + H H A + [T +T + G] + G + H 0. For C v, D h Σ + Σ Π Δ Σ + Σ + Σ Π Δ Σ Σ + Π Δ Π Σ + + [Σ ] + Δ Δ : Notation Π + Φ Σ + + [Σ ] + Γ Σ Π Δ Φ Γ Λ = 0 4 Λ Λ = Λ Λ + (Λ + Λ ) Λ Λ = Σ + + [Σ ] + (Λ). Oxford University Press, 006. All rights reserved.

. The Full Rotation Group (SU and R ) Γ (j) Γ (j ) = Γ (j + j ) + Γ (j + j ) + + Γ ( j j ) Γ (j) Γ (j) = Γ (j) + Γ (j ) + + Γ (0) + [Γ (j ) + + Γ () ] Oxford University Press, 006. All rights reserved.

Extended rotation groups (double groups): Character tables and direct product tables * D E R C (z) C (y) C (x) E / 0 0 0 * D E R C C R C C R E / 0 0 E / i i i i D 4 E R C 4 C 4 R C 4C 4C E / E / 0 0 0 0 0 0 * D E R C 6 6 C 6 R C C R C 6C 6C E / 0 0 0 E / 0 0 0 E 5/ 0 0 0 0 0 * T E R 8C d 8C R 6C 6S 4 6S 4 R σ d * O E R 8C 8C R 6C 6C 4 6S 4 R C E / 0 0 E 5/ 0 0 G / 4 4 0 0 0 0 E / E / = [A] +B +B + B E / E / E / [A ] + A + E E E / [A ] + A + A E / E / E / [A ] + A + E BB + B + E E / [A ] + A + E Oxford University Press, 006. All rights reserved. 4

E / E / E 5/ E / [A ] +A + E BB + B + E E + E E / [A ] +A + E E + E E 5/ [A ] + A +B + B E / E 5/ E / E / [A ] + T A + T E + T + T E 5/ [A ] + T E + T + T G / [A + E + T ] + A + T + T ] * Direct products of ordinary and extended representations for T d and O* A A E T T E / E / E 5/ G / E / + G / E 5/ + G / E 5/ E 5/ E / G / E 5/ + G / E / + G / G / G / G / E / + E 5/ + G / E / + E 5/ + G / E / + E 5/ + G / Oxford University Press, 006. All rights reserved. 5

Descent in symmetry and subgroups The following tables show the correlation between the irreducible representations of a group and those of some of its subgroups. In a number of cases more than one correlation exists between groups. In C s the σ of the heading indicates which of the planes in the parent group becomes the sole plane of C s ; in C v it becomes must be set by a convention); where there are various possibilities for the correlation of C axes and σ planes in D 4h and D 6h with their subgroups, the column is headed by the symmetry operation of the parent group that is preserved in the descent. C v C C s σ(zx) C s σ(yz) A A A A A A A" A" BB B A A BB B A" A" C v C C s A A A A A A" E E A + A" C 4v C v σ v C v σ d A A A A A A BB A A BB A A B B B + B E + [Other subgroups: C 4, C, C 6 ] D h C h C v σ h σ ν C ν A A A A A A A A A BB A A" E' E' E A + B A' A'+ A" A A" A A A" A" A A" A BB A" A E" E" E A + B A" A'+ A" [Other subgroups: D, C, C ] C s σ h C s σ ν Oxford University Press, 006. All rights reserved. 6

D 4h Dd Dd C ( C ) C ( C ) D C h D h C D C D C C 4h C 4v C v C, σ v C v C, σ d A g A A A g A g A A A g A A A A g A A BBg BBg BB BB A g A A A BBg BB BB A g BBg A B BBg BB A A BBg BB BB BBg A g BB A B g BB A A E g E E BBg + B g BBg + B g BB + B BB + B E g E BB + B BB + B A u BB BB A u A u A A A u A A A A u BB BB BBu BBu BB BB A u A A A BBu A A A u BBu A B BBu BB A A BBu A A BBu A u BB A B u BB A A E u E E BBu + B u BBu + B u BB + B BB + B E u E BB + B BB + B Other subgroups:d 4, C 4, S 4, C h, C s,c,c i, (C v ) D 6 DdC DdC D h σ h σ(xy) σ v σ(yz) C 6v C v σ v C v C C v C C h C C h C C h C A g A g A g A g A A A A A g A g A g A g A g A g BBg A A BB BB A g BBg BBg BBg A g A g BBg BB A A BB BBg A g BBg BBg A g A g BBg BB A BB A BBg BBg A g E g E g E g BBg + B g E E A + B A + B B g A g + B g A g + B g E g E g E g A g + B g E E A + B A + B A g A g + B g A g + B g A u A u A g A u A A A A A u A u A u A u A u A g BBu A A BB BB A u BBu BBu BBu A u A u BBu BB A BB BB BBu A u BBu BBu A u A u BBu BB A A A BBu BBu A u E u E u E u BBu + B u E E A + B A + B B u A u + B u A u + B u E u E u E u A u + B u E E A + B A + B A u A u + B u A u + B u Other subgroups: D 6, D h, C 6h, C 6, C h, D, S 6, D, C, C, C g, C i T d T D d C v C v A A A A A A A B A A E E A + B E A +A T T A + E A + E A + B + B T T BB + E A + E A + B + B Other subgroups: S 4, D, C, C, C s. Oxford University Press, 006. All rights reserved. 7

O h O T d T h D 4h D d A g A A A g A g A g A g A A A g BBg A g E g E E E g A g + B g E g T g T T T g A g + E g A g + E g T g T T T g BBg + E g A g + E g A u A A A u A u A u A u A A A u BBu BBu E u E E E u A u + B u E u T u T T T u A u + E u A u + E u T u T T T u BBu + E u A u + E u Other subgroups: T, D 4, D d, C 4h, C 4v, D h, D, C v, S 6, C 4, S 4, C v, D, C h, C, C, S, C s R O D 4 D S A A A P T A + E A + E D E + T A + B +B + E A + E F A + T +T A + B +B + E A + A + E G A + E + T + T A + A +B + B + E A + A + E H E + T + T A + A + B + B + E A + A + 4E Oxford University Press, 006. All rights reserved. 8

Notes and Illustrations General Formulae (a) Notation h Γ (i) Χ (i) (R) () i D ( R) μν Atkins, Child, & Phillips: Tables for Group Theory the order (the number of elements) of the group. labels the irreducible representation. the character of the operation R in Γ (i). the μv element of the representative matrix of the operation R in the irreducible representation Γ (i). l i the dimension of Γ (i). (the number of rows or columns in the matrices D (i) ) (b) Formulae (i) Number of irreducible representations of a group = number of classes. (ii) l i i = h (iii) () i () i χ ( R) = Dμμ ( R) μ (iv) Orthogonality of representations: () i * ( i' ) D ( R) D ( R) = ( h/ l) δ δ μν μ' ν' i ii' μμ' νν' (δ ij = if i = j and δ ij = 0 if i j δ (v) Orthogonality of characters: () i * () i χ ( R) χ ( R) = hδii' R (vi) Decomposition of a direct product, reduction of a representation: If Γ = i and the character of the operation R in the reducible representation is χ(r), then the coefficients a t are given by () i * a = (/ l h) χ ( R) χ( R). i R a Γ i () i Oxford University Press, 006. All rights reserved. 9

(vii) Projection operators: The projection operator Atkins, Child, & Phillips: Tables for Group Theory P = ( l / h) χ ( R) R () i () i * i R when applied to a function f, generates a sum of functions that constitute a component of a basis for the representation Γ (i) ; in order to generate the complete basis P (i) must be applied to l i distinct functions f. The resulting functions may be made mutually orthogonal. When l i = the function generated is a basis for Γ (i) without ambiguity: () i () i P f = f (viii) Selection rules: If f (i) is a member of the basis set for the irreducible representation Γ (i), f {k) a member of that for Γ (k), and ˆΩ (j) an operator that is a basis for Γ (j), then the integral ()* i ˆ ( j) ( k) dτ f Ω f is zero unless Γ (i) occurs in the decomposition of the direct product Γ (j) Γ (k) (ix) The symmetrized direct product is written s () i () i Γ Γ, and its characters are given by () i s () () () R i R = i i R + R χ ( ) χ ( ) χ ( ) χ ( ) a () i () i The antisymmetrized direct product is written Γ Γ and its characters are given by () i a () () () R i R = i i R + R χ ( ) χ ( ) χ ( ) χ ( ) Oxford University Press, 006. All rights reserved. 0

Worked examples Atkins, Child, & Phillips: Tables for Group Theory. To show that the representation Γ based on the hydrogen s-orbitals in NH (C v ) contains A and E, and to generate appropriate symmetry adapted combinations. A table in which symmetry elements in the same class are distinguished will be employed: C v E C + C σ σ σ A A E 0 0 0 D(R) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x(r) 0 0 Rh h h h h h h Rh h h h h h h The representative matrices are derived from the effect of the operation R on the basis (h, h, h ); see the figure below. For example 0 0 + C( h, h, h) = ( h, h, h) = ( h, h, h) 0 0 0 0 According to the general formula (b)(iii) the character χ(r) is the sum of the diagonal elements of D(R). For example, χ(σ ) = 0 + + 0 =. The decomposition of Γ follows from the formula (b)(vi): Γ = a A + a A + a E E where a = 6 { + 0+ } = a = 6 { + 0+ ( )} = 0 a = + ( ) 0+ 0 = E 6 { } Oxford University Press, 006. All rights reserved.

Therefore Γ = A + E Symmetry adapted combinations are generated by the projection operator in (b)(vii). Using the last two rows of the table, ( A ) 6 φ ( A ) = h = ( h + h + h + h + h + h ) = ( h + h + h ) ( E ) φ ( E ) = h = 6 ( h h h + 0 h + 0 h + 0 h ) = ( h h h ) ( E ) ( ) φ E = h = 6 ( h h h + 0 h 0 h + + 0 h ) = ( h + h h ) φ(e) and φ'(e) provide a valid basis for the E representation, but the orthogonal combinations φ ( E) = (/6) ( h h h ) = (/) φ( E) a { φ φ } φ ( E) = (/ ) ( h h ) = (/ ) ( E ) + '( E) b would be a more useful basis in most applications.. To determine the symmetries of the states arising from the electronic configurations e and e t for a tetrahedral complex (T d ), and to determine the group theoretical selection rules for electric dipole transitions between them. The spatial symmetries of the required states are given by the direct products in Table 7. E E = A + [A ] + E E T = T + T Combination of the electron spins yields both singlet and triplet states, but for the e configuration some possibilities are excluded. Since the total (spin and orbital) state must be antisymmetric under electron interchange, the antisymmetrized spatial combination [A ] must be a triplet, and the symmetrized combinations A and E are singlets. For the e t configuration there are no exclusions. The required terms are therefore e A + A + E e t T + T + T + T The selection rules are obtained from formula (b)(viii). For electric dipole transitions the operator Ω (j) has the symmetry of a vector (x, y, z), which from the character table for T d transforms as T. From the table of direct products, Table 7, A T = T A T = T E T = E T = T + T Assuming the spin selection rule ΔS = 0, the allowed transitions are e A e t T e A e t T e E e t T, T Oxford University Press, 006. All rights reserved.

. To determine the symmetries of the vibrations of a tetrahedral molecule AB 4, and to predict the appearance of its infrared and Raman spectra. The molecule is depicted in the figure below and the character table for the point group T d is given on page 5. The representations spanned by the vibrational coordinates are based on the 5 cartesian displacements less the representations T and T, which are accounted for by the rotations (R x, R y, R z ) and the translations (x, y, z). The stretching vibrations are the subset based on the 4 bonds of the molecule. For a particular symmetry operation, only atoms (or bonds) that remain invariant can contribute to the character of the cartesian displacement representation, Γ (all) (or the stretching representation, Γ (stretch) ). C : Two atoms invariant, x, y, z, interchanged χ (all) (C ) = 0 One bond invariant χ (stretch) (C ) = C (z): Central atom invariant; x, y, sign reversed, z invariant χ (all) (C ) = 0 No bonds invariant χ (stretch) (C ) = 0 S 4 (z): Central atom invariant; x, y, interchanged, z sign reversed x (all) (S 4 ) = No bonds invariant χ (stretch) (S 4 ) = 0 σ d (z): Three atoms invariant; x, y, interchanged, z invariant x (all) (σ d ) = Two bonds invariant χ (stretch) (σ d ) = The characters of the representations Γ (all) and Γ (stretch) are therefore E 8C C 6S 4 6σ d Γ (all) 5 0 = A + E + T + T Γ (stretch) 4 0 0 = A + T Γ (ali) and Γ (stretch) have been decomposed with the help of formula (b)(vi) (compare Example ). Allowing for the rotations and translations contained in Γ (all) there are therefore four fundamental vibrations, conventionally labelled ν (A ), ν (E), ν (T ), and ν 4 (T ). ν and v are stretching and bending vibrations respectively, ν and ν 4 involve both stretching and bending motions. Oxford University Press, 006. All rights reserved.

The selection rule (b)(viii) gives the spectroscopic properties of the vibrations. Infrared activity is induced by the dipole moment (a vector with symmetry T, according to the character table for ( ) T d ) as the operator Ωˆ j ( ) In the case of the Raman effect, Ω ˆ j is the component of the polarizability tensor (A + E + T ). f (i) is the ground vibrational state (A ), and f (k) is the excited state (with the same symmetry as the vibration in the case of the fundamental; as the direct product of the appropriate representations in the case of an overtone or a combination band). v (A )and v (E) are therefore Raman active and ν (T ) and ν 4 (T ) are infrared and Raman active. The following overtone and combination bands are allowed in the infrared spectrum: ν + ν, ν + ν 4, ν + ν, ν + ν 4, ν, ν + ν 4, ν 4 Oxford University Press, 006. All rights reserved. 4

Examples of bases for some representations The customary bases polar vector (e.g. translation x), axial vector (e.g. rotation R x ), and tensor (e.g. xy) are given in the character tables. It may be of some assistance to consider other types of bases and a few examples are given here. Base Irreducible Representation A in T d x()y() x()y() A in C 4v The normal vibration of an octahedral molecule represented by A lg in O h 4 The three equivalent normal vibrations of an octahedral molecule, one of which is represented by T u in O h Oxford University Press, 006. All rights reserved. 5

5 The π-orbital of the benzene molecule represented by A u in D 6h 6 The π-orbital of the benzene molecule represented by BBg in D 6h 7 The π-orbital of the naphthalene molecule represented by A u in D h Oxford University Press, 006. All rights reserved. 6

Illustrative Examples of Point Groups I Shapes The character tables for (a), C n, are on page 4; for (b), D n, on page 6; for (c), C nv, on page 7; for (d), C nh, on page 8; for (e), D nh, on page 0; for (f), D nd, on page ; and for (g), S n, on page 4. Oxford University Press, 006. All rights reserved. 7

C s C i. T d O h tetrahedron cube O h I h octahedron dodecahedron I h R icosahedron sphere The character table for C s is on page, for C i on page, for T d on page 5, for O h on page 6, for I h on page 7, and for R on page 9. Oxford University Press, 006. All rights reserved. 8

II Molecules Point group Example Page number for character table C CHFClBr C s BFClBr (planar), quinoline C i meso-tartaric acid C H O, S C (skew) 4 C v H O, HCHO, C 6 H 5 C 7 C v NH (pyramidal), POC 7 C 4v SF 5 Cl, XeOF 4 7 C h trans-dichloroethylene 8 C h H O H 8 O B O H (in planar configuration) D h trans-ptx Y, C H 4 0 D h BF (planar), PC 5 (trigonal bipyramid), :: 5 trichlorobenzene 0 D 4h AuCl 4 (square plane) 0 D 5h ruthenocene (pentagonal prism), IF 7 (pentagonal bipyramid) D 6h benzene D d CH =C=CH D 4d S 8 (puckered ring) D 5d ferrocene (pentagonal antiprism) S 4 tetraphenylmethane 4 T d CCl 4 5 O h SF 6, FeF 6 6 I h BBH 7 C v HCN, COS 8 D h CO, C H 8 R any atom (sphere) 9 Oxford University Press, 006. All rights reserved. 9