Grand Orthogonality on Representations (Gi)

III-19

Grand Orthogonality on Representations (Gi)

Great restriction on form of irreducible representation

also on number: n = number of irreducible representations

Easier with Characters

all elements of a class, same C

Nc = number in class

Now: Characters of each irreducible representation form a vector orthogonal to other irreducible representations. These are h-dimensional vectors but have only n independent elements thus these vectors can define an n-dimensional space. That then is the number of irreducible representations.

or number of classes = number of irreducible representations

Character Table generation

Number of irreducible representations = number of classes

C2n h = 4, n = 4

E, C2, s2, s2'

all in class by self

analogy: get set of 4 orthogonal (linearly independent)

4-vectors over 4-D vector - space

C3u h = 6, n = 3

E - class by self

C3, C32 – same class

s1, s2, s3 – same class

C4u h = 8, n = 5

E – self

C4, C43 (= C4-1) – class

C2 – class

s1, s3 – same class = su

s2, s4 – same class = sd

(must be 2d)

Reducible Representations → similarity transfer → same character so sum them

Cr (R) = ai Ci (R)

(reducible) (sum. irreducible)

setup Cr (R) Cj* (R) = ai Ci (R) Cj* (R)

= ai [Ci (R) Cj* (R)] dij h

Cr (R) Cj* (R) = aj h

by classes aj = Nk Cr (Ck) Cj* (Ck)

Means of reducing reducible representation to sum of irreducible representation

Effect of symmetry operation or w/f is ±1 for 1D

eigen value Rf = ±f

Since symmetry leaves molecule as it was → H ~ A1 total symmetry

RH = HR (total symmetry not affect R)

[R,H = 0] Þ R + H simultaneous eigen function

Integrals

or

to be non-zero f(x) = fi Hfj (full ???) must by symmetrical

in group theory – product must contain A1

problem – determine representation functions, products

Reducing reducible representation → irreducible representation

Cr (R) = ai Ci (R)

ai = Cr (R) Ci* (R) or ai = Cr (Ck) Ci (Ck)

examples – Consider

Consider interchange matrix of H’s in NH3

GH is reducible, what is it composed of?

aj* = Nk CH (Ck) Cj (Ck)

a1 = [ 1 • 3 • 1 + 2 • 0 • 1 + 3 • 1 • 1 ] = = 1

a2 = [ 1 • 3 • 1 + 2 • 0 • -1 + 3 • 1 • -1 ] = = 1

a3 = [ 1 • 3 • 2 + 2 • 0 • 1 + 3 • 1 • 0 ] = = 1

thus GH = G1 + G3

1D + 2D = 3D

Note free G(E) being identity matrix this always works

or in H2O → x y z form bases as does matrices

Cxyz = 3, -1, 1, 1

aG1 = [ 3 • 1 • 1 + (-1) • 1 • 1 + 1 • 1 • 1 + 1 • 1 • 1 ] = = 1

aG2 = [ 3 • 1 + (-1)(-1) + 1 • 1 + 1 • (-1) ] = 1

aG3 = [ 3 • 1 + (-1)(-1) + 1(-1) + 1(-1) ] = 1

aG4 = [ 3 • 1 + (-1)1 + 1(-1) + 1 (-1) ] = 0

so Gxyz = G1 + G2 + G3 = A1 + B1 + B2

In general this particular decomposition will prove to be very useful for analyses of normal modes and for A orbitals → MO’s

Xxyz often in character tables

recall m ~ (x, y, z) – important for determining dipole transitions

General behavior in multiplying representations

1D x 1D → 1D → irreducible trial by inspection

1D x 2D (3D) → 2(3)D → usually irreducible trial by inspection

2D x 2D (or higher) → dimension is product→need to reduce

GEE = 4 1 0 Þ A1 + A2 + E → character table

Characters only determined by diagonal / diagonal reducible only dependent on diagonal irreducible

Bottom line can just use character multiply from Z-1GAB Z = G'AB – same characters (???)

ai Ci (R) = CAB (R) = CA(R) CB(R)

example XE • XE = 2 • 2, (-1)(-1), 0 • 0 = 4, 1, 0 → same

XE • XA2 = 2 • 1, (-1)(1), 0(-1) = 2, -1, 0 → E

Goal of integrals: ∫ fE fE dt → A1 + A2 + E Ì A1 ¹ 0

∫ fE a fA2 → GE Ä Ga Ä GA2 if a ~ E

Can show aA1 = 1 when GA = GB A1 Î GAB if GA = GB

aA1 = 0 when GA ¹ GB

So for 0 ≠ ∫ ¦A ¦B ¦C … dt need

a) f’s all A1 symmetry

b) product function of same symmetry eg: GA = GB ∫ ¦A ¦B

or: GA Ä GB Ì GC for ∫ ¦A ¦B ¦C etc.

Subgroup: C3: E, C3, C32 s3: E, sn

combine operators → s • C3 times: E C3 C32 sn sn C3 sn C32 s3 s32

Direct Product Group

Problem Sets

We’re finishing up “technique” development before going on to applications

Last time talked about products of representations to learn how to evaluate ∫ fi fj … dt .

general:

a) if GA Ä GB = GAB GAB Î GA1 (totally symmetrical)

b) Nature of GAB from reducing it to GAB = linear combination irreducible representation by using characters

Now if know what symmetries is some combined representation wait to know form as applied to case of interest.

Projection operators used to give linear combination of (in our case, functions) that give a representation “Symmetry Adapted Linear Combination”

{¦i} forms function space if all g in space: g = å ci ¦i

n-dimensional: {¦i} i = ℓ – n, ¦i independent

scalar product: = ∫ ¦*g dt

Direct Product groups (PJS notes)

This same multiplication can be seen to apply to groups

Consider C2h – E C2 i sn

subgroup C2 (E, C2)

Ci (E, i)

Now take the product – C2 • Ci

E • (E, i) = E, i C2 • (E, i) = C2, sn

so sum is C2h, same for characters

analogous to matrix direct product done for reducible.

Note symmetry in Character table

Projection Operators

Now that we know how to find which irreducible representations in reducible representation – how find what they are?

i.e. if symmetry adapted wave functions can make simpler integrals ( ≠ or = 0) what are they?

eg: H2O stretch O-H (R1 + R2) + (R1 – R2)

??? are representation A1 B1

This by inspection – what about more complex?

NH3 2px, 2py

C3 interconnects px, py must belong to same representation Þ E but how?

C3px = cx px + cy py

What are Cx + Cy / how make p1 = Cx1 px + Cy1 py, p2 = Cx2 px + Cy2 py ^?

Symmetry Adapted Linear Combinations

function space – need to collect all functions interconnected by symmetry operations

h ≠ S cj ¦j then h independent {¦j} , ¦j span space

math: ¦(x) = S an sinn x + bn cosn x – fourier space ¥ d n

= a + b 2 vector space

function space: {¦i} n = 1 … n ¦i linear independent basis function

g = S ci ¦i

a(g + h) = S a(ci + di) ¦i

if orthogonal normal ci = =

Solution to Schrödinger Equation → set of orthogonal basis function

these span a function space

Subspaces – functions that interconnect under an operation or have a common characteristic

e.g. Fourier d = S bn cosn x → all even function

Schrödinger Eq yn = S cj fnj → all function same energy

Hyn = En ym

Conversely → any linear combination of eigen function H same eigen value

also an eigen function of H

Subsets equivalent energy: R fnj = S cj fnj no change E

from [R, H] = 0

e.g. atoms 2p + 2s in H, 2p in Ng

Method – symmetry operation yield linear combination

1-D Rfi = ±fi (dependent on irreducible representation)

multi Rfi =

generally set of {fj} will be irreducible representation (unless ??? degeneration)

general Rftk = Cst(R) m x m matrix

set of matrices Cstk (R) (all R’s) form representation of group

reformulate: Rfti =

multiply by sum over R

Projection operation:

= dij dtt' fsi