Symmetry, Group Theory And Applications

Front 1

Define symmetry operation?

Back 1

operation which, when applied to an object, leaves it indistinguishable from the original object

Front 2

The five symmetry operations + symbols

Back 2

Identity (E), proper rotation(Cn), reflection(σ), inversion(i) and improper rotation(Sn)

Front 3

What coordinate system are we using

Back 3

right hand, thumb upwards is z axis, x is the index finger pointing forwards and y is the middle finger pointing sideways

Front 4

Types of mirror planes?

Back 4

vertical - contains the principle axis

horizontal - perpendicular to the principal axis

dihedral mirror- special case of a vertical plane where it bisects two C₂ axis

Front 5

S1 and S2?

Back 5

S1 = σh

S2 = i

Front 6

What operations does a chiral compound not have?

Back 6

i, σ, Sn

Front 7

Combining symmetry operations?

Back 7

the product of a symmetry operation must itself be a symmetry operation

Front 8

Discuss associative multiplication?

Back 8

PQR = P(QR) = (PQ)R

Front 9

What is a point group?

Back 9

complete set of symmetry operations exhibited by an isolated molecule

Front 10

finding the representation of product function such as xy for C2v?

Back 10

E(xy) = (Ex)(Ey) = (x)(y) = xy . (1)

C₂(xy) = (C₂x)(C₂y) = (-x)(-y) = xy . (1)

σxz(xy) = (σxyx)(σxyy) = (x)(-y) = xy (-1)

σyz(xy) = (σyzx)(σyzy) = (-x)(y) = xy (-1)

1 1 -1 -1

Front 11

Character for operation R in representation Γ? (notation)

Back 11

Χ^Γ(R)

Where Γ is the row (symmetry species e.g. A, Eg)

R is the columen (operation e.g. E, C₂)

Front 12

what do u and g mean?

Back 12

gerade and ungerade, g for those symmetric with respect to inversion and ungerade antisymmetric

Front 13

How to get reducible representation?

Back 13

-form basis set

-apply symmetry operation

-if orbital is transformed into a different orbital then the contribution is 0

-if the sign changes = -1

-if the same = 1

Front 14

How to get irreducible representation?

Back 14

n = 1/h * ∑ Χ^Γ(R)*Χ(R)

Front 15

determine SALCs?

Back 15

SALC with Γ symmetry =
∑Χ^Γ(R)(Rs1)

e.g.

Es1 = s1

C2s1 = s2

σxzs1 = s2

σyzs1 = s1

Salc = 2(s1 + s2)

Front 16

Problem with C₃ symmetry operation?

Back 16

it mixes x and y giving two-dimensional irrep but we can still get a value out of it

e.g. pg 60 handout

Front 17

Why are degenerate representations important?

Back 17

If two orbitals transform at E then if one satisfies the Schrodinger equation then the other will also satisfy it, with the same energy

Front 18

Integrating over a function?

Back 18

If a function forms a basis for an irrudecible representation then an integral over the function is zero unless it transforms as the totally symmetric representation

Front 19

Discuss the product of irreducible representations? and integrals?

Back 19

product with itself always contains the totally symmetric representation which has a none zero integral

product of two different irreducible representations never contains the totally symmetric representation so the integral is always zero

Front 20

How do we construct MO diagrams using Group theory?

Back 20

e.g. H₂O - consider the interaction between the atomic orbitals on O and the SALCs of H 1s orbitals assigning symmetry to all the orbitals

Front 21

How do we know if two orbitals will interact?

Back 21

Hamiltonian integral is zero unless the orbitals have the same symmetry

Front 22

What basis set do we use for p orbitals on atoms that are not the centre atom?

Back 22

radial, tagential and out of plane

Front 23

Walsh diagram?

Back 23

Shows how a MO diagram changes when the geometry is changed e.g. AH₂ from linear to bend (D∞h to C2v)

Front 24

How does π donor and acceptors affect crystal field sylitting?

Back 24