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Chimica Inorganica 3

A symmetry operation carries the system into an equivalent configuration, which is, by definition physically indistinguishable from the original configuration. Clearly then, the energy of the system must be the same before and after carrying out the symmetry operation!

RH =H R

[R,H ] = RH -H R = 0

H cΨ = cHΨ

H RΨi= RHΨ

i= RE

iΨi= E

iRΨ

i

RΨi = ±Ψi

By applying each of the operations in the group to an eigenfunction Ψi belonging to a non degenerate eigenvalue, we generate a representation of the group with each matrix, Γi(R), equal to ±1. Since the representations are one dimensional, they are irreducible!

Chimica Inorganica 3

If Ei is k-fold degenerate:

H RΨi = EiRΨi

RΨi = rjj=1

k

∑ Ψij

For some other operation

SΨij = smjm=1

k

∑ Ψim

T = S ⋅ R TΨi = tmm=1

k

∑ Ψim S ⋅ RΨi = S rjj=1

k

∑ Ψij = smjm=1

k

∑ rjj=1

k

∑ Ψim

tm = smjrjj=1

k

∑

Chimica Inorganica 3

Matrices describing the transformation of a set of k eigenfunctions corresponding to a k-fold degenerate eigenvalue are a k-dimensional irreducible representation for the group

Let us consider 2px and 2py nitrogen AOs of NH3.

px = R2,1N r( )sinθ cosϕ

py = R2,1N r( )sinθ sinϕ

None of the operation of the C3v group will affect θ so that:

sinθ1 = sinθ2

but

ϕ2 =ϕ1 +2π3

Chimica Inorganica 3

If we reflect in the xz plane, we have

ϕ2 = −ϕ1; cosϕ2 = cosϕ1; sinϕ2 = −sinϕ1

cosϕ2 = cos ϕ1 +2π3

⎛⎝⎜

⎞⎠⎟ = cosϕ1 cos

2π3

− sinϕ1 sin2π3

= − 12cosϕ1 −

32sinϕ1

sinϕ2 = sin ϕ1 +2π3

⎛⎝⎜

⎞⎠⎟ = sinϕ1 cos

2π3

+ cosϕ1 sin2π3

= − 12sinϕ1 +

32cosϕ1

Chimica Inorganica 3

E:

E px= E sinθ1 cosϕ1( ) = sinθ2 cosϕ2 = sinθ1 cosϕ1 = px

E py= E sinθ1 sinϕ1( ) = sinθ2 sinϕ2 = sinθ1 sinϕ1 = py

⎧⎨⎪

⎩⎪C3 :

C3px = C3 sinθ1 cosϕ1( ) = sinθ2 cosϕ2 = sinθ1 − 12

⎛⎝⎜

⎞⎠⎟cosϕ1 + 3 sinθ1( ) = − 1

2p x −

32py

C3py = C3 sinθ1 sinϕ1( ) = sinθ2 sinϕ2 = sinθ1 − 12

⎛⎝⎜

⎞⎠⎟sinϕ1 − 3 cosθ1( ) = 3

2px− 12py

⎧

⎨

⎪⎪

⎩⎪⎪

σv:

σvpx= σ

vsinθ1 cosϕ1( ) = sinθ2 cosϕ2 = sinθ1 cosϕ1 = px

σvpy= σ

vsinθ1 sinϕ1( ) = sinθ2 sinϕ2 = −sinθ1 sinϕ1 = −p

y

⎧⎨⎪

⎩⎪

Chimica Inorganica 3

E:

1 00 1

⎡

⎣⎢

⎤

⎦⎥

px

py

⎛

⎝⎜⎜

⎞

⎠⎟⎟= E

px

py

⎛

⎝⎜⎜

⎞

⎠⎟⎟

χ E( ) = 2C3 :

− 12 − 3

2

32 − 1

2

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

px

py

⎛

⎝⎜⎜

⎞

⎠⎟⎟= C3

px

py

⎛

⎝⎜⎜

⎞

⎠⎟⎟

χ C3( ) = −1

σv:

1 00 −1

⎡

⎣⎢

⎤

⎦⎥

px

py

⎛

⎝⎜⎜

⎞

⎠⎟⎟= s v

px

py

⎛

⎝⎜⎜

⎞

⎠⎟⎟

χ σv( ) = 0

The characters are seen to be those of the E representation of C3v. Thus we see that the px and py orbitals, as a pair, provide a basis for the E representation.

Chimica Inorganica 3

The Scalar Product

AB =

a11 a12 ! ! a1n! ! ! ! !ai1 ai2 ! ! ain" " " " "am11 am12 ! ! am1n

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

b11 b12 ! b1 j ! b1m2! ! ! ! ! !bn1 bn2 ! bnj ! bnm2

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=!

" AB( )ij!

"

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Besides the scalar product between an m1 × n matrix A by an n × m2 matrix B (same n different m values) to give an m1 × m2 matrix C

Chimica Inorganica 3

The Direct Product

A =a11 a12a21 a22

⎡

⎣⎢⎢

⎤

⎦⎥⎥

B =

b11 b12 b13b21 b22 b23b31 b32 b33

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

A× B =a11B a12Ba21B a22B

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

a11b11 a11b12 a11b13 a12b11 a12b12 a12b13a11b21 a11b22 a11b23 a12b21 a12b22 a12b23a11b31 a11b32 a11b33 a12b31 a12b32 a12b33a21b11 a21b12 a21b13 a22b11 a22b12 a22b13a21b21 a21b22 a21b23 a22b21 a22b22 a22b23a21b21 a21b32 a21b33 a22b31 a22b32 a22b33

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

There is a second type of matrix multiplication that we will encounter in applications. It is called the direct product of two matrices and is written A × B. We shall define this operation by example

In general, the direct product is not commutative. There are no restrictions on the number of rows or columns in either matrix.

Chimica Inorganica 3

The Direct Product

tr A× B( ) = tr B× A( ) = tr A( )tr B( )

When A and B are both square (but not necessarily of the same order) the following important result applies:

Let us see which are the consequences

Chimica Inorganica 3

The Direct Product

RΧi = x jij=1

m

∑ Χ j

RYk = yℓkℓ=1

n

∑ Yℓ

RΧiYk = x jiyℓkℓ=1

n

∑j=1

m

∑ Χ jYℓ = z jℓ,ikℓ=1

n

∑j=1

m

∑ Χ jYℓ

Let us suppose that R is an operation in the symmetry group of a molecule and X and Y are two sets of functions, which are bases for representations of the group.

The set of functions XjYℓ, the direct product of Xj and Yℓ, also form a representation of the group. The zjℓ,ik are the elements of a matrix Z of order (mn) × (mn). Moreover, it is possible to demonstrate that characters of the representation of a direct product are equal to the product of characters of the representations based on the individual sets of functions

Chimica Inorganica 3

χZ R( ) = z j, jj∑ = x jj y =

=1

n

∑j=1

m

∑ χX R( )χY R( )

A1A1 = A1; A2A2 = A1; B1B1 = A1; B2B2 = A1 A1A2 = A2; A1B1 = B1; A1B2 = B2 A1E = E; A2E = E; B1E = E; B2E = E E2 = A1 + A2 + B1 + B2

RΧiYk = x jiyℓkℓ=1

n

∑j=1

m

∑ Χ jYℓ = z jℓ,ikℓ=1

n

∑j=1

m

∑ Χ jYℓ

Chimica Inorganica 3

The Kh (Kugel group) point group contains an infinite number of C∞ axes and a center of inversion i. It also contains elements generated from these. This is the point group to which a sphere belongs and therefore the point group to which all atoms belong. Although an atom with partially filled orbitals may not be spherically symmetrical, the electronic wave function is classified according to the Kh point group

The spherically symmetrical point group Kh contains all symmetry operations. The pure rotational subgroup K contains all rotations. In either of these groups, a complete set of orbitals is a basis for an irreducible representation of the group. The s orbital is the totally symmetric one-dimensional representation, the p orbitals give a 3 dimensional representation, d orbitals are five-dimensional, etc.

Chimica Inorganica 3

The character for rotation by the angle Φ is given by the following equation:

χ Φ( ) = sin ℓ+ 12( )Φ

sinΦ2

Chimica Inorganica 3

χ E( ) = 2ℓ+ 1

χ Φ( ) = sin ℓ+ 12( )Φ

sinΦ2

χ σ( ) = ±sin ℓ+ 12( )π = −1( )ℓ χ C2( )

χ i( ) = ± 2ℓ+ 1( ) = ±χ E( ) = −1( )ℓ χ E( )

Chimica Inorganica 3

n,ℓ,mℓ = Rn,ℓPℓ

mℓ cosθ( ) eimℓϕ

2πIf we carry a rotation of Φ around z

ϕ →ϕ +Φ

Rn,ℓPℓ

mℓ cosθ( ) eimℓϕ

2π→ Rn,ℓPℓ

mℓ cosθ( ) eimℓ ϕ+Φ( )

2π

The rotation is thus given by the matrix

eiΦ 0 00 ei −1( )Φ 0 0 0 0 e−iΦ

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

χ Φ( ) = eiℓΦ + ei ℓ−1( )Φ +…+ e− iℓΦ = e− iℓΦ eiΦ( )n

n=0

2ℓ

∑ =sin ℓ+ 1

2( )Φsin

Φ2

Chimica Inorganica 3

χ Φ( ) = eiΦ + ei −1( )Φ +…+ e−iΦ = e−iΦ eiΦ( )nn=0

2

∑ =sin + 1

2( )ΦsinΦ

2

Geometrical series from e-iℓΦ to eiℓΦ having path eiΦ

a + ar + ar2 +…+ arn−1 + arn = a rn+1 −1r −1

a = e− iΦ ; r = eiΦ ; n = 2

e− iΦeiΦ( )2+1 −1eiΦ −1

= e− iΦ ei 2+1( )Φ −1eiΦ −1

=

eiΦeiΦ − e− iΦ

eiΦ −1= e

i +1( )Φ − e− iΦ

eiΦ −1=

sin + 12( )Φ

sinΦ2

cosθ = eiθ + e−iθ

2; sinθ = e

iθ − e−iθ

2i

cos2θ = ei2θ + e−i2θ

2; sinθ = e

i2θ − e−i2θ

2i

ei +1( )Φ − e−iΦ = ei +1

2⎛⎝⎜

⎞⎠⎟Φe

i12Φ− e

−i +12

⎛⎝⎜

⎞⎠⎟Φe

i12Φ=

2iei12Φsin + 1

2⎛⎝⎜

⎞⎠⎟ Φ

eiΦ −1= ei12Φei12Φ− e

−i12Φ⎛

⎝⎜⎞⎠⎟= 2ie

i12ΦsinΦ

2

Chimica Inorganica 3

The characters of the reducible representation Γ spanned by the five d orbitals (ℓ = 2) in the point group D4h (we limit our attention to the pure rotational group D4)

χ Φ( ) = sin +12( )Φ

sinΦ2

Φ = π χ C2( ) = sin +12( )Φ

1= −1( )

Φ = π2

χ C4( ) = 1 = 0,1, 4, 5…−1 = 2,3, 6, 7…

⎧⎨⎩

E 2C4 C2 2 ′C2 2 ′′C25 −1 1 1 1

The irreducible components of Γ are

n a1( ) =1dz2

; n a2( ) = 0; n b1( ) =1

dx2−y2

; n b2( ) =1

dxy

; n e( ) =1dxz ,dyz

Chimica Inorganica 3

All other groups are subgroups of Kh, and all rotational groups are subgroups of K. If we lower the symmetry to O, we find that the representation for the s orbital is A1 and the representation for the p orbitals in O is T1. There is no five-dimensional representation, so the representation obtained for the d orbitals is a reducible representation in the O group. It reduces to E + T2. The representation in K for the f orbitals reduces to A2 + T1 + T2 in the O group.

Chimica Inorganica 3

Orbitals of even parity (ℓ even) are g and those of odd parity (ℓ odd) are u. Hence, in Oh symmetry the representations are s, A1g; p, T1u; d, Eg + T2g; and f , A2u + T1u + T2u. We can find the representations for the rotational group using for χ(ω) and add the g or u subscripts for the corresponding centrosymmetric group

In general

Chimica Inorganica 3

Why is the direct product so important?

f x( )−∞

∞

∫ dx = 0 if f x( ) is odd, f x( ) = − f −x( )

fa fb−∞

∞

∫ dτ ≠ 0If the integrand is invariant under all operation of the symmetry group to which the molecule belongs or unless some term in it remains invariant.

The representation of a direct product, Γab, will contain the totally symmetric representation iff the irreducible Γa = the irreducible Γb.

ai =1h

χab R( )χ i R( )R=1

h

∑ a1 =1h

χab R( )R=1

h

∑ a1 =1h

χa R( )χb R( ) = δabR=1

h

∑

Chimica Inorganica 3

fa fb fc−∞

∞

∫ dτ ≠ 0 if the direct product of the representations of fa, fb, and fc is or contains the totally symmetric representation.

ψ aPψ b−∞

∞

∫ dτψ aHψ b dτ

−∞

∞

∫

ψ aψ b dτ−∞

∞

∫= E

An energy integral may be nonzero only if ψa and ψb belong to the same irreducible representation of the molecular point group.

I ∝ ψ aµψ b dτ−∞

∞

∫

µ is a transition moment operator corresponding to changes in electric or magnetic dipoles, higher electric or magnetic multipoles, polarizability tensors.

Chimica Inorganica 3

The electric dipole operator ha the form

µ = eixii∑ + eiyi

i∑ + eizi

i∑

Ix ∝ ψ a xψ b dτ−∞

∞

∫

Iy ∝ ψ a yψ b dτ−∞

∞

∫

Iz ∝ ψ azψ b dτ−∞

∞

∫

An electric dipole transition will be allowed with x, y, or z polarization if the direct product of the representations of the two states is or contains the irreducible representation to which x, y, or z, respectively, belongs.

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