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! ˆ R ˆ H = ˆ H ˆ R [ ˆ R , ˆ H ] = ˆ R ˆ H - ˆ H ˆ R = 0 ˆ H cΨ = c ˆ H Ψ ˆ H ˆ RΨ i = ˆ R ˆ H Ψ i = ˆ RE i Ψ i = E i ˆ RΨ i ˆ RΨ i =± Ψ i By applying each of the operations in the group to an eigenfunction Ψ i belonging to a non degenerate eigenvalue, we generate a representation of the group with each matrix, Γ i (R), equal to ±1. Since the representations are one dimensional, they are irreducible!
Transcript
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
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.
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
The Kh 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 ω is given by the following equation:
χ ω( ) = sin ℓ+ 12( )ω
sinω2
Chimica Inorganica 3
χ E( ) = 2ℓ+1χ ω( ) = sin ℓ+ 1
2( )ωsinω2
χ σ( ) = ±sin ℓ+ 12( )π = −1( )ℓ χ C2( )
χ i( ) = ±2ℓ+1= ±χ E( ) = −1( )ℓ χ E( )
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
How direct products are used
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.
![Inorganica Chimica Acta - iris.unina.it · Inorganica Chimica Acta 453 (2016) 330–338 Contents lists available at ScienceDirect Inorganica Chimica Acta ... [10,11]. In addition,](https://static.documents.pub/doc/80x56/5c65ec8509d3f230488b5a47/inorganica-chimica-acta-irisuninait-inorganica-chimica-acta-453-2016-330338.jpg)
![Inorganica Chimica Acta - unige.ch · ics of the relaxation process, resulting in highly non-exponential relaxationcurves[18].Insomecompounds,areversephotophysical processwithirradiationat830](https://static.documents.pub/doc/80x56/5b7aaed07f8b9a460c8c59ed/inorganica-chimica-acta-unigech-ics-of-the-relaxation-process-resulting.jpg)
