CHEM601. Sample Exam / Homework Problems October **, 2009

CHEM601. Sample Exam / Homework Problems
October **, 2009
1) Using the flowchart attached, assign the following molecules to the appropriate symmetry point group.
a) cis-ClCH=CHF;
Cs
H
b) 1,2-dichloroethane, anti-conformation;
H
Cl
H
Cl
CH2h
H
c) hydrazine, N2H4, (gauche-conformation);
d) 2-chlorobutane;
H
H
H
C2
C1
2)
Much attention has been paid to the distinction of stereoisomers by vibration spectroscopy. One of the two cis-/trans- isomeric
1,2-difluoroethenes exhibits two C-F stretching bands in its IR spectrum, while another isomer shows only single IR C-F
stretching absorption.
a) Assign cis-1,2-difluoroethene to the appropriate symmetry point group;
C2v point group
b) Use two vectors of the C-F bonds of the cis-isomer as a two-dimensional basis function to build the reducible
representation Γr corresponding to the C-F stretching modes of this molecule. (Note: only unshifted vectors contribute
to characters of available symmetry operations).
Γr = 2E + 2σv’
c) Decompose Γr into a linear combination of irreducible representations of the symmetry point group.
= A1 + B2 ; both are IR active
d) How many C-F stretching bands will be seen in IR spectrum of cis-1,2-difluoroethene?
Two
3) Assign a Mulliken label to 4fxyz orbital of a central atom in a C2v symmetric species. Note that 4fxyz orbital transforms as xyz
cubic function.
The characters for xyz basis function can be obtained as a product of corresponding characters of x, y and z vectors:
χ(E)= 1·1·1=1, χ(C2)= (-1)·(-1)·1=1, χ(σv)= 1·(-1)·1= -1, χ(σv’)= (-1)·1·1= -1. This set of characters corresponds to A2
irreducible representation. So, the symmetry of 4fxyz orbital in C2v species is a2.
4) How many symmetry operations can be generated by the following symmetry elements: C2, C4, S2, S4? For each
symmetry element list all related symmetry operations.
C2: two, C21 and C22 ≡ E
C4: four, C41, C42 ≡ C2, C43 and C44 ≡ E
S2: two, S21 ≡ i and S22 ≡ E
S4: four, S41, S42 ≡ C2, S43 and S44 ≡ E
5) Consider the symmetry point group C2v which includes four symmetry operations, E, C2, σv(xz) and σ'v(yz), and fill in
corresponding group multiplication table. Using matrix representations of these symmetry operations prove that all the
products of the symmetry operations belong to the group.
C2v
E
C2
σv(xz)
σv’(yz)
E
E
C2
σv(xz)
σv’(yz)
C2
C2
E
σv’(yz)
σv(xz)
σv(xz)
σv(xz)
σv’(yz)
E
C2
Matrix representations of E, C2, σv and σv’ with vector r as a basis function are:
⎡1 0 0 ⎤
⎢0 1 0⎥ for E,
⎢
⎥
⎢⎣0 0 1⎥⎦
⎡− 1 0 0⎤
⎢ 0 − 1 0⎥ for C (z),
2
⎢
⎥
⎢⎣ 0 0 1⎥⎦
⎡1 0 0 ⎤
⎢0 − 1 0⎥ for σ , and
xz
⎢
⎥
⎢⎣0 0 1⎥⎦
⎡− 1 0 0⎤
⎢ 0 1 0⎥ for σ .
yz
⎢
⎥
⎢⎣ 0 0 1⎥⎦
Therefore,
⎡ − 1 0 0 ⎤ ⎡1 0 0 ⎤ ⎡ − 1 0 0 ⎤
⎥ ⎢
⎢
⎥⎢
⎥
C2σv(xz) = 0 − 1 0 0 − 1 0 = 0 1 0 which is σv’(yz). Etc.
⎥ ⎢
⎢
⎥⎢
⎥
⎢⎣ 0 0 1⎥⎦ ⎢⎣0 0 1⎥⎦ ⎢⎣ 0 0 1⎥⎦
6) Using a flow chart find out to which symmetry point group the following species belong:
a) WF6 (octahedron);
Oh
b) ethane, eclipsed conformation;
D3h
c) BrCCl3 (a distorted tetrahedron);
C3v
d) O=SF2 (a distorted trigonal pyramid);
Cs
e) Cl2;
D∞h
f) HCl;
C∞v
g) HOF (a non-linear molecule);
Cs
h) H2C=C=CH2 (a non-planar molecule);
D2d
i) trans-ClCH=CHCl;
C2h
j) trans-[PtCl2Br2]2- (planar complex);
D2h
k) IF5 (tetragonal pyramid);
C4v
l) cyclohexane (chair conformation);
D3d
m) 1,2-dichloroethane, gauche-conformation;
C2
n) B(OH)3 (planar conformation of the highest possible symmetry).
C3h
7) Consider the character table of the Td symmetry point group (see below). Find the order h
h = 1+8+3+6+6 = 24
and the number of classes of symmetry operations in the group.
Five
Prove that:
σv’(yz)
σv’(yz)
σv(xz)
C2
E
1) for all its irreducible representations the sum of the squares of the characters χ of the available symmetry operations is
equal to the group order h;
For A1: 12+8•12+3•12+6•12+6•12=24
For A2: 12+8•12+3•12+6•(-1)2+6•(-1)2=24
For E: 22+8•(-1)2+3•22+6•02+6•02=24
For T1: 32+8•02+3•(-1)2+6•12+6•(-1)2=24
For T2: 32+8•02+3•(-1)2+6•(-1)2+6•12=24
2) the sum of the squares of the orders (dimensions) of all irreducible representations constituting the group is equal to h
12+12+22+32+32=24
3) show that two arbitrarily chosen by you irreducible representations of the group are orthogonal to each other.
∑ χ ( X )χ
i
j
(X ) = 0
X
For A1 and A2 : 1•1•1+1•1•8+1•1•3+1•(-1)•6+1•(-1)•6 = 0
For A2 and E : 1•2•1+1•(-1)•8+1•2•3+(-1)•0•6+(-1)•0•6 = 0
For E and T1 : 2•3•1+(-1)•0•8+2•(-1)•3+0•(-1)•6+0•1•6 = 0
For T1 and T2 : 3•3•1+0•0•8+(-1)•(-1)•3+(-1)•1•6+1•(-1)•6 = 0
Td
E
8C3
3C2
6
d
6S4
A1
1
1
1
1
1
A2
1
1
1
-1
-1
E
2
-1
2
0
0
T1
3
0
-1
-1
1
T2
3
0
-1
1
-1
With center of symmetry?
D
h
C
v
8
Linear molecules?
+
+
-
8
-
-
Cn available?
σ available?
+
-
i available?
Ci
-
C1
+
Cs
+
+
+
6C5 available?
-
i available?
Ih
I
+
+
3C4 available?
-
+
+
4C3 available?
-
i available?
Oh
O
Th
i available?
+
-
σ available?
-
Td
T
Rotational or Dihedral groups
Presence of nC2
Cn?
+
+ D nh
σh available?
Presence of S2n?
+
S2n
+
Cnh
+
Cnv
Presence of σh?
Presence of nσv?
Cn
-
nσv available?
+ D nd
-
Dn