group theory and symmetry1 group theory and symmetry point groups, symmetry elements, matrix...
TRANSCRIPT
1
Group Theory and Symmetry
point groups, symmetry elements,
matrix representations and
properties of groups
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‘dj today’
• Name and show on a diagram five (5)
symmetry elements in SF6.
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symmetry elements and
associated operations
• Proper axis of Rotation
Proper rotation, Cn: One or more rotations by an
angle of 2 n radians. The "n" value is often
referred to as the order of the rotation axis. Note
that since the bonds around the proper axis are
identical, their bond moments must cancel and thus
a dipole moment (if any)lies along the proper axis of
rotation
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examples
• H2O
• CH3Cl
• Pt(NH3)2Cl2
• PtCl42-
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symmetry elements and
associated operations
• Plane of reflection
Reflection, : reflection through a plane to interchange objects
on either side of the plane. Three distinct types.
• v: reflection through a plane which contains the highest order
rotation axis (Cn with the greatest n) and atoms of the molecule.
• d: similar to v but does not contain atoms of the molecule.
• h: reflection through a plane which is normal to (perpendicular
to) the highest order rotation axis.
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examples
• H2O
• CH3Cl
• cis and trans Pt(NH3)2Cl2
• PtCl42-
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symmetry elements and
associated operations
• Improper axis of rotation
improper rotation, Sn: This symmetry operation
consists of two consecutive operations, a proper
rotation, Cn, followed by a horizontal reflection, h.
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examples
• H2O
• CH3Cl
• Pt(NH3)2Cl2
• PtCl42-
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how to categorize
molecular symmetries
• certain groups of symmetry operations form a
mathematical group
• these closed sets of symmetry operations form
what is called a point group
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Today
• Exam is on Friday…
• HW 2 due Wednesday-solutions posted shortly
after class…
• Today- point groups and matrix representations
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point groups
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group work: assign point
groups
SiCl3(CH3)
trans-SF4Cl2
OXeF4
CHClF2
trans-Cl2C2H2
CO2
PF3
cis-PtCl2Br22- (square planar geometry)
cis- Cl2C2H2
SiClHBrF
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Today’s question
• Sketch the ion PSO33- (thiophosphate) and
indicate the symmetry elements present and
state the point group
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Up to date
• Exam Friday: quantum theory, periodicity, lewis
structures and point groups are the main topics.
• Readings today…
87-100
• For Monday
109-128 (we’ll come back to vibrations…)
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Whaddya know?
• Draw the symmetry elements in 1,2
dichloroethane in the trans configuration.
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mathematical group
properties
• closure
subsequent application of any
series of symmetry operations
is equivalent to one of the
other operations
• identity
the do-nothing operation is
important
H
O
H
C2
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C2v group multiplication
table
op2 op 1 E C2 v v’
E E C2 v v’
C2 C2 E v’ v
v v v’ E C2
v’ v’ v C2 E
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matrix representations of
symmetry operations
• Use the C2v point group as an example
• Consider the effect of of performing a C2 operation on
an object. The new position of a point x,y,z after the
operation , x’,y’,z’, can be found by using a matrix form
for the operation
• the point x,y,z is found by using the three orthogonal
unit vectors
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matrix representation of
C2 on a vector, x,y,z
• C2 (rotation)
• x -x
• y -y
• z z
• the ‘reducible
representation’ is the
sum of the diagonal
elements
• -1 + (-1) + 1 = -1
C2 =
1 0 0
0 1 0
0 0 1
x
y
z
=
x
y
z
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how about v on a unit
vectors, x, y, z
• v (contains the xz
plane):
• x x
• y -y
• z z
• reducible representation
is trace of the matrix,
1+(-1) + 1 = 1
v =
1 0 0
0 1 0
0 0 1
x
y
z
=
x
y
z
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v’ (contains yz plane) : x
- x; y y and z z
• reducible representation
= -1+ 1 + 1 = 1v =
1 0 0
0 1 0
0 0 1
x
y
z
=
x
y
z
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identity operation on a
vector, x,y,z
• reducible representation
= 1+1+1 = 3 E =
1 0 0
0 1 0
0 0 1
x
y
z
=
x
y
z
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summarizing how (x,y,z)
is transformed
• operation reducible representation
• E 3
• C2 -1
• v(xz) 1
• v(yz) 1
• Note that these vectors behave like the p
orbitals
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character table for C2v
E C2 v(xz) v(yz)
A1 1 1 1 1
A2 1 1 -1 -1
B1 1 -1 1 -1
B2 1 -1 -1 1
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character tables
• the top row which lists the symmetry operations of the point group
• the leftmost column lists the Mulliken symbols for the irreducible
representations of that point group. For A and B (1 dimensional,
non-degenerate irreducible representations) the difference lies in
the character for the highest order rotation (for A, = 1 and for B,
= -1). The labels for the E and T irreducible reps can be
considered arbitrarily assigned.
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character table for C2v
E C2 v(xz) v(yz)
A1 1 1 1 1
A2 1 1 -1 -1
B1 1 -1 1 -1
B2 1 -1 -1 1
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note several features
• number of symmetry types is equal to the number of
operations
• Mullikan symbols (left column) indicate fundamental
symmetry types, irreducible representations
• these are orthogonal, with cross products = 0
• These irreducible representations are analogous to unit
vectors in a “symmetry space”
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reducible vs. irreductible
reps
• reducible representation for the vector x,y,z is
not the same as any of those for the symmetry
types in the character table.
• How do the individual unit vectors along the x,
y and z axes transform (i.e., what are their
characters)
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group work: find characters for:
(no looking at books) (C2V)
2s 1 1 1 1
3dz2 1 1 1 1
3dx2-y2 1 1 1 1
3dyz 1 -1 -1 1
Rotation(x)` 1 -1 1 -1
Rotation(y) 1 -1 - 1 1
Rotation(z) 1 1 -1 -1
E C2 v(xz) v(yz)
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today
• Reducible to irreducible representations
• What can be done with the irreducible
representations…
• Next time: bonding theory and applications of
group theory
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HE 1
ave 12.1 9.1 14.3 10.3 12.3 54.7
stdev 3.7 5.5 2.5 5.7 3.4 18.9
max 79.0
overall ave/q11.6
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add a column to
character table
C2v E C2 v(xz) v(yz)
A1 1 1 1 1 z, z2, x2,y2
A2 1 1 -1 -1 Rz, xy
B1 1 -1 1 -1 x, Ry,xz
B2 1 -1 -1 1 y, Rx, yz
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symmetry of orbitals and
vibrations
• how do things that are part of a molecule
change under allowed symmetry operations?
• how can we determine the symmetry types of
more complex systems than (x,y,z)
that is, how can one get the irreducible
representations from reducible representations?
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Orbitals: two methods
• by inspection
• by use of group theory and the following
formula
N =1
h rx• i
x
x
•nx
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add a column to
character table
C2v E C2 v(xz) v(yz)
A1 1 1 1 1 z, z2, x2,y2
A2 1 1 -1 -1 Rz, xy
B1 1 -1 1 -1 x, Ry,xz
B2 1 -1 -1 1 y, Rx, yz
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• N: number of times irred rep, x, appears in the
reducible representation
• h is the order of the group(sum of all E characters)
• r is the character of the reducible representation for
the operation, x
• i is the character of the irreducible representation
for the operation, x
• n is the number of operations in the class, x
N =1
h rx• i
x
x
•nx
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add a column to
character table
C2v E C2 v(xz) v(yz)
A1 1 1 1 1 z, z2, x2,y2
A2 1 1 -1 -1 Rz, xy
B1 1 -1 1 -1 x, Ry,xz
B2 1 -1 -1 1 y, Rx, yz
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two examples: (x,y,z) in C2v
s orbitals in PtCl42- (D4h)
• the first example will confirm what may be
deduced by inspection
• the second example will illustrate doubly
degenerate irreducible representations
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(x,y,z) in C2v
• E, identity, character = 3
• C2, proper rotation,
character = -1 (note
negative 1matrix
elements)
1 0 0
0 1 0
0 0 1
x
y
z
=
x
y
z
C2 =
1 0 0
0 1 0
0 0 1
x
y
z
=
x
y
z
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v’s
• v (xz),character is +1
(note negative 1)
• v (yz), character is +1
(note negative 1)
1 0 0
0 1 0
0 0 1
x
y
z
=
x
y
z
1 0 0
0 1 0
0 0 1
x
y
z
=
x
y
z
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character table for C2v
E C2 v(xz) v(yz)
A1 1 1 1 1
A2 1 1 -1 -1
B1 1 -1 1 -1
B2 1 -1 -1 1
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Characters for Reducible
representations
• E = 3
• C2 = -1
• v= 1
• v’= 1
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• N: number of times irred rep, x, appears in the
reducible representation
• h is the order of the group(sum of all E characters)
• r is the character of the reducible representation for
the operation, x
• i is the character of the irreducible representation
for the operation, x
• n is the number of operations in the class, x
C4 would be a class (not in C2v)
In C4v there are 2 C4
N =1
h rx• i
x
x
•nx
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N =1
h rx• i
x
x
•nx
• N(A1) =(1/4) {3*1*1+- 1* 1*1 +1* 1*1 +1* 1*1} =
1
• N(A2) = (1/4){3* 1*1 +-1* 1*1 +1*- 1*1 +1* -1*1)
= 0
• N(B1) = (1/4){3*1*1+-1*1*1+1*1*1+-1*1*1}=1
• N(B2) = (1/4){3*1*1+-1*-1*1+-1*1*1+1*1*1}=1
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irreducible
representations for x,y,z
in C2v
• A1, B1, B2
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C2v E C2 v(xz) v(yz)
A1 1 1 1 1 z, z2, x2,y2
A2 1 1 -1 -1 Rz, xy
B1 1 -1 1 -1 x, Ry,xz
B2 1 -1 -1 1 y, Rx, yz
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s orbitals in PtCl42- (D4h)
• some symmetry elements shown to
right
• all the ops are listed in the table on
the following slide
• need to see how the 5 orbitals
transform
• know E will have red rep char = 5
PtCl
Cl Cl
Cl
C4,C2
C2'
C2''
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character table for D4h
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use a matrix form:
C4
• red rep for;
E is 5
C4 is 1
1 0 0 0 0
0 0 0 0 1
0 0 0 1 0
0 0 1 0 0
0 1 0 0 0
sPt
sCl1
sCl2
sCl3
sCl4
=
sPt
sCl4
sCl3
sCl2
sCl1
1
2 Pt
1 2Pt
C4
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how about v
1 0 0 0 0
0 1 0 0 0
0 0 0 0 1
0 0 0 1 0
0 0 1 0 0
sPt
sCl1
sCl2
sCl3
sCl4
=
sPt
sCl1
sCl4
sCl3
sCl2
v1 2
3
4
51 Group work-write matrices and find
the rest of the characters for the
reducible representations of the Sym.
Ops.
• E, character = 5
• C4, character = 1
• C2, character = 1
• C2’, character = 3
• C2”, character = 1
• i, character = 1
• S4, character = 1
• h , character = 5
• v , character = 3
• d , character =1
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• N: number of times irred rep, x, appears in the
reducible representation
• h is the order of the group(sum of all E characters)
• r is the character of the reducible representation for
the operation, x
• i is the character of the irreducible representation
for the operation, x
• n is the number of operations in the class, x
N =1
h rx• i
x
x
• nx
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now to find irred reps in the
reducible representation
• h =1+2+1+2+2+1+2+1+2+2 = 16
• N(A1g)=
1/16{1•5•1+1•1•2+1•1•1+1•3•2+1•1•2+1•1•1+1•
1•1+1•5•1+1•3•2+1•1•2}
• N(A1g) = 2
• N(A2g) = 1/16{1•5•1+1•1•2+1•1•1+(-1•3•2)+
(-1•1•2)+1•1•1+1•1•1+1•5•1+(-1•3•2)+(-1•1•2)}= 0
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ctd.
• N(B1g) = 1/16 {1•5•1+ (-1•1•2) +1•1•1+1•3•2+
1•1•2+(-1•1•1)+(-1•1•1)+1•5•1+ (1•3•2)+ (-
1•1•2)}= 1
• N(B2g) = 0
• N(Eg) = 1/16 {2•5•1+(0•1•2)+ -2•1•1+0•3•2+
0•1•2+ (2•1•1)+0•1•1)+-2•5•1+ (0•3•2)
+(0•1•2)}=0
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finally
• N(A1u) = 1/16
{1•5•1+(1•1•2)+1•1•1+1•3•2+1•1•2+(-1•1•1)+(-
1•1•1)+(-1•5•1)+ (-1•3•2)+(-1•1•2)}= 0
• N(A2u) =0
• N(B1u) = 0
• N(B2u) =0
• N(Eu) = 1
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irreducible representations
are
• 2 A1g irreducible reps
• 1 B1g irreducible rep
• 1 Eu irred rep
the Eu rep is 2 fold degenerate (identity character is
2)
indicates that some “s” orbitals are interchanged
under the symmetry operations of the D4h point
group
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irreducible
representations are useful
• used in determination of molecular orbitals
orbitals that combine must be of same symmetry
• used in determining allowed transitions
symmetries of vibrations are important
use vectors to describe motion