Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix...

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1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ Name and show on a diagram five (5) symmetry elements in SF 6 .

Transcript of Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix...

Page 1: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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Group Theory and Symmetry

point groups, symmetry elements,

matrix representations and

properties of groups

2

‘dj today’

• Name and show on a diagram five (5)

symmetry elements in SF6.

Page 2: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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-

Page 3: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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-

Page 4: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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-

Page 5: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 6: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 7: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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…)

Page 8: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 9: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 10: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 11: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 12: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 13: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 14: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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)

Page 15: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 16: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 17: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 18: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 19: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 20: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 21: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 22: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 23: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 24: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 25: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 26: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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

Page 27: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 28: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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

Page 29: Group Theory and Symmetry1 Group Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ‘dj today’ • Name and show on a diagram

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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