### group theory for mathematiciansx

```How Chemists Use Group Theory
Created by Anne K. Bentley, Lewis & Clark College ([email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */) and posted on VIPEr (www.ionicviper.org) on
Why do chemists care about symmetry?
It allows the prediction of
• chirality
• IR and Raman spectroscopy
• bonding
Which objects share the same
symmetry as a water molecule?
How can we “quantify” symmetry?
Symmetry can be described by
symmetry operations and
elements.
• rotation, Cn
• reflection, σ
• inversion, i
• improper rotation, Sn
• identity, E
Objects that share the same set of
symmetry elements belong to the
same point group.
= C2v (E, C2, two σv)
The operations in a group follow the
requirements of a mathematical group.
• Closure
• Identity
if AB
= C, then C is also in the group
• Associativity
• Reciprocality
The operations in a group follow the
requirements of a mathematical group.
•
•
•
•
Closure
Identity
Associativity
Reciprocality
AE = EA = A
The operations in a group follow the
requirements of a mathematical group.
•
•
•
•
Closure
Identity
Associativity
Reciprocality
(AB)C = A(BC)
The C2v point group is an Abelian group – ie, all
operations commute (AB = BA). Most point groups
are not Abelian.
The operations in a group follow the
requirements of a mathematical group.
•
•
•
•
Closure
Identity
Associativity
Reciprocality
AA-1 = E
In the C2v point group, each operation is its own
inverse.
The operations in a group follow the
requirements of a mathematical group.
•
•
•
•
Closure
Identity
Associativity
Reciprocality
Each operation can be represented by
a transformation matrix.
é –1
ê
ê 0
êë 0
0
0
–1
0
0
1
ù
ú
ú
úû
transformation matrix
éxù
ê ú
y
ê ú
êë z úû
original
coordinates
=
é
ê
ê
êë
–x
–y
z
ù
ú
ú
úû
new
coordinates
Which operation is represented by this transformation matrix?
The transformation matrices also follow
the rules of a group.
é –1
ê
ê 0
êë 0
0
0
–1
0
0
1
C2
ùé
úê
úê
úû êë
1
0
0
0
–1
0
0
0
1
σxz
ù
é –1
ú
ê
0
=
ú
ê
úû
êë 0
0
0
1
0
0
1
σyz
ù
ú
ú
úû
Irreducible representations can be
generated for x, y, and z
C2v
σv(xz) σv(yz)
E
C2
1
–1
1
–1
x
1
–1
–1
1
y
1
1
1
1
z
A complete set of irreducible representations for a
given group is called its character table.
C2v
σv(xz) σv(yz)
E
C2
1
–1
1
–1
x
1
–1
–1
1
y
1
1
1
1
z
?
A complete set of irreducible representations for a
given group is called its character table.
C2v
σv(xz) σv(yz)
E
C2
1
–1
1
–1
x
1
–1
–1
1
y
1
1
1
1
z
1
1
–1
–1
xy
More complicated molecules…
ammonia, NH3
C3v
methane, CH4
Td
Applications of group theory
• IR spectroscopy
• Molecular orbital theory
Gases in Earth’s atmosphere
nitrogen (N2)
78%
oxygen (O2)
21%
argon (Ar)
carbon dioxide (CO2)
0.93%
400 ppm
(0.04%)
carbon dioxide stretching modes
not IR active
IR active
Are the stretching modes of
methane IR active?
Td
E
8C3
3C2
Γ
4
1
0
Γ = A1 + T2
6S4 6σd
0
2
methane’s A1 vibrational mode
not IR active
methane T2
stretching vibrations
• all at the same energy
• T2 irreducible rep transforms as (x, y, z)
• together, they lead to one IR band
Molecular Orbital Theory
How and why does something
like this form?
Bonding Basics
• Atoms have electrons
• Electrons are found in orbitals, the shapes
of which are determined by wavefunctions
Bonding Basics
• A bond forms between two atoms when
their electron orbitals combine to form one
mutual orbital.
+
+
=
=
Bonding is Determined by Symmetry
+
+
+
=
no bond forms
= bond forms
=
bond forms
Use group theory to assign symmetries
and predict bonding.
A1g
T1u
(and two more)
SF6
and
T2g
Eg
Outer atoms are treated as a group.
A1g
T1u
Eg
Which types of bonds will form?
central sulfur
six fluorine
A1g
A1g
T1u
T1u
T2g
Eg
Eg
Concluding Thoughts
Recommended Resources
Cotton, F. Albert. Chemical Applications of Group Theory,
Wiley: New York, 1990.
Carter, Robert L. Molecular Symmetry and Group Theory,
Wiley: 1998.
Harris, Daniel C. and Bertolucci, Michael D. Symmetry and
Spectroscopy, Dover Publications: New York, 1978.
Vincent, Alan, Molecular Symmetry and Group Theory,
Wiley: New York, 2001.
http://symmetry.otterbein.edu