Molecular Spectroscopy
January 29, 2008
Molecular Groups, Projection Diagrams, and PointGroups
Element
Operation
Example
E, the identity
Does nothing
All objects posses E
Cn, n-fold rotation
•Rotate object by 2π/n about the n-fold axis.
•The axis with the highest n is called the principle
axis.
•For n>2 the sense is important…Cn+ - clockwise
rotation…Cn--counterclockwise rotation
•H2O posses a C2 axis
•NH3 posses a C3 axis
•Benzene posses a C2, C3, C6, and two sets of
three C2 axes perpendicular to the C6 axis
σ, reflection plane
Reflects through the plane
Three types
σv, vertical plane
Plane containing the principle axis
σh, horizontal plane
Plane perpendicular to the principle axis
•Benzene
σd, dihedral plane
Vertical plane which bisects two C2 axes perpendicular
to the principal axis
i, inversion center
Sends each atom through the origin along a straight
line so that x→-x, y→-y, and z→-z
•H2
•CO2
•Benzene
•SF6
Rotate object by 2π/n, then reflects through plane
perpendicular to the axis of rotation
•CH4
Sn, n-fold improper rotation axis
•H2O
•Benzene
Let’s take the ECLIPSED conformation of ethane
• One three-fold axis coincident with the C-C bond
•Three two-fold axes perpendicular to the C-C bond and intersecting its midpoint
•Three reflection planes, each containing the C-C bond and a pair of C-H bonds
•One reflection plane perpendicular to the C-C bond and bisecting it
•NO CENTER OF INVERSION IS PRESENT
•One three-fold improper axis coincident with the C-C bond
Let’s take the STAGGERED conformation of ethane
• One three fold axis coincident with the C-C bond
•Three two-fold axes perpendicular to the C-C bond and intersecting its midpoint
•Three reflection planes, each containing the C-C bond and a pair of C-H bond
•No reflection plane perpendicular to the C-C bond…we lost it!
•One point of inversion at the midpoint of the C-C bond…we gained it!
•One six-fold improper axis coincident with the C-C bond…changed its order!
Stereographic Projections
Let’s consider the symmetry elements of the C2v point group and
the resulting symmetry operations
Still considering the symmetry elements of the C2v point group
and the resulting symmetry operations
Let’s compare our results with the C2v character table…
C2v
E
C2
σv
σv'
E
E
C2
σv
σv'
C2
C2
E
σv'
σv
σv
σv
σv'
E
C2
σv'
σv'
σv
C2
E
Lets try this with another point group and build the character table…
C2h
E
C2
σh
i
E
C2
σh
i
⎧
⎪
⎪
⎪
⎪
⎪
⎪
→⎨
⎪
⎪
⎪
⎪
⎪
⎪⎩
→
→
→
→
This gives us the following for the Character Table…
C2h
E
E
C2
σh
i
i
C2
σh
σh
C2
i
E
Complete the rest as we have done and hand it in on Tuesday
Point Groups
•
Symmetry operations may be organized into groups that obey the four properties
of mathematical groups…namely…
1.
2.
3.
4.
•
There exists an identity operator that commutes with all other members
The product of any two members must also be a member
Multiplication is associative
The existence of an inverse for each member
It will be possible to assign ANY molecule to one of these so-called Point Groups
¾ So named because the symmetry operation leaves one point
unchanged
Point Groups
• Point Group C1
¾Trivial group with contains all molecules having no symmetry
¾There is no center of inversion, plane of reflection, improper axis of
rotation, etc.
¾The molecule HNClF (below) is such a molecule
Point Groups
• Point Group Cs
¾Only molecules which have to them a plane of reflection as a symmetry
element
¾Formyl Chloride belongs to this point group
Point Groups
• Point Group Ci
¾Molecules whose sole symmetry element is an inversion center
¾As an example take the rotamer of 1,2-dichloro-1,2-diflouroethane in
which the fluorine atoms are off-set to one another
¾The only operations generated by this inversion are i and i2=E…this is a
group of order 2 and is designated Ci
Important to note that i is the same as an S2 axis
Point Groups
• Point Group Cn
¾Molecules with only an n-fold axis of rotation
¾Boric acid with the hydrogen atoms lying above the BO3 plane is an
example
Point Groups
• Point Group Cnv
¾Molecules with an n-fold Cn axis of rotation and n vertical mirror planes
which are colinear with the Cn axis
¾Water belongs to such a point group…the C2V
Note that if you find a Cn axis and one σv plane then you are
gauranteed to find n σv planes
Point Groups
• Point Group Cnh
¾This point group is generated by a Cn axis of rotation AND a horizontal
mirror plane, σh, which is perpendicular to the Cn axis
¾Trans-butadiene (trans-1,3-butene) is an example of the C2h point
group
Point Groups
• Point Group Dn
¾This point group is generated by a Cn axis of rotation and a C2 axis which
is perpendicular to the Cn axis…no mirror planes
¾Tris(ethylenediamine)cobalt(III) posses idealized D3 symmetry
Build this molecule and convince yourself that this in indeed the case!!
Point Groups
• Point Group Dnd
¾This point group is generated by a Cn axis of rotation and a C2 axis which
is perpendicular to the Cn axis and a dihedral mirror plane, σd
¾The dihedral plane is colinear with the principal axis and bisects the
perpendicular C2 axis
¾Crystals of Cs2CuCl4 have a squashed tetrahedral D2d structure
Point Goups
• Point Group Dnh
¾This point group is generated by a Cn axis of rotation and n C2 axis which
is perpendicular to the Cn axis and a horizontal mirror plane which is by
definition perpendicular to the principal axis of rotation
¾Benzene is the classic example, possesing D6h symmetry
Point Goups
• Point Groups Sn
¾These point groups are generated by an Sn axis.
¾For example, the spirononane…tetraflourospirononane belongs to the
S4 point group and only elements generated by S4
¾S4
¾S42 = C2
¾S43
¾S44 = E
¾The S2 point group is just Ci since S2 = i
¾When n is odd the point groups are just the same as the Cnh point groups
and so are designated as such
¾As a direct result, only S4, S6, and S8, …have a separate existence
Special Point Goups
• Tetrahedral molecules belong to the point group Td
•Octahedral molecules belong to the point group Oh
•Molecules with icosahedral (20 triangular faces) or dodecahedral (12 pentagonal
faces) belong to the point group Ih
• Atoms…which have spherical symmetry, belong to the point group Kh
• The point group Th, which may be obtained by adding to Td as set of σh planes,
which contain pairs of C2 axis…as opposed to the σd planes which contain one C2
axis and bisect another pair, giving Td)