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--Beginning Group Theory for Chemistry, Paul H. Walton, Oxford. 1998, University press
--2008
--2008
)
--
60 )
: Exercise
1H-NMR spectra
-- 2-chlorophenol → several peaks split by a complicated pattern
--Qualitatively (): symmetry has something to do with the energy levels of a molecule
− all benzene hydrogen atoms are in the same chemical environment
Quantitatively (): How important is symmetry in determining the energy levels of
molecules?
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We must provide some type of rules to define the symmetry of molecules
Symmetry: ()
bisects the H-O-H angle
original one
by 1800 about this axis
Furthermore: If the molecule is symmetric (the same) by rotation of x degree about an axis,
then we have a rotation axis of order (360/x).
ex. H2O → 3600/1800 = 2 second order axis C2
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TERMINOLOGY
The axis is know as the “symmetry element”
The actual transformation of the molecule to its symmetrical equivalent position is known as
the “symmetry operation”
The rotation is clockwise
For water molecule, the C2 axis is the symmetry element and clockwise rotation of 1800 about
this axis is the symmetry operation.
Exercise Sketch what happens to the water molecule (labeling the
hydrogen atoms as above), when C2 is carried out on it twice in
succession.
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Doing nothing (E) is an important operation in group theory
1. C1 = E; 2. C2 1C2
1 = C2 2 = E; 3. Cn
n = E
Exercise Sketch ammonia molecule and work out if it has a rotation axis (axes) and the order of
the axis (axes). Draw the axis (axes) on your sketch of molecule. (Only one rotation axis of
symmetry)
Exercise Figure out all of the rotation axes that can be
found for benzene, and sketch the axes on your picture
of benzene.
symmetry)
In molecules which have more than one rotational axis of symmetry, then the axis of highest
order is defined as “principle axis”.
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--C6 contains a C3 axis and C2 axis; 1 3
2 6 CC , 1
2 3 6 CC
Even-order axis greater than order 2 contain lower-order even axes.
The other two type of C2 axes are called C2’, C2”
Reflection Planes
The reflection plane is another symmetry element of the water molecule.
and
In group theory a reflection plane is denote by . These reflection planes contain the principle
axis within the plane, and can be said to be “vertical” along with the principle axis, are denoted
by v
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Exercise List all of the symmetry elements present in the water molecule
Reflection planes of NH3
σv in Ha-N plane
σv in Hb-N plane
σv in Hc-N plane
Reflection planes of - 4AuBr
× 2 d, d stands for “dihedral”.
d bisects the Br-Au-Br angles
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× 1 h, h stands for “horizontal”.
Center of inversion /inversion center (i)
Good thing : a single molecule can only have one such centre, and we only need to decided
whether a molecule has got one or not.
Bad thing : The actual symmetry element it self is sometimes difficult to “see”
The first way -- by inspection
Inversion center lies on Au atom
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The second way -- mathematical description if the inversion center is placed on the origin of
three-dimensional Cartesian coordinate set of
axes, and atoms are at (x,y,z), then an inversion
centre is a symmetry element if one same atom
at (-x,-y,-z). (x,y,z) (-x,-y,-z); (xa,ya,za) =
(-xc,-yc,-zc); (xb,yb,zb) = (-xd,-yd,-zd)
The third way -- combination of specific rotation axis and a specific reflection plane.
--rotation 1800 followed by reflection in a reflection plane perpendicular to the rotation axis.
--If a molecule contains a C2 axis and a reflection plane perpendicular to this axis it has
inversion center.
--The inversion center element is placed at the meeting point of the axis and the reflection
plane.
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The presence of an inversion centre of symmetry does not necessary need that the molecule
has a C2 axis and a perpendicular reflection plane as individual symmetry element.
Exercise Using the third method, reason whether or not H2O has an inversion centre as a
symmetry element? Do the same thing for AuBr4 -.
Improper rotation axes (Sn)
-- It is impossible to find a “single” symmetry element which
will allow us to transform each of the hydrogen atom position
into all of the other atom positions.
-- We need another symmetry element which can describe a
symmetry operation to relate all hydrogen atoms position to
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Check C2 axes of methane
-- Carbon atom is in the plane of the
paper
interchange
and Hb
operation)
plane of the paper.
→ Combine the rotation by 900 with
reflection in a reflection plane
perpendicular to the axis of rotation
The molecule does not necessarily have a either C4 axis of rotation, or a reflection plane as
individual symmetry elements.
The combination of a rotation and reflection in a perpendicular reflection plane gives a new
symmetry element, which is called an improper rotation axis. C4 + reflection → S4
Exercise Check S4 2 and S4
3 of methane
1 operation
Only S4 1 and S4
3 are unique
An S4 axis must have a coaxial C2 axis
For any even-order Sn axis, there must be a coaxial Cn/2 axis.
S6 2 is the same as C3
1; S6 3 is the same as i; S6
4 is the same as C3 2; S6
6 is the same as E
Only S6 1 and S6
5 are unique.
How about odd-order improper rotation axes
S3 3 is not the same as E operation; S3
3 corresponds to operation
S3 6 is the same as the E operation; S3
2is the same as C3 2; S3
4 is the same as C3
Only S3 1 and S3
5 are unique operations
Many of improper operations can be described by other symmetry operations.
With odd-order improper rotation axes; Sn 2n is the same as E.
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Exercise PF5 has a trigonal bipyramidal structure. Sketch the molecule and indentify the
improper rotation axis.
Exercise Identify the S6 axis in CoCl6 3- (an octahedral complex).
This can be difficult to “see”.
This is an even-order improper rotation
axis and it must have a coaxial Cn/2 axis.
Let’s look for C3 axis first.
Exercise Identify and list as many
symmetry elements as you can in the following molecules: PCl5, (E)-1,2-dibromoethane, CH4.
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[PCl5 : E, C3, 3C2, h, S3 and 3v; (E)-1,2-dibromoethane: E, C2, i, h; CH4: E, C3, C2, S4, d]
The properties of symmetry operation
1. Successive operation
(1) Is the order in which we perform operation important?
PH3
(v × C3) ≠ (C3 × v) (non-commutative)
(2) Any combination of symmetry operations can be described by a single symmetry
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operation.
v × C3 = v’, C3 × v = v”
(3) What about performing three successive symmetry operations C3 × v’ × v?
i. Check v’ × v
ii. v’ × v = C3 2; C3 × v’ × v = C3 × C3
2 = E
Exercise Check v’ × v × C3, (C3 × v) × v’, C3 × (v × v’) for PH3. [E, C3 2, C3
2]
2. Identity
E : doing nothing
E can be seen as the result of combinations of other symmetry operations.
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A-1 : the inverse of the A operation. A-1A = E
Any symmetry operation (A) performed on a molecule can be “reversed” or “undone”, by
another operation to give back the original molecule. The A-1 operation is called the inverse of
the A operation.
Exercise Write the inverse of the following symmetry operations in PH3. E, C3 1, C3
2, v. [E, C3 2,
C3 1, v]
Each symmetry operation has an inverse operation which is also a symmetry operation of
the molecule.
4. Class
E, C3 2, C3
1, v, v’, and v’’ are symmetry operations in NH3. C3 2 and C3
1 are “similar”, v,
v’,and v’ are “similar”. Similar perations can be related by mathematical procedure, called a
“similarity transformation”.
The similar symmetry operation can be group into “class”. NH3 : E, 2C3, 3v.
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Point groups
E, C3 1, C3
5, v, v’,
v” Too time-consuming
then define a particular group, which has
several different symmetry elements.
stands for a collection of symmetry operation. “point group”
Point: all the symmetry elements associated with the symmetry operations pass through a single
point in space. This point is not changed in position by any of the symmetry operations.
The point in PCl5 molecule lies directly on the phosphorus atom. (Note: The point is not
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Group: we have a group of symmetry operation
Classify a molecule into a point group by answering some simple question about the
molecule.
Yes: Octahedral Oh
linear, and does not have an symmetry element C∞v
linear, and has an symmetry element D∞h
No: Go to Q2
Q2: Does the molecule possess a rotation axis of order 2
No: If it has no other symmetry C1
If it has one reflection plane of symmetry Cs
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Yes: Go to Q3
Q3: Has the molecule more than one rotation axis?
No: If it has no other symmetry element Cn (n = the order of principle axis)
If it also has one h Cnh
If it has n v Cnv
If it also has an S2n coaxial with the principle axis S2n
Yes: Go Part 4.
Part 4: The molecule can be assigned a point group.
If it has no other symmetry elements Dn
If it has got n d reflection planes bisecting C2 axes Dnd
If it also has one h Dnh
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PCl5
Q2 Yes, there is C3 axis, go to Q3
Q3 Yes, go to Part 4
Part4 It has three v planes, it also has a h plane
D3h
NH3
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Q3 No, it has three v planes C3v
The point groups are shorthand for the symmetry elements within a molecule. Ex. C3v E,
2C3, 3v
Exercise Assign point groups to the following molecules: H2O, PH3, SO2, HCl, AuBr4 -, CoCl6
3-,
-, HCCH, B2H6,
Co(en)3 3+ (en = 1,2-diaminoethane).[ H2O (C2v), PH3 (C3v), SO2 (C2v), HCl (C∞v), AuBr4
- (D4h),
trichloromethane (C3v), NO3 - (D3h), SO4
- (Td), HCCH (D∞h), B2H6 (D2h), Co(en)3 3+ (D3).]