Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Symmetry Operations and Elements
•

The goal for this section of the course is to understand how symmetry arguments can be
applied to solve physical problems of chemical interest
...


•

A symmetry element is an imaginary geometrical construct about which a symmetry
operation is performed
...


•

A symmetry
t operation
ti carries
i every point
i t in
i the
th object
bj t into
i t an equivalent
i l t point
i t or the
th
identical point
...


•

The symmetry of a molecule or ion can be described in terms of the complete collection of
symmetry operations it possesses
...
The more
symmetry operations a molecule has, the higher its symmetry is
...


The Identity Operation (E)
•

The simplest of all symmetry operations is identity, given the symbol E
...
If it possesses no other symmetry, the object is said to be
asymmetric
...
It exists for every object, because the
object itself exists
...


•

In addition,
addition identity is often the result of carrying out a particular operation successively a
certain number of times,
i
...
, if you keep doing the same operation repeatedly, eventually you may bring the object
back to the identical (not simply equivalent) orientation from which was started
...


•

Thus, if a series of repeated operations carries the object back to its starting point, the result
would be identified simply as identity
...


•

If a molecule has rotational symmetry Cn , rotation by
2/n = 360°/n brings the object into an equivalent
position
...


•

If the molecule has one or more rotational axes, the
one with the highest value of n is the principal axis of
rotation
...


•

Multiple iterations are designated by a superscript,
e
...
three successive C4 rotations are identified as C43

•

The C42 and C44 operations are preferably identified as
the simpler C2 and E operations,
operations respectively
...


•

The C2' and C2" axes of a planar MX4 molecule
...


•

Only two notations are needed for the four axes,
axes because both C2’ axes are said to belong to
the same class, while the two C2’’ axes belong to a separate class
...
e
...


•

In listing the complete set of symmetry
operations for a molecule, operations of the
same class are designated by a single notation
preceded by a coefficient indicating the number
of equivalent operations comprising the class
...
g
...

The C2' and C2" axes of a planar MX4 molecule
...
)

Cnm = Cn/m

(n/m = 2, 3, 4, 5…etc
...


•

Remember, the rotational operation Cnm
operation where m/n is an integer value
...


•

For every point a distance r along a normal to a mirror plane there exists an equivalent point
at –r
...


•

For a point (x,y,z), reflection across a mirror plane σxy takes the point into (x,y,–z)
...


Horizontal, Vertical, and Dihedral Mirror Planes
•

A h plane is defined as perpendicular
to the principal axis of rotation
...


•

v and d planes are defined so as to
contain a principal axis of rotation and
to be perpendicular to a h plane
...


•

Any d planes typically will contain
bond angle bisectors
...


The Inversion Operation ( i )
•

The operation of inversion is defined relative to the central point within the molecule,
through which all symmetry elements must pass,
e g typically the origin of the Cartesian coordinate system (x,y,z
e
...
,
(x y z = 0,0,0)
...


•

Molecules or ions that have inversion symmetry are said to be centrosymmetric
...


Effect of inversion (i) on an octahedral MX6 molecule (XA = XB = XC = XD = XE = XF)
...
The inversion center is at the midpoint along the C‐C
y g atoms related byy inversion are connected byy dotted lines,, which intersect at
bond
...
The two carbon atoms are also related by inversion
...


•

Rotation‐reflection consists of a proper rotation followed by reflection in a plane
perpendicular to the axis of rotation
...


•

Sn exists
i t if the
th movements
t Cn followed
f ll
d by
b σh (or
( vice
i versa)) bring
b i the
th object
bj t to
t an equivalent
i l t
position
...

e
...
, S4 collinear with C4 in planar MX4
...

e
...
, S4 collinear with C2 in tetrahedral MX4
...

A C2 axis, collinear with an S4 axis, passes through
the centers of each pair of opposite cube faces
and through the center of the molecule
...
e
...

S4 improper rotation of a tetrahedral MX4 molecule
(XA = XB = XC = XD)
...
Rotation is arbitrarily taken in a
clockwise direction
...


•

Successive S4 operations on a tetrahedral
MX4 molecule (XA = XB = XC = XD)
...


•

Successively carrying out two S4 operations
is identical to the result of a single C2
operation about the same axis
i
...
,, S42 = C2

•

Similarly, S44 = E
...


•

In the highly symmetric tetrahedral system
there
are
three
equivalent
and
indistinguishable S4 axes
...

)

Non‐Genuine Sn Operations:
•

S1 = σh

•

S2 = i

General Relations of Sn
•

Equivalences of successive Sn operations:
 If n is even, Snn = E
 If n is odd, Snn = σ and Sn2n = E
m n when
 If m is
i even, Snm = Cnm = Cn/m when
h m < n and
d Snm = Cnm–n
h m>n

 If Sn with even n exists, then Cn/2 exists
 If Sn with odd n exists, then both Cn and σ perpendicular to Cn exist
...


The origin of the coordinate system is located at the central atom or the center of the
molecule
...


The z axis is collinear with the highest‐order rotational axis (the principal axis)
...


However, for a tetrahedral molecule, the x, y, and z axes are
defined as collinear with the three C2 axes (collinear with the three S4 axes)
...
)
3
...


If the z axis lies in the plane of the molecule, then the x axis stands perpendicular to the
plane
...
)
4
...
If there are two or more such
planes containingg identical sets of atoms,, anyy one mayy be taken as the xz p
p
plane
...


Combining Symmetry Operations (Multiplication)
•

Multiplication of symmetry operations is the successive performance of two or more
operations to achieve an orientation that could be reached by a single operation
e
...
, i 2 = E ; S4 S4 = S42 = C2 ; C4 h = S4 etc
...


•

Multiplication of symmetry operations is not commutative in general, although certain
combinations may be
...
"



We cannot assume that reversing the order will have the same result
...

AB

Multiplication of symmetry operations is associative:
C(BA) = (CB)A

The order of performing S
The
order of performing S4 and σ
and σv , shown here for a tetrahedral MX
shown here for a tetrahedral MX4 molecule, affects the result
...

S4 σv ≠ σv S4
but
C2σv S4 = S4 σv

•

We will now consider the complete set of symmetry operations for a particular molecule
and determine all the binary combinations of the symmetry operations it possesses
...
This molecule has a
tetrahedral geometry
Note: tetrahedral geometry does not automatically imply tetrahedral symmetry !

•

The complete set of symmetry operations are E, C2 , v , v‘

Matrix Notation of the Effects of the Operations
•

Rather than depict the effect of each operation on the molecule, let us introduce a column
matrix notation to indicate the positions of atoms before and after each operation
...

matrix

•

The symbols [E], [C2], v and [v’] represent operator matrices
...


•

Combination order is "top" then "side"; e
...
,

Multiplication Table for the Operations of CBr2Cl2
•

Now let us consider the results for binary combinations of these
operations
...


•

We will begin with combinations including the identity operation
...

•

Step 2: Binary self‐combinations
...

•

Let us consider the combination of C2 σv
...


•

This result is the same as that achieved by σv' alone:

•

Thus, we can write the following relationship:

C2 σv = σv'

Complete Multiplication Table

•

General Features:


The first row of results duplicates the list of operations in the header row
...




Every row shows every operation once and only once
...




The order of resultant operations in every row is different from any other row
...


Symmetry Point Groups
•

The complete set of symmetry operations (not symmetry elements) for a molecule or ion
satisfies the requirements of a mathematical group
...


•

The total number of symmetry operations comprising the group is the order of the group, h
...

4

•

The group of symmetry operations must satisfy the four requirements of a mathematical
group, i
...
, closure, identity, associativity, and reciprocality
...




All groups have a self‐contained multiplication table, whose products are members of
the group
...




These four symmetry operations constitute the complete set of elements of a point
group called C2v , the order of which is four
...




This requirement explains the need to define the symmetry operation of identity, which
functions at the identity element for every symmetry group
...


Requirements of a Mathematical Group (3)
•

Associativity:

The associative law of combination is valid for all combinations of elements
of the group
...
g
...




For example, in the C2v point group
C2 vv‘ ) = (C2v )v‘
for the first combination we see
C2 vv‘) = C2C2 = E
for the second combination we see
(C2v )v‘ = v‘v‘ = E



Groups in which all elements do commute are called Abelian (e
...
, C2v)
...
g
...




For the C2v point group every symmetry operation is its own inverse
...

note: one exception is the trivial asymmetric group,
group C1 = {E}

•

The order of any subgroup, g, relative to the order of its parent group, h, must be
h / g  =  n

( n =  1,2,3,4,…
...
)

•

Not every allowed value of g is always represented among a group’s subgroups
...


•

From the multiplication table of C2v we can identify the following subgroups

Point Groups of Molecules
•

Chemists in general and spectroscopists in particular use the Schönflies notation
...


•

Familiar Schönflies labels and their corresponding Hermann‐Mauguin notation are

•

All of the chemically important point groups fall within one of four general categories:
1
...


Single‐axis rotational

3
...


Cubic

Non‐Rotational Point Groups
•

With their low orders (h = 1,2) and lack of an axis of symmetry, the non‐rotational point groups
represent the lowest symmetry point groups
...




The Cs point group describes the symmetry of bilateral objects that lack any symmetry
other than E and h
...


Single‐Axis Rotational Point Groups
•

The simplest family of this group are the Cn point groups, which consist of operations
generated by an n‐fold rotation Cn applied successively n times
...

g
p
These p

•

A cyclic group of order h is generated by taking a single element X through all its powers up
to Xh = E
...
 , Xh = E }

•

All cyclic groups are Abelian, since all of their multiplications commute
...
g
...
g
...

planes,
axis

•

The point group Cv , which has a infinite‐fold C rotational axis, is an important member of this
family
...

e g H‐Cl,
e
...
,
H Cl CO
...


•

Since Cnh = Sn and C2h = S2 = i , these groups also have n‐fold improper axes when n > 2,
and they are centrosymmetric when n is even
...


Dihedral Point Groups
•

The dihedral groups have n twofold axes perpendicular to the principal n‐fold axis
...


•

The number and arrangement of the dihedral axes are dictated by the n‐fold order of the
principal axis
...
g
...


•

There are three families of dihedral groups:

Dn , Dnd , Dnh

1) The Dn groups may be thought of as Cn groups to which n dihedral C2 operations have
been added
...

2) Similarly
Similarly, the Dnd groups may be thought of as Cnv groups to which n dihedral C2
operations have been added
...

3) The Dnh groups may be thought of as Cnh groups to which n dihedral C2 operations have
been added
...


Cubic Point Groups
•

The cubic groups are associated with polyhedra that are geometrically related to the cube
...


•

p of this type,
yp , three of which are frequently
q
y encountered and highly
g y
There are seven ggroups
relevant in chemistry

Cube (Oh)

Tetrahedron (T d )

Octahedron (Oh)

Icosahedron (I h)

•

The perfect tetrahedron defines the Td group, comprised of the following 24 operations, listed
b classes:
by
l
E , 8C3 (= 4C3 , 4C32 ), 3C2 , 6S4 (= 3S4 , 3S43 ), 6d
with h = 24 , Td represents one of the higher symmetries encountered in chemistry
...


•

pp
g
When a tetrahedron is inscribed inside a cube a C2 axis collinear with the bisector of opposing
bond angles emerges from each pair of apposite cube faces
...


Tetrahedron (T d )

•

The octahedron and cube both belong to the point group Oh , which is comprised of the
following 48 operations (h = 48)
E , 8C3(= 4C3 , 4C32 ), 6C4(= 3C4 , 3C43 ), 6C2 , 3C2(= 3C42), i , 6S4(= 3S4 , 3S43 ), 8S6(= 4S6 , 4S65 ),
h(= xy , yz , xz), 6d

•

In the octahedron a fourfold axis emerges from each pair of opposite apices, whereas a
threefold
h f ld axis
i emerges from
f
each
h pair
i off opposite
i triangular
i
l faces
...

cube

Cube (Oh)

•

Both the regular icosahedron and dodecahedron belong to the point group Ih , composed of
120 symmetry operations
E , 8C3(= 4C3 , 4C32 ), 6C4(= 3C4 , 3C43 ), 6C2 , 3C2(= 3C42), i , 6S4(= 3S4 , 3S43 ), 8S6(= 4S6 , 4S65 ),
h((= xy , yz , xz), 6d

•

Aside from the Cv and Dh point groups which have an order of h = , Ih represents the
highest symmetry one is likely to encounter in structural chemistry
...


•

A fivefold axis emerges from the face of each five‐membered ring and a threefold axis emerges
from the face of each six‐membered ring
...


Examples for point group classification

---