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Title: Molecular geometry notes
Description: Chemistry notes of molecular geometry
Description: Chemistry notes of molecular geometry
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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
Title: Molecular geometry notes
Description: Chemistry notes of molecular geometry
Description: Chemistry notes of molecular geometry