Download Molecular Symmetry

Topics
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Introduction
Molecular Structure and Bonding
Molecular Symmetry
Coordination Complexes
Electronic Spectra of Complexes
Reactions of Metal Complexes
Organometallic Chemistry
Symmetry
• Powerful mathematical tool for understanding
structures and properties
• Use symmetry to help us with:
– Detecting optical activity and dipole
moments
– Forming MO’s
– Predicting and understanding spectroscopy
of inorganic compounds
• Infrared, Raman and UV-visible
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Symmetry and Your Text(s)
• Symmetry tools and language
– Sec 4.1-4.4 (Atkins)
3.1-3.6, 4.4 (Housecroft)
– Flow chart p.122 (Atkins)
p.81 (Housecoft)
• Bonding Theory
– Sec 4.5-4.7 (Atkins)
– Box 4.1, sec 4.5-4.7 (Housecroft)
Symmetry and the Exam
• Recognize symmetry elements
• Identify the important elements present in a
molecule
• Assign the point group of an object or
molecule
• Read a character table
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Symmetry Elements
• A symmetry element is present if the
operation is performed and the object is
indistinguishable from its original state
Element
Name
Operation
Cn
n-fold rotation
rotate by 360°/n
σ
mirror plane
Reflection through a plane
i
Center of inversion
Inversion through the center of
the object
Sn
Improper rotation axis
Rotation as Cn followed by
reflection in perpendicular mirror
plane
E
Identity
Do nothing
Center of Inversion: I
• Inverts all atoms through the centre of the
object
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Inversion vs Rotation (C2)
Mirror Planes: σ
• Reflection of object through a mirror plane
• Objects in the plane are reflected onto
themselves, objects on either side of the plane
are reflected to the other side
• Three types
– σh : Horizontal, perpendicular to principal axis
– σv : Vertical, parallel to principal axis
– σd : Dihedral, same plane as σv related by half a
rotation of the principal axis
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Dihedral vs. Vertical
• Typically σd and σv are related by rotation of
180/n
• Labelling:
– Exception: when n=2, label is σv’ not σd
• Rule of Thumb: dihedral planes pass through
fewer atoms (i.e. is dihedral to the angle of
the bonds)
Improper Rotation Axis: Sn
• Rotate 360/n followed by reflection in mirror plane
perpendicular to axis of rotation
• All planar molecules have an Sn
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Special Cases: S1 and S2
• S1= σh and S2 = i
Comparing Symmetry
• Compare NH3 and BF3
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Point Groups
• collections of symmetry elements are
summarized into Point Groups
– these are groups as strictly defined by
mathematical group theory
• short form method for identifying all of the
symmetry elements present in a molecule
OC
OC
CO
Mo
CO
CO
CO
Oh
8C3, 6C2, 6C4, i,
3C2, 6S4, 8S6,
3σh, 6σd
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Identifying Molecular Symmetry
• Requires knowing molecular geometry
– First draw a Lewis structure
– Use VSEPR to predict molecular geometry
– Use VSEPR geometry to identify symmetry
elements present
– Classify molecule according to its point group
Lewis Structures
1. Count valence electrons available, include
net charges.
2. Write skeleton structure, drawing bonds
between atoms using up two valence
electrons for each bond.
3. Distribute remaining electrons to most
electronegative species first to fill electron
shells.
4. Satisfy unfilled octets where possible by
drawing multiple bonds
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Resonance Structure and Formal Charge
• Resonance allows for non-integer bond order
and delocalized electron distribution
• Formal Charge
– Predicts which of multiple possible structures are
more favourable
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FC = group #− bonded electrons − unshared electrons
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VSEPR
• Relies on the electron distribution around the
central atom
– Bonded pairs
• Typically ignore bond order (1,2 etc)
• Ignore what atom it is bound to
• Occupy less volume than lone pair
– Lone pairs (on the central atom)
• Bulkier than bonds
• Have largest impact on molecular geometry
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Examples
• trans-CrCl4O2
• PCl5
Character Tables
• Character tables are “tell-all” manuals of
symmetry, tabulated in your text
• Using group theory, lists of behaviour under
the symmetry elements in a point group are
tabulated, these are called “irreducible
representations” and they are said to “span”
the group
– Therefore, a portion of a molecule can be
described by some linear combination of
these irreducible representations.
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Reading the Character Tables
C3v
E
2C3
3σv
A1
1
1
1
z
A2
1
1
-1
Rz
E
2
-1
0
(x,y), (Rx,Ry)
x2+y2+z2
(x2-y2,xy)
Character values:
1 means no change
-1 means change of sign
2, 0 sum of multicomponent behaviour
Mulliken Labels
• A,B,E,T – indicating degeneracy
• A vs B : symmetric or antisymmetric wrt highest order
rotation axis
• 1,2 : symmetric or antisymmetric wrt C2 axis or σv
• g,u : symmetric or antisymmetric wrt i
• ‘,” : symmetric or antisymmetric wrt σv
• MO are labeled with the Mulliken labels using lower
case to differentiate them from irreducible
representations
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D3h Character Table
D3h
E
2C3
3C2
σh
2S3
3σv
A1’
1
1
1
1
1
1
A2’
1
1
-1
1
1
-1
Rz
E’
2
-1
0
2
-1
0
(x,y)
A1”
1
1
1
-1
-1
-1
A2”
1
1
-1
-1
-1
1
Z
E”
2
-1
0
-2
1
0
(Rx, Ry)
x2+y2, z2
(x2-y2, xy)
(zx, yz)
Basis Functions
• Shows how standard components transform (are
described by which irreducible representation)
• x,y,z :corresponds to the behaviour of the px, py,
pz orbitals
• x2+y2+z2 : behaviour of an s orbital
• xy, xz, yz, z2, x2: behaviour of respective d orbitals
• Ri : behaviour of Rotation around axis i
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Degeneracy
• When two basis functions are shown in
parenthesis these objects transform together
and are degenerate
– eg. (x,y)
• They cannot be considered individually and
must be treated as a pair
Applications of Symmetry
• Chirality
– means non-superimposable on its mirror image
– presence of Sn, mirror plane or i all rule out
chirality
• Dipole moment (polarity)
– a permanent dipole cannot exist if i is present
– dipole moment cannot be perpendicular to any
mirror plane or Cn
• IR/Raman Spectroscopy
• Constructing MO’s
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Constructing MO’s
• typical structure of inorganic compounds involves a
central atom with some arrangement of surrounding
atoms attached to it (ligands)
• Build Symmetry Adapted Linear Combinations of
Atomic Orbitals (SALC’s) out of groups of atomic
orbitals (from the ligands)
– Shown in Appendix 4 of Shriver and Atkins
• Combine SALC’s with orbitals on the central atom
– Recall that only orbitals of the same symmetry can be
combined
• We use symmetry to assign and build orbitals and
combinations of orbitals to make complete MO’s
Identifying the MO Components
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it is helpful to determine in advance how many MO’s
of each type and what their symmetry should be
use vectors to represent bonds
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–
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use vectors between the atoms for the sigma orbitals
use vectors perpendicular to the sigma orbitals for the pi
orbitals
determine how these vectors transform under the
symmetry of the molecule
determine the irreducible representations which
combine to produce this behaviour
these irreducible representations are the MO labels
now all we have to do is figure out what orbital
combinations they are formed from ....
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Identifying Symmetry Adapted Orbitals
Steps:
1. Identify the point group
2. Arrange the group of ligand orbitals,
determine their transformation under the
symmetry operations of the point group.
3. Assign a symmetry label to the group orbital.
4. Match the labeled group orbitals with atomic
orbitals on the central atom.
Energy Levels of MO’s
• Can be roughly estimated by
– energies of atomic orbitals
– degree of overlap
• Determine computationally
• Recall that our biggest concern is
understanding possible interactions and
populating existing MOELD’s
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Conclusions
Symmetry
Identifying symmetry elements
Assigning point groups
Reading character tables
transforms
basis functions
Constructing MO’s
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