ki2141-2011-molecular_symmetry_2013_11_29
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Kimia, Chemistry, Struktur dan Ikatan Kimia, ITB, Ahmad RochyadiTRANSCRIPT
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Molecular Symmetry Achmad Rochliadi, MS., PhD.
and Dr. Veinardi suendo
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Molecular Symmetry 2
Contents
What and Why?
The symmetry elements and operations
The symmetry classification
Consequences of symmetry
Linear algebra in symmetry
Group representation and character tables
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Molecular Symmetry 3
What is symmetry ?
According to Webster Dictionary
Correspondence in size, shape and relative
position of parts that are on opposite sides of a
dividing line or median plane or that are
distributed about a center of axis.
Molecular symmetry
If a molecule has two or more orientation that
are indistinguishable then the molecule
possesses symmetry.
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Molecular Symmetry 4
Why symmetry important in chemistry
The symmetry of the molecule tells us whether
the molecule is chiral, and whether it has a
dipole moment.
Symmetry will allow us to interpret
spectroscopic measurements on molecules. It is
particularly important when we come to
interpreting the infrared (vibrational) spectra of
molecules.
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Molecular Symmetry 5
Symmetry is important in interpreting the crystal
structures of molecules. Modern X-ray diffraction
methods use symmetry in order to interpret the spectra
obtained and determine the absolute position of atoms
within a crystalline solid, and hence its structure.
Symmetry is crucial both in understanding the
electronic structure of molecules (Molecular orbital, or
MO theory). It is crucial in simplifying the otherwise
computationally intensive calculations that need to be
carried out in order to find the energies of molecules
and hence predict their structure and the chemical
reactions that can be carried out on them
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Molecular Symmetry 6
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Molecular Symmetry 7
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Molecular Symmetry 8
Symmetry Elements and Operations
Symmetry Elements
A point, line or plane in the molecule about
which the symmetry operation take out. There is
only 5 symmetry elements related to molecule
symmetry.
Symmetry Operations
Some transformations of the molecule such as a
rotation or reflection which leaves the molecule
in a configuration in space that is
indistinguishable from its initial configuration.
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Molecular Symmetry 9
Symmetry Elements
Elements Symbol Operation
Identity E Leaves each particles in its
original position
N-fold proper
axis
Cn Rotation about the axis by
3600/n (or by multiply)
Plane Reflexion in plane
Inversion
center
i Inversion through center
N-fold
improper axis
Sn Rotation by 3600/n followed
by reflexion in a plane
perpendicular to the axis
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Molecular Symmetry 10
1. The identity operators, E
The simplest symmetry operation is known as the
identity operator, given the symbol E (E from the
German word Einheit, meaning unity). The E operator
basically means do nothing to the molecule.
Evidently, if you do nothing to the molecule it will look
the same as when you started. The identity operation
will thus work on all molecules.
Many non-symmetrical molecules,
such as the amino acid alanine
shown here, contain only the
E operator.
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Molecular Symmetry 11
2. The n-fold rotation operators, Cn
Rotation axes have the nomenclature Cn which means rotate the molecule around the specified axis through an angle of 360/n. Thus, a C2 axis means rotate by 180, C3 by 120, C4 by 90 and so on.
An easy way to remember this is that n is the number of times you would have to rotate the molecule before you would get back to the beginning.
Note that some molecules can contain more than one rotation axis. When this occurs we refer to the axis with the highest degree of rotational symmetry as the principle axis.
Operation C1 is a rotation through 360, and it is equivalent to E Operator.
Operation Cnm
, corresponding to m successive Cn rotations.
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Molecular Symmetry 12
Some of the rotation symmetry elements of a cube. The twofold, threefold and fourfold axes are labeled with the conventional symbols.
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Molecular Symmetry 13
(a) An NH3 molecule has a threefold (C3) axis and (b) H2O molecule has a twofold (C2) axis.
Symmetry modeling : http://www.ch.ic.ac.uk/local/symmetry/
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Molecular Symmetry 14
Successive rotations
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Molecular Symmetry 15
Benzene have C6, C2, the principal axis is the sixfold
axis that perpendicular to the hexagonal ring.
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Molecular Symmetry 16
3. The reflections operators,
A mirror plane (symbol ) is a symmetry
element that results in the reflections of
the molecule through a mirror plane.
If the plane is parallel to the principal
axis, it is called vertical and denoted v.
If the plane is perpendicular to the
principal axis, it is called horizontal and
denoted h
A vertical mirror plane that bisect
(divide) the angle between two C2 axes is
called a dihedral plane and denoted d.
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Molecular Symmetry 17
An H2O molecule has two mirror planes. They are both
vertical so denoted v and v.
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Molecular Symmetry 18
Dihedral mirror planes (d) bisect the C2 axes
perpendicular to the principal axis.
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Molecular Symmetry 19
3. The inversion operators, i
Imagine taking each point in a molecule, moving it to the centre of the molecule, and then moving it out the same distance on the other side.
Move every atom at position (x,y,z) to position (-x,-y,-z). If the molecule still looks the same, then it contains a centre of inversion.
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Molecular Symmetry 20
A Regular octahedron has a centre of inversion
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Molecular Symmetry 21
Inversion of benzene, notice that the three of the C-H
groups have been color coded. When the inversion is
performed, these groups move to the mirrored side.
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Molecular Symmetry 22
5. The n-fold improper rotation, Sn
An improper is the most complex symmetry element to
understand. An improper rotation consist of TWO steps
and neither operation alone needs to be a symmetry
operation:
The rotation like Cn, where the molecule is rotate around the
axis.
Reflection through a plane perpendicular to the axis of that
rotation,
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Molecular Symmetry 23
(a) A CH4 molecule has a fourfold improper rotation axis (S4); the
molecule is indistinguishable after a 90 rotation followed by a
reflection across the horizontal plane, (b) The staggered form of
ethane has an S6 axis composed of a 60 rotation followed by a
reflections.
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Molecular Symmetry 24
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Molecular Symmetry 26
To classify the molecules according to their
symmetries, the molecule symmetry elements is
listed and collect together the molecules with
the same list of elements.
The name of the group is determined by the
symmetry elements it possesses.
Two system of notation
The Schoenflies system, more common.
The Hermann-Mauguin system/International
system, exclusively used in crystal symmetry.
The symmetry classification (Group Theory)
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Molecular Symmetry 27
What is point group
A point group is a collection of symmetry
operations that together are specific to a wide
number of different molecules. These
molecules are from a symmetry viewpoint,
equivalent. For example, both water and cis-
dichloroethene are members of the C2v point
group. Once you have learned about the various
symmetry operations, go to the link on point
groups to find out more about this concept.
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Molecular Symmetry 28
Group Theory
In essence, group theory is a set of
mathematical relationships that allow us to
study symmetry. An in depth and rigorous
study of group theory requires an extensive
knowledge of matrix algebra. As chemists, we
can usually concern ourselves less with the
details of the math, and more on visualizing
how symmetry operations transform molecules
in three dimensional space.
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Molecular Symmetry 29
The diagram for
determining the point
group of a molecule
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Molecular Symmetry 30
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Molecular Symmetry 31
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Molecular Symmetry 32
Summary of shapes corresponding to
different point groups.
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Molecular Symmetry 33
Group C1 Ci Cs
Molecule belong to C1 if has no other element
than the identity (1). Ci if has identity and
inversion (3), and Cs if it has identity and a
mirror plane alone (4).
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Molecular Symmetry 34
Group Cn Cnv Cnh
Cn : possess n-fold axis (5)
Cnv : possess n-fold + v (H2O; NH3)
Cnh : possess n-fold + h (6) - (7)
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Molecular Symmetry 35
Group Dn Dnv Dnh
Dn : possess n-fold axis ntwofold axes
perpendicular to Cn
Dnh : possess n-fold axis n-twofold axes
perpendicular to Cn + h (8, 9, 10, 11)
Dnd : possess Dn + n dihedral mirror planes d
(12. 13)
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Molecular Symmetry 36
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Molecular Symmetry 37
Group S2n
Its the molecule that has not classified into one
of the group above but possess one S2n axis.
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Molecular Symmetry 38
The cubic groups
Molecule that possess more than one principal
axis belong to the cubic groups.
Tetrahedral groups : Td (a), T (a), and Th (a)
Octahedral groups : Oh (b) and O (b)
Icosahedral group : Ih (c) and I (c)
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Molecular Symmetry 39
(a) T (b) O
(c) I (a) Th
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Molecular Symmetry 40
The full rotation group
The molecule rotational group, R3, consists of
infinite number of rotation axes with all
possible values of n.
Sphere and an atom belong to R3
C C
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Molecular Symmetry 41
Cv and Dh point groups
Asymmetrical diatomics (e.g. HF, CO and [CN]-) and linear polyatomics that do not possess a centre of symmetry (e.g. OCS and HCN) possess an infinite number of sv planes but no sh plane or inversion centre. These species belong to the Cv point group.
Symmetrical diatomics (e.g. H2, [O2]2-) and linear polyatomics that
contain a centre of symmetry (e.g. [N3]-, CO2, HCCH) possess a sh
plane in addition to a C axis and an infinite number of sv planes. These species belong to the Dh point group.
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Molecular Symmetry 42
Groups of high symmetry
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Molecular Symmetry 43
D5h D5d
Exercise 1
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Molecular Symmetry 44
Consequences of symmetry
Polarity
If the molecule belongs to group Cn with n > 1,
it cannot possess a charge distribution with a
moment dipole perpedicular to the symmetry
axis but it may have one parallel to the axis.
Molecule belong to Cn, Cnv, Cs may be polar.
All other group such C3h, D, ets there are
symmetry operations that take one end ot
molecule into the other.
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Molecular Symmetry 45
(a) A molecule with a Cn
axis cannot have a dipole
perpendicular to the axis,
but (b) it may have one
parallel to the axis.
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Molecular Symmetry 46
Consequences of symmetry
Chirality
A chiral molecule is a molecule that cannot be superimposed on its mirror image. A chiral molecule is an optic active molecule.
A molecule may be chiral only if it does not posses an axis of improper rotation, Sn.
Take notice that Sn operation could be present under different symmetry element. Example: molecules belonging to the groups Cnh posses an Sn axis implicity because the possess both Cn and h, which are the two components of an improper rotation axis. So as the molecule have i elements.
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Molecular Symmetry 47
Some symmetry elements are implied by the other symmetry elements in a group. Any molecule containing an inversion also possesses at least an S2 element because i and S2 are equivalent
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Molecular Symmetry 48
S4
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Molecular Symmetry 49
Exercise 2
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Molecular Symmetry 50
Linear algebra in symmetry: Dictionary
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Molecular Symmetry 51
The effect on a matrix of a change in coordinate system
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Molecular Symmetry 52
Traces and determinants
The trace of a matrix is defined as the sum of the
diagonal elements.
The determinant of a matrix is a value associated
with a square matrix that can be computed from
the entries of the matrix by a specific arithmetic
expression:
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Molecular Symmetry 53
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Molecular Symmetry 54
Group representation and character tables
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Molecular Symmetry 55
Example: C2v point group
C2 operation
sv(xz) operation
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Molecular Symmetry 56
Example 1: C2v point group (H2O)
C2 sv(xz) operation
Proof the following statements:
C2 C2 = E
sv(xz) sv(yz) = C2
sv(yz) sv(yz) = E
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Molecular Symmetry 57
Characters
The character, defined only for a square matrix, is the trace of the matrix, or the sum of the numbers on the diagonal from upper left to lower right. For the C2v point group, we can obtained the following characters:
We can say that this set of characters also forms a representation, which is an alternate shorthand version of the matrix representation.
Whether in matrix or character format, this is called a reducible representation, a combination of more fundamental irreducible representations.
Reducible representations are designated with a capital gamma (G).
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Molecular Symmetry 58
Reducible and irreducible representations
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Molecular Symmetry 59
Character tables
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Molecular Symmetry 60
Properties of characters of irreducible
representations in point groups
Also
i
i Eh )(
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Molecular Symmetry 61
Properties of characters of irreducible
representations in point groups
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Molecular Symmetry 62
Example 2: C3v point group (NH3)
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Molecular Symmetry 63
Transformation matrices
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Molecular Symmetry 64
In the C3v point group (C32) = (C3), which means that
they are in the same class and described as 2C3 in character table.
In addition, the three reflections have identical characters and are in the same class, as described as 3sv.
The transformation matrices for C3 and C32cannot be
block diagonalized into 1 1 matrices because the C3 matrix has off-diagonal entries. However, they can be block diagonalized into 2 2 and 1 1 matrices, with all other matrix elements equal to zero.
Transformation matrices
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Molecular Symmetry 65
Character tables of C3v point group
The C3 matrix must be blocked this way because the (x,y) combination is needed for the new x and y, while the other matrices must follow the same pattern for consistency across the representation.
The set 2 2 matrices has the characters corresponding to the E representation, while the set of 1 1 matrices matches the A1 representation.
The A2 representation can be found using the defining properties of a mathematical group as in previous example.
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Molecular Symmetry 66
Character tables of C3v point group: A1
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Molecular Symmetry 67
Character tables of C3v point group: E
Antisymmetric Symmetric disymmetric
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Molecular Symmetry 68
Properties of the characters for C3v point group
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Molecular Symmetry 69
Addition feature of character tables
The expression listed to the right of the characters indicate the symmetry of mathematical functions of the coordinates x, y and z and of rotation about the axes (Rx, Ry, Rz).
This can be used to find the orbitals that match the representation. For example: x with (+) and (-) direction matches the px orbital with (+) and (-) lobes in the quadrants in the xy plane; the product xy with alternating signs on the quadrants matches lobes of the dxy orbital.
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Molecular Symmetry 70
Addition feature of character tables
In all cases, the totally symmetric s orbital matches the first representation of in the group, one of the A set.
The rotational functions are used to describe the rotational motion of the molecule.
In the C3v example, the x and y coordinates appeared together in the E irreducible representation with notation (x,y). This means that x and y together have the same symmetry properties as the E irreducible representation. Consequently, the px and py orbitals together have the same symmetry as the E irreducible representation in this point group.
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Molecular Symmetry 71
Addition feature of character tables
Matching the symmetry operations of a molecule with those listed in the top row of the character table will confirm any point group assignment.
Irreducible representations are assigned labels according to the following rules, in which symmetric means a character of 1 and antisymmetric a character of -1.
Letter are assigned according to the dimension of the irreducible representation.
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Molecular Symmetry 72
Addition feature of character tables
This might also give us information about degeneracies as follows:
A and B (or a and b) indicate non-degenerate
E (or e) refers to doubly degenerate
T (or t) means triply degenerate
Subscript 1 designates a representation symmetric to a C2 rotation perpendicular to the principal axis, and subscript 2 designates a representation of antisymmetric to the C2. If there are no perpendicular C2 axes, 1 designates a representation symmetric to vertical plane, and 2 designates a representation antisymmetric to a vertical plane.
Subscript g (gerade) designates symmetric to inversion, and subscript u (ungerade) designates antisymmetric to inversion.
Single prime () are symmetric to sh and double prime () are antisymmetric to sh when a distinction between representations is needed (C3h, C5h, D3h, D5h).
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Molecular Symmetry 73
Molecular vibration: H2O (C2v)
Degree of freedom
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Molecular Symmetry 74
Molecular vibration: H2O (C2v)
Full C2 operation of H2O
The Ha and Hb entries are not on the principal diagonal
because Ha and Hb exchange each other in a C2 rotation, and x(Ha) = -x(Hb), y(Ha) = -y(Hb) and z(Ha) = z(Hb).
Only oxygen contribute to the character for this operation, for total of -1.
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Molecular Symmetry 75
Molecular vibration: H2O (C2v)
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Molecular Symmetry 76
Reducing representations to irreducible representations
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Molecular Symmetry 77
Symmetry molecular motion of water
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Symmetry molecular motion of water
Molecular Symmetry 78
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Molecular Symmetry 79
IR Spectra of H2O
Experimental values are 3756, 3657 and 1595 cm1
Calculated IR spectrum of gaseous H2O
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Molecular Symmetry 80
A bent triatomic: H2O (C2v)
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Molecular Symmetry 81
A bent triatomic: H2O (C2v)
2s atomic orbital of the O (a1)
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Molecular Symmetry 82
A bent triatomic: H2O (C2v)
2px atomic orbital of the O (b1)
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Molecular Symmetry 83
A bent triatomic: H2O (C2v)
2py atomic orbital of the O (b2)
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Molecular Symmetry 84
A bent triatomic: H2O (C2v)
2pz atomic orbital of the O (a1)
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Molecular Symmetry 85
A bent triatomic: H2O (C2v)
1s atomic orbital of the H (a1 and b2)
and
a1 b2
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Molecular Symmetry 86
A bent triatomic: H2O (C2v)
1s atomic orbital of the H (a1 and b2)
summation
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Molecular Symmetry 87
A bent triatomic: H2O (C2v)
1s atomic orbital of the H (a1 and b2)
Normalization
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Molecular Symmetry 88
A bent triatomic: H2O (C2v)
1s atomic orbital of the H (a1 and b2)
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A bent triatomic: H2O (C2v)
Molecular Symmetry 89
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Molecular Symmetry 90
The MO Diagram of Water with the 2a1 Showing Bonding
and Antibonding Character
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The MO Diagram of Water with the 2a1 Labeled as
Non-bonding
Molecular Symmetry 91
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The MO Diagram of Water with the 1a1 Labeled as Non-
bonding
Molecular Symmetry 92
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Further on character tables
Molecular Symmetry 93
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Further on character tables
Molecular Symmetry 94
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Further on character tables
Molecular Symmetry 95
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Further on character tables
Molecular Symmetry 96
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Further on character tables
Molecular Symmetry 97
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Further on character tables
Molecular Symmetry 98
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Molecular Symmetry 99
References and further readings
P. Atkins and J. de Paula, Atkins Physical Chemistry, 8th Edition, Oxford University Press, Oxford, 2006.
S.F.A. Kettle, Symmetry and Structure: Readable Group Theory for Chemists, 3rd Edition, John Wiley & Sons, Chichester, 2007.
A.M. Lesk, Introduction to Symmetry and Group Theory for Chemists, Kluwer Academic Publishers, Dordrecht, 2004.
C.E. Housecroft and A.G. Sharpe, Inorganic Chemistry, 3rd Edition, Pearson Education Limited, Harlow, 2008.
G.L. Miessler and D.A. Tarr, Inorganic Chemistry, 4th Edition, Pearson Education Limited, Harlow, 2010.