Lecture 37: Symmetry Orbitals

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Lecture 37 Symmetry Orbitals The material in
this lecture covers the following in Atkins. 15
Molecular Symmetry Character
Tables 15.4 Character tables
and symmetry labels (a) The
structure of character tables
(b) Character tables and orbital degeneracy
(c) Characters and operators
(d) The classification of linear
combinations of orbitals 15.5
Vanishing integrals and orbital overlaps
(a) The criteria for vanishing
integrals (b) Orbitals with
nonzero overlaps (c)
Symmetry-adapted linear combinations Lecture
on-line Symmetry Orbitals (PowerPoint)
Symmetry Orbitals (PowerPoint) Handouts
for this lecture
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One dimensional irreducible representations have
the character 1 for E.
They are termed A or B.
A is used if the character of the principle
rotation is 1.
B is used if the character of the principle
rotation is -1
A1 has the character 1 for all operations
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Irreducible representations with dimension 2 are
denoted E
Irreducible representations with dimension 3 are
denoted T
Number of symmetry species (irreducible
representations) Number of classes
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A px orbital on the central atom of a C2v
molecule and the symmetry elements of the group.
Epx 1px C2px -1 px
svpx 1px sv px -1 px
The irrep. is B1 and The symmetry b1
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A py orbital on the central atom of a C2v
molecule and the symmetry elements of the group.
Epy 1py C2py -1 py
svpy - 1py sv py 1 py
The irrep. is B2 and The symmetry b2
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A pz orbital on the central atom of a C2v
molecule and the symmetry elements of the group.
Epz 1pz C2pz 1 pz
svpz 1pz sv pz 1 pz
The irrep. is A1 and The symmetry a1
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A dxy orbital on the central atom of a C2v
molecule and the symmetry elements of the group.
Edxy 1dxy C2dxy 1 dxy
svdxy -1dxy sv dxy - 1 dxy
The irrep. is A2 and The symmetry a2
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A 1s orbital on the two terminal atoms of a
C2v molecule and the symmetry elements of the
group.
E 1s 1 1s C2 1s 1 1s
sv 1s 1 1s sv 1s 1 1s
The irrep. is A1 and The symmetry a1
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A 1s- orbital on the two terminal atoms of a
C2v molecule and the symmetry elements of the
group.
E 1s- 1 1s- C2 1s- -1 1s-
sv 1s- -1 1s- sv 1s- 1 1s-
The irrep. is B2 and The symmetry b2
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A 2p- orbital on the two terminal atoms of a
C2v molecule and the symmetry elements of the
group.
E 2p- 1 2p- C2 2p- 1 2p-
sv 2p- -1 2p- sv 2p- -1 2p-
The irrep. is A2 and The symmetry a2
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The value of an integral I (for example, an area)
is independent of the coordinate system used to
evaluate it.
That is, I is a basis of a representation of
symmetry species A1 (or its equivalent).
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1. Decide on the symetry species of
the Individual functions f1 and f2 by reference
to the character table, and write their
characters in two rows in the same same order as
in the table
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1. Decide on the symmetry species of
the individual functions f1 and f2 by reference
to the character table, and write their
characters in two rows in the same same order as
in the table
2. Multiply the numbewrs in each column, Writing
the results in the same order
2py 1 -1 -1 1 1s 1 1
1 1
3. The new character must be A1 For the integral
to be non-zero
2py1s 1 -1 -1 1
The symmetry species is B2
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2py 1 -1 -1 1 1s- 1 -1
-1 1
2py1s- 1 1 1 1
The symmetry species is A1
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The character table of a group is the list of
characters of all its irreducible
representations.
Names of irreducible representations
A1,A2,B1,B2.
Characters of irreducible representations
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The integral of the function f xy over the
tinted region is zero. In this case, the result
is obvious by inspection, but group theory can
be used to establish similar results in less
obvious cases.
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The integration of a function over a pentagonal
region.
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Two symmetry-adapted linear combinations of the
p-basis orbitals. The two combinations each
span a one-dimensional irreducible
representation, and their symmetry species are
different.
Typical symmetry-adapted linear combinations
of orbitals in a C 3v molecule.
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Construct a table showing the effect of each
operation on each orbital of the original basis
To generate the combination of a Specific
symmetry species, take Each column in turn and
(I) Multiply each member of the Column by the
character of the Corresponding operator
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(I) Multiply each member of the Column by the
character of the Corresponding operator
(2) Add and divide by group order
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(I) Multiply each member of the Column by the
character of the Corresponding operator
(2) Add and divide by group order
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(I) Multiply each member of the Column by the
character of the Corresponding operator
(2) Add and divide by group order
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(I) Multiply each member of the Column by the
character of the Corresponding operator
(2) Add and divide by group order
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(I) Multiply each member of the Column by the
character of the Corresponding operator
(2) Add and divide by group order
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(I) Multiply each member of the Column by the
character of the Corresponding operator
(2) Add and divide by group order
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What you should learn from this course
1. Be able to assign symmetries to orbitals from
character tables.
2. Be able to use character tables to determine
whether the overlap between two functions
might be different from zero.
3. Be able to use character table to construct
symmetry orbitals as linear combination of
symmetry equivalent atomic orbitals