Point Groups and Character Tables

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17 Terms

1

point group

describes a set of symmetry operations that share at least one common point and can be performed on a molecule or object

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2

how to assign point group

1. what is the highest fold rotation axis?
2. are there n C2 axes perpendicular to the principle Cn axis? if yes, D, if no, C
3. is there a σh plane? if yes, h
4. Are there n σv planes containing Cn?

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3

what is the point group for carbon monoxide?

C∞v

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4

what is the point group for O2?

D∞h

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5

special cases: high symmetry

tetrahedral, Td
octahedral, Oh

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6

special cases: low symmetry

Ci if it contains an inversion center but C1 axis, no C2 axes, no mirror planes
Cs if C1 and a σh plane
C2 if the C2 and no mirror planes

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7

character table

a complete set of irreducible representations for a point group, contains info about the structural and spectroscopic behavior for members of the point group

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8

order of a point group

the total number of symmetry operations in a group, add all coefficients of operations

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9

class of a point group

number of columns of operations

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10

how to determine the character for irreducible representation

apply the symmetry operations to the orbital(s) in question, if the phase changes, contributes -1, if not, 1

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11

what is special about s orbitals in character tables?

always transform as the fully symmetric representation

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12

reducible representation

the sum of the irreducible representations in the character table

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13

irreducible representation

the row of characters for the given symmetry species

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14

how to reduce a reducible representation

with the equation, do that for each row to determine number of reducible representations

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15

how to generate a reducible representation for a vibration

assign x,y and z vector to each atom, apply symmetry operation, if the atom moves: 0, if the atom stays put, any vector that does not change is 1 and those that change is -1

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16

IR active mode

results in a change in molecular dipole, symmetry labels with x y or z

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17

Raman active mode

results in a change in polarizability, symmetry labels associated with a product term like x^2, xy

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