Group Theory - Inorganic Chemistry

Front 1

The identity symmetry element gives rise to the ____ operation.

Back 1

The identity symmetry element gives rise to the identity operation.

Front 2

The proper axis of rotation symmetry element gives rise to the ____ operation.

Back 2

The proper axis of rotation symmetry element gives rise to the rotation operation.

Front 3

The mirror plane symmetry element gives rise to the ____ operation.

Back 3

The mirror plane symmetry element gives rise to the reflection operation.

Front 4

The inversion centre symmetry element gives rise to the ____ operation.

Back 4

The inversion centre symmetry element gives rise to the inversion operation.

Front 5

The improper axis of rotation symmetry element gives rise to the ____ and ____ operations.

Back 5

The improper axis of rotation symmetry element gives rise to the rotation and reflection operations.

Front 6

The ____ operation consists of doing nothing to a molecule. It is given the symbol ____.

Back 6

The identity operation consists of doing nothing to a molecule. It is given the symbol E.

Front 7

Molecular shapes before and after an operation must be ____ from each other.

Back 7

Molecular shapes before and after an operation must be indistinguishable from each other.

Front 8

The rotation axis is given the symbol ____. A 90 degree rotation would be given the symbol ____.

Back 8

The rotation axis is given the symbol C. A 90 degree rotation would be given the symbol C[4].

Front 9

What is the rotation operation of this molecule?

Back 9

C[2]

Front 10

What is the rotation operation of this molecule?

Back 10

C[3]

Front 11

What is the rotation operation of this molecule?

Back 11

C[4]

Front 12

What is the rotation operation of this molecule?

Back 12

C[5]

Front 13

What rotation operation has been carried out on this molecule?

Back 13

C[4]{1}

Front 14

What other symmetry operation is a C[n]{n} operation identical to?

Back 14

E (identity)

Front 15

If a C[4] axis is present, a ____ axis must also be present.

Back 15

If a C[4] axis is present, a C[2] axis must also be present.

Front 16

The highest order axis present is called the ____ axis.

Back 16

The highest order axis present is called the principle axis.

Front 17

The plane responsible for reflection operations is called a plane of ____. It is given the symbol ____.

Back 17

The plane responsible for reflection operations is called a plane of symmetry. It is given the symbol σ.

Front 18

____ symmetry planes contain the principle axis.

Back 18

σ[v] symmetry planes contain the principle axis.

Front 19

Symmetry through the σ(xz) plane is given the symbol ____.

Back 19

Symmetry through the σ(xz) plane is given the symbol σ[v].

Front 20

Symmetry through the σ(yz) plane is given the symbol ____.

Back 20

Symmetry through the σ(yz) plane is given the symbol σ[v]'.

Front 21

____ symmetry planes are perpendicular to the principle axis.

Back 21

σ[h] symmetry planes are perpendicular to the principle axis.

Front 22

What are the symmetry operations of this molecule?

Back 22

σ[v] and σ[v]'.

Front 23

What is the symmetry operation of this molecule?

Back 23

σ[h]

Front 24

____ planes are vertical symmetry planes which bisect the ____ axes perpendicular to the principal axis. They are given the symbol ____.

Back 24

Dihedral planes are vertical symmetry planes which bisect the C[2] axes perpendicular to the principal axis. They are given the symbol σ[d].

Front 25

If a molecule has no C[n] operations its symmetry plane is labelled ____.

Back 25

If a molecule has no C[n] operations its symmetry plane is labelled σ.

Front 26

What other symmetry operation is a σ[v]{2} operation identical to?

Back 26

E (identity)

Front 27

If one operation of a C[8] axis is carrier out on a molecule, through how many degrees is it rotated?

Back 27

45 degrees

Front 28

What is the order of the principal axis for this molecule?

Back 28

3

Front 29

C[6]{4} is equivalent to which other operation?

Back 29

C[3]{2}

Front 30

Give an example of an operation which is equivalent to E.

Back 30

e.g. C[4]{4} (any C[n]{n}), σ{2}

Front 31

List the rotation axes of this molecule.

Back 31

8C[3] and 3C[2]

Front 32

List the rotation axes of this complex.

Back 32

8C[3] and 3C[2]

Front 33

Inversion operations are ____ operations involving a ____ operation and a ____ operation. They are given the symbol ____.

Back 33

Inversion operations are composite operations involving a rotation operation and a reflection operation. They are given the symbol i.

Front 34

The inversion centre of a molecule is always located at the ____. The inversion operation changes the coordinates of every point from (x,y,z) to ____.

Back 34

The inversion centre of a molecule is always located at the origin. The inversion operation changes the coordinates of every point from (x,y,z) to (-x,-y,-z).

Front 35

Molecules that possess a centre of symmetry are said to be ____.

Back 35

Molecules that possess a centre of symmetry are said to be centrosymmetric.

Front 36

What other symmetry operation is a i{2} operation identical to?

Back 36

E (identity)

Front 37

Inversion operations involve a ____ degree rotation about the ____ axis followed by a reflection in the ____ plane.

Back 37

Inversion operations involve a 180 degree rotation about the z axis followed by a reflection in the xy plane.

Front 38

If a molecule possesses a ____ principal axis and a ____ plane then it must also possess an inversion centre.

Back 38

If a molecule possesses a C[2] principal axis and a σ[h] plane then it must also possess an inversion centre.

Front 39

Improper rotations are ____ operations involving a ____ operation and a ____ operation. They are given the symbol ____. ____ operations are a special case of improper rotations.

Back 39

Improper rotations are composite operations involving a rotation operation and a reflection operation. They are given the symbol S. Inversion operations are a special case of improper rotations.

Front 40

If a molecule possesses C[n] and σ[h] axes then it must also have a ____ axis.

Back 40

If a molecule possesses C[n] and σ[h] axes then it must also have a S[n] axis.

Front 41

What is the improper rotation operation of this molecule?

Back 41

S[3]

Front 42

What is the improper rotation operation of this molecule?

Back 42

S[6]

Front 43

What is the improper rotation operation of this molecule?

Back 43

S[4]

Front 44

What is the improper rotation operation of this molecule?

Back 44

S[5]

Front 45

What is the improper rotation operation of this molecule?

Back 45

S[6]

Front 46

How many S[4] operations are required to return to E?

Back 46

4

Front 47

How many S[3] operations are required to return to E?

Back 47

6

Front 48

The minimum value of n for an improper axis S[n] is ____.

Back 48

The minimum value of n for an improper axis S[n] is 3.

Front 49

Describe the symmetry of this molecule. Give the correct symbols for each of the symmetry operations/elements you describe.

Back 49

E - Identity

4C[3] - each along a C-H bond

3C[2] - each bisecting H-C-H bond angle

3S[4] - colinear with C[2] axes

8σ planes - each contains 2 H atoms and the C. Two unique planes per C[3] axis.

Front 50

Sketch the S[4] axis of this molecule. How many of these axes does the molecule possess?

Back 50

3

Front 51

Does this molecule have a centre of inversion? Does it have a S[4] axis?

Back 51

i: Yes

S[4]: No

Front 52

Does this molecule have a centre of inversion? Does it have a S[4] axis?

Back 52

i: Yes

S[4]: No

Front 53

Does this molecule have a centre of inversion? Does it have a S[4] axis?

Back 53

i: No

S[4]: No

Front 54

Does this molecule have a centre of inversion? Does it have a S[4] axis?

Back 54

i: No

S[4]: Yes

Front 55

The collection of symmetry elements and operations for a particular molecule is called a ____ ____.

Back 55

The collection of symmetry elements and operations for a particular molecule is called a point group.

Front 56

What is the point group of this molecule?

Back 56

C[2v]

Front 57

What is the point group of this molecule?

Back 57

C[3v]

Front 58

What is the point group of this molecule?

Back 58

C[4v]

Front 59

What is the point group of this molecule?

Back 59

D[4h]

Front 60

What is the point group of this molecule?

Back 60

D[2d]

Front 61

What is the point group of this molecule?

Back 61

T[d]

Front 62

What is the point group of this molecule?

Back 62

O[h]

Front 63

What is the point group of this molecule?

Back 63

C[∞v]

Front 64

What is the point group of this molecule?

Back 64

D[∞h]

Front 65

Combination of multiple symmetry operations is called ____. Results can depend on the order of operations - operations do not always ____.

Back 65

Combination of multiple symmetry operations is called multiplication. Results can depend on the order of operations - operations do not always commute.

Front 66

Complete this multiplication table.

Back 66

Front 67

Define closure.

Back 67

Combining any 2 operations results in another operation of the group.

Front 68

Define associativity.

Back 68

Operations can be grouped in any manner but the order cannot change i.e. if C(BA)=D then (CB)A=D.

Front 69

Define reciprocality.

Back 69

Every element has an inverse i.e. AA{-1}=E. Some elements are self-inverse e.g. i{2}=E, σ{2}=E.

Front 70

What is the point group of this molecule?

Back 70

C[3v]

Front 71

What is the point group of this molecule?

Back 71

C[2v]

Front 72

What is the point group of this molecule?

Back 72

T[d]

Front 73

What is the point group of this molecule?

Back 73

C[3v]

Front 74

What is the point group of this molecule?

Back 74

D[3h]

Front 75

What is the point group of this molecule?

Back 75

T[d]

Front 76

Determine the symmetry elements and assign the point group of this molecule.

Back 76

E, σ

C[s]

Front 77

Determine the symmetry elements and assign the point group of this molecule.

Back 77

E, C[3], 3C[2], S[3], σ[h],σ[v]

D[3h]

Front 78

Determine the symmetry elements and assign the point group of this molecule.

Back 78

E, 4C[3], 3C[2], 3S[4], 6σ[d]

T[d]

Front 79

Determine the symmetry elements and assign the point group of this molecule.

Back 79

E only

C[1]

Front 80

Determine the symmetry elements and assign the point group of this molecule.

Back 80

E, 4C[3], 3C[2], 3S[4], 6σ[d]

T[d]

Front 81

Molecules with limited symmetry are described as ____. Molecules with no symmetry except for the identity are described as ____.

Back 81

Molecules with limited symmetry are described as dissymmetric. Molecules with no symmetry except for the identity are described as asymmetric.

Front 82

If a molecule contains a mirror plane or an inversion centre it cannot be ____.

Back 82

If a molecule contains a mirror plane or an inversion centre it cannot be chiral.

Front 83

Which point groups can chiral molecules only be?

Back 83

C[1], C[n] and D[n]

Front 84

State Neumann's Principle.

Back 84

The physical and chemical properties of a molecule are invariant to the symmetry elements it possesses.

Front 85

Which point groups allow for a non-zero dipole moment?

Back 85

C[s], C[n], C[nv] and C[1]

Front 86

Label each part of this character table.

Back 86

A) Point group code (Schönflies symbol)

B) Symmetry elements and number of unique operations

C) Quadratic functions

D) Order

E) Irreducible representations (Mulliken symbols)

F) Characters of the irreducible representations

G) Linear functions

Front 87

Which Mulliken symbol describes a completely symmetric irreducible representation?

Back 87

A[1]

Front 88

Define the Mulliken symbol A.

Back 88

Non-degenerate representation symmetric with respect to C[n].

Front 89

Define the Mulliken symbol B.

Back 89

Non-degenerate representation of a finite order point group which is antisymmetric with respect to C[n].

Front 90

Define the Mulliken symbol E.

Back 90

Doubly degenerate representation.

Front 91

Define the Mulliken symbol T.

Back 91

Triply degenerate representation of a finite order point group.

Front 92

Define the Mulliken symbol subscript 1.

Back 92

Representation is symmetric with respect to σ[v].

Front 93

Define the Mulliken sumbol subscript 2.

Back 93

Representation is antisymmetric with respect to σ[v].

Front 94

Define the Mulliken symbol suffix ' (prime).

Back 94

Representation is symmetric with respect to σ[h].

Front 95

Define the Mulliken symbol suffix '' (double prime).

Back 95

Representation is antisymmetric with respect to σ[h].

Front 96

Define the Mulliken symbol subscript g.

Back 96

Representation is symmetric with respect to i.

Front 97

Define the Mulliken symbol subscript u.

Back 97

Representation is antisymmetric with respect to i.

Front 98

Does this molecule possess a dipole moment?

Back 98

C[3v] - Yes

Front 99

Does this molecule possess a dipole moment?

Back 99

C[2v] - Yes

Front 100

Does this molecule possess a dipole moment?

Back 100

T[d] - No

Front 101

Does this molecule possess a dipole moment?

Back 101

D[4h] - No

Front 102

Determine the crystal field splitting pattern from this character table.

Back 102

Front 103

Determine the crystal field splitting pattern from this character table.

Back 103

Front 104

State the rule of mutual exclusion.

Back 104

In a centrosymmetric molecule, vibrations which are IR active are Raman forbidden, and vice versa.

Front 105

Determine which representations are IR active and which are Raman active.

Back 105

IR active: A[2u] and E[u]

Raman active: A[1g], B[1g], B[2g] and E[g]

Front 106

Determine the order of this group. Why are the x and y vectors bracketed together? What splitting pattern would you expect from a transition metal complex with this point group? Which Mulliken symbol describes an s orbital?

Back 106

h = 12

They are degenerate

Splitting pattern: pictured left

s orbital: A[1]'

Front 107

Identify the symmetry species of all five of the d-orbitals of the central S atom in this molecule.

Back 107

z{2} = A[1]

xy = A[2]

xz = B[1]

yz = B[2]

Front 108

What is the maximum possible degree of degeneracy of the orbitals in this complex?

Back 108

3

Front 109

Determine the point group of this molecule. What is the maximum possible degree of degeneracy of the orbitals in this complex? Which 3p orbitals (on P) contribute to a molecular orbital of this degeneracy?

Back 109

Point group: D[3h]

Maximum degeneracy: 2

p orbitals: p[x] and p[y]

Front 110

The starting position of an atom is given as a ____ ____ (pictured left).

Back 110

The starting position of an atom is given as a column matrix.

Front 111

Complete this matrix calculation to show E.

Back 111

Front 112

Complete this matrix calculation to show C[2].

Back 112

Front 113

Complete this matrix calculation to show i.

Back 113

Front 114

Complete this matrix calculation to show σ[h].

Back 114

Front 115

Rotations with orders 2 or 3 tend to give rise to simple matrices with non-zero entries on the ____ ____.

Back 115

Rotations with orders 2 or 3 tend to give rise to simple matrices with non-zero entries on the lead diagonal.

Front 116

Higher order rotations can lead to ____-____ terms off the diagonal which in turn leads to ____ and ____ ____ irreducible representations. They may also have non-____ values.

Back 116

Higher order rotations can lead to non-zero terms off the diagonal which in turn leads to doubly and triply degenerate irreducible representations. They may also have non-integral values.

Front 117

Reducible representations are given the symbol ____.

Back 117

Reducible representations are given the symbol Γ.

Front 118

Write an expression for the number of times a particular irreducible representation occurs in a reducible representation.

Back 118

Where:

h = order

χ[R] = character of reducible representation

χ[I] = character of irreducible representation

g[c] = no. of unique operations in class

Front 119

Reduce this reducible representation.

Back 119

Γ = A[u] + 2B[u]

Front 120

Reduce this reducible representation.

Back 120

Γ = A[1] + T[2]

Front 121

Determine the order of this group. What is the degeneracy of the E[2g] irreducible representation? What splitting pattern would you expect for the d-orbitals of the central atom in this molecule?

Back 121

h = 20

Degeneracy: 2

Splitting pattern: pictured left

Front 122

Using the character table, reduce Γ to a sum of irreducible representations.

Back 122

Γ = A[1]' + E'

Front 123

Outline the three stages of applying group theory.

Back 123

1) Find and use a basis to generate a reducible representation of the point group.

2) Reduce the representation to irreducible representations of the point group.

3) Interpret the results with respect to the system being investigated.

Front 124

Outline a quick method for getting the character for the symmetry matrix for a reducible representation.

Back 124

Count the number of vectors (bonds) that are unmoved by the symmetry operation.

Front 125

Using the bonds as basis, generate a reducible representation for the σ-bonding in this molecule.

Back 125

Front 126

Using the bonds as basis, generate a reducible representation for the σ-bonding in this molecule.

Back 126

Front 127

What does SALC stand for?

Back 127

Symmetry Adapted Linear Combination

Front 128

Write an expression relating the molecular orbital wavefunction with the SALC wavefunction.

Back 128

Front 129

Write an expression defining the SALC wavefunction.

Back 129

Front 130

Combinations occur where SALC and central atom have the same ____.

Back 130

Combinations occur where SALC and central atom have the same symmetry.

Front 131

In the delocalised model, atomic orbitals and SALCs with the same symmetry combine to form ____ and ____ molecular orbitals.

Back 131

In the delocalised model, atomic orbitals and SALCs with the same symmetry combine to form bonding and antibonding molecular orbitals.

Front 132

In the delocalised model, atomic orbitals on the central atom with unique symmetry form ____ orbitals.

Back 132

In the delocalised model, atomic orbitals on the central atom with unique symmetry form non-bonding orbitals.

Front 133

For ammonia (C[3v]), applying the delocalised MO approach gives rise to a reducible representation for the SALCs that can be expressed as Γ = A[1] + E. Which orbitals will overlap to form σ-bonds? Which orbital will contain the lone pair? Sketch an MO diagram for ammonia.

Back 133

σ-bonding orbitals: A[1] combines with N2s, E combines with N2p[x] and N2p[y].

Lone pair: N2p[z]

MO diagram: pictured left

Front 134

There are various ways in which electrons can be arranged according to ____ and ____ ____ quantum numbers. This is known as ____-____ Coupling.

Back 134

There are various ways in which electrons can be arranged according to spin and angular momentum quantum numbers. This is known as Russell-Saunders Coupling.

Front 135

How many distinct ways can one electron be placed in to the 3d orbitals?

Back 135

10

Front 136

How many distinct ways can two electrons be placed in to the 3d orbitals?

Back 136

45

Front 137

List all the microstates the d{2} system can exist in.

Back 137

{1}S, {3}P, {1}D, 3{F}, 1{G}

Front 138

What is the ground term symbol of Cr{3+}?

Back 138

{4}F

Front 139

What is the ground term symbol of Fe{2+}?

Back 139

{5}D

Front 140

Apply Hund's 1st rule to Russell-Saunders states.

Back 140

The lowest energy term has the largest spin multiplicity.

Front 141

Apply Hund's 2nd rule to Russell-Saunders states.

Back 141

If two terms have the same multiplicity, the largest Σ(m[l]) has the lowest energy.

Front 142

What are Racah Parameters?

Back 142

Racah parameters summarise the effects of electron-electron repulsion on the energies of terms arising from a particular configuration. They are a quantitative expression of the ideas expressed by Hund's rule and account for deviations.

Front 143

In an octahedral field, the Russell-Saunders terms split in to two ____ ____ terms.

Back 143

In an octahedral field, the Russell-Saunders terms split in to two ligand field terms.

Front 144

____-____ diagrams contain all Russell-Saunders and ligand field terms. They can be used to determine Racah parameter ____ and the value of ____.

Back 144

Tanabe-Sugano diagrams contain all Russell-Saunders and ligand field terms.They can be used to determine Racah parameter B and the value of Δ[o].

Front 145

Define the nephelauxetic parameter.

Back 145

The nephelauxetic parameter is a measure of the extent of d-electron delocalisation on to the ligands; the softer the ligand, the smaller the parameter.

Front 146

What is the formula for the nephelauxetic parameter, β?

Back 146

Where:

B = Racah parameter B

Front 147

If B for this complex is 657 cm{-1} and B for the free Cr{3+} ion is 1027 cm{-1}, what is the value of the nephelauxetic parameter, β?

Back 147

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