Post on 10-Feb-2020
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Symmetry ElementsSymmetry Operations
Symmetry Elements and Operations
Christopher C. Cummins
Massachusetts Institute of Technology
ccummins@mit.edu
5.03 Lecture 1 Symmetry Elements and Operations
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Symmetry ElementsSymmetry Operations
Outline
1 Symmetry ElementsPlanes of Reflection, σAxes of Rotation, Cn
The Inversion Center, iImproper Axes of Rotation, SnThe Identity, E
2 Symmetry Operations
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Outline
1 Symmetry ElementsPlanes of Reflection, σAxes of Rotation, Cn
The Inversion Center, iImproper Axes of Rotation, SnThe Identity, E
2 Symmetry Operations
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
A Molecule with Two Mirror PlanesThe H2O Molecule
The O and H atoms lie in the same plane
The plane of the molecule is a mirror plane, σ
The plane ⊥ to the molecular plane is a second σ
The molecular plane is taken as the yz plane
The mirrors are σv (xz) and σ′v (yz)
The subscript v is for “vertical”
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
A Molecule with Two Mirror PlanesThe H2O Molecule
The O and H atoms lie in the same plane
The plane of the molecule is a mirror plane, σ
The plane ⊥ to the molecular plane is a second σ
The molecular plane is taken as the yz plane
The mirrors are σv (xz) and σ′v (yz)
The subscript v is for “vertical”
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
A Molecule with Two Mirror PlanesThe H2O Molecule
The O and H atoms lie in the same plane
The plane of the molecule is a mirror plane, σ
The plane ⊥ to the molecular plane is a second σ
The molecular plane is taken as the yz plane
The mirrors are σv (xz) and σ′v (yz)
The subscript v is for “vertical”
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
A Molecule with Two Mirror PlanesThe H2O Molecule
The O and H atoms lie in the same plane
The plane of the molecule is a mirror plane, σ
The plane ⊥ to the molecular plane is a second σ
The molecular plane is taken as the yz plane
The mirrors are σv (xz) and σ′v (yz)
The subscript v is for “vertical”
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
A Molecule with Two Mirror PlanesThe H2O Molecule
The O and H atoms lie in the same plane
The plane of the molecule is a mirror plane, σ
The plane ⊥ to the molecular plane is a second σ
The molecular plane is taken as the yz plane
The mirrors are σv (xz) and σ′v (yz)
The subscript v is for “vertical”
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
A Molecule with Two Mirror PlanesThe H2O Molecule
The O and H atoms lie in the same plane
The plane of the molecule is a mirror plane, σ
The plane ⊥ to the molecular plane is a second σ
The molecular plane is taken as the yz plane
The mirrors are σv (xz) and σ′v (yz)
The subscript v is for “vertical”
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
A Molecule with Two Mirror PlanesThe H2O Molecule
The O and H atoms lie in the same plane
The plane of the molecule is a mirror plane, σ
The plane ⊥ to the molecular plane is a second σ
The molecular plane is taken as the yz plane
The mirrors are σv (xz) and σ′v (yz)
The subscript v is for “vertical”
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Some Other MoleculesDoes the molecule have a mirror plane?
Ammonia, NH3
Sulfur tetrafluoride, SF4
Dioxygen, O2
White phosphorus, P4
Diborane, B2H6
Myoglobin
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Some Other MoleculesDoes the molecule have a mirror plane?
Ammonia, NH3
Sulfur tetrafluoride, SF4
Dioxygen, O2
White phosphorus, P4
Diborane, B2H6
Myoglobin
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Some Other MoleculesDoes the molecule have a mirror plane?
Ammonia, NH3
Sulfur tetrafluoride, SF4
Dioxygen, O2
White phosphorus, P4
Diborane, B2H6
Myoglobin
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Some Other MoleculesDoes the molecule have a mirror plane?
Ammonia, NH3
Sulfur tetrafluoride, SF4
Dioxygen, O2
White phosphorus, P4
Diborane, B2H6
Myoglobin
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Some Other MoleculesDoes the molecule have a mirror plane?
Ammonia, NH3
Sulfur tetrafluoride, SF4
Dioxygen, O2
White phosphorus, P4
Diborane, B2H6
Myoglobin
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Some Other MoleculesDoes the molecule have a mirror plane?
Ammonia, NH3
Sulfur tetrafluoride, SF4
Dioxygen, O2
White phosphorus, P4
Diborane, B2H6
Myoglobin
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Some Other MoleculesDoes the molecule have a mirror plane?
Ammonia, NH3
Sulfur tetrafluoride, SF4
Dioxygen, O2
White phosphorus, P4
Diborane, B2H6
Myoglobin
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Outline
1 Symmetry ElementsPlanes of Reflection, σAxes of Rotation, Cn
The Inversion Center, iImproper Axes of Rotation, SnThe Identity, E
2 Symmetry Operations
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Axes of Rotation, CnThese have n-fold axes of rotation
What is n?
Ammonia, NH3
Cubane, (CH)8
Water, H2O
Buckminsterfullerene, C60
Tick-Borne Encephalitis Virus
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Axes of Rotation, CnThese have n-fold axes of rotation
What is n?
Ammonia, NH3
Cubane, (CH)8
Water, H2O
Buckminsterfullerene, C60
Tick-Borne Encephalitis Virus
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Axes of Rotation, CnThese have n-fold axes of rotation
What is n?
Ammonia, NH3
Cubane, (CH)8
Water, H2O
Buckminsterfullerene, C60
Tick-Borne Encephalitis Virus
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Axes of Rotation, CnThese have n-fold axes of rotation
What is n?
Ammonia, NH3
Cubane, (CH)8
Water, H2O
Buckminsterfullerene, C60
Tick-Borne Encephalitis Virus
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Axes of Rotation, CnThese have n-fold axes of rotation
What is n?
Ammonia, NH3
Cubane, (CH)8
Water, H2O
Buckminsterfullerene, C60
Tick-Borne Encephalitis Virus
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Axes of Rotation, CnSome observations based upon symmetry
Ammonia, NH3, has a single “higher-order” axis of rotationdenoted C3
Molecules with a Cn axis where n ≥ 3 have degenerateelectronic energy levels
Molecules with a Cn axis where n ≥ 3 have degeneratevibrational energy levels
Water, H2O, has a single C2 axis and thus has nodegeneracies required by symmetry
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Axes of Rotation, CnSome observations based upon symmetry
Ammonia, NH3, has a single “higher-order” axis of rotationdenoted C3
Molecules with a Cn axis where n ≥ 3 have degenerateelectronic energy levels
Molecules with a Cn axis where n ≥ 3 have degeneratevibrational energy levels
Water, H2O, has a single C2 axis and thus has nodegeneracies required by symmetry
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Axes of Rotation, CnSome observations based upon symmetry
Ammonia, NH3, has a single “higher-order” axis of rotationdenoted C3
Molecules with a Cn axis where n ≥ 3 have degenerateelectronic energy levels
Molecules with a Cn axis where n ≥ 3 have degeneratevibrational energy levels
Water, H2O, has a single C2 axis and thus has nodegeneracies required by symmetry
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Axes of Rotation, CnSome observations based upon symmetry
Ammonia, NH3, has a single “higher-order” axis of rotationdenoted C3
Molecules with a Cn axis where n ≥ 3 have degenerateelectronic energy levels
Molecules with a Cn axis where n ≥ 3 have degeneratevibrational energy levels
Water, H2O, has a single C2 axis and thus has nodegeneracies required by symmetry
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Axes of Rotation, CnSome observations based upon symmetry
Ammonia, NH3, has a single “higher-order” axis of rotationdenoted C3
Molecules with a Cn axis where n ≥ 3 have degenerateelectronic energy levels
Molecules with a Cn axis where n ≥ 3 have degeneratevibrational energy levels
Water, H2O, has a single C2 axis and thus has nodegeneracies required by symmetry
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Outline
1 Symmetry ElementsPlanes of Reflection, σAxes of Rotation, Cn
The Inversion Center, iImproper Axes of Rotation, SnThe Identity, E
2 Symmetry Operations
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Centers of InversionThese have identical atoms with inverted coordinates
Is there an inversion center, i?
Ethylene, C2H4
Methane, CH4
Water, H2O
Diborane, B2H6
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Centers of InversionThese have identical atoms with inverted coordinates
Is there an inversion center, i?
Ethylene, C2H4
Methane, CH4
Water, H2O
Diborane, B2H6
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Centers of InversionThese have identical atoms with inverted coordinates
Is there an inversion center, i?
Ethylene, C2H4
Methane, CH4
Water, H2O
Diborane, B2H6
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Centers of InversionThese have identical atoms with inverted coordinates
Is there an inversion center, i?
Ethylene, C2H4
Methane, CH4
Water, H2O
Diborane, B2H6
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with Centers of InversionThese have identical atoms with inverted coordinates
Is there an inversion center, i?
Ethylene, C2H4
Methane, CH4
Water, H2O
Diborane, B2H6
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Outline
1 Symmetry ElementsPlanes of Reflection, σAxes of Rotation, Cn
The Inversion Center, iImproper Axes of Rotation, SnThe Identity, E
2 Symmetry Operations
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with an Improper Axis of RotationA Combination of Rotation Axis and Mirror Plane
Consider Pt(SH)42−
Pt-S-H bonds are bent
H atoms are located alternately above and below the PtS4
plane
There is not a C4 axis
There is not a mirror plane containing the four S atoms
There is an S4 axis passing through Pt and ⊥ to sulfur plane
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with an Improper Axis of RotationA Combination of Rotation Axis and Mirror Plane
Consider Pt(SH)42−
Pt-S-H bonds are bent
H atoms are located alternately above and below the PtS4
plane
There is not a C4 axis
There is not a mirror plane containing the four S atoms
There is an S4 axis passing through Pt and ⊥ to sulfur plane
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with an Improper Axis of RotationA Combination of Rotation Axis and Mirror Plane
Consider Pt(SH)42−
Pt-S-H bonds are bent
H atoms are located alternately above and below the PtS4
plane
There is not a C4 axis
There is not a mirror plane containing the four S atoms
There is an S4 axis passing through Pt and ⊥ to sulfur plane
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with an Improper Axis of RotationA Combination of Rotation Axis and Mirror Plane
Consider Pt(SH)42−
Pt-S-H bonds are bent
H atoms are located alternately above and below the PtS4
plane
There is not a C4 axis
There is not a mirror plane containing the four S atoms
There is an S4 axis passing through Pt and ⊥ to sulfur plane
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Molecules with an Improper Axis of RotationA Combination of Rotation Axis and Mirror Plane
Consider Pt(SH)42−
Pt-S-H bonds are bent
H atoms are located alternately above and below the PtS4
plane
There is not a C4 axis
There is not a mirror plane containing the four S atoms
There is an S4 axis passing through Pt and ⊥ to sulfur plane
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
Outline
1 Symmetry ElementsPlanes of Reflection, σAxes of Rotation, Cn
The Inversion Center, iImproper Axes of Rotation, SnThe Identity, E
2 Symmetry Operations
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
The Identity Symmetry ElementThis is present by default
Rotation by 360◦ about an arbitrary axis returns an equivalentconfiguration
This axis is referred to as the identity symmetry element, E
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
The Identity Symmetry ElementThis is present by default
Rotation by 360◦ about an arbitrary axis returns an equivalentconfiguration
This axis is referred to as the identity symmetry element, E
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
mirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry element
The Identity Symmetry ElementThis is present by default
Rotation by 360◦ about an arbitrary axis returns an equivalentconfiguration
This axis is referred to as the identity symmetry element, E
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
Symmetry OperationsThese are carried out with respect to symmetry elements
A mirror plane, σ, generates a single reflection operation
Two consecutive reflections with respect to a given σ isequivalent to the identity operation
A C2 axis generates a single two-fold rotation operation
A C3 axis generates two operations: rotation by 2π3 and
rotation by 4π3
The latter operations are called C3 and C 23 , respectively
note that C 33 = E
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
Symmetry OperationsThese are carried out with respect to symmetry elements
A mirror plane, σ, generates a single reflection operation
Two consecutive reflections with respect to a given σ isequivalent to the identity operation
A C2 axis generates a single two-fold rotation operation
A C3 axis generates two operations: rotation by 2π3 and
rotation by 4π3
The latter operations are called C3 and C 23 , respectively
note that C 33 = E
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
Symmetry OperationsThese are carried out with respect to symmetry elements
A mirror plane, σ, generates a single reflection operation
Two consecutive reflections with respect to a given σ isequivalent to the identity operation
A C2 axis generates a single two-fold rotation operation
A C3 axis generates two operations: rotation by 2π3 and
rotation by 4π3
The latter operations are called C3 and C 23 , respectively
note that C 33 = E
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
Symmetry OperationsThese are carried out with respect to symmetry elements
A mirror plane, σ, generates a single reflection operation
Two consecutive reflections with respect to a given σ isequivalent to the identity operation
A C2 axis generates a single two-fold rotation operation
A C3 axis generates two operations: rotation by 2π3 and
rotation by 4π3
The latter operations are called C3 and C 23 , respectively
note that C 33 = E
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
Symmetry OperationsThese are carried out with respect to symmetry elements
A mirror plane, σ, generates a single reflection operation
Two consecutive reflections with respect to a given σ isequivalent to the identity operation
A C2 axis generates a single two-fold rotation operation
A C3 axis generates two operations: rotation by 2π3 and
rotation by 4π3
The latter operations are called C3 and C 23 , respectively
note that C 33 = E
5.03 Lecture 1 Symmetry Elements and Operations
university-logo
Symmetry ElementsSymmetry Operations
Symmetry OperationsThese are carried out with respect to symmetry elements
A mirror plane, σ, generates a single reflection operation
Two consecutive reflections with respect to a given σ isequivalent to the identity operation
A C2 axis generates a single two-fold rotation operation
A C3 axis generates two operations: rotation by 2π3 and
rotation by 4π3
The latter operations are called C3 and C 23 , respectively
note that C 33 = E
5.03 Lecture 1 Symmetry Elements and Operations