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university-logo Symmetry Elements Symmetry Operations Symmetry Elements and Operations Christopher C. Cummins Massachusetts Institute of Technology [email protected] 5.03 Lecture 1 Symmetry Elements and Operations

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Page 1: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

university-logo

Symmetry ElementsSymmetry Operations

Symmetry Elements and Operations

Christopher C. Cummins

Massachusetts Institute of Technology

[email protected]

5.03 Lecture 1 Symmetry Elements and Operations

Page 2: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

university-logo

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

Page 3: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 4: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 5: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 6: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 7: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 8: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 9: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 10: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 11: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 12: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 13: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 14: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 15: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 16: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 17: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 18: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 19: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 20: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 21: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 22: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 23: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 24: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 25: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 26: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 27: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 28: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 29: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 30: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 31: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 32: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 33: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 34: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 35: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 36: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 37: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 38: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 39: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 40: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 41: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 42: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 43: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 44: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 45: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 46: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 47: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 48: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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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

Page 49: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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

Page 50: Symmetry Elements and Operations - MITweb.mit.edu/5.03/www/notes/01_elements_operations_p.pdfuniversity-logo Symmetry Elements Symmetry Operations mirror planes rotation axes equivalent

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