Transcription
Symmetry ElementsSymmetry OperationsSymmetry Elements and OperationsChristopher C. CumminsMassachusetts Institute of Technologyccummins@mit.eduuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry OperationsOutline1Symmetry ElementsPlanes of Reflection, σAxes of Rotation, CnThe Inversion Center, iImproper Axes of Rotation, SnThe Identity, E2Symmetry Operationsuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementOutline1Symmetry ElementsPlanes of Reflection, σAxes of Rotation, CnThe Inversion Center, iImproper Axes of Rotation, SnThe Identity, E2Symmetry Operationsuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementA Molecule with Two Mirror PlanesThe H2 O MoleculeThe O and H atoms lie in the same planeThe plane of the molecule is a mirror plane, σThe plane to the molecular plane is a second σThe molecular plane is taken as the yz planeThe mirrors are σv (xz) and σv0 (yz)The subscript v is for “vertical”university-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementA Molecule with Two Mirror PlanesThe H2 O MoleculeThe O and H atoms lie in the same planeThe plane of the molecule is a mirror plane, σThe plane to the molecular plane is a second σThe molecular plane is taken as the yz planeThe mirrors are σv (xz) and σv0 (yz)The subscript v is for “vertical”university-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementA Molecule with Two Mirror PlanesThe H2 O MoleculeThe O and H atoms lie in the same planeThe plane of the molecule is a mirror plane, σThe plane to the molecular plane is a second σThe molecular plane is taken as the yz planeThe mirrors are σv (xz) and σv0 (yz)The subscript v is for “vertical”university-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementA Molecule with Two Mirror PlanesThe H2 O MoleculeThe O and H atoms lie in the same planeThe plane of the molecule is a mirror plane, σThe plane to the molecular plane is a second σThe molecular plane is taken as the yz planeThe mirrors are σv (xz) and σv0 (yz)The subscript v is for “vertical”university-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementA Molecule with Two Mirror PlanesThe H2 O MoleculeThe O and H atoms lie in the same planeThe plane of the molecule is a mirror plane, σThe plane to the molecular plane is a second σThe molecular plane is taken as the yz planeThe mirrors are σv (xz) and σv0 (yz)The subscript v is for “vertical”university-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementA Molecule with Two Mirror PlanesThe H2 O MoleculeThe O and H atoms lie in the same planeThe plane of the molecule is a mirror plane, σThe plane to the molecular plane is a second σThe molecular plane is taken as the yz planeThe mirrors are σv (xz) and σv0 (yz)The subscript v is for “vertical”university-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementA Molecule with Two Mirror PlanesThe H2 O MoleculeThe O and H atoms lie in the same planeThe plane of the molecule is a mirror plane, σThe plane to the molecular plane is a second σThe molecular plane is taken as the yz planeThe mirrors are σv (xz) and σv0 (yz)The subscript v is for “vertical”university-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementSome Other MoleculesDoes the molecule have a mirror plane?Ammonia, NH3Sulfur tetrafluoride, SF4Dioxygen, O2White phosphorus, P4Diborane, B2 H6Myoglobinuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementSome Other MoleculesDoes the molecule have a mirror plane?Ammonia, NH3Sulfur tetrafluoride, SF4Dioxygen, O2White phosphorus, P4Diborane, B2 H6Myoglobinuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementSome Other MoleculesDoes the molecule have a mirror plane?Ammonia, NH3Sulfur tetrafluoride, SF4Dioxygen, O2White phosphorus, P4Diborane, B2 H6Myoglobinuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementSome Other MoleculesDoes the molecule have a mirror plane?Ammonia, NH3Sulfur tetrafluoride, SF4Dioxygen, O2White phosphorus, P4Diborane, B2 H6Myoglobinuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementSome Other MoleculesDoes the molecule have a mirror plane?Ammonia, NH3Sulfur tetrafluoride, SF4Dioxygen, O2White phosphorus, P4Diborane, B2 H6Myoglobinuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementSome Other MoleculesDoes the molecule have a mirror plane?Ammonia, NH3Sulfur tetrafluoride, SF4Dioxygen, O2White phosphorus, P4Diborane, B2 H6Myoglobinuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementSome Other MoleculesDoes the molecule have a mirror plane?Ammonia, NH3Sulfur tetrafluoride, SF4Dioxygen, O2White phosphorus, P4Diborane, B2 H6Myoglobinuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementOutline1Symmetry ElementsPlanes of Reflection, σAxes of Rotation, CnThe Inversion Center, iImproper Axes of Rotation, SnThe Identity, E2Symmetry Operationsuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Axes of Rotation, CnThese have n-fold axes of rotationWhat is n?Ammonia, NH3Cubane, (CH)8Water, H2 OBuckminsterfullerene, C60Tick-Borne Encephalitis Virusuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Axes of Rotation, CnThese have n-fold axes of rotationWhat is n?Ammonia, NH3Cubane, (CH)8Water, H2 OBuckminsterfullerene, C60Tick-Borne Encephalitis Virusuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Axes of Rotation, CnThese have n-fold axes of rotationWhat is n?Ammonia, NH3Cubane, (CH)8Water, H2 OBuckminsterfullerene, C60Tick-Borne Encephalitis Virusuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Axes of Rotation, CnThese have n-fold axes of rotationWhat is n?Ammonia, NH3Cubane, (CH)8Water, H2 OBuckminsterfullerene, C60Tick-Borne Encephalitis Virusuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Axes of Rotation, CnThese have n-fold axes of rotationWhat is n?Ammonia, NH3Cubane, (CH)8Water, H2 OBuckminsterfullerene, C60Tick-Borne Encephalitis Virusuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Axes of Rotation, CnSome observations based upon symmetryAmmonia, NH3 , has a single “higher-order” axis of rotationdenoted C3Molecules with a Cn axis where n 3 have degenerateelectronic energy levelsMolecules with a Cn axis where n 3 have degeneratevibrational energy levelsWater, H2 O, has a single C2 axis and thus has nodegeneracies required by symmetryuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Axes of Rotation, CnSome observations based upon symmetryAmmonia, NH3 , has a single “higher-order” axis of rotationdenoted C3Molecules with a Cn axis where n 3 have degenerateelectronic energy levelsMolecules with a Cn axis where n 3 have degeneratevibrational energy levelsWater, H2 O, has a single C2 axis and thus has nodegeneracies required by symmetryuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Axes of Rotation, CnSome observations based upon symmetryAmmonia, NH3 , has a single “higher-order” axis of rotationdenoted C3Molecules with a Cn axis where n 3 have degenerateelectronic energy levelsMolecules with a Cn axis where n 3 have degeneratevibrational energy levelsWater, H2 O, has a single C2 axis and thus has nodegeneracies required by symmetryuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Axes of Rotation, CnSome observations based upon symmetryAmmonia, NH3 , has a single “higher-order” axis of rotationdenoted C3Molecules with a Cn axis where n 3 have degenerateelectronic energy levelsMolecules with a Cn axis where n 3 have degeneratevibrational energy levelsWater, H2 O, has a single C2 axis and thus has nodegeneracies required by symmetryuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Axes of Rotation, CnSome observations based upon symmetryAmmonia, NH3 , has a single “higher-order” axis of rotationdenoted C3Molecules with a Cn axis where n 3 have degenerateelectronic energy levelsMolecules with a Cn axis where n 3 have degeneratevibrational energy levelsWater, H2 O, has a single C2 axis and thus has nodegeneracies required by symmetryuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementOutline1Symmetry ElementsPlanes of Reflection, σAxes of Rotation, CnThe Inversion Center, iImproper Axes of Rotation, SnThe Identity, E2Symmetry Operationsuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Centers of InversionThese have identical atoms with inverted coordinatesIs there an inversion center, i?Ethylene, C2 H4Methane, CH4Water, H2 ODiborane, B2 H6university-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Centers of InversionThese have identical atoms with inverted coordinatesIs there an inversion center, i?Ethylene, C2 H4Methane, CH4Water, H2 ODiborane, B2 H6university-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Centers of InversionThese have identical atoms with inverted coordinatesIs there an inversion center, i?Ethylene, C2 H4Methane, CH4Water, H2 ODiborane, B2 H6university-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Centers of InversionThese have identical atoms with inverted coordinatesIs there an inversion center, i?Ethylene, C2 H4Methane, CH4Water, H2 ODiborane, B2 H6university-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with Centers of InversionThese have identical atoms with inverted coordinatesIs there an inversion center, i?Ethylene, C2 H4Methane, CH4Water, H2 ODiborane, B2 H6university-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementOutline1Symmetry ElementsPlanes of Reflection, σAxes of Rotation, CnThe Inversion Center, iImproper Axes of Rotation, SnThe Identity, E2Symmetry Operationsuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with an Improper Axis of RotationA Combination of Rotation Axis and Mirror PlaneConsider Pt(SH)4 2 Pt-S-H bonds are bentH atoms are located alternately above and below the PtS4planeThere is not a C4 axisThere is not a mirror plane containing the four S atomsThere is an S4 axis passing through Pt and to sulfur planeuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with an Improper Axis of RotationA Combination of Rotation Axis and Mirror PlaneConsider Pt(SH)4 2 Pt-S-H bonds are bentH atoms are located alternately above and below the PtS4planeThere is not a C4 axisThere is not a mirror plane containing the four S atomsThere is an S4 axis passing through Pt and to sulfur planeuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with an Improper Axis of RotationA Combination of Rotation Axis and Mirror PlaneConsider Pt(SH)4 2 Pt-S-H bonds are bentH atoms are located alternately above and below the PtS4planeThere is not a C4 axisThere is not a mirror plane containing the four S atomsThere is an S4 axis passing through Pt and to sulfur planeuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with an Improper Axis of RotationA Combination of Rotation Axis and Mirror PlaneConsider Pt(SH)4 2 Pt-S-H bonds are bentH atoms are located alternately above and below the PtS4planeThere is not a C4 axisThere is not a mirror plane containing the four S atomsThere is an S4 axis passing through Pt and to sulfur planeuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementMolecules with an Improper Axis of RotationA Combination of Rotation Axis and Mirror PlaneConsider Pt(SH)4 2 Pt-S-H bonds are bentH atoms are located alternately above and below the PtS4planeThere is not a C4 axisThere is not a mirror plane containing the four S atomsThere is an S4 axis passing through Pt and to sulfur planeuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementOutline1Symmetry ElementsPlanes of Reflection, σAxes of Rotation, CnThe Inversion Center, iImproper Axes of Rotation, SnThe Identity, E2Symmetry Operationsuniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementThe Identity Symmetry ElementThis is present by defaultRotation by 360 about an arbitrary axis returns an equivalentconfigurationThis axis is referred to as the identity symmetry element, Euniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementThe Identity Symmetry ElementThis is present by defaultRotation by 360 about an arbitrary axis returns an equivalentconfigurationThis axis is referred to as the identity symmetry element, Euniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry Operationsmirror planesrotation axesequivalent atoms at inverted coordinatesrotation plus reflectionall molecules have this symmetry elementThe Identity Symmetry ElementThis is present by defaultRotation by 360 about an arbitrary axis returns an equivalentconfigurationThis axis is referred to as the identity symmetry element, Euniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry OperationsSymmetry OperationsThese are carried out with respect to symmetry elementsA mirror plane, σ, generates a single reflection operationTwo consecutive reflections with respect to a given σ isequivalent to the identity operationA C2 axis generates a single two-fold rotation operationA C3 axis generates two operations: rotation byrotation by 4π32π3andThe latter operations are called C3 and C32 , respectivelynote that C33 Euniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry OperationsSymmetry OperationsThese are carried out with respect to symmetry elementsA mirror plane, σ, generates a single reflection operationTwo consecutive reflections with respect to a given σ isequivalent to the identity operationA C2 axis generates a single two-fold rotation operationA C3 axis generates two operations: rotation byrotation by 4π32π3andThe latter operations are called C3 and C32 , respectivelynote that C33 Euniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry OperationsSymmetry OperationsThese are carried out with respect to symmetry elementsA mirror plane, σ, generates a single reflection operationTwo consecutive reflections with respect to a given σ isequivalent to the identity operationA C2 axis generates a single two-fold rotation operationA C3 axis generates two operations: rotation byrotation by 4π32π3andThe latter operations are called C3 and C32 , respectivelynote that C33 Euniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry OperationsSymmetry OperationsThese are carried out with respect to symmetry elementsA mirror plane, σ, generates a single reflection operationTwo consecutive reflections with respect to a given σ isequivalent to the identity operationA C2 axis generates a single two-fold rotation operationA C3 axis generates two operations: rotation byrotation by 4π32π3andThe latter operations are called C3 and C32 , respectivelynote that C33 Euniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry OperationsSymmetry OperationsThese are carried out with respect to symmetry elementsA mirror plane, σ, generates a single reflection operationTwo consecutive reflections with respect to a given σ isequivalent to the identity operationA C2 axis generates a single two-fold rotation operationA C3 axis generates two operations: rotation byrotation by 4π32π3andThe latter operations are called C3 and C32 , respectivelynote that C33 Euniversity-logo5.03 Lecture 1Symmetry Elements and Operations
Symmetry ElementsSymmetry OperationsSymmetry OperationsThese are carried out with respect to symmetry elementsA mirror plane, σ, generates a single reflection operationTwo consecutive reflections with respect to a given σ isequivalent to the identity operationA C2 axis generates a single two-fold rotation operationA C3 axis generates two operations: rotation byrotation by 4π32π3andThe latter operations are called C3 and C32 , respectivelynote that C33 Euniversity-logo5.03 Lecture 1Symmetry Elements and Operations
mirror planes rotation axes equivalent atoms at inverted coordinates rotation plus re ection all molecules have this symmetry element A Molecule with Two Mirror Planes The H 2O Molecule The O and H atoms lie in the same plane The plane of the molecule is a mirror plane, The plane ?to the molecular plane is a second
