Koofers

Chapter 4 Symmetry of Molecules - Flashcards

Flashcard Deck Information

Class:CHM 3610 - Intermediate Inorganic Chem
Subject:Chemistry
University:University of South Florida
Term:Fall 2010
- of -
INCORRECT CORRECT
- INCORRECT     - CORRECT     - SKIPPED
Shuffle Remaining Cards Show Definitions First Take Quiz (NEW)
Hide Keyboard shortcuts
Next card
Previous card
Mark correct
Mark incorrect
Flip card
Start Over
Shuffle
      Mode:   CARDS LIST       ? pages   PRINT EXIT
Symmetry Uses Symmetry and its concepts are also relevant at the molecular level: we can use symmetry to predict IR spectra, dipole moments, the orbitals used in bonding, predict optical activity, and interpret electronic spectra.
Symmetry elements Are used to describe the symmetry of a molecule or ion even if no symmetry is present. They include mirror planes, axes of rotation and inversion centers.
Symmetry Operation The act of conducting the symmetry element is called a symmetry operation and the molecule/ion must have exactly the same apperance after the operation as it had before the operation.
Identity Abbreviated to E, from the German 'Einheit' meaning Unity. This symmetry element simply consits of no change. Every molecule has this element. While this element seems physically trivial, its consideration is necessary for group theory to work properly. It is so called because it is analogous to multiplying by one (unity).
Generated by Koofers.com
Symmetry axis or rotation axis Is an aixs around which a 360 degree/n rotation results in a molecule, indistinguishable from the original. This is also called an n-fold rotational axis and abbreviated Cn.
Principal axis By convention is assigned the z-axis and is the one with the highest n. A moelcule can have more than one symmetry axis exists in molecules with more than one rotation axis.
Plane of symmetry Is a plane of reflection through which an identical copy of the original molecule is given. This is also called a mirror plane and abbreviated sigma.
Mirror Planes A symmetry plane parallel with the principal axis is dubbed vertical (sigma v) and one perpendicular to it horizontal (sigma h). A third type of symmetry plane exists: if a vertical symmetry pane additionally bisects the angle between the two 2-fold rotation axes perpendicular to the principal axis, the plane is dubbed dihedral (sigma d).
Generated by Koofers.com
Center of symmetry or inversion center Abbreviated i, A molecule has a center of symmetry, when, for any atom in the molecule, an identical atom exists diametrically opposite this center an equal distance from it. There may or may not be an atom at the center.
Rotation-reflection axis An axis around which a rotation by 360 degrees/n followed by a relfection in a plane perpendicular to it, leaves the molecule unchanged. Also called an n-fold improper rotation axis, it is abbreviated Sn, with n necessarily even.
Point groups A set of symmetry operations that describes the molecule's overall symmetry. This mathematical group has one point that remains fixed under all operations of the group.
Group theory The mathematical treatment of the properties of the groups and can be used to determine molecular orbitals, vibrations (i.e. spectroscopic properties) and other properties.
Generated by Koofers.com
Matrix representations A set of matrices that each correspond to symmetry operations in the group. Point groups can be defined by using matrix representation.
Reducible representation A shorthand version of matrix representation that lists characters obtained by summing diagonals of square matrix.
Irreducible representation A long hand version of matrix representation and defines how x, y, z coordinates are independently affected by symmetry operations.
Character Table The complete set of irreducible representations for a point group. For C2v there are 4.
Generated by Koofers.com
Labes to irreducible representation A = symmetry to principal rotation B = antisymmetric E = 2 dimensions T = 3 dimensions Subscripts: 1 = symmetric to C2's or sigma v (If no C2) 2 = antisymmetric g = symmetric to inversion u = antisymmetric
Expressions on the character tables Expressions to the right indicate the symmetry of the mathematical functions of the coordinates x, y, and z and of rotation around the axes Rx, Ry, and Rz. Are used to match orbitals (e.g. x with +ve and -ve directions matches Px, xy with alternating signs in quadrants matched with dxy.) Rotational functions describe rotations, other motions of a molecule.
Chirality Molecules that are disymmetric (asymmetric is a subset). A molecule is chiral if it has no symmetry operations other than E or only a proper rotation axis.
Vibrational modes A molecular vibration is infrared active if a vibration changes the dipole moment (x, y, or z symmetry). Generally, 3N-6 vibrational modes for a molecule (N = # of atoms) Water has 3 atoms, 9 degrees of freedom: 3 translations, 3 rotations, 3 vibrations.
Generated by Koofers.com
Rotation Operation (Cn) Also called proper rotation, is rotation through 360 degrees/n about a rotation axis.
Reflection operation The molecule contains a mirror plane in the reflection operation (sigma). Many molecules have mirror planes. The reflection operation exchanges left and right, as if each point had moved perpendicularly through the plane to a position exactly as far from the plane as when it started. When the plane is perpendicular to the principal axis fo rotation, it is called sigma h (horiontal). Other planes, which contain the principal axis of rotation, are labeled sigma v or sigma d.
Inversion Inversion is a more complex operation. Each point moves thrugh the center of the molecule to a position oppsoite the original position and as far from the central point as when it started.
Rotation-reflection Operation (Sn) Sometimes called improper rotation, requires a rotation of 360 degrees/n, followed by reflection through a plane perpendicular to the axis of rotation. Two Sn operations in succession generate a Cn/2 operation. Molecules sometimes have an Sn axis that is coincident with a Cn axis.
Generated by Koofers.com
Rotation angles vs Symmetry Operation Rotation angle Symmetry Operation 90 degrees S4 180 degrees C2 (= S42) 270 degrees S43 360 degrees E (= S44)
Rules for assigning a molecule to a point group 1. Determine whether the molecule belongs to one of the cases of very low symmetry (C1, Cs, or Ci) or high symmetry (Td, Oh, Cv, Dh, or Ih). 2. For all remaining molecules, find the rotation axis with the highest n, the highest order Cn axis for the molecule. 3. Does the molecule have any C2 axes perpendicular to the Cn axis? If it does, there will be n of such C2 axes, and the molecule is in the D set of groups If not, it is in the C or S set. 4. Does the molecule have a mirror plane (sigma h) perpendicular to the Cn axis? If so, it is classified as Cnh or Dnh. If not, continue with step 5. 5. Does the molecule have any mirror planes that contain the Cn axis (sigma v or sigma d)? If so, it is classified as Cnv or Dnd. If not, but it is in the D set, it is classified as Dn. If the molecule is in the C or S set, continue with step 6. 6. Is there an S2n, axis collinear with the Cn axis? If so, it is classified as S2n. If not, the molecule is classified as Cn.
Groups of Low and High Symmetry 1. Determine whether the molecule belongs to one of the special cases of low or high symmetry. Group Symmetry C1 No symmetry other than the identity operation. Cs Only one mirror plane. Ci Only an inversion center; few molecular examples Molecules with many symmetry operations may fit one of the high-symmetry cases of linear, tetrahedral, octahedral, or iscosahedral symmetry. Molecules with very high symmetry are of two types, linear and polyhedral. Linear molecules having a center of inversion have Dh symmetry; those lacking an inversion center have Cv symmetry.
Cv These molecules are linear, with an infinite number of rotations and an infinite number of reflection planes containing the rotation axis. They do not have a center of inversion.
Generated by Koofers.com
Dh These molecules are linear, with an infinite number of rotations and an infinite number of reflection planes containing the rotation axis. They also have perpendicular C2 axes, a perpendicular reflection plane, and an inversion center.
Td Most (but not all) molecules in this point group have the familiar tetrahedral geometry. They have four C3 axes, three C2 axes, three S4 axes, and six or sigma d planes. They have no C4 axes.
Oh These molecules include those of octahedral structure, although some other geometrical forms, such as the cube share the same set of symmetry operations. Among their 48 symmetry operations are four C3 rotations, three C4 rotations, and an inversion.
Ih Icosahedral structures are best recognized by their six C5 axes, as well as many other symmetry operations - 120 in all.
Generated by Koofers.com
Other groups 2. Find the rotation axis with the highest n, the highest order Cn axis for the molecule. This is the principle axis of the molecule. 3. Does the molecule have any C2 axes perpendicular to the Cn axis? Yes then D Groups - Molecules with C2 axes perpendicular to the principal axis are in one of the groups designated by the letter D; there are n C2 axes. No then C or S Groups - Molecules with no perpendicular C2 axes are in one of the groups designated by the letters C or S. No final assignments of point groups have been made, but the molecules have now been divided into two major cateogries, the D set, and the C or S set.
Other groups (continued) 4. Does the molecule have a mirror plane (sigma h horizontal plane) perpendicular to the Cn axis? D groups - Yes = Dnh No = Dn or Dnd C and S groups - Yes = Cnh No = Cn, Cnv or S2n 5. Does the molecule have any mirror planes that contain the Cn axis? D groups - Yes = Dnd No = Dn C and S Groups - Yes = Cnv No = Cn or S2n 6. Is there an S2n axis collinear with the Cn axis? D Groups - An molescules in this category that have S2n axis have already been assigned to groups. There are no additional groups to be considered here C and S groups - Yes = S2n No = Cn
Cnh Point group = C 2h (example = difluorodiazine) or C 3h (example: b(OH)3, planar)
Cnv Point group = C2v (example: Water, H2O) Point group = C3v (example: PCl3) Point Group = C4v (example: BrF5, square pyramid) Point Group: Cinfinityv (example: HF, CO, HCN)
Generated by Koofers.com
Cn Point group: C2 (Example: N2H4, which hasa gauche conformation) C3 (Example: P(C6H5)3, which is like a three-bladed proeller distorted out of the planar shape by a lone pair on the P)
Dnh Point Group = D3h (example: BF3) D4h (Example: PtCl4 2-) D5h (Example: Os(C5H5)2 (eclipsed)) D6h (Example: benzene) Dinfinityh (example: F2, N2, acetylene C2H2)
Dnd Point Group: D2d (H2C=C=CH2, allene) D4d (Ni(cyclobutadiene)2 (staggered) D5d (Fe(C5H5)2 (Staggered)
Dn D3 [Ru(NH2CH2CH2NH2)3 2+] (treating the NH2CHCH2NH2 group as a planar ring)
Generated by Koofers.com
Determining point groups
Generated by Koofers.com

List View: Terms & Definitions

  Hide All 41 Print
 
Front
Back
 Symmetry UsesSymmetry and its concepts are also relevant at the molecular level: we can use symmetry to predict IR spectra, dipole moments, the orbitals used in bonding, predict optical activity, and interpret electronic spectra.
 Symmetry elementsAre used to describe the symmetry of a molecule or ion even if no symmetry is present. They include mirror planes, axes of rotation and inversion centers.
 Symmetry OperationThe act of conducting the symmetry element is called a symmetry operation and the molecule/ion must have exactly the same apperance after the operation as it had before the operation.
 Identity Abbreviated to E, from the German 'Einheit' meaning Unity. This symmetry element simply consits of no change. Every molecule has this element. While this element seems physically trivial, its consideration is necessary for group theory to work properly. It is so called because it is analogous to multiplying by one (unity).
 Symmetry axis or rotation axisIs an aixs around which a 360 degree/n rotation results in a molecule, indistinguishable from the original. This is also called an n-fold rotational axis and abbreviated Cn.
 Principal axisBy convention is assigned the z-axis and is the one with the highest n. A moelcule can have more than one symmetry axis exists in molecules with more than one rotation axis.
 Plane of symmetryIs a plane of reflection through which an identical copy of the original molecule is given. This is also called a mirror plane and abbreviated sigma.
 Mirror PlanesA symmetry plane parallel with the principal axis is dubbed vertical (sigma v) and one perpendicular to it horizontal (sigma h). A third type of symmetry plane exists: if a vertical symmetry pane additionally bisects the angle between the two 2-fold rotation axes perpendicular to the principal axis, the plane is dubbed dihedral (sigma d).
 Center of symmetry or inversion centerAbbreviated i, A molecule has a center of symmetry, when, for any atom in the molecule, an identical atom exists diametrically opposite this center an equal distance from it. There may or may not be an atom at the center.
 Rotation-reflection axisAn axis around which a rotation by 360 degrees/n followed by a relfection in a plane perpendicular to it, leaves the molecule unchanged. Also called an n-fold improper rotation axis, it is abbreviated Sn, with n necessarily even.
 Point groupsA set of symmetry operations that describes the molecule's overall symmetry. This mathematical group has one point that remains fixed under all operations of the group.
 Group theoryThe mathematical treatment of the properties of the groups and can be used to determine molecular orbitals, vibrations (i.e. spectroscopic properties) and other properties.
 Matrix representationsA set of matrices that each correspond to symmetry operations in the group. Point groups can be defined by using matrix representation.
 Reducible representationA shorthand version of matrix representation that lists characters obtained by summing diagonals of square matrix.
 Irreducible representationA long hand version of matrix representation and defines how x, y, z coordinates are independently affected by symmetry operations.
 Character TableThe complete set of irreducible representations for a point group. For C2v there are 4.
 Labes to irreducible representationA = symmetry to principal rotation
B = antisymmetric
E = 2 dimensions
T = 3 dimensions
Subscripts: 1 = symmetric to C2's or sigma v
(If no C2) 2 = antisymmetric
g = symmetric to inversion
u = antisymmetric
 Expressions on the character tablesExpressions to the right indicate the symmetry of the mathematical functions of the coordinates x, y, and z and of rotation around the axes Rx, Ry, and Rz. Are used to match orbitals (e.g. x with +ve and -ve directions matches Px, xy with alternating signs in quadrants matched with dxy.)
Rotational functions describe rotations, other motions of a molecule.
 ChiralityMolecules that are disymmetric (asymmetric is a subset). A molecule is chiral if it has no symmetry operations other than E or only a proper rotation axis.
 Vibrational modesA molecular vibration is infrared active if a vibration changes the dipole moment (x, y, or z symmetry).
Generally, 3N-6 vibrational modes for a molecule (N = # of atoms)
Water has 3 atoms, 9 degrees of freedom: 3 translations, 3 rotations, 3 vibrations.
 Rotation Operation (Cn)Also called proper rotation, is rotation through 360 degrees/n about a rotation axis.
 Reflection operationThe molecule contains a mirror plane in the reflection operation (sigma). Many molecules have mirror planes. The reflection operation exchanges left and right, as if each point had moved perpendicularly through the plane to a position exactly as far from the plane as when it started. When the plane is perpendicular to the principal axis fo rotation, it is called sigma h (horiontal). Other planes, which contain the principal axis of rotation, are labeled sigma v or sigma d.
 InversionInversion is a more complex operation. Each point moves thrugh the center of the molecule to a position oppsoite the original position and as far from the central point as when it started.
 Rotation-reflection Operation (Sn)Sometimes called improper rotation, requires a rotation of 360 degrees/n, followed by reflection through a plane perpendicular to the axis of rotation. Two Sn operations in succession generate a Cn/2 operation. Molecules sometimes have an Sn axis that is coincident with a Cn axis.
 Rotation angles vs Symmetry OperationRotation angle Symmetry Operation
90 degrees S4
180 degrees C2 (= S42)
270 degrees S43
360 degrees E (= S44)
 Rules for assigning a molecule to a point group1. Determine whether the molecule belongs to one of the cases of very low symmetry (C1, Cs, or Ci) or high symmetry (Td, Oh, Cv, Dh, or Ih).
2. For all remaining molecules, find the rotation axis with the highest n, the highest order Cn axis for the molecule.
3. Does the molecule have any C2 axes perpendicular to the Cn axis? If it does, there will be n of such C2 axes, and the molecule is in the D set of groups If not, it is in the C or S set.
4. Does the molecule have a mirror plane (sigma h) perpendicular to the Cn axis? If so, it is classified as Cnh or Dnh. If not, continue with step 5.
5. Does the molecule have any mirror planes that contain the Cn axis (sigma v or sigma d)? If so, it is classified as Cnv or Dnd. If not, but it is in the D set, it is classified as Dn. If the molecule is in the C or S set, continue with step 6.
6. Is there an S2n, axis collinear with the Cn axis? If so, it is classified as S2n. If not, the molecule is classified as Cn.
 Groups of Low and High Symmetry1. Determine whether the molecule belongs to one of the special cases of low or high symmetry.
Group Symmetry
C1 No symmetry other than the identity operation.
Cs Only one mirror plane.
Ci Only an inversion center; few molecular examples
Molecules with many symmetry operations may fit one of the high-symmetry cases of linear, tetrahedral, octahedral, or iscosahedral symmetry. Molecules with very high symmetry are of two types, linear and polyhedral. Linear molecules having a center of inversion have Dh symmetry; those lacking an inversion center have Cv symmetry.
 CvThese molecules are linear, with an infinite number of rotations and an infinite number of reflection planes containing the rotation axis. They do not have a center of inversion.
 DhThese molecules are linear, with an infinite number of rotations and an infinite number of reflection planes containing the rotation axis. They also have perpendicular C2 axes, a perpendicular reflection plane, and an inversion center.
 TdMost (but not all) molecules in this point group have the familiar tetrahedral geometry. They have four C3 axes, three C2 axes, three S4 axes, and six or sigma d planes. They have no C4 axes.
 OhThese molecules include those of octahedral structure, although some other geometrical forms, such as the cube share the same set of symmetry operations. Among their 48 symmetry operations are four C3 rotations, three C4 rotations, and an inversion.
 IhIcosahedral structures are best recognized by their six C5 axes, as well as many other symmetry operations - 120 in all.
 Other groups2. Find the rotation axis with the highest n, the highest order Cn axis for the molecule. This is the principle axis of the molecule.
3. Does the molecule have any C2 axes perpendicular to the Cn axis?
Yes then D Groups - Molecules with C2 axes perpendicular to the principal axis are in one of the groups designated by the letter D; there are n C2 axes.
No then C or S Groups - Molecules with no perpendicular C2 axes are in one of the groups designated by the letters C or S.
No final assignments of point groups have been made, but the molecules have now been divided into two major cateogries, the D set, and the C or S set.
 Other groups (continued)4. Does the molecule have a mirror plane (sigma h horizontal plane) perpendicular to the Cn axis?
D groups - Yes = Dnh
No = Dn or Dnd
C and S groups - Yes = Cnh
No = Cn, Cnv or S2n
5. Does the molecule have any mirror planes that contain the Cn axis?
D groups - Yes = Dnd
No = Dn
C and S Groups - Yes = Cnv
No = Cn or S2n
6. Is there an S2n axis collinear with the Cn axis?
D Groups - An molescules in this category that have S2n axis have already been assigned to groups. There are no additional groups to be considered here
C and S groups - Yes = S2n
No = Cn
 CnhPoint group = C 2h (example = difluorodiazine) or C 3h (example: b(OH)3, planar)
 CnvPoint group = C2v (example: Water, H2O)
Point group = C3v (example: PCl3)
Point Group = C4v (example: BrF5, square pyramid)
Point Group: Cinfinityv (example: HF, CO, HCN)
 CnPoint group: C2 (Example: N2H4, which hasa gauche conformation)
C3 (Example: P(C6H5)3, which is like a three-bladed proeller distorted out of the planar shape by a lone pair on the P)
 DnhPoint Group = D3h (example: BF3)
D4h (Example: PtCl4 2-)
D5h (Example: Os(C5H5)2 (eclipsed))
D6h (Example: benzene)
Dinfinityh (example: F2, N2, acetylene C2H2)

 DndPoint Group: D2d (H2C=C=CH2, allene)
D4d (Ni(cyclobutadiene)2 (staggered)
D5d (Fe(C5H5)2 (Staggered)
 DnD3 [Ru(NH2CH2CH2NH2)3 2+] (treating the NH2CHCH2NH2 group as a planar ring)
 Determining point groups 
36, "/var/app/current/tmp/"