Chapter 4 Symmetry of Molecules - Flashcards

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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.

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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.

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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.

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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).

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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.

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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.

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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.

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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).

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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.

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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.

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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.

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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.

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Matrix representations

A set of matrices that each correspond to symmetry operations in the group. Point groups can be defined by using matrix representation.

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Reducible representation

A shorthand version of matrix representation that lists characters obtained by summing diagonals of square matrix.

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Irreducible representation

A long hand version of matrix representation and defines how x, y, z coordinates are independently affected by symmetry operations.

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Character Table

The complete set of irreducible representations for a point group. For C2v there are 4.

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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

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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.

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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.

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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.

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Rotation Operation (Cn)

Also called proper rotation, is rotation through 360 degrees/n about a rotation axis.

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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.

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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.

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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.

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Rotation angles vs Symmetry Operation

Rotation angle Symmetry Operation
90 degrees S4
180 degrees C2 (= S42)
270 degrees S43
360 degrees E (= S44)

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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.

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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.

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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.

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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.

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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.

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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.

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Ih

Icosahedral structures are best recognized by their six C5 axes, as well as many other symmetry operations - 120 in all.

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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.

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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

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Cnh

Point group = C 2h (example = difluorodiazine) or C 3h (example: b(OH)3, planar)

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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)

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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)

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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)

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Dnd

Point Group: D2d (H2C=C=CH2, allene)
D4d (Ni(cyclobutadiene)2 (staggered)
D5d (Fe(C5H5)2 (Staggered)

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Dn

D3 [Ru(NH2CH2CH2NH2)3 2+] (treating the NH2CHCH2NH2 group as a planar ring)

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Determining point groups

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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).

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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).

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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.

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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.

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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.

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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.

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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.

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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.

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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)

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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)

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Determining point groups

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