Symmetry and group theory

Symmetry and Symmetry and Group TheoryGroup Theory

The symmetry properties of The symmetry properties of molecules and how they can molecules and how they can be used to predict vibrational be used to predict vibrational spectra, hybridization, optical spectra, hybridization, optical activity, etc.activity, etc.

Point GroupsPoint Groups

Molecules are classified and Molecules are classified and grouped based on their symmetry. grouped based on their symmetry. Molecules with similar symmetry are Molecules with similar symmetry are but into the same but into the same point grouppoint group. A . A point group contains all objects that point group contains all objects that have the same have the same symmetry elementssymmetry elements..

Symmetry ElementsSymmetry Elements

Symmetry elementsSymmetry elements are mirror are mirror planes, axis of rotation, centers of planes, axis of rotation, centers of inversion, etc. inversion, etc.

A molecule has a given symmetry A molecule has a given symmetry element if the operation leaves the element if the operation leaves the molecule molecule appearingappearing as if nothing has as if nothing has changed (even though atoms and changed (even though atoms and bonds may have been moved.)bonds may have been moved.)

Symmetry ElementsSymmetry ElementsElementElement Symmetry OperationSymmetry Operation SymbolSymbol

IdentityIdentity EE

nn-fold axis-fold axis Rotation by 2Rotation by 2ππ//nn CCnn

Mirror planeMirror plane ReflectionReflection σσCenter of in-Center of in- InversionInversion ii

version version nn-fold axis of-fold axis of Rotation by 2 Rotation by 2ππ//nn

SSnn

improper rotationimproper rotation followed by reflectionfollowed by reflectionperpendicular to the perpendicular to the axis of rotationaxis of rotation

Identity, EIdentity, E

All molecules have Identity. All molecules have Identity. This operation leaves the entire This operation leaves the entire molecule unchanged. A highly molecule unchanged. A highly asymmetric molecule such as a asymmetric molecule such as a tetrahedral carbon with 4 different tetrahedral carbon with 4 different groups attached has only identity, groups attached has only identity, and no other symmetry elements.and no other symmetry elements.

nn-fold Rotation-fold Rotation

Water has a 2-fold axis of rotation. When rotated by 180o, the hydrogen atoms trade places, but the molecule will look exactly the same.

nn-fold Axis of Rotation-fold Axis of Rotation

Ammonia has a CAmmonia has a C3 3 axis. Note that there axis. Note that there are are twotwo operations associated with the C operations associated with the C33 axis. Rotation by 120axis. Rotation by 120oo in a clockwise or a in a clockwise or a counterclockwise direction provide two counterclockwise direction provide two different orientations of the molecule.different orientations of the molecule.

Mirror PlanesMirror Planes

The reflection of The reflection of the water molecule in the water molecule in either of its two either of its two mirror planes results mirror planes results in a molecule that in a molecule that looks unchanged. looks unchanged.


The subscript The subscript “v” in “v” in σσvv, indicates a , indicates a vertical plane of vertical plane of symmetry. This symmetry. This indicates that the indicates that the mirror plane includes mirror plane includes the principal axis of the principal axis of rotation (Crotation (C22).).


The benzene ring The benzene ring has a Chas a C66 axis as its axis as its principal axis of principal axis of rotation.rotation.

The molecular The molecular plane is perpendicular plane is perpendicular to the Cto the C66 axis, and is axis, and is designated as a designated as a horizontal plane, horizontal plane, σσhh. .

C6.


The vertical The vertical planes, planes, σσvv, go through , go through the carbon atoms, and the carbon atoms, and include the Cinclude the C6 6 axis. axis.

The planes that The planes that bisect the bonds are bisect the bonds are called called dihedral dihedral planes, planes, σσdd..

C6.

InversionInversion

The inversion operation projects The inversion operation projects each atom through the center of each atom through the center of inversion, and across to the other inversion, and across to the other side of the molecule. side of the molecule.

Improper RotationImproper Rotation

An improper rotation is rotation, An improper rotation is rotation, followed by reflection in the plane followed by reflection in the plane perpendicular to the axis of rotation.perpendicular to the axis of rotation.


The The staggered staggered conformation of conformation of ethane has an Sethane has an S66 axis that goes axis that goes through both through both carbon atoms.carbon atoms.


Note that Note that an San S11 axis axis doesn’t exist; it doesn’t exist; it is same as a is same as a mirror plane.mirror plane.


Likewise, an Likewise, an SS22 axis is a axis is a center of center of inversion.inversion.


Molecules with the same Molecules with the same symmetry elements are placed into symmetry elements are placed into point groupspoint groups. .

∞ ∞


In general, you will not need to In general, you will not need to assign a molecule to its point group. assign a molecule to its point group. Recognition of the features of some Recognition of the features of some common point groups is useful.common point groups is useful.

Point Point GroupsGroups

Water and Water and ammonia both ammonia both belong to the Cbelong to the Cnvnv class of class of molecules. These molecules. These have vertical have vertical planes of planes of reflection, but no reflection, but no horizontal planes.horizontal planes.

Point Point GroupsGroups

The DThe Dnhnh groups have a groups have a horizontal plane horizontal plane in addition to in addition to vertical planes. vertical planes. Many inorganic Many inorganic complexes belong complexes belong to these to these symmetry groups.symmetry groups.

XX

XX

Y

Y


Highly symmetrical molecules, Highly symmetrical molecules, such as identically substituted such as identically substituted tetrahedrons or octahedrons belong tetrahedrons or octahedrons belong to their own point groups (Tto their own point groups (Tdd or O or Ohh respectively). respectively).


In assigning a point group, we In assigning a point group, we typically ignore the fine detail, such typically ignore the fine detail, such as conformation isomers, of the as conformation isomers, of the ligands. ligands.

In working problems using In working problems using group theory, the point group of the group theory, the point group of the molecule will usually be provided to molecule will usually be provided to you.you.

Character TablesCharacter Tables

The symmetry properties of The symmetry properties of each point group are summarized on each point group are summarized on a a character tablecharacter table. The character . The character table lists all of the symmetry table lists all of the symmetry elements of the group, along with a elements of the group, along with a complete set of complete set of irreducible irreducible representationsrepresentations. .

http://mutuslab.cs.uwindsor.ca/macdonald/250-LectureNotes/Fall2002/Bonding-Notes5.pdf

Character Table (CCharacter Table (C2v2v))

Character Table (CCharacter Table (C2v2v))

The functions to the right are called The functions to the right are called basis functionsbasis functions. They represent . They represent mathematical functions such as orbitals, mathematical functions such as orbitals, rotations, etc.rotations, etc.

The pThe pxx orbital orbital

If a pIf a pxx orbital orbital on the central on the central atom of a molecule atom of a molecule with Cwith C2v2v symmetry symmetry is rotated about is rotated about the Cthe C22 axis, the axis, the orbital is reversed, orbital is reversed, so the character so the character will be -1.will be -1.


If a pIf a pxx orbital on the central atom of orbital on the central atom of a molecule with Ca molecule with C2v2v symmetry is rotated symmetry is rotated about the Cabout the C22 axis, the orbital is reversed, axis, the orbital is reversed, so the character will be -1.so the character will be -1.


If a pIf a pxx orbital orbital on the central on the central atom of a molecule atom of a molecule with Cwith C2v2v symmetry symmetry is reflected in the is reflected in the yz plane, the yz plane, the orbital is also orbital is also reversed, and the reversed, and the character will be -character will be -1.1.


If a pIf a pxx orbital on the central atom of orbital on the central atom of a molecule with Ca molecule with C2v2v symmetry is reflected symmetry is reflected in the yz plane, the orbital is also in the yz plane, the orbital is also reversed, and the character will be -1.reversed, and the character will be -1.


If a pIf a pxx orbital on the central atom of orbital on the central atom of a molecule with Ca molecule with C2v2v symmetry is reflected symmetry is reflected in the xz plane, the orbital is unchanged, in the xz plane, the orbital is unchanged, so the character is +1.so the character is +1.

Character Table Character Table RepresentationsRepresentations

1. Characters of +1 indicate that the 1. Characters of +1 indicate that the basis function is unchanged by the basis function is unchanged by the symmetry operation.symmetry operation.

2. Characters of -1 indicate that the 2. Characters of -1 indicate that the basis function is reversed by the basis function is reversed by the symmetry operation.symmetry operation.

3. Characters of 0 indicate that the 3. Characters of 0 indicate that the basis function undergoes a more basis function undergoes a more complicated change.complicated change.

Character Table Character Table RepresentationsRepresentations

1. An 1. An AA representation indicates that the representation indicates that the functions are symmetric with respect to functions are symmetric with respect to rotation about the principal axis of rotation.rotation about the principal axis of rotation.

2. 2. BB representations are asymmetric with representations are asymmetric with respect to rotation about the principal axis.respect to rotation about the principal axis.

3. 3. EE representations are doubly degenerate. representations are doubly degenerate.4. 4. TT representations are triply degenerate. representations are triply degenerate.5. Subscrips 5. Subscrips u u and and gg indicate asymmetric indicate asymmetric

((ungeradeungerade) or symmetric () or symmetric (geradegerade) with ) with respect to a center of inversion.respect to a center of inversion.

Applications of Group Applications of Group TheoryTheory

1. Predicting polarity of molecules. A 1. Predicting polarity of molecules. A molecule cannot have a permanent molecule cannot have a permanent dipole moment if itdipole moment if it

a) has a center of inversiona) has a center of inversion

b) belongs to any of the D point b) belongs to any of the D point groupsgroups

c) belongs to the cubic groups T c) belongs to the cubic groups T or Oor O


2. Predicting chirality of molecules. 2. Predicting chirality of molecules. Chiral molecules lack an improper Chiral molecules lack an improper axis of rotation (axis of rotation (SSnn), a center of ), a center of symmetry (symmetry (ii) or a mirror plane () or a mirror plane (σσ).).


3. Predicting the orbitals used in 3. Predicting the orbitals used in σσ bonds. Group theory can be used to bonds. Group theory can be used to predict which orbitals on a central predict which orbitals on a central atom can be mixed to create hybrid atom can be mixed to create hybrid orbitals.orbitals.


4. Predicting the orbitals used in 4. Predicting the orbitals used in molecular orbitalsmolecular orbitals. Molecular . Molecular orbitals result from the combining or orbitals result from the combining or overlap of atomic orbitals, and they overlap of atomic orbitals, and they encompass the entire molecule.encompass the entire molecule.


5. Determining the symmetry 5. Determining the symmetry properties of all molecular motion properties of all molecular motion (rotations, translations and (rotations, translations and vibrations). Group theory can be vibrations). Group theory can be used to predict which molecular used to predict which molecular vibrations will be seen in the vibrations will be seen in the infrared or Raman spectra.infrared or Raman spectra.

Determining Determining HybridizationHybridization

1. Determine the point group of the molecule.1. Determine the point group of the molecule.2. Consider the 2. Consider the σσ bonds as vectors, and bonds as vectors, and

determine how they are transformed by the determine how they are transformed by the symmetry operations of the group.symmetry operations of the group.

3. Obtain the characters for the bonds. For 3. Obtain the characters for the bonds. For each symmetry operation, a bond which each symmetry operation, a bond which remains in place contributes a value of +1. If remains in place contributes a value of +1. If the bond is moved to another position, it the bond is moved to another position, it contributes a value of 0.contributes a value of 0.

4. 4. ReduceReduce the set of characters to a linear the set of characters to a linear combination of the character sets of the point combination of the character sets of the point group.group.

HybridizationHybridization

Determine the hybridization of boron in Determine the hybridization of boron in BFBF33. The molecule is trigonal planar, . The molecule is trigonal planar, and belongs to point group Dand belongs to point group D3h3h..

1. Consider the 1. Consider the σσ bonds as vectors. bonds as vectors.

FFaa

B FB Fcc

FFbb

HybridizationHybridization

Determine how each vector (Determine how each vector (σσ bond) bond) is transformed by the symmetry is transformed by the symmetry operations of the group.operations of the group.


EE 2C2C33 3C3C22 σσhh 2S2S33 33σσvv

ГГredred



ГГredred 33



ГГredred 33 0 0



ГГredred 33 0 0 1 1



ГГredred 33 0 0 1 1 33



ГГredred 33 0 0 1 1 33 0 0



ГГredred 33 0 0 1 1 33 0 0 1 1

Reducing a Reducing a RepresentationRepresentation

nnii = the number of times an irreducible = the number of times an irreducible representation representation ii occurs in the reducible occurs in the reducible representationrepresentationh = the order of the group (the total number of h = the order of the group (the total number of

operations in the point group)operations in the point group)c = the class (type) of operationc = the class (type) of operationggcc= the number of operations in the class= the number of operations in the classχχii = the character of the irreducible = the character of the irreducible representationrepresentationχχrr = the character of the reducible representation = the character of the reducible representation

Reducing a Reducing a RepresentationRepresentation

The order of the group, h, is the total number of operations.

h = 1+2+3+1+2+3=12

Hybridization of BFHybridization of BF33

ГГredred reduces to A reduces to A11′ + E ′. The orbitals ′ + E ′. The orbitals used in hybridization must have this used in hybridization must have this symmetry.symmetry.



The s orbital and the dz2 orbitals on boron have A1 ′ symmetry. The 3 ′ symmetry. The 3 dz2 orbital is too high in energy to hybridize.



The px and py orbitals and the dthe dxx22-y-y22 and dand dxyxy orbitals orbitals have E ′ symmetry. Since E ′ symmetry. Since the the d d orbitals on boron are too high in orbitals on boron are too high in energy, they will not be used.energy, they will not be used.



The hybridization of boron will spThe hybridization of boron will sp22

or, more specifically, spor, more specifically, spxxppy.y.. .

Molecular VibrationsMolecular Vibrations

Molecular motion includes Molecular motion includes translations, rotations and translations, rotations and vibrations. The total number of vibrations. The total number of degrees of freedom (types of degrees of freedom (types of molecular motion) is equal to 3N, molecular motion) is equal to 3N, where N is the number of atoms in where N is the number of atoms in the molecule.the molecule.


Of the 3N types of motion, three Of the 3N types of motion, three represent molecular translations in represent molecular translations in the x, y or z directions. Linear the x, y or z directions. Linear molecules have two rotational molecules have two rotational degrees of freedom, and non-linear degrees of freedom, and non-linear molecules have three rotational molecules have three rotational degrees of freedom.degrees of freedom.


For linear molecules, the For linear molecules, the number of molecular vibrations = number of molecular vibrations = 3N-3-2 = 3N-5.3N-3-2 = 3N-5.

For non-linear molecules, the For non-linear molecules, the number of molecular vibrations = number of molecular vibrations = 3N-3-3= 3N-6.3N-3-3= 3N-6.


To obtain To obtain ГГredred for for allall molecular molecular motion, we must consider the symmetry motion, we must consider the symmetry properties of the three cartesian properties of the three cartesian coordinates on coordinates on allall atoms of the molecule. atoms of the molecule.


The molecule lies in the xz The molecule lies in the xz plane. The x axis is drawn in blue, plane. The x axis is drawn in blue, and the y axis is drawn in black. The and the y axis is drawn in black. The red arrows indicate the z axis.red arrows indicate the z axis.

x

yz


The molecule lies in the xz The molecule lies in the xz plane. The x axis is drawn in blue, plane. The x axis is drawn in blue, and the y axis is drawn in black. The and the y axis is drawn in black. The red arrows indicate the z axis.red arrows indicate the z axis.

yx


If a symmetry operation If a symmetry operation changes the position of an atom, all changes the position of an atom, all three cartesian coordinates three cartesian coordinates contribute a value of 0. contribute a value of 0.

x

yz


For operations that leave an atom in For operations that leave an atom in place, the character is +1 for an axis that place, the character is +1 for an axis that remains in position, -1 for an axis that is remains in position, -1 for an axis that is reversed, and 0 for an axis that has been reversed, and 0 for an axis that has been moved. moved.

x

yz


EE CC22 σσvv(xz)(xz) σσv′(yz)v′(yz)

Identity leaves Identity leaves all 3 atoms in all 3 atoms in position, so the position, so the character will character will be 9.be 9.



99

Identity leaves Identity leaves all 3 atoms in all 3 atoms in position, so the position, so the character will character will be 9.be 9.



99

The CThe C22 axis goes axis goes through the through the oxygen atom, and oxygen atom, and exchanges the exchanges the hydrogen atoms.hydrogen atoms.



99

The z axis (The z axis (redred) ) on oxygen stays on oxygen stays in position. This in position. This axis contributes axis contributes +1 towards the +1 towards the character for Ccharacter for C22..



99

The y axis (black) The y axis (black) on oxygen is on oxygen is rotated by 180rotated by 180oo. . This reverses the This reverses the axis, and axis, and contributes -1 to contributes -1 to the character for the character for CC22..



99

The x axis (blue) on The x axis (blue) on oxygen is also oxygen is also rotated by 180rotated by 180oo. . This reverses the This reverses the axis, and axis, and contributes -1 to the contributes -1 to the character for Ccharacter for C22..



99

The character for the CThe character for the C22 operation will be +1 (z axis on operation will be +1 (z axis on oxygen) -1 (y axis on oxygen) -1 oxygen) -1 (y axis on oxygen) -1 (x axis on oxygen) = -1(x axis on oxygen) = -1



99 -1-1

The character for the CThe character for the C22 operation will be +1 (z axis on operation will be +1 (z axis on oxygen) -1 (y axis on oxygen) -1 oxygen) -1 (y axis on oxygen) -1 (x axis on oxygen) = -1(x axis on oxygen) = -1



99 -1-1

The xz mirror The xz mirror plane is the plane is the molecular plane, molecular plane, and all three and all three atoms remain in atoms remain in position.position.

x



99 -1-1

The z axis and the The z axis and the x axis both lie x axis both lie within the xz within the xz plane, and remain plane, and remain unchanged.unchanged.

x



99 -1-1

Each unchanged Each unchanged axis contributes axis contributes +1 to the +1 to the character for the character for the symmetry symmetry operation.operation.

x



99 -1-1

For 3 atoms, the For 3 atoms, the contribution to contribution to the character will the character will be:be:

3(1+1) =63(1+1) =6

x



99 -1-1

The y axis will be The y axis will be reversed by the reversed by the mirror plane, mirror plane, contributing a value contributing a value of -1 for each of the of -1 for each of the three atoms on the three atoms on the plane.plane.

xy



99 -1-1 33

The character for The character for the xz mirror the xz mirror plane will be:plane will be:

6-3 = 36-3 = 3

xy



99 -1-1 33

The yz mirror The yz mirror plane bisects the plane bisects the molecule. Only molecule. Only the oxygen atom the oxygen atom lies in the plane.lies in the plane.

xy



99 -1-1 33

The y and z axis The y and z axis lie within the yz lie within the yz plane, and each plane, and each contributes +1 to contributes +1 to the character.the character.

xy



99 -1-1 33

The x axis on The x axis on oxygen is reversed oxygen is reversed by the reflection, by the reflection, and contributes a -and contributes a -1 towards the 1 towards the character.character.

xy



99 -1-1 33 11

The character for The character for reflection in the reflection in the yz plane is:yz plane is:

1+1-1=11+1-1=1

xy



99 -1-1 33 11

The above reducible representation is The above reducible representation is sometimes called sometimes called ГГ3N3N, because it reduces , because it reduces to all (3N) modes of molecular motion.to all (3N) modes of molecular motion.

ГГ3N3N for water reduces to: for water reduces to:

3A3A11 + A + A22 + 3B + 3B11 + 2B + 2B22


ГГ3N3N for water = 3A for water = 3A11 + A + A22 + 3B + 3B11 + + 2B2B22

Note that there are 9 modes Note that there are 9 modes of motion. These include of motion. These include vibrations, rotations and vibrations, rotations and translations.translations.


ГГ3N3N for water = 3A for water = 3A11 + A + A22 + 3B + 3B11 + 2B + 2B22

Translations have the same Translations have the same symmetry properties as x, y and z.symmetry properties as x, y and z.




2




2 2




2 2 1


ГГrot & vibrot & vib = 2A = 2A11 + A + A22 + 2B + 2B11 + 1B + 1B22


ГГrot & vibrot & vib = 2A = 2A11 + A + A22 + 2B + 2B11 + 1B + 1B22

Rotations have the same Rotations have the same symmetry as Rsymmetry as Rxx, R, Ryy and R and Rzz..


ГГrot & vibrot & vib = 2A = 2A11 + 2B + 2B11 + 1B + 1B22


1


ГГrot & vibrot & vib = 2A = 2A11 + 1B + 1B11 + 1B + 1B22


Rotations and Rotations and TranslationsTranslations

Rz

Rx

Ry

Transz

Transy

Transx

Molecular VibrationsMolecular VibrationsГГvibvib = 2A = 2A11 + B + B11

The three vibrational modes The three vibrational modes remain. Two have Aremain. Two have A11 symmetry, symmetry, and one has Band one has B11 symmetry. symmetry.


ГГvibvib = 2A = 2A11 + B + B11

Two vibrations are symmetric Two vibrations are symmetric with respect to all symmetry with respect to all symmetry operations of the group.operations of the group.


One vibration is asymmetric with One vibration is asymmetric with respect to rotation and reflection respect to rotation and reflection perpendicular to the molecular plane.perpendicular to the molecular plane.


A1 symmetric stretch

A1 bend

B1 asymmetric stretch


For a molecular vibration to be For a molecular vibration to be seen in the infrared spectrum (IR seen in the infrared spectrum (IR active), it must change the dipole active), it must change the dipole moment of the molecule. The dipole moment of the molecule. The dipole moment vectors have the same moment vectors have the same symmetry properties as the symmetry properties as the cartesian coordinates x, y and z.cartesian coordinates x, y and z.


Raman spectroscopy measures the Raman spectroscopy measures the wavelengths of light (in the IR range) wavelengths of light (in the IR range) scatted by a molecule. Certain scatted by a molecule. Certain molecular vibrations will cause the molecular vibrations will cause the frequency of the scattered radiation to frequency of the scattered radiation to be less than the frequency of the incident be less than the frequency of the incident radiation. radiation.


For a molecular vibration to be For a molecular vibration to be seen in the Raman spectrum (Raman seen in the Raman spectrum (Raman active), it must change the active), it must change the polarizability of the molecule. The polarizability of the molecule. The polarizability has the same polarizability has the same symmetry properties as the symmetry properties as the quadratic functions: quadratic functions:

xy, yz, xz, xxy, yz, xz, x22, y, y22 and z and z22

Molecular Vibrations of Molecular Vibrations of WaterWaterГГvibvib = 2A = 2A11 + B + B11

The two vibrations with A1 symmetry have z as a basis function, so they will be seen in the infrared spectrum of water. This will result in two peaks (at different frequencies) in the IR spectrum of water.


The two vibrations with A1 symmetry also have quadratic basis functions, so they will be seen in the Raman spectrum of water as well.


The two vibrations with A1 symmetry will appear as two peaks in both the IR and Raman spectra. The two frequencies observed in the IR and Raman for these vibrations will be the same in both spectra.

Molecular Vibrations of Molecular Vibrations of WaterWater


The vibration with B1 symmetry has x and xz as basis functions. This vibration will be both IR active and Raman active. This vibration will appear as a peak (at the same frequency) in both spectra.

Molecular Vibrations of Molecular Vibrations of WaterWater


Both the IR and Raman spectra should show three different peaks.

SummarySummary

1. 1. Obtain the point group of the Obtain the point group of the molecule.molecule.

2. 2. Obtain Obtain ГГ3N3N by considering the three by considering the three cartesian coordinates on all atoms that cartesian coordinates on all atoms that aren’t moved by the symmetry operation.aren’t moved by the symmetry operation.

3. 3. Reduce Reduce ГГ3N3N . .

4.4. Eliminate translations and rotations.Eliminate translations and rotations.

5.5. Determine if remaining vibrations are IR Determine if remaining vibrations are IR and/or Raman active.and/or Raman active.

Application: Carbonyl Application: Carbonyl StretchesStretches

Can IR and Raman spectroscopy Can IR and Raman spectroscopy determine the difference between determine the difference between two square planar complexes: two square planar complexes: ciscis--MLML22(CO)(CO)22 and and transtrans-ML-ML22(CO)(CO)22??

ciscis and and trans trans MLML22(CO)(CO)22

OC

OC

OC

CO

cis isomer – C2v

trans isomer – D2h

ciscis - ML - ML22(CO)(CO)22

OC

OC

C2v: E C2 σxz σyz

ГCO: 2 0 2 0


ГCO reduces to A1 + B1.

A1 is a symmetric stretch, and B1 is an asymmetric stretch.

OC

OC



The symmetric stretch (A1) is IR and Raman active.

OC

OC



The asymmetric stretch (B1) is both IR and Raman active.

OC

OC

trans trans MLML22(CO)(CO)22

OC

CO


The trans isomer lies in the xy plane. The point group D2h has the following symmetry elements:DD2h2h EE CC22(z(z

))CC22((y)y)

CC22(x)(x) ii σσxyxy σσxzxz σσyzyz

yx


OC

CO


The trans isomer lies in the xy plane. ГCO is obtained by looking only at the two C-O bonds.DD2h2h EE CC22(z(z

))CC22((y)y)

CC22(x)(x) ii σσxyxy σσxzxz σσyzyz

ГCO 22 00 00 22 00 22 22 00

yx


OC

CO


ГCO reduces to Ag

(a symmetric stretch) and B3u(an asymmetric stretch).

yx


OC

CO


ГCO reduces to Ag

(a symmetric stretch) and B3u(an asymmetric stretch).

Ag has x2, y2 and z2 as basis functions, so this vibration is Raman active.

yx


OC

CO



B3u has x as a basis function, so this vibration is IR active.

yx



B3u has x as a basis function, so this vibration is IR active.

The IR and Raman spectra will each show one absorption at different frequencies.

Exclusion RuleExclusion Rule

If a molecule has a center of If a molecule has a center of symmetry, symmetry, nonenone of its modes of of its modes of vibration can be both infrared and vibration can be both infrared and Raman active.Raman active.

Exclusion RuleExclusion Rule

If a molecule has a center of If a molecule has a center of symmetry, symmetry, nonenone of its modes of of its modes of vibration can be both infrared and vibration can be both infrared and Raman active.Raman active.The cis and trans isomers of square planar MLML22(CO)(CO)22, can be, can be easily distinguished using easily distinguished using spectroscopy. The spectroscopy. The ciscis isomer has isomer has absorptions that are seen in both absorptions that are seen in both the IR and Raman spectra, whereas the IR and Raman spectra, whereas the the trans trans isomer does not.isomer does not.

Symmetry and group theory

Science

Transcript of Symmetry and group theory