Chapter 5 Molecular Symetry-CHEM 221 PDF

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DEPARTMENT OF CHEMISTRY FACULTY OF SCIENCES

CHEMISTRY 221: Inorganic Chemistry (I)

5 Molecular Symmetry Dr. Adnan S. Abu-S Abu-Surrah urrah Professor of Inorganic Chemistry (Office: 204, Tel. 4315), E-mail: [email protected] https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/symmetry/symmtry.htm http://symmetry.otterbein.edu/

Symmetry Symmetry is all around us and is a fundamental property of nature

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Molecular Symmetry • The term symmetry is derived from the Greek word “symmetria” which means “measured together”.

• An object is symmetric if one part (e.g. one side) of it is the same as all of the other parts.

• Group theory (point groups) is a powerful tool that

allows us to rationalize and simplify many problems in Chemistry.

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Molecule

Hybridization

VSEPR

Geometry

CH4 CH3Cl CH2Cl2 CHCl3

sp3

AX4

Tetrahedral

NH3

sp3

AX3E

T. Pyramidal

H2 O

sp3

AX2E2

V-Shape (Bent)

A point group consists of a set of symmetry elements (e.g. H2O: C2V) and associated symmetry operations – (see next slide) that completely describe the symmetry of an object. 4

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Symmetry operations and symmetry elements

• A symmetry operation is defined as: – An operation performed on a molecule that leaves it apparently unchanged. – Example: if a water molecule is rotated by 180° around a line that cuts the molecule in half, the resulting molecule is indistinguishable from the original.

• A symmetry element is the point, line (symmetry axis) or plane (mirror) with respect to which a symmetry operation is performed. • The symmetry element associated with the rotation axis, around which the molecule is rotated: 5 • That object is said to possess this symmetry element.

Symmetry Elements and Symmetry Operations

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The identity operation: E • Every molecule possesses at least one symmetry element: the identity -- E • The identity operation is doing nothing to a molecule –leaves any molecule completely unchanged.

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n-Fold Rotation => Cn – where n = 2, 180o rotation  C2 –

n = 3, 120o rotation  C3

–

n = 4, 90o rotation  C4

–

n = 6, 60o rotation  C6

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n-fold rotation a rotation of 360°/n about the Cn axis (n = 1 to )

180°

O(1)

H(3)

H(2)

O(1)

H(3)

H(2)

In water there is a C2 axis so we can perform a 2-fold (180°) rotation to get the identical arrangement of atoms. 9

n-fold rotation H(3)

H(2)

H(4)

120°

120° N(1)

N(1)

H(4)

H(3)

H(2)

N(1)

H(4)

H(3)

H(2)

In ammonia there is a C3 axis so we can perform 3-fold (120°) rotations to get identical arrangement of atoms.

Check BF3 Vs BF2Cl

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n-fold rotation

- Each possible rotation operation is assigned using a superscript integer m of the form Cnm. - The rotation Cnn is equivalent to the identity operation (nothing is moved). 11

3-Fold Axis of Rotation •

Example:The BF3 molecule is left unchanged by a rotation of 120° around an axis perpendicular to the molecular plane:

• • •

(360°/n) = 120° so n = 3 The operation is a threefold or C3 rotation. The symmetry element is a C3 rotation axis. Actually two rotations are possible about this axis as rotation clockwise and anti-clockwise are not the same: – Thus, the BF3 molecule possesses two C3 axes and this is written 2C3.

•

Any rotation axis with n > 2 generates two rotation axes corresponding to clockwise and anticlockwise rotation. – Note that rotation by 180° clockwise or anticlockwise is the same. For example, H2O 12 only has one C2 axis.

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3-Fold Axis of Rotation • Example: In addition to the 2C3 axes, BF3 also possesses C2 axes:

• There is an equivalent C2 axis along each bond so that BF3 possess three C2 axes: 3C2. BF3 thus possesses 2C3 and 3C2 axes. The 3-fold C3 is considered the principal rotation axis (axis of rotation with highest n value). 13

n-fold rotation Cl(2)

Cl(3)

Cl(4)

C41 Cl(3)

Ni(1)

Cl(4)

C42 = C21

Cl(5)

Ni(1)

Cl(5)

Ni(1)

Cl(4)

Cl(2)

Cl(3)

Cl(2)

Cl(5)

C43 Cl(5)

Cl(2)

Ni(1)

Cl(3)

Cl(4)

- If n/m is an integer, then that rotation operation is equivalent to an n/m fold rotation. e.g. C42 = C21, C62 = C31, C63 = C21, etc. (identical to simplifying fractions) 14

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Symmetry Elements and Symmetry Operations

Mirror planes sv => mirror plane containing principal axis of rotation sh => mirror plane perpendicular to a principal axis of rotation sd => mirror plane bisects dihedral angle made by the principal axis of rotation and two adjacent C2 axes perpendicular to principal rotation axis

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Rotations and Mirrors in a Bent Molecule

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Rotations Axis Dihedral or σd planes. –

•

These bisect two C2 axes.

Example: The principal rotation axis of the benzene molecule is a C6 axis running perpendicular to the molecule

•

It also possess 3C2 axes running through opposite carbon atoms:

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Improper axis of rotation => Sn – Improper rotations actually consist of two separate operations: (1) an n-fold rotation (rotation by 360°/n) about an axis followed by (2) reflection in a plane perpendicular to that axis. OR – rotation about n axis followed by inversion (discussed later) through center of symmetry

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Improper Rotation (Rotation followed by a reflection)

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Center of symmetry OR Inversion Center => i Inversion operation • Every atom is moved in a straight line to the center of a molecule and then moved out the same distance on the other side. – In this operation, every part of the object is reflected through the inversion center, which must be at the center of mass of the object.

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Center of symmetry OR Inversion Center => i

Center of Inversion

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Points Groups We need to be able to specify the symmetry of molecules clearly. F

H

No symmetry – CHFClBr F Cl

H

H

Br

H

Some symmetry – CHFCl2 Cl

Cl

More symmetry – CH2Cl2 Cl Cl

H

More symmetry – CHCl3

Cl

Cl Cl

What about ?

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Point groups provide us with a way to indicate the symmetry unambiguously.

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Symmetry and Point Groups • The symmetry of a molecule can be completely specified by listing all the symmetry elements (E, Cn, σ, i and Sn) it possesses. • Point groups have symmetry about a single point at the center of mass of the system. • In a point group, all symmetry elements must pass through the center of mass (the point). 23

Selection of Point Group from Shape 1. First determine shape using Lewis Structure and VSEPR Theory 2. Next use models to determine which symmetry operations are present 3. Then use the flow chart (next slide or slides after next slide) to determine the point group – Determine if molecule is linear or non-linear – Determine the highest axis of rotation • Check for other non-coincident axis of rotation

– Check for mirror planes

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Identifying point groups

You should know about C1, Cs, Ci, Cn, Cnv, Cnh, Dnh and Sn groups

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Example – Find point group of trans CHBrFCl H

• Draw the structure:

C Br

Cl F

Linear molecules (Dh) Multiple high order axes (Td, Oh)

• Does molecule belong to special group? -- no •

Is there a Cn axis? -- no

• Is there a mirror plane? – no • Is there a center of inversion? -- no •

The point group is C1

•

Only E

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Identifying point groups Determine the point group of water •It does not belong to a special group •There is a C2 axis C2v •There are no other C2 axes •There is no σh plane •There is a σv plane

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Identifying point groups • cis-dichloroethene possesses the following symmetry elements:

• Symmetry elements: E, C2, σv and σv’. – Same as for H2O --- as a result, the two molecules have many physical properties in common despite their very different chemistries.

• Water, cis-dichloroethene and other apparently diverse molecules such as ClF3 and SO2Cl2 all possess the same set of symmetry elements --- They belong to the same point group which is called ‘C2v’. 31

http://symmetry.otterbein.edu/tutorial/reflection.html http://www.chemtube3d.com/Sym-D3hBF3new.htm

Example – Find point group of POCl3 • Draw the structure: Cl Cl

O P

Cl

• Does molecule belong to special group? -- no •

C1, Cs, Ci, Cn, Cnv, Cnh, Dnh Sn groups

Is there a Cn axis? -- Yes -- one C3

• Are there three C2 axis perpendicular to the C3? -- No • Is there a sh plane? -- no • Are there n sv planes containing principal axis? -- yes – The point group is C3v

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Example – Find point group of trans CO

C

O

C

O

• Draw the structure: • Does molecule belong to special group? -- no •

Is there a Cn axis? -- Yes -- one Ca

• Are there nC2 axis perpendicular to the Cn? -- No • Is there a sh plane? -- no • Are there n sv planes containing principal axis? -- yes – The point group is Cav –

E, C2, 2C, sV

Identifying point groups

D5h This point group contains the following symmetry operations: E the identity operation C5 a fivefold principal symmetry axis 5 * C2 five twofold symmetry axes orthogonal to the principal axis σh a horizontal mirror plane intersecting the principal symmetry axis 5 * σv five vertical mirror planes aligned with the principal symmetry axis S5 a fivefold improper rotation axis 34

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Introduction to a Group & Multiplication of Symmetry Operations

C2 35

C3

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Applications of Point Groups

 Point Groups can be used to:  determine the polarity of a molecule  determine the optical activity of a molecule  determine the linear combination of atomic orbitals needed to create MO’s  determine allowed molecular vibrations

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Applications of Point Groups  Polar Molecule: Permanent dipole moment  Symmetry elements may be used to find polarity of a molecule  Molecules with center of inversion (i) can’t be polar  Molecules belonging to D group can’t be polar  It implies matching charge distribution --

Nonpolar

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Nonpolar

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Applications of Point Groups  Dipole moment can’t lie perpendicular to any mirror plane or axis of rotation – BF3

Nonpolar

 In general, molecules with D, Td, and Oh point groups are non polar. 39

Cl(2)

Applications of Point Groups

Ni(1)

Cl(4)

Cl(5)

achiral

 Chiral molecules:

Cl(3)

 Non-superimposable mirror images - Molecule and mirror images are optically active F

 Molecules not chiral (achiral) if:  Belongs to Dnh, or Dnd

F F

F

 Belongs to Td or Oh point groups

achiral H C

chiral

Br

Cl F

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Enantiomer Pairs

http://symmetry.otterbein.edu/tutorial/reflection.html

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Next slides Additional Reading not included in Exams

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Vibrational Spectroscopy • Spectroscopy is the study of the interaction of matter with energy (radiation in the electromagnetic spectrum).” • A molecular vibration is a periodic distortion of a molecule from its equilibrium

DAB

geometry. • The energy required for a molecule to vibrate is quantized (not continuous) • generally in the infrared region of the 0

electromagnetic spectrum.

rAB re

re = equilibrium distance between A and B DAB = energy required to dissociate into A and B atoms 43

Vibrational Spectroscopy For a diatomic molecule (A-B), the bond between the two atoms can be approximated by a spring that restores the distance between A and B to its equilibrium value. The bond can be assigned a force constant, k (in Nm-1; the stronger the bond, the larger k ) and the relationship between the frequency of the vibration, , is given by the relationship: 

k 

or, more typically

2c 

k 

where , c is the speed of light,  is the frequency in “wave numbers” (cm-1) and  is the reduced mass (in amu) of A and B given by the equation: 

mA  mB mA  mB 44

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Vibrational Spectroscopy 2 c  

k 

can be rearranged to solve for k (in N/m):

k  5.89  10 5 2 

Molecule

 (cm-1)

k (N/m)

 (amu)

HF

3962

878

19/20

HCl

2886

477

35/36 or 37/38

HBr

2558

390

79/80 or 81/82

HI

2230

290

127/128

Cl2

557

320

17.5

Br2

321

246

39.5

CO

2143

1855

6.9

NO

1876

1548

7.5

N2

2331

2240

7

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Vibrational Spectroscopy

• For a vibration to be active (observable) in an infrared (IR) spectrum: • the vibration must change the dipole moment of the molecule • the vibrations for Cl2, Br2, and N2 will not be observed in an IR experiment • For a vibration to be active in a Raman spectrum: • the vibration must change the polarizability of the molecule. • polarizibility is the ability to distort with electric field like a neighboring ion -- more easy if the empty orbitals are close to the highest energy filled orbitals. -- typically found for large, heavy atoms and ions 47

Vibrational Spectroscopy -- polyatomic moleucles

• The situation is more complicated • more possible types of motion. • Each set of possible atomic motions is known as a mode . • There are a total of 3N possible motions for a molecule containing N atoms because each atom can move in one of the three orthogonal directions (i.e. in the x, y, or z direction). A mode in which all the atoms are moving in the same direction is called a translational mode because it is equivalent to moving the molecule there are three translational modes for any molecule.

Translational modes

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Vibrational Spectroscopy -- polyatomic moleucles

A mode in which the atoms move to rotate (change the orientation) the molecule called a rotational mode there are three rotational modes for any non-linear molecule and only two for linear molecules.

(2 if linear, 3 otherwise) Rotational modes

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Vibrational Spectroscopy -- polyatomic moleucles • The 3 N-6 modes for non-linear (or 3N-5 modes for a linear) molecule correspond to vibrations that we might be able to observe experimentally. • We must use symmetry to figure out how many signals we expect to see and what atomic motions contribute to the particular vibrational modes . 50

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Vibrational Modes in CO2

For linear molecules: 3N - 5 IR fundamentals

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Vibrational Modes in SO2

For non-linear molecules: 3N - 6 IR fundamentals Vibrational Modes for CH4

For non-linear molecules: 3N - 6 IR fundamentals 52

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