Molecular symmetry
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Molecular Symmetry. Symmetry Elements Group Theory Photoelectron Spectra Molecular Orbital (MO) Diagrams for Polyatomic Molecules. The Symmetry of Molecules. The shape of a molecule influences its physical properties, reactivity, and its spectroscopic behavior

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Molecular Symmetry

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Molecular symmetry

Molecular Symmetry

Symmetry Elements

Group Theory

Photoelectron Spectra

Molecular Orbital (MO) Diagrams for Polyatomic Molecules


The symmetry of molecules

The Symmetry of Molecules

  • The shape of a molecule influences its physical properties, reactivity, and its spectroscopic behavior

  • Determining the symmetry of a molecule is fundamental to gaining insight into these characteristics of molecules

  • The chemist’s view of symmetry is contained in the study of group theory

    • This branch of mathematics classifies the properties of a molecule into groups, defined by the symmetry of the molecule

    • Each group is made up of symmetry elements or operations, which are essentially quantum operators disguised as matrices

  • Our goal is to use group theory to build more complex molecular orbital diagrams

Symmetry


Symmetry elements

Symmetry Elements

  • You encounter symmetry every day

    • A ball is spherically symmetric

    • Your body has a mirror image (the left and right side of your body)

    • Hermite polynomials are either even (symmetric on both sides of the axis) or odd (symmetric with a twist).

  • Symmetry operations are movements of a molecule or object such that after the movement the object is indistinguishable from its original form

  • Symmetry elements are geometric representations of a point, line, or plane to which the operation is performed

    • Identity element (E)

    • Plane of reflection (s)

    • Proper rotation (Cn)

    • Improper rotation (Sn)

    • Inversion (i)

Symmetry


Symmetry elements ii

Symmetry Elements II

  • Identity

    • If an object (O) has coordinates (x,y,z), then the operation E(x,y,z) (x,y,z)

    • The object is unchanged

E

  • Plane of reflection

    • s(xz) (x,y,z) = (x,-y,z)

    • s(xy) (x,y,z) = (x,y,-z)

    • s(yz) (x,y,z) = (-x,y,z)

s(xz)

z

s(xy)

y

x

Symmetry


Symmetry elements iii

Symmetry Elements III

  • Proper Rotation

    • Cn where n represents angle of rotation out of 360 degrees

      • C2 = 180°, C3 = 120°, C4 = 90° …

    • C2(z) (x,y,z) = (-x,-y,z)

C6

C6

C3

How many C2 operations are there for benzene?

C6

C2

Symmetry


Symmetry elements iv

Symmetry Elements IV

  • Inversion

    • Takes each point through the center in a straight line to the exact distance on the other side of center

    • i (x,y,z) = (-x,-y,-z)

1

4

6

2

3

5

5

3

2

6

4

1

  • Improper Rotation

    • A two step operation that first does a proper rotation and then a reflection through a mirror plane perpendicular to the rotational axis.

    • S4(z) (x,y,z) = (y,-x,-z)

      • Same as (s)(C4) (x,y,z)

    • Note, symmetry operations are just quantum operators: work from right to left

1

6

6

2

5

1

sh C6 = S6

5

3

4

2

4

3

Symmetry


A few to try

A Few To Try

  • Determine which symmetry elements are applicable for each of the following molecules

Symmetry


Point groups

Point Groups

  • We can systematically classify molecules by their symmetry properties

    • Call these point groups

    • Use the flow diagram to the right

Special groups:

a) Cv,Dh (linear groups)

b) T, Th, Td, O, Oh, I, Ih

(1)

Start

(2)

No proper or improper axes

(3)

Only Sn (n = even) axis: S4, S6…

Cn axis

(4)

(5)

No C2’s  to Cn

n C2’s  to Cn

sh

n sv’s

No s’s

sh

n sd’s

No s’s

Cnh

Cnv

Cn

Dnh

Dnv

Dn

Symmetry


Some common groups

Some Common Groups

AB3

  • D3h Point Group

    • C3, C32, 3C2, S3, S35, 3sv, sh

    • Trigonal planar

  • C3v Point Group

    • C3, 3sv

    • Trigonal pyramid

  • D3h Point Group

    • C3, C32, 3C2, S3, S35, 3sv, sh

    • Trigonal bi-pyramid

  • C4h Point Group

    • C2, 2C2’, 2C2’’, C4, S4, S42, 2sv, 2sd, sh, i

    • Square planar

  • C4v Point Group

    • C2, C4, C42, 2sv, 2sd

    • Square pyramid

AB3

AB5

AB4

AB5

Symmetry


Character tables

Character Tables

Point Group

  • Character tables hold the combined symmetry and effects of operations

    • For example, consider water (C2v)

Symmetry operations available

Effect of this operation on an orbital of this symmetry.

Coordinates and rotations of this symmetry

Symmetry of states

A, B = singly degenerate

E = doubly degenerate

T = triply degenerate

1 = symmetric to C2 rotation*

2 = antisymmetric to C2 rotation*

Symmetry


The oxygen s p x orbital

The Oxygen’s px orbital

  • For water, we can look at any orbital and see which symmetry it is by applying the operations and following the changes made to the orbital

    • If it stays the same, it gets a 1

    • If it stays in place but gets flipped, -1

    • If it moves somewhere else, 0

1

-1

1

-1

The px orbital has B1 symmetry

Symmetry


The projection operator

The Projection Operator

  • Since we want to build MO diagrams, our symmetry needs are simple

    • Only orbitals of the same symmetry can overlap to form bonds

  • Each point group has many possible symmetries for an orbital, and thus we need a way to find which are actually present for the particular molecule of that point group

    • We’ll also look at collections of similar atoms and their collective orbitals as a group

  • The projection operator lets us find the symmetry of any orbital or collection of orbitals for use in MO diagrams

    • It will also be of use in determining the symmetry of vibrations, later

    • We just did this for the px orbital for water’s oxygen atom

    • It’s functional form is

  • But it’s easier to use than this appears

Symmetry


Ammonia

Ammonia

  • Let’s apply this mess to ammonia

    • First, draw the structure and determine the number of s and p bonds

    • Then look at how each bond changes for the group of hydrogen atoms, building a set of “symmetry adapted linear combinations of atomic orbitals” (SALC) to represent the three hydrogen's by symmetry (not by their individual atomic orbitals)

    • Finally, we’ll compare these symmetries to those of the s and p atomic orbitals of the nitrogen to see which overlap, thus building our MO diagram from the SALC

3 sigma bonds and no pi bonds, thus we’ll build our SALC’s from the projection operator and these 3 MO’s

Note, ammonia is in C3v point group (AB3)

Symmetry


Ammonia ii

Ammonia II

  • The Character Table for C3v is to the right

  • Take the s bonds through the operations & see how many stay put

  • We now have a representation (G) of this group of orbitals that has the symmetry

(3)

(0)

(1)

Symmetry


Ammonia iii

Ammonia III

  • This “reducible representation” of the hydrogen’s s-bonds must be a sum of the symmetries available

    • Only one possible sum will yield this reducible representation

    • By inspection, we see that Gs = A1 + E

  • From the character table, we now can get the symmetry of the orbitals in N

    • N(2s) = A1 -- x2 + y2 is same as an s-orbital

    • N(2pz) = A1

    • N(2px) = N(2py) = E

  • So, we can now set up the MO diagram and let the correct symmetries overlap

Symmetry


Ammonia the mo diagram

Ammonia, The MO Diagram

3A1

2Ex

2Ey

s*

2p’s

A1, Ex, Ey

2A1

Ex, Ey

A1

2s

1A1

Symmetry


Methane

Methane

  • The usual view of methane is one where four equivalent sp3 orbitals are necessary for the tetrahedral geometry

sp3-s overlap for a s MO

sp3-s overlap for a s MO

Photoelectron spectrum (crude drawing) adapted from Roy. Soc. Chem., Potts, et al.

  • However, the photoelectron spectrum shows two different orbital energies with a 3:1 population ratio

    • Maybe the answer lies in symmetry

    • Let’s build the MO diagram using the SALC method we saw before

Symmetry


Methane salc approach

Methane: SALC Approach

  • Methane is a tetrahedral, so use Td point group

    • The character table is given below

  • To find the SALC’s of the 4H’s, count those that do not change position for each symmetry operation and create the reducible representation, GSALC.

GSALC = A1 + T2

  • The character table immediately gives us the symmetry of the s and p orbitals of the carbon:

    • C(2s) = A1

    • C(2px, 2py, 2pz) = T2

Symmetry


The mo diagram of methane

The MO Diagram of Methane

  • Using the symmetry of the SALC’s with those of the carbon orbitals, we can build the MO diagram by letting those with the same symmetry overlap.

s7*

s8*

s9*

T2

A1

s6*

2px

2py

2pz

T2

1s

A1 + T2

2s

s3

s4

s5

A1

T2

s2

A1

s1

1s

A1

A1

C CH4 4H’s

Symmetry


An example bf 3

An Example: BF3

  • BF3 affords our first look at a molecule where p-bonding is possible

    • The “intro” view is that F can only have a single bond due to the remaining p-orbitals being filled

    • We’ll include all orbitals

  • The point group for BF3 is D3h, with the following character table:

  • We’ll begin by defining our basis sets of orbitals that do a certain type of bonding

Symmetry


F 3 residue basis sets

F3 Residue Basis Sets

  • Looking at the 3 F’s as a whole, we can set up the s-orbitals as a single basis set:

Regular character table

Worksheet for reducingGss

Symmetry


The other basis sets

The Other Basis Sets

Symmetry


The mo diagram for bf 3

The MO Diagram for BF3

No s-orbital interaction from F’s is included in this MO diagram!

Symmetry


The mo diagram for bf 31

The MO Diagram for BF3

s-orbital interaction from F’s allowed

Symmetry


Using hybrid orbitals for bf 3

Using Hybrid Orbitals for BF3

  • If we use sp2hybrids and the remaining p-orbital (pz) of the boron, we see how hybridization yields the same exact picture.

    • Build our sp2hybrids and take them through the operations

    • Find the irreducible representations using the worksheet method and the reducible representation

    • pz is found in the character table to be A2”.

  • Result: identical symmetries for boron’s orbital’s in both cases

  • This is how it should be, since hybridization is an equivalent set of orbitals that are simply oriented in space differently.

Symmetry


P bonding in aromatic compounds

p-Bonding in Aromatic Compounds

The -bonding in C3H3+1 (aromatic)

1 node

0 nodes

The -bonding in C4H4+2 (aromatic)

2 nodes

1 node

Aromatic compounds must have a completely filled set of bonding -MO’s.

This is the origin of the Hückel (4N+2) -electron definition of aromaticity.

0 nodes

Symmetry


Cyclopentadiene

Cyclopentadiene

  • As the other examples showed, the actual geometric structure of the aromatic yields the general shape of the p-MO region

The -bonding in C5H5-1 (aromatic)

2 nodes

1 node

0 nodes

Symmetry


Benzene

Benzene

3 nodes

The -bonding in C6H6 (aromatic)

2 nodes

1 node

0 nodes

Symmetry


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