Symmetry and MolecularOrbitals (II)

Rules of Molecular Orbitals

Rules for forming bonding and antibonding MOsnumber of total molecular orbitals = number of total atomic orbitalsatomic orbitals have the right symmetryatomic orbitals overlap wellatomic orbitals have similar energy

Rules for filling electrons are the same for MOs and AOs.start filling from the lowest energy orbitalfollows Pauli exclusion principle and Hund's rule

Provides molecular energy informationExplains magnetic behaviorCan provide molecular structural explanation, but moredifficult to comprehend than VB theory

LCAO-MO

•LCAO –linear combination of atomic orbitals

= cii

MO spreads over ALL molecule (delocalized)More nodes, less stablesame sign, bondingdifferent sign, antibondingAO lower in E, MO lower in E

LCAO-MOHeteronuclear Diatomic Molecules

•A more electronegative•B less electronegative

+

+

+ +

+ -

SALCSALC –symmetry adapted linear

combinations, the linearcombinations of atomic orbitalsof specified symmetry

Linear combination among orbitalswith wrong symmetry notconsidered

Linear combination among ALLorbitals (AO, MO) with rightsymmetry are allowed (calledorbital mixing)

MO Symmetrybond: no nodal plane passing through internuclear axisbond: 1 nodal plane passing through internuclear axisbond: 2 nodal planes passing through internuclear axis

+ +

+ +

+ +

+ +

-

++

+-

-

--

-- -

MO Symmetry

MO Symmetry

B.MO

A.MO

g: geradeu: ungerade

+ +

+ +

+

+ +-

- -

-

-

Symmetry of C3v Group Orbitals

Linear H3

3= 1SA - 21/2 1SB + 1SC

2= 1SA - 1SC

1= 1SA + 21/2 1SB + 1SC+ + +

+ +

+-

-

Node

2

1

0

The Particle in a Box

+ + +

+ +

+-

-

++

+ +

--

Node

2

1

0

Triangular H3

a1 = 1SA + 1SB + 1SC

e = 1SA - 1SC

1SA - 21/2 1SB + 1SC

+

+

+ + ++-

-

a, b : nondegeneratee: doubly degeneratet: triply degenerate

Node 0 1 1

Walsh Diagram –Correlation Diagram

+ + +

+ +

+-

-

+

+

+ + ++-

-

H3+ stable

Simplest3 center2 e bond

Character Tables

I: Mulliken symbol.A, B: 1D E: 2D T: 3DA: 1D symmetric about the principal axis (1)B: 1D unsymmetric about the principal axis (1)

II: Irreducible representations for the groupIII: Transformation properties of vectors and

rotations along the x, y and z axisIV: Transformation properties of squares and binary

products of the coordinates

(yz)

Total Representation for C2vIndividually block diagonalized matrices

Reduced to 1D matricesx [ 1] [-1] [ 1] [-1]y [ 1] [-1] [-1] [ 1]

z [ 1] [ 1] [ 1] [ 1]

irreducible representationΓx = 1 -1 1 -1Γy = 1 -1 -1 1Γz = 1 1 1 1ΓRz = 1 1 -1 -1z

Symmetry of OOrbitals in H2O

•2px - B1

•2py - B2

•2pz - A1

•2s - A1

(spherical, highest symmetry)

(yz)

(yz)

Symmetry of H Group Orbitals in H2O

= HA1s - HB1s B2

= HA1s + HB1s A1

MO of H2O

O orbitals 2s 2py 2px

2pz

H group orbitals + -

A1 B2 B1

H2O has 3A1 2B2 1B1 MOs

3A1 bonding, nonbonding, antibonding

2B2 bonding, antibonding

1B1 nonbonding

MO of H2O 1a1

•BondingO2s + O2pz + +

MO of H2O 2a1

•NonbondingO2s - O2pz + +

MO of H2O 3a1

•AntibondingO2s + O2pz - +

MO of H2O 1b2

•BondingO2py + -

MO of H2O 2b2

•AntibondingO2py - -

MO of H2O 1b1

•NonbondingO2px

Localization of Bonds –MO View

MO =>delocalized view

Sum and differenceof MOs =>localized view

Localized View ~Hybridizeddescription in VB

Molecular Shape and MO

8 e7 e6 e5 e4 e

105o103o136o131o180o

OH2NH2CH2BH2BeH2

Molecular Shape and MO

Molecular Shape and MOBeH2 NH2

OH2

BH2CH2

Localized vs DelocalizedDescription of MO

Localized Appropriate•Bond strengths•Bond lengths•VSEPR description

of moleculargeometry

•Bronsted acidity

Delocalized Appropriate•Electronic spectra•Photoionization•Electron attachment•Magnetism•Walsh description of

molecular geometry•Standard potentials

Symmetry of N Orbitals in NH3

•2px 2py - E

•2pz - A1

•2s - A1

(spherical,highestsymmetry)

Symmetry of H Group Orbitals in NH3

a1 = 1SA + 1SB + 1SC

e = 1SA - 1SC

1SA - 21/2 1SB + 1SC

+

+

+ + ++-

-

a, b : nondegeneratee: doubly degeneratet: triply degenerate

Node 0 1 1

MO of NH3

N orbitals 2s 2py 2px

2pz

H group orbitals a1 e

A1 E

NH3 has 3A1 2E MOs

3A1 bonding, nonbonding, antibonding

2E bonding, antibonding (doubly degenerate)

MO of NH3 1a1

•BondingN2s + N2pz + a1

MO of NH3 2a1

•NonbondingN2s - N2pz + a1

MO of NH3 3a1

•AntibondingN2s + N2pz - a1

MO of NH3 1e

•Bonding(N2px,N2py) + e

MO of NH3 2e

•Antibonding(N2px,N2py) - e

NH3 MO

NH3 MO

Steps to Construct MO of ABn

•Determine the molecular symmetry•Determine the symmetry of the orbitals of

central atom (A) and the group orbitals (Bn)•MOs can be constructed from A orbitals and

B orbitals of same symmetry•Consider the relative energy and the overlap

among the orbitals to decide if the interactionis bonding, nonbonding, or antibonding

MO of SF6

S orbitals

•2px 2py 2pz -T1u

•2s -A1g

F orbitals

•T1u

•Eg

•A1g

4 pairs of electrons in A1g & T1u are shared by 7 Atoms

Electron DeficiencyAntibonding

Nonbonding

Bonding+

+

++

+ +-

-

B BH

MO Theory of Solids

N AO gives N MOBand

MO Bands

N AO gives N MO withvery small energy gap

S and P Bands in 1D Solid

Discrete & Overlapping Bands

Weak AOinteractions

Narrow bands

Strong AOinteractions

Wide bands

Fermi Level - HOMO

Band with empty levels conduct electric current

Conductor Insulator Semiconductor

ConductionBand

ValenceBand

Fermi Distribution and Semiconductor

Intrinsic semiconductor

Fermi Distribution

n-type p-type

Extrinsic Semiconductor

ConductionBand

ValenceBand

p-n Junction Diode

Density of States (DOS)

Metal Semimetal

X-ray Emission

Empty state generated bye bombardment