Molecular OrbitalsMolecular Orbitals

An approach to bonding in which orbitals An approach to bonding in which orbitals encompass the entire molecule, rather than encompass the entire molecule, rather than

being localized between atoms.being localized between atoms.

Molecular OrbitalsMolecular Orbitals

Molecular orbitals result from the Molecular orbitals result from the combination of atomic orbitals. combination of atomic orbitals.

Since orbitals are wave functions, they can Since orbitals are wave functions, they can combine either constructively (forming a combine either constructively (forming a bonding molecular orbital), or destructively bonding molecular orbital), or destructively (forming an antibonding molecular orbital).(forming an antibonding molecular orbital).

Molecular OrbitalsMolecular Orbitals

Molecular orbitals form when atomic orbitals Molecular orbitals form when atomic orbitals with similar energies and proper symmetry can with similar energies and proper symmetry can overlap.overlap.

Atomic orbitals with differing energies or the Atomic orbitals with differing energies or the wrong spatial orientation (orthogonal) do not wrong spatial orientation (orthogonal) do not combine, and are called combine, and are called non-bondingnon-bonding orbitals. orbitals.

Molecular Orbital TheoryMolecular Orbital Theory

In order to simplify things, we’ll consider the In order to simplify things, we’ll consider the interaction of the orbitals containing valence interaction of the orbitals containing valence electrons to create molecular orbitals.electrons to create molecular orbitals.

The wave functions of hydrogen atom A and The wave functions of hydrogen atom A and hydrogen atom B can interact either hydrogen atom B can interact either constructively or destructively.constructively or destructively.

Rules for linear combinationRules for linear combination

1. Atomic orbitals must be roughly of the same energy.

2. The orbital must overlap one another as much as possible- atoms must be close enough for effective overlap.

3. In order to produce bonding and antibonding MOs, either the symmetry of two atomic orbital must remain unchanged when rotated about the internuclear line or both atomic orbitals must change symmetry in identical manner.

Linear combination of atomic orbitalsLinear combination of atomic orbitals

Typical molecular energy levels diagram of an octahedral complex showing the frontier orbitals in the tinted box

Singly degenerate s a1g

Triply degenerate p t1u

Doubly degenerate d eg

Triply degenerate d t2g

A

Bg- identical under inversion

u- not identical

Rules for the use of MOsRules for the use of MOs

* When two AOs mix, two MOs will be produced * Each orbital can have a total of two electrons (Pauli principle) * Lowest energy orbitals are filled first (Aufbau principle) * Unpaired electrons have parallel spin (Hund’s rule)

Bond order = ½ (bonding electrons – antibonding electrons)

Molecular Orbital TheoryMolecular Orbital Theory

Constructively:Constructively:

ΨΨ((σσ)) or or ΨΨ++ = ( = (1/√2 ) 1/√2 ) [[φφ(1s(1saa) ) + + φφ(1s(1sbb) ) ]]

Destructively:Destructively:

ΨΨ((σσ*)*) or or ΨΨ-- = ( = (1/√2 ) 1/√2 ) [[φφ(1s(1saa) ) - - φφ(1s(1sbb) ) ]]

cA = cB = 1

+. +. . .+

bondingψg

Amplitudes of wave functions added

ψg = N [ψA + ψB]

Constructive interferenceConstructive interference

The accumulation of electron density between the nuclei put the electron in a position where it interacts strongly with both nuclei.

The energy of the molecule is lower

Nuclei are shielded from each other

Amplitudes of wave functions

subtracted.

Destructive interferenceDestructive interferenceNodal plane perpendicular to the H-H bond Nodal plane perpendicular to the H-H bond axis (en density = 0) axis (en density = 0) Energy of the en in this orbital is higher.Energy of the en in this orbital is higher.

+. -. ..

node

antibondingψu = N [ψA - ψB]

cA = +1, cB = -1 ψu

+ -

ΨA-ΨB

The electron is excluded from internuclear region The electron is excluded from internuclear region destabilizing destabilizing

AntibondingAntibonding

When 2 atomic When 2 atomic orbitalsorbitals combine there are 2 combine there are 2resultant resultant orbitalsorbitals..

low energy bonding orbitallow energy bonding orbital

high energy high energy antibondingantibonding orbital orbital1sb 1sa

σ1s

σ*

E1s

MolecularMolecular orbitalsorbitals

EgEg. s . s orbitalsorbitals

Molecular Orbital TheoryMolecular Orbital Theory

The bonding orbital is The bonding orbital is sometimes given the sometimes given the notation notation σσgg, where the , where the gg

stands for stands for geradegerade, or , or symmetric with respect symmetric with respect to a center of inversion. to a center of inversion.

+

+

-

The signs on the molecular orbitals indicate the sign of the wave function, not ionic charge.

Molecular Orbital TheoryMolecular Orbital Theory

The anti-bonding orbital The anti-bonding orbital is sometimes given the is sometimes given the notation notation σσuu, where the , where the u u

stands for stands for ungeradeungerade, or , or asymmetric with respect asymmetric with respect to a center of inversion. to a center of inversion.

+

+

-

The signs on the molecular orbitals indicate the sign of the wave function, not ionic charge.

11.4 eV

109 nm

HH22

Location ofBonding orbital4.5 eV

LCAO of n A.O ⇒ n M.O.

Period 2 Diatomic MoleculesPeriod 2 Diatomic Molecules

For the second period, assume that, due to a For the second period, assume that, due to a better energy match, better energy match, ss orbitals combine with orbitals combine with ss orbitals, and orbitals, and pp orbitals combine with orbitals combine with pp orbitals. orbitals.

The symmetry of The symmetry of pp orbitals permits end-on- orbitals permits end-on-end overlap along the bond axis, or side-by-side end overlap along the bond axis, or side-by-side overlap around, but not along, the internuclear overlap around, but not along, the internuclear axis.axis.

dx2-dy2 and dxy

Cl4Re ReCl42-

A

Bg- identical under inversion

u- not identical

MOs using MOs using pp orbitals orbitals

Some texts will use the symmetry designations of Some texts will use the symmetry designations of gg (gerade) or (gerade) or uu (ungerade) instead of indicating bonding or anti-bonding. (ungerade) instead of indicating bonding or anti-bonding.

For these orbitals, the anti-bonding orbital is asymmetric about the For these orbitals, the anti-bonding orbital is asymmetric about the bond axis, and is designated as bond axis, and is designated as σσuu. Note that the designations of . Note that the designations of uu or or gg do notdo not correlate with bonding or anti-bonding. correlate with bonding or anti-bonding.

+

++ -

--

-

σg

σu

ππ Molecular Orbitals Molecular Orbitals

The orbital overlap side-by-side is less than The orbital overlap side-by-side is less than that of overlap along the bond axis (end-on-that of overlap along the bond axis (end-on-end). As a result, the bonding orbital will be end). As a result, the bonding orbital will be higher in energy than the previous example.higher in energy than the previous example.

side-by-side overlap

++

+

--

-

Molecular Orbital DiagramMolecular Orbital Diagram

This is a molecular This is a molecular orbital energy level orbital energy level diagram for the diagram for the pp orbitals. Note that the orbitals. Note that the σσ bonding orbital is bonding orbital is lowest in energy due to lowest in energy due to the greater overlap the greater overlap end-on-end.end-on-end.

2p 2p

σg

πu

πg

σu

Place labels Place labels gg or or uu in this diagram in this diagram

σg

π∗g

σ∗u

πu

Molecular Orbital DiagramsMolecular Orbital Diagrams

1.1. Electrons preferentially occupy molecular Electrons preferentially occupy molecular orbitals that are lower in energy.orbitals that are lower in energy.

2.2. Molecular orbitals may be empty, or contain Molecular orbitals may be empty, or contain one or two electrons.one or two electrons.

3.3. If two electrons occupy the same molecular If two electrons occupy the same molecular orbital, they must be spin paired.orbital, they must be spin paired.

4.4. When occupying degenerate molecular When occupying degenerate molecular orbitals, electrons occupy separate orbitals orbitals, electrons occupy separate orbitals with parallel spins before pairing.with parallel spins before pairing.

Molecular Orbital DiagramsMolecular Orbital Diagrams

Although molecular orbitals form from inner Although molecular orbitals form from inner (core) electrons as well as valence electrons, (core) electrons as well as valence electrons, many molecular orbital diagrams include only many molecular orbital diagrams include only the valence level.the valence level.

First period diatomic moleculesFirst period diatomic molecules

σ1s2HE

nerg

yHH2

1s 1s

σg

σu*

Bond order = ½ (bonding electrons – antibonding electrons)

Bond order: 1

σ1s2, σ*1s2 HeE

nerg

yHeHe2

1s 1s

σg

σu*

Molecular Orbital theory is powerful because it allows us to predict whether molecules should exist or not and it gives us a clear picture of the of the electronic structure of any hypothetical molecule that we can imagine.

Diatomic molecules: The bonding in He2

Bond order: 0

Second period diatomic moleculesSecond period diatomic molecules

σ1s2, σ*1s2, σ2s2

Bond order: 1

Li

Ene

rgy

LiLi2

1s 1s

1σg

1σu*

2s 2s

2σg

2σu*

σ1s2, σ*1s2, σ2s2,

σ*2s2

Bond order: 0

Be

Ene

rgy

BeBe2

1s 1s

1σg

1σu*

2s 2s

2σg

2σu*

Diatomic molecules: Homonuclear Molecules of the Second Period

SimplifiedSimplified

SimplifiedSimplified

Diamagnetic??

2σg

2σu*

3σg

1πu

1πg*

3σu*

MO diagram for BMO diagram for B22

Li : 200 kJ/molF: 2500 kJ/mol

Same symmetry, energy mix-the one with higher energy moves higher and the one with lower energy moves lower

2σg

2σu*

3σg

1πu

1πg*

3σu*

B BB2

2s 2s

2σg

2σu*

2p

2p

3σg

3σu*

1πu

1πg*

(px,py)

HOMO

LUMO

MO diagram for BMO diagram for B22

Paramagnetic

1σg

1πu

1πg

1σ g

1πu

1π g

C2

DiamagneticParamagnetic ?X

1σg

1πu

1πg

1σg

1πu

1πg

Li2 to N2O2 and F2

General MO diagrams

Distance between b-MO and AO

Heteronuclear Diatomics….

� The energy level diagram is not symmetrical. � The bonding MOs are

closer to the atomic orbitals which are lower in energy.

� The antibonding MOs are closer to those higher in energy.

c – extent to which each atomicorbitals contribute to MO

If cA>cB the MO is composed principally of φA

Rules for Combining Atomic Rules for Combining Atomic OrbitalsOrbitals

For heteronuclear molecules:For heteronuclear molecules:

1. The bonding orbital(s) will reside 1. The bonding orbital(s) will reside predominantly on the atom of lower orbital predominantly on the atom of lower orbital energy (the more electronegative atom).energy (the more electronegative atom).

2. The anti-bonding orbital(s) will reside 2. The anti-bonding orbital(s) will reside predominantly on the atom with greater orbital predominantly on the atom with greater orbital energy (the less electronegative atom).energy (the less electronegative atom).

HFHF

The 2s and 2pThe 2s and 2pxx orbitals orbitals

on fluorine interact with on fluorine interact with the 1s orbital on hydrogen. the 1s orbital on hydrogen.

The pThe pyy and p and pzz orbitals orbitals

on fluorine lack proper on fluorine lack proper symmetry to interact with symmetry to interact with hydrogen, and remain as hydrogen, and remain as non-bonding orbitals.non-bonding orbitals.

HFHF

The anti-bonding The anti-bonding orbital resides primarily on orbital resides primarily on the less electronegative the less electronegative atom (H).atom (H).

Note that the Note that the subscripts subscripts gg and and uu are not are not used, as the molecule no used, as the molecule no longer has a center of longer has a center of symmetry.symmetry.

Carbon monoxideCarbon monoxide

In carbon In carbon monoxide, the bonding monoxide, the bonding orbitals reside more on orbitals reside more on the oxygen atom, and the oxygen atom, and the anti-bonding the anti-bonding orbitals reside more on orbitals reside more on the carbon atom. the carbon atom.

Carbon monoxideCarbon monoxide

CO is a highly CO is a highly reactive molecule with reactive molecule with transition metals. transition metals. Reactivity typically Reactivity typically arises from the arises from the hhighest ighest ooccupied ccupied mmolecular olecular oorbital (HOMO), when rbital (HOMO), when donating electrons.donating electrons.

Carbon monoxideCarbon monoxide

When acting as an When acting as an electron pair acceptor, electron pair acceptor, the the llowest owest uunoccupied noccupied mmolecular olecular oorbital (LUMO), is rbital (LUMO), is significant.significant.

Carbon monoxideCarbon monoxide

When acting as an When acting as an electron pair donor, electron pair donor, the the hhighest ighest ooccupied ccupied mmolecular olecular oorbital rbital (HOMO), is (HOMO), is significant.significant.

The highest occupied molecular orbital of CO is a molecular orbital which puts significant electron density on the carbon atom.

The lowest unoccupied molecular orbital of CO is the π* orbitals. The lobes of the LUMO are larger on the carbon atom than on the oxygen atom.

CO as a LigandCO as a Ligand

Carbon monoxide is known as a Carbon monoxide is known as a σσ donor and donor and a a ππ acceptor ligand. It donates electrons from its acceptor ligand. It donates electrons from its HOMO to form a sigma bond with the metal.HOMO to form a sigma bond with the metal.

CO as a LigandCO as a Ligand

Carbon monoxide accepts electrons from Carbon monoxide accepts electrons from filled filled dd orbitals on the metal into its antibonding orbitals on the metal into its antibonding (LUMO) orbital.(LUMO) orbital.

CO as a LigandCO as a Ligand

This phenomenon is called This phenomenon is called back bondingback bonding. The . The increased electron density in the antibonding orbitals of increased electron density in the antibonding orbitals of CO causes an increase in the C-O bond length and a CO causes an increase in the C-O bond length and a decrease in its stretching frequency.decrease in its stretching frequency.

MOs for Larger MoleculesMOs for Larger Molecules

Group theory is usually used to develop Group theory is usually used to develop molecular orbital diagrams and drawings of molecular orbital diagrams and drawings of more complicated molecules. When a central more complicated molecules. When a central atom is bonded to several atoms of the same atom is bonded to several atoms of the same element (Helement (H22O, BFO, BF33, or PtCl, or PtCl44

2-2-], group theory can ], group theory can

be used to analyze the symmetry of the orbitals be used to analyze the symmetry of the orbitals of the non-central atoms, and then combine of the non-central atoms, and then combine them with the appropriate orbitals of the central them with the appropriate orbitals of the central atom.atom.

MOs for Larger MoleculesMOs for Larger Molecules

The orbitals of the non-central atoms are The orbitals of the non-central atoms are called called group orbitalsgroup orbitals. In considering a simple . In considering a simple example, Hexample, H22O, we obtain group orbitals using O, we obtain group orbitals using

the two the two 1s1s orbitals on the hydrogen atoms. orbitals on the hydrogen atoms.

Group Orbitals of WaterGroup Orbitals of Water

The AThe A11 representation has both representation has both 1s1s orbitals with orbitals with

positive wave functions: Hpositive wave functions: Haa+H+Hbb..

The BThe B11 representations is H representations is Haa+H+Hbb..

Molecular Orbitals of WaterMolecular Orbitals of Water

Since the Since the 2p2pyy orbital on oxygen doesn’t match orbital on oxygen doesn’t match

the symmetry of the group orbitals of hydrogen, the symmetry of the group orbitals of hydrogen, it will remain non-bonding. The other orbitals it will remain non-bonding. The other orbitals on oxygen will combine with the appropriate on oxygen will combine with the appropriate group orbitals to form bonding and antibonding group orbitals to form bonding and antibonding molecular orbitals.molecular orbitals.

4p

4s

3d

x2-y2 z2 xy xz yzNB MOs

Bonding MOs

Six donor orbitals

6NH3 each donating 2 e-s

Antibonding MOs

Atomic orbitals in metal ion Atomic orbitals in ligand ionMolecular orbitals

Molecular Orbital diagram for [CoIII(NH3)6]3+

∆ο

Oh σ bonding

4p

4s

3d

x2-y2 z2 xy xz yz NB MOs

Bonding MOs

Six donor orbitals

6 F- each donating 2 e-s

Antibonding MOs

Atomic orbitals in metal ion Atomic orbitals in ligand ionMolecular orbitals

Molecular Orbital diagram for CoF63-

∆ο

Oh σ bonding

Clearly good σ donor ligandResult in good M-L overlap Strongly antibonding eg set

t2g

eg

t2g

ML6

σ-onlyML6

σ + π

Stabilization

(empty π-orbitals on ligands)

∆o∆’o ∆o has increased

Case 1 (CN-, CO, C2H4)empty π-orbitals on the ligandsM→L π-bonding (π-back bonding)

t2g (π)

t2g (π*)

eg

t2g

eg

t2g

ML6

σ-onlyML6

σ + π

Case 2 (Cl-, F-) filled π-orbitals on the ligandsL→ M π-bonding

(filled π-orbitals)

Stabilization

Destabilization

t2g (π)

t2g (π*)

eg

∆’o

∆o∆ o has decreased

Strong field Weak field

Putting it all on one diagram.

Summary:strong σ- or π-donor weak field ligands.

π-acceptors strong field ligands.

π donor ligands lower in E than t2g.

π acceptor ligands higher in E than t2g.

Or