molecular orbital theory lecture

Molecular Molecular OrbitalsOrbitals

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

being localized between being localized between atoms.atoms.

Molecular OrbitalsMolecular Orbitals

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

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

Molecular OrbitalsMolecular Orbitals

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

Atomic orbitals with differing Atomic orbitals with differing energies or the wrong spatial energies or the wrong spatial orientation (orthogonal) do not 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 In order to simplify things, we’ll consider the interaction of the consider the interaction of the orbitals containing valence electrons orbitals containing valence electrons to create molecular orbitals.to create molecular orbitals.

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


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

+. +. . .+

bondingg

Amplitudes of wave functions added

g = N [A + B]

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

+. -. ..

node

antibondingu = N [A - B]

cA = +1, cB = -1 u

+ -

A-B

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

1s

*

E1s

MolecularMolecular orbitalsorbitals

EgEg. s . s orbitalsorbitals


The bonding The bonding orbital is orbital is sometimes given sometimes given the notation the notation σσgg, , where the where the gg stands stands for for geradegerade, or , or symmetric with symmetric with respect to a center respect to a center of inversion. of inversion.

+

+

-

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


The anti-bonding The anti-bonding orbital is orbital is sometimes given sometimes given the notation the notation σσuu, , where the where the u u stands stands for for ungeradeungerade, or , or asymmetric with asymmetric with respect to a center respect to a center of inversion. 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 Period 2 Diatomic MoleculesMolecules

For the second period, assume For the second period, assume that, due to a better energy match, that, due to a 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 orbitals permits end-on-end overlap along permits end-on-end overlap along the bond axis, or side-by-side the bond axis, or side-by-side overlap around, but not along, the overlap around, but not along, the internuclear axis.internuclear 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 (ungerade) instead of indicating bonding or anti-bonding. bonding or anti-bonding.

For these orbitals, the anti-bonding orbital is For these orbitals, the anti-bonding orbital is asymmetric about the bond axis, and is designated asymmetric about the bond axis, and is designated as 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 The orbital overlap side-by-side is less than that of overlap along the bond axis than that of overlap along the bond axis (end-on-end). As a result, the bonding (end-on-end). As a result, the bonding orbital will be higher in energy than the orbital will be higher in energy than the previous example.previous example.

side-by-side overlap

+

+

+

-

-

-

Molecular Orbital Molecular Orbital DiagramDiagram

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

g

g

u

u

Molecular Orbital Molecular Orbital DiagramsDiagrams

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

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

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

Molecular Orbital Molecular Orbital DiagramsDiagrams

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

First period diatomic moleculesFirst period diatomic molecules

1s2HE

nerg

yHH2

1s 1s

g

u*

Bond order = ½ (bonding electrons – antibonding electrons)

Bond order: 1

1s2, *1s2 HeE

nerg

yHeHe2

1s 1s

g

u*

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

1s2, *1s2, 2s2

Bond order: 1

Li

Ene

rgy

LiLi2

1s 1s

1g

1u*

2s 2s

2g

2u*

1s2, *1s2, 2s2, *2s2

Bond order: 0

Be

Ene

rgy

BeBe2

1s 1s

1g

1u*

2s 2s

2g

2u*

Diatomic molecules: Homonuclear Molecules of the Second Period

SimplifiedSimplified

Diamagnetic??

2g

2u*

3g

1u

1g*

3u*

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

2g

2u*

3g

1u

1g*

3u*

B BB2

2s 2s

2g

2u*

2p

2p

3g

3u*

1u

1g*

(px,py)

HOMO

LUMO

MO diagram for BMO diagram for B22

Paramagnetic

1g

1u

1g

1g

1u

1g

C2

DiamagneticParamagnetic ?X

1g

1u

1g

1g

1u

1g

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 cAcB the MO is composed principally of A

Rules for Combining Rules for Combining Atomic OrbitalsAtomic Orbitals

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 predominantly on the atom of lower orbital energy (the more orbital energy (the more electronegative atom).electronegative atom).

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

HFHF

The 2s and 2pThe 2s and 2pxx orbitals on fluorine orbitals on fluorine interact with the 1s interact with the 1s orbital on hydrogen. orbital on hydrogen.

The pThe pyy and p and pzz orbitals on fluorine orbitals on fluorine lack proper symmetry lack proper symmetry to interact with to interact with hydrogen, and remain hydrogen, and remain as non-bonding as non-bonding orbitals.orbitals.

HFHF

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

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

Carbon monoxideCarbon monoxide

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


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


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


When When acting as an acting as an electron pair electron pair donor, the donor, the hhighest ighest ooccupied ccupied mmolecular olecular oorbital (HOMO), rbital (HOMO), is significant.is 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 a donor and a ππ acceptor ligand. It acceptor ligand. It donates electrons from its HOMO to donates electrons from its HOMO to form a sigma bond with the metal.form a sigma bond with the metal.


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


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

MOs for Larger MOs for Larger MoleculesMolecules

Group theory is usually used to Group theory is usually used to develop molecular orbital diagrams and develop molecular orbital diagrams and drawings of more complicated molecules. drawings of more complicated molecules. When a central atom is bonded to When a central atom is bonded to several atoms of the same element (Hseveral atoms of the same element (H22O, O, BFBF33, or PtCl, or PtCl44

2-2-], group theory can be used ], group theory can be used to analyze the symmetry of the orbitals of to analyze the symmetry of the orbitals of the non-central atoms, and then combine the non-central atoms, and then combine them with the appropriate orbitals of the them with the appropriate orbitals of the central atom.central atom.

MOs for Larger MOs for Larger MoleculesMolecules

The orbitals of the non-central The orbitals of the non-central atoms are called atoms are called group orbitalsgroup orbitals. In . In considering a simple example, Hconsidering a simple example, H22O, O, we obtain group orbitals using the we obtain group orbitals using the two two 1s1s orbitals on the hydrogen orbitals on the hydrogen atoms.atoms.

Group Orbitals of WaterGroup Orbitals of Water

The AThe A11 representation has both representation has both 1s1s orbitals with positive wave orbitals with positive wave functions: Hfunctions: Haa+H+Hbb..

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

Molecular Orbitals of Molecular Orbitals of WaterWater

Since the Since the 2p2pyy orbital on oxygen orbital on oxygen doesn’t match the symmetry of the doesn’t match the symmetry of the group orbitals of hydrogen, it will group orbitals of hydrogen, it will remain non-bonding. The other remain non-bonding. The other orbitals on oxygen will combine with orbitals on oxygen will combine with the appropriate group orbitals to the appropriate group orbitals to form bonding and antibonding 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+

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-

Oh σ bonding

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

t2g

eg

t2g

ML6

-onlyML6

+ π

Stabilization

(empty π-orbitals on ligands)

o’o o has increased

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

t2g (π)

t2g (π*)

eg

t2g

eg

t2g

ML6

-onlyML6

+ 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

oo 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

molecular orbital theory lecture

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