Molecular Orbital Theory Valence Bond Theory: …biewerm/3-MO theory.pdfMolecular Orbital Theory!...

Molecular Orbital Theory

Valence Bond Theory: Electrons are located in discrete pairs between specific atoms

Molecular Orbital Theory: Electrons are located in the molecule, not held in discrete regions between two bonded atoms

φ1 φ2 φ3 φn ΨMOL c1 c2 c3 cn = + + +

Thus the main difference between these theories is where the electrons are located, in valence bond theory we predict the electrons are always held between two bonded atoms

and in molecular orbital theory the electrons are merely held “somewhere” in molecule

Mathematically can represent molecule by a linear combination of atomic orbitals (LCAO)

Where Ψ2 = spatial distribution of electrons

If the ΨMOL can be determined, then where the electrons are located can also be determined

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Building Molecular Orbitals from Atomic Orbitals

Similar to a wave function that can describe the regions of space where electrons reside on time average for an atom, when two (or more) atoms react to form new bonds, the region

where the electrons reside in the new molecule are described by a new wave function

This new wave function describes molecular orbitals instead of atomic orbitals

Mathematically, these new molecular orbitals are simply a combination of the atomic wave functions (e.g LCAO)

Hydrogen 1s atomic orbital

H-H bonding molecular orbital

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An important consideration, however, is that the number of wave functions (molecular orbitals) resulting from the mixing process must equal the number of

wave functions (atomic orbitals) used in the mixing

In the case of H2, in addition to the new bonding molecular orbital obtained by adding the two atomic 1s orbitals, an antibonding orbital is obtained by subtracting the two atomic orbitals

H-H antibonding molecular orbital

node

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Electronic Configuration for H2

Each Hydrogen 1s atomic orbital has one electron When two atomic orbitals mix, they produce two molecular orbitals

As the number of nodes increases, the energy of the orbital increases

The molecule has a total of two electrons and follow Aufbau principle and Pauli principle to fill electrons in molecule 69

Bond Strength

The bond strength for H2 is considered the amount of energy required to break the bond and produce two hydrogen atoms

Called the bond dissociation energy (BDE)

Homolytic bond cleavage

X Y X Y

Heterolytic bond cleavage

X Y X Y

Eσ* > Eσ

-due to electron repulsion

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The σ and σ* orbitals can be written mathematically thus as a combination of atomic orbitals Ψσ = c1φ1 + c2φ2

Ψσ* = c1φ1 - c2φ2

The size of coefficients (c1 and c2) is related to the electron density as the CN

2 is a measure of the electron density in the neighborhood of the atom in question

By normalization, for each MO ΣCN2 = 1

Thus for the only filled orbital in H2, because the molecule is symmetric |C1| = |C2|

Therefore C1 = C2 and C12 = 1/2

C1 = C2 = 1/√2 = 0.707

Also if all the MOs are filled, there must be one electron in each spin state on each atom

Therefore ΣCN2 = 1 (for each atom)

c1 For H2: c2

σ σ*

0.707 0.707 0.707 -0.707

ΣC2 (for orbital)

ΣC2 (for atom)

1 1

1 1 71


The electron location in H2 is identical between valence bond theory and molecular orbital theory (due to there only being one bond in H2 and thus the electrons must be located on the two atoms)

What happens however if there is more than one bond in the molecule, how do the bonding theories differ in describing the location of electrons?

Consider methane

Valence bond theory predicts four identical C-H bonds in methane formed by the carbon hybridizing to an sp3 hybridization

energy

1s

2s

2p hybridization

1s

sp3

Each sp3 hybridized orbital would thus form a bond with the 1s orbital from each hydrogen to

form four identical energy C-H bonds 72


Molecular orbital theory would not use the concept of hybridization (hybridization is entirely a concept developed with valence bond theory)

Instead of hybridizing the atomic orbitals first before forming bonds, molecular orbital theory would instead treat the molecular orbitals used to form the bonds

as a result of mixing the atomic orbitals themselves

For methane thus would have 4 1s orbitals from each hydrogen and four second shell orbitals from the carbon atom (2s, 2px, 2py, 2pz)

The 8 valence electrons would need to placed in bonds formed from the combination of these atomic orbitals

CH

H HH C

H

H HH

Valence Bond Theory Molecular Orbital Theory

Each bond is a result of two electrons being shared

between sp3 hybridized carbon and hydrogen

Where are the electrons located and what orbitals

are being used?

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To visualize where the electrons are located and what molecular orbitals the electrons are located, consider the four hydrogen 1s orbitals and the four outer shell orbitals of carbon

Then mix the outer shell atomic orbitals to find the bonding patterns

H 1s C 2s C 2px C 2py C 2pz

0 nodes 1 node 1 node 1 node Molecular orbital theory predicts there are 4 bonding MOs, 1 with 0 nodes and 3 with 1 node

(therefore they must be at different energy levels if different number of nodes!) 74


The bonding pattern in methane is thus different using either valence bond or molecular orbital theory

Inner shell C 1s

Csp3-H bond

Each sp3 hybridized orbital would thus form a bond with the 1s orbital from each hydrogen to

form four identical energy C-H bonds

energy

Inner shell C 1s

MO with 0 nodes

MO with 1 node

The bonding MOs for methane would not be of identical energy

Molecular Orbital Theory Valence Bond Theory

How to know which model is correct if either?

75


Can first compare what orbitals look like computationally (obtained with Spartan 08, DFT B3LYP 6-31G*)

0 nodes 1 node 1 node 1 node

Orbitals obtained computationally are identical to those qualitatively determined

If the computer theory was established using molecular orbital theory, it is not surprising the obtained MOs resemble the qualitative picture 76


Is there an experimental method to test which bonding theory matches reality?

Can use photoelectron spectroscopy (PES) and electron spectroscopy for chemical analysis (ESCA) which measure the ionization potential of electrons expelled from orbitals

Difference between PES (<~20 eV) and ESCA (>~20 eV) is ionization potential range

In short for these experiments the gas phase sample of compound under analysis is irradiated and the binding energy for an electron can be calculated by knowing the energy of ionizing

irradiation and subtracting the kinetic energy of the detected emitted electrons

Chem. Phys. Lett. (1968), 613-615

Inner shell electrons Bonded electrons have 2 different

energy levels The experiment confirms the MO description of bonding

Methane does have two different energy levels for the four C-H bonds

Even though valence bond theory is not correct, it is still widely used by organic chemists as a guide to predict reactions

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Using molecular orbital theory therefore the electrons are located in regions of space on time average (orbitals) that are formed by the mixing of atomic orbitals

Have already seen a simple orbital description with forming H2 MOs from mixing atomic 1s orbitals from each hydrogen atom

Remember also that Eσ* > Eσ

-due to electron repulsion

Eσ

Eσ*

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When forming molecular orbitals of H2 from atomic orbitals from each hydrogen atom, hydrogen only has one electron in a 1s orbital (or a 1,0,0 orbital using n,l,m designation)

When building molecular orbitals from a 2nd row atom (like carbon) can use either the 2s (2,0,0) or 2p (2,1,-1; 2,1,0; or 2,1,1) atomic orbitals to form the bonds

When different orbitals interact, the overlap of the orbitals changes depending upon the direction of bond formation (both in degree of overlap and symmetry of the bond)

When two s orbitals interact, due to symmetry of orbital the direction of approach

is irrelevant

Overlap between orbitals of same phase leads to bonding region

Bonding MO (called σ bond)

When two p orbitals interact, if lobe with same phase is pointing toward each other a

bonding region can occur

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Due to the unsymmetrical orientation of a p orbital, however, there are other possible orientations of approach

One bonding approach direction has the p orbitals on both atoms directed opposite to the approach direction

Still a bonding MO, but electron density is not symmetric about

internuclear axis (called π bond)

If s orbital approaches p orbital from side, however, there is no net overlap

The positive overlap (blue with blue) is exactly canceled with the negative overlap (blue with red),

thus there is no net overlap

The orbitals are said to be “orthogonal” to each other and thus do not mix

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Building New Molecular Orbitals from Molecular Orbitals

In addition to building new molecular orbitals from adding atomic orbitals, new molecular orbitals can result from combining orbitals from two different molecules

using their molecular orbitals (the result of a reaction between two molecules)

For a given molecule there might be a multitude of molecular orbitals (the total number are due to the number of atoms in the molecule)

Hypothetical molecule that contains 6 molecular orbitals and 6 electrons

Would fill the orbitals by following Pauli exlusion (only 2 electrons per

orbital) and filling the lowest energy orbitals first

The orbitals are classified by whether they are “filled” or

“unfilled”

Occupied molecular orbitals (OMOs)

Unoccupied molecular orbitals (UMOs)

In addition the molecular orbital that is occupied that is highest in

energy is called the HOMO

The molecular orbital the is unoccupied that is lowest in energy

is called the LUMO

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When two molecules react (each with their own set of molecular orbitals) it is important to recognize which molecular orbital interaction determines the reaction

(if a chemist knows this then they can predict reactions)

Compound A Compound B

The number of OMOs and UMOs and energy placement of orbitals is dependent upon the

compound

Each molecular orbital in compound A however will react with each molecular

orbital in compound B

Whenever any two nonorthogonal orbitals interact they will create two new MOs, one higher in energy and one lower in energy

When two UMOs react, there is no change in energy as there are no electrons in the orbitals

When two OMOs react, there is an increase in energy due to the two higher energy electrons

outweighing any energy gain

There is only an energy gain when a OMO of one molecule interacts with an UMO on the

other molecule 82


The amount of energy gain is also dependent upon how close in energy the two orbitals are before mixing

Consider mixing of two orbitals, one filled (OMO) and one unfilled (UMO)

If the OMO is identical in energy to the UMO there will be the maximum energy gain due to the best possible mixing of the orbitals

ΔE

As the OMO has a greater difference in energy to the UMO, the mixing will be less

and the energy gain will thus be lower

ΔE

Thus the energy gain is greatest in a reaction when the HOMO of one compound is closest in energy to the LUMO of the second compound

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When mixing any two orbitals therefore the two important considerations are the overlap between the two orbitals and the match in energy of the two orbitals before mixing

Considerations between mixing of orbitals are therefore: -when two nonorthogonal orbitals overlap and mix, they generate two new orbitals

(one higher in energy and one lower in energy) -the amount of energy shift upon mixing is greater with more overlap of the orbitals and

lower the further apart in energy the orbitals are before mixing -average energy of two new orbitals is slightly higher than average of original orbitals

(partly an artifact of electron-electron repulsion in higher energy orbital)

Compound A Compound B

Consider the original hypothetical compound A reacting with compound B

The most important interaction to consider is the HOMO of A reacting with the LUMO of B

(largest energy gain) The energy gain from this interaction must be large

enough to overcome the energy loss of each OMO mixing with another OMO (which causes an energy loss)

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Frontier Molecular Orbital Theory

Since the majority of energy gain in a reaction between two molecules is a result of the HOMO of one molecule reacting with the LUMO of a second molecule this interaction is

called a Frontier Molecular Orbital (FMO) interaction

A reaction is thus favored when the HOMO (nucleophile) is unusually high in energy and the LUMO (electrophile) is unusually low in energy

What does unusually high HOMO or unusually low LUMO mean?

Must be compared relative to something -usually compare energy levels with a known unreactive C-H (or C-C) single bond

If the HOMO of a new compound is higher in energy than the HOMO of the C-H bond, then it will be more reactive as a nucleophile

If the LUMO of a new compound is lower in energy than the LUMO of the C-H bond, then it will be more reactive as an electrophile

How much higher or lower in energy will determine the relative rates of reactions 85


We can compare the placement of HOMO and LUMO levels relative to placement of C-H bonds

σC-H

σ*C-H

sp3C 1s H

A sp3 hybridized carbon atom and a 1s orbital of hydrogen have similar energy levels and strong overlap,

therefore high mixing

Factors that can adjust MO energy levels: 1) Unmixed valence shell atomic orbitals

H+

No electrons in atomic orbital, therefore very electrophilic

:NH3

:OH2

Lone pair of electrons placed in atomic orbital

Because nitrogen is more electronegative than carbon,

orbital is lower in energy (likewise oxygen is lower than

nitrogen)

Both are very nucleophilic, ammonia more than water

Very low HOMO, therefore poor nucleophile

Very high LUMO, therefore poor electrophile

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σC-H

σ*C-H

sp3C 1s H




H+

:NH3

:OH2

2) Electric charge

OH

CH3

Negative charge will raise the energy of orbital, therefore make compound more nucleophilic



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σC-H

σ*C-H

sp3C 1s H




The degree of mixing of two orbitals is related to the amount of overlap between the orbitals

When two p orbitals overlap to form a π bond, the orbitals begin higher in energy than a

hybridized orbital and the amount of overlap is less

2) Electric charge



3) Poor overlap of atomic orbitals

2p C 2p C

πC-C

π*C-C This makes HOMO into a good nucleophile

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σC-H

σ*C-H

sp3C 1s H




2) Electric charge



3) Poor overlap of atomic orbitals 4) Poor energy match of orbitals

πC-O

π*C-O 2p C

2p O

Since the oxygen 2p orbital is much lower in energy, the

energy match with carbon 2p is worse and therefore less mixing

This makes LUMO into a good electrophile

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Frontier Molecular Orbital Theory Frontier Molecular Orbital Theory


σC-H

σ*C-H

sp3C 1s H




2) Electric charge



3) Poor overlap of atomic orbitals

sp3C

sp3Cl

σC-Cl

ο*C-Cl

A C-Cl bond is good electrophile

4) Poor energy match of orbitals

σC-Mg

σ*C-Mg

sp3C

spMg

A C-Mg bond is good nucleophile

Can also use orbital energy levels to understand differences in reactivity for C-X bonds

90


Frontier molecular orbital (FMO) theory allows a chemist to make predictions about a reaction by knowing the placement of the HOMO and LUMO energy levels

A high HOMO level represents a compound that is a good nucleophile

A low LUMO level represents a compound that is a good electrophile

The energy level of the HOMO and LUMO can be predicted by knowing that when two atomic orbitals mix they form two new molecular orbitals,

one lower in energy and one higher in energy The amount of mixing is dependent upon:

1)  The amount of overlap between the mixing orbitals (e.g., the overlap for a σ bond is greater than the overlap for a π bond)

2) The closer in energy are two orbitals, the greater the amount of mixing that occurs

OHCH3 NH2> >

C OR

RH3C Cl

Anything that will raise energy level of HOMO will

increase nucleophilicity

Anything that will lower energy level of LUMO will

increase electrophilicity

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FMO will also allow prediction about where a reaction will occur (regiochemistry) and direction of approach (stereochemistry)

Consider a reaction with a carbonyl compound

FMO predicts that a carbonyl should react as an electrophile due to the low energy LUMO

The regio- and stereochemistry can also be predicted by considering the interacting frontier orbital (the LUMO)

rotate

LUMO of formaldehyde

The coefficient on carbon is larger than the coefficient on oxygen, therefore nucleophile

reacts at carbon

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What direction should a nucleophile approach the carbonyl?

NUC

NUC

Expect this direction to be highly disfavored due to orthogonal

interaction with orbitals

Direction appears better, but still not optimal interaction

NUC

Optimal interaction (best overlap of

interacting orbitals)

Could there possibly be a method to test the angle of approach of nucleophile to carbonyl?

X-ray structures come to the solution once again!

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What direction should a nucleophile approach the carbonyl?

NUC

Optimal interaction (best overlap of

interacting orbitals)

Could there possibly be a method to test the angle of approach of nucleophile to carbonyl? X-ray structures come to the solution once again!

Bürgi, H.B., Dunitz, J.D., Shefter,E., J. Am. Chem. Soc. (1973), 95, 5065-5067

Studied a variety of X-ray structures where a N

reacts with a carbonyl intramolecularly

As the N came closer to carbonyl, the C-O bond

lengthened and the carbonyl carbon becomes

pyramidalized

The angle of <N-C-O averaged 107˚ (α)

Called the “Bürgi-Dunitz” angle

α

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What about the stereochemistry for a reaction with an alkyl halide?

Since alkyl halide is reacting as the electrophile, need to look at the LUMO

LUMO of methyl halide

Largest coefficient is on the backside of the carbon

NUC

So called “inversion of configuration”

Nucleophile thus reacts with a methyl halide in a SN2 reaction

with backside attack

LUMO of 2˚ alkyl halide

base Bonds that break

New π bond

The base will abstract the hydrogen that is anticoplanar

to leaving group

Base thus reacts by abstracting hydrogen anticoplanar to

leaving group and form new π bond in E2 reaction

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Molecular Orbital Theory Valence Bond Theory: …biewerm/3-MO theory.pdfMolecular Orbital Theory!...

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