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

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)

ΨMOL = c1 φ1 + c2 φ2 + c3 φ3 + cn φn

Where Ψ2 = spatial distribution of electrons

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

66 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 H-H bonding atomic orbital molecular orbital

67 Building Molecular Orbitals from Atomic Orbitals

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

node

H-H antibonding molecular orbital

68 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

Eσ* > Eσ -due to electron repulsion

Called the bond dissociation energy (BDE)

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

X Y X Y X Y X Y

Homolytic bond cleavage Heterolytic bond cleavage 70 Molecular Orbital Theory 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 2 as the CN is a measure of the electron density in the neighborhood of the atom in question

2 By normalization, for each MO ΣCN = 1

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

2 Therefore C1 = C2 and C1 = 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

2 Therefore ΣCN = 1 (for each atom)

For H2: 2 c1 c2 ΣC (for orbital)

σ 0.707 0.707 1 * σ 0.707 -0.707 1 71 2 ΣC (for atom) 1 1 Molecular Orbital Theory

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

2p sp3 hybridization energy 2s

1s 1s 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

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

Each bond is a result of Where are the electrons two electrons being shared H H 3 located and what orbitals between sp hybridized C C H H H H are being used? carbon and hydrogen H H

Valence Bond Theory Molecular Orbital Theory

73 Molecular Orbital Theory 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

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

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

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 Molecular Orbital Theory

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

Valence Bond Theory Molecular Orbital Theory

Csp3-H MO with 1 node bond energy MO with 0 nodes

Inner shell C 1s Inner shell C 1s

Each sp3 hybridized orbital would thus form a The bonding MOs for methane would not bond with the 1s orbital from each hydrogen to be of identical energy form four identical energy C-H bonds

How to know which model is correct if either?

75 Molecular Orbital Theory

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 Molecular Orbital Theory

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 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

77 Chem. Phys. Lett. (1968), 613-615 Molecular Orbital Theory

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

Eσ*

Eσ

Remember also that Eσ* > Eσ -due to electron repulsion 78 Building Molecular Orbitals from Atomic Orbitals

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 When two p orbitals interact, if lobe with symmetry of orbital the direction of approach same phase is pointing toward each other a is irrelevant bonding region can occur

Overlap between orbitals of same phase leads to bonding region Bonding MO (called σ bond) 79 Building Molecular Orbitals from Atomic Orbitals 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 80 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)

The molecular orbital the is unoccupied that is lowest in energy Hypothetical molecule that contains is called the LUMO 6 molecular orbitals and 6 electrons Unoccupied molecular orbitals (UMOs) Would fill the orbitals by following Pauli exlusion (only 2 electrons per orbital) and filling the lowest Occupied molecular orbitals energy orbitals first (OMOs)

The orbitals are classified by In addition the molecular orbital whether they are “filled” or that is occupied that is highest in “unfilled” energy is called the HOMO 81 Building New Molecular Orbitals from Molecular Orbitals

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)

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 Compound A Compound B energy due to the two higher energy electrons outweighing any energy gain The number of OMOs and UMOs and energy placement of orbitals is dependent upon the There is only an energy gain when a OMO of compound one molecule interacts with an UMO on the other molecule 82 Building New Molecular Orbitals from Molecular Orbitals

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)

ΔE ΔE

If the OMO is identical in energy to the As the OMO has a greater difference in UMO there will be the maximum energy gain energy to the UMO, the mixing will be less due to the best possible mixing of the orbitals and the energy gain will thus be lower

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 83 Building New Molecular Orbitals from Molecular Orbitals

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)

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) Compound A Compound B 84 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 Frontier Molecular Orbital Theory

We can compare the placement of HOMO and LUMO levels relative to placement of C-H bonds Very high LUMO, therefore poor electrophile Lone pair of electrons placed in σ*C-H atomic orbital No electrons in atomic orbital, therefore very electrophilic Because nitrogen is more electronegative than carbon, orbital is lower in energy sp3 1s H+ C H (likewise oxygen is lower than

nitrogen) :NH3 Both are very nucleophilic, :OH2 ammonia more than water σC-H Very low HOMO, Factors that can adjust MO energy levels: therefore poor nucleophile 1) Unmixed valence shell atomic orbitals A sp3 hybridized carbon atom and a 1s orbital of hydrogen have similar energy levels and strong overlap, therefore high mixing 86 Frontier Molecular Orbital Theory

We can compare the placement of HOMO and LUMO levels relative to placement of C-H bonds Very high LUMO, therefore poor electrophile Negative charge will raise the energy of orbital, σ*C-H therefore make compound more nucleophilic

CH3 sp3 1s H+ C H OH

:NH3

:OH2

σC-H Very low HOMO, Factors that can adjust MO energy levels: therefore poor nucleophile 1) Unmixed valence shell atomic orbitals A sp3 hybridized carbon atom and a 1s orbital of hydrogen have similar 2) Electric charge energy levels and strong overlap, therefore high mixing 87 Frontier Molecular Orbital Theory

We can compare the placement of HOMO and LUMO levels relative to placement of C-H bonds Very high LUMO, therefore poor electrophile The degree of mixing of two orbitals is related to the amount σ*C-H of overlap between the orbitals

π*C-C This makes HOMO

into a good nucleophile 3 2p C 2p C sp C 1s H

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

σC-H hybridized orbital and the amount of overlap is less Very low HOMO, Factors that can adjust MO energy levels: therefore poor nucleophile 1) Unmixed valence shell atomic orbitals A sp3 hybridized carbon atom and a 1s orbital of hydrogen have similar 2) Electric charge energy levels and strong overlap, 3) Poor overlap of atomic orbitals therefore high mixing 88 Frontier Molecular Orbital Theory

We can compare the placement of HOMO and LUMO levels relative to placement of C-H bonds Very high LUMO, therefore poor electrophile Since the oxygen 2p orbital is σ* C-H much lower in energy, the This makes LUMO energy match with carbon 2p is into a good worse and therefore less mixing electrophile 2p C π*C-O 3 sp C 1s H

2p O

πC-O

σC-H Very low HOMO, Factors that can adjust MO energy levels: therefore poor nucleophile 1) Unmixed valence shell atomic orbitals A sp3 hybridized carbon atom and a 1s orbital of hydrogen have similar 2) Electric charge energy levels and strong overlap, 3) Poor overlap of atomic orbitals therefore high mixing 4) Poor energy match of orbitals 89 Frontier Molecular Orbital Theory

We can compare the placement of HOMO and LUMO levels relative to placement of C-H bonds Very high LUMO, Can also use orbital energy levels to understand therefore poor electrophile differences in reactivity for C-X bonds σ*C-H A C-Cl bond is good σ*C-Mg

electrophile spMg ο*C-Cl 3 3 3 sp C 1s H sp C sp C σC-Mg

3 A C-Mg bond is sp Cl good nucleophile σC-Cl σC-H Very low HOMO, Factors that can adjust MO energy levels: therefore poor nucleophile 1) Unmixed valence shell atomic orbitals A sp3 hybridized carbon atom and a 1s orbital of hydrogen have similar 2) Electric charge energy levels and strong overlap, 3) Poor overlap of atomic orbitals therefore high mixing 4) Poor energy match of orbitals 90 Frontier Molecular Orbital Theory

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 Anything that will raise CH3 > NH2 > OH energy level of HOMO will increase nucleophilicity

A low LUMO level represents a compound that is a good electrophile R Anything that will lower C O H3C Cl energy level of LUMO will R increase electrophilicity

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 91 Frontier Molecular Orbital Theory

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)

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

LUMO of

formaldehyde 92 Frontier Molecular Orbital Theory What direction should a nucleophile approach the carbonyl?

NUC NUC Direction appears better, Optimal interaction but still not optimal interaction (best overlap of interacting orbitals)

NUC Expect this direction to be highly disfavored due to orthogonal interaction with 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!

93 Frontier Molecular Orbital Theory What direction should a nucleophile approach the carbonyl?

Studied a variety of X-ray NUC structures where a N reacts with a carbonyl Optimal interaction α (best overlap of intramolecularly interacting orbitals) As the N came closer to carbonyl, the C-O bond lengthened and the carbonyl carbon becomes pyramidalized

Called the “Bürgi-Dunitz” angle The angle of

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

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 Largest coefficient is on the backside of the carbon Nucleophile thus reacts with a NUC methyl halide in a SN2 reaction with backside attack So called “inversion of configuration” LUMO of methyl halide Bonds that break base Base thus reacts by abstracting hydrogen anticoplanar to The base will abstract the leaving group and form new π hydrogen that is anticoplanar bond in E2 reaction to leaving group New π bond

LUMO of 2˚ alkyl halide 95