Frontier Molecular Orbital Theory_Handout 1_Compact

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Chemistry 261 2009

Frontier Molecular Orbital Theory

Handout #1

Prepared by John Taylor

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Frontier Molecular Orbital (FMO) Theory

Most organic chemists use a very simplified version of molecular

orbital theory to help understand and predict chemical

interactions and reactions.

FMO theory is concerned with the interaction between the frontier molecular orbitals, that is, the highest occupied molecular orbital

(HOMO) of one molecule (or part of a molecule) and the lowest

unoccupied molecular orbital (LUMO) of another.

HOMOs are electron-rich (nucleophilic, Lewis basic) and interact

strongly with LUMOs which are electron deficient (electrophilic,

Lewis acidic).

To understand and apply FMO theory, we first must understand

some basic principles of MO theory. The following pages recap

what was presented in class and provide problems to help you

better understand MOs, hybridization, and frontier molecular

orbital interactions.

In the following orbital interaction diagrams we explore what

happens when we bring atomic, hybridized, or bonding and

antibonding orbitals together within bonding distance. In MO

theory when we interact n atomic orbitals, we must generate n

molecular orbitals. When we interact 2 orbitals, we get two MOs

one of which goes down in energy (constructive interference, the

bonding orbital), and one goes up in energy (destructive

interference, the antibonding orbital).



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Orbital Energy Level Diagrams

Before embarking on orbital interaction diagrams used in the construction of

molecular orbitals and discussion of frontier molecular orbital interactions we

need to review some of the basics of energy level diagrams. Consider the

energy level diagram below. On the left is an energy axis, with increasing

energy in the direction of the arrow. On the diagram are horizontal lines

representing energy levels. In this particular energy level diagram, each

horizontal line represents the energy level of an orbital, such as an atomic

orbital, a hybridized orbital or a molecular orbital. Zero, one or two electronscan be in one such orbital at a time, which we represent with an arrow

indicating the spin state (+ ½ or – ½). If two electrons are in one orbital at the

same time, they have to be spin paired (i.e., have opposite spins). Electrons

can be placed in a set of orbitals any number of ways, so long as they meet

these rules (an electron configuration). If there are many orbitals, and only a

few electrons there are many electron configurations possible. The total

energy of a particular electronic configuration will be the sum of the energy

levels of the individual electrons plus a small additional amount of positive

energy if the spins are paired. If two orbitals have the same energy level,

they are said to be degenerate. In this case, the energy of the system will be

lower (more favorable) if the two electrons are added unpaired to separate

orbitals.

E(au)

Degenerate

orbitals

Oxygen

-.6

-1.3

-21

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4

2nd Row Atomic Orbital Energies

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

Li Be B C N O F

Element

E

( a . u . ) 1s

2s

2p

3rd Row Atomic Oribital Energies

-2

-2

-2

-1

-1

-1

-1

-1

0

0

0

Na Mg Al Si P S Cl

Element

E (

a . u . )

1s

2s

2p

3s

3p

Valence Shell Orbital Energies

H = -0.5 a.u.



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Orbital Energies (a.u.)

Main group elements:H He

1s -0.50 -0.92Li Be B C N O F Ne

1s -2.5 -4.7 -7.7 -11.3 -15.7 -20.7 -26.4 -32.8

2s -0.20 -0.31 -0.49 -0.71 -0.96 -1.25 -1.57 -1.93

2p -0.31 -0.41 -0.51 -0.62 -0.73 -0.85

Na Mg Al Si P S Cl Ar

1s -40 -49 -59 -69 -80 -92 -105 -119

2s -2.8 -3.8 -4.9 -6.2 -7.5 -9.0 -10.6 -12.3

2p -1.5 -2.3 -3.2 -4.3 -5.4 -6.7 -8.1 -9.57

3s -0.18 -0.25 -0.39 -0.54 -0.71 -0.88 -1.07 -1.28

3p -0.21 -0.28 -0.35 -0.43 -0.51 -0.59

K Ca Ga Ge As Se Br Kr

1s -134 -149 -379 -405 -433 -461 -490 -520

2s -14 -17 -48 -52 -56 -61 -65 -702p -12 -14 -42 -46 -50 -54 -59 -63

3s -1.7 -2.2 -6.4 -7.2 -8.0 -8.9 -9.9 -11

3p -1.0 -1.3 -4.5 -5.2 -5.9 -6.7 -7.5 -8.3

3d -1.2 -1.6 -2.1 -2.7 -3.2 -3.8

4s -0.15 -0.20 -0.42 -0.56 -0.70 -0.84 -0.99 -1.2

4p -0.21 -0.27 -0.33 -0.39 -0.46 -0.52

Adapted from: Joseph B. Mann, "Atomic Structure Calculations I. Hartree-Fock Energy Results for the

Report LA-3690 (Los Alamos National Laboratory,

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Orbital interaction diagrams.

When dealing with single atoms, the possible energy

levels for the electrons correspond to the atomic orbitals.

When atoms interact to form molecules, n (some number)

of their atomic orbitals interact (overlap) to form n (same

number) molecular orbitals. Insight into the nature of

these molecular orbitals and their origin comes fromorbital interaction diagrams in which the results of

interacting (overlapping) selected atomic orbitals is

diagrammed. We will be primarily interested in two types

of molecular orbital interactions, sigma (σ) type, and pi (π)type. The sigma type results from the interaction of s or p

type orbitals (or hybrids) aligned along the interatomic axis

(orbital density along bond axis), whereas the pi type

results from interaction of p-type orbitals oriented

perpendicular to the interatomic axis (node along bondaxis).



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Let us first consider what happens to the orbital energies

when the s orbitals of two hydrogen atoms approach each

other from far away. We chose the vertical axis to

represent the energy of the system, and the horizontal axis

(along the x-coordinate) to represent distance. At infinite

distance there is no interaction between the orbitals, and

the energy levels correspond to that of two isolated sorbitals, and are therefore degenerate. As the two

hydrogen atoms approach each other, however, the

energy levels split. One energy level goes down as the

result of a favorable interaction between the s orbitals

which have the same phase (positive overlap, constructive

interference) indicated by the same color. This

constructive interference represents a bonding molecular

orbital. On the other hand, the other energy level goes up

as a result of an unfavorable interaction between orbitals of different phases (plus and minus, zero overlap, destructive

interference) indicated by different colors and a node

(change in phase) occurs between the two lobes of the

orbital. This destructive interference represents an

antibonding molecular orbital. The bonding combination is

a sigma type, as there is orbital density along the bond

axis that is forming. Interestingly, when the two nucleii are

on top of each other the two molecular orbitals turn into the

atomic orbitals of helium. The bonding combination intothe 1s orbital of He, and the antibonding combination into

the 2px orbital.

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2 H atoms

far apart

H2 molecule

at bonding

distance

He atom

2px orbital

He atom

1s orbital

2 1s orbitals

bonding MO

(1s + 1s)

Antibonding

MO (1s – 1s)

E n e r

g y

When one brings together two hydrogen atoms together, two molecular

orbitals form that eventually turn into Helium atomic orbitals. One goes

up and the other down.

node

node

r 0

r = internuclear

separation



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At first glance one might expect that the two hydrogen atoms

should simply merge together to form a helium atom as the

energy of the bonding orbital keeps on decreasing, and

indeed this would be a favorable process (nuclear fusion)

except that there is a tremendous barrier to doing so

resulting from the huge coulombic repulsion from bring thetwo positively charged protons close together. When one

adds all the energy terms together, one finds that there is a

low energy intermediate corresponding to a hydrogen

molecule in which the two hydrogen nucleii are separated by

a “bond length”.

When we construct orbital energy interaction diagrams, we

will generally construct them at a bonding distance. For

frontier molecular orbital interaction diagrams (later), we willbe constructing them at an interatomic distance in a

transition state structure.

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2 H atoms

far apart

2 1s orbitals

bonding MO

(1s + 1s)

Antibonding

MO (1s – 1s)

E n e r

g y

The total energy of the system goes way up when the

two atoms get closer than the bond length due to

nuclear nuclear repulsion (black dot = nucleus).

r

r = internuclear

separation

Equilibrium

Bond length



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Basic MO Interaction Diagram for Orbitals

of Equal Energy. Example, H2

1. 2 orbitals interact to give 2 molecular orbitals (MOs): a lower energy

bonding σ orbital, and a higher energy antibonding orbital σ*.

2. The MO are filled according to the Aufbau system. The highest filled

(occupied) orbital (HOMO) is the σ orbital, and the lowest unoccupied

(unfilled) orbital is the σ* orbital.

3. E* > E. This is because when filled, there will be unfavorable spin pairing

energy which raises the level of the σ* orbital more than the σ orbital isstabilized (according to I. Fleming)

4. The σ* orbital differs from the σ orbital by having an extra node.

5. Each atomic orbital contributes equally to the σ and σ* orbitals becausethe energy of the starting orbitals are equal.

6. The covalent bonding energy for H2 ΔEσ

= -(2*E) (2 e a lowered by ΔE)

H(1s)1 H(1s)1

σ *

σ ΔE

ΔE*LUMO

HOMO

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σ

σ*

Molecular Orbitals for H2



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Two filled orbitals will repel other at

bonding distances. For example He2

1. The interaction diagram is the same as for H2, except that 4 electrons

are involved.

2. Since only 2 electrons can be in any one orbital, both the σ and σ*orbitals become filled

3. The HOMO in this case is the σ* orbital. The LUMO in this case

would be the σ MO formed between the empty He2s orbitals (not

shown).

4. The energy of bringing two He together is unfavorable since ΔEtotal >0

because ΔEtotal = -2ΔE + 2ΔE* and ΔE*>ΔE.

5. Conclusion: Bringing together any two filled (i.e., 2 electron) orbitals

within bonding distance is unfavorable.

He(1s)2

σ *

σ ΔE

ΔE∗

LUMO

HOMO

He(1s)2

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S orbital overlap with S orbitals: Always favorable

S orbital overlap with P orbitals: Favorability depends

on p orbital orientation (x = internuclear axis).

positive

overlap

zero

overlap

zerooverlap

positive

overlap

Only some combinations of orbitals leadto favorable orbital overlap.

+

s s sigma

px

py

pz

sigma

Non-Bonding

Bonding



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P orbitals can combine to form sigma and pi type

bonds, but only if positive overlap can occur.

Shown below are the bonding and non-bonding

combinations.

Bonding Non-bonding

px

py

pz

py

pz

px

pzpy

px py

pzpx

sigma

pi

pi

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Molecular Orbital Interaction Diagram for Cl2Note: bonding in Cl2 is due solely to σpx, since σ2s bonding is cancelled

by σ*2s, and πy & πz bonding is cancelled by π*y and π*z. Thus the Cl-Cl

bond is a weak bond with a 58 kcal/mole bond dissociation energy, and

can easily be cleaved by heat and light.

2px 2py 2pz2px2py2pz

2s 2s

σ2s

σ*2s

σpx

σ*pz

πyπz

π∗yπ∗z



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MO pictures for Cl2

σ2s

σ∗px

σ∗2s

σpx

π∗z

πz πy

π∗y

Atom

centers

shown as balls.

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1. When one orbital is lower in energy (ΔE) than the other orbital (e.g.,

fluorine p < hydrogen 1s), Eσ does not decrease as much as when the

orbitals are of equal energy. Likewise Eσ* does not increase as much.

2. The σ MO looks more like the lower energy orbital, and the σ* looks morelike the higher energy orbital. In this case the s orbital looks mostly like

the fluorine p orbital, and the σ* looks more like the hydrogen 1s orbitalas indicated by the relative size of the contributing orbitals. This is also

emphasized by the horizontal placement of the s energy level near the

lower energy orbital, and the σ* energy near the higher energy orbital.

3. As a result of #2, the electron supplied by the atom with the higher

energy orbital resides more on the atom with the lower energy orbital and

takes on ionic character, which in this case corresponds to the H+ F-

resonance form.

4. The bonding energy of the 2 electrons = -2*ΔE - ΔEorbital.

MO Interaction diagram for sigmabond between orbitals of differentenergies (example, HF)

H(1s)1

F(2px)1

σ *

σ ΔE

ΔE*

node

ΔEorbital

-0.5 au

-0.7 au



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σ*s-px

nb

σs-px

F2s

Complete molecular orbitals for HF

2 lone pairs

in py and pz

orbitals

3rd lone pair

sigma bonding

orbital

sigma*

Antibonding

orbital

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MO interaction diagram for a pi bond

formed between orbitals of different

energy. Formaldehyde (H2C=O).

O(2pz)1

π*

πΔE

ΔE*

ΔEorbital

C(2pz)1

OH

HO

H

H

Electrondeficient CNucleophilewill attack here

A nucleophilewill attack here(Nu: = HOMO)

LUMO



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π*

π

top viewside view

4.0 ev

-14.7 ev

Pictures of π and π* MOs of formaldehyde

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FMO Theory.

For the most part we will be interested in molecular

orbitals that help explain particular phenomena, such as

absorption of light, bond breaking, stabilization of a

particular conformation, intermolecular interactions, and

reactions between molecules that lead to bond making.

As a result we will only focus on interactions betweenselected orbitals, and ignore the rest, as they are usually

not important.

As you recall the strongest interaction occurs between

orbitals are those that are closest in energy, and that the

energy lowering occurs when a filled or half-filled orbital

interacts with an empty or half-filled orbital. Interactions

between filled orbitals are destabilizing while interactions

between empty orbitals have no effect on the energy of the system. The pairs of orbitals that fulfill these criteria

are the frontier molecular orbitals, the highest occupied

(filled) orbitals (HOMOs), singly occupied MOs (SOMOs)

and lowest unoccupied MOs (LUMOs).

What type of orbitals generally make up HOMOs, LUMOs

and SOMOs? Taking the MO picture of a diatomic as a

guide, it would appear that HOMOs could be σ, π, or

filled non-bonding orbital (lone pair orbital), and LUMOscould be an empty p-type orbital, π*, or σ*. A SOMO

could be any half-filled orbital of these types.



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Interactions between the LUMO on one molecule

and the HOMO of another molecule leads to a

lowering of the energy. This can be used to

predict and explain where two molecules will

react.

Examples of LUMOs (electrophile, Lewis acid):

• empty p orbital (in BX3, CR3+)• σ* orbital (R-Halide)

• π* orbital (C=O, C=N, C=N)

Examples of HOMOs (nucleophile, Lewis base):• Lone pair orbital (O:, N:, S:)

• π orbital (C=C)

ΔE

LUMO

HOMO

ΔEorb

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Molecule 1 Molecule 2

Possible pairwise HOMO – LUMO

interactions between two molecules

σ

σ

π

π

nb

nb

π*π*

σ*

σ* p

p

LUMO

LUMO

HOMO

HOMO

Lower E

orbitalsLower E

orbitals

Because molecule 2 has lower energy orbitals, the

HOMOs of molecule 1 match better in energy with the

LUMOs of molecule 2 and lead to stronger interactions.

Which HOMO-LUMO interaction takes place depends

on what the HOMO and LUMO are. Some molecules

may not have a lone pair or pi bond, and in that case,

the HOMO will be a sigma bond.



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ΔE

LUMO

SOMO

ΔEorb

ΔE

SOMO

HOMO

ΔEorb

SOMO’s (singly occupied molecular orbitals) can

also interact with either a HOMO or LUMO and

lead to a lowering of the energy. This can be

used to predict and explain where two molecules

will react (Chapter 4).

Examples of SOMOs (nucleophilic or electrophilic)• Radicals (CR3

., Br .)

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Frontier Molecular Orbital Interaction:

Protonation.

1. Protonation can by understood by FMO theory.

2. The proton has no electrons in its 1s orbital and can only

function as a LUMO (Bronsted acid, Lewis acid, electrophile).

3. The oxygen has two lone pairs and one must function as the

HOMO (Bronsted base, Lewis Base, nucleophile).

4. When the HOMO and LUMO interact, there is a net lowering of

energy (bond formation).

5. The Lewis structure version is similar to the FMO picture. The

electron pair of oxygen becomes shared between the oxygenand hydrogen, but as the no bond resonance structure indicates,

the electron pair will reside on the oxygen to some degree due to

its electronegativity and the H will become positively charged.

H(1s)0

O(2sp3)2LUMO

HOMO

node

H

H

O

H

H

H

O

H

H

H

O

H H

O

H

H

H

HO O

O

O

H



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6. Identify the HOMO and LUMO in the following molecules (i.e., a lone

pair, pi bond, or sigma bond), and indicate the likely site of reaction on

a pi bond or sigma bond (site of largest possible positive orbital

overlap on HOMO or LUMO). Draw a pi or sigma resonance form

that would also predict the site of reaction (i.e., the most nucleophilic

or electrophilic site).

CH3 Cl C C CH3CH3

H2C

O

CH2

CH3

NCH3

CH3

H

O

NH

H

Br Br

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Study Problem Answers

1. Consider two atoms within bonding distance. Which pairs of p orbitals

can interact (i.e., can lead to positive orbital overlap)? Consider the

bond vector to lie along the x axis. Draw pictures. How do these

compare with π and π* molecular orbitals shown for Cl2?

px

py py

pzpz

px

px ± px yields the σ px and σ∗ px MOs of Cl2.

py ± py yields the πy and π*y MOs.

pz ± pz yields the πz and π*z MOs



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2. Two helium atoms can’t bond together, but can a H+ atom bond to He? YES, and

exists in space. Can H·? YES in principle based on molecular orbital

energies, but it is not a stable molecule. Draw the appropriate MO

interaction diagram to explain your answer. (He orbital is lower in energy

than the H orbital).

H+(1s)o

He(1s)2

σ *

σ

ΔE*

ΔEσ = -2 ΔE ΔE

H(1s)1

He(1s)

2

σ *

σ

ΔE*

ΔEσ = -ΔE ΔE

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3. See earlier part of handout and lecture notes.

4. (Harder) Lone pairs on water. One approach that works is to take linear

combinations of the two sp3 hybridized atomic orbitals to generate two molecular

orbitals as shown below. Another way is to recall that an sp3 orbital is 25% s +

75% p. The two sp3 orbitals used to make the lone pairs therefore are composed

of 50% s + 150% p. The p orbitals involved can be seen to be 100% pz

and 50%

px . Thus we could remix these orbitals to end up with a pure pz orbital (100% pz)

and an sp orbital (50% px

+ 50% s). See next page.

+

-- -

-

+++

+

+ +

+---

-

+

+

+

+

-

-

=

=

pz

sp

σ−type

lone pair

+

-

π-typelone pair

+-

H

H

H

H

sp3



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σ−typelone pair

+

-

π-typelone pair

+-H

H

H

H

sp3

xy

xz

orbital s px py pzsp3 25% 25% 0% 50%sp3 25% 25% 0% 50%total 50% 50% 0% 100%

σ lp (spx) 50% 50% 0% 0%

π lp (pz) 0% 0% 0% 100%

total 50% 50% 0% 100%

Individual sp3 don’t have

xy mirror plane symmetry

σ and p-type l.p. do have

xy mirror plane symmetry

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π lone pair -12.5 eV

HOMO

σ lone pair

-15 eVHomo -1

(one level

below

HOMO)

Side view

Top view

MOs for σ and π lone pairs

Note: sp orbitalmixes with both

H1s orbitals:

H1s

H1s

Osp



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What frontier molecular interaction would take place between BH3 and water OH2 (i.e., what

HOMO-LUMO pair is involved)? Draw a simple FMO interaction diagram to explain your

answer.

a) How would the diagram change if BF3 was used instead of BH3 (Hint: What might F do to

the orbital energies of B?).

One prediction would be that the orbital energy of the LUMO (vacant p) drops due to

withdrawal of electrons away from the boron by the fluorines, and because it will be

closer in energy to the HOMO of H2O, Eσ will be greater, and hence the complex will

be more stable. Alternatively, the fluorine lone pairs could interact with the vacant p

orbital, raising its energy, making the complex less stable.

b) Draw an electron pushing mechanism for the reaction, and the Lewis structures for the

products together with charges. (see below)

c) How does the FMO picture compare to the Lewis structural representation of the

interaction?

The no-bond resonance structure best represents the bonding orbital.

B(2px)

O(2sp3)2

σ *

σ Eσ

Eσ∗

ΔE

B

H

H H

O

H

H

B O

H

H

H

H

H

B

H

H

H

O

H

H

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More detailed discussion

B

H

H H

BH3 no possibility for pi bonding. H only has a 1 s orbital available for

bonding, the 2p orbital is too high in energy to form pi b onds.

B

F

F F

Fluorine is very electronegative and capable of withdrawing electronsfrom the boron through the C-F sigma bond, which would lower theenergy of the empty p orbital, and hence would make it interact better with the oxygen orbital (more Lewis acidic).

Fluorine also has lone pairs which can pi bond to the boron (as seenby resonance structures), that would make it less electrophilic, i.e.,less desirous of an electron pair as it now has a filled octet. This canalso be seen in an orbital interaction diagram, whereby the LUMO of BF3 is now a pi* orbital, of higher energy than the original p orbital.B

F

F F

F(2pz)1

π*

πEπ

Eπ∗

ΔE

B(2pz)1

The water lonepair will coordinatewith the larger orbitalon the B

LUMO



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BF3 π*1.76 ev

BH3 pz

1.6 ev

Pictures of the boron LUMO MOs

Note: All 3 fluorine p orbitals interact with the boron p orbital to generate 4

MOs. Only the π* is shown which looks like 3 π* B-F orbitals. It is the onlyone that is unfilled. If your are interested you can see what the phases of the

other 3 MOs should look like by using the simple MO calculator

(http://www.chem.ucalgary.ca/SHMO/).

H

H

H

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6. Identify the HOMO and LUMO in the following molecules (i.e., a lone

pair, pi bond, or sigma bond), and indicate the likely site of reaction

with the HOMO of a nucleophile (electron pair donor) or the LUMO of

an electrophile (electron pair acceptor). The site will that of largest

possible positive orbital overlap between the HOMO and LUMO.

Ans. See below. Recall the relative energies of filled orbitals

that could be potential HOMOs, σ < π < n.b. For empty orbitals

that could be potential LUMOs: empty p, π∗, σ*.

Draw a pi or sigma resonance form that would also predict the site of

reaction (i.e., the most nucleophilic or electrophilic site). This should

agree with your answer above. Try on your own.

C C CH3CH3

H2C

O

CH2H

O

NH

H

Br Br

HOMO - lone pair LUMO - σ* CN

HOMO - lone pair LUMO - σ* CCl

HOMO - π orbitalLUMO - π* orbital

HOMO - lone pair

LUMO - σ* CO

HOMO - lone pair

LUMO -σ

* Br-Br

HOMO - o xygen lone pair

(consider resonancestructure)

LUMO - π* orbital

CH3

NCH3

CH3

CH3 Cl

Nu

E

Nu

Nu

NuNu

E

E

E

E

N,E

Frontier Molecular Orbital Theory_Handout 1_Compact

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