L5-lecture4b

7/28/2019 L5-lecture4b

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Chemistry 250:Lecture 4

Bonding Goes Quantum

Valence Bond Theory

MO Theory


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-Lewis model (1916) fails to predict energy levels in CH4, or that O2 has 2unpaired electrons (attracted to magnetic field).

- 1925 A quantum description of hydrogen in terms of n and s,p,d orbitals

We need a new model of bonding. One that utilized what we learned from thequantum mechanical solution for the H-atom would be great.

We cant solve quantum mechanical multi-electron systems (all of chemistry!)

Lets approximate!

OO O Ooxygen atoms

6 electrons eachpair electronshappy octets


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Attempt 1: Valence Bond Theory

The description of the wavefunction as a product of hydrogen-like orbitals

correctly describes the lack of bonding at distance

The energy is the sum of the energy of the two individual atoms, Ea and Eb

Note: Valence bond theory is mostly taught for historical reasons. Still getsused in its most simple form

Start with the simplest covalent bond: H2


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Valence-Bond Theory

Wavefunction different from non-interacting because electrons cannotbe distinguished

Covalent bonding can be rationalized by the pairing of electrons invalence orbitals, and that the energy gained is from exchange

Note:a) because this is not antisymmetric with respect to electron exchange,

we should need a spin componentb) we could also add ionic components where both electrons are in oneorbital.... We won't


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Orbital Overlap (S) Criterion of Bond Strength

For H2

, VBT describes wavefunctionas a superposition of the 1s orbitals.

Qualitatively, the bonding interaction inVBT can be thought of in terms of

overlap integrals

Integral describes how well functionsoverlap. (e.g. an orbital integrated

against itself gives S=1)

A -bonds has orbital overlap thatdoes not change sign as we rotatearound the bond axis.

More overlap = stronger bond

= dSBA

d

is integral over all space


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Valence Bond Theory as a Back of the Napkin Approach

Valence bond theory treatment of bonding in H2 and F2 the way we will use it.

HA 1s1 HB 1s

1

A B

This gives a 1s-1s

bond

between the two H atoms.

All the methods we will discuss can be used to make quantitive predictions. Wewill just use them in a visual, intuitive manner.

F

2s 2p

F

2s 2p

2pz 2pz

Z axis

This gives a 2p-2p

bond between the

two F atoms.


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Orbital Overlap and -Bonding

A -bond changes signwhen rotated around thebond axis

There are symmetryrequirements to get S >0

S > 0 results in -bonding interaction

Net S = 0 implies no netbonding interaction


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Valence bond theory treatment of bonding in O2

This gives a 2p-2p bondbetween the two O atoms.O

2s 2p

O

2s 2p2pz 2pz

Z axis

Z axis

2py

2py

This gives a 2p-2p

bond between

the two O atoms. In VBT,

bonds

are predicted to be weaker than bonds because there is less overlap.VBT predicts that O2 is

diamagnetic not good!

(the choice of 2py is arbitrary)

O OLewis structure

Double bond:

bond +

bondTriple bond:

bond + 2

bond


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2s 2p

C

C*

Valence Bond Theory Treatment of CH4

11

4

1

4

1

4

1

4= + + +

s p p px y z

21

4

1

4

1

4

1

4= +

s p p px y z

31

4

1

4

1

4

1

4

= + s p p px y z

41

4

1

4

1

4

1

4= +

s p p px y z

This gives four sp3 orbitals that are orientedin a tetrahedral fashion.

sp3

C* (sp3)

+x,+y,+z

-x,-y,+z

-x,+y,-z

+x,-y,+z

Promotion

Energy2s2p

hybridize


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CH4 with sp3 hybrids:

Failure to predict Energy

Levels

Key assumption is that bonding is localized between twoatoms; this must be incorrect, because any model with

localized molecular orbitals will only have 1 ionizationenergy for the valence electrons transition

VBT predicts that CH4 shouldhave 1 ionization energy for

valence electrons not good!


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New Theory of Bonding; Molecular Orbital Theory

Lesson from O2:

Perhaps we need to consider overlap of all atomic orbitals in amolecule, not just ones with unpaired electrons?

Lesson from CH4:

In molecules larger than diatomics, these molecular orbitalscannot be localized between two atoms

Subtle point: This appears to contradict Lewis and VSEPR (electronslocalized in bonding pairs between atoms and non-bonding pairslocalized on atoms), but that depends on whether on not youbelieve there is any physical interpretation of the wavefunction (2is the electron density, an observable)


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New Approach: Molecular Orbitals

Valence bond theory took halfoccupied states and allowedsuperposition to form bonds

+ +++

++

_

++ _

1

2

In molecular orbital (MO)theory we will take linearcombinations of atomicorbitals (LCAO)

Next step is to fill theseMOs with electrons, similarto Aufbau principle used

with atomic orbitals


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Molecular Orbital Theory

Some basic rules for making MOs using the LCAO method:

1) Conservation of orbitals; n atomic orbitals must produce n molecularorbitals (e.g. 8 AOs must produce 8 MOs).

2) To combine, the AOs must be of the appropriate symmetry.

3) To combine, the atomic AOs must be of similar energy.

4) Each MO must be normal (integrate to 1 electron total):

5) Each MO must be orthogonal to every other MO.

021 = d

111 = d


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In terms of 3-dimensional

vectors this is like choosingunit vectors x, y and z vectorsto span 3-D space: Theprojection of one on the other

is 0.

Any other vector can beexpressed as combinations ofthe x, y and z vectors

In molecular orbital space,orbitals are orthogonal when

the overlap integral betweendifferent molecular orbitals iszero.

Orthogonality

x x xy

xy x y

021 = d


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This produces an MO over themolecule with a node betweenthe atoms (it is also symmetricalabout the H-H axis). This isknown as an antibonding MO

and is given the label u*because of its symmetry. Thestar indicates antibonding.


Diatomic molecules: The bonding in H2HA HB

Consider the addition of the two 1s functions (with the same phase):

1sA 1sB

+

This produces an MO around

both H atoms and has the samephase everywhere and issymmetrical about the H-H axis.

This is known as a bonding MO

and is given the label g

becauseof its symmetry.

Consider the subtraction of the two 1s functions (with the same phase):

1sA 1sB

-

Remember that: - +is equivalent to:

g =0.5 (1sA + 1sB)

u* =0.5 (1sA - 1sB)


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You may ask Why is g called bonding and u* antibonding? How do you know

the relative energy ordering of these MOs?Remember that each 1s orbital is an atomic wavefunction (1s) and each MO is alsoa wave function, , so we can also write LCAOs like this:

Remember that the square of a wavefunction gives us a probability density function,so the density functions for each MO are:

g =1 =0.5 (1sA +1sB) u* =2 =0.5 (1sA - 1sB)

(1)2 = 0.5 [(1sA 1sA) + 2(1sA 1sB) +(1sB 1sB)](2)2 = 0.5 [(1sA 1sA) - 2(1sA 1sB) +(1sB 1sB)]and

The only difference between the two probablility functions is in the cross term (in bold),which is attributable to the kind and amount ofoverlap between the two 1s atomic

wavefunctions (the integral (1sA 1sB)

is known as the overlap integral, S). In-

phase overlap makes bonding orbitals and out-of-phase overlap makes antibondingorbitalswhy?


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(1)2 = 0.5 [(1sA 1sA) + 2(1sA 1sB) +(1sB 1sB)](2)2 = 0.5 [(1sA 1sA) - 2(1sA 1sB) +(1sB 1sB)]



The increase of electron density between the nuclei from the in-phase overlap

reduces the amount of repulsion between the positive charges. This also increasesnuclear-electron attraction. This means that a bonding MO will be lower in energy(more stable) than the corresponding antibonding MO or two non-bonded H atoms.

(1sA)2 (1sB)2

(1 )

2

(2 )2

[(1sA)2 +(1sB)2]/2

Note: looks likethese arentquitnormalized;

overlap wasignored innormalization


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Molecular Orbital Theory: Determining Bond Order


MO diagram:

H

Energy

HH2

1s 1s

g

u*

To clearly identify the symmetry of thedifferent MOs, we add the appropriate

subscripts g (symmetric with respect tothe inversion center) and u (anti-symmetric with respect to the inversioncenter) to the labels of each MO.

The electrons are then added to the MOdiagram using the Aufbau principle.

Note:

The amount of stabilization of the g MO (indicated by the red arrow) is slightly less thanthe amount of destabilization of the u* MO (indicated by the blue arrow). For H2, thestabilization energy is 432 kJ /mol and the bond order is 1.

Bond Order =(# of e 's in bonding MO's) - (# of e 's in antibonding MO's)

2

- -


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Diatomic molecules: He2He also has only 1s AO, so the MO diagram for the molecule He2 can be formed in

an identical way, except that there are two electrons in the 1s AO on He.

He

Energy

HeHe2

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 theelectronic structure of any hypothetical molecule that we can imagine.

The bond order in He2 is (2-2)/2 = 0, sothe molecule will not exist.

However the cation [He2]+, in which one

of the electrons in the u* MO isremoved, would have a bond order of(2-1)/2 = , so such a cation might be

predicted to exist. The electronconfiguration for this cation can bewritten in the same way as we writethose for atoms except with the MO

labels replacing the AO labels:[He2]+=g2u1


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Diatomic molecules: Homonuclear Molecules of the Second Period

Li has both 1s and 2s AOs, so the MO diagram for the molecule Li2 can be formed

in a similar way to the ones for H2 and He2. The 2s AOs are not close enough inenergy to interact with the 1s orbitals, so each set can be considered independently.

Li

Energy

LiLi2

1s 1s

1g

1u*

The bond order in Li2 is (4-2)/2 = 1, so

the molecule could exists. In fact, abond energy of 105 kJ /mol has beenmeasured for this molecule.

Notice that now the labels for the MOshave numbers in front of them - this is todifferentiate between the molecularorbitals that have the same symmetry.

2s 2s2g

2u*


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Diatomic molecules: Homonuclear Molecules of the Second Period

Be

Energy

BeBe2

1s 1s

1g

1u*

2s 2s2g

2u*

Be also has both 1s and 2s AOs, so the MOs for the MO diagram of Be2 are

identical to those of Li2. As in the case of He2, the electrons from Be fill all of thebonding and antibonding MOs so the molecule will not exist.

The bond order in Be2 is (4-4)/2 = 0, so

the molecule can not exist.

Note:The shells below the valence shell willalways contain an equal number ofbonding and antibonding MOs so youonly have to consider the MOs formedby the valence orbitals when you wantto determine the bond order in a

molecule!


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This produces an MO over themolecule with a node betweenthe F atoms. This is thus an

antibonding MO ofu symmetry.

Diatomic molecules: The bonding in F2Each F atom has 2s and 2p valence orbitals, so to obtain MOs for the F 2 molecule,

we must make linear combinations of each appropriate set of orbitals. In addition tothe combinations ofns AOs that weve already seen, there are now combinations ofnp AOs that must be considered. The allowed combinations can result in the

formation of either

or

type bonds.

2pzA

+

This produces an MO aroundboth F atoms and has the samephase everywhere and issymmetrical about the F-F axis.

This is thus a bonding MO ofgsymmetry.

u* =0.5 (2pzA + 2pzB)2pzB

-2pzA

+

g =0.5 (2pzA - 2pzB)2pzB

The combinations of symmetry:


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This produces an MO over themolecule with a node on the

bond between the F atoms. Thisis thus a bonding MO ofusymmetry.


Diatomic molecules: The bonding in F2

2pyA

+

This produces an MO aroundboth F atoms that has two nodes:one on the bond axis and oneperpendicular to the bond. This

is thus an antibonding MO ofgsymmetry.

u =0.5 (2pyA + 2pyB)2pyB

-

g* =0.5 (2pyA - 2pyB)

The first set of combinations of symmetry:

2pyA 2pyB


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The second set of combinations with symmetry (orthogonal to the first set):This produces an MO over themolecule with a node on the

bond between the F atoms. Thisis thus a bonding MO ofusymmetry.


Diatomic molecules: The bonding in F2

2pxA

+

This produces an MO aroundboth F atoms that has two nodes:one on the bond axis and oneperpendicular to the bond. This

is thus an antibonding MO ofgsymmetry.

u =0.5 (2pxA + 2pxB)

2pxB

-

g* =0.5 (2pxA - 2pxB)

2pxA 2pxB

M l l O bit l Th


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F

Ene

rgy

FF2

2s 2s

2g

2u*

2p

2p

3g

3u*

1u

1g*


MO diagram for F2

(px,py)pz

You will typically see the diagramsdrawn in this way. The diagram is

only showing the MOs derived fromthe valence electrons because thepair of MOs from the 1s orbitals aremuch lower in energy and can beignored.

Although the atomic 2p orbitals aredrawn like this: they areactually all the same energy andcould be drawn like this:

at least for two non-interacting Fatoms.

Notice that there is no mixing ofAOs of the same symmetry from a

single F atom because there is asufficient difference in energybetween the 2s and 2p orbitals in F.

Also notice that the more nodes anorbital of a given symmetry has, the

higher the energy.

Note: For the sake of simplicity, electronsare not shown in the atomic orbitals.

M l l O bit l Th F ti O bit l


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F

Ene

rgy

FF2

2s 2s

2g

2u*

2p

2p

3g

3u*

1u

1g*

Molecular Orbital Theory: Frontier OrbitalsMO diagram for F2

(px,py)pz

Another key feature of suchdiagrams is that the -type MOsformed by the combinations of the pxand py orbitals make degeneratesets (i.e. they are identical in

energy).

The highest occupied molecular

orbitals (HOMOs) are the 1g* pair -

this is where F2 will react as anelectron donor.

The lowest unoccupied molecular

orbital (LUMO) is the 3u* orbital -this is where F2 will react as anelectron acceptor.

HOMO

LUMO

Bond order = ?

MO diagram for O


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SUCCESS! MO theoryexplains why oxygen isparamagnetic (2 unpairedelectrons)

Not predicted from Lewisdiagram or valence bondtheory

O

Ene

rgy

OO2

2s 2s

2g

2u*

2p

2p

3g

3u*

1u

1g*

MO diagram for O2

(px,py)pz

HOMO

LUMO

Bond order =

(8 bonding-4 antibonding)/2 = 2

OO O O

oxygen atoms6 electrons each

pair electronshappy octets


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Summary

This provides an intro to MO theory. We willcontinue over the next class to further developthese ideas

Rotatable representations of molecular orbitalsfor some diatomics are on the class website

Tutorial Friday will focus mainly on Lewisdiagrams and VSEPR review, for those who feelthey could use the practice.

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