Chem 125 Lecture 1310/3/08

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It is not readily understood without

reference to notes from the lecture.

Overlap&

Energy-Match

Consider how theOverlap Integral

(the “sum” of A x B over all space)

depends on the Distancebetween two Carbon Atomsand on Hybridization

of their Atomic Orbitals

2s 2s

C Overlap Scale

Diameter of node for 2sC is 0.7 Å

Sliding together to1.4Å

(~CC bond distance)superimposesthe two 'X's

xx

2s

x

C Overlap Scale

2s

x

2s

x

2s

x

2s

x

2s

x

2s

x

2s

x

Sliding together to1.4Å

superimposesthe two 'X's

Overlap Integral = 0.41!

Guess the overlap integral, A B(remember that A A = 1)

C Overlap

C-C Orbital Overlap (Clementi)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.15 1.25 1.35 1.45 1.55Angstroms

Orbital Overlap Integral

1.0

0.8

0.6

0.4

0.2

0.0

Overlap Integral

1.2 1.3 1.4 1.5 Å

s-

p

2s2p

2s2p

2p2p

+ x -

+ x +

2p

xx

s-sp-

p

C C C C C C

and are“orthogonal”(net overlap = 0)

to -1 atD = 0

to 0 atD = 0

to 1 atD = 0

p-

p

(sigma) is Greek “s” MO

analogue of s AO.(no node

through nuclei)

(pi) is Greek “p”

MO analogue of p AO.

(nodal planethrough nuclei)

C-C Orbital Overlap (Clementi)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.15 1.25 1.35 1.45 1.55Angstroms

Orbital Overlap Integral

Curiosity:Over most of this range 2s overlaps with 2p

better than either 2s with 2s or 2p with 2p

1.0

0.8

0.6

0.4

0.2

0.0

Overlap Integral

1.2 1.3 1.4 1.5 Å

s-

p

p-

p

s-sp-

p

sp3-sp3

s2p-s2p

C C C C C C

sp3-sp3

sp2-sp2

sp-sp

xx

sp2-sp2

sp-sp

C-C Orbital Overlap (Clementi)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.15 1.25 1.35 1.45 1.55Angstroms

Orbital Overlap Integral

1.0

0.8

0.6

0.4

0.2

0.0

Overlap Integral

1.2 1.3 1.4 1.5 Å

s-

p

p-

p

s-sp-

p

sp3-sp3

s2p-s2p

C C C C C C

sp2-sp2

sp-sp

Hybrids overlap about twice as much as pure atomic orbitals.

sp gives best overlap, but only allows two orbitals (50% s in each)

sp3 gives four orbitals with nearly as much overlap (25% s in each) (because they allow nearly full measure of s with p overlap plus s with s, and

p with p.)

Influence of Overlapon “MO” Energy ofa One-DimensionalDouble Minimum

Case I:

Perfect Energy Match

Degenera

te

EnergyRising

EnergyFallingIncreasing Overlap

No SignificantEnergy Difference

Creates Splitting

Overlap Holds Atoms Together

A B

Electron Energy

separate separate

1/√2 (A+B)

1/√2 (A-B)

together

<

>

with greateroverlap

Electron Count and Bond Strength

•

•

•

•

A B

Electron Energy

separate separatetogether

•

•

•

•# Effect1 Bonding2 Strongly Bonding3 Weakly Bonding4 Antibonding

Why Doesn’t Increasing Overlap

Make MolecularPlum Puddings

Collapse?H2 He

?

Electrons do become 55% more stable (~650 kcal/mole)

But proton-proton repulsion increases much more dramatically (1/r)

(already increases by 650 kcal/mole from H-H to 0.3 Å)

Unless one uses neutron “glue” D2 He fusion fuels the Sun (200 million kcal/mole)

Finally we understand

the atom-atom ….

force law! … ….

Bonding Potential

Electron pair becomes more stable with increasing overlap.

Nuclear repulsio

n becomes dominant

All from Coulomb’s Lawand

Schrödinger Kinetic Energy of Electrons(This curve provides the potential for studying

molecular vibration.)

Atom-Atom Distance

Energy

Newton Opticks (1717)

Query 31There are therefore Agents in Nature able to make the Particles of Bodies stick together by very strong Attractions. And it is the business of experimental Philosophy to find them out.

shop.rpg.net

Overlap&

Energy-Match

What if partner is lower in energy than A?

A B

Electron Energy

separate separate

1/√2 (A+B)

1/√2 (A-B)

together

<

>

“Splitting” Overlap?B

*

*) approximately

Why use any of an“Inferior” Orbital?

The 1s “core” AOs did indeed remain pure and unmixed during creation of molecular orbitals

for CH3CHFOH :

1 1s (F)Core1

2 1s(O)Core2

3 1s(C1)Core3

4 1s(C2)Core4

Why use any of an“Inferior” Orbital?

but the valence-level AOs were heavily mixed.

The compact 1s “core”

AOs did indeed remain pure and unmixed during creation of

molecular orbitals for CH3CHFOH,

5 “1s(valence)”

2s of F

2sp hybrid of O

2s of C

(aA + bB)2 = a2 A2 + b2

B2 + 2abAB

Why use any of an“Inferior” Orbital?Suppose the energy of the A orbital

is muchhigher (less favorable) than that of

the B orbital.

Can one profit from shifting electron density toward

the internuclear AB region (from the “outside” region)

without paying too much of the high-energy“cost” of A?

Yes, because for a small amount (a) of A in the MO,

the amount of A2 probability density (a2) is REALLY small,

while the amount of AB shifting (2ab) is much larger.

e.g. a = 0.03, b = 0.98 means a2 = 0.001, b2 = 0.96, 2ab = 0.06(Incidentally, this is normalized, since the integral of AB is ~0.6, and 0.6 x 0.06 is ~0.04 = 1 - 0.96)

Influence of Overlapon “MO” Energy ofa One-Dimensional Double Minimum

Case II:

Poor Energy Match

Degenera

te

EnergyRising

EnergyFallingIncreasing Overlap

Splitting dueonly toOriginal

Well Offset

Fights Well Difference

Note Small Energy Mismatch

still

Mixing non-

degenerate AOsNegligible

Mixing

StillBiased

What if partner is lower in energy than A?

What are the ultimate energies?

A B

Electron Energy

separate separate

1/√2 (A+B)

1/√2 (A-B)

together

<

>

?C

A-C

A+C

largerenergyshifts

smallerenergyshifts

looks mostly like C inshape & energy

looks mostly like A

B

A given overlapyields thissplitting forperfect E-match

How much smaller is the bonding shift when

energy is mismatched?

C

A

Electron Energy

separate separatetogether

Averageof A and C

Energy-mismatch

B

How much smaller is the bonding shift when

energy is mismatched?

C

A

Electron Energy

separate separatetogether

With E-mismatch larger

splittingfor same overlap

A given overlapyields thissplitting forperfect E-matchEnergy-

mismatch

B

How much smaller is the bonding shift when

energy is mismatched?

C

A-C

A+C

A

Electron Energy

separate separatetogether

(shift up a bit for >,< normalization)Splitting is less sensitive

to lesser contributor of mismatch / overlap

For a given overlap,bonding shift is reduced

by energy mismatch.(Still A+C ends lower than

A+B, because C starts lower.)

e.g. when mismatch is relatively

large, a given amount of

overlap doesn’t make much difference

Important Generalizations

Mixing two overlapping orbitals gives one composite orbital that is lower in

energy than either parent and one that is higher in energy than

either parent.The lower-energy combination looks

more like the lower-energy parent,

both in shape and in energy (ditto for higher-).

For a given overlap, increasing energy mismatch

decreases the amount of mixing and decreases the magnitude of energy

shifts.

Which Bond is Stronger A-B or A-C?

A B

Electron Energy

separate separate

C

Compared to What?

••

••

••

••

A-B stronger if forming Ions (A+ B-)

together

A-C electrons clearly lower in energy,

but…

Which Bond is Stronger A-B or A-C?

A B

Electron Energy

separate separate

C

Compared to What?

••

••

A-B stronger if forming Ions (A+ B-)

••

A-C stronger if forming Atoms (A C)• •

together

mismatch aids Heterolysis

mismatch hinders Homolysis

•

Experimental Evidence

Is All This True?

H-H vs. H-F

*

Homolysis to A• •Bkcal/mole

136104 HF Bondis Stronger

Heterolysis to A+ B-

kcal/mole (gas phase)

400 373HF Bondis Weaker

BigonF

BigonH

"Hydrofluoric Acid "

antibondingmolecular orbital

:

empty

(match) (mismatch)

End of Lecture 13Oct. 3, 2008