Chemistry 125: Lecture 13

Overlap and Energy-Match Covalent bonding depends primarily on two factors: orbital overlap and energy-match.

Overlap depends on hybridization. Bond strength depends on the number of shared

electrons. In these terms quantum mechanics shows that Coulomb’s law answers Newton’s

query about what “makes the Particles of Bodies stick together by very strong Attractions.”

Energy mismatch between the constituent orbitals weakens the influence of their overlap.

The predictions of this theory are confirmed experimentally by measuring the bond strengths

of H-H and H-F during heterolysis and homolysis.

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Overlap&

Energy-Match

Consider how theOverlap Integral

(the “sum” of A x B over all space)

depends on the Distancebetween two Carbon Atoms

and on Hybridizationof 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

Ove

rlap

Inte

gra

l

1.2 1.3 1.4 1.5 Å

s-p

2s2p

2s2p

2p

2p

+ x -

+ x +

2p

xx

s-sp-p

C C C C C C

and are“orthogonal”(net overlap = 0)

to -1 at

D = 0

to 0 at

D = 0

to 1 at

D = 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

Ove

rlap

Inte

gra

l

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-sp3sp2-sp2sp-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

Ove

rlap

Inte

gra

l

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

Degenerate

EnergyRising

EnergyFallingIncreasing Overlap

No SignificantEnergy Difference

Creates Splitting

Overlap Holds Atoms Together

A B

Ele

ctro

n E

nerg

y

separate separate

1/√2 (A+B)

1/√2 (A-B)

together

<

>

with greateroverlap

Electron Count and Bond Strength

•

•

•

•

A B

Ele

ctro

n E

nerg

y

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 repulsion becomes

dominant

All from Coulomb’s Lawand

Schrödinger Kinetic Energy of Electrons

(This curve provides the potential for studying molecular vibration.)

Atom-Atom Distance

Ene

rgy

Newton Opticks (1717)

Query 31

There 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

Ele

ctro

n E

nerg

y

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)Core 1

2 1s(O)Core 2

3 1s(C1)Core 3

4 1s(C2)Core 4

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 much

higher (less favorable) than that of the B orbital.

Can one profit from shifting electron density towardthe 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

Degenerate

EnergyRising

EnergyFallingIncreasing Overlap

Splitting dueonly to

OriginalWell 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

Ele

ctro

n E

nerg

y

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 this

splitting forperfect E-match

How much smaller is the bonding shift when energy is mismatched?

C

A

Ele

ctro

n E

nerg

y

separate separatetogether

Averageof A and C

Energy-mismatch

B

How much smaller is the bonding shift when energy is mismatched?

C

A

Ele

ctro

n E

nerg

y

separate separatetogether

With E-mismatch larger splitting

for same overlapA given overlap

yields thissplitting for

perfect E-match Energy-mismatch

B

How much smaller is the bonding shift when energy is mismatched?

C

A-C

A+C

A

Ele

ctro

n E

nerg

y

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

Ele

ctro

n E

nerg

y

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

Ele

ctro

n E

nerg

y

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)

Hybridization Reality Check:

Structure and Dynamics of

XH3BH3 CH3 NH3

End of Lecture 13Oct. 3, 2008

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