Chem 125 Lecture 1510/5/2005

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What gives Orbitals their Shape?

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Potential Energy

Kinetic Energy

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4d

2s

e-densitycontours

of H2

Single “United Atom”

distorted by afragmented nucleus

Whichcontourshould

we use?

The Plum-Pudding View of Molecular Orbitals Shows Generality of Kinetic-Energy-Based Clouds

Atom-PairBonding

But One Must Probe Harder to Gain a Qualitative Understanding of Chemical Bonds

Reality: Structure (Nuclear Arrangement )

Stability/Reactivity (Energy)Total Electron Density

LCAO Atom-Pair Localized Bond Orbitals(cf. H2 When necessary, mix Localized Bond Orbitals to give MOs)

MO Plum Pudding(MO-to-Atom analogy ; useful for one-electron phenomena)

NOT a "sphere of uniform density” à la J.J. Thomson Appearance depends on chosen contour.Models help Understand Electrons:

Orbitals Simplify xi,yi,zi)(2 for total e-density)

i=1

n

i i

Molecules from Atoms:LCAO MO

1√2

( AOa + AOb)(x1,y1,z1) =SUM of AOs

(like “hybridization” but with two atoms)

Why is this sensible?

H2 at Great Distance

1√2

( AOA + AOB)(x1,y1,z1) =

H2 at Bonding Distance

Overlap Creates BondingIf we approximate a molecular orbital as a sum of atomic orbitals:

€

12A+B( )

and square to find electron density:

€

12A2+B2+2AB( )

then subtract the average of the atom electron densities:

€

12A2+B2

( )

we find bonding, the difference electron density due to overlap:

€

AB

<(normalization)

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QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.A

2 B

2

Where is A B significant?Where is A significant?

no

yes a littleno!

b small yes!

At the center AB is as large as A2 !

“Overlap Integral” is AB

Region of Significant Overlap

92.9% of Total Electronic Energy

(almost all of which wasalready present in the atoms)

Great accuracy required to calculate correct value of bond energy (a difference).

(Cf. X-ray difference density)

Total e-Density Difference Density

1s (atomic)

52%

BondEnergy

0.02e/ao

3Coutoured at

0.025 e/ao3

Coutoured at

0.004 e/ao3

State-of-the-art 40 years ago

Total e-Density Difference Density

1s (atomic)

52%0.02

1s (optimize exponent)

73%0.04

BondEnergy

Total e-Density Difference Density

Hybridized + SCF(96.7% 1s; 0.6% 2s; 2.7% 2p)

76%

BondEnergy

0.11

1s (expanded)

73%0.04

100% 1sHybrid: 96.7% 1s 0.6% 2s 2.7%2p

Helps overlapbut at the cost of 3% n=2 character

Total e-Density Difference Density

Hybridized + SCF(96.7% 1s; 0.6% 2s; 2.7% 2p)

76%

BondEnergy

0.11

+ some correlation

90%0.11

LCAO-MO

Looks like atoms (especially near nuclei) (the Main Event for electrons; ~100 bond)

<1√2

( AOA + AOB)(x1,y1,z1) =

Virtues:

Builds up e-density between nuclei (through Overlap - the source of Bonding)

Hybridizing AOs provides flexibility (unlimited if you use all H-like AOs in hybrid)

Easy to formulate and understand

(but keep it simple - valence shell is fairly good)

LCAO-MO<1√2

( AOA + AOB)(x1,y1,z1) =

<12

(AOA2 + AOB

2 + 2 AOA AOB)=

Atoms Bond(overlap / product)

>1

>1

Anti

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

SCALE:At node of 2s orbital = 2

2 = = r * 2Z/nao

or rnode = nao/Zn for 2s is 2ao = 0.58Å

Zeff for C 2s is 3.2

Diameter of node is 0.7 Å

0.7 Ånode

diameter

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

(~CC bond distance)

superimposesthe two 'X's

Overlap Integral = 0.41

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

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

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

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

2s2p

2s2p

+ x -

+ x +

and areorthogonal

2p

2p

+ x -

+ x +

and areorthogonal

2p

xx

s-sp-p

C C C C C C

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

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

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

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

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 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-sp3sp2-sp2

sp-sp

s2p-s2p

C C C C C C

sp3-sp3sp2-sp2sp-sp

xx

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-sp3sp2-sp2

sp-sp

s2p-s2p

C C C C C C

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)

Influence of Overlapon “MO” Energy ofa Double Minimum

Case I:

Perfect Energy Match

Degenerate

EnergyRising

EnergyFallingIncreasing Overlap

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 by much more (1/r)

(increases by 650 kcal/mole already by 0.3 Å)

Unless one uses neutron “glue” (200 million kcal/mole; D2He fusion fuels the Sun)

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

<

>

?B

Why use any of an“Inferior” Orbital?

€

aA+bB( )2 =a2A2+b2B2+2abAB

Suppose the energy of the A orbital is muchhigher (less favorable) than that of the B orbital.

Can you 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 orbital,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 Double Minimum

Case II:

Poor Energy Match

Energy MismatchNote Energy MismatchIncreasing Overlap

TinyEnergyShifts

Mixing non-degenerate

AOs

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

<

>

?B

A-B

A+B

largerenergyshifts

smallerenergyshifts

B

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

C

A

Ele

ctro

n E

nerg

y

separate separatetogether

Splitting forperfect match

mismatch

B

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

C

A

Ele

ctro

n E

nerg

y

separate separatetogether

Splitting forperfect match

mismatch

Splitting withmismatch

(shift up for >,<normalization)

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

Splitting withmismatch

Splitting not very sensitive to lesser

contributor of mismatch / overlap

(shift up for >,<normalization)

Important Generalizations

Mixing two orbitals gives one new orbitallower in energy than either parent and

one higher in energy than either parent.

The lower-energy combination looks mostly like the lower-energy parent,

both in shape and in energy (and vice versa).

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

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

Heterolysis

Homolysis