After applying the united-atom “plum-pudding” view of molecular orbitals, introduced in the

previous lecture, to more complex molecules, this lecture introduces the more utilitarian

concept of localized pairwise bonding between atoms. Formulating an atom-pair orbital as a

sum of atomic orbitals creates an electron difference density by means of the cross product

that enters upon squaring a sum. This “overlap” term is the key to bonding. The hydrogen

molecule is used to illustrate how close a simple sum of atomic orbitals comes to matching

reality, especially when the atomic orbitals are allowed to hybridize.

Synchronize when the speaker finishes saying

“…looked at methane and ammonia…”Synchrony can be adjusted by using the pause(||) and run(>) controls.

Chemistry 125: Lecture 12

Overlap and Atom-Pair Bonds

For copyright notice see final page of this file

Ethane&

Methanol

(Spartan 6-31G*)

7 Pairs ofValence

Electrons

C

C

HH

H H

H

H

OC

HH H

H

Compare MOsto AOs of Ar

(7 electron pairs)

2s

CH3CH3

OrbitalEnergy

Occupied

Vacant

HOMO-6CH3OHOrbitalEnergy

Occupied

Vacant

Rotated 90°

Pedantic Note: with two “heavy” atoms there are two boring “core” orbitals. For the purpose of making atomic analogies to study valence-level molecular orbitals, we’ll use the atomic 1s orbital to stand for the set of molecular core orbitals. Thus we start with 2s rather than 1s for valence-level MOs, which will in truth include tiny nodes around the heavy nuclei.

HOMO-5

2pz

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

HOMO-4

2px

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

HOMO-3

2py

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

HOMO-2

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

3s

HOMO-1

3dxz

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

HOMO

3dyz

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

LUMO

LUMO

3dz2

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

LUMO+1

LUMO+1

3pz

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

LUMO+3

LUMO+2

3py

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

LUMO+2

LUMO+3

3px

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

LUMO+4

3dxy

CH3CH3

OrbitalEnergy

LUMO+5

3dx2-y2

CH3CH3

OrbitalEnergy

LUMO+6

LUMO+4

4f

CH3CH3

OrbitalEnergy

CH3OHOrbitalEnergy

1-Fluoroethanol

Wire

1s (F)Core 1

1s(O)Core 2

1s(C1)Core 3

1s(C2)Core 4

1s(valence)

2px

2py

rotate

2py

rotate

2spz (up)

2spz(down)

3dxy

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

Atoms withweak bonding

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

Pairwise LCAO MOs1√2

( AOa + AOb)(x1,y1,z1) =

SUM (Linear Combination) of AOs(like hybridization, but with two atoms)

Why is this form sensible?

“True” molecular orbitalsextend over entire molecules, but we want to understand local bonds as

H2 at Great Distance

12

( AOA2 + AOB

2)(x1,y1,z1) =

H2 at Bonding Distance?

1√2

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

+ AOA AOBerror?negligible!

Overlap (A B) Creates Bonding

If we approximate a molecular orbital as a sum of atomic orbitals:

€

12A+B( )

and square to find electron density:

€

1

2A2 + B2 + 2A × B( )

then subtract the average of the atom electron densities:

€

12A2+B2

( )

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

€

A × B

Looks very goodnear nuclei

(A near A, B near B)

“By-product” of squaring a sum.

A completely differentinstance of multiplying!

(NOT two electrons)

“By-product” of squaring a sum.

<(normalization)

<

Shifts e-density from atoms_

to overlap region.

in A in B

Wells farapart

Wells farapartT

otal

Ene

rgy

of P

arti

cle

"Mixing" localized s for double minimum

Wells closetogether

in AB

Antibonding

HoldsA & B

together

Black line is energy

Blue line is

Bonding!Sta

bilz

atio

nof

Par

ticl

ee-Density Grows

e-Density Shrinks

A

2 B

2

Where is A B significant?

no

yes a littleno!

b small yes!

Where is A2 significant?

At the center 2AB is as large as A2 +B

2

Electron Density nearly Doubled!

“Overlap Integral” ( AB)measures net change from atoms.

Region of Significant Overlap

92.9% of Total Electronic Energy

(almost all of which wasalready present in the atoms)

High accuracy is required to calculate correct value of theBond Energy, the difference between atoms and molecule.

(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 agoLaws & Lipscomb, Isr. J. Chem. 10, 77 (1970)

Total e-Density Difference Density

1s (atomic)

52%0.02

1s (optimized exponent)

73%0.04

BondEnergy

Very crudest model shows most of bond.

General spread increases bonding density/stabilization.

shift fromatom to bond

largershift from

atom to bond

Adjust molecular orbital to lower the energy.This makes it more realistic, because the true energy is the lowest possible

according to the “variational principle”.)

1s (optimized exponent)

73%0.04

General spread increases bonding density/stabilization.

Directed spread improves bonding density.

largershift from

atom to bond

Total e-Density Difference Density

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

76%

BondEnergy

0.11

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

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

shift from beyondnucleus to bond

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

Density ~unchanged

much betterenergy

Directed spread improves bonding density.

(How so?)

Pairwise LCAO-MO

Looks like atoms (especially near nuclei) (the Main Event for electrons; ~ 6x larger than 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)

Smooths to lower kinetic energy [though ultimate contraction toward nuclei raises it again]

Pairwise 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

End of Lecture 12Oct. 1, 2008

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