Transcript Chem 125 Lecture 10 9/26/07 Preliminary

Chemistry 125: Lecture 12
Overlap and
Atom-Pair Bonds
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…”
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notice see final
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Ethane
&
Methanol
(Spartan 6-31G*)
H
H C H
H C H
7 Pairs of
Valence
Electrons
H
Compare MOs
to AOs of Ar
(7 electron pairs)
O H
H C H
H
Vacant
Rotated 90°
CH3CH3
Orbital
Energy
Occupied
HOMO-6
CH3OH
Orbital
Energy
Pedantic Note: with two
“heavy” atoms there are two
boring “core” orbitals.
For the purpose of making
atomic analogies to study
valence-level molecular
2s orbitals, we’ll use the atomic
1s orbital to stand for the set
Vacant of molecular core orbitals.
Thus we start with 2s rather
than 1s for valence-level
MOs, which will in truth
Occupied include tiny nodes around the
heavy nuclei.
CH3CH3
Orbital
Energy
2pz
HOMO-5
CH3OH
Orbital
Energy
CH3CH3
Orbital
Energy
2px
HOMO-4
CH3OH
Orbital
Energy
CH3CH3
Orbital
Energy
2py
CH3OH
Orbital
Energy
HOMO-3
CH3CH3
Orbital
Energy
3s
HOMO-2
CH3OH
Orbital
Energy
CH3CH3
Orbital
Energy
3dxz
HOMO-1
CH3OH
Orbital
Energy
CH3CH3
Orbital
Energy
3dyz
CH3OH
Orbital
Energy
HOMO
LUMO
CH3CH3
Orbital
Energy
3dz2
LUMO
CH3OH
Orbital
Energy
LUMO+1
CH3CH3
Orbital
Energy
3pz
LUMO+1
CH3OH
Orbital
Energy
LUMO+2
CH3CH3
Orbital
Energy
3py
CH3OH
Orbital
Energy
LUMO+3
LUMO+3
CH3CH3
Orbital
Energy
3px
LUMO+2
CH3OH
Orbital
Energy
LUMO+4
CH3CH3
Orbital
Energy
3dxy
LUMO+5
CH3CH3
Orbital
Energy
3dx2-y2
LUMO+6
CH3CH3
Orbital
Energy
4f
LUMO+4
CH3OH
Orbital
Energy
1-Fluoroethanol
Wire
Core 1
1s (F)
Core 2
1s(O)
Core 3
1s(C1)
Core 4
1s(C2)
1s(valence)
2px
2py
rotate
2py
rotate
2spz (up)
2spz(down)
3dxy
The Plum-Pudding View of Molecular Orbitals
Shows Generality of Kinetic-Energy-Based Clouds
But One Must Probe Harder to Gain a
Qualitative Understanding of Chemical Bonds
Single “United Atom”
e-density
contours
of H2
distorted by a
Atoms with
fragmented nucleus
weak bonding
Which
contour
should
we use?
“True” molecular orbitals
extend over entire molecules, but we
want to understand local bonds as
Pairwise LCAO MOs
1
Y(x1,y1,z1) = √2 ( AOa + AOb)
SUM (Linear Combination) of AOs
(like hybridization, but with two atoms)
Why is this form sensible?
H2 at Great Distance

Y (x
1,y1,z1)
=
1
2
2
2
( AOA + AOB )
negligible!
error? + AOA  AOB
Y(x1,y1,z1) =
1
√2
( AOA + AOB)
H2 at Bonding Distance?
Overlap (A  B) Creates Bonding
If we approximate a molecular orbital as a sum of atomic orbitals:
Looks very good
(normalization) < 1
A  B
near nuclei
2
(A near A, B near B)
and square to find electron density:
A

2
<1
2
 B  2A  B
2
then subtract the average of the atom electron densities:
_

1 2
A  B2
2

Shifts e-density
from atoms
to overlap region.
we 
find bonding, the difference electron density due to overlap:
A completely different
instance of multiplying!
(NOT two electrons)
AB
“By-product” of
squaring a sum.
Black line is energy e-Density Shrinks
Blue line is Y
Antibonding
e-Density Grows
in A
Stabilzation
of Particle
Total Energy of Particle
"Mixing" localized Y s for double minimum
in B
Bonding!
Wells far
apart
in AB
Wells far
apart
Holds
Wells close A & B
together together
2 significant?
Where
Where
is Y
is AYAY
B significant?
At the center 2YAYB is as large as YA2 +YB2
Electron Density nearly Doubled!
2
a little
YB
2
YA
no!
yes
Yb small
yes!
Region of Significant Overlap
“Overlap Integral” (  YAYB)
measures net change from atoms.
no
Total e-Density
Difference Density
1s (atomic)
Bond
Energy
52%
Coutoured at
0.025 e/ao3
92.9% of
Total Electronic Energy
0.02 Coutoured at
e/ao3 0.004 e/ao3
(almost all of which was
already present in the atoms)
High accuracy is required to calculate correct value of the
Bond Energy, the difference between atoms and molecule.
(Cf. X-ray difference density)
State-of-the-art 40 years ago
Laws & Lipscomb, Isr. J. Chem. 10, 77 (1970)
Total e-Density
Difference Density
Very crudest model
shows most of bond.
1s (atomic)
Bond
Energy
shift from
atom to bond
52%
0.02
General spread increases
Adjust
molecular orbital to lower
thedensity/stabilization.
energy.
1s
(optimized exponent)
bonding
This makes it more realistic, because
the true energy is the lowestlarger
possible
shift from
according to the “variational
atomprinciple”.)
to bond
73%
0.04
Total e-Density
Hybridized + SCF
(96.7% 1s; 0.6% 2s; 2.7% 2p)
Difference Density
Directed spread improves
Helps
overlapdensity.
bonding
larger
but at the cost
of 3% n=2 character
shift from beyond
Bond nucleus to bond
Energy
76%
0.11
General spread increases
bonding density/stabilization.
1s (optimized exponent)
larger
shift from
Hybrid: 96.7%
100%1s
1s 0.6% 2s 2.7%2p
atom
to bond
73%
0.04
Total e-Density
Difference Density
Directed spread improves
bonding density.
Hybridized + SCF
(96.7% 1s; 0.6% 2s; 2.7% 2p)
Bond
Energy
76%
0.11
+ some correlation
Density ~unchanged
(How so?)
much
better
energy
90%
0.11
Pairwise LCAO-MO
Y(x1,y1,z1) =
<1
√2
( AOA + AOB)
Virtues: Easy to formulate and understand
Looks like atoms (especially near nuclei)
(the Main Event for electrons; ~ 6x larger than bond)
Builds up e-density between nuclei
(through Overlap - the source of Bonding)
Smooths Y to lower kinetic energy
[though ultimate contraction toward nuclei raises it again]
Hybridizing AOs provides flexibility
(unlimited if you use all H-like AOs in hybrid)
(but keep it simple - valence shell is fairly good)
Pairwise LCAO-MO
Y(x1,y1,z1) =
Y
=
>1
<1
2
>1
<1
√2
( AOA + AOB)
(AOA2 + AOB2 + 2 AOA AOB)
Atoms
Anti Bond
(overlap / product)
Overlap
&
Energy-Match
End of Lecture 12
Oct. 1, 2008
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J. M. McBride, Chem 125. License: Creative Commons BY-NC-SA 3.0