after applying the united-atom “plum-pudding” view of molecular orbitals, introduced in the...
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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
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.
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
End of Lecture 12Oct. 1, 2008
Copyright © J. M. McBride 2009. Some rights reserved. Except for cited third-party materials, and those used by visiting speakers, all content is licensed under a Creative Commons License (Attribution-NonCommercial-ShareAlike 3.0).
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The following attribution may be used when reusing material that is not identified as third-party content: J. M. McBride, Chem 125. License: Creative Commons BY-NC-SA 3.0
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