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Chem 125 Lecture 1510/5/2005
Projected material
This material is for the exclusive use of Chem 125 students at Yale and may not
be copied or distributed further.
It is not readily understood without reference to notes from the lecture.
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
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