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Molecular Orbital Theory
Valence Bond Theory: Electrons are located in discrete pairs between specific atoms
Molecular Orbital Theory: Electrons are located in the molecule, not held in discrete regions between two bonded atoms
Thus the main difference between these theories is where the electrons are located, in valence bond theory we predict the electrons are always held between two bonded atoms
and in molecular orbital theory the electrons are merely held “somewhere” in molecule
Mathematically can represent molecule by a linear combination of atomic orbitals (LCAO)
ΨMOL
=
c1
φ1
+
c2
φ2
+
c3
φ3
+
cn
φn
Where Ψ2 = spatial distribution of electrons
If the ΨMOL can be determined, then where the electrons are located can also be determined
66 Building Molecular Orbitals from Atomic Orbitals
Similar to a wave function that can describe the regions of space where electrons reside on
time average for an atom, when two (or more) atoms react to form new bonds, the region
where the electrons reside in the new molecule are described by a new wave function
This new wave function describes molecular orbitals instead of atomic orbitals
Mathematically, these new molecular orbitals are simply a combination of the atomic wave functions (e.g LCAO)
Hydrogen 1s atomic orbital
H-H bonding
molecular orbital
67 Building Molecular Orbitals from Atomic Orbitals
An important consideration, however, is that the number of wave functions (molecular orbitals) resulting from the mixing process must equal the number of wave functions (atomic orbitals) used in the mixing
In the case of H2, in addition to the new bonding molecular orbital obtained by adding the two
atomic 1s orbitals, an antibonding orbital is obtained by subtracting the two atomic orbitals
node
H-H antibonding
molecular orbital
68 Electronic Configuration for H2
Each Hydrogen 1s atomic orbital has one electron
When two atomic orbitals mix, they produce two molecular orbitals
As the number of nodes increases, the energy of the orbital increases
The molecule has a total of two electrons and follow Aufbau principle and Pauli principle to fill electrons in molecule
69 Bond Strength
Eσ* > Eσ
-due to electron repulsion
Called the bond dissociation energy
(BDE)
The bond strength for H2 is considered the amount of energy required to break the bond and produce two hydrogen atoms X
Y
X
Y
Homolytic bond cleavage
X
Y
X
Heterolytic bond cleavage
Y
70 Molecular Orbital Theory
The σ and σ* orbitals can be written mathematically thus as a combination of atomic orbitals
Ψσ = c1φ1 + c2φ2
Ψσ* = c1φ1 - c2φ2
The size of coefficients (c1 and c2) is related to the electron density as the CN2 is a measure of the electron density in the neighborhood of the atom in question
By normalization, for each MO ΣCN2 = 1
Thus for the only filled orbital in H2, because the molecule is symmetric |C1| = |C2|
Therefore C1 = C2 and C12 = 1/2
C1 = C2 = 1/√2 = 0.707
Also if all the MOs are filled, there must be one electron in each spin state on each atom
Therefore ΣCN2 = 1 (for each atom)
For H2:
σ
σ*
ΣC2 (for atom)
c1
c2
0.707
0.707
1
0.707
-0.707
1
ΣC2 (for orbital)
1
1
71 Molecular Orbital Theory
The electron location in H2 is identical between valence bond theory and molecular orbital theory
(due to there only being one bond in H2 and thus the electrons must be located on the two atoms)
What happens however if there is more than one bond in the molecule, how do the bonding theories differ in describing the location of electrons?
Consider methane
Valence bond theory predicts four identical C-H bonds in methane formed by the carbon hybridizing to an sp3 hybridization
2p
energy
2s
1s
sp3
hybridization
1s
Each sp3 hybridized orbital would thus form a
bond with the 1s orbital from each hydrogen to
form four identical energy C-H bonds
72 Molecular Orbital Theory
Molecular orbital theory would not use the concept of hybridization (hybridization is entirely a concept developed with valence bond theory)
Instead of hybridizing the atomic orbitals first before forming bonds, molecular orbital theory would instead treat the molecular orbitals used to form the bonds as a result of mixing the atomic orbitals themselves
For methane thus would have 4 1s orbitals from each hydrogen and four second shell orbitals from the carbon atom (2s, 2px, 2py, 2pz)
The 8 valence electrons would need to placed in bonds formed from the combination of these atomic orbitals
Each bond is a result of
two electrons being shared
between sp3 hybridized
carbon and hydrogen
Where are the electrons
located and what orbitals
are being used?
H
C H
H
H
H
C H
H
H
Valence Bond Theory
Molecular Orbital Theory
73 Molecular Orbital Theory
To visualize where the electrons are located and what molecular orbitals the electrons are
located, consider the four hydrogen 1s orbitals and the four outer shell orbitals of carbon
H 1s
C 2s
C 2px
C 2py
C 2pz
Then mix the outer shell atomic orbitals to find the bonding patterns
0 nodes
1 node
1 node
1 node
Molecular orbital theory predicts there are 4 bonding MOs, 1 with 0 nodes and 3 with 1 node
(therefore they must be at different energy levels if different number of nodes!)
74 Molecular Orbital Theory
The bonding pattern in methane is thus different using either valence bond or molecular orbital theory
Valence Bond Theory
energy
Csp3-H
bond
Inner shell C 1s
Each sp3 hybridized orbital would thus form a
bond with the 1s orbital from each hydrogen to
form four identical energy C-H bonds
Molecular Orbital Theory
MO with 1 node
MO with 0 nodes
Inner shell C 1s
The bonding MOs for methane would not
be of identical energy
How to know which model is correct if either?
75 Molecular Orbital Theory
Can first compare what orbitals look like computationally
(obtained with Spartan 08, DFT B3LYP 6-31G*)
0 nodes
1 node
1 node
1 node
Orbitals obtained computationally are identical to those qualitatively determined
If the computer theory was established using molecular orbital theory, it is not surprising the obtained MOs resemble the qualitative picture
76 Molecular Orbital Theory
Is there an experimental method to test which bonding theory matches reality?
Can use photoelectron spectroscopy (PES) and electron spectroscopy for chemical analysis
(ESCA) which measure the ionization potential of electrons expelled from orbitals
Difference between PES (<~20 eV) and ESCA (>~20 eV) is ionization potential range
In short for these experiments the gas phase sample of compound under analysis is irradiated
and the binding energy for an electron can be calculated by knowing the energy of ionizing
irradiation and subtracting the kinetic energy of the detected emitted electrons
Bonded electrons
Inner shell electrons
have 2 different
The experiment confirms the MO
energy levels
description of bonding
Methane does have two different energy
levels for the four C-H bonds
Even though valence bond theory is not
correct, it is still widely used by organic
chemists as a guide to predict reactions
Chem. Phys. Lett. (1968), 613-615
77 Molecular Orbital Theory
Using molecular orbital theory therefore the electrons are located in regions of space on time average (orbitals) that are formed by the mixing of atomic orbitals
Have already seen a simple orbital description with forming H2 MOs from mixing atomic 1s orbitals from each hydrogen atom
Eσ*
Eσ
Remember also that Eσ* > Eσ
-due to electron repulsion
78 Building Molecular Orbitals from Atomic Orbitals
When forming molecular orbitals of H2 from atomic orbitals from each hydrogen atom,
hydrogen only has one electron in a 1s orbital (or a 1,0,0 orbital using n,l,m designation)
When building molecular orbitals from a 2nd row atom (like carbon) can use either the 2s (2,0,0) or 2p (2,1,-1; 2,1,0; or 2,1,1) atomic orbitals to form the bonds
When different orbitals interact, the overlap of the orbitals changes depending upon the
direction of bond formation (both in degree of overlap and symmetry of the bond)
When two s orbitals interact, due to
symmetry of orbital the direction of approach
is irrelevant
When two p orbitals interact, if lobe with
same phase is pointing toward each other a
bonding region can occur
Overlap between orbitals of same phase leads to bonding region
Bonding MO
(called σ bond)
79 Building Molecular Orbitals from Atomic Orbitals
Due to the unsymmetrical orientation of a p orbital, however, there are other possible orientations of approach
One bonding approach direction has the p orbitals on both atoms directed opposite to the approach direction Still a bonding MO, but electron density is not symmetric about internuclear axis
(called π bond)
If s orbital approaches p orbital from side, however, there is no net overlap
The positive overlap (blue with blue) is exactly
canceled with the negative overlap (blue with red),
thus there is no net overlap
The orbitals are said to be “orthogonal” to each
other and thus do not mix
80 Building New Molecular Orbitals from Molecular Orbitals
In addition to building new molecular orbitals from adding atomic orbitals, new molecular orbitals can result from combining orbitals from two different molecules
using their molecular orbitals (the result of a reaction between two molecules)
For a given molecule there might be a multitude of molecular orbitals
(the total number are due to the number of atoms in the molecule)
Hypothetical molecule that contains
6 molecular orbitals and 6 electrons
The molecular orbital the is
unoccupied that is lowest in energy
is called the LUMO
Unoccupied molecular orbitals
(UMOs)
Would fill the orbitals by following
Pauli exlusion (only 2 electrons per
orbital) and filling the lowest
energy orbitals first Occupied molecular orbitals
(OMOs)
The orbitals are classified by
whether they are “filled” or
“unfilled”
In addition the molecular orbital
that is occupied that is highest in
energy is called the HOMO
81 Building New Molecular Orbitals from Molecular Orbitals
When two molecules react (each with their own set of molecular orbitals) it is important to
recognize which molecular orbital interaction determines the reaction (if a chemist knows this then they can predict reactions)
Each molecular orbital in compound A
however will react with each molecular
orbital in compound B
Whenever any two nonorthogonal orbitals
interact they will create two new MOs, one
higher in energy and one lower in energy
When two UMOs react, there is no change in
energy as there are no electrons in the orbitals
Compound A
Compound B
When two OMOs react, there is an increase in
energy due to the two higher energy electrons
outweighing any energy gain
The number of OMOs and UMOs and energy
placement of orbitals is dependent upon the There is only an energy gain when a OMO of
compound
one molecule interacts with an UMO on the
other molecule
82 Building New Molecular Orbitals from Molecular Orbitals
The amount of energy gain is also dependent upon how close in energy the two orbitals are before mixing
Consider mixing of two orbitals, one filled (OMO) and one unfilled (UMO)
ΔE
ΔE
If the OMO is identical in energy to the
UMO there will be the maximum energy gain
due to the best possible mixing of the orbitals
As the OMO has a greater difference in
energy to the UMO, the mixing will be less
and the energy gain will thus be lower
Thus the energy gain is greatest in a reaction when the HOMO of one compound is closest in energy to the LUMO of the second compound
83 Building New Molecular Orbitals from Molecular Orbitals
When mixing any two orbitals therefore the two important considerations are the overlap
between the two orbitals and the match in energy of the two orbitals before mixing
Considerations between mixing of orbitals are therefore:
-when two nonorthogonal orbitals overlap and mix, they generate two new orbitals (one higher in energy and one lower in energy)
-the amount of energy shift upon mixing is greater with more overlap of the orbitals and
lower the further apart in energy the orbitals are before mixing
-average energy of two new orbitals is slightly higher than average of original orbitals (partly an artifact of electron-electron repulsion in higher energy orbital)
Consider the original hypothetical compound A reacting with compound B
The most important interaction to consider is the HOMO of A reacting with the LUMO of B (largest energy gain)
The energy gain from this interaction must be large
enough to overcome the energy loss of each OMO mixing
with another OMO (which causes an energy loss)
Compound A
Compound B
84 Frontier Molecular Orbital Theory
Since the majority of energy gain in a reaction between two molecules is a result of the
HOMO of one molecule reacting with the LUMO of a second molecule this interaction is
called a Frontier Molecular Orbital (FMO) interaction
A reaction is thus favored when the HOMO (nucleophile) is unusually high in energy and the LUMO (electrophile) is unusually low in energy
What does unusually high HOMO or unusually low LUMO mean?
Must be compared relative to something -usually compare energy levels with a known unreactive C-H (or C-C) single bond
If the HOMO of a new compound is higher in energy than the HOMO of the C-H bond,
then it will be more reactive as a nucleophile
If the LUMO of a new compound is lower in energy than the LUMO of the C-H bond, then it will be more reactive as an electrophile
How much higher or lower in energy will determine the relative rates of reactions
85 Frontier Molecular Orbital Theory
We can compare the placement of HOMO and LUMO levels relative to placement of C-H bonds
Very high LUMO, therefore poor electrophile
Lone pair of electrons placed in
σ*C-H
atomic orbital
No electrons in atomic orbital,
therefore very electrophilic
sp3C
1s H
H+
Because nitrogen is more
electronegative than carbon,
orbital is lower in energy
(likewise oxygen is lower than
nitrogen)
:NH
3
σC-H
Very low HOMO, therefore poor nucleophile
Both are very nucleophilic,
ammonia more than water
A sp3 hybridized carbon atom and a 1s
orbital of hydrogen have similar
energy levels and strong overlap,
therefore high mixing
:OH2
Factors that can adjust MO energy levels:
1) Unmixed valence shell atomic orbitals
86 Frontier Molecular Orbital Theory
We can compare the placement of HOMO and LUMO levels relative to placement of C-H bonds
Very high LUMO, therefore poor electrophile
Negative charge will raise the energy of orbital, σ*C-H
therefore make compound more nucleophilic
CH3
sp3C
1s H
H+
OH
:NH3
:OH2
σC-H
Very low HOMO, therefore poor nucleophile
A sp3 hybridized carbon atom and a 1s
orbital of hydrogen have similar
energy levels and strong overlap,
therefore high mixing
Factors that can adjust MO energy levels:
1) Unmixed valence shell atomic orbitals
2) Electric charge
87 Frontier Molecular Orbital Theory
We can compare the placement of HOMO and LUMO levels relative to placement of C-H bonds
Very high LUMO, The degree of mixing of two
therefore poor electrophile
orbitals is related to the amount
σ*C-H
of overlap between the orbitals
π*C-C
sp3C
1s H
2p C
This makes HOMO into a good nucleophile
2p C
πC-C
When two p orbitals overlap to form a π bond, the orbitals begin higher in energy than a
hybridized orbital and the amount of overlap is less
σC-H
Very low HOMO, therefore poor nucleophile
A sp3 hybridized carbon atom and a 1s
orbital of hydrogen have similar
energy levels and strong overlap,
therefore high mixing
Factors that can adjust MO energy levels:
1) Unmixed valence shell atomic orbitals
2) Electric charge
3) Poor overlap of atomic orbitals
88 Frontier Molecular Orbital Theory
sp3C
We can compare the placement of HOMO and LUMO levels relative to placement of C-H bonds
Very high LUMO, therefore poor electrophile
Since the oxygen 2p orbital is
σ*C-H
much lower in energy, the
This makes LUMO energy match with carbon 2p is
into a good
worse and therefore less mixing
electrophile
2p C
π*C-O
1s H
2p O
πC-O
σC-H
Very low HOMO, therefore poor nucleophile
A sp3 hybridized carbon atom and a 1s
orbital of hydrogen have similar
energy levels and strong overlap,
therefore high mixing
Factors that can adjust MO energy levels:
1) Unmixed valence shell atomic orbitals
2) Electric charge
3) Poor overlap of atomic orbitals
4) Poor energy match of orbitals
89 Frontier Molecular Orbital Theory
We can compare the placement of HOMO and LUMO levels relative to placement of C-H bonds
Very high LUMO, Can also use orbital energy levels to understand
therefore poor electrophile
differences in reactivity for C-X bonds
σ*C-H
σ*C-Mg
A C-Cl bond is good
electrophile
sp3C
1s H
sp3C
ο*C-Cl
spMg
sp3C
σC-Mg
sp3Cl
σC-H
Very low HOMO, therefore poor nucleophile
A sp3 hybridized carbon atom and a 1s
orbital of hydrogen have similar
energy levels and strong overlap,
therefore high mixing
σC-Cl
A C-Mg bond is
good nucleophile
Factors that can adjust MO energy levels:
1) Unmixed valence shell atomic orbitals
2) Electric charge
3) Poor overlap of atomic orbitals
4) Poor energy match of orbitals
90 Frontier Molecular Orbital Theory
Frontier molecular orbital (FMO) theory allows a chemist to make predictions about a
reaction by knowing the placement of the HOMO and LUMO energy levels
A high HOMO level represents a compound that is a good nucleophile
CH3
>
NH2
>
Anything that will raise
OH energy level of HOMO will
increase nucleophilicity
A low LUMO level represents a compound that is a good electrophile
Anything that will lower
R
energy level of LUMO will
C O
H3C Cl
increase electrophilicity
R
The energy level of the HOMO and LUMO can be predicted by knowing that when two atomic orbitals mix they form two new molecular orbitals, one lower in energy and one higher in energy
The amount of mixing is dependent upon:
1)  The amount of overlap between the mixing orbitals (e.g., the overlap for a σ bond is greater than the overlap for a π bond)
2) The closer in energy are two orbitals, the greater the amount of mixing that occurs
91 Frontier Molecular Orbital Theory
FMO will also allow prediction about where a reaction will occur (regiochemistry) and direction of approach (stereochemistry)
Consider a reaction with a carbonyl compound
FMO predicts that a carbonyl should react as an electrophile due to the low energy LUMO
The regio- and stereochemistry can also be predicted by considering the interacting frontier orbital (the LUMO)
The coefficient on carbon is
larger than the coefficient on
oxygen, therefore nucleophile
reacts at carbon
rotate
LUMO of
formaldehyde
92 Frontier Molecular Orbital Theory
What direction should a nucleophile approach the carbonyl?
NUC
Optimal interaction (best overlap of
interacting orbitals)
NUC
Direction appears better, but still not optimal interaction
NUC
Expect this direction to be highly
disfavored due to orthogonal
interaction with orbitals
Could there possibly be a method to test the angle of approach of nucleophile to carbonyl?
X-ray structures come to the solution once again!
93 Frontier Molecular Orbital Theory
What direction should a nucleophile approach the carbonyl?
NUC
Optimal interaction α
(best overlap of
interacting orbitals)
Studied a variety of X-ray
structures where a N
reacts with a carbonyl
intramolecularly
As the N came closer to
carbonyl, the C-O bond
lengthened and the
carbonyl carbon becomes
pyramidalized
Called the “Bürgi-Dunitz” angle
The angle of <N-C-O
averaged 107˚ (α)
Could there possibly be a method to test the angle of approach of nucleophile to carbonyl?
X-ray structures come to the solution once again!
Bürgi, H.B., Dunitz, J.D., Shefter,E., J. Am. Chem. Soc. (1973), 95, 5065-5067
94 Frontier Molecular Orbital Theory
What about the stereochemistry for a reaction with an alkyl halide?
Since alkyl halide is reacting as the electrophile, need to look at the LUMO
Largest coefficient is on the
backside of the carbon
Nucleophile thus reacts with a
methyl halide in a SN2 reaction
with backside attack
NUC
So called “inversion of
configuration”
LUMO of methyl halide
Bonds that break
Base thus reacts by abstracting
hydrogen anticoplanar to
leaving group and form new π
bond in E2 reaction
base
The base will abstract the
hydrogen that is anticoplanar to leaving group
New π bond
LUMO of 2˚ alkyl halide
95