Rethinking Hybridization
For more than 60 years, one of the most used concepts to come out of the valence bond
model developed by Pauling was that of hybrid orbitals. The ideas of hybridization seemed
to be consistent with many experimental observations. Hybrid orbitals were simple to
envision, they predicted geometries most of the time for simple p-block compounds and
they made the distinction between sigma and pi bonding easy to understand. However, it
has always been true that when molecular structures and properties are probed more deeply,
the hybrid orbital model – particularly the extreme limits of the model presented in
introductory and organic chemistry texts – presents many difficulties. Some of these
difficulties include:
1.
Hybridization schemes as typically discussed represent extremes of orbital mixing.
3
An sp orbital set on a carbon atom for example implies that all four hybrid orbitals are
constituted identically, and that each has 25% s character and 75% p character. In addition,
the directional properties of those orbitals – one of the features that make them attractive to
chemists – imply that all bond angles around the sp3 hybridized atom will be the same.
While this particular argument works fine for methane, it does not work for
monochloromethane, where the bond angles are not all the same. The HCH angle is less
than the tetrahedral angle and the only way to rationalize this in hybridization terms is to
have more p character and less s character in the carbon orbitals interacting with the
hydrogen atoms. These fractional hybridization schemes have been used, but have never
gained wide acceptance.
More dramatic deviations from idealized hybridization are found in the hydrides of the
group VI (16) elements. The bond angle in water is 105.4o, implying that the oxygen
orbitals used to bond with the hydrogens have more than 75% p character. By the time you
reach H2Se, the bond angle is essentially 90o, implying that only p orbitals are being used by
the Se atom to form bonds. A detailed picture of the bonding in water shows that the OH
bonds have predominately O p orbital and H s orbital parentage, with some O s character
mixed in, while the lone pairs, typically represented as being equivalent (and in equivalent
orbitals in a hybridization scheme) are quite different, one being purely O p in character and
the other predominately O s in character with a little bit of p character mixed in.
2. The strict hybrid orbital model is inconsistent with the results of photoelectron
spectroscopy. This is observed for many molecules. One of the most dramatic examples is
that of methane. Hybrid orbital theory predicts four equivalent bonds in methane.
Consequently, the photoelectron spectrum of methane in the bonding region should show a
single peak (with associated vibrational structure). This is not the case – two peaks are
clearly present, and the integrated intensities of those peaks are very close to 3:1. Likewise
for water, where hybrid orbital theory would predict two ionizations – one from the two
equivalent bonding orbitals and one from the equivalent lone pairs – four ionizations are
observed, consistent with detailed molecular orbital calculations.
3.
Hybrid orbital models are inconsistent with group theoretical predictions. The
previous examples of methane and water are useful here. In the Td point group of methane
the maximum degeneracy is three (a T representation). Rather than four equivalent bonds,
molecular orbital theory and group theory predict that the bonding molecular orbitals fall
into two sets – a triply degenerate T2 set and a singly degenerate A1 orbital. This is
certainly consistent with the photoelectron spectral results for methane. The four bonds,
which arise from the A1 and T2 molecular orbital are equivalent, but they arise from
molecular orbitals that differ in symmetry and energy.
A similar situation is found in water. There are no degenerate irreducible
representations in the C2v point group of the water molecule. Rather the four molecular
orbitals in the bonding region have four distinct energies. One of these orbitals, of B1
symmetry, is a pure p orbital on oxygen, and is therefore one of the lone pairs of electrons.
Another orbital, of A1 symmetry, is composed predominately of oxygen s character, and is
best described as the second lone pair. The two orbitals that produce the O-H bonds are of
A1 and B2 symmetry. As is the case in methane these two orbitals, when taken together,
produce two equivalent bonds, but the orbitals themselves are of different symmetry and
energy.
In the molecular orbital model there is orbital mixing, but that orbital mixing must be
based on symmetry. In some molecules that orbital mixing can produce results that appear
very similar to a hybrid orbital picture. For example, in carbon monoxide, a hybrid orbital
model would invoke sp hybridization on both C and O with the unhybridized p orbitals
forming the pi bonds. An examination of the wavefunctions for the molecular orbitals in
this molecule shows that the degenerate pi molecular orbitals are indeed formed from C and
O p orbitals only, and the singly-degenerate highest occupied molecular orbital (the sigma
bond) is formed from s and p orbitals. However, this agreement is purely a consequence of
symmetry. An examination of the coefficients in the wavefunctions shows the pi orbitals to
be more than 70% oxygen in character and the sigma orbital to be more than 80% carbon in
character, and to have essentially no oxygen s character. This latter orbital is best described
as being predominately a lone pair located on the carbon atom (and it the electronic
rationale for the fact that CO bonds to metal atoms almost exclusively through the C atom
as is the case in binding to hemoglobin and in metal carbonyls).
So what are we to do? One option is to abandon the hybrid orbital model completely
and to make descriptions based solely on the molecular orbital coefficients. Another, and
perhaps one that is more palatable to many chemists, is to rethink what we mean by
hybridization, and realize that all it really means is that orbitals of the same symmetry have
been involved in forming a molecular wavefunction. This approach requires discussing
fractional orbital mixing from the outset, and makes use of the results of detailed molecular
calculations that though once prohibitive for all but the simplest molecules can down be
done in minutes for fairly complicated systems. The importance of symmetry concepts in
these arguments cannot be underestimated.
From http://www.wellesley.edu/Chemistry/chem341/hybridization.html
Photoelectron Spectroscopy
How do we know if the energy level diagrams
have any meaning ?
The actual energy levels of the MO’s in molecules can be determined
experimentally by a technique called photoelectron spectroscopy. Such
experiments show that the MO approach to the bonding in molecules
provides an excellent description of their electronic structure.In the UV-PES
experiment, a molecule is bombarded with high energy ultraviolet photons
(usually Ephoton= hν= 21.1 eV). When the photon hits an electron in the
molecule it transfers all the energy to the electron. Part of the energy (equal
to the ionization potential, I, of the MO in which the electron was located) of
the photoelectron is used to leave the molecule and the rest is left as kinetic
energy (KE). The kinetic energy of the electrons are measured so I can be
calculated from the equation:
PES Spectrum of Methane
2.4
2.2
2
1.8
Ionization Energy (eV)
1.6
1.4
1.2
How to Read The Spartan Output File
The pictorial representations of orbitals, the charges on the atoms, the shapes, bond
lengths and bond angles that we determine using a program like Spartan are merely the
visual manifestation of a complex set of calculations (remember we are attempting to
solve the Schrodinger equation for the molecule). Some of the numerical results of those
calculations are stored in the output file.
Of particular interest to us are the eigenvalues (the energies of the molecular orbitals) and
the eigenvectors (the wavefunctions for the molecular orbitals). The eigenvalues are self
explanatory, and you can compare them to the orbital energies in the energy level
diagram resulting from a particular calculation. Remember that our molecular orbital
model constructs model orbitals by taking linear combinations of atomic orbitals (think
back to making the MO's of homonuclear diatomics). The eigenfunctions are those linear
combinations.
The portion of the output file that is of interest to us at this time is the section labeled
eigenvalues and eigenvectors.
Here is the relevant section from the output file for CO.
We infer the origins of any of the molecular orbitals by reading down the appropriate
column. For example, orbital 5 is the highest occupied molecular orbital in CO (why do
we know this is the case). We would conclude that the wavefunction for orbital 5 was
given by:
In this case Spartan has chosen the z axis as the internuclear axis. Hence, the negative
lobe of pz on one atom is pointing toward the positive lobe of pz on the other atom, so if
the pz orbitals have opposite signs, as they do here, the result is a bonding interaction
(draw yourself some pictures if this is confusing).
Remember that it is the square of the coefficients that tell us about the fraction of a
particular atomic orbital in the molecular orbital. So in this case we would say that the
molecular orbital we are looking at comes has about 44% C 2s character, 35% C 2pz
character, less than 1% O 2s character and about 22% O 2pz character. So this orbital is
mostly C in nature (about 80%) and is the reason we consider this highest occupied
orbital to be similar to a lone pair of electrons on the C atom.
The sum of the squares must add to one, as each AO is used completely in forming the
MO’s.
0.318186
0
0 0.436406
0
0 0.080435
0 0.004036 0.262042
0 0.719952 0.013964
0
0 0.262042 0.004036
0 0.013964 0.719952
0
0.005971
0
0 0.345626
0
0 0.533586
0.234488
0
0 0.001388
0
0 0.089814
0 0.01113 0.722789
0 0.261019 0.005062
0
0 0.722789 0.01113
0 0.005062 0.261019
0
0.441348
0
0 0.216569
0
0 0.296165
1
2
3
4
5
6
7
8
1 SI
2 SI
1 PI
1 PI
3 SI
2 PI
2 PI
4 SI
-40.0287 -20.6846 -16.1531 -16.1531 -13.0279 1.00008 1.00008 6.08184
0.164974
0
0
0.114813
0.674304
0
0
0.045907
Mol. Orbital
Symmetry:
Eigenvalue
squares
S C 1
Px C 1
Py C 1
Pz C 1
S O 2
Px O 2
Py O 2
Pz O 2
31.81862
0
0 43.64056
0 0.403606 26.20416
0
0 26.20416 0.403606
0
0.597065
0
0 34.56264
23.44884
0
0 0.138831
0 1.113025 72.2789
0
0 72.2789 1.113025
0
44.13476
0
0 21.65692
0 8.043463
1.396415
0
71.99523
0
0 53.35864
0 8.98141
0.506232
0
26.10188
0
0 29.61645
16.49741
0
0
11.48125
67.43037
0
0
4.590735
0
71.99523
1.396415
0
0
26.10188
0.506232
0
% squares
S C 1
Px C 1
Py C 1
Pz C 1
S O 2
Px O 2
Py O 2
Pz O 2
sum squares
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
C
C*
C* (sp3)
2s
2p
Valence bond theory treatment of a tetrahedral molecule: the bonding in CH4
sp3
The overlap of the sp3 hybrid orbitals on C with the 1s orbitals on the H atoms gives four C-H (sp3)1s s bonds oriented 109.47° from each other. This provides the tetrahedral geometry predicted by
VSEPR theory.
C*
3
C
C*
4H
1s
2s
1s
sp3
1s
2p
1s
H
H
C H
H
MO and Valence Bond
MOLECULAR ORBITALS of METHANE
Molecular Orbital
Coefficients
MO:
Eigen
1 H0
2 C1
3 C1
4 C1
5 C1
6 H2
7 H3
8 H4
values:
(ev):
S
S
PX
PY
PZ
S
S
S
1
2
3
4
-1.06145 -0.48908 -0.48908 -0.48908
-28.88346 -13.3084 -13.3084 -13.3084
A1
T2
0.31023 0.43746
0.78425
0
0
0.01279
0
-0.49444
0
0.51121
0.31023 -0.48614
0.31023 -0.15614
0.31023 0.20481
T2
-0.42317
0
-0.0016
-0.51131
-0.4945
-0.21685
0.14235
0.49767
T2
0.00882
0
-0.7112
-0.00774
0.01031
-0.29524
0.57086
-0.28443
5
0.17127
4.66049
6
0.17127
4.66049
T2
T2
-0.43909 -0.43195
0
0
0.01424 0.00158
-0.49277 0.50113
0.501
0.49285
0.49311 -0.21527
0.15813 0.14528
-0.21215 0.50194
7
0.17127
4.66049
T2
0.00987
0
0.70273
0.00886
-0.01126
-0.29997
0.57738
-0.28728
8
0.19048
5.1831
A1
-0.39212
0.62045
0
0
0
-0.39212
-0.39212
-0.39212
SQUARES
MO:
Eigen
1 H0
2 C1
3 C1
4 C1
5 C1
6 H2
7 H3
8 H4
values:
(ev):
S
S
PX
PY
PZ
S
S
S
1
2
3
4
-1.06145 -0.48908 -0.48908 -0.48908
-28.88346 -13.3084 -13.3084 -13.3084
A1
0.0962427
0.6150481
0
0
0
0.0962427
0.0962427
0.0962427
T2
0.191371
0
0.000164
0.244471
0.261336
0.236332
0.02438
0.041947
T2
0.179073
0
2.56E-06
0.261438
0.24453
0.047024
0.020264
0.247675
T2
7.78E-05
0
0.505805
5.99E-05
0.000106
0.087167
0.325881
0.0809
5
0.17127
4.66049
6
0.17127
4.66049
7
0.17127
4.66049
8
0.19048
5.1831
T2
0.1928
0
0.000203
0.242822
0.251001
0.243157
0.025005
0.045008
T2
0.186581
0
2.5E-06
0.251131
0.242901
0.046341
0.021106
0.251944
T2
9.74E-05
0
0.493829
7.85E-05
0.000127
0.089982
0.333368
0.08253
A1
0.153758
0.384958
0
0
0
0.153758
0.153758
0.153758
% SQUARES
MO:
Eigen
1 H0
2 C1
3 C1
4 C1
5 C1
6 H2
7 H3
8 H4
values:
(ev):
1
2
3
4
5
6
7
8
-1.06145 -0.48908 -0.48908 -0.48908 0.17127 0.17127 0.17127 0.19048
-28.88346 -13.3084 -13.3084 -13.3084 4.66049 4.66049 4.66049 5.1831
S
S
PX
PY
PZ
S
S
S
A1
9.6242653
61.504806
0
0
0
9.6242653
9.6242653
9.6242653
T2
19.13713
0
0.016358
24.44709
26.13357
23.63321
2.43797
4.194714
T2
17.90728
0
0.000256
26.14379
24.45303
4.702392
2.026352
24.76754
T2
0.007779
0
50.58054
0.005991
0.01063
8.716666
32.58811
8.090042
T2
19.28
0
0.020278
24.28223
25.1001
24.31575
2.50051
4.500762
T2
18.65808
0
0.00025
25.11313
24.29011
4.634117
2.110628
25.19438
T2
0.009742
0
49.38295
0.00785
0.012679
8.9982
33.33677
8.25298
A1
15.37581
38.49582
0
0
0
15.37581
15.37581
15.37581
Methane Molecular Orbital Diagram (AM1 Level)
Walsh Diagrams
We can use a Walsh diagram to compare and assess the relative energies of
different possible structures. In a Walsh diagram, the relative energies of important
MO’s are plotted as the value of a metrical parameter (e.g. bond lengths or angles) is
changed. The amount of stabilization or destabilization of the MO’s is based on the
amount of increase or decrease in the in-phase overlap of the AO’s used to make
each molecular orbital.
10
Eu
Au
5
A 1g
0
B1g
Energy (eV)
-5
A 2u
-10
å åå
-15
Eu
-20
-25
å
-30
å
Au
WHAT ABOUT WATER?
You can think of this spectrum as having 3 major peaks. The fine structure within
each peak arises from vibrational energy, and is not something we'll worry about now.
The energies along the x-axis refer to the electron binding energies. The light
source used in this experiment is not sufficiently energetic to ionize electrons from
the lowest lying molecular
orbital.
BeH2 and H2O
If you are trying to estimate the appropriate geometry for a triatomic molecule
using a Walsh diagram, all that is necessary is to correctly determine the
number of electrons that will populate the orbitals in the diagram. Then you
can estimate which electron configuration will provide the lowest overall
energy (and thus the most stable geometry).
Orbital overlap analyses such as these allow for the prediction of molecular
geometry using the delocalized model for covalent bonding in the same way
that VSEPR is used in the localized approach.
B2H6
A really strange molecule
D2h
What is going on here?????????????
Hydrogen can form more than one bond??????? More “Lies”!!!
YES it can!!!
Hydrogen can not form more than one “standard” (2 center-2 electron) bond.
But it can form two 3 center – 2 electron (3c-2e) bonds.