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. An sp3 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

1.21.41.61.822.22.4

Ionization Energy (eV)

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.

Mol. O

rbital 1

23

45

6 7

8Sym

metry:

1 SI 2 SI

1 PI 1 PI

3 SI 2 PI

2 PI 4 SI

Eigenvalue -40.0287

-20.6846-16.1531

-16.1531-13.0279

1.00008 1.00008

6.08184

squares

sum

squares S C

1 0.164974

0.3181860

00.436406

0 0

0.0804351.00

Px C 1

0 0

0.0040360.262042

00.719952

0.0139640

1.00Py C

1 0

00.262042

0.0040360

0.013964 0.719952

01.00

Pz C 1

0.114813 0.005971

00

0.3456260

00.533586

1.00S O

2 0.674304

0.2344880

00.001388

0 0

0.0898141.00

Px O 2

0 0

0.011130.722789

00.261019

0.0050620

1.00Py O

2 0

00.722789

0.011130

0.005062 0.261019

01.00

Pz O 2

0.045907 0.441348

00

0.2165690

00.296165

1.00

% squares

S C 1

16.49741 31.81862

00

43.640560

08.043463

100.00Px C

1 0

00.403606

26.204160

71.99523 1.396415

0100.00

Py C 1

0 0

26.204160.403606

01.396415

71.995230

100.00Pz C

1 11.48125

0.5970650

034.56264

0 0

53.35864100.00

S O 2

67.43037 23.44884

00

0.1388310

08.98141

100.00Px O

2 0

01.113025

72.27890

26.10188 0.506232

0100.00

Py O 2

0 0

72.27891.113025

00.506232

26.101880

100.00Pz O

2 4.590735

44.134760

021.65692

0 0

29.61645100.00

V

alence bond theory treatment of a tetrahedral m

olecule: the bonding in CH

4 The overlap of the sp 3 hybrid orbitals on C

with the 1s orbitals on the H

atoms gives four C

-H (sp 3)-

1s s bonds oriented 109.47° from each other. This provides the tetrahedral geom

etry predicted by V

SE

PR

theory.

2s2p

C

C*

sp3

C* (sp

3)

2s2p

1s

1s

1s

1s

C

4 H C*

HC H

HH

sp3

C* 3

MO and Valence Bond

MOLECULAR ORBITALS of METHANE

Molecular Orbital Coefficients

MO: 1 2 3 4 5 6 7 8Eigen values: -1.06145 -0.48908 -0.48908 -0.48908 0.17127 0.17127 0.17127 0.19048 (ev): -28.88346 -13.3084 -13.3084 -13.3084 4.66049 4.66049 4.66049 5.1831 A1 T2 T2 T2 T2 T2 T2 A1 1 H0 S 0.31023 0.43746 -0.42317 0.00882 -0.43909 -0.43195 0.00987 -0.392122 C1 S 0.78425 0 0 0 0 0 0 0.62045 3 C1 PX 0 0.01279 -0.0016 -0.7112 0.01424 0.00158 0.70273 0 4 C1 PY 0 -0.49444 -0.51131 -0.00774 -0.49277 0.50113 0.00886 0 5 C1 PZ 0 0.51121 -0.4945 0.01031 0.501 0.49285 -0.01126 0 6 H2 S 0.31023 -0.48614 -0.21685 -0.29524 0.49311 -0.21527 -0.29997 -0.392127 H3 S 0.31023 -0.15614 0.14235 0.57086 0.15813 0.14528 0.57738 -0.392128 H4 S 0.31023 0.20481 0.49767 -0.28443 -0.21215 0.50194 -0.28728 -0.39212 SQUARES MO: 1 2 3 4 5 6 7 8Eigen values: -1.06145 -0.48908 -0.48908 -0.48908 0.17127 0.17127 0.17127 0.19048 (ev): -28.88346 -13.3084 -13.3084 -13.3084 4.66049 4.66049 4.66049 5.1831 A1 T2 T2 T2 T2 T2 T2 A1 1 H0 S 0.0962427 0.191371 0.179073 7.78E-05 0.1928 0.186581 9.74E-05 0.1537582 C1 S 0.6150481 0 0 0 0 0 0 0.3849583 C1 PX 0 0.000164 2.56E-06 0.505805 0.000203 2.5E-06 0.493829 0 4 C1 PY 0 0.244471 0.261438 5.99E-05 0.242822 0.251131 7.85E-05 0 5 C1 PZ 0 0.261336 0.24453 0.000106 0.251001 0.242901 0.000127 0 6 H2 S 0.0962427 0.236332 0.047024 0.087167 0.243157 0.046341 0.089982 0.1537587 H3 S 0.0962427 0.02438 0.020264 0.325881 0.025005 0.021106 0.333368 0.1537588 H4 S 0.0962427 0.041947 0.247675 0.0809 0.045008 0.251944 0.08253 0.153758 % SQUARES MO: 1 2 3 4 5 6 7 8Eigen values: -1.06145 -0.48908 -0.48908 -0.48908 0.17127 0.17127 0.17127 0.19048 (ev): -28.88346 -13.3084 -13.3084 -13.3084 4.66049 4.66049 4.66049 5.1831 A1 T2 T2 T2 T2 T2 T2 A1 1 H0 S 9.6242653 19.13713 17.90728 0.007779 19.28 18.65808 0.009742 15.375812 C1 S 61.504806 0 0 0 0 0 0 38.495823 C1 PX 0 0.016358 0.000256 50.58054 0.020278 0.00025 49.38295 0 4 C1 PY 0 24.44709 26.14379 0.005991 24.28223 25.11313 0.00785 0 5 C1 PZ 0 26.13357 24.45303 0.01063 25.1001 24.29011 0.012679 0 6 H2 S 9.6242653 23.63321 4.702392 8.716666 24.31575 4.634117 8.9982 15.375817 H3 S 9.6242653 2.43797 2.026352 32.58811 2.50051 2.110628 33.33677 15.375818 H4 S 9.6242653 4.194714 24.76754 8.090042 4.500762 25.19438 8.25298 15.37581

Methane M

olecular Orbital D

iagram (A

M1 L

evel)

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.

-30

-25

-20

-15

-10

-5

0

5

10

Ener

gy (e

V)

åååå

åååå åååå åååå åååå

A2u

Eu

B1g

A1g

Eu

Au

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.