CHEM0019 Computational Experiment 1: Hückel Theory and Molec-

ular Orbitals
Molecular orbitals (MOs) are solutions to the electronic Schrödinger equation and provide well-defined ener-
gies for the electrons in a molecule. They are written as a Linear Combination of Atomic Orbitals (LCAO)
X
ψi = cai ϕa
a

where cai coefficient representing the contribution of AO ϕa to the MO, ψi . The Schrödinger equation can
then be written in a matrix form
(H − ϵS) c = 0
and solved to get energies (eigenvalues) and MO coefficients (eigenvectors).
The Hückel Method
This is a simple way of solving the Schrödinger equation for aromatic molecules. In its basic form, the
following approximations are made:

1. Look at π-orbitals only.

2. Each carbon 2pπ electron has the same energy, α
3. Interactions on adjacent atoms equal, energy β

4. Interactions on non-adjacent atoms zero

Ethene
The Hückel form of the Schrödinger equation for ethene is solved as followed:
1. Set up secular equations   
α−ϵ β c1
=0
β α−ϵ c2

2. To obtain eigenvalues (energies) solve secular determinant

α−ϵ β
=0
β α−ϵ

3. Put energies back in to secular equations and solve to get the coeffients that describe the MOs in terms
of the AOs.
The resulting energies and orbitals can be plotted as an MO Diagram

α−β ψ2 = √1 (pz,1 − pz,2 )

2

√1 (pz,1
α+β ψ1 = 2
+ pz,2 )

α is energy of an electron in an isolated p-orbital

β is interaction energy of electrons in neighbouring p-orbitals
α + β is energy of a π-bond (β < 0)
The factor of √12 on the orbital coefficients is from the normalisation condition a c2ai = 1
P

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For Butadiene, the Hückel secular determinant to be solved is

α−ϵ β 0 0
β α−ϵ β 0
=0
0 β α−ϵ β
0 0 β α−ϵ

which results in the following MO diagram.

α−1.62β

α−0.62β
α
α+0.62β

α+1.62β

The Butadiene MOs from Hückel theory have the following composition.

ψ4 = −0.37pz,1 + 0.6pz,2 − 0.6pz,3 + 0.37pz,4

ψ3 = −0.60pz,1 + 0.37pz,2 + 0.37pz,3 − 0.60pz,4

ψ2 = 0.60pz,1 + 0.37pz,2 − 0.37pz,3 − 0.60pz,4

ψ1 = 0.37pz,1 + 0.6pz,2 + 0.6pz,3 + 0.37pz,4

Analysis
Using the MO energies and occupations, the following quantities are useful for describing a molecule:
1. Total π-energy X
E= n i ϵi (1)
i
2. Delocalisation Energy
Edeloc = E − N (α + β) (2)

3. π-electron charge on atom a (net charge Qa = 1 − qa )

X
qa = ni c2ai (3)
i
4. π bond order between atoms a and b X
pab = ni cai cbi (4)
i

where the indices a, b refers to an AO and i to an MO, ni are the number of electrons in an MO and cai are
the AO coefficients.
Note that a bond order of 1 denotes a double bond, and 0 a single.

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Ethene Analysis
1. E = 2(α + β)
2. Edeloc = 0

3. q1 = q2 = 1 Q1 = Q2 = 0
4. p12 = 1
Butadiene Analysis
1. E = 4α + 4.48β

2. Edeloc = 0.48β
3. q1 = q2 = q3 = q4 = 1 i.e. no charge migration has occurred.
4. p12 = p34 = 0.90 p23 = 0.45 i.e. conjugation has occurred.

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In this practical we shall be using WebMO. This is an interface to quantum chemistry programs that are
able to calculate the electronic wavefunction and energies using a variety of methods. Answer the questions
by filling in the tables and spaces in the workbook. All energies should be reported in eV unless otherwise
asked for.

1. Start internet browser

2. Open webmo.chem.ucl.ac.uk/

This will open the login page for WebMO. Please log in with the username and password which have been
provided to you. Note that for this to work on your computer, you must either be connected to eduroam,
using your VPN or using Desktop@UCL.

The first calculations will use the Extended Hückel method: an extension to Hückel theory that includes
the σ orbitals. We still want to use the analysis and pictures of the simple Hückel theory, so after each
calculation the π orbitals need to be located and the orbital energies found.
Note that the programs will continue to use the term ”Hückel”, when it is in fact the extended Ḧuckel
method. Use the correct terminology in your writing.

Ethene
1. Set up ethene
– click on “New Job” Menu and select “Create New Job”
– click on “build icon”
– you can select atom types with the periodic table icon. Carbon is the default. Click twice to add
2 carbon atoms.
– click on first atom and drag to second to form a bond.
– right-click on the bonds to change to a double bond.
– click on “Cleanup” −→ “Add Hydrogens”
– click on “Cleanup” −→ “Mechanics Optimize” to get a sensible geometry.
– click on “Calculate” −→ “Symmetry”−→ “Symmetrize”. Check molecule has D2h symmetry. If
not, click on “Symmetrize”
2. Click on bottom right arrow to go on to calculations. If you are asked to “Choose Computational
Engine”, then select “Gaussian” and click on the bottom right arrow to continue. Otherwise just
proceed with the next step.
3. Configure job.
– For “Calculation” choose “Geometry Optimisation”
– For “Theory” choose “Other” and enter “Huckel”
– For “Basis Set” choose “Other” and remove what is entered to leave this blank.
4. Submit the job by clicking on the bottom right arrow.
5. When the job has finished, click on its name (or the magnifying glass icon) to open the results. Click
on “New job using this geometry”
6. Go through the pages using the bottom right arrow as before to “Configure Gaussian Job Options”.
This time choose “Molecular Orbitals” as the “Calculation” and submit as before.

When this second job is finished, open the results (by clicking on the name or magnifying glass icon in the
job manager). We are now ready for the analysis.
Questions 1.

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 (a) To view the orbitals go to the bottom of the results where there is a table of orbitals and energies.
Click on one of the magnifying glass icons to open the viewing panel. Search through orbitals (select
in turn) and find the 2 π-orbitals. There will be 1 occupied and 1 unoccupied (occupation numbers are
in the table). Make a note below of which number MOs these are and their energies.
πocc πunocc
MO no. E (au) MO no. E (au)

(b) In the notes above you will see that in Hückel theory the orbitals have energies in terms of the Hückel
Parameters Eocc = α + β and Eunocc = α − β. From the orbital energies, calculate the values of α and
β? Values in the table are in atomic units (Hartree). Convert the result to eV (1 Hartree = 27.21 eV).

α=

β=

Butadiene
Following the prescription for ethene, set up 1,3-butadiene in a trans conformation. Check that the molecule
has C2h symmetry and do the 2 step calculation as before
1. Geometry optimisation
2. Molecular orbitals
starting the second calculation from the result of the first as a “New job using this geometry”. Choose the
same options for “Theory” and “Basis set”, i.e. “Other(Huckel)” and “Other()”.
When the second job is finished, open the results (by clicking on the name or magnifying glass icon in the
job manager).
Questions 2.

(a) Search through orbitals and find the 4 π-orbitals. There will be 2 occupied and 2 unoccupied. Make a
note of which orbitals these are, and draw out an MO diagram (like those in the notes) with the orbital
energies and a sketch of the orbitals.
MO no. E (au) MO diagram

(b) In the notes above, the butadiene orbital energies are given in terms of the Hückel α and β parameters.
Calculate the expected values using your values of α and β calculated above. Tabulate these values

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 with the ones calculated by WebMO. How do they compare?
MO no. WebMO E (eV) Extended Hückel E (eV)

(c) Open the calculation data file by clicking on “Raw output” on the left hand tool bar. Scroll down the
output to find the “Molecular Orbital Coefficients”. This is a table with the MO numbers at the top of
columns with the coefficients of the atomic orbitals (AOs) used to build them. Find the four columns
for the four π orbitals and tabulate the coefficients of the carbon 2PZ orbitals. Compare these to the
values in the notes above. Check the pattern of signs is the same.
MO no. Orbital Coefficients

(d) Calculate the total energy in the π system using Eq. (1).
Etot =

(e) Calculate the delocalisation energy in the π system using Eq. (2).
Edeloc =
(f) Using the AO coefficients, calculate the charges on the atoms using Eq. (3)
q1 = q2 = q3 = q4 =

(g) Using the AO coefficients, calculate the bond orders for the 3 bonds using Eq. (4). Sketch the molecule
with the orders labeling the appropriate bonds.
p12 = p23 = p34 =

Cyclo-butadiene
Again following the prescription for ethene, now set up cyclo-butadiene. Check that the molecule has D4h
symmetry and do the 2 step calculation as before, choosing the same options for “Calculation”, “Theory”
and “Basis set”:
1. Geometry optimisation
2. Molecular orbitals
starting the second calculation from the result of the first as a “New job using this geometry”.
Open the results and answer the following questions. Questions 3.

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 (a) Search through orbitals and find the 4 π-orbitals. Make a note of which orbitals these are, and draw
out an MO diagram with the orbital energies and a sketch of the orbitals.
MO no. E (au) MO diagram

(b) Using the Aufbau principle, add electrons to the MO diagram. Which is the likely to be the most
stable configuration? Is this a singlet or a triplet? (Use your chemical knowledge rather the occupation
numbers in calculation results).
(c) Calculate the total energy in the π system using Eq. (1).
Etot =

(d) Calculate the delocalisation energy in the π system using Eq. (2).
Edeloc =
(e) Is the cyclic molecule more or less stable than the linear?

Cyclo-pentadienyl Radical
In the final exercise we will look at the cyclopentadienyl radical, C5 H5 . Set this up, check that the molecule
has D5h symmetry and do the 2 step calculation as before, choosing the same options for “Calculation”,
“Theory” and “Basis set”:
1. Geometry optimisation
2. Molecular orbitals

You will have to select that the multiplicity is a “Doublet”

Open the results and answer the following questions.
Questions 4.

(a) Search through orbitals and find the 5 π-orbitals. Make a note of which orbitals these are, and draw
out an MO diagram with the orbital energies and a sketch of the orbitals.

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 MO no. E (au) MO diagram

(b) We have calculated the neutral molecule and specified that this is a doublet. What does this mean?

Using the Aufbau principle, add the π electrons to the MO diagram.

(c) Repeat the MO diagram and fill it with the π electrons for the cyclopentadienyl anion C5 H−
5 and cation
C5 H +
5 .
MO diagram anion MO diagram cation

(d) What is the multiplicity of the anion and the cation?

(e) Calculate the total energy in the π systems using Eq. (1) for the neutral, anion and cation. Calculate
the delocalisation energy in the π system using Eq. (2) for the neutral, anion and cation.
Etot Edeloc

Neutral

Cation

Anion

(f) Estimate the energy for electron attachment and ionisation of the cyclopentadienyl radical.

Eattach = Eionise =

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 (g) Which of the three charge states will be most stable? Why?

Accurate MO calculations
More accurate calculations of MOs is done using Hartree-Fock theory, with Møller-Plesset 2nd order pertur-
bation theory required for good energies. Repeat the cyclopentadienyl radical calculations (optimisation +
orbitals) for the neutral molecule, but this time use the options
• Theory −→ Møller-Plesset 2
• Basis Set −→ Routine: 6-31G(d)
• Charge −→ 0
• Multiplicity −→ Doublet
Do the same again, now for the anion
• Theory −→ Møller-Plesset 2
• Basis Set −→ Routine: 6-31G(d)
• Charge −→ -1
• Multiplicity −→ Singlet
and the cation
• Theory −→ Møller-Plesset 2
• Basis Set −→ Routine: 6-31G(d)
• Charge −→ +1
• Multiplicity −→ Singlet
You may be able to get the ions to have D5h symmetry, but the neutral will have C2v symmetry.
Questions 5.
(a) Search through orbitals from the 3 calculations and find the 5 π-orbitals. Tabulate the orbital energies
below.
(b) Calculate the total energy in the π systems using Eq. (1) and add them to the table.
EM O (neutral) EM O (anion) EM O (cation)

Π1

Π2

Π3

Π4

Π5

Etot = Etot = Etot =

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(c) Estimate the energy for electron attachment and ionisation using the Hückel analysis of the π orbitals
(i.e. use Etot in the above table).

Eattach = Eionise =

(d) In the output data, the overall MP2 energy is given. Use the differences in energies between the three
molecules to get a better estimate the energy of electron attachment and ionisation of the cyclopenta-
dienyl radical.

Eattach = Eionise =

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Comment on the difference between the 3 estimates of the ionisation and electron attachment energies.

Discuss briefly how well the extended Hückel approach compares to the Hartree-Fock (Møller-Plesset 2)
results (both orbital energies and energy differences).

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