CM3109 Practical 3.
CM4113 Years Work
Using Symmetry to simplify the calculation of the Huckel molecular orbitals of naphthalene.
Calculating the Huckel molecular orbitals of naphthalene does not involve solving a 10 x 10 secular determinant. The molecule only has three symmetry distinct carbon atoms:
carbons 1, 5 and 9. If the LCAO coefficient of carbon 1 has been determined to be c1, then clearly the coefficient of carbons 2, 3 and 4 are ±c1. Similarly carbons 5, 6, 7 an 8 are related and carbons 9 and 10.
All possible relationships can be generated by using the mirror planes perpendicular to the molecular plane:
c1 = c2 = c3 = c4
c5 = c6 = c7 = c8
c9 = c10
a 3x3 problem
c1 = c2 = -c3 = -c4
c5 = c6 = -c7 = -c8
c9 = c10 = 0
a 2x2 problem
c1 = -c2 = -c3 = c4
c5 = -c6 = -c7 = c8
c9 = -c10
a 3x3 problem
c1 = -c2 = c3 = -c4
c5 = -c6 = c7 = -c8
c9 = c10 = 0
a 2x2 problem
Notice that in 2 cases it is only possible to satisfy the left/right mirror symmetry if the coefficients on carbons 9 and 10 are zero. From these 4 diagrams 10 values of the orbital energy and the corresponding molecular orbitals can be calculated.
The three secular equations needed are:
n = 1: x c1 + c5 + c9 = 0
n = 5: c1 + x c5 + c6 = 0
n = 9: c1 + c4 + xc9 +c10 = 0 (this one only needed in two cases).
For each set of symmetry conditions solve the secular equations for the values of x and hence the molecular orbital energy ε. For each value of x < 0 (bonding orbitals) calculate the molecular orbital coefficients. NOTE, the normalization requires that
c12 + c22 + … + c102 = 1
which reduces to
4c12 + 4c52 + 2c92 = 1
Show that the net charges on carbon atoms 1, 5 an 9 are zero. Calculate the bond order of the 4 distinct C-C bonds in the molecule and estimate their bond lengths. Compare with experimental data.
Use the web site: http://www.chem.ucalgary.ca/SHMO/ to check your results.
The report should contain a good presentation of the 10 energy levels, the 5 bonding molecular orbitals (chemdraw can help with this) and charge and bond order calculations. Comment on the molecular orbital energies.