Problem Set #6
Note:Relevant sections in Mortimer are Chapter 18 and Appendix H.
Valence Bond Method; Slater Determinants
1. Use the following spin functions
to write the four complete wave functions for H2 (including the spin component) using the following Heitler-London valence bond wave functions and . What are the degeneracies of the energy levels described by these wave functions? Hint: Your wave functions for H2 should be in terms of
2. Express as Slater determinants the 4 valence bond wave functions for H2 (found in the previous problem).
3. Consider the ClF molecule to consist of two ions of opposite charge separated by the bond length of 0.163 nm. Calculate the dipole moment for this model.
4. The Cl-F, Cl-Cl, and F-F bond energies are 255 kJ/mol, 243 kJ/mol, 159 kJ/mol, respectively. Using the Pauling definition of electronegativity (Mortimer, Chapter 18), calculate the difference in electronegativities of Cl and F.
5. An alternative method for calculating electronegativities is the Mulliken definition given by , where EN is the electronegativity, IE is the ionization energy, and EA is the electron affinity. The ionization of Cl is 1251.1 kJ/mol and its electron affinity is kJ/mol. Using the Mulliken definition, calculate the electronegativity of Cl.
Huckel Method; Secular Determinants
6. What are the 3 approximations of the Huckel method?
7. Using the Huckel approximations, write the secular determinant for hexatriene. If we were to expand this determinant, we would obtain where . The general solutions for x for this type of determinant are given by where j labels different energy states with ; while is the number of carbons in the molecule. Find expressions for each of the six energy states of hexatriene in terms of and .
8. Trial wave functions for hexatriene are constructed by taking a linear combination of the basis functions as follows where r = 1,2,3,4,5,6. The coefficients can be determined from the equation
where j = 1,2,3,4,5,6.
Write the Huckel molecular orbital wave functions for hexatriene in terms of
9. The 6 electrons of hexatriene are placed in the bonding orbitals (the first three energy states). Determine the total energy and the delocalization energy for hexatriene. Express both answers in terms of and/or . Hint: The delocalization energy is the energy by which the molecule is stabilized relative to isolated double bonds. The total energy for one molecule of ethylene is .
10. Using the Huckel approximations, write the secular determinant for butadiene, and determine the energy states of the four LCAO-MOs.
11. Using the Huckel approximations, write the secular determinant for cyclobutadiene. (Please note that there is no need to determine the energy states for this problem.)