Self Consistent Field Theory

Examples

1. (a) Sketch the normalised H 1s orbital.

(b) Calculate the ratio $\sigma_g^2$(midpoint of bond)/ $\sigma_g^2$(on atom) for H₂, using a simple form for the $\sigma_g$ orbital (R=1.4a_o)

2. Write down the form of the molecular orbitals for N₂, O₂ and HF. Give the form of the molecular orbital wavefunction for N₂.

3. Use sp³ hybrid orbitals to write down a valence bond wavefunction for ethane, and sp² hybrids for a VB wavefunction for ethylene.

4. Find the symmetry groups of CO₂,O₃,NH₃,C₂H₄,HC$\ell$,UF₆,H₂O₂,B₂H₆,(C₅H₅)₂Fe.

5. For three H atoms on a line, construct the secular matrix $\langle\eta_i\vert F\vert\eta_j\rangle$, where $\eta_i$ is a 1s function on atom i (use the values $\alpha=\langle\eta_i\vert F\vert\eta_i\rangle,\beta=\langle\eta_1\vert F\vert\e......\langle\eta_2\vert F\vert\eta_3\rangle,\langle\eta_1\vert F\vert\eta_3\rangle=0$ and neglect overlap). Find the eigenvalues and the molecular orbitals.

6. For three H atoms in an equilateral triangle, construct three orthonormal symmetry orbitals $\phi_1,\phi_2,\phi_3$ expressed in terms of the atomic 1s orbitals $\eta_1,\eta_2,\eta_3$. Evaluate the secular matrix $\langle\phi_i\vert F\vert\phi_j\rangle$. Hence determine the molecular orbitals, their energies and draw an orbital energy level diagram.

7. Use the answers to questions 5 and 6 to predict whether the reactions
H+H $_2\rightarrow$H₂+H
H+H $_2^+\rightarrow$H₂⁺+H
have linear or equilateral transition states. (Remember these are approximate theories)

8. Determine the symmetry orbitals for CH₄ and use them to construct a molecular orbital energy level diagram.

9. Derive all the symmetry orbitals that can be constructed from the H(1s) and C(2s,2p_x,2p_y,2p_z) orbitals of C₂H₄ (molecule lies in xy plane with CC being the x axis). Sketch the occupied molecular orbitals of C₂H₄ given that the orbital energy level ordering is 1a_g<b_3u<b_2u<2a_g<b_1g<b_1u<b_2g.

10. Construct an orbital energy level digram for BH₃.

11. Start from question 10 to construct a Walsh diagram for the distortion of an AH₃ molecule from planar to pyramidal geometry (D $_{3h}\rightarrow$C_3v). Use the descent-in-symmetry tables to obtain the symmetry species in C_3v from those in D_3h. What do the results tell you about the shapes of BH₃, CH₃ and NH₃? What features would you expect to see in the first band of the NH₃ photoelectron spectrum?

12. Find the symmetry orbitals of C₂H₂. Sketch the molecular orbitals whose energy ordering is $\Sigma_g^{+}<\Sigma_u^{+}<\Sigma_g^{+}<\Pi_u<\Pi_g<\Sigma_u^{+}$. Why do the lowest excited states formed by a $\pi_u\rightarrow\pi_g$ excitation have either C_2v or C_2h symmetry and not D $_{\infty h}$ symmetry? Hence give the symmetries of the four lowest triplet states.

13. Find linear combinations of the atomic $\pi$ orbitals on C atoms at the corners of a square which have symmetries a_2u, e_g and either b_1u or b_2u depending on your choice of axes. Then use Huckel theory to calculate the delocalisation energy of tetramethylene-cyclobutane.

14. (a) Obtain the molecular orbital coefficients and energies for butadiene by making use of symmetry orbitals. Use these molecular orbitals and first order perturbation theory to find the approximate orbital energies of N$\equiv$C-C$\equiv$N. Take $\alpha_N=\alpha+0.5\beta,\beta_{CN}=\beta$.

(b) Using orbital symmetry, solve the Huckel secular equations to find the `exact' orbital energies.

15. Construct orbital correlation diagrams for the conrotatory and disrotatory ring opening of cyclohexadiene. Determine which mode is favoured for thermal and for photochemical reaction.

16. Draw an orbital correlation diagram for the reaction of two ethane molecules forming cyclobutane via a rectangular transition state. Will the reaction proceed thermally? Is there an alternative geometry for the transition state that is more likely to lead to thermal reaction?

17. Interpret in as much detail as possible the attached SCF outputs for CO₂ and O₃. Walsh's rules for triatomic molecules predict that those with 16 valence electrons are linear, while those with 17,18,19 or 20 valence electrons are bent. Use symmetry arguments and the SCF outputs to explain the derivation of these rules.

18. (a) Show that the $\pi$ molecular orbital coefficients c_ra for a cyclic polyolefin, with N atoms, $\phi_r=\sum c_{ra}\pi_a$, may be given by $c_{ra}=\sin 2\pi ra/N$ or $c_{ra}=\cos2\pi ra/N$. Hence obtain the molecular orbital energies, and comment on the ground state configuration of (4n+2) $\pi$-electron systems and 4n $\pi$-electron systems. For benzene give explicit forms for the molecular orbitals and their energies. Discuss the approximations involved in Huckel theory, their application to benzene, and the probable accuracy of these orbitals and energies.

(b) Use perturbation theory to calculate the energy of the lowest orbital of naphthalene, starting from the orbitals in (a) above.

19. What are the symmetries of the vibrations of formaldehyde? (Take the x axis to be perpendicular to the plane of the molecule.) Specify (by giving the number of quanta in each mode) a vibrational state with A₂ symmetry.

20. Determine the symmetries of the vibrations of ethene. (let CC be the z axis, and the molecule lie in the yz plane). Which vibrations are ir active?

  
    Self Consistent Field Theory

1998-09-23