# 22.2.3: iii. Problems

## Q1

Given the following orbital energies (in hartrees) for the N atom and the coupling elements between two like atoms (these coupling elements are the Fock matrix elements from standard ab-initio minimum-basis SCF calculations), calculate the molecular orbital energy levels and 1-electron wavefunctions. Draw the orbital correlation diagram for formation of the $$N_2$$ molecule. Indicate the symmetry of each atomic and molecular orbital. Designate each of the molecular orbitals as bonding, non-bonding, or antibonding.

$N_{1s} = -15.31^{\text{*}}$ $N_{2s} = -0.86^{\text{*}}$

$N_{2p} = -0.48^{\text{*}}$

$N_2 \sigma_g \text{Fock matrix}^{\text{*}}$

\begin{bmatrix} -6.52 & & \\ -6.22 & -7.06 & \\ 3.61 & 4.00 & -3.92 \end{bmatrix}

$N_2 \pi_g \text{ For Matrix}^{\text{*}}$

$[0.28]$

$N_2 \sigma_u \text{ For Matrix}^{\text{*}}$

\begin{bmatrix} 1.02 & & \\ -0.60 & -7.59 & \\ 0.02 & 7.42 & -8.53 \end{bmatrix}

$N_2 \pi_u \text{ Fock matrix}^{\text{*}}$

$[-0.58]$

*The Fock matrices (and orbital energies) were generated using standard STO3G minimum basis set SCF calculations. The Fock matrices are in the orthogonal basis formed from these orbitals.

## Q2

Given the following valence orbital energies for the C atom and $$H_2$$ molecule draw the orbital correlation diagram for formation of the $$CH_{2}$$ molecule (via a $$C_{2v}$$ insertion of C into $$H_2$$ resulting in bent $$CH_2$$). Designate the symmetry of each atomic and molecular orbital in both their highest point group symmetry and in that of the reaction path ($$C_{2v}$$).

\begin{align} C_{1s}=-10.91^{\text{*}} & & H_2 \sigma_g = -0.58^{\text{*}} \\ C_{2s}=-0.60^{\text{*}} & & H_2 \sigma_u = 0.67^{\text{*}} \\ C_{2p}=-0.33^{\text{*}} & & \end{align}

*The orbital energies were generated using standard STO3G minimum basis set SCF calculations.

## Q3

Using the empirical parameters given below for C and H (taken from Appendix F and "The HMO Model and its Applications" by E. Heilbronner and H. Bock, Wiley- Interscience, NY, 1976), apply the Hückel model to ethylene in order to determine the valence electronic structure of this system. Note that you will be obtaining the 1-electron energies and wavefunctions by solving the secular equation (as you always will when the energy is dependent upon a set of linear parameters like the MO coefficients in the LCAO- MO approach) using the definitions for the matrix elements found in Appendix F.

$C\alpha_{2p\pi} = -11.4 eV$

$C \alpha_{sp^2} = -14.7 eV$

$H \alpha_s = -13.6 eV$

$C-C\beta_{2p\pi -2\pi} = -1.2 eV$

$C-C\beta_{sp^2-2p^2} = -5.0 eV$

$C-H\beta_{sp^2-s} = -4.0 eV$

1. Determine the C=C $$(2\pi )$$ 1-electron molecular orbital energies and wavefunctions. Calculate the $$\pi \rightarrow \pi^{\text{*}}$$ transition energy for ethylene within this model.
2. Determine the C-C ($$sp^2$$) 1-electron molecular orbital energies and wavefunctions.
3. Determine the C-H ($$sp^2$$-s) 1-electron molecular orbital energies and wavefunctions (note that appropriate choice of symmetry will reduce this 8x8 matrix down to 4 2x2 matrices; that is, you are encouraged to symmetry adapt the atomic orbitals before starting the Hückel calculation). Draw a qualitative orbital energy diagram using the HMO energies you have calculated.

## Q4Q5

Using the empirical parameters given below for B and H (taken from Appendix F and "The HMO Model and its Applications" by E. Heilbronner and H. Bock, Wiley- Interscience, NY, 1976), apply the Hückel model to borane ($$BH_3$$) in order to determine the valence electronic structure of this system.

$B \alpha_{2p\pi} = -8.5 eV$

$B \alpha_{sp^2} = -10.7 eV$

$H\alpha_s = -13.6 eV$

$B-H \beta_{sp^2-s} = -3.5 eV$

Determine the symmetries of the resultant molecular orbitals in the $$D_{3h}$$ point group. Draw a qualitative orbital energy diagram using the HMO energies you have calculated.

## Q5

Qualitatively analyze the electronic structure (orbital energies and 1-electron wavefunctions) of $$PF_5$$. Analyze only the 3s and 3p electrons of P and the one 2p bonding electron of each F. Proceed with a $$D_{3h}$$ analysis in the following manner:

1. Symmetry adapt the top and bottom F atomic orbitals.
2. Symmetry adapt the three (trigonal) F atomic orbitals.
3. Symmetry adapt the P 3s and 3p atomic orbitals.
4. Allow these three sets of $$D_{3h}$$ orbitals to interact and draw the resultant orbital energy diagram. Symmetry label each of these molecular energy levels. Fill this energy diagram with 10 "valence" electrons.