Project 2: Constructing Walsh Diagrams for a Small Polyatomic Molecule

Depict an ab initio Walsh diagram with respect to the specified geometrical parameter for an assigned molecule using the Hartree-Fock 6-31G** level of theory. Include only the valence orbitals: you may omit core orbitals. One molecule will be assigned per individual.

One of the following

1.

H₂S (¹A₁) with respect to the bond angle

2.

CH₂ (¹A₁) with respect to the bond angle

3.

SO₂ (¹A₁) with respect to the bond angle

4.

CO₂ ( $^1\Sigma_g^+$) with respect to the bond angle

5.

CS₂ ( $^1\Sigma_g^+$) with respect to the bond angle

6.

HCN ( $^1\Sigma^+$) with respect to the bond angle

7.

NNO ( $^1\Sigma^+$) with respect to the bond angle

8.

OCS ( $^1\Sigma^+$) with respect to the bond angle

9.

C₂H₂ ( $^1\Sigma_g^+$) with respect to the cis and trans bends

10.

Si₂H₂ ( $^1\Sigma_g^+$) with respect to the cis and trans bends

11.

HNNH ( $^1\Sigma_g^+$) with respect to the cis and trans bends

12.

H₂O₂ (¹A) with respect to the torsional angle

13.

H₂S₂ (¹A) with respect to the torsional angle

14.

NH₃ (¹A) with respect to the bond angle

15.

BH₃ (¹A_g) with respect to the bond angle

16.

H₂CO (¹A₁) with respect to the out-of-plane angle

17.

H₂CS (¹A₁) with respect to the out-of-plane angle

To prepare the Walsh diagram, first you will need to optimize the geometry at the given (6-31G** HF) level of theory. Then, you will need to perform single-point energies at a series of geometries (10$^{\rm o}$ increments are suggested) displaced from the equilibrium geometry. Bond lengths should be fixed at their equilibrium values; only the given parameter (usually an angle) should be allowed to change from its equilibrium value. At each geometry, perform the 6-31G** HF single-point computation and tabulate the orbital energies of all the valence orbitals (both occupied and unoccupied). If in doubt, tabulate more orbitals than you need, and cut them out later. This will keep you from having to re-run anything. You might be able to cut and paste the relevant orbital energies into a file to avoid having to write them all down. For at least one geometry point, you will need to record the orbital energies for all occupied orbitals, along with the total Hartree-Fock energy, to answer one of the questions in the discussion.

Orbitals should be labeled with their appropriate symmetry designations (e.g., 5a₁) in the diagram; refer to the Q-Chem output or the Walsh diagrams in Herzberg. Note that for some molecules the Q-Chem program may choose a different axis convention than Herzberg, making some irrep labels swap. For example, for a C_2v molecule, what Q-Chem calls the 1b₁ orbital might be called 1b₂ in Herzberg, depending on whether the molecule lies in the xz or yzplane. In such cases, Herzberg's choice is usually more acceptable.

Questions for Discussion:

1.

Is the Walsh diagram consistent with the theoretically determined equilibrium geometry of the given state?

2.

Consider a different electronic state obtained by moving an electron from the highest-occupied MO (HOMO) to the lowest-unoccupied MO (LUMO). Without doing any more computations, what would you predict for the geometry of this electronic state based on the Walsh diagram?

3.

An implicit assumption of Walsh diagrams is that the total energy is the sum of the ``orbital binding energies.'' Is the total electronic energy from Hartree-Fock equal to the sum of the occupied orbital energies?