The oxygen molecule has 16 electrons, but the highest occupied molecular orbital is an antibonding orbital. This means that two electrons occupy the pair of molecular orbitals and by Hund's rule have their spins parallel. Thus we have 9 electrons of spin and 7 electrons of spin. The state is a triplet state; the multiplicity is 3. The simplest conceptual method is the Restricted Open shell Hartree-Fock (ROHF) method. The data for oxygen is:
# ROHF STO-3G POP=REGULAR Oxygen molecule - STO-3G at R=1.208 0 3 O O 1 R R=1.208The ROHF method is obtained by replacing HF by ROHF on the command line. The triplet state is specified by a 3 on the charge-multiplicity line. Otherwise the data is fairly similar to that for the nitrogen molecule.
To run these open-shell systems you need the very general Gaussian interface.
This allows you to enter the charge and multiplicity and also to select the ROHF method. Run this data and examine the orbital energies and coefficients. Do these agree with the simple qualitative molecular orbital method for diatomic molecules?
Although the ROHF is conceptually simple, it is not as computationally convenient as another approach to the open shell problem. In particular it is not readily extended to include correlation effects beyond the molecular orbital approach. The other method is the unrestricted Hartree-Fock (UHF) method. This is a "Different Orbitals for Different Spin" method. This means that the and electrons occupy different orbitals, each singly. For the oxygen molecule there are 9 electrons of spin occupying 9 orbitals. There are 7 electrons of spin occupying a quite different set of 7 orbitals.
To run the UHF method, just change ROHF to UHF on the command line, or of course on the method menu. The data should be:
# UHF STO-3G POP=REGULAR Oxygen molecule - STO-3G at R=1.208 0 3 O O 1 R R=1.208Again use the very general Gaussian interface.
Consider how these ROHF and UHF results can be used to interpret the experimental spectrum.
Now carry out ROHF and UHF STO-3G calculations using the very general Gaussian interface on NO (use the experimental bond length of 1.15Å) and consider how they can be used to interpret the experimental spectrum. In the UHF case, there can be some convergence difficulties in the SCF iterations. Put SCFCYC=200 in the command line to ensure that the maximum number of iteration cycles is increased beyond the default of 64 to 200. A better basis set such as 6-31G* gives significantly better results for an open shell system such as NO.
These brief runs should give you good experience for the next part.
However later, a very much better molecular orbital calculation was carried out on the nitrogen molecule. This calculation, by P. E. Cade, K. D. Sales and A. C. Wahl (J. Chem. Phys., 44, 1973, 1966), showed a different result. The orbitals here lie above the orbital in apparent contradiction with the experimental result. There is no doubt that this result is correct. The best molecular orbital calculations do not seem to predict the experimental result. These results are shown in the table below.
The experimental results for the difference in energy between the nitrogen molecule and its ion states, the ionisation energies, are 0.580 for the state and 0.624 for the state. These are in the theoretical unit of Hartree, as are the orbital energies in the table.
One of the few books that gives a very good account of this problem is Atoms and Molecules, M. Karplus and R. N. Porter, W. A. Benjamin, Inc., 1970, pages 347 - 351.
Orbital energies can be used to predict ionisation energies through Koopmans' Theorem which identifies the negative of the orbital energy with the ionisation energy. We can now see that while Ransil's result gives the correct order for the two states, the numerical agreement with experiment is poor. Koopmans' Theorem would be exact if two conditions hold and they do not.
# HF STO-3G Nitrogen molecule - STO-3G at R=1.098 0 1 N N 1 R R=1.098Other basis sets can be used by just replacing STO-3G by another basis set code. Run the nitrogen molecule with the following basis sets:
STO-6G 3-21G 6-31G 6-31G* 6-311G*Again use the very general Gaussian interface.
6-311G is a triply split valence set with the addition of d polarisation functions. Note that the basis set 6-31G** is identical to 6-31G* as there are no hydrogen atoms. You will find that only a slight improvement in the basis set is sufficient to change the order of the orbitals from the Ransil order to the Cade et al order. If you were to run the following basis sets:
6-311G(2d) 6-31G(2df) 6-311G(3df)which add further d and f polarisation functions to the basis set, you would confirm that the Cade et al order still holds as the molecular orbitals are improved even further. These results are too time consuming. If you are interested, the results are available for your inspection.
The second part of the exercise looks at the relaxation and correlation effects. Here it is best to make a compromise between accuracy and economy, so we shall restrict the calculations to the 6-31G* basis set. The orbitals here are clearly in the Cade et al. order with the level highest. However inspection of the order of the orbitals at the beginning of the SCF iterative process where the program starts from the semi-empirical INDO orbitals, shows that here the level is highest. For the cation, the UHF method therefore removes a electron and this orbital occupancy will continue throughout the SCF process. Thus the following data will do a calculation on the ion in the state, not the state:
# UHF 6-31G* Nitrogen molecule ion - sigma state - 6-31G at R=1.098 1 2 N N 1 R R=1.098There are two changes from the molecule data. First, we use the unrestricted Hartree-Fock (UHF) method since the ion is open shell. Second, the charge-multiplicity line is altered to 1 2 since the charge is now +1 and the state is a doublet.
Since the MP2 method carries out a full molecular orbital calculation before calculating the correlation correction to the energy, we can get at the relaxation and correlation effects in one hit. We amend the data to:
# UMP2 6-31G* Nitrogen molecule ion - sigma state - 6-31G* at R=1.098 1 2 N N 1 R R=1.098This will do a UHF molecular orbital calculation followed by the MP2 correction for the ion. Run this data and note both the molecular orbital and MP2 total energies. To get the MP2 energy for the nitrogen molecule, go back and amend the molecule data for the 6-31G* basis set by replacing HF by MP2. This will repeat the molecular orbital calculation done earlier and add the MP2 correction. Again note both the molecular orbital and the MP2 total energies. The difference between the molecule and ion energies gives the ionisation energy for the state.
Again use the very general Gaussian interface.
The state is a little more tricky and we have to introduce one other feature of the Gaussian programs. We now have a state for the ion where the electron is removed not from the highest occupied molecular orbital of the initial INDO guess, but from a lower orbital. The UHF method assumes the electron is from the highest level, so we have to fool the program by exchanging the order of the molecular orbitals. We do this with GUESS=ALTER on the command line. The data is:
# UMP2 6-31G* GUESS=ALTER Nitrogen molecule ion - pi state - 6-31G* at R=1.098 1 2 N N 1 R R=1.098 5 7GUESS=ALTER tells the program that the orbitals that are used as the initial guess for the iterative SCF process are to be altered. The UHF method uses separate orbitals for the electrons of each spin. The orbital is the 7th orbital, while the orbitals are the 5th and 6th orbitals. We need to change 5 and 7 for the electrons only. The program assumes that the number of electrons is always greater than the number of electrons, so the ionised electron is a electron. Thus we have to have two blank lines after the variable line defining R. The first terminates the variable section. The second says we are not altering the orbitals, only the orbitals. Note that there still has to be a final blank line.
Again use the very general Gaussian interface.
It is easier to enter the altered orbitals in the Z-matrix window as it is clearer how many blank lines have been entered. This does not alter the result. The second window is just added to the first one.
For the state of the ion, the UHF method removes an electron from one of the degenerate molecular orbitals. The result is that the degeneracy is removed. The UHF method is not the proper method to use; we should be using a technique that preserves this degeneracy. Nevertheless we do get a reasonable estimate of the total energy, so do not worry about the rather odd orbital energies.
Having obtained energies for the ion and the molecule at both Hartree-Fock
and MP2 level, you can work out the ionisation energies as energy differences.
Does taking relaxation and correlation into account give a correct theoretical
prediction of the experimental result? Which effect is most important? How
good are the quantitative predictions?