Problem 3: Ionisation energies and photoelectron spectra

Ionisation energies and photoelectron spectra of nitrogen, oxygen and nitric oxide, with particular reference to a case study on the ion states of nitrogen

Objectives

The purpose of this exercise is to look at photo-electron spectroscopy for three diatomic molecules and compare these against the results of molecular orbital calculations. Since two of the molecules are open shell, you will gain some experience in this aspect of ab initio molecular orbital theory. The very simple experimental results for nitrogen are not easily predicted by simple theory, so as a case study we explore how a variety of methods of increasing sophistication have to be used in order to get agreement, even qualitatively, with experiment. In doing so it introduces several new features of the Gaussian programs, such as the introduction of correlation by perturbation methods. 

Section 1: A photoelectron spectrum

Study the attached spectra for the molecules nitrogen, nitric oxide and oxygen. Draw out the molecular orbital energy diagrams that appear to fit these spectra. (Ignore the peak marked A for oxygen.) Comment on the values of the ionisation energies for the core electrons. 

Section 2: Calculations on O2 and NO

You already have a calculation for N2 from the simple examples section. Use this to see if the theory agrees with the experiment for the top few photoelectron bands.

The oxygen molecule has 16 electrons, but the highest occupied molecular orbital is an antibonding [pi] 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 [alpha] spin and 7 electrons of [beta] 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=1.208
The 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 [alpha] and [beta] electrons occupy different orbitals, each singly. For the oxygen molecule there are 9 electrons of [alpha] spin occupying 9 orbitals. There are 7 electrons of [beta] 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=1.208
Again use the very general Gaussian interface.
You will notice that the energy is slightly lower than in the ROHF case. You will also see reference in the output to <S2>. This is the average value of S2, which should be S(S + 1) or 1(1 + 1) = 2.0 in this case, since S = 1 (S = the number of unpaired electrons x 1/2). The value is actually slightly higher. The UHF does not give a pure spin state. The triplet is slightly contaminated by other spin states. The difference however is usually small and one should not trust the results if it is not.

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. 

Section 3: Background on nitrogen problem

In the earlier example you should have found that the highest occupied molecular orbital is a [sigma] orbital lying just above the doubly occupied [pi] orbitals. This is consistent with the experimental evidence from photoelectron spectroscopy that shows that the lowest state of the ion N2+ is a [SIGMA] state and the next lowest state is a [PI] state. It seems reasonable that the lowest state of the ion arises from removal of an electron in the highest occupied molecular orbital of the parent molecule. This result was obtained in the earliest set of ab initio calculations on diatomic molecules by B. J. Ransil (Rev. Mod. Phys., 32, 245, 1960). He used a minimal basis set of Slater orbitals directly, not Slater orbitals fitted by a number of Gaussian functions.

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 [pi] orbitals here lie above the [sigma] 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.

Table goes here!

The experimental results for the difference in energy between the nitrogen molecule and its ion states, the ionisation energies, are 0.580 for the [SIGMA] state and 0.624 for the [PI] 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.

  1. The molecular orbital wave function would have to be an exact solution of the Schrödinger equation. However good we make the molecular orbitals, this is never so. Pairing electrons in the same molecular orbital neglects the tendency of these electrons to keep apart. We only include an average repulsion. The molecular orbital total energy always lies above the exact energy by an amount called the correlation energy. We now have methods for calculating at least part of this correlation energy.

    2.  
  2. The molecular orbitals would have to remain the same on removal of the electron. In other words, the wave function for the ion would have to be constructed from the same molecular orbitals as that for the molecule. This also does not hold. On ionisation the orbitals relax and are different for the ion from the molecule. This is called the relaxation effect and it can be calculated by carrying out a separate calculation of the total energy of each ion state and comparing this with the total energy of the molecule. This method is usually called the [ Greek Delta ]SCF method and it avoids the use of the orbital energies entirely. To get at the correlation effect, we again carry out separate calculations on the ion states and the molecule, but now we must do a calculation that is better than the molecular orbital method. Gaussian allows us to do this using the Møller-Plesset perturbation method to second (MP2), third (MP3) or fourth (MP4) order. Although MP2 gives us only a small part of the correlation energy, it is sufficient in this case.

Section 4: A case study on molecular nitrogen

The first part of the exercise looks at when the change over from the Ransil to the Cade et al. order occurs. We have already seen that the STO-3G basis set gives the Ransil order. As a reminder, the data for the nitrogen molecule is: 
        # HF STO-3G

        Nitrogen molecule - STO-3G at R=1.098

        0 1
        N
        N 1 R

        R=1.098
Other 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 [pi] 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 [sigma] level is highest. For the cation, the UHF method therefore removes a [sigma] electron and this orbital occupancy will continue throughout the SCF process. Thus the following data will do a calculation on the ion in the [SIGMA] state, not the [PI] state: 

        # UHF 6-31G*

        Nitrogen molecule ion - sigma state - 6-31G at R=1.098

        1 2
        N
        N 1 R

        R=1.098
There 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=1.098
This 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 [SIGMA] state.

Again use the very general Gaussian interface.

The [PI] 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 7
GUESS=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 [sigma] orbital is the 7th orbital, while the [pi] orbitals are the 5th and 6th orbitals. We need to change 5 and 7 for the [beta] electrons only. The program assumes that the number of [alpha] electrons is always greater than the number of [beta] electrons, so the ionised electron is a [beta] 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 [alpha] orbitals, only the [beta] 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 [beta] 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 [PI] state of the ion, the UHF method removes an electron from one of the degenerate [pi] 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?


This exercise has been published by Brian O'Leary and Brian Salter-Duke in the Journal of Chemical Education, 1995, 72, 501 - 504.

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