Friday, June 5, 2009

Am I HOMO- and LUMO-phobic?

Molecular orbital theory is a wonderful theory which has provided many qualitative insights into the electronic structure, optical and magnetic properties, and reactivity of organic molecules. Schematics such as that below can be extremely useful.

However, it is important to appreciate its limitations. In its simplest version it completely neglects interactions between electrons. Bear in mind molecular orbitals are just a theoretical construct. They do not actually exist. One can certainly calculate theoretically molecular orbital energies. However, these energies (e.g., of the HOMO and LUMO) can not be measured. What one can measure (and also calculate theoretically) are the energy of quantum states that do exist and obtain quantities such as
  • ionisation energy, I
  • electron affinity, A
  • electrochemical oxidation potential
  • electrochemical reduction potential
  • energy of the lowest lying singlet excited state, E(S1)
  • energy of the lowest lying triplet excited state, E(T1)
Molecular orbital theory predicts I-A=E(S1)-E(S0)=E(T1)-E(S0). Often these quantities differ by energies of the order of 0.1-1 eV.
Hence, we should never expect molecular orbital theory to be quantitatively reliable and the figure above to be able to give a quantitative description of quantities such as open circuit voltages of solar cells based on donor-acceptor complexes.


  1. The states which come out of quantum chemistry calculations are statistical ensembles defined over the set of exact (in the basis) solutions of the Born-Oppenheimer electronic structure problem. In some useful cases the convergence criteria can be interpreted as maximum entropy ensembles subject to constraints, and these ans├Ątze are useful because they are easy to interpret if the constraints can be compactly expressed. The molecular orbital picture results from a factorization of the N-body ensembles, and are dispersion-free in the context of the ensemble averaged state and operator, not in the context of the exact solution for a single molecule.

    Everything I have said derives from a principle of statistical complementarity which is built into the structure of quantum mechanics (see ET Jaynes, Phys. Rev. 1957 vol. 108 (2) pp. 171-190).

    For a discussion of maximum entropy interpretations of SCF states, see Tishby & Levine, Chem. Phys. Lett., 1984 vol. 104 (1) pp. 4-8. The coupled cluster density matrix can trivially shown to be of maximum entropy form, but we do not offer a proof here...

  2. In defence of HOMOs/LUMOs

    At the risk of sounding much more like a deconstructionist than I'd ever want to, it all depends what those words mean.

    If we take HOMO and LUMO to be the objects that appear in molecular orbital theory then I agree with everything you've said.

    But following this week's COPE meeting where we discussed Peter I. Djurovich, Elizabeth I. Mayo, Stephen R. Forrest, Mark E. Thompson, Organic Electronics 10 (2009) 515–520 "Measurement of the lowest unoccupied molecular orbital energies of molecular organic semiconductors" it occurs to me that there may be a "better" way. [Better, perhaps, in the sense of a fight that's easier to win!]

    What they do, essentially (although I don't think they'd put it like this, and I don't know the literature well enough to say this is really new - but it was new to me), is to propose new definitions of the words HOMO and LUMO, so that they way they are typically used in the field of organic electronics is actually correct.

    HOMO = energy required to remove one electron from a molecule (starting in its neutral state) in the solid state

    LUMO = energy required to add one electron to a molecule (starting in its neutral state) in the solid state

    excition binding energy = difference between LUMO energy (as defined above) and the HOMO energy + the optical absorption. In an MO picture we can then think of this as the "binding energy" as being due to the interaction of the "hole in the HOMO" and the electrons.

    This makes be kind of happy. All the energies here are now defined in terms of many-body states that really exist and can be measured and so we don't don't have to rely on theoretical fictions.

    On the other hand this does mean that one needs to be careful about redefining the terms HOMO and LUMO every time you use them - so in my ideal world I'd prefer some other term. (But perhaps that's just asking for the moon on a stick).

    I asked the room why people would want to keep the terms HOMO and LUMO rather than, say, talking about solid state ionisation energies and electron affinities.
    - BK suggested that perhaps one would like to retain these terms for in vacuo or in solution measurements.
    - Paul B suggested that these terms are so ingrained in the literature that it's best just to redefine them.
    I guess that both these reasons are probably true of a sociological level, but I, personally, would rather get rid of the words HOMO and LUMO.

    Nevertheless three things are clear: (i) If you want to use the terms HOMO and LUMO you need to be very clear about what you mean (and make sure that your audience is too), (ii) if you're going to quote values for HOMO/LUMO energies in a talk/paper you need to make absolutely clear where that number came from, and (iii) if you are talking in terms of HOMOs and LUMOs you absolutely need to think carefully about the exciton binding energy as well.

    Anyway thanks to Ellen for suggesting the paper. I really enjoyed reading it and yesterday's discussion.

  3. The problem isn't really in the "MO" part, but in the "HO/LU" part. An MO doesn't exist in the same way that your average family with 1.75 adults and 1.5 kids doesn't really exist. They are statistical factorizations (quantum-statistical, but the idea is analogous).

    The problem really is that the terms "Highest Occupied" and "Lowest Unoccupied" implies that one has in hand a set of orthogonal projectors which differentiate occupied and unoccupied spaces. This won't correspond to physical reality because specifying the many-body information will destroy 1-body information - in other words the "HOMO" and "LUMO" aren't necessarily going to be mutually exclusive (orthogonal), because they are quantum-statistical factorizations on different states.

    This may not become apparent if one studies a homologous series of molecules, because as long as the 'topology' of chemical space is respected (i.e. the bonding or pair information, does not change too much), then these problems may cancel out in the 1-body analysis. It may be a reasonable empirical relationship, in other words.

  4. I don't agree. The problem is MOs. They don't exist. If they did then there wouldn't be all the problems described in Ross' post. I think this is clear from how far you have to move the goal post of what an orbital is in order to maintain the fiction at all.

  5. Keeping the MO's either completely occupied or unoccupied (0 or 1 eigenvalue of the appropriate 1-electron RDM) and referring to many-body states will generally not be possible. If the state is truly many-body ('strongly correlated') then it cannot be represented as a direct product of 1 body states with integer occupation (you can't span the right space with the unitary equivalence class generated by the outer products of pure 1-RDMs). If one insists on keeping integer occupation, and a set of pure 1-RDMs, then the only way to generate enough freedom is to allow the 1-RDMs to be non-orthogonal. However, this is just a hack fix because that space could be spanned just as well by orthogonal 1-RDMs with non-equal coefficients in the Hilbert-Schmidt space. In a wavefunction representation, this means destroying normalization...

  6. Regarding the argument about existence, in a "real" sense, of pure states of a system at ANY level, this is a question that quantum mechanics can never answer, due to a principle of statistical complementarity built into its structure. See, for example, Jaynes. Information theory and statistical mechanics. II. Physical review (1957) vol. 108 (2) pp. 171-190

    Obviously, I am speaking from quite a hard deconstructionist perspective. Gustavus non disputandum est (sic).