Wednesday, July 1, 2009

Orbitals that do exist

Previously I discussed why I don't like the way some people talk about LUMO's and HOMO's. They don't really exist, i.e., they have no measurable properties.
They are very useful for qualitative discussions though.

Is there something better, that respects the fact that the ground state of molecules is really
a highly correlated quantum many-body state?

Yes. Natural orbitals. These are the eigenfunctions of the full one-particle density matrix.
The corresponding eigenvalues are the occupation of these orbitals.
Note these can have non-integer values.
There is a connection to Fermi liquid theory, which I hope to come back to....

A nice introduction can be found in
Sections 1.5 & 1.6 of the wonderful book, Valency and Bonding: A Natural Bond Orbital Donor-Acceptor Perspective, by Frank Weinhold and Clark Landis.
The book shows how to ground chemical intuition (orbitals, lewis pairs, bonding) in a rigorous theoretical framework.


  1. I'm not sure that I understand how natural orbitals exist more than canonical ones. Hartree-Fock orbitals are also natural (in that case, there is no distinction). If non-idempotency is allowed in the 1-matrix (as implied by fractional occupations), then the reduction of an arbitrary state can be represented. This should also be true of the canonical orbitals, local orbitals, what-have-you... one can choose the representation. The most meaningful orbital basis would be one which is leads to a formally idempotent 1-matrix for the state upon reduction. If the state is strongly correlated, though, this may not be compatible with the restriction that the 1-body basis be orthonormal. Then, formal idempotency can be achieved, but at the cost of inducing a non-flat metric on the space of density matrices.

  2. Another good explanation of Natural Orbitals is given at the NBO homepage

    Since the Canonical Molecular Orbitals (e.g., HOMO and LUMO) are related to Natural Bond Orbitals by a unitary transformation, it seems to me that neither is more real than the other.

    The orbitals are each determined through different criterion, which are doubly-occupying the lowest energy orbitals for CBO's and maximizing two-center orbitals for NBO's (very roughly speaking).

    But they each have different and useful interpretations. CMO's are often related to energy differences observed in spectra, while NBO's are related to "Lewis bonding" within molecules.

    Personally, I find NBO's very useful in the teaching of General and Organic Chemistry, as they quantitatively and visually support the concept of forming molecular bonding orbitals from atomic hybrid orbitals (which in turn are formed from atomic orbitals). They also quantitatively support the intuition that chemists use to understand chemical bonding in large molecules.