Tuesday, August 2, 2011

What is a Pauling point?

I first heard of the term "Pauling point" in a chemistry seminar that Fritz Schaefer gave at UQ several years ago. He asked the audience if anyone knew what it was. Only one person knew (and they were not from UQ! It was David Sholl who was on sabbatical). I think the term is originally due to Per Lowdin who here explains what it is:
At the Valadalen symposium in 1958, the author pointed out [Ref. 26, p. 23] that a characteristic feature of quantum chemistry was that even a fairly simple theory could sometimes give excellent agreement with experimental experience, but that this agreement may disappear whenever one tries to improve the theory. The point of excellent agreement was coined the "Pauling point" in honour of one of the great pioneers in our field who is also present here in Dubrovnik, not only because he could construct simple theories built on physical and chemical insight, but also because of his mastership in predicting figures which had not yet been measured. 
In the beginning of the 1930s one had constructed theories of chemical reactivity based on the properties of the valence electrons only to find that the good agreement disappeared when one included the inner shells leading to the concept of the "nightmare of the inner shells". In the MO-LCAO treatment of large molecules , one could get very good results without including the atomic overlap integrals, whereas in solid-state theory the inclusion may lead to the famous "non- orthogonality catastrophe". In the treatment of metal complexes, the original crystal- field theory for some reason seemed to give better agreement than the improved ligand- field theories. In the treatment of magnetic phenomena, the Hartree method seemed to give better results than the Hartree-Fock method, simply because the errors in treating parallel and antiparallel spins were better balanced in the former. Let me quickly add that my own doctoral thesis in 1948 treating the properties of ionic crystals by means of the independent-particle model is a typical example of a "Pauling point", where the good agreement with the experiments would disappear when one tries to include e.g. correlation in an unbalanced way. 
It goes without saying that, if one improves the theory more and more, the good agreement is expected to come back, but the simplicity of the theory is usually lost in this connection.
One should hence be somewhat suspicious, if a low-order perturbation theory seems to give excellent results - one may be at a "Pauling point".
The painful reality is that in quantum many-body theory we often do perturbation theory in dimensionless coupling constants that are of order one. This is not just in quantum chemistry. Another case is in spin-wave theory for quantum Heisenberg models where one expands in powers of 1/S where S is the total spin (usually S=1/2).

In lattice QCD (Quantum ChromoDynamics) one should worry about the size of the lattice a that one is using to approximate the space-time continuum. This PRL is one example of how a judicious choice of an effective Hamiltonian (action in field theory) [the O(a) technique] can give quite reasonable results for a relatively coarse lattice.

In quantum chemistry, one is often truncating other things such as the basis set for atomic orbitals or the size of the active space in CAS methods. One is always a long way from the asymptotic limit at which one expects to get the exact result. Yet there are many people who seem to assume that the bigger the basis set or the larger the active space the better. i.e., one is necessarily getting closer to the exact answer. The experience of "Pauling points" clearly shows this is not necessarily true and caution is in order.


  1. It seems to me that being able to identify "Pauling Points" is a good thing... if you ask the right questions. For example, what does the occurrence of a Pauling point tell you? Finding a set of Pauling points which give similar answers at different levels (but for which intermediate levels fail) seems to be telling one about the "irreducible" subspaces of the system.

    Let's say you have a particular low-rank model (say a CASSCF model) that seems to be providing a good description. Now assume that there is a disjoint subspace (not in the original active space) which interacts strongly with itself but not with the original space because the cross-space couplings only vanish due to inclusion of sufficient cross-terms. Then at intermediate levels, where the second space is only partially included, one could see spurious "unphysical" interaction emerging between the two spaces that vanishes when enough operators are included so as to make the coupling zero again.

  2. I agree that identifying a "Pauling point" may be a good thing. One could argue that one is finding an effective Hamiltonian (and reduced Hilbert space) which describes the essential physics and chemistry.
    But, how does one actually show that this is the case?
    Indeed, this is the whole point of O(a) methods in lattice QCD.

    Fritz Schaefer (not surprisingly) has a different view. He says "The Pauling point is where you most easily get the right answer for the wrong reason."