Seth Olsen and I had a nice discussion this week about Monkhorst's paper, Chemical physics without the Born-Oppenheimer approximation: The molecular coupled-cluster method, who emphasizes that corrections to the Born-Oppenheimer approximation can be as large as ten per cent. This raises important questions about the dreams and dogmas of computational quantum chemists. The goal is to calculate energies (especially heats of reaction, binding energies, and activation energies) to "chemical accuracy" which is of the order of kBT, about 1 kcal/mol or 0.03 eV. This is much better than most methods can do.

The claim of computational chemists is that it just a matter of more computational power. In principle, if one uses a large enough basis set (for the atomic orbitals) and a sophisticated enough treatment of the electronic correlations, then one will converge on the correct answer. However, this is all done assuming the Born-Oppenheimer approximation and "clamped" nuclei.

But what about corrections to Born-Oppenheimer? Are there any specific cases of molecules for which any deviations between experiment and computations might be due to non-BOA corrections?

Furthermore, there is a tricky issue of "parametrising" and "benchmarking" functionals, pseudo-potentials, and results against experiment. How can one separate out effects due to correlations and those due to corrections to BOA?

The claim of computational chemists is that it just a matter of more computational power. In principle, if one uses a large enough basis set (for the atomic orbitals) and a sophisticated enough treatment of the electronic correlations, then one will converge on the correct answer. However, this is all done assuming the Born-Oppenheimer approximation and "clamped" nuclei.

But what about corrections to Born-Oppenheimer? Are there any specific cases of molecules for which any deviations between experiment and computations might be due to non-BOA corrections?

Furthermore, there is a tricky issue of "parametrising" and "benchmarking" functionals, pseudo-potentials, and results against experiment. How can one separate out effects due to correlations and those due to corrections to BOA?

Indeed, the presumption of monotonic convergence is routinely made in cases where there is no clear justification. This is particularly riven with this in the excited-state field, when state-averaged techniques are used. In this case, although the bracketing theorem provides some justification in the sense that each state is an upper bound to an exact state, and the state beneath it a corresponding lower bound, this does not provide that much help. It is still very possible to converge to many different model spaces with different choices of active space at any given rank, and the choice becomes "which is correct"? It is not clear that the state-averaged energy is as useful a guide as a ground-state energy (although even the latter does not guarantee a low dispersion to the state nor good estimation for non-energy properties, as we have previously discussed many times).

ReplyDeleteThe observation that there are multiple accessible solutions for active spaces of a given electron and orbital rank implies that monotonic convergence should NOT be expected. Nevertheless, this assumption is implicitly made throughout the literature.

It is very interesting to note that the 10% error that Monkhorst suggests can arise from first order BOA errors is entirely consistent with the observation that no electronic structure method in existence can systematically predict excitation energies of common organic chromophores to better than ~0.1eV.

Hi Seth,

ReplyDeleteWhat do you mean by monotonic convergence?

Is this the notion that as one increases the basis set or active space or number of reference states that the result should uniformly converge to the correct answer?

Well, a little more severe; the assumption that the spectral representation of the model Hamiltonian converges in a monotonic fashion on the exact lowest energy subspace as more basis vectors are appended to the set used for the variational estimate. This is pretty clearly untrue. It may also be untrue for any exact target space, but this seems like a less obvious assertion.

ReplyDeleteOn second thought, this is wrong becausevwhen the active orbital space is enlarged, it is less simple than appending vectors to a space. It appends excitations to the operator algebra, and then solution of the casscf can move in a complicated way. The problem is that the casscf community doesn't have a good language to describe how the solutions evolve along different series of orbital additions, nor a consistent way of comparing solutions along different branches. If such a language is known in physics, then it should be brought to the attention of the quantum chemistry community.

ReplyDelete