Saturday, February 27, 2010

Getting to the heart of quantum chemistry

For physicists like me struggling to understand quantum chemistry it often seems a plethora of acronyms, software, and approximations that appear to be detached from my knowledge of quantum many-body theory and chemical intuition. However, Seth Olsen recently recommended a review article by Michael Schmidt and Mark Gordon to me. I have started reading it and am fin
ding it understandable and helpful. I loved the introductory paragraph:

The essence of chemistry involves processes such as the formation and dissociation of chemical bonds, the excitation of an atom or molecule into a higher electronic state, and atomic or molecular ionization, in which electron pairs are separated. Although the end points of such processes (that is, the reactants and products) may frequently be reasonably well described using a simple wavefunction that corresponds to a single Lewis structure, this often cannot be said for the key species in between, such as transition structures, reactive intermediates, and excited electronic states. Such species often must be described with more complex wavefunctions in which several different arrangements of the electrons (electronic configurations) are taken into account.
[Aside: practically all chemical research concerning new materials for energy and information technologies is concerned with "key species in between". Density functional theory based methods work for the starting and ending species but generally fail miserably for the in between species.]
The key problem the article addresses is
The general form of a MCSCF [Multi-Configuration Self-Consistent Field] wavefunction is

which is a linear combination of several configurations [referred to as configuration state functions (CSFs), ΦK] [i.e. Slater determinants]. Each CSF differs in how the electrons are placed in the MOs [Molecular Orbitals], i. The MOs are usually expanded in a basis of AOs [Atomic Orbitals], χμ. A MCSCF wavefunction is one in which both the configuration mixing coefficients AK and the MO expansion coefficients Cμiare variationally optimized. Such a wavefunction is therefore distinct from a configuration interaction (CI) wavefunction, in which only the configuration mixing coefficients are variationally optimized.


  1. That's right, but it's important to understand that there are linear dependencies in the problem, between the orbital coefficients and the CI coefficients. The number of linearly independent number of constraints satisfied at convergence is less than the number of derivatives of the orbitals and CI coefficients combined.

  2. This is one of the benefits of active-space approaches as opposed to a general MCSCF approach, the ability to readily identify/remove these linear dependencies (with the obvious downside of active space approaches being the limited number of orbitals one can include in the active space). Another important thing to note is that there are a vast number of molecular systems that exhibit multireference effects even at equilibrium, so you're not limited to the "species in between".