The Klyachko paradigm in theoretical chemistry

It is well known that the problem of determining the energy of molecules and other quantum many-body systems reduces in the standard approximation to optimizing a simple linear functional of a twelve-variable object, the two-electron reduced density matrix (2-RDM). The difficulty is: the variation ensemble for that functional has never been satisfactorily determined. This is known as the $N$-representability problem of quantum chemistry (which to a large extent is a problem of quantum information theory), and remains without satisfactory solution.

The situation has given rise to competing research programs, typically trading more complicated functionals for simpler representability conditions. Chief among them, and historically the first, is density functional theory, based on a three-variable object for which $N$-representability is trivial, whereas the exact functional is very strange indeed, and probably forever unknowable. An intermediate position is occupied by 1-RDM functional theory.

Ensemble representability for 1-RDMs was solved 50 years ago. However, only recently, thanks to outstanding work by Klyachko on generalized Pauli constraints, real progress has been made on pure representability for 1-RDM. These constraints determine small polytopes of admissible pure $N$-representable sets of 1-RDMs. Somewhat mysteriously, physical states seem to cling to the boundary of the polytopes. We speak of pinning when there are saturated constraints, implying strong selection rules which drastically simplify the configurations. Quasi-pinning appears to be ubiquitous. We review recent numerical evidence and theoretical justification for this phenomenon.