A-representability problem

Then, in the Old Ages (1940 or 1951-1967) some ingenious people became aware that, in the case of two-body interactions, it is the two-particle reduced density matrix (2-RDM) that carries in a compact way all the relevant information about the system (energy, correlations, etc.). Early insight by Husimi (1940) and challenges by Charles Coulson were completed by a clear realization and formulation of the A-representability problem by John Coleman in 1951 (for the history, see his book [1] and Chapters 1 and 17 of the present book). Then a series of theorems on A-representability followed, by John Coleman and many... 

The 2-RDM, the 2-HRDM, and the G-matrix are the only three second-order matrices which (by themselves) are Hermitian and positive semidefinite thus they are at the center of the research in this field. Recently, a formally exact solution of the A -representability problem was published [12] but this solution is unfeasable in practice [40]. 

C. Valdemoro, L. M. Tel, and E. Perez-Romero, A-representability problem within the framework of the contracted Schrodinger equation. Phys. Rev. A 61, 032507 (2000). 

Although a formal solution of the A-representability problem for the 2-RDM and 2-HRDM (and higher-order matrices) was reported [1], this solution is not feasible, at least in a practical sense [90], Hence, in the case of the 2-RDM and 2-HRDM, only a set of necessary A-representability conditions is known. Thus these latter matrices must be Hermitian, Positive semidefinite (D- and Q-conditions [16, 17, 91]), and antisymmetric under permutation of indices within a given row/column. These second-order matrices must contract into the first-order ones according to the following relations ... 

As an extension of the A-representability problem, Valdemoro and co-workers introduced the 5-representability problem [14], that is, the incomplete knowledge of the set of necessary and sufficient conditions that a p-RDM must fullfil in order to ensure that it derives from an A-electron wavefunction having well-defined spin quantum numbers ... 

Unfortunately, the A-representability constraints from the orbital representation are not readily generalized to the spatial representation. A first clue that the A-representability problem is more complicated for the spatial basis is that while every A-representable Q-density can be written as a weighted average of Slater determinantal Q-densities in the orbital resolution (cf. Eq. (54)), this is clearly not true in the spatially resolved formulation. For example, the pair density (Q = 2) of any real electronic system will have a cusp where electrons of opposite spin coincide but a weighted average of Slater determinantal pair densities,... 

P. W. Ayers and M. Levy, Generalized density-functional theory conquering the A-representability problem with exact functionals for the electron pair density and the second-order reduced density matrix. J. Chem. Set 117, 507-514 (2005). 

H. Kummer, A-representability problem for reduced density matrices. / Math. P/iyi. 8,2063-2081 (1967). 

The two-electron reduced density matrix is a considerably simpler quantity than the N-electron wavefunction and again, if the A -representability problem could be solved in a simple and systematic manner the two-matrix would offer possibilities for accurate treatment of very large systems. The natural expansion may be compared in form to the expansion of the electron density in terms of Kohn-Sham spin orbitals and it raises the question of the connection between the spin orbital space and the -electron space when working with reduced quantities, such as density matrices and the electron density. 

Another conceptual problem with the HK theorem, much better known and more studied than nonuniqueness, is representability. To understand what representability is about, consider the following two questions (i) How does one know, given an arbitrary function n(r), that this function can be represented in the form (8), i.e., that it is a density arising from an antisymmetric A-body wave function T(rx... rN) (ii) How does one know, given a function that can be written in the form (8), that this density is a ground-state density of a local potential u(r) The first of these questions is known as the A-representability problem, the second is called u-representability. Note that these are quite important questions if one should find, for example, in a... 

Most of the better approximate functionals give reasonable, albeit unspectacular, results for the kinetic energy when accurate atomic and molecular densities are used. Flowever, when the functionals are used in the variational principle for the electron density, the results deteriorate catastrophically, giving kinetic energies, total energies, and electron densities that are qualitatively incorrect. These failings may be attributed to the A-representability problem for density functionals the kinetic energy functional does not correspond to an acceptable A-electron system. 

Pistol, M. E. Investigations of random pair densities and the application to the A-representability problem. 2007, 449, 208-211. 

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