Representation problem

Harriman (20) has shown that this map is "onto" i.e. any element of comes from at least one element of S. Note that this jH-operty does not rule out the possibility that an element of can also come from operators not in. This "onto" property should be compared to the case that arises in the N-representability problem (28) where not every positive two-particle operator comes from a state in fj so the contraction map in that case does not have the onto property. 

Levy, M., 1979, Universal Variational Functionals of Electron Densities, First Order Density Matrices, and Natural Spin Orbitals and Solution of the v-Representability Problem , Proc. Natl. Acad. Sci. USA, 16, 6062. 

The density functional (DF) method has been successful and quite useful in correlating experimental results when model densities are used in the calculations. In fact, the equations characteristic of the DF method can be derived from a variational approach as Kohn and Sham showed some time ago. In this approach, when model densities are introduced, it is not always possible to relate such densities to corresponding wave functions this is the N-representability problem. Fortunately, for any normalized well behaved density there exists a Slater single determinant this type of density is then N-representable. The problem of approximately N-representable density functional density matrices has been recently discussed by Soirat et al. [118], In spite... 

However, it is indeed fortunate that the IV-representability problem for the electron density p(r) greatly simplifies itself. In fact, the necessary and sufficient conditions that a given p(r) be /V-representable are actually given by Equation 4.5 above. Nevertheless, question remains Can the single-particle density contain all information about a many-electron system, at least in its ground state An affirmative answer to this question can be given from Kato s cusp condition for a nuclear site in the ground state of any atom, molecule, or solid, viz.,... 

But there is a more basic difficulty in the Hohenberg-Kohn formulation [19-21], which has to do with the fact that the functional iV-representability condition on the energy is not properly incorporated. This condition arises when the many-body problem is presented in terms of the reduced second-order density matrix in that case it takes the form of the JV-representability problem for the reduced 2-matrix [19, 22-24] (a problem that has not yet been solved). When this condition is not met, an energy functional is not in one-to-one correspondence with either the Schrodinger equation or its equivalent variational principle therefore, it can lead to energy values lower than the experimental ones. 

We have already introduced the notion of orbit in Sect. 2.6. Here we formalize this concept and show its implications for the functional iV-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... 

A quantum system of N particles may also be interpreted as a system of (r — N) holes, where r is the rank of the one-particle basis set. The complementary nature of these two perspectives is known as the particle-hole duality [13, 44, 45]. Even though we treated only the iV-representability for the particles in the formal solution, any p-hole RDM must also be derivable from an (r — A)-hole density matrix. While the development of the formal solution in the literature only considers the particle reduced Hamiltonian, both the particle and the hole representations for the reduced Hamiltonian are critical in the practical solution of N-representability problem for the 1-RDM [6, 7]. The hole definitions for the sets and are analogous to the definitions for particles except that the number (r — N) of holes is substituted for the number of particles. In defining the hole RDMs, we assume that the rank r of the one-particle basis set is finite, which is reasonable for practical calculations, but the case of infinite r may be considered through the limiting process as r —> oo. 

H. Kummer, iV-representability problem for reduced density matrices. J. Math. Phys. 8, 2063 (1967). 

F. Weinhold and E. Bright Wilson, Jr., Reduced density matrices of atoms and molecules. 11. On the V-representability problem. J. Chem. Phys. 47, 2298 (1967). 

The Ai-representability problem was defined in a remarkable paper by Coleman in 1963 [27]. This problem asks about the necessary and sufficient conditions that a matrix represented in a p-electron space must satisfy in order to be N-representable that is, the conditions that must be imposed to ensure that there exists an /-electron wavefunction from which this matrix may be obtained by integration over N-p electron variables. All the relations and properties that will now be described are the basis of a set of important necessary... 

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 ... 

These last two contractions (Eqs. (121) and (122)) are also very important since they lead to the 1-RDM and therefore the corresponding 2-RDM and 2-HRDM [24, 25, 83]. Hence it follows that the 2-RDM M-representability problem may be studied equivalently by focusing on the M-representability conditions for the 2-CM matrix [71, 83]. Thus the set of relations given above constitutes a set of M-representability conditions—strongly exacting and necessary conditions—not only for the 2-CM matrix but also for the 2-RDM as well as for the 2-HRDM. 

Any of the four conditions has an infinity of solutions. Actually, the energy is stationary for any eigenstate of the Hamiltonian, so one has to specify in which state one is interested. This will usually be done at the iteration start. Moreover, the stationarity conditions do not discriminate between pure states and ensemble states. The stationarity conditions are even independent of the particle statistics. One must hence explicitly take care that one describes an n-fermion state. The hope that by means of the CSE or one of the other sets of conditions the n-representability problem is automatically circumvented has, unfortunately, been premature. 

We may conclude that the 1-RDM and the functional 7/-representability problems are entirely different. The former is trivially solved since ONs sum up to the number of electrons N and lie between 0 and 1, assuring an N-representable 1-RDM. The latter refers to the conditions that guarantee the... 

M. Levy, Universal variational functionals of electron-densities, Ist-order density-matrices, and natural spin-orbitals and solution of the v-representability problem. Pmc. Natl. Acad. Sci. U.S.A. 76(12), 6062-6065 (1979). 

The importance of N-representability for pair-density functional theory was not fully appreciated probably because most research on pair-density theories has been performed by people from the density functional theory community, and there is no W-representability problem in conventional density functional theory. Perhaps this also explains why most work on the pair density has been performed in the first-quantized spatial representation (p2(xi,X2) = r2(xi,X2 xi,X2)) instead of the second-quantized orbital representation... 

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). 

W. T. Yang, P. W. Ayers, and Q. Wu, Potential functionals dual to density functionals and solution to the upsilon-representability problem. Phys. Rev. Lett. 92, 146404 (2004). 

The efforts by several very able quantum scientists in four countries in the period preceding 1972 had failed to obtain a complete solution of the N-representability problem. It was assumed that we would never find one. My announcement of the solution in June of that year at a Conference in Boulder was therefore greeted with incredulity except by Ernie Davidson who understood my argument immediately. 

---