V-representability problems

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. 

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). 

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

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 expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

Related to the v-representability problem is the Kohn-Sham v-representability problem. That is, given a system of interest, can one always find an internal potential, w(r), such that the ground-state electron density of the Kohn-Sham model system is the same as that of the state of interest Again, the answer seems to be no [87], but if one allows fractional occupation numbers of the Kohn Sham orbitals, then no essential difficulties arise [3,88,89]. We note that in this case, the idempotency constraint, Eq. (71), is no longer appropriate, and the less stringent Eq. (70) should be used instead. 

Yet, there appears the so-called V-representability problem signaling the impossibility of an a priori selection of the external potentials t)q)es that are in bi-univocal relation with ground state of an electronic system (Chen Stott, 1991a,b Kryachko Ludena, 1991a,b). The problem was revealed as very difficult at mathematical level due to the equivocal... 

Electron Densities, First-Order Density Matrices, and Natural Spin-Orbitals and Solution of the v-Representability Problem. 

W. Kohn and L. H. Sham, Phys. Rev. A, 140, 1133 (1965). Self-Consistent Equations Including Exchange and Correlation Effects. M. Levy, Proc. Natl. Acad. Sci. U.S.A., 76, 6062 (1979). Universal Variational Functionals of Electron Densities, First-Order Density Matrices, and Natural Spin-Orbitals and Solution of the v-Representability Problem. R. G. Parr and W. Yang, Eds., Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1988. J. Labanowski and J. Andzelm, Eds., Density Functional Methods in Chemistry, Springer Verlag, Heidelberg, 1991. L.J. Bartolotti and K. Flurchick, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1995, Vol. 7, pp. 187-216. An Introduction to Density Functional Theory. 

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