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Physical Review A

v-representability for systems with low degeneracy

Jiqiang Chen, Malcolm J. Stott

Published Sep 1, 1991• 34 citations
Publisher

Abstract

We consider the v-representability of the particle density for a noninteracting system of spinless fermions by introducing the idea of proper order of a set of energy levels. It is shown that if ${\mathit{E}}_{1}$(\ensuremath{\lambda}), ${\mathit{E}}_{2}$(\ensuremath{\lambda}), and ${\mathit{E}}_{3}$(\ensuremath{\lambda}) are three energy levels associated with some local potential ${\mathit{V}}_{\ensuremath{\lambda}}$(r) that is a continuous function of \ensuremath{\lambda}=(${\ensuremath{\lambda}}_{1}$,${\ensuremath{\lambda}}_{2}$,${\ensuremath{\lambda}}_{3}$) over all possible points \ensuremath{\lambda}, where ${\ensuremath{\lambda}}_{\mathit{i}}$ is the occupation number of the ith state and M=${\ensuremath{\lambda}}_{1}$+${\ensuremath{\lambda}}_{2}$+${\ensuremath{\lambda}}_{3}$ is the total number of particles distributed over the three levels, then there must be at least one \ensuremath{\lambda} for which the three levels are in so-called proper order, in which the levels below the highest occupied level are filled. This result provides a basis for the proof of ensemble v-representability of some N-particle density for which the ground-state degeneracy of the system is no more than three. As examples, three- and two-dimensional central systems are examined, and an N-particle central density is shown to be ensemble v-representable for small N (N\ensuremath{\le}14 and N\ensuremath{\le}9 for three- and two-dimensional cases, respectively). The implications for density-functional theory are discussed.

DOI
10.1103/physreva.44.2816

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•Jan 1, 2015•4 citations
We consider the v-representability of the particle density for a noninteracting system of spinless fermions by introducing the idea of proper order of a set of energy levels. It is shown that if E1{\mathit{E}}_{1}E1​(\ensuremath{\lambda}), E2{\mathit{E}}_{2}E2​(\ensuremath{\lambda}), and E3{\mathit{E}}_{3}E3​(\ensuremath{\lambda}) are three energy levels associated with some local potential V\ensuremathλ{\mathit{V}}_{\ensuremath{\lambda}}V\ensuremathλ​(r) that is a continuous function of \ensuremath{\lambda}=(\ensuremathλ1{\ensuremath{\lambda}}_{1}\ensuremathλ1​,\ensuremathλ2{\ensuremath{\lambda}}_{2}\ensuremathλ2​,\ensuremathλ3{\ensuremath{\lambda}}_{3}\ensuremathλ3​) over all possible points \ensuremath{\lambda}, where \ensuremathλi{\ensuremath{\lambda}}_{\mathit{i}}\ensuremathλi​ is the occupation number of the ith state and M=\ensuremathλ1{\ensuremath{\lambda}}_{1}\ensuremathλ1​+\ensuremathλ2{\ensuremath{\lambda}}_{2}\ensuremathλ2​+\ensuremathλ3{\ensuremath{\lambda}}_{3}\ensuremathλ3​ is the total number of particles distributed over the three levels, then there must be at least one \ensuremath{\lambda} for which the three levels are in so-called proper order, in which the levels below the highest occupied level are filled. This result provides a basis for the proof of ensemble v-representability of some N-particle density for which the ground-state degeneracy of the system is no more than three. As examples, three- and two-dimensional central systems are examined, and an N-particle central density is shown to be ensemble v-representable for small N (N\ensuremath{\le}14 and N\ensuremath{\le}9 for three- and two-dimensional cases, respectively). The implications for density-functional theory are discussed.
A particle-density distribution is said to be v-representable if it is the ground-state density for some potential v(r). By developing the idea of proper order, proposed earlier [J. Chen and M. J. Stott, Phys. Rev. A 44, 2816 (1991)], to any number of energy levels which are involved in possible ground-state configurations, we show that any reasonable density with an arbitrary number of spinless fermions is either pure-state or ensemble v-representable. This result supports the general applicability of the Kohn-Sham method [Phys. Rev. 140, A1133 (1965)] in density-functional theory.
The v-representability of the particle density for a system of noninteracting fermions is considered. It is shown that any reasonable spherical density for two spinless fermions is either the nondegenerate ground-state density or a linear combination of degenerate ground-state densities for some effective local potential, and is therefore ensemble v-representable. The results of model calculations are given that illustrate various possible situations for two-particle systems. An extension of this result for systems of more particles is made and the implications for density-functional theory are discussed.
A method is presented for relating a ground-state density distribution to the effective single-particle potential which has this density for its ground state for a noninteracting system of particles. The method is useful for systems consisting of a small number of spinless fermions and in the case of two particles a second-order, nonlinear differential equation relates the density and the effective potential. The question of vvv representability is addressed through examples of densities for one- and three-dimensional systems. It is found that densities of the form ${e}^{\ensuremath{-}ar}$ for two-particle systems are not ground-state vvv representable. The added flexibility of three dimensions over one allows an effective potential which does not necessarily have the chosen density as its ground state, but as its lowest nondegenerate state.
We show that the particle density $\ensuremath{\rho}(\mathbf{r})$ and the paramagnetic current density jp(r){\mathbf{j}}^{p}(\mathbf{r})jp(r) are not sufficient to determine the set of degenerate ground-state wave functions. This is a general feature of degenerate systems where the degenerate states have different angular momenta. We provide a general strategy for constructing Hamiltonians that share the same ground-state density pair yet differ in degree of degeneracy. We then provide a fully analytical example for a noninteracting system subject to electrostatic potentials and uniform magnetic fields. Moreover, we prove that when $(\ensuremath{\rho},{\mathbf{j}}^{p})$ is ensemble (v,A)(v,\mathbf{A})(v,A)-representable by a mixed state formed from rrr degenerate ground states, then any Hamiltonian $H({v}^{\ensuremath{'}},{\mathbf{A}}^{\ensuremath{'}})$ that shares this ground-state density pair must have at least rrr degenerate ground states in common with H(v,A)H(v,\mathbf{A})H(v,A). Thus, any set of Hamiltonians that shares a ground-state density pair $(\ensuremath{\rho},{\mathbf{j}}^{p})$ by necessity has to have at least one joint ground state.
A fermion ground state energy functional is set up in terms of particle density, relative pair density, and kinetic energy tensor density. It satisfies a minimum principle if constrained by a complete set of compatibility conditions. A partial set, which thereby results in a lower bound energy under minimization, is obtained from the solution of model systems, as well as a small number of exact sum rules. Prototypical application is made to several one-dimensional spinless noninteracting models. The effectiveness of ``atomic'' constraints on model ``molecules'' is observed, as well as the structure of systems with only finitely many bound states.
It is well known that determining the energy of molecules and other quantum many-body systems reduces in the standard approximation to optimizing a simple linear functional of a 12-variable object, the two-electron reduced density matrix (2-RDM). The difficulty is that 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). The situation has given rise to competing research programs, typically trading more complicated functionals for simpler representability conditions. Chief among them is density functional theory, based on a three-variable object for which the 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. Postulated by Pauli to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for the fermionic natural occupation numbers, accordingly allowing no more than one electron in each quantum state. This is a necessary and sufficient condition for a 1-RDM to be the contraction of an ensemble N-body density matrix. The fermionic one-body quantum marginal problem asks whether given natural occupation numbers can arise from an antisymmetric (ensemble or pure) N-particle state. The configuration interaction method affords optimal descriptions of quantum states of atoms and molecules by expanding the wave function in terms of orbital-based configurations of Slater determinants. For these systems, the dimension of the Hilbert space grows binomially with m, the number of spin-orbitals in the basis of the one-particle Hilbert space, and N, the number of electrons of the system. It has been observed that for the rank-six approximation of a pure-state N=3 system, the occupation numbers satisfy some additional constraints, stronger than the Pauli exclusion principle. The recent analysis by Alexander Klyachko and coworkers of the one-body marginal problem of the pure N-fermion state establishes a systematic approach to this type of constraints. In fact, for a pure quantum system of N electrons arranged in m spin-orbitals the occupation numbers satisfy a set of linear inequalities, known as generalized Pauli constraints (GPC). These inequalities define a convex polytope of allowed states in R^m. They are conditions for a 1-RDM to be the contraction of a N-body state. When one of the GPC is completely saturated, the system is said to be pinned, and it lies on one of the facets of the polytope. The nature of those conditions has been explored till now only in a few systems: a model of three spinless fermions confined to a one-dimensional harmonic potential, the lithium isoelectronic series and ground and excited states of some three- and four-electron molecules for the rank being at least twice the number of electrons. For all these systems the inequalities are (quite often) nearly saturated. This is the so-called quasipinning phenomenon. In this PhD thesis we have taken up the challenge of using numerical and analytical methods to examine pinned and quasipinned states, for atoms and molecules, starting from scratch with configuration-interaction and multiconfiguration self-consistent methods. This procedure serves to study the occurrence of quasipinning in realistic systems. A second goal is to show how the subsets of pinned states defined by GPC give rise to the most efficient approach from a computational viewpoint, yielding the leading order of the electron-electron correlations. As a consequence, we underline in this thesis the theoretical and practical importance of Klyachko's approach to the quantum marginal problem and its impact on the competing research programs to determine feasible electronic densities, 1-RDM, 2-RDM, intracular distributions or Wigner density quasiprobabilities. In relation with the above, our research provides a new variational optimization method for few-fermion ground states. We quantitatively confirm its high accuracy for quasipinned systems and derive an upper bound on the error of the correlation energy given by the ratio of the numerical value of the Klyachko inequality and the distance to the Hartree-Fock point. Depending on the details of the algorithm, we are able to reach 98%-99% of the correlation energy for such systems.