The electronic ground-state energy problem: A new reduced density matrix approach

Cancès, Éric; Stoltz, Gabriel; Lewin, Mathieu

doi:10.1063/1.2222358

Cited by 115 publications

(88 citation statements)

References 27 publications

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“…[32], the solution of the physical Hamiltonian by optimizing the second-order reduced density matrix has been tackled with a suitable dual problem. The use of Legendre transform techniques for the simplification of minimizations involving permutations in the many-electron problem has also been stressed and has been applied in Refs.…”

Section: Discussionmentioning

confidence: 99%

Optimal-transport formulation of electronic density-functional theory

2012

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The most challenging scenario for Kohn-Sham density-functional theory, that is, when the electrons move relatively slowly trying to avoid each other as much as possible because of their repulsion (strong-interaction limit), is reformulated here as an optimal transport (or mass transportation theory) problem, a well-established field of mathematics and economics. In practice, we show that to solve the problem of finding the minimum possible internal repulsion energy for N electrons in a given density ρ(r) is equivalent to find the optimal way of transporting N − 1 times the density ρ into itself, with the cost function given by the Coulomb repulsion. We use this link to set the strong-interaction limit of density-functional theory on firm ground and to discuss the potential practical aspects of this reformulation.

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Section: Discussionmentioning

confidence: 99%

Optimal-transport formulation of electronic density-functional theory

2012

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“…(The assignation of a problem as "primal" or "dual" is largely matter of convention. In [21], e.g., the opposite convention is used.) The properties of the present primal-dual method can lead to a serious reduction in computation time since we can stop the algorithm at a prescribed error estimate.…”

Section: Primal-dual Semidefinite Programmentioning

confidence: 99%

A primal–dual semidefinite programming algorithm tailored to the variational determination of the two-body density matrix

Verstichel¹,

Aggelen²,

Neck³

et al. 2011

Computer Physics Communications

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The quantum many-body problem can be rephrased as a variational determination of the twobody reduced density matrix, subject to a set of N -representability constraints. The mathematical problem has the form of a semidefinite program. We adapt a standard primal-dual interior point algorithm in order to exploit the specific structure of the physical problem. In particular the matrix-vector product can be calculated very efficiently. We have applied the proposed algorithm to a pairing-type Hamiltonian and studied the computational aspects of the method. The standard N -representability conditions perform very well for this problem. * brecht.verstichel@ugent.be

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“…Direct calculation of the reduced variables, however, requires that they and their functionals be consistent with a realistic N -electron quantum system; in other words, the reduced variables and functionals must be representable by the integration of an N -electron density matrix. Such consistency relations are known as the N -representability conditions [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][20][21][22][23][24]. These conditions are particularly important to 2-RDM methods where they enable the direct calculation of the 2-RDM without the wavefunction, but they are also implicit in the design of realistic approximations to the density functional in density functional theory [31,32].…”

Section: Introductionmentioning

confidence: 99%

Significant conditions for the two-electron reduced density matrix from the constructive solution ofNrepresentability

Mazziotti

2012

Phys. Rev. A

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We recently presented a constructive solution to the N -representability problem of the twoelectron reduced density matrix (2-RDM)-a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic N -electron quantum system [D. A. Mazziotti, Phys. Rev. Lett. 108, 263002 (2012)]. In this paper we provide additional details and derive further N -representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the (2, q)-positivity conditions where the q indicates their derivation from the nonnegativity of q-body operators. In addition to the known T1 and T2 conditions, we derive a new class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) lifting conditions, that is conditions which arise from lifting lower (2, q)-positivity conditions to higher (2, q + 1)-positivity conditions and (ii) pure conditions, that is conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure (2, q)-positivity conditions for q > 3 require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the new N -representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation.

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The electronic ground-state energy problem: A new reduced density matrix approach

Cited by 115 publications

References 27 publications

Optimal-transport formulation of electronic density-functional theory

Optimal-transport formulation of electronic density-functional theory

A primal–dual semidefinite programming algorithm tailored to the variational determination of the two-body density matrix

Significant conditions for the two-electron reduced density matrix from the constructive solution ofNrepresentability

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