Variational determination of the second-order density matrix for the isoelectronic series of beryllium, neon, and silicon

Verstichel, Brecht; Aggelen, Helen van; Neck, Dimitri Van; Ayers, Paul W.; Bultinck, Patrick

doi:10.1103/physreva.80.032508

Cited by 65 publications

(79 citation statements)

References 25 publications

(32 reference statements)

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“…We remark that with this choice of the basis sets, especially for ionic systems, our HF and CCSD(T) results differ from benchmark results 88,89 . Nevertheless, because the main goal of the present work is to perform a relative (and mostly qualitative) comparison between different methods and because all the exchange-only as well as all the XC methods considered here have a similar basis set convergence behaviour (almost linear for exchange and cubic for correlation), the analysis of the different results is expected to be only slightly influenced by this issue.…”

Section: Basis-setscontrasting

confidence: 71%

A density difference based analysis of orbital-dependent exchange-correlation functionals

et al. 2013

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We present a density difference based analysis for a range of orbital-dependent Kohn-Sham functionals. Results for atoms, some members of the neon isoelectronic series and small molecules are reported and compared with ab initio wave-function calculations. Particular attention is paid to the quality of approximations to the exchange-only optimized effective potential (OEP) approach: we consider both the Localized Hartree Fock as well as the Krieger-Li-Iafrate methods. Analysis of density differences at the exchange-only level reveals the impact the approximations have on the resulting electronic densities. These differences are further quantified in terms of the ground state energies, frontier orbital energy differences and highest occupied orbital energies obtained. At the correlated level an OEP approach based on a perturbative second-order correlation energy expression is shown to deliver results comparable with those from traditional wave function approaches, making it suitable for use as a benchmark against which to compare standard density-functional approximations.

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Section: Basis-setscontrasting

confidence: 71%

A density difference based analysis of orbital-dependent exchange-correlation functionals

et al. 2013

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“…Even better, at any point during the optimization, the error on the current value is limited from above by the primal-dual gap. Note that in our previous implementation [20], a dual-only algorithm was used. (The assignation of a problem as "primal" or "dual" is largely matter of convention.…”

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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Variational determination of the second-order density matrix for the isoelectronic series of beryllium, neon, and silicon

Cited by 65 publications

References 25 publications

A density difference based analysis of orbital-dependent exchange-correlation functionals

A density difference based analysis of orbital-dependent exchange-correlation functionals

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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