Explicit approximate relation between reduced two- and one-particle density matrices

Müller, Anja

doi:10.1016/0375-9601(84)91034-x

Cited by 314 publications

(252 citation statements)

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“…For instance, if one approximates a two-body RDM in terms of a one-body RDM in the first equation, one arrives at the time-dependent version of reduced density matrix functional theory (TD-RDMFT). Similar to the TDDFT, most of the approximations used in the TD-RDMFT are adiabatic extensions of the existing ground-state ones, [15][16][17] and even though they can successfully describe the ground state of some strongly correlated systems, 18,19 they suffer from flaws, such as lack of memory when we apply them in the time domain. As an example, assuming that these approximations have even the right form, the sign adopted in each term generally becomes a time-dependent phase when we extend them over the time domain.…”

Section: Introductionmentioning

confidence: 99%

Challenges in truncating the hierarchy of time-dependent reduced density matrices equations

et al. 2012

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In this work, we analyze the Born, Bogoliubov, Green, Kirkwood, and Yvon (BBGKY) hierarchy of equations for describing the full time evolution of a many-body fermionic system in terms of its reduced density matrices (at all orders). We provide an exhaustive study of the challenges and open problems linked to the truncation of such a hierarchy of equations to make them practically applicable. We restrict our analysis to the coupled evolution of the one-and two-body reduced density matrices, where higher-order correlation effects are embodied into the approximation used to close the equations. We prove that within this approach, the number of electrons and total energy are conserved, regardless of the employed approximation. Further, we demonstrate that although most of the truncation schemes available in the literature give acceptable ground-state energy, when applied to describe driven electron dynamics, they exhibit undesirable and unphysical behavior, e.g., violation and even divergence in local electronic density, both in weakly and strongly correlated regimes. We illustrate and analyze these problems within the few-site Hubbard model. The model can be solved exactly and provides a unique reference for our detailed study of electron dynamics for different values of interaction, different initial conditions, and the large set of approximations considered here. Moreover, we study the role of compatibility between two hierarchical equations and positive semidefiniteness of reduced density matrices in the instability of electron dynamics. We show that even if the used approximation holds the compatibility, electron dynamics can still diverge when positive definitiveness is violated. We propose some partial solutions of such problems and point out the main paths for future work in order to make this approach applicable for the description of the correlated electron dynamics in complex systems.

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Section: Introductionmentioning

confidence: 99%

Challenges in truncating the hierarchy of time-dependent reduced density matrices equations

et al. 2012

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“…However, the goal was to calculate 1RDMs with a specific expectation value for the total spin which requires a functional of Ŝ2 in terms of the 1RDM. For the Müller functional, the energy functional was derived using an ansatz for the second-order reduced density matrix Γ (2) in terms of the 1RDM [8]. Using this ansatz for Γ (2) we can calculate Ŝ 2 from Eq.…”

Section: Resultsmentioning

confidence: 99%

Conditions for Describing Triplet States in Reduced Density Matrix Functional Theory

Θεοφίλου

Lathiotakis

Helbig

2016

J. Chem. Theory Comput.

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We consider necessary conditions for the one-body-reduced density matrix (1RDM) to correspond to a triplet wave-function of a 2-electron system. The conditions concern the occupation numbers and are different for the high spin projections, Sz = ±1, and the Sz = 0 projection. Hence, they can be used to test if an approximate 1RDM functional yields the same energies for both projections. We employ these conditions in reduced density matrix functional theory calculations for the triplet excitations of two electron systems. In addition, we propose that these conditions can be used in the calculation of triplet states of systems with more than two electrons by restricting the active space. We assess this procedure in calculations for a few atomic and molecular systems. We show that the quality of the optimal 1RDMs improves by applying the conditions in all the cases we studied.

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“…4 fulfilled and the natural orbitals orthonormal. Unfortunately, the exact correlation functional E c [ ] is not known, so many approximations to it, starting from Müller's functional [29], have been proposed in the literature [30]. New functionals continue to appear [27,31,32].…”

Section: Methods and Computational Details 21 One-electron Reduced Dmentioning

confidence: 99%

“…• 10 lowest bands instead of 7 for 1D lattices, • 24 k-divisions instead of 16 for 1D lattices, • 12 k-divisions instead of 8 for 3D lattices, and • Müller's functional [29] with = 0.5 instead of the power functional already described [13] functional for 1D lattices have been performed and evaluated.…”

Section: Methods and Computational Details 21 One-electron Reduced Dmentioning

confidence: 99%

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Electron sharing and localization in real space for the Mott transition from 1RDMFT periodic calculations

Baranov

Pendás

2017

Theor Chem Acc

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of the electronic structure of matter. The most important family of such compounds is found for systems containing d-and f-electrons, many of which are known to possess fascinating properties, for instance, high-temperature superconductivity or colossal magnetoresistance. After many decades of intensive investigations, Mott insulators still stay in the focus of active research, 1 being especially difficult objects to deal with from the theoretical viewpoint. The reason behind that difficulty is that they are essentially governed by strong electron correlation which makes them complicated interacting many-body systems for which one-electron theories often turn out to be even qualitatively wrong [4].Due to this complexity, lots of efforts have been put on the study of prototypical systems: from analytical models, e.g., Hubbard models [5,6], allowing even exact solutions under certain circumstances [7] to hydrogen lattices handled by a number of advanced numerical techniques like dynamic mean-field theory (DMFT), [8], density matrix embedding theory (DMET) [9,10] Mott transitions (MTs) are tightly related to the paradigmatic homolytic dissociation of covalent bonds in molecules [14]. Finite clusters from hydrogen lattices have been therefore also extensively used as model systems for studying the MT, with a wide variety of methods ranging from full CI, used ubiquitously as a reference, to density matrix renormalization group [15], auxiliary-field QMC Abstract One-particle reduced density matrix functional theory (1RDMFT) has been applied for the investigation of the real-space picture of the Mott transitions in archetypal hydrogen lattices. The evolution of the degree of electron localization during the transition has been evaluated using electron sharing indices for QTAIM atoms, and the connection between the strength of electronic correlation expressed by the 1RDMFT correlation energy and the degree of electron localization has been analyzed at various stages of transition. The spatial decay of the electron sharing indices in the course of transition has also been evaluated and compared with the results obtained for the Hubbard model and finite cluster calculations.

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Explicit approximate relation between reduced two- and one-particle density matrices

Cited by 314 publications

References 3 publications

Challenges in truncating the hierarchy of time-dependent reduced density matrices equations

Challenges in truncating the hierarchy of time-dependent reduced density matrices equations

Conditions for Describing Triplet States in Reduced Density Matrix Functional Theory

Electron sharing and localization in real space for the Mott transition from 1RDMFT periodic calculations

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