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#### One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures

##### External Resource

https://arxiv.org/abs/1710.08805

(Preprint)

https://dx.doi.org/10.1016/j.physrep.2019.01.010

(Publisher version)

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##### Fulltext (public)

vRepr.pdf

(Postprint), 417KB

##### Supplementary Material (public)

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##### Citation

Giesbertz, K. J. H., & Ruggenthaler, M. (2019). One-body reduced density-matrix
functional theory in finite basis sets at elevated temperatures.* Physics Reports: Review Section of
Physics Letters,* *806*, 1-47. doi:10.1016/j.physrep.2019.01.010.

Cite as: https://hdl.handle.net/21.11116/0000-0001-AFE2-B

##### Abstract

In this review we provide a rigorous and self-contained presentation of one-body reduced density-matrix (1RDM) functional theory. We do so for the case of a finite basis set, where density-functional theory (DFT) implicitly becomes a 1RDM functional theory. To avoid non-uniqueness issues we consider the case of fermionic and bosonic systems at elevated temperature and variable particle number, i.e, a grand-canonical ensemble. For the fermionic case the Fock space is finite-dimensional due to the Pauli principle and we can provide a rigorous 1RDM functional theory relatively straightforwardly. For the bosonic case, where arbitrarily many particles can occupy a single state, the Fock space is infinite-dimensional and mathematical subtleties (not every Hermitian Hamiltonian is self-adjoint, expectation values can become infinite, and not every self-adjoint Hamiltonian has a Gibbs state) make it necessary to impose restrictions on the allowed Hamiltonians and external non-local potentials. For simple conditions on the interaction of the bosons a rigorous 1RDM functional theory can be established, where we exploit the fact that due to the finite one-particle space all 1RDMs are finite- dimensional. We also discuss the problems arising from 1RDM functional theory as well as DFT formulated for an infinite-dimensional one-particle space.