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

Klaas J.H. Giesbertz*, Michael Ruggenthaler

*Corresponding author for this work

Research output: Contribution to JournalReview articleAcademicpeer-review

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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.

Original languageEnglish
Pages (from-to)1-47
Number of pages47
JournalPhysics Reports
Early online date12 Feb 2019
Publication statusPublished - 10 May 2019


  • Finite basis set DFT
  • Finite temperature
  • One-body reduced density matrix
  • v-representability


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