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.
| Original language | English |
|---|---|
| Pages (from-to) | 1-47 |
| Number of pages | 47 |
| Journal | Physics Reports |
| DOIs | |
| Publication status | E-pub ahead of print - 12 Feb 2019 |
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Keywords
- Finite basis set DFT
- Finite temperature
- One-body reduced density matrix
- v-representability
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One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures. / Giesbertz, Klaas J.H.; Ruggenthaler, Michael.
In: Physics Reports, 12.02.2019, p. 1-47.Research output: Contribution to Journal › Review article › Academic › peer-review
TY - JOUR
T1 - One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures
AU - Giesbertz, Klaas J.H.
AU - Ruggenthaler, Michael
PY - 2019/2/12
Y1 - 2019/2/12
N2 - 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.
AB - 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.
KW - Finite basis set DFT
KW - Finite temperature
KW - One-body reduced density matrix
KW - v-representability
UR - http://www.scopus.com/inward/record.url?scp=85062675452&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85062675452&partnerID=8YFLogxK
U2 - 10.1016/j.physrep.2019.01.010
DO - 10.1016/j.physrep.2019.01.010
M3 - Review article
SP - 1
EP - 47
JO - Physics Reports
JF - Physics Reports
SN - 0370-1573
ER -