because the Runge–Gross proof [138] cannot straightforwardly be extended to
1-RDMs and non-local potentials. Not only is the mathematical foundation of
TD-RDMFT more challenging than in TDDFT, but also the formulation of a
satisfactory adiabatic approximation has turned out to be rather involved. The
standard adiabatic approximation (same as in TDDFT) leads to a mismatch between
the static response equations and the frequency-dependent response equations in
their ω ! 0 limit. This problem can be mitigated by assuming an instantaneous
response of the natural occupation numbers. Nevertheless, important diagonal
double excitations are still missing and a justification for the use of the PILS
functional, which is not a proper 1-RDM functional, is still lacking. All these
problems are solved by augmenting the time-dependent 1-RDM with explicit
phase-factors for the natural spinorbitals. The TD-PINO equations have all the
desired properties of a decent adiabatic approximation. In contrast to TDDFT, even
in the adiabatic approximation, the TD-PINO response equations are able to
describe excitations of double and bond-breaking character and charge transfer
excitations are also recovered without difficulty. A confirmation of this statement
has been delivered by the results for the H 2 and HeH
+ molecules, obtained within
the adiabatic TD-PINO formalism with the PILS functional and the extended RPA
results with the APSG density matrices. The latter approach has been shown to be
equivalent to the adiabatic TD-PINO if the APSG functional is employed [165] and
has been tested on several small molecules. Even though it recovers certain double
excitations, its overall accuracy is not satisfactory. The main challenges in the timedependent direction are to formulate a general definition for the PINO phase factors
to develop functionals for N-electron systems and to establish a proper mathematical foundation.
Acknowledgements The authors wish to acknowledge Dr. Ewa Pastorczak for critically reading
the manuscript. KJHG gratefully acknowledges a VENI grant by the Netherlands Foundation for
Research NWO (722.012.013). KP acknowledges support by the National Science Centre of
Poland under grant DEC-2012/07/E/ST4/03023.
References
1. Gilbert TL (1975) Hohenberg–Kohn theorem for nonlocal external potentials. Phys Rev B 12
(6):2111–2120. doi:10.1103/PhysRevB.12.2111
2. Coleman AJ (1963) Structure of fermion density matrices. Rev Mod Phys 35(3):668–687.
doi:10.1103/RevModPhys.35.668
3. Smith DW (1966) N-representability problem for fermion density matrices. II. The first-order
density matrix with N even. Phys Rev 147(4):896–898. doi:10.1103/PhysRev.147.896
4. Levy M (1979) Universal variational functionals of electron-densities, first-order densitymatrices, and natural spin-orbitals and solution of the v-representability problem. Proc Natl
Acad Sci U S A 76(12):6062–6065. doi:10.1073/pnas.76.12.6062
5. Levy M (1987) Correlation energy functionals of one-matrices and Hartree–Fock densities.
In: Erdahl R, Smith VHJ (eds) Density matrices and density functionals. Reidel, Dordrecht,
pp 479–498
Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time. . .
175
1-RDMs and non-local potentials. Not only is the mathematical foundation of
TD-RDMFT more challenging than in TDDFT, but also the formulation of a
satisfactory adiabatic approximation has turned out to be rather involved. The
standard adiabatic approximation (same as in TDDFT) leads to a mismatch between
the static response equations and the frequency-dependent response equations in
their ω ! 0 limit. This problem can be mitigated by assuming an instantaneous
response of the natural occupation numbers. Nevertheless, important diagonal
double excitations are still missing and a justification for the use of the PILS
functional, which is not a proper 1-RDM functional, is still lacking. All these
problems are solved by augmenting the time-dependent 1-RDM with explicit
phase-factors for the natural spinorbitals. The TD-PINO equations have all the
desired properties of a decent adiabatic approximation. In contrast to TDDFT, even
in the adiabatic approximation, the TD-PINO response equations are able to
describe excitations of double and bond-breaking character and charge transfer
excitations are also recovered without difficulty. A confirmation of this statement
has been delivered by the results for the H 2 and HeH
+ molecules, obtained within
the adiabatic TD-PINO formalism with the PILS functional and the extended RPA
results with the APSG density matrices. The latter approach has been shown to be
equivalent to the adiabatic TD-PINO if the APSG functional is employed [165] and
has been tested on several small molecules. Even though it recovers certain double
excitations, its overall accuracy is not satisfactory. The main challenges in the timedependent direction are to formulate a general definition for the PINO phase factors
to develop functionals for N-electron systems and to establish a proper mathematical foundation.
Acknowledgements The authors wish to acknowledge Dr. Ewa Pastorczak for critically reading
the manuscript. KJHG gratefully acknowledges a VENI grant by the Netherlands Foundation for
Research NWO (722.012.013). KP acknowledges support by the National Science Centre of
Poland under grant DEC-2012/07/E/ST4/03023.
References
1. Gilbert TL (1975) Hohenberg–Kohn theorem for nonlocal external potentials. Phys Rev B 12
(6):2111–2120. doi:10.1103/PhysRevB.12.2111
2. Coleman AJ (1963) Structure of fermion density matrices. Rev Mod Phys 35(3):668–687.
doi:10.1103/RevModPhys.35.668
3. Smith DW (1966) N-representability problem for fermion density matrices. II. The first-order
density matrix with N even. Phys Rev 147(4):896–898. doi:10.1103/PhysRev.147.896
4. Levy M (1979) Universal variational functionals of electron-densities, first-order densitymatrices, and natural spin-orbitals and solution of the v-representability problem. Proc Natl
Acad Sci U S A 76(12):6062–6065. doi:10.1073/pnas.76.12.6062
5. Levy M (1987) Correlation energy functionals of one-matrices and Hartree–Fock densities.
In: Erdahl R, Smith VHJ (eds) Density matrices and density functionals. Reidel, Dordrecht,
pp 479–498
Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time. . .
175