functional and a fixed set of the natural occupation numbers, a one-electron
Hamiltonian can be constructed in a self-consistent way such that its eigenfunctions
minimize the functional (for fixed occupancies) [103, 110, 112]. The problem is
that the self-consistent procedure of calculating orbitals from diagonalization of the
effective Hamiltonian is highly divergent [103]. Moreover, the spectrum of this
Hamiltonian is infinitely degenerate if it is constructed from optimal natural orbitals
and occupation numbers. However, by proper combination of level-shifting and
scaling off-diagonal elements of the Hamiltonian matrix, optimal orbitals can be
obtained from iterative diagonalization [103, 111]. Such an approach does not,
however, seem to surpass gradient methods significantly in terms of speed of
convergence.
It has recently been proposed to employ an optimal effective potential (OEP)
method formulated originally for optimization of orbital-dependent density functionals [114, 115] in RDMFT [104]. For a given density matrix functional, a local
potential is sought such that its orbitals minimize the functional for fixed occupation numbers. The main difference from the above-mentioned scheme which
employs a nonlocal Hamiltonian is that in local-RDMFT the potential is
constrained to be local and to possess proper asymptotic behavior. Replacing a
nonlocal potential with a local one and employing the OEP approach formulated
originally for DFT (cf. [116]) leads to an efficient optimization method which
enlarges scopes of applicability of the density matrix functionals to larger molecules and provides good estimations for the ionization potentials [105]. These
advantages notwithstanding, it should also be noted that there is no theoretical
justification for local-RDMFT. Moreover, by definition, the method does not
provide a solution to the original variational problem given in (11) and for a
given functional the optimal energy resulting from the local method is higher
(although not much) than that obtained by solving the “nonlocal” RDMFT optimization problem [104].
5 Time-Dependent RDMFT
The extension of ground state RDMFT to the time domain was recently considered
[117–121]. The main motivation to develop time-dependent RDMFT
(TD-RDMFT) is the poor performance of time-dependent DFT (TDDFT) in the
adiabatic approximation in combination with the approximate ground state density
functionals. The best known failure is the inability of approximate TDDFT to
capture charge transfer excitations [122, 123], though this deficiency has been
remedied with some success using range-separated hybrid functionals [123, 124]
and by an explicit reconstruction of the spatial divergence in the kernel [125,
126]. Other failures of adiabatic TDDFT are bond-breaking excitations which are
predicted to be too low in energy upon dissociation (they can even go to zero) [127,
128] and a lack of double excitations [128–130]. All these failures are connected to
the inability of approximate adiabatic density functionals to deal with static
Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time. . .
157
Hamiltonian can be constructed in a self-consistent way such that its eigenfunctions
minimize the functional (for fixed occupancies) [103, 110, 112]. The problem is
that the self-consistent procedure of calculating orbitals from diagonalization of the
effective Hamiltonian is highly divergent [103]. Moreover, the spectrum of this
Hamiltonian is infinitely degenerate if it is constructed from optimal natural orbitals
and occupation numbers. However, by proper combination of level-shifting and
scaling off-diagonal elements of the Hamiltonian matrix, optimal orbitals can be
obtained from iterative diagonalization [103, 111]. Such an approach does not,
however, seem to surpass gradient methods significantly in terms of speed of
convergence.
It has recently been proposed to employ an optimal effective potential (OEP)
method formulated originally for optimization of orbital-dependent density functionals [114, 115] in RDMFT [104]. For a given density matrix functional, a local
potential is sought such that its orbitals minimize the functional for fixed occupation numbers. The main difference from the above-mentioned scheme which
employs a nonlocal Hamiltonian is that in local-RDMFT the potential is
constrained to be local and to possess proper asymptotic behavior. Replacing a
nonlocal potential with a local one and employing the OEP approach formulated
originally for DFT (cf. [116]) leads to an efficient optimization method which
enlarges scopes of applicability of the density matrix functionals to larger molecules and provides good estimations for the ionization potentials [105]. These
advantages notwithstanding, it should also be noted that there is no theoretical
justification for local-RDMFT. Moreover, by definition, the method does not
provide a solution to the original variational problem given in (11) and for a
given functional the optimal energy resulting from the local method is higher
(although not much) than that obtained by solving the “nonlocal” RDMFT optimization problem [104].
5 Time-Dependent RDMFT
The extension of ground state RDMFT to the time domain was recently considered
[117–121]. The main motivation to develop time-dependent RDMFT
(TD-RDMFT) is the poor performance of time-dependent DFT (TDDFT) in the
adiabatic approximation in combination with the approximate ground state density
functionals. The best known failure is the inability of approximate TDDFT to
capture charge transfer excitations [122, 123], though this deficiency has been
remedied with some success using range-separated hybrid functionals [123, 124]
and by an explicit reconstruction of the spatial divergence in the kernel [125,
126]. Other failures of adiabatic TDDFT are bond-breaking excitations which are
predicted to be too low in energy upon dissociation (they can even go to zero) [127,
128] and a lack of double excitations [128–130]. All these failures are connected to
the inability of approximate adiabatic density functionals to deal with static
Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time. . .
157