1-RDM, it is evident that retaining solely Coulomb and exchange integrals in the
functional and reconstructing N-representable 2-RDM would lead to a variational
functional deficient in accounting for dynamic electron correlation. However, Nrepresentability conditions for 2-RDM employed in developing novel functionals
are of necessary but insufficient character. Consequently, functionals resulting from
employing a reconstructional approach are not necessarily variational. As a result,
they can yield lower energy than the best “JK-only” functional if the limited “JKonly” form is properly compensated by the lack of N-representability of the
underlying 2-RDM.
BBC, ELS, or PNOF functionals are orbital-dependent and, by incorporating a
subtle orbital structure of the exact two-electron functional, they are more appropriate for molecular systems than for solids. A functional proposed to work mainly
for extended systems is the power functional (80) with the value of the power α
found empirically. Because of a simple form of the power functional its optimization is highly efficient. Taking into account that its form has been proposed rather
ad hoc without imposing any exact conditions, it is remarkable how well it works
for solids. The most spectacular application of the power functional is for Mott
insulators which are properly predicted to be nonmetallic [84]. In general, power
functional cannot compete with the BBC3 or the recent PNOF functionals in
describing the electronic structure of molecular systems.
It should be mentioned that most of the functionals have been proposed in spinrestricted formulation but extensions to open-shell systems are also available
[167]. So far, RDMFT for high-spin systems has been tested for only a limited
set of systems.
Size-consistency is another property that a useful functional should possess.
Apart from the BB or power functionals, most of the other available approximations
are, in principle, not size-consistent. However, in [168] it has been shown that
violation of size-consistency is negligible for BBC, AC3, and ML functionals.
Undoubtedly there has been significant progress in the last 10 years in the
development of methods in RDMFT. More accurate and versatile functionals
have been proposed. Surprisingly, the “JK-only” form has not yet been fully
exploited and the most recent functionals, ELS [42] and PNOF6 [48], still stay
within this form. As has been discussed, future functionals can either include other
integrals than Coulomb and exchange or stay within the “JK-only” form at the price
of abandoning variationality from the start. Unfortunately, development of
RDMFT-based methods is hindered by slow advances in improving computational
efficiency of optimization algorithms for density matrix functionals. The lack of
sufficiently fast methods has not allowed for application of the existing functionals
to systems consisting of more than a few tens of electrons. Only the very recently
proposed local-RDMFT approach [104] holds any promise of extending limits of
the size of systems that can be treated with RDMFT by at least one order of
magnitude.
The practical use of a time-dependent version of RDMFT has recently been
explored to calculate excitation energies and other frequency-dependent response
properties. A rigorous mathematical foundation for TD-RDMFT is still lacking,
174
K. Pernal and K.J.H. Giesbertz
functional and reconstructing N-representable 2-RDM would lead to a variational
functional deficient in accounting for dynamic electron correlation. However, Nrepresentability conditions for 2-RDM employed in developing novel functionals
are of necessary but insufficient character. Consequently, functionals resulting from
employing a reconstructional approach are not necessarily variational. As a result,
they can yield lower energy than the best “JK-only” functional if the limited “JKonly” form is properly compensated by the lack of N-representability of the
underlying 2-RDM.
BBC, ELS, or PNOF functionals are orbital-dependent and, by incorporating a
subtle orbital structure of the exact two-electron functional, they are more appropriate for molecular systems than for solids. A functional proposed to work mainly
for extended systems is the power functional (80) with the value of the power α
found empirically. Because of a simple form of the power functional its optimization is highly efficient. Taking into account that its form has been proposed rather
ad hoc without imposing any exact conditions, it is remarkable how well it works
for solids. The most spectacular application of the power functional is for Mott
insulators which are properly predicted to be nonmetallic [84]. In general, power
functional cannot compete with the BBC3 or the recent PNOF functionals in
describing the electronic structure of molecular systems.
It should be mentioned that most of the functionals have been proposed in spinrestricted formulation but extensions to open-shell systems are also available
[167]. So far, RDMFT for high-spin systems has been tested for only a limited
set of systems.
Size-consistency is another property that a useful functional should possess.
Apart from the BB or power functionals, most of the other available approximations
are, in principle, not size-consistent. However, in [168] it has been shown that
violation of size-consistency is negligible for BBC, AC3, and ML functionals.
Undoubtedly there has been significant progress in the last 10 years in the
development of methods in RDMFT. More accurate and versatile functionals
have been proposed. Surprisingly, the “JK-only” form has not yet been fully
exploited and the most recent functionals, ELS [42] and PNOF6 [48], still stay
within this form. As has been discussed, future functionals can either include other
integrals than Coulomb and exchange or stay within the “JK-only” form at the price
of abandoning variationality from the start. Unfortunately, development of
RDMFT-based methods is hindered by slow advances in improving computational
efficiency of optimization algorithms for density matrix functionals. The lack of
sufficiently fast methods has not allowed for application of the existing functionals
to systems consisting of more than a few tens of electrons. Only the very recently
proposed local-RDMFT approach [104] holds any promise of extending limits of
the size of systems that can be treated with RDMFT by at least one order of
magnitude.
The practical use of a time-dependent version of RDMFT has recently been
explored to calculate excitation energies and other frequency-dependent response
properties. A rigorous mathematical foundation for TD-RDMFT is still lacking,
174
K. Pernal and K.J.H. Giesbertz