proposed in constructions given in (9) or (10), and would be competing in computational efficiency with one-electron methods. Over the years different approaches
to construct approximate functionals have been explored, some of them leading to
successful functionals. The first generation of functionals such as BB (20), GU (22),
or CHF (23) has turned out to be insufficiently accurate for molecules and extended
systems. Their failure in predicting potential energy curves of diatomic molecules
has led to realizing the importance of incorporating orbital structure of the exact
two-electron functional into approximate N-electron functionals. A series of BBC
(41)–(43) functionals and the recent ELS (44) functional have emerged as a result
of a careful analysis of the orbital structure of the energy expression obtained from a
size-consistent CI ansatz. The orbital structure of the most accurate BBC functional
– BBC3 – leads to accurate potential energy curves of simple molecules. Because of
numerical problems with selecting bonding and antibonding orbitals, assumed in a
definition of BBC3 [see (43)], an “automated” version has been proposed – the AC3
functional [40]. The orbital structures of the BBC3, AC3, and ELS functionals
account for that of the exact two-electron functional necessary to provide a correct
description of electron-pair dissociation.
Almost all approximate electron–electron interaction functionals proposed so far
are the so-called “JK-only” functionals, i.e., they include only two-electron integrals of the Coulomb and exchange type. In [76] Kollmar addressed the question of
accuracy of the most general “JK-only” variational energy expression. Based on his
findings, one is driven to a conclusion of fundamental importance for functional
development. Namely, the limits of accuracy of the variational “JK-only” functionals are set by a pair-excited CI ansatz (67) that leads to the best “JK-only”
energy expression [41]. This ansatz is known to be insufficiently accurate for
chemical problems. It has been shown in [76] that variational (bounded from
below by an exact ground state energy) “JK-only” functionals unavoidably miss a
significant portion of the dynamic electron correlation. Therefore, successful variational functionals should include other than Coulomb and exchange integrals or
one should not try to impose variationality in developing accurate and versatile
“JK-only” functionals.
Another class of functionals – Piris natural orbital functionals (PNOF’s)
[cf. (51)] – are also of “JK-only” type. They have been proposed by employing a
cumulant expansion given in (28) and approximating two-electron reduced density
matrix elements in terms of the natural occupation numbers. Reconstruction of
2-RDM in terms of 1-RDM is guided by N-representability conditions for 2-RDM.
PNOFs, especially one of the latest ones, PNOF5, have been extensively tested for
predicting energy and different properties of molecules of diversified electronic
structure. PNOF5 is particularly successful in describing systems for which static
electron correlation is nonnegligible. At the same time, it has become apparent that
this functional misses an important part of dynamic correlation, which seriously
plagues its performance for some systems. These findings are perfectly understandable because, as a variational “JK-only” functional, PNOF5 inherits the aforementioned limitations of the best “JK-only” functional. Thinking about the possible
ways of developing functionals based on reconstructing 2-RDM in terms of
Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time. . .
173
to construct approximate functionals have been explored, some of them leading to
successful functionals. The first generation of functionals such as BB (20), GU (22),
or CHF (23) has turned out to be insufficiently accurate for molecules and extended
systems. Their failure in predicting potential energy curves of diatomic molecules
has led to realizing the importance of incorporating orbital structure of the exact
two-electron functional into approximate N-electron functionals. A series of BBC
(41)–(43) functionals and the recent ELS (44) functional have emerged as a result
of a careful analysis of the orbital structure of the energy expression obtained from a
size-consistent CI ansatz. The orbital structure of the most accurate BBC functional
– BBC3 – leads to accurate potential energy curves of simple molecules. Because of
numerical problems with selecting bonding and antibonding orbitals, assumed in a
definition of BBC3 [see (43)], an “automated” version has been proposed – the AC3
functional [40]. The orbital structures of the BBC3, AC3, and ELS functionals
account for that of the exact two-electron functional necessary to provide a correct
description of electron-pair dissociation.
Almost all approximate electron–electron interaction functionals proposed so far
are the so-called “JK-only” functionals, i.e., they include only two-electron integrals of the Coulomb and exchange type. In [76] Kollmar addressed the question of
accuracy of the most general “JK-only” variational energy expression. Based on his
findings, one is driven to a conclusion of fundamental importance for functional
development. Namely, the limits of accuracy of the variational “JK-only” functionals are set by a pair-excited CI ansatz (67) that leads to the best “JK-only”
energy expression [41]. This ansatz is known to be insufficiently accurate for
chemical problems. It has been shown in [76] that variational (bounded from
below by an exact ground state energy) “JK-only” functionals unavoidably miss a
significant portion of the dynamic electron correlation. Therefore, successful variational functionals should include other than Coulomb and exchange integrals or
one should not try to impose variationality in developing accurate and versatile
“JK-only” functionals.
Another class of functionals – Piris natural orbital functionals (PNOF’s)
[cf. (51)] – are also of “JK-only” type. They have been proposed by employing a
cumulant expansion given in (28) and approximating two-electron reduced density
matrix elements in terms of the natural occupation numbers. Reconstruction of
2-RDM in terms of 1-RDM is guided by N-representability conditions for 2-RDM.
PNOFs, especially one of the latest ones, PNOF5, have been extensively tested for
predicting energy and different properties of molecules of diversified electronic
structure. PNOF5 is particularly successful in describing systems for which static
electron correlation is nonnegligible. At the same time, it has become apparent that
this functional misses an important part of dynamic correlation, which seriously
plagues its performance for some systems. These findings are perfectly understandable because, as a variational “JK-only” functional, PNOF5 inherits the aforementioned limitations of the best “JK-only” functional. Thinking about the possible
ways of developing functionals based on reconstructing 2-RDM in terms of
Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time. . .
173