occupied (n p < 1/2), of the corresponding phase factors f p ¼ À1 [61, 67]. As already
mentioned in Sect. 2.1 for two-electron singlet systems (described within the APSG
theory by one geminal), exceptions to this rule have been observed [36, 39], but
they occur for very weakly occupied orbitals. Practically, fixing the phases in the
APSG functional given in (62) according to the aforementioned rule, instead of
finding them variationally, has only a small effect on the APSG energy. The APSG
functional with the phase factors fixed can be seen as a density matrix functional.
Comparison of the PNOF5 functional defined by (58) and (55) with (62) immediately reveals that they are identical if the dimension of the expansion space for each
geminal in the APSG approach is limited to 2 and the phase factors of the two
orbitals which form a given geminal are opposite, i.e., f 1 ¼ Àf 2 [60]. Because
PNOF5 is equivalent to such constrained APSG approximation, it inherits its
features from the latter, which explains the good performance of the PNOF5
functional for predicting dissociation energy curves of molecules [47] and the
localized character of its optimal orbitals [68].
Lifting the restriction on the dimensionality of expansion spaces for the geminals
in PNOF5 functional should allow one to recover a part of the correlation energy
missing in this functional. This procedure has been proposed in [69] but clearly
such extended PNOF5 functional (PNOF5e) is identical to the APSG functional
(62) with fixed phases. For PNOF5 and PNOF5e functionals a systematic reconstruction of the 2-RDM in terms of the 1-RDM has merged with a theory based on
the ansatz for the wavefunction [49]. On one hand this may seem to be a desirable
result – the functionals are N-representable and bound by the exact ground state
energy, but the drawback is that the functionals suffer from the same deficiencies as
the APSG approximation.
An interesting idea that leads to incorporating the dynamic correlation that
PNOF5 lacks has been proposed in [48]. The intrapair correlation is included in
PNOF5 by proposing the elements Δ pq (57a) and Π pq (57b) corresponding to
uncoupled orbitals p and q (belonging to different pairs) to be nonzero and
expressing them as functions of the occupation numbers. The new functional,
PNOF6, employs, similarly to PNOF5, a paired-orbitals picture. Compared to
PNOF5, the PNOF6 functional underestimates the dissociation energies to a lesser
degree. Unlike its predecessor, PNOF6 yields delocalized orbitals and it avoids
spatial symmetry breaking of the benzene equilibrium geometry [48].
An ongoing development of natural orbital functionals, PNOF, originating from
reconstruction of 2-RDM in terms of 1-RDM, has already resulted in functionals
competing in accuracy with MP2 method for single-reference systems. Unlike the
MP2 method, the PNOF4, PNOF5, and PNOF6 functionals are useful in describing
potential energy surface also when bonds are stretched and dissociation potential
energy curves are often of the quality of the much more expensive CASSCF
approach.
Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time. . .
143
mentioned in Sect. 2.1 for two-electron singlet systems (described within the APSG
theory by one geminal), exceptions to this rule have been observed [36, 39], but
they occur for very weakly occupied orbitals. Practically, fixing the phases in the
APSG functional given in (62) according to the aforementioned rule, instead of
finding them variationally, has only a small effect on the APSG energy. The APSG
functional with the phase factors fixed can be seen as a density matrix functional.
Comparison of the PNOF5 functional defined by (58) and (55) with (62) immediately reveals that they are identical if the dimension of the expansion space for each
geminal in the APSG approach is limited to 2 and the phase factors of the two
orbitals which form a given geminal are opposite, i.e., f 1 ¼ Àf 2 [60]. Because
PNOF5 is equivalent to such constrained APSG approximation, it inherits its
features from the latter, which explains the good performance of the PNOF5
functional for predicting dissociation energy curves of molecules [47] and the
localized character of its optimal orbitals [68].
Lifting the restriction on the dimensionality of expansion spaces for the geminals
in PNOF5 functional should allow one to recover a part of the correlation energy
missing in this functional. This procedure has been proposed in [69] but clearly
such extended PNOF5 functional (PNOF5e) is identical to the APSG functional
(62) with fixed phases. For PNOF5 and PNOF5e functionals a systematic reconstruction of the 2-RDM in terms of the 1-RDM has merged with a theory based on
the ansatz for the wavefunction [49]. On one hand this may seem to be a desirable
result – the functionals are N-representable and bound by the exact ground state
energy, but the drawback is that the functionals suffer from the same deficiencies as
the APSG approximation.
An interesting idea that leads to incorporating the dynamic correlation that
PNOF5 lacks has been proposed in [48]. The intrapair correlation is included in
PNOF5 by proposing the elements Δ pq (57a) and Π pq (57b) corresponding to
uncoupled orbitals p and q (belonging to different pairs) to be nonzero and
expressing them as functions of the occupation numbers. The new functional,
PNOF6, employs, similarly to PNOF5, a paired-orbitals picture. Compared to
PNOF5, the PNOF6 functional underestimates the dissociation energies to a lesser
degree. Unlike its predecessor, PNOF6 yields delocalized orbitals and it avoids
spatial symmetry breaking of the benzene equilibrium geometry [48].
An ongoing development of natural orbital functionals, PNOF, originating from
reconstruction of 2-RDM in terms of 1-RDM, has already resulted in functionals
competing in accuracy with MP2 method for single-reference systems. Unlike the
MP2 method, the PNOF4, PNOF5, and PNOF6 functionals are useful in describing
potential energy surface also when bonds are stretched and dissociation potential
energy curves are often of the quality of the much more expensive CASSCF
approach.
Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time. . .
143