demonstrated that only ten PINOs are sufficient to give a quantitative prediction of
the double ionization yield [162], so an accurate three-dimensional calculation
should now come within reach.
The main challenge is to make the PINO approach applicable to systems with
more than two electrons. For the two-electron system it is clear how to define the
PINO phase factors which gives a very simple expression for the two-body effects,
W, and is even exact. For systems with more electrons, it is not so clear what a
suitable and convenient definition for the PINO phase would be and which preferably reduces to the exact functional for two electrons. For multi-electron systems
which only have one electron pair constituting a chemical bond (Li 2 and LiH for
example), one can try to use a Hartree–Fock (HF) functional for the core electrons
and the PILS functional for the “HONO” which is the highest strongly occupied
natural orbital and some encouraging results have already been obtained for
diatomic molecules with a single chemical bond [163].
A different route is also explored by combining features from the PINO response
equations to the extended RPA equations [164, 165] obtained from Rowe’s equation of motion framework [166]. The advantage is that the response matrices are
now formulated as partial contractions of the 1-RDM and 2-RDM instead of
functional derivatives with respect to PINOs and occupation numbers. This
makes it easier to use other sources for approximate 2-RDMs such as the APSG
wavefunction or other correlated methods. However, the APSG wavefunction can
also be used to construct a PINO functional (62). The adiabatic PINO response
equations with the APSG functional are actually identical to those obtained by
applying time-dependent response theory to the APSG wavefunction directly
[165]. Calculations on small molecular systems have demonstrated that the lowest
excitation energies for the APSG functional (62) are in very good agreement with
more sophisticated approaches. Higher excitation energies seem to be less reliable.
Experiments using a range-separated version of the APSG functional indicate a
shortcoming of the APSG functional rather than an inherent limitation of the
adiabatic TD-PINO linear response equations [165]. However, more evidence
needs to be gathered before we can make any conclusive statement.
6 Summary and Outlook
Reduced density matrix functional theory is a promising approach to the problem of
electron correlation based on the existence of a functional of the one-electron
reduced density matrix (1-RDM). One-electron components of the total energy,
i.e. the kinetic part and the external potential interaction, are explicitly given in
terms of 1-RDM. The electron–electron interaction functional, the two-electron
part of the energy, is well defined, cf. (10), but its practical exact realization remains
unknown. A formalism that would lead to systematically more accurate and efficient approximations to E ee [γ] is not available. By “efficient” we mean approximations that would avoid searching for minimizing wavefunctions or ensembles
172
K. Pernal and K.J.H. Giesbertz
the double ionization yield [162], so an accurate three-dimensional calculation
should now come within reach.
The main challenge is to make the PINO approach applicable to systems with
more than two electrons. For the two-electron system it is clear how to define the
PINO phase factors which gives a very simple expression for the two-body effects,
W, and is even exact. For systems with more electrons, it is not so clear what a
suitable and convenient definition for the PINO phase would be and which preferably reduces to the exact functional for two electrons. For multi-electron systems
which only have one electron pair constituting a chemical bond (Li 2 and LiH for
example), one can try to use a Hartree–Fock (HF) functional for the core electrons
and the PILS functional for the “HONO” which is the highest strongly occupied
natural orbital and some encouraging results have already been obtained for
diatomic molecules with a single chemical bond [163].
A different route is also explored by combining features from the PINO response
equations to the extended RPA equations [164, 165] obtained from Rowe’s equation of motion framework [166]. The advantage is that the response matrices are
now formulated as partial contractions of the 1-RDM and 2-RDM instead of
functional derivatives with respect to PINOs and occupation numbers. This
makes it easier to use other sources for approximate 2-RDMs such as the APSG
wavefunction or other correlated methods. However, the APSG wavefunction can
also be used to construct a PINO functional (62). The adiabatic PINO response
equations with the APSG functional are actually identical to those obtained by
applying time-dependent response theory to the APSG wavefunction directly
[165]. Calculations on small molecular systems have demonstrated that the lowest
excitation energies for the APSG functional (62) are in very good agreement with
more sophisticated approaches. Higher excitation energies seem to be less reliable.
Experiments using a range-separated version of the APSG functional indicate a
shortcoming of the APSG functional rather than an inherent limitation of the
adiabatic TD-PINO linear response equations [165]. However, more evidence
needs to be gathered before we can make any conclusive statement.
6 Summary and Outlook
Reduced density matrix functional theory is a promising approach to the problem of
electron correlation based on the existence of a functional of the one-electron
reduced density matrix (1-RDM). One-electron components of the total energy,
i.e. the kinetic part and the external potential interaction, are explicitly given in
terms of 1-RDM. The electron–electron interaction functional, the two-electron
part of the energy, is well defined, cf. (10), but its practical exact realization remains
unknown. A formalism that would lead to systematically more accurate and efficient approximations to E ee [γ] is not available. By “efficient” we mean approximations that would avoid searching for minimizing wavefunctions or ensembles
172
K. Pernal and K.J.H. Giesbertz