giving rise (via (26)) to the exchange-correlation power functional (80) satisfies the
sum rule (48) only for α ¼ 1/2. Application of the power functional to the homogeneous electron gas revealed that momentum distribution resulting from the
functional optimization not only lacks a step structure but is very different from
the exact distributions in general [83, 88]. Even though the power functional does
not recover the exact high-density limit of the correlation energy of the HEG [83,
89], it has been shown that with a carefully chosen value of α it is possible to obtain
rather accurate values of the correlation energy for this system in the broad range of
densities [86]. Moreover, the power functional performs remarkably well in
predicting accurate band gaps of semiconductors and insulators [84]. The test set
included materials of covalent or ionic character with band gaps ranging from 1 to
14.2 eV. It is striking that all these systems are incorrectly predicted to be metallic if
described with the α ¼ 1/2 power functional, whereas choosing α ¼ 0.65 or α ¼ 0.7
results in obtaining nonzero gaps deviating on average from experimental values by
less than 10%. Reducing α below the value 0.65 leads to zero energy gap for some
materials, so it seems the range of admissible values of α is quite narrow.
Performance of the simple power functional (80) with α 2 (0.65, 0.7) when
applied to transition metal oxides (TMO) is even more impressive. TMOs can be
regarded as prototypes of strongly correlated Mott insulators, the nonzero band gap
of which is a result of strong Mott–Hubbard correlations. Most approximate density
functionals incorrectly predict TMO to be metallic. The power density matrix
functional, however, yields finite values for band gaps of nonmagnetic TMOs,
although deviations of the computed gaps from their experimental counterparts
are larger than in the case of conventional insulators [84]. Sharma et al. also showed
that apart from band gaps the power functional is capable of accurately predicting
other properties of solids such as equilibrium lattice constants. Another successful
application of the power functional includes predicting the photoelectron spectra of
strongly correlated Mott insulators within a density matrix functional method
proposed in [90]. Despite its very simplistic form, the power functional has been
shown to be a useful tool for studying solids, including those for which most density
functionals provide unreliable results.
3 Predicting Properties of Electronic Systems with Density
Matrix Functionals
Reduced density matrix functionals give immediate access to total energies of
systems under investigation and, because the 1-RDM is known, to expectation values
of local or nonlocal one-electron operators. However, in recent years a number of
methods have been formulated within RDMFT allowing one to obtain properties of
systems which are not mere traces of products of 1-RDM with one-electron operators.
The properties accessible within static RDMFT include second- and higher-order
static response properties, photoelectron spectra, or fundamental gaps.
Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time. . .
151
sum rule (48) only for α ¼ 1/2. Application of the power functional to the homogeneous electron gas revealed that momentum distribution resulting from the
functional optimization not only lacks a step structure but is very different from
the exact distributions in general [83, 88]. Even though the power functional does
not recover the exact high-density limit of the correlation energy of the HEG [83,
89], it has been shown that with a carefully chosen value of α it is possible to obtain
rather accurate values of the correlation energy for this system in the broad range of
densities [86]. Moreover, the power functional performs remarkably well in
predicting accurate band gaps of semiconductors and insulators [84]. The test set
included materials of covalent or ionic character with band gaps ranging from 1 to
14.2 eV. It is striking that all these systems are incorrectly predicted to be metallic if
described with the α ¼ 1/2 power functional, whereas choosing α ¼ 0.65 or α ¼ 0.7
results in obtaining nonzero gaps deviating on average from experimental values by
less than 10%. Reducing α below the value 0.65 leads to zero energy gap for some
materials, so it seems the range of admissible values of α is quite narrow.
Performance of the simple power functional (80) with α 2 (0.65, 0.7) when
applied to transition metal oxides (TMO) is even more impressive. TMOs can be
regarded as prototypes of strongly correlated Mott insulators, the nonzero band gap
of which is a result of strong Mott–Hubbard correlations. Most approximate density
functionals incorrectly predict TMO to be metallic. The power density matrix
functional, however, yields finite values for band gaps of nonmagnetic TMOs,
although deviations of the computed gaps from their experimental counterparts
are larger than in the case of conventional insulators [84]. Sharma et al. also showed
that apart from band gaps the power functional is capable of accurately predicting
other properties of solids such as equilibrium lattice constants. Another successful
application of the power functional includes predicting the photoelectron spectra of
strongly correlated Mott insulators within a density matrix functional method
proposed in [90]. Despite its very simplistic form, the power functional has been
shown to be a useful tool for studying solids, including those for which most density
functionals provide unreliable results.
3 Predicting Properties of Electronic Systems with Density
Matrix Functionals
Reduced density matrix functionals give immediate access to total energies of
systems under investigation and, because the 1-RDM is known, to expectation values
of local or nonlocal one-electron operators. However, in recent years a number of
methods have been formulated within RDMFT allowing one to obtain properties of
systems which are not mere traces of products of 1-RDM with one-electron operators.
The properties accessible within static RDMFT include second- and higher-order
static response properties, photoelectron spectra, or fundamental gaps.
Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time. . .
151