Catalytic Properties of Selected Transition Metal Oxides—Computational Studies
347
Coulombic repulsion in the pure DFT functionals tends to delocalise the unpaired
electron in the radicals [15] and also affects the electronic structure of the narrow
band semiconductors [16].
The hybrid functionals are preferable in the systems with strong correlation at the
expense of the necessity of calculation of the Hartree–Fock four-centre integrals and
hence the significant rise of the computational cost. For such systems, the non-hybrid
functionals have difficulty in the description of the properties stemming explicitly
from electron Coulombic correlation, like spin density localisation or the band gap
width. Such weakness leads to, e.g., metallisation of the Mott insulators [17, 18]. The
hybrid functionals cancel out some of the self-interaction, i.e. effect of the interaction
of given electron with the electron density to which the given electron contributes. The
self-interaction problem is particularly pronounced in the d- or f -electron systems,
which are of special importance in the solid-state catalysis. The post-Hartree–Fock
methods, like CCSD(T) often regarded as the high-accuracy method of choice can,
at the time of writing, be applied to relatively small clusters (vide infra) only.
Generally, one of the methods of simplification of the extraordinarily difficult
problem of application of quantum mechanics to the chemical problems is the simplification of the Hamiltonian: instead of the system with many electron, many
nuclei interactions, the simplified Hamiltonian with only selected degrees of freedom, appropriate for the description of the desired class of problems is applied. For
instance, the Hubbard model is particularly applicable in the systems with strong
electronic correlation.
The Hubbard DFT+U level of theory [19–21] can be applied for the strongly
correlated d- or f -electron system of the same size as in the case of pure functionals.
The DFT+U method accounts for the correlation effects arising from the strong onsite Coulomb repulsion and exchange interactions and, in the formulation of Dudarev
et al. [22], incorporates the (semi-empirical) parameter U eff = U − J , where U is
a parameter describing the energy increase for an extra electron on a particular site
and J is a parameter representing the screened exchange energy.
If possible, the computationally very challenging, but giving excellent accuracy,
methods can be used. These are the methods based on the analysis of the poles of
the Green’s function for the system with screened Coulombic kernel, derived from
many-body perturbation theory (MBPT), i.e. the G (0) W (0) methods. For detailed
discussion of GW methods, see, e.g., [23, 24].
Recently, a plethora of computational studies was published on reducible oxides—
both on bulk and surfaces—with use of DFT+U [25, 26], hybrid DFT [27, 28], or GW
method [29]. In these articles, it is shown that the known shortcomings of pure DFT
(e.g. metallisation of isolating ceria) can be significantly improved thus correcting
the underestimated band gap. The same roots has the issue of the charge or spin
density localisation at the defects [30].
The DFT-KS is the theory of the ground state so the other methods, beyond
the DFT, must be used to precisely describe the excited states. One of the most
successful and widely used is the time-dependent density functional theory (TDDFT), founded by the Runge–Gross theorem, the analogue to the Hohenberg-Kohn
theorem, but concerning the evolving many-body system [31, 32]; according to it,
347
Coulombic repulsion in the pure DFT functionals tends to delocalise the unpaired
electron in the radicals [15] and also affects the electronic structure of the narrow
band semiconductors [16].
The hybrid functionals are preferable in the systems with strong correlation at the
expense of the necessity of calculation of the Hartree–Fock four-centre integrals and
hence the significant rise of the computational cost. For such systems, the non-hybrid
functionals have difficulty in the description of the properties stemming explicitly
from electron Coulombic correlation, like spin density localisation or the band gap
width. Such weakness leads to, e.g., metallisation of the Mott insulators [17, 18]. The
hybrid functionals cancel out some of the self-interaction, i.e. effect of the interaction
of given electron with the electron density to which the given electron contributes. The
self-interaction problem is particularly pronounced in the d- or f -electron systems,
which are of special importance in the solid-state catalysis. The post-Hartree–Fock
methods, like CCSD(T) often regarded as the high-accuracy method of choice can,
at the time of writing, be applied to relatively small clusters (vide infra) only.
Generally, one of the methods of simplification of the extraordinarily difficult
problem of application of quantum mechanics to the chemical problems is the simplification of the Hamiltonian: instead of the system with many electron, many
nuclei interactions, the simplified Hamiltonian with only selected degrees of freedom, appropriate for the description of the desired class of problems is applied. For
instance, the Hubbard model is particularly applicable in the systems with strong
electronic correlation.
The Hubbard DFT+U level of theory [19–21] can be applied for the strongly
correlated d- or f -electron system of the same size as in the case of pure functionals.
The DFT+U method accounts for the correlation effects arising from the strong onsite Coulomb repulsion and exchange interactions and, in the formulation of Dudarev
et al. [22], incorporates the (semi-empirical) parameter U eff = U − J , where U is
a parameter describing the energy increase for an extra electron on a particular site
and J is a parameter representing the screened exchange energy.
If possible, the computationally very challenging, but giving excellent accuracy,
methods can be used. These are the methods based on the analysis of the poles of
the Green’s function for the system with screened Coulombic kernel, derived from
many-body perturbation theory (MBPT), i.e. the G (0) W (0) methods. For detailed
discussion of GW methods, see, e.g., [23, 24].
Recently, a plethora of computational studies was published on reducible oxides—
both on bulk and surfaces—with use of DFT+U [25, 26], hybrid DFT [27, 28], or GW
method [29]. In these articles, it is shown that the known shortcomings of pure DFT
(e.g. metallisation of isolating ceria) can be significantly improved thus correcting
the underestimated band gap. The same roots has the issue of the charge or spin
density localisation at the defects [30].
The DFT-KS is the theory of the ground state so the other methods, beyond
the DFT, must be used to precisely describe the excited states. One of the most
successful and widely used is the time-dependent density functional theory (TDDFT), founded by the Runge–Gross theorem, the analogue to the Hohenberg-Kohn
theorem, but concerning the evolving many-body system [31, 32]; according to it,