systems (Szabo and Ostlund 2012). Moreover, the complicated N-electron wave
function has proved to be unnecessary, and the total electron density can check the
ground-state properties (Hohenberg and Kohn 1964; Kohn and Sham 1965). The
DFT method proposed by Hohenberg, Kohn, and Sham decreases the complexity of
the many-body Schrödinger equation into a series of Kohn–Sham single particle
(Li et al. 2017). In the Kohn–Sham Hamiltonian, everything is known, apart from the
exchange–correlation functional (Li et al. 2017). Therefore, the accuracy of DFT is
due to its consideration of the exchange–correlation functional. Among the several
types of the exchange–correlation functional that have been used in photocatalysis,
the generalized gradient approximation (Perdew 1986) and the local density
approximation (Kohn and Sham 1965) are the most common ones. The local
density approximation works well for metallic systems but not for insulators and
semiconductors, for which band gaps are underestimated (Li et al. 2017). The
generalized gradient approximation (Becke 1988) is more accurate than the local
density approximation since the generalized gradient approximation considers the
inhomogeneity of electron density. Higher-order derivatives of electron density
with the meta-generalized gradient approximation functionals can enhance the
chemical accuracy of photocatalyst systems with severe variations of electron
density (Li et al. 2017). Nonetheless, the delocalized effects of the generalized
gradient approximation and meta-generalized gradient approximation are intrinsically localized (Li et al. 2017). The delocalized effects are treated by replacing the
exchange–correlation energy with the exact Hartree–Fock exchange–correlation
energy. The hybrid Heyd–Scuseria–Ernzerhof functional (Heyd et al. 2003) mixes
the nonlocal Hartree–Fock exchange with the local/semi-local density functional
theory exchange which enhances the accuracy of electronic structures (Muscat
et al. 2001). Moreover, adding a Hubbard parameter (U) to the generalized gradient
approximation and the local density approximation (DFT + U) calculation also
improves the band structure depending on the choice of several empirical parameters (Anisimov et al. 1997). The DFT + U and hybrid Heyd–Scuseria–Ernzerhof
functional have been more robust and reliable with progress in solid-state materials
science, solid-state physics, chemistry, and chemical engineering (Lejaeghere et al.
2016). Hence, new photocatalyst materials with several structural characteristics,
electronic structure (band structure, electron density distributions, the density of
state, and charge population), and compositions are continuously studied using
DFT calculations. For example, Geng et al. (2013b) carried out DFT calculations
to understand the influence of interface structure on the photocatalytic property of
ZnO/graphene composites. The authors established that the weak interactions
between ZnO monolayer and graphene sheets have no influence on the electronic
properties of graphene. Moreover, stronger binding energies and larger charge
transfers were observed for thick ZnO slabs with polarized surfaces on the
graphene sheet. Coupling graphene sheets on the surface of ZnO with O termination showed lower work function and p-type conductivity, while graphene sheets
on the surface of ZnO with Zn termination exhibited higher work function and
n-type conductivity. Through the conjugate gradient minimization method with the
norm-conserving pseudopotential and the double-zeta plus polarization basis sets,
22
F. Opoku et al.
function has proved to be unnecessary, and the total electron density can check the
ground-state properties (Hohenberg and Kohn 1964; Kohn and Sham 1965). The
DFT method proposed by Hohenberg, Kohn, and Sham decreases the complexity of
the many-body Schrödinger equation into a series of Kohn–Sham single particle
(Li et al. 2017). In the Kohn–Sham Hamiltonian, everything is known, apart from the
exchange–correlation functional (Li et al. 2017). Therefore, the accuracy of DFT is
due to its consideration of the exchange–correlation functional. Among the several
types of the exchange–correlation functional that have been used in photocatalysis,
the generalized gradient approximation (Perdew 1986) and the local density
approximation (Kohn and Sham 1965) are the most common ones. The local
density approximation works well for metallic systems but not for insulators and
semiconductors, for which band gaps are underestimated (Li et al. 2017). The
generalized gradient approximation (Becke 1988) is more accurate than the local
density approximation since the generalized gradient approximation considers the
inhomogeneity of electron density. Higher-order derivatives of electron density
with the meta-generalized gradient approximation functionals can enhance the
chemical accuracy of photocatalyst systems with severe variations of electron
density (Li et al. 2017). Nonetheless, the delocalized effects of the generalized
gradient approximation and meta-generalized gradient approximation are intrinsically localized (Li et al. 2017). The delocalized effects are treated by replacing the
exchange–correlation energy with the exact Hartree–Fock exchange–correlation
energy. The hybrid Heyd–Scuseria–Ernzerhof functional (Heyd et al. 2003) mixes
the nonlocal Hartree–Fock exchange with the local/semi-local density functional
theory exchange which enhances the accuracy of electronic structures (Muscat
et al. 2001). Moreover, adding a Hubbard parameter (U) to the generalized gradient
approximation and the local density approximation (DFT + U) calculation also
improves the band structure depending on the choice of several empirical parameters (Anisimov et al. 1997). The DFT + U and hybrid Heyd–Scuseria–Ernzerhof
functional have been more robust and reliable with progress in solid-state materials
science, solid-state physics, chemistry, and chemical engineering (Lejaeghere et al.
2016). Hence, new photocatalyst materials with several structural characteristics,
electronic structure (band structure, electron density distributions, the density of
state, and charge population), and compositions are continuously studied using
DFT calculations. For example, Geng et al. (2013b) carried out DFT calculations
to understand the influence of interface structure on the photocatalytic property of
ZnO/graphene composites. The authors established that the weak interactions
between ZnO monolayer and graphene sheets have no influence on the electronic
properties of graphene. Moreover, stronger binding energies and larger charge
transfers were observed for thick ZnO slabs with polarized surfaces on the
graphene sheet. Coupling graphene sheets on the surface of ZnO with O termination showed lower work function and p-type conductivity, while graphene sheets
on the surface of ZnO with Zn termination exhibited higher work function and
n-type conductivity. Through the conjugate gradient minimization method with the
norm-conserving pseudopotential and the double-zeta plus polarization basis sets,
22
F. Opoku et al.