92
K. B. Be´ c et al.
CC approaches still capture a higher fraction of the correlation energy than their CI
counterparts do, albeit at a higher computational cost. The CCSD(T) variant is highly
valued for its high accuracy achieved at a relatively acceptable computational cost,
and is often considered as the “golden standard of quantum chemistry.”
5.3.2.3 Density-Functional Theory
Inclusion of electron correlation in wavefunction-based methods leads to a steep
increase in their computational complexity. This gave an impulse for the development of a fundamentally different concept known as density-functional theory (DFT)
[7, 8]. It is based on the Hohenburg–Kohn theorems postulating that the state of a
many-electron system can be described based on a unique functional (i.e., a function
acting on another function), which in this case is the spatially dependent electron
density function. The benefit of this formalism is a reduction of the dimensionality of the problem from that of a multidimensional (3N) N-electron wavefunction
to a three-dimensional electron density function. The practical implementation of
DFT became possible due to the formulation of the Kohn–Sham equations, which
enabled the reduction of an intractable problem of interacting electrons in a static
external potential to a tractable problem of non-interacting electrons in a local effective potential, i.e., the Kohn–Sham potential. The latter is constituted by the external
potential plus electron exchange and correlation effects expressed via the associated
exchange–correlation functional E xc . Unfortunately, the exact E xc is unknown except
for the limiting case of a free electron gas, which became known as the local-density
approximation (LDA; E
LDA
xc ). While this formalism is applicable in case of metals
and simple ionic solids, the LDA approximation fails to deliver satisfactory results
for more complex systems. The meaningful development of DFT within the regime
of chemistry started with the introduction of the generalized-gradient approximation
(GGA) level, which was followed by more advanced approximations, such as metaGGA functionals. A significant progress in the underlying theory was marked with
the introduction of hybrid Kohn–Sham theory and the resulting hybrid formulation
for E xc . A hybrid E xc is constructed as a linear combination of GGA and/or LDA
(explicit) functionals and a HF ‘exact’ exchange functional (implicit functional).
This inclusion of an ab initio electron exchange term in hybrid functionals greatly
improved the accuracy and applicability of DFT. Popular hybrid functionals include
the B3LYP, PBE0, HSE, and M06 functionals. Further advancement was achieved
with the development of double-hybrid functionals. These approaches represent a
natural progression from hybrid functionals, as in addition to the exchange term,
ab initio correlation is included as well. The correlation is calculated similar to postHF methods, e.g., MP2 correlation is employed in B2PLYP, mPW2PLYP, PBE0DH,
or PBEQIDH double-hybrids. In addition to a much improved treatment of electron correlation, double-hybrid functionals also enable a better implementation of
HF exchange; however, they are significantly more expensive than single-hybrid
functionals.
K. B. Be´ c et al.
CC approaches still capture a higher fraction of the correlation energy than their CI
counterparts do, albeit at a higher computational cost. The CCSD(T) variant is highly
valued for its high accuracy achieved at a relatively acceptable computational cost,
and is often considered as the “golden standard of quantum chemistry.”
5.3.2.3 Density-Functional Theory
Inclusion of electron correlation in wavefunction-based methods leads to a steep
increase in their computational complexity. This gave an impulse for the development of a fundamentally different concept known as density-functional theory (DFT)
[7, 8]. It is based on the Hohenburg–Kohn theorems postulating that the state of a
many-electron system can be described based on a unique functional (i.e., a function
acting on another function), which in this case is the spatially dependent electron
density function. The benefit of this formalism is a reduction of the dimensionality of the problem from that of a multidimensional (3N) N-electron wavefunction
to a three-dimensional electron density function. The practical implementation of
DFT became possible due to the formulation of the Kohn–Sham equations, which
enabled the reduction of an intractable problem of interacting electrons in a static
external potential to a tractable problem of non-interacting electrons in a local effective potential, i.e., the Kohn–Sham potential. The latter is constituted by the external
potential plus electron exchange and correlation effects expressed via the associated
exchange–correlation functional E xc . Unfortunately, the exact E xc is unknown except
for the limiting case of a free electron gas, which became known as the local-density
approximation (LDA; E
LDA
xc ). While this formalism is applicable in case of metals
and simple ionic solids, the LDA approximation fails to deliver satisfactory results
for more complex systems. The meaningful development of DFT within the regime
of chemistry started with the introduction of the generalized-gradient approximation
(GGA) level, which was followed by more advanced approximations, such as metaGGA functionals. A significant progress in the underlying theory was marked with
the introduction of hybrid Kohn–Sham theory and the resulting hybrid formulation
for E xc . A hybrid E xc is constructed as a linear combination of GGA and/or LDA
(explicit) functionals and a HF ‘exact’ exchange functional (implicit functional).
This inclusion of an ab initio electron exchange term in hybrid functionals greatly
improved the accuracy and applicability of DFT. Popular hybrid functionals include
the B3LYP, PBE0, HSE, and M06 functionals. Further advancement was achieved
with the development of double-hybrid functionals. These approaches represent a
natural progression from hybrid functionals, as in addition to the exchange term,
ab initio correlation is included as well. The correlation is calculated similar to postHF methods, e.g., MP2 correlation is employed in B2PLYP, mPW2PLYP, PBE0DH,
or PBEQIDH double-hybrids. In addition to a much improved treatment of electron correlation, double-hybrid functionals also enable a better implementation of
HF exchange; however, they are significantly more expensive than single-hybrid
functionals.