116
3 Fundamentals of the Analysis Tools
It is noted that most of the functionals used in the DFT method are named taken
from the initials of the developers like VWN, LYP (see below), and so on. The
namings are sometimes accompanied by the year number of the development
like B88 (also see below).
(2) A sophisticated way of dealing with functionals is to include the gradient of
density, ∇ρ(r), in the functional to correct the behavior of the density ρ(r)
(generalized gradient approximation: GGA). The GGA concept is employed
in the BP86 (Perdew 1986), B88 (Becke 1988), LYP (Lee et al. 1988), PW91
(Perdew and Wang 1992), PBE (Perdew et al. 1996) functionals, and so on.
(3) Further refinement of the GGA functional includes the kinetic energy density τ
and the higher order derivative of the density, ∇
2
ρ(r). This procedure leads to the
meta-GGA functionals such as VS98 (van Voorhis and Scuseria 1998), PKZB
(Perdew et al. 1999), KCIS (Krieger et al. 1999), and TPSS (Tao et al. 2008).
(4) The inclusion of contribution from the exchange energy based on the HF method
in the functional with a certain ratio is often adopted to make up, what is called,
hybrid exchange functional. This incldes B97 (Becke 1997), PBE0 (Adamo and
Barone 1999) and HSE (Heyd et al. 2003). One of the most often used hybrid
functionals is the B3LYP consisting of a linear combination of three-parameter
hybrid exchange functional B3 (Becke 1993) and the GGA-based correlation
functional LYP with the coefficients decided by considering the experimental
values of miscellaneous molecules as a benchmark (Curtiss et al 1991). Also, it
is noted that self-interaction error (SIE) owing to the overcount of the Coulomb
interaction of an electron itself is removed in the LYP-type functional.
(5) It is noted that the DFT method is not necessarily appropriate to describe weak
interactions such as hydrogen bond and long-range interactions involving van
der Waals interactions due to a lack of the exchange interactions between the
long-separated electrons. To effectively deal with those within the DFT method,
several frameworks such as the long-range correction (LC)-DFT (Iikura et al.
2001), Coulomb attenuating method (CAM)-B3LYP (Yanai et al. 2004), and
ωB97X (long-range correction to B97) (Chai and Head-Gordon 2008a) functionals have been proposed. Moreover, correction including the dispersion interaction becomes possible by using B97-D (Antony and Grimme 2006) and
ωB97X-D (Chai and Head-Gordon 2008b).
Rigorously speaking, the KS orbitals have no physical meaning by themselves, in
principle, and hence they are different from the MO’s obtained by the HF method.
Hence, the KS determinant is alluded to merely the wavefunction in the “reference
system without interaction”. Nonetheless, both the KS orbitals and the HF MO’s
seem rather similar as seen in Fig. 3.7 (Stowasser and Hoffmann 1999), for instance,
and such coincidence frequently happens. Therefore, these days, the KS orbitals
are often called MO’s as well but this naming ought to be considered with certain
reservations due to the above reason.
Some features of the DFT method are listed in Table 3.1 in comparison with those
of the HF method. It is noted that incorporation of the exchange and correlation
effects into the DFT functionals with good balance is the most important to describe
3 Fundamentals of the Analysis Tools
It is noted that most of the functionals used in the DFT method are named taken
from the initials of the developers like VWN, LYP (see below), and so on. The
namings are sometimes accompanied by the year number of the development
like B88 (also see below).
(2) A sophisticated way of dealing with functionals is to include the gradient of
density, ∇ρ(r), in the functional to correct the behavior of the density ρ(r)
(generalized gradient approximation: GGA). The GGA concept is employed
in the BP86 (Perdew 1986), B88 (Becke 1988), LYP (Lee et al. 1988), PW91
(Perdew and Wang 1992), PBE (Perdew et al. 1996) functionals, and so on.
(3) Further refinement of the GGA functional includes the kinetic energy density τ
and the higher order derivative of the density, ∇
2
ρ(r). This procedure leads to the
meta-GGA functionals such as VS98 (van Voorhis and Scuseria 1998), PKZB
(Perdew et al. 1999), KCIS (Krieger et al. 1999), and TPSS (Tao et al. 2008).
(4) The inclusion of contribution from the exchange energy based on the HF method
in the functional with a certain ratio is often adopted to make up, what is called,
hybrid exchange functional. This incldes B97 (Becke 1997), PBE0 (Adamo and
Barone 1999) and HSE (Heyd et al. 2003). One of the most often used hybrid
functionals is the B3LYP consisting of a linear combination of three-parameter
hybrid exchange functional B3 (Becke 1993) and the GGA-based correlation
functional LYP with the coefficients decided by considering the experimental
values of miscellaneous molecules as a benchmark (Curtiss et al 1991). Also, it
is noted that self-interaction error (SIE) owing to the overcount of the Coulomb
interaction of an electron itself is removed in the LYP-type functional.
(5) It is noted that the DFT method is not necessarily appropriate to describe weak
interactions such as hydrogen bond and long-range interactions involving van
der Waals interactions due to a lack of the exchange interactions between the
long-separated electrons. To effectively deal with those within the DFT method,
several frameworks such as the long-range correction (LC)-DFT (Iikura et al.
2001), Coulomb attenuating method (CAM)-B3LYP (Yanai et al. 2004), and
ωB97X (long-range correction to B97) (Chai and Head-Gordon 2008a) functionals have been proposed. Moreover, correction including the dispersion interaction becomes possible by using B97-D (Antony and Grimme 2006) and
ωB97X-D (Chai and Head-Gordon 2008b).
Rigorously speaking, the KS orbitals have no physical meaning by themselves, in
principle, and hence they are different from the MO’s obtained by the HF method.
Hence, the KS determinant is alluded to merely the wavefunction in the “reference
system without interaction”. Nonetheless, both the KS orbitals and the HF MO’s
seem rather similar as seen in Fig. 3.7 (Stowasser and Hoffmann 1999), for instance,
and such coincidence frequently happens. Therefore, these days, the KS orbitals
are often called MO’s as well but this naming ought to be considered with certain
reservations due to the above reason.
Some features of the DFT method are listed in Table 3.1 in comparison with those
of the HF method. It is noted that incorporation of the exchange and correlation
effects into the DFT functionals with good balance is the most important to describe