5.3.1 Introduction‣ 5.3 Overview of Available Functionals ‣ Chapter 5 Density Functional Theory ‣ Q-Chem 6.0 User’s Manual

Q-Chem currently has more than 30 exchange functionals as well as more than 30 correlation functionals, and in addition over 150 exchange-correlation (XC) functionals, which refer to functionals that are not separated into exchange and correlation parts, either because the way in which they were parameterized renders such a separation meaningless (e.g., B97-D ⁴²⁴ Grimme S.
J. Comput. Chem.
(2006), 27, pp. 1787. Link or $\omega$ B97X ¹⁹⁰ Chai J.-D., Head-Gordon M.
J. Chem. Phys.
(2008), 128, pp. 084106. Link ) or because they are a standard linear combination of exchange and correlation (e.g., PBE ⁹²³ Perdew J. P., Burke K., Ernzerhof M.
Phys. Rev. Lett.
(1996), 77, pp. 3865. Link or B3LYP ⁸² Becke A. D.
J. Chem. Phys.
(1993), 98, pp. 5648. Link ^, ¹¹⁴⁷ Stephens P. J. et al.
J. Phys. Chem.
(1994), 98, pp. 11623. Link ). User-defined XC functionals can be created as specified linear combinations of any of the 30+ exchange functionals and/or the 30+ correlation functionals.

KS-DFT functionals can be organized onto a ladder with five rungs, in a classification scheme (“Jacob’s Ladder”) proposed by John Perdew in 2001. ⁹³¹ Perdew J. P. et al.
J. Chem. Phys.
(2005), 123, pp. 062201. Link The first rung contains a functional that only depends on the (spin-) density $\rho_{\sigma}$ , namely, the local spin-density approximation (LSDA). These functionals are exact for the infinite uniform electron gas (UEG), but are highly inaccurate for molecular properties whose densities exhibit significant inhomogeneity. To improve upon the weaknesses of the LSDA, it is necessary to introduce an ingredient that can account for inhomogeneities in the density: the density gradient, $\hat{\nabla}\rho_{\sigma}$ . These generalized gradient approximation (GGA) functionals define the second rung of Jacob’s Ladder and tend to improve significantly upon the LSDA. Two additional ingredients that can be used to further improve the performance of GGA functionals are either the Laplacian of the density $\nabla^{2}\rho_{\sigma}$ , and/or the kinetic energy density,

\tau_{\sigma}=\displaystyle\sum_{i}^{n_{\sigma}}\left|\nabla\psi_{i,\sigma}% \right|^{2}\;.

(5.10)

While functionals that employ both of these options are available in Q-Chem, the kinetic energy density is by far the more popular ingredient and has been used in many modern functionals to add flexibility to the functional form with respect to both constraint satisfaction (non-empirical functionals) and least-squares fitting (semi-empirical parameterization). Functionals that depend on either of these two ingredients belong to the third rung of the Jacob’s Ladder and are called meta-GGAs. These meta-GGAs often further improve upon GGAs in areas such as thermochemistry, kinetics (reaction barrier heights), and even non-covalent interactions.

Functionals on the fourth rung of Jacob’s Ladder are called hybrid density functionals. This rung contains arguably the most popular density functional of our time, B3LYP, the first functional to see widespread application in chemistry. “Global” hybrid (GH) functionals such as B3LYP (as distinguished from the “range-separated hybrids" introduced below) add a constant fraction of “exact” (Hartree-Fock) exchange to any of the functionals from the first three rungs. Thus, hybrid LSDA, hybrid GGA, and hybrid meta-GGA functionals can be constructed, although the latter two types are much more common. As an example, the formula for the B3LYP functional, as implemented in Q-Chem, is

E_{xc}^{\textrm{B3LYP}}=c_{x}E_{x}^{\textrm{HF}}+\left(1-c_{x}-a_{x}\right)E_{% x}^{\textrm{Slater}}+a_{x}E_{x}^{\textrm{B88}}+\left(1-a_{c}\right)E_{c}^{% \textrm{VWN1RPA}}+a_{c}E_{c}^{\textrm{LYP}}

(5.11)

where $c_{x}=0.20$ , $a_{x}=0.72$ , and $a_{c}=0.81$ .

A more recent approach to introducing exact exchange into the functional form is via range separation. Range-separated hybrid (RSH) functionals split the exact exchange contribution into a short-range (SR) component and a long-range (LR) component, often by means of the error function (erf) and complementary error function ( $\rm erfc\equiv 1-erf$ ):

\frac{1}{r_{12}}=\underbrace{\frac{\mbox{erfc}(\omega r_{12})}{r_{12}}}_{\text% {SR}}+\underbrace{\frac{\mbox{erf}(\omega r_{12})}{r_{12}}}_{\text{LR}}\;

(5.12)

The first term on the right in Eq. (5.12) is singular but short-range, and decays to zero on a length scale of $\sim 1/\omega$ , while the second term constitutes a non-singular, long-range background. An RSH XC functional can be expressed generically as

E_{xc}^{\textrm{RSH}}=c_{x,{\rm SR}}E_{x,{\rm SR}}^{\textrm{HF}}+c_{x,\rm LR}E% _{x,\rm LR}^{\textrm{HF}}+(1-c_{x,\rm SR})E_{x,\rm SR}^{\textrm{DFT}}+(1-c_{x,% \rm LR})E_{x,\rm LR}^{\textrm{DFT}}+E_{c}^{\textrm{DFT}}\;,

(5.13)

where the SR and LR parts of the Coulomb operator are used, respectively, to evaluate the HF exchange energies $E_{x,\rm SR}^{\textrm{HF}}$ and $E_{x,\rm LR}^{\textrm{HF}}$ . The corresponding DFT exchange functional is partitioned in the same manner, but the correlation energy $E_{c}^{\textrm{DFT}}$ is evaluated using the full Coulomb operator, $r_{12}^{-1}$ . Of the two linear parameters in Eq. (5.13), $c_{x,\rm LR}$ is usually either set to 1 to define long-range corrected (LRC) RSH functionals (see Section 5.6) or else set to 0, which defines screened-exchange (SE) RSH functionals. On the other hand, the fraction of short-range exact exchange ( $c_{x,\rm SR}$ ) can either be determined via least-squares fitting, theoretically justified using the adiabatic connection, or simply set to zero. As with the global hybrids, RSH functionals can be fashioned using all of the ingredients from the lower three rungs. The rate at which the local DFT exchange is turned off and the non-local exact exchange is turned on is controlled by the parameter $\omega$ . Large values of $\omega$ tend to lead to attenuators that are less smooth (unless the fraction of short-range exact exchange is very large), while small values of (e.g., $\omega=$ 0.2–0.3 bohr ${}^{-1}$ ) are the most common in semi-empirical RSH functionals.

The final rung on Jacob’s Ladder contains functionals that use not only occupied orbitals (via exact exchange), but virtual orbitals as well (via methods such as MP2 or the random phase approximation, RPA). These double hybrids (DH) are the most expensive density functionals available in Q-Chem, but can also be very accurate. The most basic form of a DH functional is

E_{xc}^{\textrm{DH}}=c_{x}E_{x}^{\textrm{HF}}+\left(1-c_{x}\right)E_{x}^{% \textrm{DFT}}+c_{c}E_{x}^{\textrm{MP2}}+\left(1-c_{c}\right)E_{c}^{\textrm{DFT% }}\;.

(5.14)

As with hybrids, the coefficients can either be theoretically motivated or empirically determined. In addition, double hybrids can use exact exchange both globally or via range-separation, and their components can be as primitive as LSDA or as advanced as in meta-GGA functionals. More information on double hybrids can be found in Section 5.9.

Finally, the last major advance in KS-DFT in recent years has been the development of methods that are capable of accurately describing non-covalent interactions, particularly dispersion. All of the functionals from Jacob’s Ladder can technically be combined with these dispersion corrections, although in some cases the combination is detrimental, particularly for semi-empirical functionals that were parameterized in part using data sets of non-covalent interactions, and already tend to overestimate non-covalent interaction energies. The most popular such methods available in Q-Chem are:

•

Non-local correlation (NLC) functionals (Section 5.7.2), including those of Vydrov and Van Voorhis ¹²³⁸ Vydrov O. A., Van Voorhis T.
Phys. Rev. Lett.
(2009), 103, pp. 063004. Link ^, ¹²⁴⁰ Vydrov O. A., Van Voorhis T.
J. Chem. Phys.
(2010), 133, pp. 244103. Link (VV09 and VV10) and of Lundqvist and Langreth ²⁸³ Dion M. et al.
Phys. Rev. Lett.
(2004), 92, pp. 246401. Link ^, ²⁸⁴ Dion M. et al.
Phys. Rev. Lett.
(2005), 95, pp. 109902. Link (vdW-DF-04 and vdW-DF-10). The revised VV10 NLC functional of Sabatini and coworkers (rVV10) is also available ¹⁰⁴⁶ Sabatini R., Gorni T., de Gironcoli S.
Phys. Rev. B
(2013), 87, pp. 041108. Link .
•

Damped, atom–atom pairwise empirical dispersion potentials from Grimme and others ⁴²⁴ Grimme S.
J. Comput. Chem.
(2006), 27, pp. 1787. Link ^, ¹⁹¹ Chai J.-D., Head-Gordon M.
Phys. Chem. Chem. Phys.
(2008), 10, pp. 6615. Link ^, ⁴¹⁷ Grimme S. et al.
J. Chem. Phys.
(2010), 132, pp. 154104. Link ^, ⁴¹⁹ Grimme S., Ehrlich S., Goerigk L.
J. Comput. Chem.
(2011), 32, pp. 1456. Link ^, ¹⁰⁶⁸ Schröder H., Creon A., Schwabe T.
J. Chem. Theory Comput.
(2015), 11, pp. 3163. Link ^, ¹¹¹⁶ Smith D. G. et al.
J. Phys. Chem. Lett.
(2016), 7, pp. 2197. Link [DFT-D2, DFT-CHG, DFT-D3(0), DFT-D3(BJ), DFT-D3(CSO), DFT-D3M(0), DFT-D3M(BJ), and DFT-D3(op)]; see Section 5.7.3.
•

The exchange-dipole models (XDM) of Johnson and Becke (XDM6 and XDM10); see Section 5.7.4.
•

The Tkatchenko-Scheffler (TS) method for dispersion interactions; ¹¹⁹¹ Tkatchenko A., Scheffler M.
Phys. Rev. Lett.
(2009), 102, pp. 073005. Link see Section 5.7.5.
•

The Many-Body Dispersion (MBD) method for van der Waals interactions ¹¹⁹⁰ Tkatchenko A. et al.
Phys. Rev. Lett.
(2012), 108, pp. 236402. Link ^, ³⁸ Ambrosetti A. et al.
J. Chem. Phys.
(2014), 140, pp. 18A508. Link ; see Section 5.7.6.

Below, we categorize the functionals that are available in Q-Chem, including exchange functionals (Section 5.3.3), correlation functionals (Section 5.3.4), and exchange-correlation functionals (Section 5.3.5). Within each category the functionals will be categorized according to Jacob’s Ladder. Exchange and correlation functionals can be invoked using the $rem variables EXCHANGE and CORRELATION, while the exchange-correlation functionals can be invoked either by setting the $rem variable METHOD or alternatively (in most cases, and for backwards compatibility with earlier versions of Q-Chem) by using the $rem variable EXCHANGE. Some caution is warranted here. While setting METHOD to PBE, for example, requests the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional, ⁹²³ Perdew J. P., Burke K., Ernzerhof M.
Phys. Rev. Lett.
(1996), 77, pp. 3865. Link which includes both PBE exchange and PBE correlation, setting EXCHANGE = PBE requests only the exchange component and setting CORRELATION = PBE requests only the correlation component. Setting both of these values is equivalent to specifying METHOD = PBE.

	Single-Point	Optimization	Frequency
Ground State	LSDA	LSDA	LSDA
GGA	GGA	GGA
meta-GGA	meta-GGA	meta-GGA
GH	GH	GH
RSH	RSH	RSH
NLC	NLC	VV10
DFT-D	DFT-D	DFT-D
SRC	—	—
	XDM	—	—
TDDFT	LSDA	LSDA	LSDA
GGA	GGA	GGA
meta-GGA	meta-GGA	meta-GGA
GH	GH	GH
RSH	RSH	RSH
NLC	—	—
DFT-D	DFT-D	DFT-D
SRC	—	—
	—	—	—

Finally, Table 5.1 provides a summary, arranged according to Jacob’s Ladder, of which categories of functionals are available with analytic first derivatives (for geometry optimizations) or second derivatives (for vibrational frequency calculations). If analytic derivatives are not available for the requested job type, Q-Chem will automatically generate them via finite difference. Tests of the finite-difference procedure, in cases where analytic second derivatives are available, suggest that finite-difference frequencies are accurate to $<1$ cm ${}^{-1}$ , except for very low-frequency, non-bonded modes. ⁷⁴⁵ Liu K.-Y., Liu J., Herbert J. M.
J. Comput. Chem.
(2017), 38, pp. 1678. Link Also listed in Table 5.1 are which functionals are available for excited-state time-dependent DFT (TDDFT) calculations, as described in Section 7.3.

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5.3.1 Introduction