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
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or
$\omega $B97X
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) or because they are a standard linear
combination of exchange and correlation (e.g., PBE
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or
B3LYP
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). 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.
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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,
$\widehat{\nabla}{\rho}_{\sigma}$. 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,
$${\tau}_{\sigma}=\sum _{i}^{{n}_{\sigma}}{\left|\nabla {\psi}_{i,\sigma}\right|}^{2}.$$ | (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
$${E}_{xc}^{\text{B3LYP}}={c}_{x}{E}_{x}^{\text{HF}}+\left(1-{c}_{x}-{a}_{x}\right){E}_{x}^{\text{Slater}}+{a}_{x}{E}_{x}^{\text{B88}}+\left(1-{a}_{c}\right){E}_{c}^{\text{VWN1RPA}}+{a}_{c}{E}_{c}^{\text{LYP}}$$ | (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 ($\mathrm{erfc}\equiv 1-\mathrm{erf}$):
$$\frac{1}{{r}_{12}}=\underset{\text{SR}}{\underset{\u23df}{\frac{\text{erfc}(\omega {r}_{12})}{{r}_{12}}}}+\underset{\text{LR}}{\underset{\u23df}{\frac{\text{erf}(\omega {r}_{12})}{{r}_{12}}}}$$ | (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
$${E}_{xc}^{\text{RSH}}={c}_{x,\mathrm{SR}}{E}_{x,\mathrm{SR}}^{\text{HF}}+{c}_{x,\mathrm{LR}}{E}_{x,\mathrm{LR}}^{\text{HF}}+(1-{c}_{x,\mathrm{SR}}){E}_{x,\mathrm{SR}}^{\text{DFT}}+(1-{c}_{x,\mathrm{LR}}){E}_{x,\mathrm{LR}}^{\text{DFT}}+{E}_{c}^{\text{DFT}},$$ | (5.13) |
where the SR and LR parts of the Coulomb operator are used, respectively, to evaluate the HF exchange energies ${E}_{x,\mathrm{SR}}^{\text{HF}}$ and ${E}_{x,\mathrm{LR}}^{\text{HF}}$. The corresponding DFT exchange functional is partitioned in the same manner, but the correlation energy ${E}_{c}^{\text{DFT}}$ is evaluated using the full Coulomb operator, ${r}_{12}^{-1}$. Of the two linear parameters in Eq. (5.13), ${c}_{x,\mathrm{LR}}$ 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 (${c}_{x,\mathrm{SR}}$) 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
$${E}_{xc}^{\text{DH}}={c}_{x}{E}_{x}^{\text{HF}}+\left(1-{c}_{x}\right){E}_{x}^{\text{DFT}}+{c}_{c}{E}_{x}^{\text{MP2}}+\left(1-{c}_{c}\right){E}_{c}^{\text{DFT}}.$$ | (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 5.7.1), including those of
Vydrov and Van Voorhis
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(VV09 and VV10) and of Lundqvist and Langreth
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(vdW-DF-04 and vdW-DF-10). The revised VV10 NLC functional of Sabatini and coworkers (rVV10) is also available
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.
Damped, atom–atom pairwise empirical dispersion potentials from Grimme and
others
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[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 5.7.2.
The exchange-dipole models (XDM) of Johnson and Becke (XDM6 and XDM10); see Section 5.7.3.
Below, we categorize the functionals that are available in Q-Chem, including
exchange functionals (Section 5.3.2), correlation functionals
(Section 5.3.3), and exchange-correlation functionals
(Section 5.3.4). 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 $rem variable
EXCHANGE. Some caution is warranted here. While setting
METHOD to PBE, for example, requests the
Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional,
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which includes both PBE exchange and PBE correlation, setting
EXCHANGE = PBE requests only the exchange component and setting
CORRELATION = PBE requests only the correlation component.
Setting both of these values is equivalent to specifying
METHOD = PBE.
Single-Point | Optimization | Frequency | |
Ground State | LSDA${}^{\u2020\star}$ | LSDA${}^{\u2020\star}$ | LSDA${}^{\u2020\star}$ |
GGA${}^{\u2020\star}$ | GGA${}^{\u2020\star}$ | GGA${}^{\u2020\star}$ | |
meta-GGA${}^{\u2020\star}$ | meta-GGA${}^{\u2020}$ | meta-GGA${}^{\u2020}$ | |
GH${}^{\u2020\star}$ | GH${}^{\u2020\star}$ | GH${}^{\u2020\star}$ | |
RSH${}^{\u2020\star}$ | RSH${}^{\u2020\star}$ | RSH${}^{\u2020\star}$ | |
NLC${}^{\u2020\star}$ | NLC${}^{\u2020\star}$ | — | |
DFT-D | DFT-D | DFT-D | |
XDM | — | — | |
TDDFT | LSDA${}^{\u2020\star}$ | LSDA${}^{\u2020\star}$ | LSDA${}^{\u2020}$ |
GGA${}^{\u2020\star}$ | GGA${}^{\u2020\star}$ | GGA${}^{\u2020}$ | |
meta-GGA${}^{\u2020\star}$ | — | — | |
GH${}^{\u2020\star}$ | GH${}^{\u2020\star}$ | GH${}^{\u2020}$ | |
RSH${}^{\u2020\star}$ | RSH${}^{\u2020\star}$ | — | |
— | — | — | |
DFT-D | DFT-D | DFT-D | |
— | — | — | |
${}^{\u2020}$OpenMP parallelization available | |||
${}^{\star}$MPI parallelization available |
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 $$ cm${}^{-1}$, except for very
low-frequency, non-bonded modes.
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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. Lastly, Table 5.1
describes which functionals have been parallelized with OpenMP and/or MPI.