Q-Chem 5.1 User’s Manual

5.3 Overview of Available Functionals

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[Grimme(2006b)] or $\omega $B97X[Chai and Head-Gordon(2008a)]) or because they are a standard linear combination of exchange and correlation (e.g., PBE[Perdew et al.(1996)Perdew, Burke, and Ernzerhof] or B3LYP[Becke(1993), Stephens et al.(1994)Stephens, Devlin, Chabolowski, and Frisch]). 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[Perdew et al.(2005)Perdew, Ruzsinszky, Tao, Staroverov, Scuseria, and Csonka] in 2001. 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, $\nabla \rho _{\sigma }$. 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,

  \begin{equation} \label{eq:tau} \tau _{\sigma }=\displaystyle \sum _{i}^{n_{\sigma }}\left|\nabla \psi _{i,\sigma }\right|^{2} \; . \end{equation}   (5.17)

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

  \begin{equation} \label{eq:GHGGA} 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}} \end{equation}   (5.18)

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 ($\rm erfc \equiv 1 - erf$):

  \begin{equation} \label{eq:erf_ partition} \frac{1}{r^{}_{12}} =\frac{\mbox{erfc}(\omega r^{}_{12})}{r^{}_{12}} + \frac{\mbox{erf}(\omega r_{12}^{})}{r^{}_{12}} \; \end{equation}   (5.19)

The first term on the right in Eq. () 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

  \begin{equation} \label{eq:RSHGGA} 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}} \; , \end{equation}   (5.20)

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. eq:RSHGGA, $c^{}_{x,\rm LR}$ 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

  \begin{equation} \label{eq:DHGGA} 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}} \; . \end{equation}   (5.21)

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:

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,[Perdew et al.(1996)Perdew, Burke, and Ernzerhof] 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$^{\dagger \star }$

LSDA$^{\dagger \star }$

LSDA$^{\star }$

 

GGA$^{\dagger \star }$

GGA$^{\dagger \star }$

GGA$^{\star }$

 

meta-GGA$^{\dagger \star }$

meta-GGA$^{\dagger }$

—

 

GH$^{\dagger \star }$

GH$^{\dagger \star }$

GH$^{\star }$

 

RSH$^{\dagger \star }$

RSH$^{\dagger \star }$

RSH$^{\star }$

 

NLC$^{\dagger \star }$

NLC$^{\dagger \star }$

—

 

DFT-D

DFT-D

DFT-D

 

XDM

—

—

TDDFT

LSDA$^{\dagger \star }$

LSDA$^{\dagger \star }$

LSDA

 

GGA$^{\dagger \star }$

GGA$^{\dagger \star }$

GGA

 

meta-GGA$^{\dagger \star }$

—

—

 

GH$^{\dagger \star }$

GH$^{\dagger \star }$

GH

 

RSH$^{\dagger \star }$

RSH$^{\dagger \star }$

—

 

—

—

—

 

DFT-D

DFT-D

DFT-D

 

—

—

—

$^\dagger $OpenMP parallelization available

$^\star $MPI parallelization available
Table 5.1: Available analytic properties and parallelization for SCF calculations.

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.[Liu et al.(2017)Liu, Liu, and Herbert] 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.

5.3.1 Suggested Density Functionals

Q-Chem contains over 150 exchange-correlation functionals, not counting those that can be straightforwardly appended with a dispersion correction (such as B3LYP-D3). Therefore, we suggest a few functionals from the second through fourth rungs of Jacob’s Ladder in order to guide functional selection. Most of these suggestions come from a benchmark of over 200 density functionals on a vast database of nearly 5000 data points, covering non-covalent interactions, isomerization energies, thermochemistry, and barrier heights. The single recommended method from each category is indicated in bold.

From the GGAs on Rung 2, we recommend:

From the meta-GGAs on Rung 3, we recommend:

From the hybrid GGAs on Rung 4, we recommend:

From the hybrid meta-GGAs on Rung 4, we recommend:

5.3.2 Exchange Functionals

Note: All exchange functionals in this section can be invoked using the $rem variable EXCHANGE. Popular and/or recommended functionals within each class are listed first and indicated in bold. The rest are in alphabetical order.

Local Spin-Density Approximation (LSDA)

Generalized Gradient Approximation (GGA)

Meta-Generalized Gradient Approximation (meta-GGA)

5.3.3 Correlation Functionals

Note: All correlation functionals in this section can be invoked using the $rem variable CORRELATION. Popular and/or recommended functionals within each class are listed first and indicated in bold. The rest are in alphabetical order.

Local Spin-Density Approximation (LSDA)

Generalized Gradient Approximation (GGA)

Meta-Generalized Gradient Approximation (meta-GGA)

5.3.4 Exchange-Correlation Functionals

Note: All exchange-correlation functionals in this section can be invoked using the $rem variable METHOD. For backwards compatibility, all of the exchange-correlation functionals except for the ones marked with an asterisk can be used with the $rem variable EXCHANGE. Popular and/or recommended functionals within each class are listed first and indicated in bold. The rest are in alphabetical order.

Local Spin-Density Approximation (LSDA)

Generalized Gradient Approximation (GGA)

Meta-Generalized Gradient Approximation (meta-GGA)

Global Hybrid Generalized Gradient Approximation (GH GGA)

Global Hybrid Meta-Generalized Gradient Approximation (GH meta-GGA)

Range-Separated Hybrid Generalized Gradient Approximation (RSH GGA)

Range-Separated Hybrid Meta-Generalized Gradient Approximation (RSH meta-GGA)

Double Hybrid Generalized Gradient Approximation (DH GGA)

Note: In order to use the resolution-of-the-identity approximation for the MP2 component, specify an auxiliary basis set with the $rem variable AUX_BASIS

Double Hybrid Meta-Generalized Gradient Approximation (DH MGGA)

5.3.5 Specialized Functionals

5.3.6 User-Defined Density Functionals

Users can also request a customized density functional consisting of any linear combination of exchange and/or correlation functionals available in Q-Chem. A “general” density functional of this sort is requested by setting EXCHANGE = GEN and then specifying the functional by means of an $xc_functional input section consisting of one line for each desired exchange (X) or correlation (C) component of the functional, and having the format shown below. 

$xc_functional
   X   exchange_symbol   coefficient
   X   exchange_symbol   coefficient
   ...
   C   correlation_symbol   coefficient
   C   correlation_symbol   coefficient
   ...
   K   coefficient
$end

Each line requires three variables: X or C to designate whether this is an exchange or correlation component; the symbolic representation of the functional, as would be used for the EXCHANGE or CORRELATION keywords variables as described above; and a real number coefficient for each component. Note that Hartree-Fock exchange can be designated either as “X" or as “K". Examples are shown below.

Example 5.44  Q-Chem input for H$_2$O with the B3tLap functional.

$molecule
  0  1
  O
  H1  O  oh
  H2  O  oh  H1  hoh

  oh  =   0.97
  hoh = 120.0
$end

$rem
   EXCHANGE     gen 
   CORRELATION  none 
   BASIS        g3large   ! recommended for high accuracy
   THRESH       14        ! and better convergence
$end

$xc_functional
X    Becke     0.726
X    S         0.0966
C    PK06      1.0
K    0.1713
$end

Example 5.45  Q-Chem input for H$_2$O with the BR89B94hyb functional.

$molecule
   0  1
   O
   H1  O  oh
   H2  O  oh  H1  hoh

   oh  =   0.97
   hoh = 120.0
$end

$rem
   EXCHANGE     gen
   CORRELATION  none
   BASIS        g3large   ! recommended for high accuracy
   THRESH       14        ! and better convergence
$end

$xc_functional
   X    BR89      0.846
   C    B94hyb    1.0
   K              0.154
$end

The next two examples illustrate the use of the RI-B05 and RI-PSTS functionals. These are presently available only for single-point calculations, and convergence is greatly facilitated by obtaining converged SCF orbitals from, e.g., an LDA or HF calculation first. (LDA is used in the example below but HF can be substituted.) Use of the RI approximation (Section 6.6) requires specification of an auxiliary basis set.

Example 5.46  Q-Chem input of H$_2$ using RI-B05.

$comment
   H2, example of SP RI-B05.  First do a well-converged LSD, G3LARGE is the
   basis of choice for good accuracy. The input lines
   PURECART    2222
   SCF_GUESS   CORE
   are obligatory for the time being here.
$end

$molecule
0 1
H   0.  0.   0.0
H   0.  0.   0.7414
$end

$rem
   SCF_GUESS        core
   METHOD           lda
   BASIS            g3large
   PURECART         2222
   THRESH           14
   INCDFT           false
   SYM_IGNORE       true
   SYMMETRY         false
   SCF_CONVERGENCE  9      
$end

@@@

$comment
   For the time being the following input lines are obligatory:
   PURECART        2222
   AUX_BASIS       riB05-cc-pvtz
   DFT_CUTOFFS     0
   MAX_SCF_CYCLES  0
$end

$molecule
   read
$end

$rem
   SCF_GUESS       read
   EXCHANGE        b05   ! or set to psts for ri-psts
   PURECART        2222
   BASIS           g3large
   AUX_BASIS       rib05-cc-pvtz  ! the aux basis for both RI-B05 and RI-PSTS
   THRESH          4
   PRINT_INPUT     true
   INCDFT          false
   SYM_IGNORE      true
   SYMMETRY        false
   MAX_SCF_CYCLES  0
   DFT_CUTOFFS     0
$end