Density Functionals (XC)
************************

The density functional, also called the exchange-correlation (XC)
functional, can include an LDA part, a GGA part, a Hartree-Fock
exchange part (hybrids), and a meta-GGA part (meta-GGA or meta-
hybrid). Possibly, it also depends on virtual Kohn-Sham orbitals
through the inclusion of an orbital-dependent correlation (double-
hybrids). LDA stands for the Local Density Approximation, which
implies that the XC functional at each point in space depends only on
the (spin) density at that point. GGA stands for Generalized Gradient
Approximation and extends the LDA part by including terms that depend
on derivatives of the density. A hybrid GGA (for example B3LYP) is a
combination of a standard GGA with a fraction of Hartree-Fock
exchange. A meta-GGA (for example TPSS) has a GGA part, but also
depends on the kinetic energy density. A meta-hybrid (for example
TPSSh) has a GGA part, a fraction of Hartree-Fock exchange, and a part
that depends on the kinetic energy density. For these terms, ADF
supports a large number of formulas advocated in the literature. For
post-SCF energies only, ADF also supports various other meta-GGA
functionals and additional hybrid functionals. A double-hybrid has a
hybrid or a meta-hybrid part, but also contains a contribution from
second-order Møller-Plesset perturbation theory (MP2). Here, only the
hybrid (meta-hybrid) part is evaluated self-consistently; the MP2 part
is evaluated post-SCF and added to the hybrid (meta-hybrid) energy.

The key that controls the Density Functional is "XC". All subkeys are
optional.

   XC
     {LDA LDA {Stoll}}
     {GGA GGA}
     {MetaGGA metagga}
     {Model MODELPOT [IP]}
     {HartreeFock}
     {HYBRID hybrid {HF=HFpart}}
     {MetaHYBRID metahybrid}
     {DOUBLEHYBRID doublehybrid}
     {RPA {option}}
     {XCFUN}
     {RANGESEP {GAMMA=X} {ALPHA=a} {BETA=b}}
     {LibXC functional}
     {DISPERSION [s6scaling] [RSCALE=r0scaling] [Grimme3] [BJDAMP] [PAR1=par1] [PAR2=par2] [PAR3=par3] [PAR4=par4]}
     {Dispersion Grimme4 {s6=...} {s8=...} {a1=...} {a2=...}}
     {DISPERSION dDsC}
     {DISPERSION UFF}
   end

If the "XC" key is omitted, the program will use the Local Density
Approximation (no GGA terms). The chosen LDA form is then VWN.


LDA
===

   XC
     LDA {functional} {Stoll}
   End

"LDA"
   Defines the LDA part of the XC functional. If "functional" is
   omitted, VWN will be used (even if LYP is specified in the GGA
   part).

   Available LDA functionals:

   * **Xonly**: The pure-exchange electron gas formula. Technically,
     this is identical to the Xalpha form (see next) with a value of
     2/3 for the X-alpha parameter.

   * **Xalpha**: The scaled (parametrized) exchange-only formula. When
     this option is used, you may (optionally) specify the X-alpha
     *parameter* by typing a numerical value after the string Xalpha
     (separated by a blank). If omitted, this parameter takes the
     default value 0.7.

   * **VWN**: The parametrization of electron gas data given by Vosko,
     Wilk, and Nusair (Ref. [1], formula version V). Among the
     available LDA options, this is one of the more advanced ones,
     including correlation effects to a fair extent.

     * "Stoll": For the VWN variety of the LDA form, you may include
       Stoll's correction [2] by typing Stoll on the same line, after
       the main LDA specification. You must not use Stoll's correction
       in combination with the Xonly or the Xalpha form for the Local
       Density functional. The Stoll formula is considered to be a
       *correlation* correction to the *Local* Density Approximation.
       It is conceptually not correct to use the Stoll correction
       *and* apply gradient (GGA) corrections to the correlation. It
       is the user's responsibility, in general and also here, to
       avoid using options that are not solidly justified
       theoretically.

   * **PW92**: The parametrization of electron gas data given by
     Perdew and Wang (Ref. [3]).


GGA
===

   XC
      GGA functional
   End

"GGA"
   Specifies the GGA part of the XC functional (formerly often called
   the 'non-local' correction to the LDA part of the density
   functional). It uses derivatives (gradients) of the charge density.

   Available GGA functionals:

   * **BP86**: Exchange: Becke, Correlation: Perdew

   * **PW91**: Exchange: pw91x, Correlation: pw91c

   * **mPW**: Exchange: mPWx, Correlation: pw91c

   * **PBE**: Exchange: PBEx, Correlation: PBEc

   * **RPBE**: Exchange: RPBEx, Correlation: PBEc

   * **revPBE**: Exchange: revPBEx, Correlation: PBEc

   * **mPBE**: Exchange: mPBEx, Correlation: PBEc

   * **PBEsol**: Exchange: PBEsolx, Correlation: PBEsolc

   * **HTBS**: Exchange: HTBSx, Correlation: PBEc

   * **BLYP**: Exchange: Becke, Correlation: LYP

   * **OLYP**: Exchange: OPTX, Correlation: LYP

   * **OPBE**: Exchange: OPTX, Correlation: PBEc [4]

   * **BEE**: Exchange: BEEx, Correlation: PBEc

   * **XLYP**: Exchange: XLYPx [5] (exchange, not available separately
     from LYP) + LYP

   * **SSB-D**: Dispersion corrected functional by Swart-
     Solà-Bickelhaupt [65] [66]. The SSB-D functional by definition
     already includes a dispersion correction by Grimme (factor
     0.847455). There are some numerical issues with the GGA
     implementation in ADF of SSB-D (Ref. [65] [66]) for some systems.
     Because of this, the GGA SSB-D option is only available for
     single-point calculations (and NMR). Geometry optimizations
     (etc.) are still possible by using instead:

        XC
          METAGGA SSB-D
        END

     This METAGGA implementation requires an all-electron basis set.
     Use GGA SSB-D for NMR calculations.

   * **S12g**: Dispersion corrected (Grimme-D3) functional by Swart,
     successor of SSB-D [6].

   * **LB94**: By Van Leeuwen and Baerends [7].

   * **KT1**: By Keal and Tozer [8].

   * **KT2**: By Keal and Tozer [8].

   If only a GGA part is specified (omitting the "LDA" subkey), the
   LDA part defaults to VWN, except when the LYP correlation
   correction is used: in that case the LDA default is Xonly, i.e.
   pure exchange. The reason for this is that the LYP formulas assume
   the pure-exchange LDA form, while, for instance, the Perdew-86
   correlation correction is a correction to a *correlated* LDA form.
   The precise form of this correlated LDA form assumed in the
   Perdew-86 correlation correction is not available as an option in
   ADF, but the VWN formulas are fairly close to it.

   Separate choices can be made for the GGA exchange correction and
   the GGA correlation correction. Both specifications must be typed
   (if at all) on the same line after the "GGA" subkey.

   For the **exchange** part the options are:

   * **Becke**: Becke (1988) [10].

   * **PW86x**: Perdew-Wang (1986) [11].

   * **PW91x**: Perdew-Wang (1991) [12].

   * **mPWx**: Modified PW91 by Adamo-Barone (1998) [13].

   * **PBEx**: Perdew-Burke-Ernzerhof (1996) [14].

   * **RPBEx**: Revised PBE by Hammer-Hansen-Norskov (1999) [15].

   * **revPBEx**: Revised PBE by Zhang-Wang (1998) [16].

   * **mPBEx**: Modified PBE by Adamo-Barone (2002) [17].

   * **PBEsolx**: Perdew-Ruzsinszky-Csonka-Vydrov-Scuseria (2008)
     [18].

   * **HTBSx**: [19].

   * **OPTX**: Handy-Cohen (2001) [20].

   * **BEEx**: Mortensen-Kaasbjerg-Frederiksen-Nørskov-Sethna-Jacobsen
     (2005) [21].

   For the **correlation** part the options are:

   * **Perdew**: Perdew (1986) [22].

   * **PBEc**: Perdew-Burke-Ernzerhof (1996) [14].

   * **PBEsolc**: The PBEsol correlation correction by Perdew-
     Ruzsinszky-Csonka-Vydrov-Scuseria (2008) [18].

   * **PW91c**: Perdew-Wang (1991), see [12].

   * **LYP**: Lee-Yang-Parr (1988) correlation correction [67] [68]
     [69].

   The "GGA" specification must contain at most one exchange option
   and at most one correlation option. If options are applied for
   both, they must be separated by a blank or a comma. Example:

      XC
        GGA Becke Perdew
      End

   is equivalent to

      XC
        GGA BP86
      End

   It is questionable to apply gradient corrections to the
   *correlation* while not doing so at the same time for the exchange.
   Therefore, the program will check this and stop with an error
   message. This check can be overridden with the key ALLOW.


MetaGGA
=======

   XC
     MetaGGA functional
   End

"MetaGGA"
   Specifies that a meta-GGA should be used during the SCF. **All-
   electron basis sets should be used** (see Basis key).

   Available meta-GGA functionals:

   * **M06-L**: Functional by Yan-Truhlar [61] [62].

   * **TPSS**: Functional by Tao-Perdew-Staroverov-Scuseria [63] [64].

   * **revTPSS**: Revised TPSS functional [24].

   * **SSB-D**: Dispersion corrected GGA functional by Swart-
     Solà-Bickelhaupt [65] [66]. Use GGA SSB-D for NMR calculations.

   * **MVS**: Functional by Sun-Perdew-Ruzsinszky [25].

   * **MS0**: Functional by Sun et al. [26].

   * **MS1**: Functional by Sun et al. [27].

   * **MS2**: Functional by Sun et al. [27].

   * **SCAN**: Functional by Sun et al. [28].

   * **TASKxc**: Functional by Aschebrock et al. [34], intended for
     charge transfer systems.

   * **TASKCC**: Functional by Lebeda et al. [36], an inprovement of
     TASKxc for atomization energies.

   * **r2SCAN-3c**: Composite DFT method by Gasevic et al. [35].

   * **LAK**: Functional by Lebeda and coworkers [9]. It is a general-
     purpose functional.

The r2SCAN-3c composite method [35] uses the r^2 SCAN (r2SCAN)
exchange-correlation functional, in combination with a tailor-made
all-electron polarized basis set (mTZ2P), the semiclassical London
dispersion correction (D4), and a geometrical counterpoise (gCP)
correction. NumericalQuality should be set to Good, and ZORA should be
used. Note that LibXC will be used internally for the r2SCAN
functional, and the D4 and gCP corrections will be included
automatically. The STO-optimized r2SCAN-3c outperforms many
conventional hybrid/QZ approaches in most common applications at a
fraction of their cost.

The M06-L functional needs high integration accuracy (at least
BeckeGrid quality good) for reasonable gradients. For TPSS, moderate
integration accuracy is sufficient for reasonable gradients. For
heavier elements (Z>36), and if one uses the M06-L functional, it is
also necessary to include the following keyword:

   FragMetaGGAToten

Using this FRAGMETAGGATOTEN key, the difference in the meta-hybrid or
meta-GGA exchange-correlation energies between the molecule and its
fragments will be calculated using the molecular integration grid,
which is more accurate than the default but is much more time-
consuming. By default, the meta-GGA exchange-correlation energies for
the fragments are calculated on the numerical integration grid of the
fragments.


Hartree-Fock
============

   XC
     HartreeFock
   End

"HartreeFock"
   Specifies that the Hartree-Fock exchange should be used during the
   SCF.


Hybrid
======

   XC
     HYBRID functional {HF=HFpart}
   End

"HYBRID"
   Specifies that a hybrid functional should be used during the SCF.

   Available hybrid functionals:

   * **B3LYP**: ADF uses VWN5 in the B3LYP functional (20% HF
     exchange) by Stephens-Devlin-Chablowski-Frisch [29].

   * **B3LYP***: Modified B3LYP functional (15% HF exchange) by
     Reiher-Salomon-Hess [30].

   * **B1LYP**: Functional (25% HF exchange) by Adamo-Barone [31].

   * **KMLYP**: Functional (55.7% HF exchange) by Kang-Musgrave [32].

   * **O3LYP**: Functional (12% HF exchange) by Cohen-Handy [33].

   * **X3LYP**: Functional (21.8% HF exchange) by Xu-Goddard [5].

   * **BHandH**: 50% HF exchange, 50% LDA exchange, and 100% LYP
     correlation.

   * **BHandHLYP**: 50% HF exchange, 50% LDA exchange, 50% Becke88
     exchange, and 100% LYP correlation.

   * **B1PW91**: Functional (25% HF exchange) by Adamo-Barone [31].

   * **mPW1PW**: Functional (25% HF exchange) by Adamo-Barone [13].

   * **mPW1K**: Functional (42.8% HF exchange) by Lynch-Fast-Harris-
     Truhlar [37].

   * **PBE0**: Functional (25% HF exchange) by Ernzerhof-Scuseria [38]
     and by Adamo-Barone [39], hybrid form of PBE.

   * **OPBE0**: Functional (25% HF exchange) by Swart-Ehlers-
     Lammertsma [4], hybrid form of OPBE.

   * **S12H**: Dispersion corrected (Grimme-D3) functional (25% HF
     exchange) by Swart [6].

   "HFpart"
      Specifies the amount of HF exchange to use instead of the
      default HF exchange fraction for the given hybrid. For example,
      HF=0.25 means 25% Hartree-Fock exchange.


Meta-Hybrid
===========

   XC
     MetaHYBRID functional
   End

"MetaHYBRID"
   Specifies that a meta-hybrid functional should be used during the
   SCF.

   Available meta-hybrid functionals:

   * **M06**: Functional (27% HF exchange) by Yan-Truhlar [61] [62].

   * **M06-2X**: Functional (54% HF exchange) by Yan-Truhlar [61]
     [62].

   * **M06-HF**: Functional (100% HF exchange) by Yan-Truhlar [61]
     [62].

   * **TPSSH**: Functional (10% HF exchange) by Tao-Perdew-Staroverov-
     Scuseria [63] [64].


Range-separated hybrids
=======================

In ADF, there are two mutually exclusive ways of specifying range-
separated hybrid functionals:

* Through the "RANGESEP" and "XCFUN" keys. This will use the Yukawa
  potential as switching function, see Ref. [40];

* By specifying a range-separated functional via the "LibXC" key.

See also the advanced tutorial: Tuning the range separation in LC-wPBE
for organic electronics.


RangeSep + XCFun: Yukawa range-separated hybrids
------------------------------------------------

"RANGESEP {GAMMA=X} {ALPHA=a} {BETA=b}"
   If RANGESEP is included, by default a long-range corrected (LC)
   functional is created with a range separation parameter GAMMA of
   0.75. In this case, the switching function is the Yukawa potential,
   see Ref. [40]. Range-separated functionals require XCFUN and are
   limited to GGA, meta-GGA, and CAMY-B3LYP. The CAMY-B3LYP functional
   is not the same as the CAM-B3LYP functional, since a different
   switching function is used. No other hybrids or meta-hybrids are
   supported. The special CAMY-B3LYP functional is defined by three
   parameters, ALPHA, BETA, and the attenuation parameter GAMMA. For
   CAMY-B3LYP, by default ALPHA is 0.19, BETA is 0.46, and GAMMA is
   0.34.

Range-separated functionals make use of a modified form of the Coulomb
operator that is split into pieces for exact exchange and DFT. In this
case, the switching function is the Yukawa potential, see Ref. [40].
Global hybrids can be thought of as a special case of a range-
separated functional where the split is independent of the inter-
electronic distance and is a simple X exact and 1-X DFT in all space.

In a general RS-functional, the split depends on the inter-electronic
distance. How the split is achieved depends on the functional in
question, but it is achieved using functions that smoothly go from 1
to 0. In practical terms, this means that a range-separated functional
looks very much like a hybrid (or meta-hybrid) functional but with
additional integrals over the attenuated interaction with fit
functions on the exact exchange side and a modified functional on the
DFT side.

**DFT part of RS-functionals**

Using Hirao's approach for creating RS-functionals, the RS form of a
given exchange functional is created by multiplying the standard
energy density by a factor that depends on the energy density. The
factor is the same for all functionals and the only difference is
introduced by the individual energy densities.

The range separation comes in at the level of the integrals over the
operator with fit functions. They are very similar to the standard
Coulomb integrals.

**RS-functionals**

An RS-functional is described by a series of regions describing each
of the pieces of the Coulomb operator. The total function is built up
by looping over the regions and summing all the pieces. Currently,
simple LC functionals can be defined where the exact exchange goes
from 0 to 1 as the inter-electronic distance increases, and the DFT
part does the reverse. In addition, CAMY-B3LYP type functionals can be
defined. More general functionals are not possible with XCFun.

**Functionality and limitations**

RS functionals with XCFUN are limited to the GGA and meta-GGA
functionals and one hybrid CAMY-B3LYP. The following functionals can
be evaluated with range separation at the present time:

* LDA: VWN5, X-ALPHA PW92

* GGA exchange: Becke88, PBEX, OPTX, PW91X, mPW, revPBEX

* GGA correlation: LYP, Perdew86, PBEC

* MetaGGA: TPSS, M06L, B95

* Hybrids: CAMY-B3LYP

The following functionality has been tested: XC potential, energy,
ground state geometry, and TDDFT. Starting from ADF2018, singlet-
triplet excitation calculations and excited state geometry
optimizations are possible. For possible limitations in excitation
calculations or excited state geometry optimizations, see the
corresponding part of the ADF manual.

**Numerical stability**

The range-separated implementation requires that the range separation
parameter is not too close to the exponent of a fit function. In
practice, this means that values of the separation parameter between
1.0 and 50 can cause numerical problems. Typical useful values are in
the range 0.2 to 0.9, so this should not be too serious a limitation.

   XC
     XCFUN
     RANGESEP {GAMMA=X} {ALPHA=a} {BETA=b}
   END

Range separation is activated by putting RANGESEP in the XC block.
Inclusion of XCFUN is required; see the XCFUN description. By default,
a long-range corrected (LC) functional is created with a range
separation parameter of 0.75. The parameter can be changed by
modifying X in GAMMA=X in the RANGESEP card. Range separation is
typically used in combination with a GGA or METAGGA functional.

Range separation cannot be included with a hybrid or meta-hybrid, with
one exception: the special RS functional CAMY-B3LYP. This is entered
as HYBRID CAMY-B3LYP and must be used in combination with XCFUN (see
XCFUN description) and RANGESEP. The CAMY-B3LYP functional is defined
by three parameters, ALPHA, BETA, and the attenuation parameter GAMMA.
The GAMMA parameter can be modified as for the LC functionals. For
CAMY-B3LYP it defaults to 0.34. The ALPHA and BETA parameters can be
modified through ALPHA=a and BETA=b in the RANGESEP card. They default
to 0.19 and 0.46, respectively.

   XC
     HYBRID CAMY-B3LYP
     XCFUN
     RANGESEP GAMMA=0.34 ALPHA=0.19 BETA=0.46
   END

List of the most important functionals for which one can use range
separation:

   LDA VWN
   GGA BLYP
   GGA BP86
   GGA PBE
   HYBRID CAMY-B3LYP


Range-separated hybrids with LibXC
----------------------------------

You can simply specify a range-separated hybrid functional in the
LibXC key, *e.g.*:

   XC
     LibXC CAM-B3LYP
   End

See the LibXC section for a list of available range-separated hybrid
functionals.

For the HSE03 and HSE06 short range-separated hybrids, you can
(optionally) specify the switching parameter omega, *e.g.*:

   XC
     LibXC HSE06 omega=0.1
   End


Notes on Hartree-Fock and (meta-)hybrid functionals
===================================================

If a functional contains a part of Hartree-Fock exchange, then the
LDA, GGA, meta-GGA, or MODEL key should not be used. You should
specify only one of HartreeFock, HYBRID, or MetaHYBRID. Dispersion can
be added. Note that it is not recommended to use (part of the)
Hartree-Fock exchange in combination with frozen cores, since the
frozen core orbitals are currently not included in the Hartree-Fock
exchange operator. In ADF, one can do unrestricted Hartree-Fock (or
hybrid or meta-hybrid) calculations, as long as one has integer
occupation numbers. The default implementation in ADF for unrestricted
Hartree-Fock calculations is UHF. In case of a high-spin electron
configuration, one can do ROHF; see ROKS for high-spin open shell
molecules. You need to use the same XC potential in the create run of
the atoms, which is done automatically if you use the BASIS key.

Starting from ADF2009.01, the meta-hybrids M06, M06-2X, M06-HF, and
TPSSH can be used during the SCF. Also starting from ADF2009.01,
Hartree-Fock and the (meta-)hybrid potentials can be used in
combination with geometry optimization, TS, IRC, LT, and numerical
frequencies; hybrids can be used in calculating NMR chemical shift;
PBE0 can be used in calculating NMR spin-spin coupling; Hartree-Fock
and (meta-)hybrid can be used in calculating excitation energies, in
which the kernel consists of the Hartree-Fock percentage times the
Hartree-Fock kernel plus one minus the Hartree-Fock percentage times
the ALDA kernel (thus no (meta-)GGA kernel). Hartree-Fock and the
(meta-)hybrid potentials still cannot (or should) not be used in
combination with analytical frequencies, the (AO)RESPONSE key, EPR/ESR
g-tensor, and frozen cores. Starting from ADF2010, it is possible to
use Hartree-Fock and hybrids to calculate CD spectra.

In ADF, one can do unrestricted Hartree-Fock (or hybrid or meta-
hybrid) calculations (UHF, UKS) as long as the occupation numbers are
integers. For a high-spin electron configuration, one can use ROHF or
ROKS; see ROKS for high-spin open shell molecules.

It is possible to change the amount of HF exchange in the input for
hybrids (not for meta-hybrids and Hartree-Fock). For many hybrid
functionals, the sum of the amount of Hartree-Fock exchange and the
amount of LDA exchange (or GGA exchange) is one. If that is the case,
then if one changes the amount of Hartree-Fock exchange in the input,
the amount of LDA exchange (or GGA exchange) will also be changed such
that the sum remains one. Example:

   XC
     Hybrid B3LYP HF=0.25
   END

In this case, the amount of Hartree-Fock for the B3LYP functional will
be changed to 25% (instead of 20%), and the amount of LDA exchange to
75% (instead of 80%). The LDA correlation and GGA exchange and
correlation part will be left unaltered.

An accuracy issue is relevant for some of the meta-GGA functionals, in
particular the M06 functionals. These need high integration accuracy
(at least BeckeGrid quality good) for reasonable gradients. For TPSSH,
moderate integration accuracy is sufficient for reasonable gradients.
For heavier elements (Z>36), and if one uses one of the M06
functionals, it is also necessary to include the following keyword

   FragMetaGGAToten

Using this key "FRAGMETAGGATOTEN", the difference in the meta-hybrid
or meta-GGA exchange-correlation energies between the molecule and its
fragments will be calculated using the molecular integration grid,
which is more accurate than the default but is much more time-
consuming. By default, the meta-hybrid or meta-GGA exchange-
correlation energies for the fragments are calculated on the numerical
integration grid of the fragments.

For benchmark calculations, one would like to use a large basis set,
like the QZ4P basis set. In such cases it is recommended to use a good
numerical quality. Thus, for accurate hybrid calculations of **small**
molecules, one could use:

   basis
     type QZ4P
   end
   AddDiffuseFit
   Dependency bas=1e-4
   NumericalQuality good


MP2, Double Hybrids, RPA
========================

To treat correlation energies beyond DFT, ADF offers MP2 and random-
phase approximation (RPA) based methods. In addition, ADF offers a
large number of modern double-hybrid functionals that combine MP2
correlation with a hybrid functional.

ADF implements canonical MP2 using density fitting. Additionally, ADF
implements RPA and direct MP2 using an efficient atomic-orbital-based
algorithm. The algorithm is described in Ref. [92]. The algorithm has
been continuously improved over the last few years and currently
allows single-point calculations for systems with up to 1000 atoms on
a single modern compute node. Many of the most accurate double-hybrid
functionals use direct MP2 only.


Double Hybrid Functionals
-------------------------

   XC
     DOUBLEHYBRID functional
   End

"DOUBLEHYBRID"
   Specifies that a double-hybrid functional [71] should be used.

See also:

  The paper Double hybrid DFT calculations with Slater type orbitals
  and the page Double hybrids: recommendations for accurate
  thermochemistry and kinetics contain useful recommendations for
  Double Hybrid calculations in ADF.

Double hybrids usually yield considerably better energies than
(meta-)GGA and (meta-)hybrid functionals for (main group)
thermochemistry and kinetics, transition metal chemistry, and non-
covalent interactions. For an overview of the capabilities of double-
hybrids implemented in ADF, we refer to a review. [72]

The MP2 correlation energy consists of two terms,

   E_{MP2}           = & \frac{1}{2} (E_{\text{direct}} -
   E_{\text{ex}} )\\ E_{\text{direct}} = & \sum_{ij} \sum_{ab}\frac{(i
   a| j b) (i a| j b)}{{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b}}
   \\ E_{\text{ex}}     = & \sum_{ij} \sum_{ab}\frac{(i a| j b) (i b|
   j a)}{{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b}} \\ (i a| j b)
   = &  \iint \phi_i^\dagger(1) \phi_a(1) \frac{1}{r_{12}}
   \phi_j^\dagger(2) \phi_b(2) d1d2

For a closed-shell system, the MP2 correlation energy can also be
partitioned as

   E_{MP2} = E_{\text{OS}} + E_{\text{SS}} \\ E_{\text{OS}} =
   E_{\text{direct}} \\ E_{\text{SS}} = E_{\text{direct}} -
   E_{\text{ex}}

Here, OS (opposite spin) denotes the contribution to the correlation
energy from electrons with unpaired spins, and SS (same spin) denotes
the contribution to the correlation energy from electrons with paired
spins. In case of spin-orbit coupling, approximate SS and OS
contributions are calculated.

There are three classes of double-hybrid functionals:

* Opposite-spin-only functionals (only the OS term is used and scaled
  by an empirical factor)

* Spin-component-scaled functionals (both the OS and SS components are
  individually scaled by empirical factors)

* Standard double-hybrid functionals (both the OS and SS components
  are scaled by the same empirical factor)

We recommend using opposite-spin-only functionals for large systems
(50-100 atoms and larger), since they are computationally more
efficient than the other functionals, which also include the same-spin
contribution. **An opposite-spin-only functional calculation is always
feasible when a hybrid calculation is feasible too!**

For additional technical details of the algorithm and how to tweak the
technical parameters, see the MBPT section.

**Opposite-spin-only Double Hybrids**

Currently, ADF supports the following opposite-spin-only Double Hybrid
functionals:

* **SOS1-PBE-QIDH**:     1-parameter functional with PBE exchange, PBE
  correlation (69 % HF, 44 % OS-MP2) [73]

* **rev-DOD-BLYP**:      B88 exchange, LYP correlation, Grimme3
  dispersion (71 % HF, 62.2 % OS-MP2) [74]

* **rev-DOD-BLYP-D4**:   B88 exchange, LYP correlation, Grimme4
  dispersion (71 % HF, 63.5 % OS-MP2) [74]

* **rev-DOD-PBE**:       PBE exchange, PBE correlation, Grimme3
  dispersion (68 % HF, 61.3 % OS-MP2) [74]

* **rev-DOD-PBE-D4**:    PBE exchange, PBE correlation, Grimme4
  dispersion (68 % HF, 61.8 % OS-MP2) [74]

* **rev-DOD-PBEP86**:    PBE exchange, P86 correlation, Grimme3
  dispersion (69 % HF, 60.6 % OS-MP2) [74]

* **rev-DOD-PBEP86-D4**: PBE exchange, P86 correlation, Grimme4
  dispersion (69 % HF, 61.2 % OS-MP2) [74]

* **DOD-SCAN**:          SCAN exchange and correlation, Grimme3
  dispersion (66 % HF, 63.0 % OS-MP2) [74]

* **rev-DOD-SCAN-D4**:   SCAN exchange and correlation, Grimme4
  dispersion (66 % HF, 63.4 % OS-MP2) [74]

Except for **SOS1-PBE-QIDH**, all functionals include dispersion
correction by default which cannot be switched off.

**Standard Double Hybrids**

Currently, ADF supports the following standard double-hybrid
functionals:

* **B2PLYP**:     B88 exchange and LYP correlation (53 % HF, 27 % MP2)
  [71]

* **B2PIPLYP**:   B88 exchange and LYP correlation (60 % HF, 27 %
  MP2), parametrized for pi-pi interactions [75]

* **ROB2PLYP**:   B88 exchange and LYP correlation (59 % HF, 28 %
  MP2), restricted-open-shell version of B2PLYP [76]

* **B2TPLYP**:    B88 exchange and LYP correlation (60 % HF, 31 %
  MP2), parametrized for thermodynamics [77]

* **B2GPPLYP**:   B88 exchange and LYP correlation (65 % HF, 36 %
  MP2), 'General Purpose' parametrization [78]

* **B2KPLYP**:    B88 exchange and LYP correlation (72 % HF, 42 %
  MP2), parametrized for kinetics [77]

* **B2NCPLYP**:   B88 exchange and LYP correlation (70 % HF, 49 %
  MP2), parametrized for non-covalent interactions [79]

* **mPW2PLYP**:   mPW exchange and LYP correlation (55 % HF, 25 % MP2)
  [80]

* **mPW2KPLYP**:  mPW exchange and LYP correlation (72 % HF, 42 % MP2)
  [77]

* **mPW2NCPLYP**: mPW exchange and LYP correlation (42 % HF, 49 % MP2)
  [79]

* **DH-BLYP**:    1-Parameter functional with B88 exchange and LYP
  correlation (65 % HF, 42 % MP2) [81]

* **PBE0-DH**:    1-Parameter functional with PBE exchange and PBE
  correlation (50 % HF, 13 % MP2) [82]

* **PBE-QIDH**:   1-parameter functional with PBE exchange and PBE
  correlation (69 % HF, 33 % MP2) [73]

* **LS1-DH**:     1-Parameter functional with PBE exchange and PBE
  correlation (75 % HF, 42 % MP2) [83]

* **PBE0-2**:     1-Parameter functional with PBE exchange and PBE
  correlation (79 % HF, 50 % MP2) [84]

* **LS1-TPSS**:   1-Parameter functional with TPSS meta-GGA (85 % HF,
  61 % MP2) [85]

* **DS1-TPSS**:   1-Parameter functional with TPSS meta-GGA (73 % HF,
  53 % MP2) [85]

Empirical dispersion corrections can be requested in the XC block in
the usual way. Some functionals can be combined with Grimme's D4
empirical dispersion correction with optimized parameters: B2PLYP,
B2GPPLYP, mPW2PLYP, PBE0-DH, PBE0-2. All functionals in this category
can be combined with Grimme's D3(BJ) empirical dispersion correction
with optimized parameters. Note that for Grimme's D3(BJ) the
parameters for B2PIPLYP, ROB2PLYP, B2TPLYP, B2KPLYP, mPW2PLYP,
mPW2KPLYP, DH-BLYP, LS1-DH, PBE0-2, and DS1-TPSS are modified B2PLYP
parameters, see Ref. [72]. If no optimized empirical dispersion
parameters exist for a certain functional, default parameters are
used, which may not give the expected results.

**Spin-component-scaled functionals**

Currently, ADF supports the following spin-component scaled Double
Hybrid functionals:

* **DSD-BLYP**:          B88 exchange and LYP correlation, Grimme3
  dispersion (69 % HF, 46 % OS-MP2, 37 % SS-MP2) [86]

* **rev-DSD-BLYP**:      revised version of DSD-BLYP, Grimme3
  dispersion (71 % HF, 54.7 % OS-MP2, 19.8 % SS-MP2) [74]

* **rev-DSD-BLYP-D4**:   revised version of DSD-BLYP, Grimme4
  dispersion (71 % HF, 55.9 % OS-MP2, 19.7 % SS-MP2) [74]

* **DSD-PBEP86**:        PBE exchange and P86 correlation (69 % HF, 52
  % OS-MP2, 22 % SS-MP2) [87]

* **rev-DSD-PBEP86**:    revised version of DSD-PBEP86 (69 % HF, 57.9
  % OS-MP2, 8 % SS-MP2) [74]

* **rev-DSD-PBEP86-D4**: PBE exchange and P86 correlation, Grimme4
  dispersion (69 % HF, 59.2 % OS-MP2, 6.4 % SS-MP2) [74]

* **DSD-PBE**:           PBE exchange and PBE correlation, Grimme3
  dispersion (68 % HF, 55 % OS-MP2, 13 % SS-MP2) [87]

* **rev-DSD-PBE**:       revised version of DSD-PBE, Grimme3
  dispersion (68 % HF, 58.5 % OS-MP2, 7 % SS-MP2) [74]

* **rev-DSD-PBE-D4**:    revised version of DSD-PBE, Grimme4
  dispersion (68 % HF, 60 % OS-MP2, 4.2 % SS-MP2) [74]

* **rev-DSD-SCAN-D4**:   based on SCAN meta-GGA (66 % HF, 63.2 % OS-
  MP2, 1.3 % SS-MP2) [74]

* **SD-SCAN69**:         Based on SCAN meta-GGA, no dispersion
  correction (69 % HF, 62 % OS-MP2, 26 % SS-MP2) [74]

Except for **SD-SCAN69**, all functionals include dispersion
correction by default which cannot be switched off.


MP2
---

   XC
     MP2
     EmpiricalScaling {NONE|SOS|SCS|SCSMI}
   END

In addition to double-hybrids, ADF also implements MP2 including some
popular spin-scaled variants. Technically, they are not distinct from
double-hybrids; however, they all rely on an HF instead of a DFT
calculation. The following variants are supported.

* **SOS-MP2**:    pure HF reference (100 % HF, 130 % OS-MP2) [88]

* **MP2**:        pure HF reference (100 % HF, 100 % MP2 correlation)

* **SCS-MP2**:    pure HF reference (100 % HF, 120 % OS-MP2, 33 % SS-
  MP2) [89]

* **SOS-MI-MP2**: pure HF reference (100 % HF, 40 % OS-MP2, 129 % SS-
  MP2) [90]

In case of spin-orbit coupling, approximate SS and OS contributions
are calculated.

The spin-scaling variant can be requested in the XC block together
with the "MP2" keyword:

   XC
     MP2
     EmpiricalScaling SOS
   END

requests an SOS-MP2 calculation.

For additional technical details of the algorithm and how to tweak
parameters, see the MBPT section.


RPA
---

Note: In AMS2022, the keyword for RPA+SOX was RPASOX.

The RPA goes beyond MP2 by accounting explicitly for the
polarizability of the system, which screens the electron-electron
interaction. It can therefore be applied to large systems for which
MP2 typically diverges. [93] The following RPA-based methods are
available.

* **RPA**        : Standard (direct) RPA without exchange

* **RPA + SOX**  : Standard RPA plus statically screened second-order
  exchange [94]

* **RPA + SOSEX**: Standard RPA plus dynamically screened second-order
  exchange [95]

A detailed overview of the RPA algorithm in ADF and a detailed
assessment of the performance of second-order exchange corrections can
be found in [97].

An RPA calculation is requested in the XC block:

   XC
     RPA {NONE|DIRECT|SOSEX|SOSSX|SIGMA}
   End

An RPA calculation needs to be combined with an XC functional. For
instance,

   XC
     hybrid pbe0
     RPA DIRECT
   End

will perform a PBE0 calculation followed by a direct RPA calculation.
RPA and all of its variants can be used in conjunction with LDA, GGAs,
hybrid, and RSH functionals.

For additional technical details of the algorithm and how to tweak
parameters, see the RPA section.


sigma-functional
----------------

Starting from AMS2023, the sigma-functional by Görling and coworkers
is implemented. [96] In this method, the correlation kernel is
calculated from the adiabatic fluctuation-dissipation theorem. In
addition to the direct RPA (Hartree) kernel, higher-order
contributions to the kernel are included by the so-called sigma-
kernel, which is fitted to relative energies. Sigma-functionals are as
fast as RPA.

ADF features a σ-functional implementation which is specifically
parametrized for STO-type basis sets [23]. In particular, for the
Corr/TZ3P basis one should use the S1re parametrization of the
σ-functional, preferably with the PBE0 functional.

Like an RPA calculation, a sigma-functional calculation needs to be
combined with an XC functional. For instance,

   XC
     hybrid pbe0
     RPA sigma
   End

requests to use the sigma-functional with the W1 parametrization for
PBE0. [98] Sigma-functionals can only be used with a limited number of
exchange-correlation functionals, since they need to be explicitly
parametrized for each functional. Currently, the sigma-functional can
be used in conjunction with the GGA PBE, and the hybrids PBE0 and
B3LYP. The available parametrizations for each functional are listed
in the following table:

+----------------------------------------------------+----------------------------------------------------+
| functional                                         | available parametrizations [98] [99] [23]          |
+----------------------------------------------------+----------------------------------------------------+
| PBE                                                | W1 S1 S1re S2                                      |
+----------------------------------------------------+----------------------------------------------------+
| PBE0                                               | W1 S1 S1re S2 W2                                   |
+----------------------------------------------------+----------------------------------------------------+
| B3LYP                                              | W1                                                 |
+----------------------------------------------------+----------------------------------------------------+

Note that S1re is the default value for the parametrization, which is
optimized for the Corr/TZ3P basis set (all-electron). The
parametrization can be changed in the MBPT block, see the MBPT block.
For instance:

   MBPT
     SigmaFunctionalParametrization S1
   End


Spin-orbit coupling
-------------------

In case of spin-orbit coupling, approximate SS and OS contributions
are calculated, which is relevant for open shell molecules with double
hybrids or MP2 variants that use different scaling factors for these
contributions:

   (i a| j b)   = &  \iint \phi_i^\dagger(1) \phi_a(1)
   \frac{1}{r_{12}} \phi_j^\dagger(2) \phi_b(2) d1d2 \\ m_{ij}       =
   & \int \phi_i^\dagger(1) \vec{\sigma} \phi_i(1) d1 \cdot \int
   \phi_j^\dagger(2) \vec{\sigma} \phi_j(2) d2 \\ E_2^{SS}     = & -
   \sum_{ijab} \frac{(1+m_{ij}) (i a| j b) (a i| b
   j)}{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b} + \sum_{ijab}
   \frac{2(i a| j b) (a j| b
   i)}{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b} \\ E_2^{OS}     =
   & - \sum_{ijab} \frac{(1-m_{ij}) (i a| j b) (a i| b
   j)}{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b}

with, i,j occupied spinors, a,b virtual spinors, \epsilon spinor
energies, \vec{\sigma} Pauli spin matrices. Note that with pure \alpha
and \beta orbitals, m_{i^\alpha j^\alpha} = m_{i^\beta j^\beta} = 1,
m_{i^\alpha j^\beta} =  m_{i^\beta j^\alpha} = -1, one has the
familiar SS and OS energy expressions.


Model Potentials
================

Several asymptotically correct XC potentials have been implemented in
ADF, such as the (now somewhat outdated) LB94 potential [7], the
gradient-regulated asymptotic correction (GRAC) [41], and the
statistical average of orbital potentials (SAOP) [44] [42]. These can
currently be used only for response property calculations, not for
geometry optimizations. For spectroscopic properties, they usually
give results superior to those obtained with LDA or GGA potentials
(see Ref. [43] for applications to (hyper)polarizabilities Cauchy
coefficients, etc. of small molecules). This is particularly true if
the molecule is small and the (high-lying) virtual orbitals are
important for the property under study.

It was also shown that, simply using the orbital energies of the
occupied Kohn-Sham orbitals of a SAOP calculation, quite good
agreement with experimental vertical ionization potentials is
obtained. This is true not only for the HOMO orbital energy, which
should be identical to (minus) the experimental ionization potential
with the exact XC potential, but also for lower-lying occupied orbital
energies. The agreement becomes worse for deep-lying core orbital
energies. A theoretical explanation and practical results are given in
Ref. [45].

   XC
     Model ModelPotential [IP]
   End

"MODEL"
   Specifies that one of the less common XC potentials should be used
   during the SCF. These potentials specify both the exchange and the
   correlation part. No LDA, GGA, MetaGGA, HartreeFock, HYBRID, or
   MetaHYBRID key should be used in combination with these keys. It is
   also not advised to use any energy analysis in combination with
   these potentials. For energy analysis, we recommend using one of
   the GGA potentials. It is currently not possible to perform a
   Create run with these potentials. It is possible to do a one-atom
   regular ADF calculation with these potentials, though, using a
   regular adf.rkf (TAPE21) file from an LDA or GGA potential as
   input. Available model potentials:

   * **LB94**: This refers to the XC functional of Van Leeuwen and
     Baerends [7]. There are no separate entries for the exchange and
     correlation parts of LB94. Usually the GRAC or SAOP potentials
     give results superior to LB94.

   * **GRAC**: The gradient-regulated asymptotic correction, which in
     the outer region closely resembles the LB94 potential [41]. It
     requires a further argument: the ionization potential [IP] of the
     molecule, in Hartree units. This should be estimated or obtained
     externally, or calculated in advance from two GGA total energy
     calculations.

   * "IP": Should be supplied only if GRAC is specified.

   * **SAOP**: The statistical average of orbital potentials [44]
     [42]. It can be used for all-electron calculations only. It will
     be expensive for large molecules, but requires no further
     parameter input.

The LB94, GRAC, and SAOP functionals have only an SCF (= potential)
implementation, but **no Energy** counterpart.

The LB94, GRAC, and SAOP forms are density functionals specifically
designed to get the correct asymptotic behavior. This yields much
better energies for the highest occupied molecular orbital (HOMO) and
better excitation energies in a calculation of response properties
(time-dependent DFT). Energies for lower-lying orbitals (sub-valence)
should improve as well (in case of GRAC and SAOP, but not LB94). The
energy expression underlying the LB94 functional is very inaccurate.
This does not affect the response properties, but it does imply that
the energy and its derivatives (gradients) should not be used because
LB94-optimized geometries will be wrong; see, for instance, [46]. The
application of the LB94 functional in a runtype that involves the
computation of energy gradients is disabled in ADF. You can override
this internal check with the key ALLOW.

In a GRAC calculation, the user should be aware that the potential in
the outer region is shifted up with respect to the usual level. In
other words, the XC potential does not tend to zero in the outer
region in this case. The size of the shift is the difference between
the HOMO orbital energy and the IP given as input. In order to compare
to regular GGA orbital energies, it is advisable to subtract this
amount from all orbital energies. Of course, orbital energy
differences, which enter excitation energies, are not affected by this
shift in the potential.


XCFun
=====

Any publication employing calculations carried out with XCFun needs to
cite Ref. [U. Ekström, L. Visscher, R. Bast, A.J. Thorvaldsen, and K.
Ruud, J. Chem. Theory Comput. 6, 1971 (2010)] [47].

"XCFUN"
   XCFun is a library of approximate exchange-correlation functionals,
   see Ref. [47], for which functional derivatives can be calculated
   automatically. For example, with XCFUN the full (non-ALDA) kernel
   can be evaluated and this has been implemented in the calculation
   of TDDFT excitations. The full kernel cannot be used in combination
   with symmetry or excited state geometry optimizations. The
   following functionals can be evaluated with XCFUN at the present
   time:

   * LDA: VWN5, X-ALPHA, PW92

   * GGA exchange: Becke88, PBEX, OPTX, PW91X, mPW, revPBEX

   * GGA correlation: LYP, Perdew86, PBEC

   * MetaGGA: TPSS, M06L, B95

   * Meta-hybrids: M06, M05, M062X, M06HF

   * Hybrids: PBE0, B3LYP, BHandH, B1LYP, B3LYP*, PBEFALFX

   * Yukawa range-separated hybrids: CAMY-B3LYP and more, see Yukawa
     RS hybrids with XCFUN

Here, MetaGGA B95 means Becke88 exchange + B95c correlation. The meta-
hybrids PW6B95 and PWB6K have been removed from this list, since they
do not agree with the LibXC implementation.


LibXC
=====

Any publication employing calculations carried out with LibXC needs to
cite the current literature reference of LibXC, which is at the moment
Ref. [S. Lehtola, C. Steigemann, M.J.T. Oliveira, M.A.L. Marques,
SoftwareX 7, 1 (2018)] [49]. Benchmark papers employing functionals
from LibXC should especially include a reference to the employed DFT
library, since as discussed in Ref. [S. Lehtola, M.A.L. Marques, J.
Chem. Phys. 159, 114116 (2023)] [54], the implementations of a given
functional in various program packages may not lead to the same
result, even at the complete basis set limit.

"LibXC functional"
   LibXC is a library of approximate exchange-correlation functionals,
   see Ref. [48] [49]. **All-electron basis sets should be used** (see
   Basis key). LibXC version 7.0.0 is used. The following functionals
   can be evaluated with LibXC (incomplete list):

   * LDA: LDA, PW92, TETER93

   * GGA: AM05, BCGP, B97-GGA1, B97-K, BLYP, BP86, EDF1, GAM, HCTH-93,
     HCTH-120, HCTH-147, HCTH-407, HCTH-407P, HCTH-P14, PBEINT, HTBS,
     KT2, MOHLYP, MOHLYP2, MPBE, MPW, N12, OLYP, PBE, PBEINT, PBESOL,
     PW91, Q2D, SOGGA, SOGGA11, TH-FL, TH-FC, TH-FCFO, TH-FCO, TH1,
     TH2, TH3, TH4, XLYP, XPBE, HLE16

   * MetaGGA: M06-L, M11-L, MN12-L, MS0, MS1, MS2, MVS, PKZB, RSCAN,
     R2SCAN, REVSCAN, SCAN, TPSS, HLE17

   * Hybrids: B1LYP, B1PW91, B1WC, B3LYP, B3LYP*, B3LYP5, B3LYP5,
     B3P86, B3PW91, B97, B97-1 B97-2, B97-3, BHANDH, BHANDHLYP, EDF2,
     MB3LYP-RC04, MPW1K, MPW1PW, MPW3LYP, MPW3PW, MPWLYP1M, O3LYP,
     OPBE, PBE0, PBE0-13, REVB3LYP, REVPBE, RPBE, SB98-1A, SB98-1B,
     SB98-1C, SB98-2A, SB98-2B, SB98-2C, SOGGA11-X, SSB, SSB-D, X3LYP

   * MetaHybrids: B86B95, B88B95, BB1K, M05, M05-2X, M06, M06-2X,
     M06-HF, M08-HX, M08-SO, MPW1B95, MPWB1K, MS2H, MVSH, PW6B95,
     PW86B95, PWB6K, REVSCAN0, SCAN0, REVTPSSH, TPSSH, X1B95, XB1K

   * Range-separated: CAM-B3LYP, CAMY-B3LYP, HJS-PBE, HJS-PBESOL, HJS-
     B97X, HSE03, HSE06, LRC_WPBE, LRC_WPBEH, LCY-BLYP, LCY-PBE,
     M06-SX, M11, MN12-SX, N12-SX, TUNED-CAM-B3LYP, WB97, WB97X

One can use one of the acronyms in the list above or one can use the
functionals described at the LibXC website
https://libxc.gitlab.io/functionals (where one should include "XC_" at
the beginning). Example usage for the BP86 functional:

   XC
     LibXC BP86
   End

Alternative

   XC
     LibXC XC_GGA_X_B88 XC_GGA_C_P86
   End

When using LibXC, the ADF output will provide the reference for the
selected functional, see also the LibXC website
https://libxc.gitlab.io/functionals.

Do not use any of the subkeys LDA, GGA, METAGGA, MODEL, HARTREEFOCK,
HYBRID, METAHYBRID, XCFUN, or RANGESEP in combination with the subkey
LIBXC. One can use the DISPERSION key with LIBXC. For many XC
functionals the optimized dispersion parameters will then be used
automatically, please check the output in this case. Note that in many
cases you have to include the DISPERSION key and include the correct
dispersion parameters yourself.

Note that ADF cannot calculate VV10 dispersion-dependent LibXC
functionals, like VV10, LC-VV10, B97M-V, WB97X-V, WB97M-V. Before
ADF2025, ADF would stop if one tried to calculate a VV10 dispersion
dependent functional. Starting from ADF2025, ADF will not stop if one
includes another dispersion method (with the DISPERSION key), in which
case the VV10 dispersion-dependent part of the functional will be
replaced with the other dispersion method that is specified.

Starting with AMS2026 many of the LibXC functionals can be used with
frozen cores, NMR calculations, the (AO)RESPONSE key, or the EPR/ESR
g-tensor. However, meta-XC functionals, like meta-GGA or meta-hybrid
XC functionals cannot be used with frozen cores, NMR calculations, the
(AO)RESPONSE key, or the EPR/ESR g-tensor. Note also that the kernel
in (range-separated) hybrids in case of the (AO)RESPONSE key may be
incorrect. Most LibXC functionals can be used in combination with
geometry optimization, TS, IRC, LT, numerical frequencies, and
excitation energies (ALDA kernel used). For a few GGA LibXC
functionals, analytical frequencies can be calculated, and one can use
the full kernel in the calculation of excitation energies (if
FULLKERNEL is included as subkey of the key EXCITATIONS). In case of
LibXC (meta-)hybrids and calculating excitation energies, the kernel
consists of the Hartree-Fock percentage times the Hartree-Fock kernel
plus one minus the Hartree-Fock percentage times the ALDA kernel (thus
no (meta-)GGA kernel). For the LibXC range-separated functionals, like
CAM-B3LYP, starting from ADF2016.102, the kernel consists of range-
separated ALDA plus the kernel of the range-separated exact exchange
part. In ADF2016.101, the kernel for LibXC range-separated
functionals, like CAM-B3LYP, used a 100% ALDA plus range-separated
exact exchange kernel (the ALDA part was not range-separated
corrected). For the range-separated functionals WB97 and WB97X, one
can use the full kernel in the calculation of excitation energies.


Dispersion corrections
======================


DFT-D4 functionals
------------------

Grimme's dispersion correction, D4(EEQ) [50], was added in AMS 2019.3
and updated in AMS 2025.1. In contrast to the earlier D3 dispersion
correction, in D4(EEQ) the atomic coordination-dependent dipole
polarizabilities are scaled based on atomic partial charges obtained
from an electronegativity equilibrium model (EEQ). Compared to D3, the
introduced charge dependence improves thermochemical properties,
especially for systems containing metals. The authors recommend
D4(EEQ) as a physically improved and more sophisticated dispersion
model in place of D3. In ADF2025, parameters for Grimme4 were updated
according to version 3.7.0 of dftd4 (https://github.com/dftd4/dftd4).

"Dispersion Grimme4 {s6=...} {s8=...} {a1=...} {a2=...}"
   If "Dispersion Grimme4" is present in the "XC" block, the D4(EEQ)
   dispersion correction (with the electronegativity equilibrium
   model) by the Grimme group [50] will be added to the total bonding
   energy, gradient, and second derivatives, where applicable.

   Version 3.7.0 of dftd4 is used (https://github.com/dftd4/dftd4),
   which includes parameters for actinides. For elements for which no
   parameters are present (103 < Z < 112, Z > 118), the parameters for
   the element Z'=Z-32 will be used.

   The D4(EEQ) model has four general parameters: s_6, s_8, a_1, and
   a_2, and their value should depend on the XC functional used. For
   many functionals the D4(EEQ) parameters are **predefined**, for
   example: B1B95, B3LYP, B3PW91, BLYP, BP86, CAM-B3LYP, HartreeFock,
   OLYP, OPBE, PBE, PBE0, PW6B95, REVPBE, RPBE, TPSS, TPSSH. For these
   functionals it is enough to specify "Dispersion Grimme4" in the
   input block. E.g.:

      XC
        GGA BLYP
        Dispersion Grimme4
      END

   For many other functionals you should explicitly specify the
   D4(EEQ) parameters in the "Dispersion" key (otherwise the PBE
   parameters will be used). For example, for the PW91 functional you
   could use the following input (although this is not necessary,
   since the parameters for PW91 are predefined):

      XC
        GGA PW91
        Dispersion Grimme4 s6=1.0 s8=0.7728 a1=0.3958 a2=4.9341
      END

   The D4(EEQ) parameters for many functionals can be found in the
   supporting information of the following paper: [50], see also
   https://github.com/dftd4/dftd4.

   For double hybrids, see the Double Hybrid Functionals section of
   the user manual.


DFT-D3 functionals
------------------

The D3 dispersion correction by Stefan Grimme is available in ADF, and
updated in ADF2025.1. Grimme and his coworkers at the Universität
Münster outlined the parametrization of this correction, dubbed
DFT-D3, in Ref. [53]. A slightly improved version with a more moderate
BJ damping function appeared later, and was called DFT-D3-BJ. [51] The
authors list the advantages of this method as the following:

* It is less empirical, i.e., the most important parameters are
  computed from first principles by standard Kohn-Sham (KS)-(TD)DFT.

* The approach is asymptotically correct with all DFs for finite
  systems (molecules) or nonmetallic infinite systems. It gives the
  almost exact dispersion energy for a gas of weakly interacting
  neutral atoms and smoothly interpolates to molecular (bulk) regions.

* It provides a consistent description of all chemically relevant
  elements of the periodic system (nuclear charge Z = 1-94).

* Atom pair-specific dispersion coefficients and cutoff radii are
  explicitly computed.

* Coordination number (geometry) dependent dispersion coefficients are
  used that do not rely on atom connectivity information
  (differentiable energy expression).

* It provides similar or better accuracy for "light" molecules and a
  strongly improved description of metallic and "heavier" systems.

Parametrizations are available for: B3LYP, TPSS, BP86, BLYP, revPBE,
PBE, PBEsol, RPBE, and many more functionals, and will be
automatically set if one of these functionals is used. Otherwise, PBE
parameters will be used. The parameters can be set manually; see the
XC key block. In ADF2025, parameters for Grimme3 and Grimme3 BJDAMP
were updated according to version 1.2.1 of s-dftd3
(https://github.com/dftd3/simple-dftd3).

"DISPERSION Grimme3 BJDAMP"
   If DISPERSION Grimme3 BJDAMP is present, a dispersion correction
   (DFT-D3(BJ)) by Grimme [51] will be added to the total bonding
   energy, gradient, and second derivatives, where applicable.

   Version 1.2.1 of s-dftd3 for DFT-D3(BJ) is used
   (https://github.com/dftd3/simple-dftd3), which includes parameters
   for actinides. For elements for which no parameters are present
   (Z>103), the parameters for the element Z'=Z-32 will be used.

   For many XC functionals, parametrizations are implemented e.g. for
   B3LYP, TPSS, BP86, BLYP, PBE, PBEsol, RPBE, and SCAN. For example,
   this is the input block for specifying the PBE functional with
   Grimme3 BJDAMP dispersion correction (PBE-D3(BJ)):

      XC
        GGA PBE
        DISPERSION Grimme3 BJDAMP
      End

   The D3(BJ) dispersion correction has four general parameters that
   depend on the XC functional. One can override the default
   parametrization by using *PAR1=.. PAR2=..*, etc. In the table, the
   relation is shown between the parameters and the real parameters in
   the Dispersion correction.

   +----------------------------------------------------+----------------------------------------------------+
   | variable                                           | variable on Bonn  website                          |
   +----------------------------------------------------+----------------------------------------------------+
   | PAR1                                               | s6                                                 |
   +----------------------------------------------------+----------------------------------------------------+
   | PAR2                                               | a1                                                 |
   +----------------------------------------------------+----------------------------------------------------+
   | PAR3                                               | s8                                                 |
   +----------------------------------------------------+----------------------------------------------------+
   | PAR4                                               | a2                                                 |
   +----------------------------------------------------+----------------------------------------------------+

   For example, this is the input block for specifying the
   PBE-D3(BJ)-GP parametrization by Proppe et al. [91] (i.e. a_1=0,
   s_8=0, a_2=5.6841):

      XC
        GGA PBE
        DISPERSION Grimme3 BJDAMP PAR2=0 PAR3=0 PAR4=5.6841
      End

"DISPERSION Grimme3"
   If DISPERSION Grimme3 is present, a dispersion correction (DFT-D3)
   by Grimme [53] will be added to the total bonding energy, gradient,
   and second derivatives, where applicable.

   Version 1.2.1 of s-dftd3 for DFT-D3 is used
   (https://github.com/dftd3/simple-dftd3), which includes element-
   specific parameters for actinides. For elements for which no
   parameters are present (Z>103), the parameters for the element
   Z'=Z-32 will be used.

   Parametrizations are available for many XC functionals, e.g. for
   B3LYP, TPSS, BP86, BLYP, revPBE, PBE, PBEsol, RPBE, and SCAN, and
   many more, and will be automatically set if one of these
   functionals is used. There are also parameters directly recognized
   for S12g and S12h. For other functionals, PBE-D3 parameters are
   used as default. You can explicitly specify the three parameters.

   +----------------------------------------------------+----------------------------------------------------+
   | variable                                           | variable on Bonn  website                          |
   +----------------------------------------------------+----------------------------------------------------+
   | PAR1                                               | s6                                                 |
   +----------------------------------------------------+----------------------------------------------------+
   | PAR2                                               | sr,6                                               |
   +----------------------------------------------------+----------------------------------------------------+
   | PAR3                                               | s8                                                 |
   +----------------------------------------------------+----------------------------------------------------+


DFT-D functionals
-----------------

An implementation of dispersion corrections, called DFT-D is available
in ADF. Like DFT-D3, this implementation is easy to use and is also
supported by the GUI.

This DFT-D implementation is based on the paper by Grimme [38] and is
extremely easy to use. The correction is switched on by specifying
*DISPERSION*, possibly with parameters, in the XC input block. See
description of the XC input block for details about the DISPERSION
keyword.

Post-SCF energies for different DFT-D or GGA-D functionals are also
included in the table printed when the METAGGA keyword is specified.
These include: BLYP-D, PBE-D, BP86-D, TPSS-D, B3LYP-D, and B97-D.
NOTE: this option does not require specifying a DISPERSION keyword in
the XC block and thus there is **no correction added to the energy
gradient** in this case. Please also note that although the original
B97 functional includes HF exchange (and is thus a hybrid functional),
the B97-D is a pure GGA. B3LYP-D is, however, a hybrid functional. The
following functional-dependent global scaling factors s_6 are used:
1.2 (BLYP-D), 0.75 (PBE-D), 1.05 (BP86-D), 1.0 (TPSS-D), 1.05
(B3LYP-D), and 1.25 (B97-D). These are fixed and cannot be changed.

Regarding performance of different functionals, testing has shown that
BLYP-D gives good results for both energies and gradients involving
VdW interactions. Post-SCF energy-only calculations at fixed
geometries showed that B97-D also gives good binding energies compared
to high-level reference data. Thorough comparison of different DFT-D
functionals can be found in Ref. [70].

**Note:** The original paper by Grimme included parameters for
elements H throughout Xe. In ADF2009.01 values for dispersion
parameters for DFT-D functionals for heavier elements (Cs-Rn) have
been added. These newer values have not been tested extensively. Thus,
in this implementation, no dispersion correction is added for
interactions involving atoms heavier than Radon.

DFT-D is invoked with the XC block, for example:

   XC
     GGA BLYP
     Dispersion
   END

"DISPERSION {s6scaling} {RSCALE=r0scaling}"
   If the DISPERSION keyword is present (without the argument
   Grimme3), a dispersion correction (DFT-D) by Grimme [38] will be
   added to the total bonding energy, gradient, and second
   derivatives, where applicable. The global scaling factor with which
   the correction is added depends on the exchange-correlation
   functional used at SCF, but it can be modified using the
   *s6scaling* parameter. The following scaling factors are used (with
   the XC functional in parentheses): 1.20 (BLYP), 1.05 (BP), 0.75
   (PBE), 1.05 (B3LYP). In all other cases a factor 1.0 is used unless
   modified via the s6scaling parameter. The SSB-D functional includes
   the dispersion correction (factor 0.847455) by default.

   The van der Waals radii used in this implementation are hard coded
   in ADF. However, it is possible to modify the global scaling
   parameter for them using the *RSCALE=r0scaling* argument. The
   default value is 1.1 as proposed by Grimme [38].


dDsC: density-dependent dispersion correction
---------------------------------------------

The DISPERSION dDsC key invokes the density-dependent dispersion
correction [55], which has been parametrized for the functionals BLYP,
PBE, BP, revPBE, B3LYP, PBE0, and BHANDHLYP.

   XC
     GGA BLYP
     Dispersion dDsC
   END

For other functionals, one can set the dDsC parameters ATT0 and BTT0
with

   XC
     ...
     DISPERSION dDsC ATT0=att0 BTT0=btt0
   END

The dispersion dDsC in ADF cannot be used with fragments larger than
one atom. The reason is that ADF uses the Hirshfeld partitioning on
fragments for dDsC, which is only correct if the fragments are atoms.


DFT-ulg
-------

The DISPERSION UFF key invokes the universal correction of density
functional theory to include London dispersion (DFT-ulg) [56], which
has been parametrized for all elements up to Lr (Z=103), and for the
functional PBE, PW91, and B3LYP. For other functionals, the PBE
parameters will be used. Example:

   XC
     GGA PBE
     Dispersion UFF
   END


DFT-MBD functionals
-------------------

The DISPERSION MBD key invokes the MBD@rsSCS method [57], which is
designed to accurately describe long-range correlation (and thus
dispersion) in finite-gap systems, including at the same time a
description of the short-range interactions from the underlying DFT
computation of the electronic structure. The MBD (many-body
dispersion) method [58] obtains an accurate description of van der
Waals (vdW) interactions that includes both screening effects and
treatment of the many-body vdW energy to infinite order.  The revised
MBD@rsSCS method [57] employs a range separation (rs) of the self-
consistent screening (SCS) of polarizabilities and the calculation of
the long-range correlation energy. It has been parametrized for the
elements H-Ba, Hf-Rn, and for the functional PBE and PBE0. Note that
the MBD@rsSCS method depends on Hirshfeld charges. In calculating
forces the dependence of the Hirshfeld charges on the actual geometry
is neglected. The MBD method is implemented in case the BeckeGrid is
used for the numerical integration. Example for PBE MBD@rsSCS:

   XC
     GGA PBE
     Dispersion MBD
   END

One can use user-defined values with:

   XC
     Dispersion MBD {RSSCS|TS} {BETA=beta}
   END

"MBD {RSSCS|TS} {BETA=beta}"
   The default method for MBD is MBD@rsSCS. Optionally, one can use
   MBD@TS or change the used parameter \beta with setting beta.


Post-SCF energy functionals
===========================


GGA energy functionals
----------------------

In principle, you may specify different functionals to be used for the
*potential,* which determines the self-consistent charge density, and
for the *energy* expression that is used to evaluate the (XC part of
the) energy of the charge density. To be consistent, one should
generally apply the same functional to evaluate the potential and
energy, respectively. Two reasons, however, may lead one to do
otherwise:

* The evaluation of the GGA part in the *potential* is more time-
  consuming than LDA. The effect of the GGA term in the potential on
  the self-consistent charge density is often not very large. From the
  point of view of computational efficiency, it may therefore be
  attractive to solve the SCF equations at the LDA level (i.e. not
  including GGA terms in the potential), and to apply the full
  expression, including GGA terms, to the energy evaluation *a
  posteriori* (post-SCF).

* A particular XC functional may have only an implementation for the
  potential, but not for the energy (or vice versa). This is a rather
  special case, intended primarily for fundamental research of Density
  Functional Theory, rather than for routine production runs.

One possibility is to calculate a whole list of post-SCF energy
functionals using the METAGGA keyword; see the next section. For some
functionals the next possibility is enough. One has to specify
different functionals for potential and energy evaluations
respectively, using:

   XC
    {LDA {Apply} LDA {Stoll}}
    {GGA {Apply} GGA}
   end

"Apply"
   States whether the functional defined on the corresponding line
   will be used self-consistently (in the SCF-potential), or only
   post-SCF, i.e. to evaluate the XC energy corresponding to the
   charge density. The value of "Apply" must be SCF or Energy. A value
   postSCF will also be accepted and is equivalent to Energy. A value
   Potential will also be accepted and is equivalent to SCF. For each
   record separately, the default (if no Apply value is given in that
   record) is SCF. For each of the two terms (LDA, GGA) in the
   functional: if no record with Energy specification is found in the
   data block, the evaluation of the XC energy will use the same
   functional as is applied for the potential.

"LDA, GGA"
   See the XC potential section for all possible values.


Meta-GGA and hybrid energy functionals
--------------------------------------

The post-SCF energy calculation is an easy and cheap way to get a
reasonable guess for the bond energies for different XC functionals at
the same time. Note that post-SCF energy calculations for a certain XC
functional will not be so accurate if the functional form of the XC
functional used in the SCF is very different from the XC functional
used post-SCF. The relative accuracy of post-SCF energies may not be
so high if one looks at small energy differences. For accurate energy
calculations it is recommended to use the same XC functional during
the SCF as for the energy.

The calculation of a large, pre-specified list of LDA, GGA, and meta-
GGA energy functionals is invoked by specifying

   METAGGA

as a separate keyword. The following (incomplete) list gives an idea
of the (meta-)GGA density functionals that will then be calculated
(the t-MGGA functional is the \theta-MGGA functional of Ref. [59]):

   BP, PW91, mPW, BLYP, PBE, RPBE, revPBE, mPBE, OLYP, OPBE, KCIS, PKZB, VS98, FT97, BLAP3,
   HCTH, tau-HCTH, BmTau1, BOP, OLAP3, TPSS, KT1, KT2, B97, M06-L, t-MGGA.

The hybrid GGA and hybrid meta-GGA energy functionals are calculated
if in addition to the METAGGA key, the key

   HARTREEFOCK

is included. The following (incomplete) list gives an idea of the
extra hybrid (meta-)GGA density functionals that will then be
calculated:

   B3LYP, B3LYP*, B1LYP, KMLYP, O3LYP, X3LYP, BHandH, BHandHLYP, B1PW91, MPW1PW, MPW1K,
   PBE0, OPBE0, TPSSh, tau-HCTH-hybrid, B97, M05, M05-2X, M06, M06-2X.

The keys METAGGA and HARTREEFOCK can be used in combination with any
XC potential. Note that at the moment hybrid functionals cannot be
used in combination with frozen cores. Also most METAGGA functionals
will give wrong results if used in combination with frozen cores. Thus
it is best to use an all-electron basis set if one of the keywords
METAGGA or HARTREEFOCK is used. One should include the "HARTREEFOCK"
keyword also in the create runs of the atoms. In ADF, the hybrid
energies only make sense if the calculation is performed with
completely filled orbitals. In case of a high-spin electron
configuration, one can do ROKS; see ROKS for high-spin open shell
molecules.

The Examples document describes an application to the OH molecule for
the METAGGA option. More output, on the total XC energy of the system,
can be obtained by specifying

   PRINT METAGGA

This latter option is intended for debugging purposes mainly and is
not recommended for general use.

The implementation calculates the total XC energy for a system and
writes it to a file. This is always done in Create runs. If the basic
fragments are atoms, the keyword

   ENERGYFRAG
      ATOM [filename]
      ATOM [filename]
      ... ...
   END

specifies that different atomic fragment files are to be used in the
meta-GGA energy analysis than the regular atomic fragment files from
the create runs. This keyword cannot be used for molecular fragment
files. In order to compare meta-GGA energy differences between
molecular fragments and the total molecule, results from the various
calculations need to be combined by hand.

In such situations, it is advisable to use a somewhat higher
integration accuracy than one would normally do, at least for the
smaller fragments, as there is no error cancellation as in a regular
ADF bond energy analysis.

A general comment is that some functionals show a more stable behavior
than others (at least in our current implementation). In general, the
functionals which are dependent on the Laplacian of the density may
display a large variation with respect to basis set changes or
different numerical integration accuracy. For this reason we currently
recommend FT97 in favor of FT98. Similarly, the results with the
BmTau1 functional should still be carefully checked. In our test
calculations on the G2 set of molecules, the VS98 showed best
performance, both for the average error and for the maximum error. The
G2 set consists only of small molecules with elements up to Cl. The
relative performance for transition metals and heavy elements is
unknown and may well be very different from the ordering for the G2
set.


Post Hartree-Fock energy functionals
------------------------------------

This section is largely adapted from text by the authors of Ref. [60]:
   In the early days of DFT, non-self-consistent Kohn-Sham energy was
   often evaluated on Hartree-Fock (HF) densities as a way to test new
   approximations. This method was called HF-DFT. It has been
   discovered that in some cases, HF-DFT actually gave more accurate
   answers when compared to self-consistent DFT calculations. In Ref.
   [60], it was found that DFT calculations can be categorized into
   two different types of calculations. The error of an approximate
   functional can be decomposed into two parts: error from the
   functional (functional error), and error from the density (density-
   driven error). For most calculations, functional error is dominant,
   and here self-consistent DFT is usually better than non-self-
   consistent DFT on more accurate densities (called density-corrected
   DFT (DC-DFT)). Unlike these 'normal' calculations, there is a class
   of calculations where the density-driven error is much larger, so
   DC-DFT gives a better result than self-consistent DFT. These
   calculations can be classified as 'abnormal'. HF-DFT is a simple
   implementation of DC-DFT, and a small HOMO-LUMO gap is an indicator
   of an 'abnormal' calculation, thus HF-DFT would perform better in
   such cases.

In ADF, one can do HF-DFT with:

   XC
     HartreeFock
   END
   MetaGGA

This will produce a large, pre-specified list of LDA, GGA, meta-GGA,
hybrid, and meta-hybrid energy functionals.

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