Density Functionals (XC)¶
The Density Functional, also called the exchangeandcorrelation (XC) functional, consists of an LDA, a GGA part, a HartreeFock exchange part (hybrids), and a metaGGA part (metaGGA or metahybrid). Possibly, it also depends on virtual KohnSham orbitals through inclusion of an orbitaldependent correlation (doublehybrids). LDA stands for the Local Density Approximation, which implies that the XC functional in each point in space depends only on the (spin) density in that same point. GGA stands for Generalized Gradient Approximation and is an addition to the LDA part, by including terms that depend on derivatives of the density. A hybrid GGA (for example B3LYP) stands for some combination of a standard GGA with a part of HartreeFock exchange. A metaGGA (for example TPSS) has a GGA part, but also depends on the kinetic energy density. A metahybrid (for example TPSSh) has GGA part, a part of HartreeFock exchange and a part that depends on the kinetic energy density. For these terms ADF supports a large number of the formulas advocated in the literature. For postSCF energies only, ADF supports also various other metaGGA functionals and more hybrid functionals. A doublehybrid has a hybrid or a metahybrid part, but also contains a contribution from secondorder MøllerPlesset perturbation theory (MP2). Here, only the hybrid (metahybrid) part is evaluated selfconsistently, whereas the MP2 part is evaluated postSCF and added to the hybrid (metahybrid) 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}
{OEP fitmethod {approximation}}
{HYBRID hybrid {HF=HFpart}}
{MetaHYBRID metahybrid}
{DOUBLEHYBRID doublehybrid}
{RPA}
{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 apply only 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 (also if LYP is specified in the GGA part).Available LDA functionals:
 Xonly: The pureexchange electron gas formula. Technically this is identical to the Xalpha form (see next) with a value 2/3 for the Xalpha parameter.
 Xalpha: The scaled (parametrized) exchangeonly formula. When this option is used you may (optionally) specify the Xalpha 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 the more advanced one, 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 (in earlier times often called the ‘nonlocal’ 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
SSBD: Dispersion corrected functional by SwartSolàBickelhaupt [62] [63]. The SSBD 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 SSBD (Ref. [62] [63]) for some systems. Because of this, the GGA SSBD option is only available for singlepoints (and NMR). Geometry optimizations (etc.) are still possible by using instead:
XC METAGGA SSBD END
This METAGGA implementation is only possible with allelectron basis sets. Use GGA SSBD for NMR calculations.
S12g: Dispersion corrected (GrimmeD3) functional by Swart, successor of SSBD [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
sub key) the LDA part defaults to VWN, except when the LYP correlation correction is used: in that case the LDA default is Xonly: pure exchange. The reason for this is that the LYP formulas assume the pureexchange LDA form, while for instance the Perdew86 correlation correction is a correction to a correlated LDA form. The precise form of this correlated LDA form assumed in the Perdew86 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 respectively. 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) [9].
 PW86x: PerdewWang (1986) [10].
 PW91x: PerdewWang (1991) [11]
 mPWx: Modified PW91 by AdamoBarone (1998) [12]
 PBEx: PerdewBurkeErnzerhof (1996) [13]
 RPBEx: revised PBE by HammerHansenNorskov (1999) [14]
 revPBEx: revised PBE by ZhangWang (1998) [15]
 mPBEx: Modified PBE by AdamoBarone (2002) [16]
 PBEsolx: PerdewRuzsinszkyCsonkaVydrovScuseria (2008) [17]
 HTBSx: [18]
 OPTX: HandyCohen (2001) [19]
 BEEx: MortensenKaasbjergFrederiksenNørskovSethnaJacobsen (2005) [20]
For the correlation part the options are:
 Perdew: Perdew (1986) [21].
 PBEc: PerdewBurkeErnzerhof (1996) [13] .
 PBEsolc: The PBEsol correlation correction by PerdewRuzsinszkyCsonkaVydrovScuseria (2008) [17]
 PW91c: PerdewWang (1991), see [11].
 LYP: LeeYangParr (1988) correlation correction [64] [65] [66].
The string GGA must contain not more than one of the exchange options and not more than one of the correlation options. 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 overruled with the key ALLOW.
MetaGGA¶
XC
MetaGGA functional
End
MetaGGA
Specifies that a metaGGA should be used during the SCF. All electron basis sets should be used (see Basis key).
Available metaGGA functionals:
 M06L: Functional by YanTruhlar [58] [59]
 TPSS: Functional by TaoPerdewStaroverovScuseria [60] [61]
 revTPSS: Revised TPSS functional [22]
 SSBD: Dispersion corrected GGA functional by SwartSolàBickelhaupt [62] [63]. Use GGA SSBD for NMR calculations.
 MVS: Functional by SunPerdewRuzsinszky [23]
 MS0: Functional by Sun et al. [24]
 MS1: Functional by Sun et al. [25]
 MS2: Functional by Sun et al. [25]
 SCAN: Functional by Sun et al. [26]
 TASKxc: Functional by Aschebrock et al. Intended for charge transfer systems.
The M06L functional needs high integration accuracy (at least BeckeGrid quality good) for reasonable gradients. For TPSS moderate integration accuracy for reasonable gradients is sufficient. For heavier elements (Z>36) and if one uses the M06L functional it is also necessary to include the following keyword
FragMetaGGAToten
Using this key FRAGMETAGGATOTEN the difference in the metahybrid or metaGGA exchangecorrelation 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. Default is to calculate the metaGGA exchangecorrelation energies for the fragments in the numerical integration grid of the fragments.
HartreeFock¶
XC
HartreeFock
End
HartreeFock
 Specifies that the HartreeFock 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 B3LYP. functional (20% HF exchange) by StephensDevlinChablowskiFrisch [27].
 B3LYP*: Modified B3LYP functional (15% HF exchange) by ReiherSalomonHess [28].
 B1LYP: Functional (25% HF exchange) by AdamoBarone [29].
 KMLYP: Functional (55.7% HF exchange) by KangMusgrave [30].
 O3LYP: Functional (12% HF exchange) by CohenHandy [31].
 X3LYP: Functional (21.8% HF exchange) by XuGoddard [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 by (25% HF exchange) AdamoBarone [29].
 mPW1PW: Functional (25% HF exchange) by AdamoBarone [12].
 mPW1K: Functional (42.8% HF exchange) by LynchFastHarrisTruhlar [32].
 PBE0: Functional (25% HF exchange) by ErnzerhofScuseria [33] and by AdamoBarone [34], hybrid form of PBE.
 OPBE0: Functional (25% HF exchange) by SwartEhlersLammertsma [4], hybrid form of OPBE.
 S12H: Dispersion corrected (GrimmeD3) functional (25% HF exchange) by Swart [6].
HFpart
 Specifies the amount of HF exchange that should be used in the functional, instead of the default HF exchange percentage for the given hybrid. Example HF=0.25 means 25% HartreeFock exchange.
MetaHybrid¶
XC
MetaHYBRID functional
End
MetaHYBRID
Specifies that a metahybrid functional should be used during the SCF.
Available metahybrid functionals:
Range separated hybrids¶
In ADF there are two (mutually exclusive) ways of specifying range separated hybrids functionals:
 Through the
RANGESEP
andXCFUN
keys. This will use the Yukawa potential as switching function, see Ref. [35];  By specifying a range separated functional via the
LibXC
key.
See also the advanced tutorial: Tuning the range separation in LCwPBE for organic electronics
RangeSep + XCFun: Yukawarange separated hybrids¶
RANGESEP {GAMMA=X} {ALPHA=a} {BETA=b}
 If RANGESEP is included, by default a longrange corrected (LC) functional is created with range separation parameter GAMMA of 0.75. As switching function in ADF the Yukawa potential is utilized, see Ref. [35]. Range separated functionals require XCFUN and are limited to GGA, metaGGA, and CAMYB3LYP. The CAMYB3LYP functional is not the same as the CAMB3LYP functional, since a different switching function is used. No other hybrids or metahybrids are supported. The special CAMYB3LYP functional is defined by three parameters, ALPHA, BETA and the attenuation parameter GAMMA. For CAMYB3LYP by default ALPHA is 0.19, BETA is 0.46, and GAMMA is 0.34.
Rangeseparated functionals make use of a modified form of the Coulomb operator that is split into pieces for exact exchange and DFT. As switching function in ADF the Yukawa potential is utilized, see Ref. [35]. Global hybrids can be thought of as a special case of a rangeseparated functional where the split is independent of the interelectronic distance and is a simple X exact and 1X DFT in all space.
In a general RSfunctional the split depends on the interelectronic 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 ADF an exponential function is used (the error function is common in Gaussian based codes). In a rangeseparated function the potential goes from a Coulomb interaction to a sum of Coulomb functions attenuated by an exponential function.
In practical terms, this means that a rangeseparated functional looks very much like a hybrid (or metahybrid) 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 RSfunctionals
Using Hirao’s approach for creating RSfunctionals, 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 rangeseparation comes in at the level of the integrals over the operator with fit functions. They are very similar to the standard Coulomb integrals.
RSfunctionals
An RSfunctional 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 adding up all the pieces. Currently, simple LC functionals can be defined where the exact exchange goes from 0 to 1 as the interelectronic distance increases and the DFT part does the reverse. In addition, CAMYB3LYP type functionals can be defined. More general functionals are not possible yet.
Functionality/Limitations
RS functionals with XCFUN are limited to the GGA and metaGGA functionals and one hybrid CAMYB3LYP. The following functionals can be evaluated with rangeseparation at the present time:
 LDA: VWN5, XALPHA PW92
 GGA exchange: Becke88, PBEX, OPTX, PW91X, mPW, revPBEX
 GGA correlation: LYP, Perdew86, PBEC
 MetaGGA: TPSS, M06L, B95
 Hybrids: CAMYB3LYP
The following functionality has been tested: XC potential, energy, ground state geometry, TDDFT. Starting from ADF2018 singlettriplet excitation calculations and excited state geometry optimizations are possible. See for possible limitations in case of excitation calculations or excited state geometry optimizations the corresponding part of the ADF manual.
Numerical stability
The rangeseparated implementation requires that the rangeseparation 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 longrange corrected (LC) functional is created with range separation parameter of 0.75. The parameter can be changed by modifying X in GAMMA=X in the RANGESEP card. Range separation typically will be used in combination with a GGA or METAGGA functional.
Range separation can not be included with a hybrid or metahybrid, with one exception, the special RS functional: CAMYB3LYP. This is entered as HYBRID CAMYB3LYP and must be used in combination with XCFUN (see XCFUN description) and RANGESEP. The CAMYB3LYP 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 CAMYB3LYP 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 CAMYB3LYP
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 CAMYB3LYP
Rangeseparated hybrids with LibXC¶
One can simply specify a range separated hybrid functional in the LibXC key, e.g.:
XC
LibXC CAMB3LYP
End
See the LibXC section for a list of available range separated hybrid functionals.
For the HSE03 and HSE06 short rangeseparated hybrids you can (optionally) specify the switching parameter omega, e.g.:
XC
LibXC HSE06 omega=0.1
End
Notes on HartreeFock and (meta)hybrid functionals¶
If a functional contains a part of HartreeFock exchange then the LDA, GGA, metaGGA, or MODEL key should not be used in combination with this key, and one should only specify one of HartreeFock, HYBRID or MetaHYBRID. Dispersion can be added. Note that it is not recommended to use (part of the) HartreeFock exchange in combination with frozen cores, since at the moment the frozen core orbitals are not included in the Hartree Fock exchange operator. In ADF one can do unrestricted HartreeFock (or hybrid or metahybrid) calculations, as long as one has integer occupation numbers (ROHF is not implemented in ADF, only UHF). You also need to use the same XCpotential in the create run of the atoms, which is done automatically if you use the BASIS key.
Starting from ADF2009.01 the metahybrids M06, M062X, M06HF, and TPSSH can be used during the SCF. Also starting from ADF2009.01 HartreeFock 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 spinspin coupling; HartreeFock and (meta)hybrid can be used in calculating excitation energies, in which the kernel consists of the HartreeFock percentage times the HartreeFock kernel plus one minus the HartreeFock percentage times the ALDA kernel (thus no (meta)GGA kernel). HartreeFock and the (meta)hybrid potentials still can not or should not be used in combination with analytical frequencies, the (AO)RESPONSE key, EPR/ESR gtensor, and frozen cores. Starting from ADF2010 it is possible to use HartreeFock and hybrids to calculate CD spectra.
In ADF one can do unrestricted HartreeFock (or hybrid or metahybrid) calculations, as long as one has integer occupation numbers (ROHF is not implemented in ADF, only UHF).
It is possible to change the amount of HF exchange in the input for hybrids (not for metahybrids and HartreeFock). For many hybrid functionals the sum of the amount of HartreeFock exchange and the amount of LDA exchange (or GGA exchange) is one. If that is the case, then if one changes the amount of HartreeFock 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 HartreeFock 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 metaGGA functionals, in particular the M06 functionals. These need high integration accuracy (at least BeckeGrid quality good) for reasonable gradients. For TPSSH moderate integration accuracy for reasonable gradients is sufficient. 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 metahybrid or metagga exchangecorrelation 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. Default is to calculate the metahybrid or metaGGA exchangecorrelation energies for the fragments in 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=1e4
NumericalQuality good
PostSCF correlation methods¶
PostSCF correlation methods consist of two steps. In a first step, a DFT calculation using LDA, (meta)GGA or (most often) a (meta)hybrid calculation is performed. Based on the converged orbitals and orbital energies, a correlation energy is calculated after the SCF is completed and the result is added to the SCF XCenergy (or alternatively, in case of RPA, replaces the SCF XC energy). ADF offers RPA and MP2 methods and most importantly combinations of MP2 with various kinds of XCfunctionals (socalled doublehybrids). RPA can not be used icw relativistic effects, because it internally uses the TOTALENERGY implementation, which is not supported for relativistic effects. The RPA correlation energy contains a bug in AMS2020.101AMS2020.103. This bug is fixed in the bug fixed release of AMS2020 with revision number r95864 or later.
While traditionally, postSCF correlation methods are associated with highcomputational cost, ADF offers algorithms for RPA and some doublehybrids which offer tremendous speedups and make routine application to systems with hundreds of atoms possible. These are RPA and all doublehybrids using only the oppositespin component of the MP2 energy. In case postSCF correlation methods are to be used, we strongly recommend to use one of these methods for large systems.
DoubleHybrid¶
XC
DOUBLEHYBRID functional
End
DOUBLEHYBRID
 Specifies that a doublehybrid functional [68] should be used. They combine a selfconsistent (meta)hybrid calculation with a postSCF MP2 energy correction.
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 doublehybrid calculations in ADF.
Double hybrids usually yield considerably better ground state energies than (meta)GGA and (meta)hybrid functionals. This especially true for thermochemistry, kinetics, transition metal chemistry as well as relative isomerization energies for systems with difficult electronic structure. For an overview of the capabilities of doublehybrids implemented in ADF we refer to a recent review. [69]
The MP2 correlation energy can be partitioned into two components, correlation of electron with paired spins (SS), and correlation of electron with unpaired spins (OS). Thus, it makes sense to divide doublehybrids into three categories: doublehybrids using the OScomponent only, doublehybrids using both components, and doublehybrids using both components scaled independently.
The computational effort for the evaluation of the latter component is much lower and a very efficient, quadratic scaling algorithm for this task has been developed and implemented in ADF. Doublehybrids using all spincomponents can be treated with a standard, canonical implementation using global density fitting. Starting from AMS2020, the canonical MP2 implementation has been improved. Systems with more than 100 atoms are not a problem any more. Especially when larger QZ basis sets are used, the MP2 correction is usually evaluated faster than the preceding SCF. For systems much larger than ~100 atoms, it is advised however, to use an OSonly functional.
OSonly functionals
OSfunctionals combine a (meta)hybridfunctional with empirical dispersion correction (in most cases) and use the correlation energy of unpaired spins only. They are often referred to as DODfunctionals. With the efficient implementation in ADF [89], these calculations are routine for systems of hundreds of atoms, the computational bottleneck always being the underlying hybrid calculation during the SCF. A DODcalculation is always feasible when a hybrid calculation is feasible too!
Currently, ADF supports the following DODfunctionals:
 SOS1PBEQIDH: 1parameter functional with PBE exchange, PBE correlation (69 % HF, 44 % OSMP2 [70]
 revDODBLYP: B88 exchange, LYP correlation, Grimme3 dispersion (71 % HF, 62.2 % OSMP2) [71]
 revDODBLYPD4: B88 exchange, LYP correlation, Grimme4 dispersion (71 % HF, 63.5 % OSMP2) [71]
 revDODPBE: PBE exchange, PBE correlation, Grimme3 dispersion (68 % HF, 61.3 % OSMP2) [71]
 revDODPBED4: PBE exchange, PBE correlation, Grimme4 dispersion (68 % HF, 61.8 % OSMP2) [71]
 revDODPBEP86: PBE exchange, P86 correlation, Grimme3 dispersion (69 % HF, 60.6 % OSMP2) [71]
 revDODPBEP86D4: PBE exchange, P86 correlation, Grimme4 dispersion (69 % HF, 61.2 % OSMP2) [71]
 DODSCAN: SCAN exchange and correlation, Grimme3 dispersion (66 % HF, 63,0 % OSMP2) [71]
 revDODSCAND4: SCAN exchange and correlation, Grimme4 dispersion (66 % HF, 63.4 % OSMP2) [71]
Note, that except for SOS1PBEQIDH, all functionals include dispersion correction by default since they are an intrinsic part of the functionals.
Standard and 1parameter functionals
The functionals in this category combine a (metahybrid) functional with a scaled correlation energy from both components of the MP2energy.
Available functionals in this category:
 B2PLYP: B88 exchange and LYP correlation (53 % HF, 27 % MP2)[#ref95]_
 B2PIPLYP: B88 exchange and LYP correlation (60 % HF, 27 % MP2), parametrized for pipi interactions [72]
 ROB2PLYP: B88 exchange and LYP correlation (59 % HF, 28 % MP2), restrictedopenshell version of B2PLYP [73]
 B2TPLYP: B88 exchange and LYP correlation (60 % HF, 31 % MP2), parametrized for thermodynamics [74]
 B2GPPLYP: B88 exchange and LYP correlation (65 % HF, 36 % MP2), ‘General Purpose’ parametrization [75]
 B2KPLYP: B88 exchange and LYP correlation (72 % HF, 42 % MP2), parametrized for kinetics [74]
 B2NCPLYP: B88 exchange and LYP correlation (70 % HF, 49 % MP2), parametrized for noncovalent interactions [76]
 mPW2PLYP: mPW exchange and LYP correlation (55 % HF, 25 % MP2) [77]
 mPW2KPLYP: mPW exchange and LYP correlation (72 % HF, 42 % MP2) [74]
 mPW2NCPLYP: mPW exchange and LYP correlation (42 % HF, 49 % MP2) [76]
 DHBLYP: 1Parameter functional with B88 exchange and LYP correlation (65 % HF, 42 % MP2) [78]
 PBE0DH: 1Parameter functional with PBE exchange and PBE correlation (50 % HF, 13 % MP2) [79]
 PBEQIDH: 1parameter functional with PBE exchange and PBE correlation (69 % HF, 33 % MP2) [70]
 LS1DH: 1Parameter functional with PBE exchange and PBE correlation (75 % HF, 42 % MP2) [80]
 PBE02: 1Parameter functional with PBE exchange and PBE correlation (79 % HF, 50 % MP2) [81]
 LS1TPSS: 1Parameter functional with TPSS metaGGA (85 % HF, 61 % MP2) [82]
 DS1TPSS: 1Parameter functional with TPSS metaGGA (73 % HF, 53 % MP2) [82]
All functionals in this category can be combined with empirical dispersion correction which can be requested in the XC block in the usual way.
Spincomponentscaled functionals
The functionals in this category combine a (metahybrid) functional with a scaled correlation energy from both components of the MP2energy. Opposed to the preceding category, both spincomponents are scaled individually, allowing for more flexibility in the functional parametrization. As OSfunctionals, they also include an empirical dispersion correction term in their parametrization. They are often referred to as DSDfunctionals
Currently, ADF supports the following DSDfunctionals:
 DSDBLYP: B88 exchange and LYP correlation, Grimme3 dispersion (69 % HF, 46 % OSMP2, % 37 % SSMP2) [83]
 revDSDBLYP: revised version of DSDBLYP, Grimme3 dispersion (71 % HF, 54.7 % os, 19.8 % SSMP2) [71]
 revDSDBLYPD4: revised version of DSDBLYP, Grimme4 dispersion (71 % HF, 55.9 % OS, 19.7 % SSMP2) [71]
 DSDPBEP86: PBE exchange and P86 correlation (69 % HF, 52 % OS, % 22 % SSMP2) [84]
 revDSDPBEP86: revised version of DSDPBEP86 (69 % HF, 57.9 % OS, 8 % SSMP2) [71]
 revDSDPBEP86D4: PBE exchange and P86 correlation, Grimme4 dispersion (69 % HF, 59.2 % OS, 6.4 % SSMP2) [71]
 DSDPBE: PBE exchange and PBE correlation, Grimme3 dispersion (68 % HF, 55 % OS, % 13 % SSMP2) [84]
 revDSDPBE: revised version of DSDPBE, Grimme3 dispersion (68 % HF, 58.5 % os, 7 % SSMP2) [71]
 revDSDPBED4: revised version of DSDPBE, Grimme4 dispersion (68 % HF, 60 % os, 4.2 % SSMP2) [71]
 revDSDSCAND4: based on SCAN metaGGA (66 % HF, 63.2 % OS, 1.3 % SSMP2) [71]
 SDSCAN69: Based on SCAN metaGGA, no dispersion correction (69 % HF, 62 % OS, 26 % SSMP2) [71]
Except for SDSCAN69, all functionals mentioned include dispersion correction by default. For more technical details of the algorithm and how to tweak parameters, see the MBPT section.
MP2methods¶
XC
MP2
EmpiricalScaling {NONESOSSCSSCSMI}
END
In addition to doublehybrids, ADF also implements MP2 including some popular spinscaled variants. Technically, they are not distinct from doublehybrids, however, the all rely on a HF instead of a DFT calculation. The following variants are supported.
 SOSMP2: pure HF reference (100 % HF, 130 % OSMP2) [85]
 MP2: pure HF reference (100 % HF, 100 % MP2 correlation)
 SCSMP2: pure HF reference (100 % HF, 120 % OSMP2, 33 % SSMP2) [86]
 SOSMIMP2: pure HF reference (100 % HF, 40 % OSMP2, 129 % SSMP2) [87]
The spinscaling variant can be requested in the XC block together with the MP2
keyword:
XC
MP2
EmpiricalScaling SOS
END
requests an SOSMP2 calculation.
For more technical details of the algorithm and how to tweak parameters, see the MBPT section.
RPA¶
The RPA correlation energy contains a bug in AMS2020.101AMS2020.103. This bug is fixed in the bug fixed release of AMS2020 with revision number r95864 or later.
XC
RPA
End
RPA
 requests that the RPA exchange and correlation energies are evaluated after the SCF. the XCenergy evaluated from the functional used in the SCF is replaced by the RPA XCenergy.
RPA can be used in conjunction with LDA, GGAs and hybrid functionals (RPA@LDA
, RPA@PBE
, RPA@hybrid
, respectively). IMPORTANT: relativistic effects, metaGGAs and metahybrids are not supported yet! RPA@LDA
is the default. To specify a different functional for the SCF, simply add the RPA
keyword to the functional specification. In example, the following key specifies a RPA@PBE
calculation,
XC
GGA PBE
RPA
END
A RPA@PBE0
calculation is requested by
XC
HYBRID PBE0
RPA
END
In ADF, the RPA energy is always evaluated in the AObasis and the algorithm is similar to SOSAOPARIMP2. The main difference is that the final RPA correlation energy is evaluated along the imaginary frequency axis, while the polarizability (noninteracting KohnSham density response function, which is also evaluated in SOSMP2) is evaluated in imaginary time. Thus, two grid parameters can be tweaked:
MBPT
nTime 12
nFrequency 12
END
requests 12 points for imaginary time and imaginary frequency integration each, which is also the default (As opposed to 9 points for MP2). It is also possible to use a different number of points for imaginary frequency than for imaginary time. The computational time required for the RPA calculation increases linearly with the number of imaginary time points and sublinearly with the number of imaginary frequency points. However, memory increases linearly with the number of imaginary frequency points while it is not affected by the size of the imaginary time grid.
Notes on MP2, doublehybrid functionals and RPA¶
All postSCF correlation methods can only be used for singlepoint calculations (Although numerical gradients are available in the AMS driver). However, they all support the use of frozen cores, empirical dispersion correction, and spinunrestricted calculations.
RPA can not be used icw relativistic effects.
MP2 and doublehybrid functionals can be used icw scalar relativistic effects within the ZORA
formalism, but can not be used icw spinorbit coupling.
The basis set requirements are higher than for standard functionals. We recommend to use allelectron basis sets of at least TZP quality to obtain meaningful results. Using basis sets of quadruplezeta quality is usually better. all postSCF correlation methods rely on information from virtual orbitals which are especially prone to linear dependency issues in the basis set coming from almost linearly dependent linear combinations of AOs. These linear combinations need to be removed during the SCF. This is controlled inside the DEPENDENCY
block. While in ADF the default is 1e4, for postSCF correlation methods, a default of 1e03 is chosen. This is a safe choice, in the sense that it ensures numerical stability in most cases but does not truncate the basis set size too much. Still, one should be aware that this value already corresponds to a drastic truncation of the virtual space.
It is also recommended to use NumericalQuality GOOD
but FitSetQuality NORMAL
. Thus, recommended settings for e.g. a doublehybrid calculation using the revDODBLYPD4 functional with the QZ4P basis set is:
BASIS
Type QZ4P
Core NONE
End
XC
Doublehybrid revDODBLYPD4
End
NumericalQualtity Good
RIHartreeFock
FitSetQuality Normal
End
In case of numerical difficulties to converge the SCF, it is often a good idea to increase also the fitset during the SCF, but still use FitSetQuality NORMAL
for the postSCF correlation method.
BASIS
Type QZ4P
Core NONE
End
XC
Doublehybrid revDODBLYPD4
End
NumericalQualtity Good
MPBT
FitsetQuality Normal
End
In this case, FitSetQuality GOOD
is used during the SCF (through the NumericalQuality key). After the SCF, the fit is recalculated using FitSetQuality NORMAL
.
For more technical details of the algorithm and how to tweak parameters, see the RPA section.
Model Potentials¶
Several asymptotically correct XC potentials have been implemented in ADF, such as the (now somewhat outdated) LB94 potential [7], the gradientregulated asymptotic correction (GRAC) [36], and the statistical average of orbital potentials (SAOP) [39] [37]. 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. [38] for applications to (hyper)polarizabilities Cauchy coefficients, etc. of small molecules). This is particularly true if the molecule is small and the (highlying) virtual orbitals are important for the property under study.
It was also shown that, simply using the orbital energies of the occupied KohnSham orbitals of a SAOP calculation, quite good agreement with experiment 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 lowerlying occupied orbital energies. The agreement becomes worse for deeplying core orbital energies. A theoretical explanation and practical results are given in Ref. [40].
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 to use one of the GGA potentials. It is currently not possible to do 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 respectively of LB94. Usually the GRAC or SAOP potentials give results superior to LB94.
 GRAC: The gradientregulated asymptotic correction, which in the outer region closely resembles the LB94 potential [36]. 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 [39] [37]. 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 a 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 (subvalence) 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 LB94optimized geometries will be wrong, see for instance [41]. 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 case of 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.
Optimized effective potentials¶
XC
OEP fitmethod {approximation}
End
OEP
 Defines the optimized effective potential expanded into a set of the fit functions. The subkeyword fitmethod can be any of the following: BARTLETT [42], SCUSERIA [43]. In the case of SCUSERIA one of the following approximations needs to be specified: CEDA, KLI or SLATER. An application of OEP in ADF can be found in Ref. [44].
XCFun¶
XCFUN
XCFun is a library of approximate exchangecorrelation functionals, see Ref. [45], for which functional derivatives can be calculated automatically. For example, with XCFUN the full (nonALDA) kernel can be evaluated and this has been implemented in the calculation of TDDFT excitations. The Full kernel can not 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, XALPHA, PW92
 GGA exchange: Becke88, PBEX, OPTX, PW91X, mPW, revPBEX
 GGA correlation: LYP, Perdew86, PBEC
 MetaGGA: TPSS, M06L, B95
 MetaHybrids: M06, M05, M062X, M06HF
 Hybrids: PBE0, B3LYP, BHandH, B1LYP, B3LYP*, PBEFALFX
 Yukawa range separated Hybrids: CAMYB3LYP and more, see Yukawa RS hybrids with XCFUN
Here MetaGGA B95 means Becke88 exchange + B95c correlation. The Metahybrids PW6B95 and PWB6K have been removed from this list, since they do not agree with the LibXC implementation.
LibXC¶
LibXC functional
LibXC is a library of approximate exchangecorrelation functionals, see Ref. [46] [47]. All electron basis sets should be used (see Basis key). Version 5.0.0 of LibXC is used. The interface in ADF to LibXC is incompatible with LibXC version 2. The following functionals can be evaluated with LibXC (incomplete list):
 LDA: LDA, PW92, TETER93
 GGA: AM05, BCGP, B97GGA1, B97K, BLYP, BP86, EDF1, GAM, HCTH93, HCTH120, HCTH147, HCTH407, HCTH407P, HCTHP14, PBEINT, HTBS, KT2, MOHLYP, MOHLYP2, MPBE, MPW, N12, OLYP, PBE, PBEINT, PBESOL, PW91, Q2D, SOGGA, SOGGA11, THFL, THFC, THFCFO, THFCO, TH1, TH2, TH3, TH4, XLYP, XPBE, HLE16
 MetaGGA: M06L, M11L, MN12L, MS0, MS1, MS2, MVS, PKZB, TPSS, HLE17
 Hybrids: B1LYP, B1PW91, B1WC, B3LYP, B3LYP*, B3LYP5, B3LYP5, B3P86, B3PW91, B97, B971 B972, B973, BHANDH, BHANDHLYP, EDF2, MB3LYPRC04, MPW1K, MPW1PW, MPW3LYP, MPW3PW, MPWLYP1M, O3LYP, OPBE, PBE0, PBE013, REVB3LYP, REVPBE, RPBE, SB981A, SB981B, SB981C, SB982A, SB982B, SB982C, SOGGA11X, SSB, SSBD, X3LYP
 MetaHybrids: B86B95, B88B95, BB1K, M05, M052X, M06, M062X, M06HF, M08HX, M08SO, MPW1B95, MPWB1K, MS2H, MVSH, PW6B95, PW86B95, PWB6K, REVTPSSH, TPSSH, X1B95, XB1K
 Rangeseparated: CAMB3LYP, CAMYB3LYP, HJSPBE, HJSPBESOL, HJSB97X, HSE03, HSE06, LRC_WPBE, LRC_WPBEH, LCYBLYP, LCYPBE, M11, MN12SX, N12SX, TUNEDCAMB3LYP, WB97, WB97X
One of the acronyms in the list above can be used, or one can also use the functionals described at the LibXC website http://www.tddft.org/programs/Libxc. Note that ADF can not calculate VV10 dependent LibXC functionals, like VV10, LCVV10, B97MV, WB97XV. Example usage for the BP86 functional:
XC
LibXC BP86
End
Alternative
XC
LibXC XC_GGA_X_B88 XC_GGA_C_P86
End
In case of LibXC the output of the ADF calculation will give the reference for the used functional, see also the LibXC website http://www.tddft.org/programs/Libxc.
Do not use any of the subkeys LDA, GGA, METAGGA, MODEL, HARTREEFOCK, OEP, HYBRID, METAHYBRID, XCFUN, RANGESEP in combination with the subkey LIBXC. One can use the DISPERSION key icw LIBXC. For a selected number of functionals the optimized dispersion parameters will then be used automatically, please check the output in that case. Note that in many cases you have to include the DISPERSION key and include the correct dispersion parameters yourself.
The LibXC functionals can not be used icw frozen cores, NMR calculations, the (AO)RESPONSE key, EPR/ESR gtensor. 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 HartreeFock percentage times the HartreeFock kernel plus one minus the HartreeFock percentage times the ALDA kernel (thus no (meta)GGA kernel). For the LibXC range separated functionals, like CAMB3LYP, 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 CAMB3LYP, was using a 100% ALDA plus range separated exact exchange kernel (the ALDA part was not rangeseparated corrected). For the range separated functionals WB97 and WB97X one can use the full kernel in the calculation of excitation energies.
Dispersion corrections¶
Dispersion Grimme4 {s6=...} {s8=...} {a1=...} {a2=...}
If
Dispersion Grimme4
is present in theXC
block the D4(EEQ) dispersion correction (with the electronegativity equilibrium model) by the Grimme group [48] will be added to the total bonding energy, gradient and second derivatives, where applicable.The D4(EEQ) model has four parameters: \(s_6\), \(s_8\), \(a_1\) and \(a_2\) and their value should depend on the XC functional used. For the following functionals the D4(EEQ) parameters are predefined: B1B95, B3LYP, B3PW91, BLYP, BP86, CAMB3LYP, 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 all 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 should use the following input: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: [48].
For DoubleHybrids, see the DoubleHybrid section of the user manual.
DISPERSION Grimme3 BJDAMP
If DISPERSION Grimme3 BJDAMP is present a dispersion correction (DFTD3(BJ)) by Grimme [49] will be added to the total bonding energy, gradient and second derivatives, where applicable. Parametrizations are implemented e.g. for B3LYP, TPSS, BP86, BLYP, PBE, PBEsol, and RPBE. For SCAN parameters from Ref. [50] are used.
For example, this is the input block for specifying the PBE functional with Grimme3 BJDAMP dispersion correction (PBED3(BJ)):
XC GGA PBE DISPERSION Grimme3 BJDAMP End
The D3(BJ) dispersion correction has four parameters. 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 PBED3(BJ)GP parametrization by Proppe et.al. [88] (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 (DFTD3) by Grimme [51] will be added to the total bonding energy, gradient and second derivatives, where applicable. Parametrizations are available e.g. for B3LYP, TPSS, BP86, BLYP, revPBE, PBE, PBEsol, and RPBE, and will be automatically set if one of these functionals is used. There are also parameters directly recognized for S12g and S12h. For SCAN parameters from Ref. [50] are used. For all other functionals, PBED3 parameters are used as default. You can explicitly specify the three parameters.
variable variable on Bonn website PAR1 s6 PAR2 sr,6 PAR3 s8 DISPERSION {s6scaling]} {RSCALE=r0scaling}
If the DISPERSION keyword is present (without the argument Grimme3) a dispersion correction (DFTD) by Grimme [33] 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 exchangecorrelation 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 SSBD 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 [33]. Please also see additional documentation for more information about this topic.
DISPERSION dDsC
 The DISPERSION dDsC key invokes the density dependent dispersion correction [54], which has been parametrized for the functionals BLYP, PBE, BP, revPBE, B3LYP, PBE0 and BHANDHLYP.
DISPERSION UFF
 The DISPERSION UFF key invokes the universal correction of density functional theory to include London dispersion (DFTulg) [52], 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.
DISPERSION MBD
 The DISPERSION MBD key invokes the MBD@rsSCS method [53], which is designed to accurately describe longrange correlation (and thus dispersion) in finitegap systems, including at the same time a description of the shortrange interactions from the underlying DFT computation of the electronic structure.
DFTD4 functionals¶
Grimme’s latest dispersion correction, D4(EEQ) [48], has been added in the 2019.3 release of the Amsterdam Modeling Suite. This is the latest dispersion correction in the DFTD family. In contrast to the earlier D3 dispersion correction, in D4(EEQ) the atomic coordinationdependent 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.
DFTD3 functionals¶
The D3 dispersion correction by Stefan Grimme is available in ADF. Grimme and his coworkers at the Universität Münster outlined the parametrization of this new correction, dubbed DFTD3, in Ref. [51]. A slightly improved version with a more moderate BJ damping function appeared later, and was called DFTBD3BJ. [49] Here they list the advantages of the new method as the following:
 It is less empirical, i.e., the most important parameters are computed from first principles by standard KohnSham (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 = 194).
 Atom pairspecific 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.
DFTD3BJ is invoked with the XC block, for example
XC
GGA BLYP
Dispersion Grimme3 BJDAMP
END
Parametrizations are available for: B3LYP, TPSS, BP86, BLYP, revPBE, PBE, PBEsol, RPBE, and some 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 ADF2016 parameters for Grimme3 and Grimme3 BJDAMP were updated according to version 3.1.1 of the coefficients, available at the Bonn website
DFTD functionals¶
An implementation for dispersion corrections based, called DFTD is available in ADF. Like DFTD3 this implementation is easy to use and is also supported by the GUI.
This DFTD implementation is based on the paper by Grimme [33] 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.
Energies calculated PostSCF using different DFTD or GGAD functionals are also present in table printed when METAGGA keyword is specified. These include: BLYPD, PBED, BP86D, TPSSD, B3LYPD, and B97D. 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 B97D is a pure GGA. B3LYPD is, however, a hybrid functional. The following functionaldependent global scaling factors s_{6} are used: 1.2 (BLYPD), 0.75 (PBED), 1.05 (BP86D), 1.0 (TPSSD), 1.05 (B3LYPD), and 1.25 (B97D). These are fixed and cannot be changed.
Regarding performance of different functionals, testing has shown that BLYPD gives good results for both energies and gradients involving VdW interactions. PostSCF energyonly calculations at fixed geometries showed that also B97D gives good binding energies compared to highlevel reference data. Thorough comparison of different DFTD functionals can be found in ref. [67]
Note: The original paper by Grimme included parameters for elements H throughout Xe. In ADF2009.01 values for dispersion parameters for DFTD functionals for heavier elements (CsRn) have been added. These new values have not been tested extensively. Thus, in this implementation, no dispersion correction is added for interactions involving atoms heavier than Radon.
DFTD is invoked with the XC block, for example
XC
GGA BLYP
Dispersion
END
dDsC: density dependent dispersion correction¶
The DISPERSION dDsC key invokes the density dependent dispersion correction [54], 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 can not be used icw fragments larger than 1 atom. The reason is that ADF uses the Hirshfeld partitioning on fragments for dDsC, which is only correct if the fragments are atoms.
DFTulg¶
The DISPERSION UFF key invokes the universal correction of density functional theory to include London dispersion (DFTulg) [52], 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
DFTMBD functionals¶
The DISPERSION MBD key invokes the MBD@rsSCS method [53], which is designed to accurately describe longrange correlation (and thus dispersion) in finitegap systems, including at the same time a description of the shortrange interactions from the underlying DFT computation of the electronic structure. The MBD (manybody dispersion) method [55] obtains an accurate description of van der Waals (vdW) interactions that includes both screening effects and treatment of the manybody vdW energy to infinite order. The revised MBD@rsSCS method [53] employs a rangeseparation (rs) of the selfconsistent screening (SCS) of polarizabilities and the calculation of the longrange correlation energy. It has been parametrized for the elements HBa, HfRn, 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 {RSSCSTS} {BETA=beta}
END
MBD {RSSCSTS} {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.
PostSCF energy functionals¶
GGA energy functionals¶
In principle you may specify different functionals to be used for the potential, which determines the selfconsistent 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 timeconsuming than LDA. The effect of the GGA term in the potential on the selfconsistent 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: postSCF.
 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 runofthemill production runs.
One possibility is to calculate a whole list of postSCF energy functionals using the METAGGA keyword, see 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 pertaining line will be used selfconsistently (in the SCFpotential), or only postSCF, 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.
MetaGGA 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 postSCF 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 postSCF 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, prespecified list of LDA, GGA, and metaGGA 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 tMGGA functional is the \(\theta\)MGGA functional of Ref. [56]):
BP, PW91, mPW, BLYP, PBE, RPBE, revPBE, mPBE, OLYP, OPBE, KCIS, PKZB, VS98, FT97, BLAP3,
HCTH, tauHCTH, BmTau1, BOP, OLAP3, TPSS, KT1, KT2, B97, M06L, tMGGA.
The hybrid GGA and hybrid metaGGA 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, tauHCTHhybrid, B97, M05, M052X, M06, M062X.
The keys METAGGA and HARTREEFOCK can be used in combination with any XC potential.
Note that at the moment hybrid functionals can not 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 (ROHF is not implemented in ADF, only UHF).
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 metaGGA 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 metaGGA 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 HartreeFock energy functionals¶
 This is mostly taken from text by the authors of Ref. [57]:
 In the early days of DFT, nonselfconsistent KohnSham energy was often evaluated upon HartreeFock (HF) densities as a way to test new approximations. This method was called HFDFT. It has been discovered that in some cases, HFDFT actually gave more accurate answers when compared to selfconsistent DFT calculations. In Ref. [57], 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 (densitydriven error). For most calculations, functional error is dominant, and here selfconsistent DFT is usually better than nonself consistent DFT on more accurate densities (called density corrected DFT (DCDFT)). Unlike these ‘normal’ calculations, there is a class of calculations where the densitydriven error is much larger, so DCDFT give better a result than selfconsistent DFT. These calculations can be classified as ‘abnormal’. HFDFT is a simple implementation of DCDFT and a small HOMOLUMO gap is an indicator of an ‘abnormal’ calculation, thus, HFDFT would perform better in such cases.
In ADF one can do HFDFT with:
XC
HartreeFock
END
MetaGGA
This will produce a large, prespecified list of LDA, GGA, metaGGA, hybrid, and metahybrid energy functionals.
References
[1]  S.H. Vosko, L. Wilk and M. Nusair, Accurate spindependent electron liquid correlation energies for local spin density calculations: a critical analysis, Canadian Journal of Physics 58 (8), 1200 (1980) 
[2]  H. Stoll, C.M.E. Pavlidou, and H. Preuss, On the calculation of correlation energies in the spindensity functional formalism, Theoretica Chimica Acta 49, 143 (1978) 
[3]  J.P. Perdew and Y. Wang, Accurate and simple analytic representation of the electrongas correlation energy, Physical Review B 45, 13244 (1992) 
[4]  (1, 2) M. Swart, A.W. Ehlers and K. Lammertsma, Performance of the OPBE exchangecorrelation functional, Molecular Physics 2004 102, 2467 (2004) 
[5]  (1, 2) X. Xu and W.A. Goddard III, The X3LYP extended density functional for accurate descriptions of nonbond interactions, spin states, and thermochemical properties, Proceedings of the National Academy of Sciences 101, 2673 (2004) 
[6]  (1, 2) M. Swart, A new family of hybrid density functionals, Chemical Physics Letters 580, 166 (2013) 
[7]  (1, 2, 3) R. van Leeuwen and E.J. Baerends, Exchangecorrelation potential with correct asymptotic behavior, Physical Review A 49, 2421 (1994) 
[8]  (1, 2) T.W. Keal and D.J. Tozer, The exchangecorrelation potential in Kohn.Sham nuclear magnetic resonance shielding calculations, Journal of Chemical Physics 119, 3015 (2003) 
[9]  A.D. Becke, Densityfunctional exchangeenergy approximation with correct asymptotic behavior, Physical Review A 38, 3098 (1988) 
[10]  J.P. Perdew and Y. Wang, Accurate and simple density functional for the electronic exchange energy: generalized gradient approximation, Physical Review B 33, 8822 (1986) 
[11]  (1, 2) J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Sing and C. Fiolhais, Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation, Physical Review B 46, 6671 (1992) 
[12]  (1, 2) C. Adamo and V. Barone, Exchange functionals with improved longrange behavior and adiabatic connection methods without adjustable parameters: The mPW and mPW1PW models, Journal of Chemical Physics 108, 664 (1998) 
[13]  (1, 2) J.P. Perdew, K. Burke and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Physical Review Letters 77, 3865 (1996) 
[14]  B. Hammer, L.B. Hansen, and J.K. Norskøv, Improved adsorption energetics within densityfunctional theory using revised PerdewBurkeErnzerhof functionals, Physical Review B 59, 7413 (1999) 
[15]  Y. Zhang and W. Yang, Comment on “Generalized Gradient Approximation Made Simple”, Physical Review Letters 80, 890 (1998) 
[16]  C. Adamo and V. Barone, Physically motivated density functionals with improved performances: The modified Perdew.Burke.Ernzerhof model, Journal of Chemical Physics 1996 116, 5933 (1996) 
[17]  (1, 2) J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria, Restoring the DensityGradient Expansion for Exchange in Solids and Surfaces, Physical Review Letters 100, 136406 (2008) 
[18]  Ph. Haas, F. Tran, P. Blaha, and K.H. Schwartz, Construction of an optimal GGA functional for molecules and solids, Physical Review B83, 205117 (2011). 
[19]  N.C. Handy and A.J. Cohen, Leftright correlation energy, Molecular Physics 99, 403 (2001) 
[20]  J.J. Mortensen, K. Kaasbjerg, S.L. Frederiksen, J.K. Nørskov, J.P. Sethna, and K.W. Jacobsen, Bayesian Error Estimation in DensityFunctional Theory, Physical Review Letters 95, 216401 (2005) 
[21]  J.P. Perdew, Densityfunctional approximation for the correlation energy of the inhomogeneous electron gas, Physical Revied B 33, 8822 (1986) Erratum: J.P. Perdew, Physical Review B 34, 7406 (1986) 
[22]  J.P. Perdew, A. Ruzsinszky, G.I. Csonka, L.A. Constantin, and J. Sun, Workhorse Semilocal Density Functional for Condensed Matter Physics and Quantum Chemistry, Physical Review Letters 103, 026403 (2009). 
[23]  J. Sun, J.P. Perdew, and A. Ruzsinszky, Semilocal density functional obeying a strongly tightened bound for exchange, Proceedings of the National Academy of Sciences 112, 685 (2015) 
[24]  J. Sun, B. Xiao, A. Ruzsinszky, Communication: Effect of the orbitaloverlap dependence in the meta generalized gradient approximation, Journal of Chemical Physics 137, 051101 (2012). 
[25]  (1, 2) J. Sun, R. Haunschild, B. Xiao, I.W. Bulik, G.E. Scuseria, J.P. Perdew, Semilocal and hybrid metageneralized gradient approximations based on the understanding of the kineticenergydensity dependence, Journal of Chemical Physics 138, 044113 (2013). 
[26]  J. Sun, A. Ruzsinszky, J.P. Perdew, Strongly Constrained and Appropriately Normed Semilocal Density Functional, Physical Review Letters 115, 036402 (2015). 
[27]  P.J. Stephens, F.J. Devlin, C.F. Chabalowski and M.J. Frisch, Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields, Journal of Physical Chemistry 98, 11623 (1994) 
[28]  M. Reiher, O. Salomon and B.A. Hess, Reparameterization of hybrid functionals based on energy differences of states of different multiplicity, Theoretical Chemistry Accounts 107, 48 (2001) 
[29]  (1, 2) C. Adamo and V. Barone, Toward reliable adiabatic connection models free from adjustable parameters, Chemical Physics Letters 274, 242 (1997) 
[30]  J.K. Kang and C.B. Musgrave, Prediction of transition state barriers and enthalpies of reaction by a new hybrid densityfunctional approximation, Journal of Chemical Physics 115, 11040 (2001) 
[31]  A.J. Cohen and N.C. Handy, Dynamic correlation, Molecular Physics 99, 607 (2001) 
[32]  B.J. Lynch, P.L. Fast, M. Harris and D.G. Truhlar, Adiabatic Connection for Kinetics, Journal of Physical Chemistry A 104, 4811 (2000) 
[33]  (1, 2, 3, 4) S. Grimme, Accurate description of van der Waals complexes by density functional theory including empirical corrections, Journal of Computational Chemistry 25, 1463 (2004) 
[34]  M. Ernzerhof and G. Scuseria, Assessment of the Perdew.Burke.Ernzerhof exchangecorrelation functional, Journal of Chemical Physics 110, 5029 (1999) 
[35]  (1, 2, 3) M. Seth and T. Ziegler, RangeSeparated Exchange Functionals with SlaterType Functions, Journal of Chemical Theory and Computation 8, 901 (2012) 
[36]  (1, 2) M. Grüning, O.V. Gritsenko, S.J.A. van Gisbergen and E.J. Baerends, Shape corrections to exchangecorrelation KohnSham potentials by gradientregulated seamless connection of model potentials for inner and outer region, Journal of Chemical Physics 114, 652 (2001) 
[37]  (1, 2) P.R.T. Schipper, O.V. Gritsenko, S.J.A. van Gisbergen and E.J. Baerends, Molecular calculations of excitation energies and (hyper)polarizabilities with a statistical average of orbital model exchangecorrelation potentials, Journal of Chemical Physics 112, 1344 (2000) 
[38]  M. Grüning, O.V. Gritsenko, S.J.A. van Gisbergen and E.J. Baerends, On the required shape correction to the LDA and GGA Kohn Sham potentials for molecular response calculations of (hyper)polarizabilities and excitation energies, Journal of Chemical Physics 116, 9591 (2002) 
[39]  (1, 2) O.V. Gritsenko, P.R.T. Schipper and E.J. Baerends, Approximation of the exchangecorrelation KohnSham potential with a statistical average of different orbital model potentials, Chemical Physics Letters 302, 199 (1999) 
[40]  D.P. Chong, O.V. Gritsenko and E.J. Baerends, Interpretation of the KohnSham orbital energies as approximate vertical ionization potentials, Journal of Chemical Physics 116, 1760 (2002) 
[41]  R. Neumann, R.H. Nobes and N.C. Handy, Exchange functionals and potentials, Molecular Physics 87, 1 (1996) 
[42]  S. Ivanov, S. Hirata, R. J. Bartlett, Exact Exchange Treatment for Molecules in FiniteBasisSet KohnSham Theory, Physical Review Letters 83, 5455 (1999) 
[43]  A.F. Izmaylov, V.N. Staroverov, G.E. Scuseria, E.R. Davidson, G. Stoltz, E. Cancès, The effective local potential method: Implementation for molecules and relation to approximate optimized effective potential techniques, Journal of Chemical Physics 126, 084107 (2007) 
[44]  M. Krykunov and T. Ziegler, On the use of the exact exchange optimized effective potential method for static response properties, International Journal of Quantum Chemistry 109, 3246 (2009) 
[45]  U. Ekström, L. Visscher, R. Bast, A.J. Thorvaldsen, and K. Ruud, ArbitraryOrder Density Functional Response Theory from Automatic Differentiation, Journal of Chemical Theory and Computation 6, 1971 (2010) 
[46]  M.A.L. Marques, M.J.T. Oliveira, and T. Burnus, Libxc: a library of exchange and correlation functionals for density functional theory, Computer Physics Communications 183, 2272 (2012) 
[47]  S.Lehtola, C. Steigemann, M.J.T. Oliveira, M.A.L. Marques, Recent developments in LibXC – A comprehensive library of functionals for density functional theory, SoftwareX 7, 1 (2018) 
[48]  (1, 2, 3) E. Caldeweyher, S. Ehlert, A. Hansen, H. Neugebauer, S. Spicher, C. Bannwarth, S. Grimme, A Generally Applicable AtomicCharge Dependent London Dispersion Correction Scheme, ChemRxiv 7430216 v2 
[49]  (1, 2) S. Grimme, S. Ehrlich, and L. Goerigk, Effect of the Damping Function in Dispersion Corrected Density Functional Theory, Journal of Computational Chemistry 32, 1457 (2011). 
[50]  (1, 2) J.G. Brandenburg, J.E. Bates, J. Sun, and J.P. Perdew, Benchmark tests of a strongly constrained semilocal functional with a longrange dispersion correction, Physical Review B 94, 115144 (2016). 
[51]  (1, 2) S. Grimme, J. Anthony, S. Ehrlich, and H. Krieg, A consistent and accurate ab initio parametrization of density functional dispersion correction (DFTD) for the 94 elements HPu, Journal of Chemical Physics 132, 154104 (2010). 
[52]  (1, 2) H. Kim, J.M. Choi, W.A. Goddard, Universal Correction of Density Functional Theory to Include London Dispersion (up to Lr, Element 103), Journal of Physical Chemistry Letters 3, 360 (2012) 
[53]  (1, 2, 3) A. Ambrosetti, A.M. Reilly, Robert A. DiStasio Jr., A. Tkatchenko, Longrange correlation energy calculated from coupled atomic response functions, Journal of Chemical Physics 140, 18A508 (2014) 
[54]  (1, 2) S.N. Steinmann, and C. Corminboeuf, Comprehensive Benchmarking of a DensityDependent Dispersion Correction, Journal of Chemical Theory and Computation 7, 3567 (2011). 
[55]  A. Tkatchenko, R.A. DiStasio Jr., R. Car, M. Scheffler Accurate and Efficient Method for ManyBody van der Waals Interactions, Physical Review Letters 108, 236402 (2012) 
[56]  P. de Silva and C. Corminboeuf, Communication: A new class of nonempirical explicit density functionals on the third rung of Jacob’s ladder, Journal of Chemical Physics 143, 111105 (2015) 
[57]  (1, 2) M.C. Kim, E. Sim, and K. Burke, Understanding and Reducing Errors in Density Functional Calculations, Physical Review Letters 111, 2073003 (2013) 
[58]  (1, 2, 3, 4) Y. Zhao and D.G. Truhlar, A new local density functional for maingroup thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions, Journal of Chemical Physics 125, 194101 (2006) 
[59]  (1, 2, 3, 4) Y. Zhao and D.G. Truhlar, The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06class functionals and 12 other functionals, Theoretical Chemical Accounts 120, 215 (2008) 
[60]  (1, 2) J. Tao, J.P. Perdew, V.N. Staroverov and G.E. Scuseria, Climbing the Density Functional Ladder: Nonempirical MetaGeneralized Gradient Approximation Designed for Molecules and Solids Physical Review Letters 91, 146401 (2003) 
[61]  (1, 2) V.N. Staroverov, G.E. Scuseria, J. Tao and J.P. Perdew, Comparative assessment of a new non empirical density functional: Molecules and hydrogenbonded complexes Journal of Chemical Physics 119, 12129 (2003) 
[62]  (1, 2, 3) M. Swart, M. Solà and F.M. Bickelhaupt, A new allround DFT functional based on spin states and SN2 barriers, Journal of Chemical Physics 131, 094103 (2009) 
[63]  (1, 2, 3) M. Swart, M. Solà and F.M. Bickelhaupt, Switching between OPTX and PBE exchange functionals, Journal of Computational Methods in Science and Engineering 9, 69 (2009) 
[64]  C. Lee, W. Yang and R.G. Parr, Development of the ColleSalvetti correlationenergy formula into a functional of the electron density, Physical Review B 37, 785 (1988) 
[65]  B.G. Johnson, P.M.W. Gill and J.A. Pople, The performance of a family of density functional methods, Journal of Chemical Physics 98, 5612 (1993) 
[66]  T.V. Russo, R.L. Martin and P.J. Hay, Density Functional calculations on firstrow transition metals, Journal of Chemical Physics 101, 7729 (1994) 
[67]  S. Grimme, , J. Antony, T. Schwabe and C. MückLichtenfeld, Density Functional Theory with Dispersion Corrections for Supramolecular Structures, Aggregates, and Complexes of (Bio)Organic Molecules, Organic & Biomolecular Chemistry 5, 741 (2007) 
[68]  S. Grimme, Semiempirical hybrid density functional with perturbative secondorder J. Chem. Phys. 2006, 124, 034108 
[69] 

[70]  (1, 2) Js. Garcia, E. Bremond, M. Savarese, AJ. PerezGimenez, C. Adamo, Partnering dispersion corrections with modern parameterfree doublehybrid density functionals Phys. Chem. Chem. Phys., 19, 13481 (2017) 
[71]  (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16) G.Santra, N. Sylvetsky, and J.M.L. Martin, Minimally Empirical DoubleHybrid Functionals Trained against the GMTKN55 Database: revDSDPBEP86D4 revDODPBED4, and DODSCAND4 J. Chem. Phys. 2019, 123, 5129 
[72]  J.C. SanchoGarcía, A.J. PérezJiménez, Assessment of doublehybrid energy functionals for PIconjugated systems J. Chem. Phys. 2009, 131, 084108 
[73]  D.C. Graham, A.S. Menon, L. Goerigk, S. Grimme, L. Radom Optimization and basisset dependence of a restrictedopenshell form of B2PLYP doublehybrid density functional theory J. Phys. Chem. A 2009, 113, 9861 
[74]  (1, 2, 3) A. Tarnopolsky, A. Karton, R. Sertchook, D. Vuzman, J.M.L Martin, Doublehybrid functionals for thermochemical kinetics J. Phys. Chem. A 2008, 112, 3 
[75]  A. Karton, A. Tarnopolsky, J.F. Lamere, G.C. Schatz, J.M.L, Martin, Highly accurate firstprinciples bench mark data sets for the parametrization and validation of density functional and other approximate methods. Derivation of a robust, generally applicable, doublehybrid functional or thermochemistry and thermochemical kinetics. J. Phys. Chem. A 2008, 112, 12868 
[76]  (1, 2) Y. Feng, DoubleHybrid Density Functionals Free of Dispersion and Counterpoise Corrections for NonCovalent Interactions J. Phys. Chem. A 2014, 118, 3175 
[77]  T. Schwabe, S. Grimme, Towards chemical accuracy for the thermodynamics of large molecules: new hybrid density functionals including nonlocal correlation effects. Phys. Chem. Chem. Phys. 2006, 8, 4398 
[78]  K. Sharkas, J. Toulouse, A. Savin, Doublehybrid densityfunctional theory made rigorous J. Chem. Phys. 2011, 134, 064113 
[79]  E. Bremond, C. Adamo, Seeking for parameterfree doublehybrid functionals: The PBE0DH model J. Chem. Phys. 2011, 135, 024106 
[80]  J. Toulouse, K. Sharkas, E. Bremond, C. Adamo, Rationale for a new class of doublehybrid approximations J. Chem. Phys. 2011, 135, 101102 
[81]  J.D. Chai, S.P. Mao, Seeking for reliable doublehybrid density functionals without fitting parameters: the PBE02 functional Chem. Phys. Lett. 2012, 538, 121 
[82]  (1, 2) S.O. Souvi, K. Sharkas and J. Toulouse, Doublehybrid density functional theory with metageneralized gradient approximations J. Chem. Phys. 2014, 140, 084107 
[83]  S. Kozuch, D. Gruzman, J.M.L. Martin, DSDBLYP: a general purpose double hybrid density functional including spin component scaling and dispersion correction J. Phys. Chem. C 2010, 114, 20801 
[84]  (1, 2) S. Kozuch, J.M.L. Martin, DSDPBEP86: In search of the best doublehybrid DFT with spincomponent scaled MP2 and dispersion corrections Phys. Chem. Chem. Phys. 2011, 13, 20104 
[85]  Y. Jung, R.C. Lochan, A. D. Dutoi, M. HeadGordon, Scaled oppositespin second order Møllerplesset correlation energy: An economical electronic structure method. Journal of Chemical Physics 2004, 121, 9793 
[86]  S. Grimme, Improved secondorder MøllerPlesset perturbation theory by separate scaling of parallel and antiparallelspin pair correlation energies. Journal of Chemical Physics 2003, 118, 9095 
[87]  R.C. Lochan, Y. Jung, M. HeadGordon, Scaled opposite spin second order MøllerPlesset theory with improved physical description of longrange dispersion interactions Journal of Physical Chemistry A2005,109, 7598 
[88]  J. Proppe, S. Guglerb and M. Reiher, Gaussian ProcessBased Refinement of Dispersion Corrections https://arxiv.org/pdf/1906.09342.pdf 
[89]  A. Förster, M. Franchini, E. van Lenthe, L. Visscher, A Quadratic Pair Atomic Resolution of the Identity Based SOSAOMP2 Algorithm Using Slater Type Orbitals, Journal of Chemical Theory and Computation 16, 875 (2020) 