Density Functionals (XC)¶

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 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, 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 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.

Range-separated 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 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 ADF an exponential function is used (the error function is common in Gaussian based codes). In a range-separated 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 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 adding up 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 yet.

Functionality/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, TDDFT. Starting from ADF2018 singlet-triplet 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 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 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 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¶

One 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

Double-Hybrid¶

XC
  DOUBLEHYBRID functional
End

DOUBLEHYBRID

Specifies that a double-hybrid functional [68] should be used. They combine a self-consistent (meta-)hybrid calculation with a post-SCF MP2 energy correction.

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 double-hybrids 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 double-hybrids into three categories: double-hybrids using the OS-component only, double-hybrids using both components, and double-hybrids 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. Double-hybrids using all spin-components 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 OS-only functional.

OS-only functionals

OS-functionals combine a (meta-)hybrid-functional with empirical dispersion correction (in most cases) and use the correlation energy of unpaired spins only. They are often referred to as DOD-functionals. 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 DOD-calculation is always feasible when a hybrid calculation is feasible too!

Currently, ADF supports the following DOD-functionals:

• SOS1-PBE-QIDH: 1-parameter functional with PBE exchange, PBE correlation (69 % HF, 44 % OS-MP2 [70]
• rev-DOD-BLYP: B88 exchange, LYP correlation, Grimme3 dispersion (71 % HF, 62.2 % OS-MP2) [71]
• rev-DOD-BLYP-D4: B88 exchange, LYP correlation, Grimme4 dispersion (71 % HF, 63.5 % OS-MP2) [71]
• rev-DOD-PBE: PBE exchange, PBE correlation, Grimme3 dispersion (68 % HF, 61.3 % OS-MP2) [71]
• rev-DOD-PBE-D4: PBE exchange, PBE correlation, Grimme4 dispersion (68 % HF, 61.8 % OS-MP2) [71]
• rev-DOD-PBEP86: PBE exchange, P86 correlation, Grimme3 dispersion (69 % HF, 60.6 % OS-MP2) [71]
• rev-DOD-PBEP86-D4: PBE exchange, P86 correlation, Grimme4 dispersion (69 % HF, 61.2 % OS-MP2) [71]
• DOD-SCAN: SCAN exchange and correlation, Grimme3 dispersion (66 % HF, 63,0 % OS-MP2) [71]
• rev-DOD-SCAN-D4: SCAN exchange and correlation, Grimme4 dispersion (66 % HF, 63.4 % OS-MP2) [71]

Note, that except for SOS1-PBE-QIDH, all functionals include dispersion correction by default since they are an intrinsic part of the functionals.

Standard and 1-parameter functionals

The functionals in this category combine a (meta-hybrid) functional with a scaled correlation energy from both components of the MP2-energy.

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 pi-pi interactions [72]
• ROB2PLYP: B88 exchange and LYP correlation (59 % HF, 28 % MP2), restricted-open-shell 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 non-covalent 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]
• DH-BLYP: 1-Parameter functional with B88 exchange and LYP correlation (65 % HF, 42 % MP2) [78]
• PBE0-DH: 1-Parameter functional with PBE exchange and PBE correlation (50 % HF, 13 % MP2) [79]
• PBE-QIDH: 1-parameter functional with PBE exchange and PBE correlation (69 % HF, 33 % MP2) [70]
• LS1-DH: 1-Parameter functional with PBE exchange and PBE correlation (75 % HF, 42 % MP2) [80]
• PBE0-2: 1-Parameter functional with PBE exchange and PBE correlation (79 % HF, 50 % MP2) [81]
• LS1-TPSS: 1-Parameter functional with TPSS meta-GGA (85 % HF, 61 % MP2) [82]
• DS1-TPSS: 1-Parameter functional with TPSS meta-GGA (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.

Spin-component-scaled functionals

The functionals in this category combine a (meta-hybrid) functional with a scaled correlation energy from both components of the MP2-energy. Opposed to the preceding category, both spin-components are scaled individually, allowing for more flexibility in the functional parametrization. As OS-functionals, they also include an empirical dispersion correction term in their parametrization. They are often referred to as DSD-functionals

Currently, ADF supports the following DSD-functionals:

• DSD-BLYP: B88 exchange and LYP correlation, Grimme3 dispersion (69 % HF, 46 % OS-MP2, % 37 % SS-MP2) [83]
• rev-DSD-BLYP: revised version of DSD-BLYP, Grimme3 dispersion (71 % HF, 54.7 % os, 19.8 % SS-MP2) [71]
• rev-DSD-BLYP-D4: revised version of DSD-BLYP, Grimme4 dispersion (71 % HF, 55.9 % OS-, 19.7 % SS-MP2) [71]
• DSD-PBEP86: PBE exchange and P86 correlation (69 % HF, 52 % OS-, % 22 % SS-MP2) [84]
• rev-DSD-PBEP86: revised version of DSD-PBEP86 (69 % HF, 57.9 % OS-, 8 % SS-MP2) [71]
• rev-DSD-PBEP86-D4: PBE exchange and P86 correlation, Grimme4 dispersion (69 % HF, 59.2 % OS-, 6.4 % SS-MP2) [71]
• DSD-PBE: PBE exchange and PBE correlation, Grimme3 dispersion (68 % HF, 55 % OS-, % 13 % SS-MP2) [84]
• rev-DSD-PBE: revised version of DSD-PBE, Grimme3 dispersion (68 % HF, 58.5 % os, 7 % SS-MP2) [71]
• rev-DSD-PBE-D4: revised version of DSD-PBE, Grimme4 dispersion (68 % HF, 60 % os, 4.2 % SS-MP2) [71]
• rev-DSD-SCAN-D4: based on SCAN meta-GGA (66 % HF, 63.2 % OS-, 1.3 % SS-MP2) [71]
• SD-SCAN69: Based on SCAN meta-GGA, no dispersion correction (69 % HF, 62 % OS-, 26 % SS-MP2) [71]

Except for SD-SCAN69, all functionals mentioned include dispersion correction by default. For more technical details of the algorithm and how to tweak parameters, see the MBPT section.

MP2-methods¶

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, the all rely on a HF instead of a DFT calculation. The following variants are supported.

• SOS-MP2: pure HF reference (100 % HF, 130 % OS-MP2) [85]
• MP2: pure HF reference (100 % HF, 100 % MP2 correlation)
• SCS-MP2: pure HF reference (100 % HF, 120 % OS-MP2, 33 % SS-MP2) [86]
• SOS-MI-MP2: pure HF reference (100 % HF, 40 % OS-MP2, 129 % SS-MP2) [87]

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

RPA¶

The RPA correlation energy is calculated incorrectly in AMS2021.101-AMS2021.102. This bug is fixed in the bug fix release of AMS2021 with revision number r95877 or later.

XC
  RPA
End

RPA

requests that the RPA exchange and correlation energies are evaluated after the SCF. the XC-energy evaluated from the functional used in the SCF is replaced by the RPA XC-energy.

RPA can be used in conjunction with LDA, GGAs and hybrid functionals (RPA@LDA, RPA@PBE, RPA@hybrid, respectively). IMPORTANT: spin-orbit relativistic effects 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 AO-basis and the algorithm is similar to SOS-AO-PARI-MP2. The main difference is that the final RPA correlation energy is evaluated along the imaginary frequency axis, while the polarizability (non-interacting Kohn-Sham density response function, which is also evaluated in SOS-MP2) 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 sub-linearly 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, double-hybrid functionals and RPA¶

All post-SCF correlation methods can only be used for single-point calculations (Although numerical gradients are available in the AMS driver). However, they all support the use of frozen cores, empirical dispersion correction, and spin-unrestricted calculations. MP2, RPA, and double-hybrid functionals can be used icw scalar relativistic effects within the ZORA formalism, but can not be used icw spin-orbit coupling.

The basis set requirements are higher than for standard functionals. We recommend to use all-electron basis sets of at least TZP quality to obtain meaningful results. Using basis sets of quadruple-zeta quality is usually better. all post-SCF 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 1e-4, for post-SCF correlation methods, a default of 1e-03 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 double-hybrid calculation using the rev-DOD-BLYP-D4 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 post-SCF 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.

DFT-D4 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 DFT-D family. 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.

DFT-D3 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 DFT-D3, in Ref. [51]. A slightly improved version with a more moderate BJ damping function appeared later, and was called DFTB-D3-BJ. [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 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.

DFT-D3-BJ 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

DFT-D functionals¶

An implementation for dispersion corrections based, 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 [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 Post-SCF using different DFT-D or GGA-D functionals are also present in table printed when 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 s₆ 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 also B97-D gives good binding energies compared to high-level reference data. Thorough comparison of different DFT-D 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 DFT-D functionals for heavier elements (Cs-Rn) 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.

DFT-D 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.

DFT-ulg¶

The DISPERSION UFF key invokes the universal correction of density functional theory to include London dispersion (DFT-ulg) [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

DFT-MBD functionals¶

The DISPERSION MBD key invokes the MBD@rsSCS method [53], 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 [55] 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 [53] 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.

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 run-of-the-mill production runs.

One possibility is to calculate a whole list of post-SCF 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 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. [56]):

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 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 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 is mostly taken from text by the authors of Ref. [57]:

In the early days of DFT, non-self-consistent Kohn-Sham energy was often evaluated upon 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. [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 (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 give better a 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 metahybrid energy functionals.

References