SCM: ADF User's Guide - Exchange Correlation Functionals

Exchange Correlation Functionals

The Density Functional, also called the exchange-and-correlation (XC) functional, consists of an LDA, a GGA part, and possibly a Hartree-Fock exchange part (hybrids). 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 Hartree Fock exchange. For these terms ADF supports a large number of the formulas advocated in the literature. For post-SCF energies only, ADF supports also various meta-GGA functionals and more hybrid functionals.

The Perdew-Zunger self-interaction correction (SIC) was implemented [47-49] self-consistently using the Krieger-Li-Iafrate approximation to the optimized effective potential, and the Vosko-Wilk-Nusair (VWN) functional or gradient corrected density functionals. This approach is found to improve several properties, which are sometimes difficult to describe with standard DFT techniques, like for example some 'problematic' NMR chemical shifts, or some 'difficult' reaction barriers.

Finally, several asymptotically correct XC potentials have been implemented in ADF, such as the (now somewhat outdated) LB94 potential [15], the gradient-regulated asymptotic correction (GRAC) [16], and the statistical average of orbital potentials (SAOP) [244,17]. 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.[18] for applications to (hyper)polarizabilities Cauchy coefficients, etc. of small molecules). This is particularly true if the molecule is small and the (high-lying) virtual orbitals are important for the property under study.

It was also shown that, simply using the orbital energies of the occupied Kohn-Sham orbitals of a SAOP calculation, quite good agreement with 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 lower-lying occupied orbital energies. The agreement becomes worse for deep-lying core orbital energies. A theoretical explanation and practical results are given in Ref. [19].

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 rather time-consuming.
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.

The key that controls the Density Functional is XC, with sub keys LDA and GGA (or equivalently: gradients) to define the LDA and GGA parts of the functional, and MODEL in case one of the special 'model' XC potentials is required in stead of LDA or GGA. All subkeys are optional (need not be used) and may occur twice in the data block: if one wants to specify different functionals for potential and energy evaluations respectively, see above.

XC
 {LDA {Apply} LDA {Stoll}}
 {GGA {Apply} GGA}
 {Model MODELPOT [IP]}
 {HARTREEFOCK}
 {HYBRID hybrid}
 {DISPERSION [s6scaling] [RSCALE=r0scaling]} 
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

Defines the LDA part of the XC functional and can be any of the following:
Xonly: The pure-exchange electron gas formula. Technically this is identical to the Xalpha form (see next) with a value 2/3 for the X-alpha parameter.
Xalpha: the scaled (parameterized) exchange-only formula. When this option is used you may (optionally) specify the X-alpha parameter by typing a numerical value after the string Xalpha (separated by a blank). If omitted this parameter takes the default value 0.7
VWN: the parameterization of electron gas data given by Vosko, Wilk and Nusair (ref [20], 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 or GL variety of the LDA form you may include Stoll's correction [21] 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.

GGA

Specifies the GGA part of the XC Functional, in earlier times often called the 'non-local' correction to the LDA part of the density functional. It uses derivatives (gradients) of the charge density. 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: the gradient correction proposed in 1988 by Becke [22].
PW86x: the correction advocated in 1986 by Perdew-Wang [23].
PW91x: the exchange correction proposed in 1991 by Perdew-Wang [24]
mPWx: the modified PW91 exchange correction proposed in 1998 by Adamo-Barone [25]
PBEx: the exchange correction proposed in 1996 by Perdew-Burke-Ernzerhof [26]
RPBEx: the revised PBE exchange correction proposed in 1999 by Hammer-Hansen-Norskov [27]
revPBEx: the revised PBE exchange correction proposed in 1998 by Zhang-Wang [28]
mPBEx: the modified PBE exchange correction proposed in 2002 by Adamo-Barone [174]
OPTX: the OPTX exchange correction proposed in 2001 by Handy-Cohen [29]

For the correlation part the options are:
Perdew: the correlation term presented in 1986 by Perdew [30].
PBEc: the correlation term presented in 1996 by Perdew-Burke-Ernzerhof [26] .
PW91c: the correlation correction of Perdew-Wang (1991), see [24].
LYP: the Lee-Yang-Parr 1988 correlation correction [31-33].

Some GGA options define the exchange and correlation parts in one stroke. These are:
BP86: this is equivalent to Becke + Perdew together.
PW91: this is equivalent to pw91x + pw91c together.
mPW: this is equivalent to mPWx + pw91c together.
PBE: this is equivalent to PBEx + PBEc together
RPBE: this is equivalent to RPBEx + PBEc together
revPBE: this is equivalent to revPBEx + PBEc together
mPBE: this is equivalent to mPBEx + PBEc together
BLYP: this is equivalent to Becke (exchange) + LYP (correlation).
OLYP: this is equivalent to OPTX (exchange) + LYP (correlation).
OPBE: this is equivalent to OPTX (exchange) + PBEc (correlation) [175].
XLYP: this is equivalent to XLYPx [172] (exchange, not available separately from LYP) + LYP (correlation).
LB94: this refers to the XC functional of Van Leeuwen and Baerends [15].
KT1: this refers to the KT1 functional of Keal and Tozer [171].
KT2: this refers to the KT2 functional of Keal and Tozer [171].

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.

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, HARTREEFOCK, or HYBRID 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 TAPE21 file from an LDA or GGA potential as input.

LB94: this refers to the XC functional of Van Leeuwen and Baerends [15]. There are no separate entries for the Exchange and Correlation parts respectively of LB94. Usually the GRACLB or SAOP potentials give results superior to LB94.

GRACLB: the gradient-regulated asymptotic correction, which in the outer region closely resembles the LB94 potential [16]. 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.

SAOP: the statistical average of orbital potentials [244,17]. It can be used for all electron calculations only. It will be expensive for large molecules, but requires no further parameter input.

IP: should be supplied only if GRACLB is specified.

HARTREEFOCK

Specifies that the Hartree-Fock exchange should be used during the SCF. No LDA, GGA, MODEL, or HYBRID key should be used in combination with this key. Note that it is not recommended to use Hartree-Fock exchange in combination with frozen cores, since at the moment the frozen core orbitals are not included in the Hartree Fock exchange operator. The implementation in ADF of the calculation of exact exchange (Hartree Fock exchange), which is also needed for the hybrid functionals, is based on work by Watson et al., Ref. [138]. In ADF one can do unrestricted Hartree-Fock calculations, as long as one has integer occupation numbers (ROHF is not implemented in ADF, only UHF). Note that the DEPENDENCY key is switched on for Hartree-Fock exchange in order to circumvent numerical problems, see also the ADDDIFFUSEFIT key. You also need to include Hartree-Fock exchange in the create run of the atoms, which is done automatically if you use the BASIS key.

HYBRID

Specifies that a hybrid functional should be used during the SCF. No LDA, GGA, MODEL, or HARTREEFOCK key should be used in combination with this key. Note that it is not recommended to use Hartree-Fock 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 calculation with these hybrid functionals, as long as one has integer occupation numbers (ROHF is not implemented in ADF, only UHF). Note that the DEPENDENCY key is switched on for hybrid functionals in order to circumvent numerical problems, see also the ADDDIFFUSEFIT key. You also need to include the same hybrid functional in the create run of the atoms, which is done automatically if you use the BASIS key.

The hybrid can be one of the following:
B3LYP: ADF uses VWN5 in B3LYP. functional (20% Hartree-Fock exchange) by Stephens-Devlin-Chablowski-Frisch [176].
B3LYP*: modified B3LYP functional (15% Hartree-Fock exchange) by Reiher-Salomon-Hess [177].
B1LYP: functional (25% Hartree-Fock exchange) by Adamo-Barone [178].
KMLYP: functional (55.7% Hartree-Fock exchange) by Kang-Musgrave [179].
O3LYP: functional (12% Hartree-Fock exchange) by Cohen-Handy [180].
X3LYP: functional (21.8% Hartree-Fock exchange) by Xu-Goddard [172].
BHandH: 50% Hartree-Fock exchange, 50% LDA exchange, and 100% LYP correlation.
BHandHLYP: 50% Hartree-Fock exchange, 50% LDA exchange, 50% Becke88 exchange, and 100% LYP correlation.
B1PW91: functional by (25% Hartree-Fock exchange) Adamo-Barone [178].
mPW1PW: functional (42.8% Hartree-Fock exchange) by Adamo-Barone [25].
mPW1K: functional (25% Hartree-Fock exchange) by Lynch-Fast-Harris-Truhlar [181].
PBE0: functional (25% Hartree-Fock exchange) by Ernzerhof-Scuseria [211] and by Adamo-Barone [212], hybrid form of PBE.
OPBE0: functional (25% Hartree-Fock exchange) by Swart-Ehlers-Lammertsma [175], hybrid form of OPBE.

DISPERSION

If the DISPERSION keyword is present a dispersion correction by Grimme [226] will be added to the total bonding energy, gradient and second derivatives, where applicable. The global scaling factor with which the correction is added depends on the exchange-correlation functional used at SCF but it can be modified using the s6scaling parameter. The following scaling factors are used (with the XC functional in parantheses): 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.
Unlike the MMDISPERSION keyword, the van der Waals radii used in this implementation are hardcoded 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 [226]. Please also see additional documentation for more information about this topic.