Exchange Correlation Functionals

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

Recently 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Finally, several asymptotically correct xc potentials have been implemented in ADF, such as the (now somewhat outdated) LB94 potential [15], the gradient-regulated asymptotic correction (GRAC) [16], and the statistical average of orbital potentials (SAOP) [17].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.

Recently 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]}
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]
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:
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
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).
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.

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 or GGA 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 ressembles 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 [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.