The Density Functional, also called the exchange-and-correlation (XC) functional, consists of an LDA, a GGA part, a Hartree-Fock exchange part (hybrids), and a meta-GGA part (meta-GGA or meta-hybrid). 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. A meta-GGA (for example TPSS) has a GGA part, but also depends on the kinetic energy density. A meta-hybrid (for example TPSSh) has GGA part, a part of Hartree-Fock 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 post-SCF energies only, ADF supports also various other 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. Note that some of these problems can also be circumvented with hybrids.
Several asymptotically correct XC potentials have been implemented in ADF, such as the (now somewhat outdated) LB94 potential [15], the gradient-regulated asymptotic correction (GRAC) [16], and the statistical average of orbital potentials (SAOP) [244,17].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].
If a functional contains a part of Hartree-Fock 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) 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 (or hybrid or meta-hybrid) 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 the same XC-potential in the create run of the atoms, which is done automatically if you use the BASIS key.
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). Some may occur twice in the data block.
XC
{LDA LDA {Stoll}}
{GGA GGA}
{MetaGGA metagga}
{Model MODELPOT [IP]}
{HartreeFock}
{OEP fitmethod {approximation}}
{HYBRID hybrid {HF=HFpercentage}}
{MetaHYBRID metahybrid}
{DISPERSION [s6scaling] [RSCALE=r0scaling]}
end
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.
PW92: the parameterization of electron gas data given by Perdew and Wang (ref [288]).
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]
PBEsolx: the PBEsol exchange
correction proposed in 2008 by Perdew-Ruzsinszky-Csonka-Vydrov-Scuseria [285]
OPTX: the OPTX exchange
correction proposed in 2001 by Handy-Cohen [29]
BEEx: the BEEx exchange
correction proposed in 2005 by Mortensen-Kaasbjerg-Frederiksen-Nørskov-Sethna-Jacobsen [284]
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] .
PBEsolc: the PBEsol correlation
correction proposed in 2008 by Perdew-Ruzsinszky-Csonka-Vydrov-Scuseria [285]
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
PBEsol: this is equivalent to PBEsolx + PBEsolc 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).
BEE: this is equivalent to BEEx (exchange) + PBEc (correlation).
SSB-D: dispersion corrected functional by
Swart-Solà-Bickelhaupt [286,287].
Single point only. Use METAGGA SSB-D in other cases.
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.
Specifies that a meta-GGA should
be used during the SCF. All electron basis sets should be used.
The meta-GGA can be one of the following:
M06-L:
functional by Yan-Truhlar [223,224]
TPSS:
functional by Tao-Perdew-Staroverov-Scuseria [246,247]
SSB-D: dispersion corrected GGA functional by
Swart-Solà-Bickelhaupt [286,287].
Use GGA SSB-D for NMR calculations.
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 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.
Specifies that the Hartree-Fock exchange should be used during the SCF.
Defines the optimized effective potential expanded into a set of the fit functions. The subkeyword fitmethod can be any of the following: BARTLETT [248], SCUSERIA [249]. 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.[250].
Specifies that a hybrid functional should
be used during the SCF. The hybrid can be one of the following:
B3LYP:
ADF uses VWN5 in B3LYP.
functional (20% HF exchange) by Stephens-Devlin-Chablowski-Frisch
[176].
B3LYP*:
modified B3LYP functional (15% HF exchange) by Reiher-Salomon-Hess
[177].
B1LYP:
functional (25% HF exchange) by Adamo-Barone
[178].
KMLYP:
functional (55.7% HF exchange) by Kang-Musgrave
[179].
O3LYP:
functional (12% HF exchange) by Cohen-Handy
[180].
X3LYP:
functional (21.8% HF exchange) by Xu-Goddard
[172].
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) Adamo-Barone
[178].
mPW1PW:
functional (42.8% HF exchange) by Adamo-Barone
[25].
mPW1K:
functional (25% HF exchange) by Lynch-Fast-Harris-Truhlar
[181].
PBE0:
functional (25% HF exchange)
by Ernzerhof-Scuseria [211]
and by Adamo-Barone [212],
hybrid form of PBE.
OPBE0:
functional (25% HF exchange) by Swart-Ehlers-Lammertsma
[175],
hybrid form of OPBE.
HFpercentage
Specifies the amount of HF exchange that should be used in the functional, instead of the default HF exchange percentage for the given hybrid.
Specifies that a meta-hybrid functional should
be used during the SCF. The meta-hybrid can be one of the following:
M06:
functional (27% HF exchange) by Yan-Truhlar [223,224]
M06-2X:
functional (54% HF exchange) by Yan-Truhlar [223,224]
M06-HF:
functional (100% HF exchange) by Yan-Truhlar [223,224]
TPSSH:
functional (10% HF exchange) by Tao-Perdew-Staroverov-Scuseria [246,247]
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. The SSB-D functional includes the dispersion correction (factor 0.847455) by default.
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.
