Density Functional (DFT) Methods

DESCRIPTION

Gaussian 09 offers a wide variety of Density Functional Theory (DFT) [Hohenberg64, Kohn65, Parr89, Salahub89] models (see also [Labanowski91, Andzelm92, Becke92, Gill92, Perdew92, Scuseria92, Becke92a, Perdew92a, Perdew93a, Sosa93a, Stephens94, Stephens94a, Ricca95] for discussions of DFT methods and applications). Energies [Pople92], analytic gradients, and true analytic frequencies [Johnson93a, Johnson94, Stratmann97] are available for all DFT models.

The self-consistent reaction field (SCRF) can be used with DFT energies, optimizations, and frequency calculations to model systems in solution.

Pure DFT calculations will often want to take advantage of density fitting. See the discussion in Basis Sets for details.

The next subsection presents a very brief overview of the DFT approach. Following this, the specific functionals available in Gaussian 09 are given. The final subsection surveys considerations related to accuracy in DFT calculations.

The same optimum memory sizes given by freqmem are recommended for DFT frequency calculations.

Polarizability derivatives (Raman intensities) and hyperpolarizabilities are not computed by default during DFT frequency calculations. Use Freq=Raman to request them. Polar calculations do compute them.

Note: The double hybrid functionals are discussed with the MP2 keyword since they have similar computational cost.

BACKGROUND

In Hartree-Fock theory, the energy has the form:

    EHF = V + <hP> + 1/2<PJ(P)> - 1/2<PK(P)>

where the terms have the following meanings:

V

  

The nuclear repulsion energy.

P

  

The density matrix.

<hP>

  

The one-electron (kinetic plus potential) energy.

1/2<PJ(P)>

  

The classical coulomb repulsion of the electrons.

-1/2<PK(P)>

  

The exchange energy resulting from the quantum (fermion) nature of electrons.

In the Kohn-Sham formulation of density functional theory [Kohn65], the exact exchange (HF) for a single determinant is replaced by a more general expression, the exchange-correlation functional, which can include terms accounting for both the exchange and the electron correlation energies, the latter not being present in Hartree-Fock theory:

    EKS = V + <hP> + 1/2<PJ(P)> + EX[P] + EC[P]

where EX[P] is the exchange functional, and EC[P] is the correlation functional.

Within the Kohn-Sham formulation, Hartree-Fock theory can be regarded as a special case of density functional theory, with EX[P] given by the exchange integral -1/2<PK(P)> and EC=0. The functionals normally used in density functional theory are integrals of some function of the density and possibly the density gradient:

    EX[P] = f(ρα(r),ρβ(r),∇ρα(r),∇ρβ(r))dr

where the methods differ in which function f is used for EX and which (if any) f is used for EC. In addition to pure DFT methods, Gaussian supports hybrid methods in which the exchange functional is a linear combination of the Hartree-Fock exchange and a functional integral of the above form. Proposed functionals lead to integrals which cannot be evaluated in closed form and are solved by numerical quadrature.

KEYWORDS FOR DFT METHODS

Names for the various pure DFT models are given by combining the names for the exchange and correlation functionals. In some cases, standard synonyms used in the field are also available as keywords.

Exchange Functionals. The following exchange functionals are available in Gaussian 09. Unless otherwise indicated, these exchange functionals must be combined with a correlation functional in order to produce a usable method.

  • S: The Slater exchange, ρ4/3 with theoretical coefficient of 2/3, also referred to as Local Spin Density exchange [Hohenberg64, Kohn65, Slater74]. Keyword if used alone: HFS.

  • XA: The XAlpha exchange, ρ4/3 with the empirical coefficient of 0.7, usually employed as a standalone exchange functional, without a correlation functional [Hohenberg64, Kohn65, Slater74]. Keyword if used alone: XAlpha.

  • B: Becke’s 1988 functional, which includes the Slater exchange along with corrections involving the gradient of the density [Becke88b]. Keyword if used alone: HFB.

  • PW91: The exchange component of Perdew and Wang’s 1991 functional [Perdew91, Perdew92, Perdew93a, Perdew96, Burke98].

  • mPW: The Perdew-Wang 1991 exchange functional as modified by Adamo and Barone [Adamo98].

  • G96: The 1996 exchange functional of Gill [Gill96, Adamo98a].

  • PBE: The 1996 functional of Perdew, Burke and Ernzerhof [Perdew96a, Perdew97].

  • O: Handy’s OPTX modification of Becke’s exchange functional [Handy01, Hoe01].

  • TPSS: The exchange functional of Tao, Perdew, Staroverov, and Scuseria [Tao03].

  • RevTPSS: The revised TPSS exchange functional of Perdew et. al. [Perdew09, Perdew11].

  • BRx: The 1989 exchange functional of Becke [Becke89a].

  • PKZB: The exchange part of the Perdew, Kurth, Zupan and Blaha functional [Perdew99].

  • wPBEh: The exchange part of screened Coulomb potential-based final of Heyd, Scuseria and Ernzerhof (also known as HSE) [Heyd03, Izmaylov06, Henderson09].

  • PBEh: 1998 revision of PBE [Ernzerhof98].

Correlation Functionals. The following correlation functionals are available, listed by their corresponding keyword component, all of which must be combined with the keyword for the desired exchange functional:

  • VWN: Vosko, Wilk, and Nusair 1980 correlation functional(III) fitting the RPA solution to the uniform electron gas, often referred to as Local Spin Density (LSD) correlation [Vosko80] (functional III in this article).

  • VWN5: Functional V from reference [Vosko80] which fits the Ceperly-Alder solution to the uniform electron gas (this is the functional recommended in [Vosko80]).

  • LYP: The correlation functional of Lee, Yang, and Parr, which includes both local and non-local terms [Lee88, Miehlich89].

  • PL (Perdew Local): The local (non-gradient corrected) functional of Perdew (1981) [Perdew81].

  • P86 (Perdew 86): The gradient corrections of Perdew, along with his 1981 local correlation functional [Perdew86].

  • PW91 (Perdew/Wang 91): Perdew and Wang’s 1991 gradient-corrected correlation functional [Perdew91, Perdew92, Perdew93a, Perdew96, Burke98].

  • B95 (Becke 95): Becke’s τ-dependent gradient-corrected correlation functional (defined as part of his one parameter hybrid functional [Becke96]).

  • PBE: The 1996 gradient-corrected correlation functional of Perdew, Burke and Ernzerhof [Perdew96a, Perdew97].

  • TPSS: The τ-dependent gradient-corrected functional of Tao, Perdew, Staroverov, and Scuseria [Tao03].

  • RevTPSS: The revised TPSS correlation functional of Perdew et. al. [Perdew09, Perdew11].

  • KCIS: The Krieger-Chen-Iafrate-Savin correlation functional [Rey98, Krieger99, Krieger01, Toulouse02].

  • BRC: Becke-Roussel correlation functional [Becke89a].

  • PKZB: The correlation part of the Perdew, Kurth, Zupan and Blaha functional [Perdew99].

Specifying Actual Functionals. Combine an exchange functional component keyword with the one for desired correlation functional. For example, the combination of the Becke exchange functional (B) and the LYP correlation functional is requested by the BLYP keyword. Similarly, SVWN requests the Slater exchange functional (S) and the VWN correlation functional, and is known in the literature by its synonym LSDA (Local Spin Density Approximation). LSDA is a synonym for SVWN. Some other software packages with DFT facilities use the equivalent of SVWN5 when “LSDA” is requested. Check the documentation carefully for all packages when making comparisons.

Correlation Functional Variations. The following correlation functionals combine local and non-local terms from different correlation functionals:

  • VP86: VWN5 local and P86 non-local correlation functional.

  • V5LYP: VWN5 local and LYP non-local correlation functional.

Standalone Functionals. The following functionals are self-contained and are not combined with any other functional keyword components:

  • VSXC: van Voorhis and Scuseria’s τ-dependent gradient-corrected correlation functional [VanVoorhis98].

  • HCTH/*: Handy’s family of functionals including gradient-corrected correlation [Hamprecht98, Boese00, Boese01]. HCTH refers to HCTH/407, HCTH93 to HCTH/93, HCTH147 to HCTH/147, and HCTH407 to HCTH/407. Note that the related HCTH/120 functional is not implemented.

  • tHCTH: The τ-dependent member of the HCTH family [Boese02]. See also tHCTHhyb below.

  • M06L: The pure functional of Truhlar and Zhao [Zhao06a]. See also M06 below.

  • B97D: Grimme’s functional including dispersion [Grimme06]. B97D3 requests the same but with Grimme's D3BJ dispersion [Grimme11].

  • SOGGA11 [Peverati11], M11L [Peverati12], MN12L [Peverati12c], and N12 [Peverati12b] request these pure functionals from the Truhlar group.

Hybrid Functionals. A number of hybrid functionals, which include a mixture of Hartree-Fock exchange with DFT exchange-correlation, are available via keywords:

  • Becke Three Parameter Hybrid Functionals. These functionals have the form devised by Becke in 1993 [Becke93a]:

    A*EXSlater+(1-A)*EXHF+B*ΔEXBecke+ECVWN+C*ΔECnon-local

    where A, B, and C are the constants determined by Becke via fitting to the G1 molecule set.

    There are several variations of this hybrid functional. B3LYP uses the non-local correlation provided by the LYP expression, and VWN functional III for local correlation (not functional V). Note that since LYP includes both local and non-local terms, the correlation functional used is actually:

    C*ECLYP+(1-C)*ECVWN

    In other words, VWN is used to provide the excess local correlation required, since LYP contains a local term essentially equivalent to VWN.

    B3P86 specifies the same functional with the non-local correlation provided by Perdew 86, and B3PW91 specifies this functional with the non-local correlation provided by Perdew/Wang 91.

  • Becke One Parameter Hybrid Functionals. The B1B95 keyword is used to specify Becke’s one-parameter hybrid functional as defined in the original paper [Becke96].

    The program also provides other, similar one parameter hybrid functionals [Becke96], as implemented by Adamo and Barone [Adamo97]. In one variation, B1LYP, the LYP correlation functional is used (as described for B3LYP above). Another version, mPW1PW91, uses Perdew-Wang exchange as modified by Adamo and Barone combined with PW91 correlation [Adamo98]; the mPW1LYP, mPW1PBE and mPW3PBE variations are available.

  • Becke’s 1998 revisions to B97 [Becke97, Schmider98]. The keyword is B98, and it implements equation 2c in reference [Schmider98].

  • Handy, Tozer and coworkers modification to B97: B971 [Hamprecht98].

  • Wilson, Bradley and Tozer’s modification to B97: B972 [Wilson01a].

  • The 1996 pure functional of Perdew, Burke and Ernzerhof [Perdew96a, Perdew97], as made into a hybrid by Adamo [Adamo99a]. The keyword is PBE1PBE. This functional uses 25% exchange and 75% correlation weighting, and is known in the literature as PBE0.

  • HSEH1PBE: The recommended version of the full Heyd-Scuseria-Ernzerhof functional, referred to as HSE06 in the literature [Heyd04, Heyd04a, Heyd05, Heyd06, Izmaylov06, Krukau06, Henderson09]. Two earlier forms are also available:

    • OHSE2PBE: the first form of this functional, referred to as HSE03 in the literature.

    • OHSE1PBE: The version of the functional prior to modification to support third derivatives.

  • PBEh1PBE: Hybrid using the 1998 revised form of PBE pure functional (exchange and correlation) [Ernzerhof98].

  • O3LYP: A three-parameter functional similar to B3LYP:

    A*EXLSD+(1-A)*EXHF+B*ΔEXOPTX+C*ΔECLYP+(1-C)ECVWN

    where A, B and C are as defined by Cohen and Handy in reference [Cohen01].

  • TPSSh: Hybrid functional using the TPSS functionals [Tao03].

  • BMK: Boese and Martin’s τ-dependent hybrid functional [Boese04].

  • M06: The hybrid functional of Truhlar and Zhao [Zhao08]. The M06HF [Zhao06b, Zhao06c] and M062X [Zhao08] variations are also available, as are the earlier M05 [Zhao05] and M052X [Zhao06].

  • X3LYP: Functional of Xu and Goddard [Xu04].

  • tHCTHhyb: Hybrid functional using the tHCTH functional [Boese02].

  • Half-and-half Functionals. Implements the following functionals, which are included for backward-compatibility only. Note that these are not the same as the “half-and-half” functionals proposed by Becke [Becke93].

    • BHandH: 0.5*EXHF + 0.5*EXLSDA + ECLYP

    • BHandHLYP: 0.5*EXHF + 0.5*EXLSDA + 0.5*ΔEXBecke88 + ECLYP

  • APFD requests the Austin-Frisch-Petersson functional with dispersion [Austin12], and APF requests the same functional without dispersion.

  • HISSbPBE requests the HISS functional [Henderson08].

  • M11 [Peverati11a], SOGGA11X [Peverati11b], N12SX [Peverati12a] and MN12SX [Peverati12a] request these hybrid functionals from the Truhlar group.

Functionals including dispersion.

  • The standalone keyword EmpiricalDispersion allows you to specify a dispersion scheme (see below).

  • The APFD functional is the Austin-Frisch-Petersson functional with dispersion [Austin12].

  • The B97D and B97D3 functionals are the B97 functional with Grimme's D2 and D3 dispersion schemes (see below).

  • The wB97xD functional uses a version of Grimme's D2 dispersion model (see below).

Long range corrected functionals. The non-Coulomb part of exchange functionals typically dies off too rapidly and becomes very inaccurate at large distances, making them unsuitable for modeling processes such as electron excitations to high orbitals. Various schemes have been devised to handle such cases. Gaussian 09 offers the following functionals which include long range corrections:

  • LC-wPBE: Long range-corrected version of wPBE [Vydrov06, Vydrov06a, Vydrov07].

  • CAM-B3LYP: Handy and coworkers’ long range corrected version of B3LYP using the Coulomb-attenuating method [Yanai04].

  • wB97XD: The latest functional from Head-Gordon and coworkers, which includes empirical dispersion [Chai08a]. The wB97 and wB97X [Chai08] variations are also available. These functionals also include long range corrections.

In addition, the prefix LC- may be added to any pure functional to apply the long correction of Hirao and coworkers [Iikura01]: e.g., LC-BLYP.

User-Defined Models. Gaussian 09 can use any model of the general form:

    P2EXHF + P1(P4EXSlater + P3ΔExnon-local) + P6EClocal + P5ΔECnon-local

The only available local exchange method is Slater (S), which should be used when only local exchange is desired. Any combinable non-local exchange functional and combinable correlation functional may be used (as listed previously).

The values of the six parameters are specified with various non-standard options to the program:

  • IOp(3/76=mmmmmnnnnn) sets P1 to mmmmm/10000 and P2 to nnnnn/10000. P1 is usually set to either 1.0 or 0.0, depending on whether an exchange functional is desired or not, and any scaling is accomplished using P3 and P4.

  • IOp(3/77=mmmmmnnnnn) sets P3 to mmmmm/10000 and P4 to nnnnn/10000.

  • IOp(3/78=mmmmmnnnnn) sets P5 to mmmmm/10000 and P6 to nnnnn/10000.

For example, IOp(3/76=1000005000) sets P1 to 1.0 and P2 to 0.5. Note that all values must be expressed using five digits, adding any necessary leading zeros.

Here is a route section specifying the functional corresponding to the B3LYP keyword:

#P BLYP IOp(3/76=1000002000) IOp(3/77=0720008000) IOp(3/78=0810010000)

The output file displays the values that are in use:

 IExCor=  402 DFT=T Ex=B+HF Corr=LYP ExCW=0 ScaHFX=  0.200000
 ScaDFX=  0.800000  0.720000  1.000000  0.810000 

where the value of ScaHFX is P2, and the sequence of values given for ScaDFX are P4, P3, P6 and P5.

EMPIRICAL DISPERSION

The EmpiricalDispersion keyword enables empirical dispersion. It takes the following options:

PFD
Add the Petersson-Frisch dispersion model from the APFD functional [Austin12].

GD2
Add the D2 version of Grimme’s dispersion [Grimme06]. The table below gives the list of functionals in Gaussian 09 for which GD2 parameters are defined. The functionals highlighted in bold include this dispersion model by default when the indicated keyword is specified (e.g., B2PLYPD). For the rest of the functionals, dispersion is requested with EmpiricalDispersion=GD2.

Functional S6 SR6
B97D 1.2500 1.1000
B2PLYPD 0.5500 1.1000
mPW2PLYPD 0.4000 1.1000
PBEPBE 0.7500 1.1000
BLYP 1.2000 1.1000
B3LYP 1.0500 1.1000
BP86 1.0500 1.1000
TPSSTPSS 1.0000 1.1000

The damping function used by this model also contains a D6 parameter with a fixed value of 6.0.

You can use this empirical dispersion method with other functionals by defining the values of the SR6 and S8 parameters (the value of SR6 is always 1.1). This is done using an environment variable with the name GAUSS_DFTD3_S6. The value of the environment variable sets the corresponding parameter to value/1,000,000. For example, the command:
     export GAUSS_DFTD3_S6=1200000
sets the value of S6 to 1200000/1000000=1.2000.

The wB97xD functional—specified as an independent keyword—uses a version of this dispersion model with values of S6 and SR6 of 1.0 and 1.1, respectively. This functional uses a similar damping function to that used by the GD3 model, with D6 and IA6 having fixed values of 6.0 and 12, respectively.

GD3
Add the D3 version of Grimme’s dispersion with the original D3 damping function [Grimme10]. The table below gives the list of functionals in Gaussian 09 for which GD3 parameters are defined. For the rest of the functionals, dispersion is requested with EmpiricalDispersion=GD3.

Functional S6 SR6 S8
B2PLYPD3 [Goerigk11] 0.6400 1.4270 1.0220
B97D3 1.0000 0.8920 0.9090
B3LYP 1.0000 1.2610 1.7030
BLYP 1.0000 1.0940 1.6820
PBE1PBE 1.0000 1.2870 0.9280
TPSSTPSS 1.0000 1.1660 1.1050
PBEPBE 1.0000 1.2170 0.7220
BP86 1.0000 1.1390 1.6830
BPBE 1.0000 1.0870 2.0330
B3PW91 1.0000 1.1760 1.7750
BMK 1.0000 1.9310 2.1680
CAM-B3LYP 1.0000 1.3780 1.2170
LC-wPBE 1.0000 1.3550 1.2790
M05 1.0000 1.3730 0.5950
M052X 1.0000 1.4170 0.0000
M06L 1.0000 1.5810 0.0000
M06 1.0000 1.3250 0.0000
M062X 1.0000 1.6190 0.0000
M06HF 1.0000 1.4460 0.0000

This model also use a SR8 parameter with a fixed value of 1.0. The damping function used by this model also contains D6, IA6, D8 and IA8 parameters with fixed values of 6.0, 14, 6.0, and 16, respectively.

You can use this empirical dispersion method with other functionals by defining the values of the SR6 and S8 parameters (the value of S6 is always 1.0). This is done using environment variables with names of the form GAUSS_DFTD3_param, where param is one of the parameter names. The value of the environment variable sets the corresponding parameter to value/1,000,000. For example, the command:
     export GAUSS_DFTD3_S8=1375000
sets the value of S8 to 1375000/1000000=1.375.

GD3BJ
Add the D3 version of Grimme’s dispersion with Becke-Johnson damping [Grimme11]. The table below gives the list of functionals in Gaussian 09 for which GD3 parameters are defined. The functionals highlighted in bold include this dispersion model by default when the indicated keyword is specified (e.g., B2PLYPD3). For the rest of the functionals, dispersion is requested with EmpiricalDispersion=GD3BJ.

Functional S6 S8 ABJ1 ABJ2
B2PLYPD3 [Goerigk11] 0.6400 0.9147 0.3065 5.0570
B97D3 1.0000 2.2609 0.5545 3.2297
B3LYP 1.0000 1.9889 0.3981 4.4211
BLYP 1.0000 2.6996 0.4298 4.2359
PBE1PBE 1.0000 1.2177 0.4145 4.8593
TPSSTPSS 1.0000 1.9435 0.4535 4.4752
PBEPBE 1.0000 0.7875 0.4289 4.4407
BP86 1.0000 3.2822 0.3946 4.8516
BPBE 1.0000 4.0728 0.4567 4.3908
B3PW91 1.0000 2.8524 0.4312 4.4693
BMK 1.0000 2.0860 0.1940 5.9197
CAM-B3LYP 1.0000 2.0674 0.3708 5.4743
LC-wPBE 1.0000 1.8541 0.3919 5.0897

You can use this empirical dispersion method with other functionals by defining the values of the S8, ABJ1 and ABJ2 parameters (the value of S6 is always 1.0). This is done using environment variables with names of the form GAUSS_DFTD3_param, where param is one of the parameter names. The value of the environment variable sets the corresponding parameter to value/1,000,000. For example, the command:
     export GAUSS_DFTD3_S8=2375000
sets the value of S8 to 2375000/1000000=2.375.

ACCURACY CONSIDERATIONS

A DFT calculation adds an additional step to each major phase of a Hartree-Fock calculation. This step is a numerical integration of the functional (or various derivatives of the functional). Thus in addition to the sources of numerical error in Hartree-Fock calculations (integral accuracy, SCF convergence, CPHF convergence), the accuracy of DFT calculations also depends on the number of points used in the numerical integration.

The “fine” integration grid (corresponding to Integral=FineGrid) is the default in Gaussian 09. This grid greatly enhances calculation accuracy at minimal additional cost. We do not recommend using any smaller grid in production DFT calculations. Note also that it is important to use the same grid for all calculations where you intend to compare energies (e.g., computing energy differences, heats of formation, and so on).

Larger grids are available when needed (e.g. tight geometry optimizations of certain kinds of systems). An alternate grid may be selected by including Integral(Grid=N) in the route section (see the discussion of the Integral keyword for details).

AVAILABILITY

Energies, analytic gradients, and analytic frequencies; ADMP calculations.

Third order properties such as hyperpolarizabilities and Raman intensities are not available for functionals for which third derivatives are not implemented: the exchange functionals Gill96, P (Perdew86), BRx, PKZB, TPSS, wPBEh and PBEh; the correlation functionals PKZB and TPSS; the hybrid functionals OHSE1PBE and OHSE2PBE.

RELATED KEYWORDS

IOp, Int=Grid, Stable, TD, DenFit, B2PLYP, mPW2LYP

EXAMPLES

The energy is reported in DFT calculations in a form similar to that of Hartree-Fock calculations. Here is the energy output from a B3LYP calculation:

 SCF Done:  E(RB+HF-LYP) =  -75.3197099428     A.U. after    5 cycles

The item in parentheses following the E denotes the method used to obtain the energy. The output from a BLYP calculation is labeled similarly:

 SCF Done:  E(RB-LYP) =  -75.2867073414     A.U. after    5 cycles

QUICK REFERENCE OF AVAILABLE FUNCTIONALS

COMBINATION FORMS    STAND ALONE FUNCTIONALS
EXCHANGE RANGE-SEPARATED
EXCHANGECORRELATION    ONLYPUREHYBRID HYBRID
SVWN    HFSVSXCB3LYP HSEH1PBE
XAVWN5    XAlphaHCTHB3P86 OHSE2PBE
BLYP    HFBHCTH93B3PW91 OHSE1PBE
PW91PL     HCTH147B1B95 wB97XD
mPWP86     HCTH407mPW1PW91 wB97
G96PW91     tHCTHmPW1LYP wB97X
PBEB95     M06LmPW1PBE LC-wPBE
OPBE     B97DmPW3PBE CAM-B3LYP
TPSSTPSS     B97D3 B98 HISSbPBE
BRxKCIS     SOGGA11 B971 M11
PKZBBRC     M11LB972 N12SX
wPBEhPKZB     N12 PBE1PBE MN12SX
PBEhVP86     MN12L B1LYP  
V5LYP     O3LYP  
     BHandH  
LONG RANGE      BHandHLYP  
CORRECTION      BMK  
LC-      M06  
     M06HF  
     M062X  
     tHCTHhyb  
     APFD  
     APF  
     SOGGA11X  
     PBEh1PBE  
     TPSSh  
     X3LYP  

 


Last update: 2 August 2013