Density Functional (DFT) Methods

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

E_HF = V + 〈hP〉 + 1/2〈PJ(P)〉 - 1/2〈PK(P)〉

where the terms have the following meanings:

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:

E_KS = V + 〈hP〉 + 1/2〈PJ(P)〉 + E_X[P] + E_C[P]

where E_X[P] is the exchange functional, and E_C[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 E_X[P] given by the exchange integral -1/2 and E_C=0. The functionals normally used in density functional theory are integrals of some function of the density and possibly the density gradient:

E_X[P] = ∫f(ρ_α(r),ρ_β(r),∇ρ_α(r),∇ρ_β(r))dr

where the methods differ in which function f is used for E_X and which (if any) f is used for E_C. 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.

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*E_X^Slater+(1-A)*E_X^HF+B*ΔE_X^Becke+E_C^VWN+C*ΔE_C^non-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*E_C^LYP+(1-C)*E_C^VWN

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.

O3LYP is a three-parameter functional similar to B3LYP:

A*E_X^LSD+(1-A)*E_X^HF+B*ΔE_X^OPTX+C*ΔE_C^LYP+(1-C)E_C^VWN

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

Functionals Including Dispersion

• APFD requests the Austin-Frisch-Petersson functional with dispersion [Austin12], and APF requests the same functional without dispersion.
• The wB97XD functional uses a version of Grimme’s D2 dispersion model.

The standalone keyword EmpiricalDispersion also allows you to specify a dispersion scheme with various functionals.

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 16 offers the following functionals which include long-range corrections:

• LC-wHPBE: Recommended version [Henderson09] of the long-range-corrected ωPBE functional [Vydrov06, Vydrov06a, Vydrov07]. LC-wPBE requests the original version.
• 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 most pure functionals to apply the long correction of Hirao and coworkers [Iikura01]: e.g., LC-BLYP.

Other Hybrid Functionals

Functionals from the Truhlar Group

• MN15 requests the MN15 [Yu16] functional.
• M11 [Peverati11a], SOGGA11X [Peverati11b], N12SX [Peverati12a], and MN12SX [Peverati12a] request these hybrid functionals from the Truhlar group.
• PW6B95 and PW6B95D3 [Zhao05a].
• M08HX: The M08-HX functional [Zhao08a].
• M06 hybrid functional of Truhlar and Zhao [Zhao08]. The M06HF [Zhao06b, Zhao06c] and M062X [Zhao08] variations are also available.
• M05 [Zhao05] and M052X [Zhao06].

Functionals Employing PBE Correlation

• The 1996 pure functional of Perdew, Burke and Ernzerhof [Perdew96a, Perdew97] as made into a hybrid functional by Adamo [Adamo99a]. The keyword is PBE1PBE. This functional uses 25% exact exchange and 75% DFT exchange. It is known in the literature as PBE0 [Adamo99a] and as the PBE hybrid [Ernzerhof99].
• HSEH1PBE: The recommended version of the full Heyd-Scuseria-Ernzerhof functional, referred to as HSE06 in the literature [Heyd04, Heyd04a, Heyd05, Heyd06, Henderson09, Izmaylov06, Krukau06].
• OHSE2PBE: The initial form of the HS06 functional, referred to as HSE03 in the literature.
• OHSE1PBE: The version of the HS06 functional prior to modification to support third derivatives.
• PBEh1PBE: Hybrid using the 1998 revised form of PBE pure functional (exchange and correlation) [Ernzerhof98].

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 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.

Revisions to B97

• Becke’s 1998 revisions to B97 [Becke97, Schmider98]. The keyword is B98, and it implements fit 2c in reference [Schmider98].
• Handy, Tozer and coworkers modification to B97: B971 [Hamprecht98].
• Wilson, Bradley and Tozer’s modification to B97: B972 [Wilson01a].

Functionals with τ-Dependent Gradient-Corrected Correlation

• TPSSh: Hybrid functional using the TPSS functionals [Tao03, Staroverov03].
• tHCTHhyb: Hybrid functional using the tHCTH functional [Boese02].
• BMK: Boese and Martin’s τ-dependent 2004 hybrid functional [Boese04].

Older Functionals

• HISSbPBE requests the HISS functional [Henderson08].
• X3LYP: Functional of Xu and Goddard [Xu04].

Half-and-Half Functionals

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*E_X^HF + 0.5*E_X^LSDA + E_C^LYP
• BHandHLYP: 0.5*E_X^HF + 0.5*E_X^LSDA + 0.5*ΔE_X^Becke88 + E_C^LYP

User-Defined Hybrid Models

Gaussian 16 can use any model of the general form:

P₂E_X^HF + P₁(P₄E_X^Slater + P₃ΔE_x^non-local) + P₆E_C^local + P₅ΔE_C^non-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.

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. In order to specify a pure functional, 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.

Exchange Functionals

The following exchange functionals are available in Gaussian 16. Unless otherwise indicated, these exchange functionals must be combined with a correlation functional in order to produce a usable method.

• S: The Slater exchange, ρ^4/3 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, ρ^4/3 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].

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 Pure Functionals

The following pure 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.
• B97D: Grimme’s functional including dispersion [Grimme06]. B97D3 requests the same but with Grimme’s D3BJ dispersion [Grimme11].
• M06L [Zhao06a], SOGGA11 [Peverati11], M11L [Peverati12], MN12L [Peverati12c] N12 [Peverati12b] and MN15L [Yu16a] request these pure functionals from the Truhlar group.

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 16 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.

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 via the IOps(3/174,176) (SR6 should be 1.1).

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 16 for which GD3 parameters are defined. For these functionals, dispersion is requested with EmpiricalDispersion=GD3.

This model also uses an 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 via the IOps(3/174-176) (S6 should be 1.0).

GD3BJ

Add the D3 version of Grimme’s dispersion with Becke-Johnson damping [Grimme11]. The table below gives the list of functionals in Gaussian 16 for which GD3BJ 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.

You can use this empirical dispersion method with other functionals via the IOps(3/174-178) (S6 should be 1.0).

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 G96, P86, PKZB, wPBEh and PBEh; the correlation functional PKZB; the hybrid functionals OHSE1PBE and OHSE2PBE.

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