Density Functional (DFT) Methods

DESCRIPTION

Gaussian 09 offers a wide variety of Density Functional Theory (DFT) [Hohenberg64, Kohn65, Parr89, Salahub89] models (see also [Salahub89, 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:

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:

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<PK(P)> 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:

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.

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, 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].

• 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].

• 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].

• 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].

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*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.

• 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, Krukau06]. Two earlier forms are also available:

  • HSE2PBE: the first form of this functional, referred to as HSE03 in the literature.

  • HSE1PBE: 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*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].

• 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 [Zhao08] and M062X [Zhao06b] variations are also available.

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

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

  • 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

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 [Tawada04, 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:

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.

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 HSE1PBE and HSE2PBE.

RELATED KEYWORDS

IOp, Int=Grid, Stable, TD, DenFit

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

Last updated on: 10 May 2009