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:

    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.

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:

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:

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

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

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:

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:

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

COMBINATION FORMS    STAND ALONE FUNCTIONALS
EXCHANGE
EXCHANGECORRELATION    ONLYPUREHYBRID
SVWN    HFSVSXCB3LYP
XAVWN5    XAlphaHCTHB3P86
BLYP    HFBHCTH93B3PW91
PW91PL     HCTH147B1B95
mPWP86     HCTH407mPW1PW91
G96PW91     tHCTHmPW1LYP
PBEB95     M06LmPW1PBE
OPBE     B97DmPW3PBE
TPSSTPSS     B98
BRxKCIS     B971
PKZBBRC     B972
wPBEhPKZB     PBE1PBE
PBEhVP86     B1LYP
V5LYP     O3LYP
     BHandH
LONG RANGE      BHandHLYP
CORRECTION      BMK
LC-      M06
     M06HF
     M062X
     tHCTHhyb
     HSEh1PBE
     HSE2PBE
     HSEhPBE
     PBEh1PBE
     wB97XD
     wB97
     wB97X
     TPSSh
     X3LYP
     LC-wPBE
     CAM-B3LYP

Last updated on: 9 Jun 2009