Density Functional Methods (DFT) Keywords
Last Update: 6/26/2001

Gaussian 98 offers a wide variety of Density Functional Theory (DFT) [62-63,220-221] models (see also [220,223,224,225,226,227,228,229,230,231,232,233,237] for discussions of DFT methods and applications). Energies [65], analytic gradients, and true analytic frequencies [112-113] 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.

The next subsection presents a very brief overview of the DFT approach. Following this, the specific functionals available in Gaussian 98 are given. The final subsections discuss considerations related to accuracy and stability in DFT calculations.

Note: Polarizability derivatives (Roman intensities) and hyperpolarizabilities are not computed during DFT frequency calculations.

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

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

where: V is the nuclear repulsion energy, P is the density matrix, <hP> is the one-electron (kinetic plus potential) energy, 1/2<PJ(P)> is the classical coulomb repulsion of the electrons, and -1/2<PK(P)> is the exchange energy resulting from the quantum (fermion) nature of electrons.

In density functional theory, 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 exchange energy and the electron correlation which is omitted from 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.

Hartree-Fock theory is really 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 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 98 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.

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. There are following exchange functionals available in Gaussian 98:

The combination forms are used when one of these exchange functionals is used in combination with a correlation functional (see below).

Correlation Functionals. The following functionals are available:

All of the keywords for these correlation functionals must be combined with the keyword for the desired exchange functional. For example, BLYP requests the Becke exchange functional and the LYP correlation functional. SVWN requests the Slater exchange 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.

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

User-Defined Models. Gaussian 98 can use any model of the general form:

P₂E_x^HF + P₁(P₄E_x^Slater + P₃DE_x^non-local) + P₆E_c^local + P₅DE_c^non-local

Currently, the only available local exchange method is Slater, (S),which should be used when only local exchange is desired. The available non-local exchange corrections (which include Slater local exchange) are Becke (B), Perdew-Wang (PW91), Barone's modified PW91 (MPW) and Gill 96 (G96). One of these prefixes should be combined with the available correlation functional suffixes given in the following table:

You specify the values of the six parameters with various non-standard options to the program:

• IOp(5/45=mmmmnnnn) sets P₁ to mmmm/1000 and P₂ to nnnn/1000. P₁ is usually set to either 0.0 or 1.0, depending on whether an exchange functional is desired or not, and any scaling is accomplished using P₃ and P₄.

• IOp(5/46=mmmmnnnn) sets P₃ to mmmm/1000 and P₄ to nnnn/1000.

• IOp(5/47=mmmmnnnn) sets P₅ tommmm/1000 and P₆ to nnnn/1000.

For example, IOp(5/45=10000500) sets P₁ to 1.0 and P₂ to 0.5. Note that all values must be expressed using four digits, adding any necessary leading zeros.

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

# BLYP IOp(5/45=10000200) IOp(5/46=07200800) IOp(5/47=08101000)

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 number of points used in the numerical integration.

The "fine" integration grid (corresponding to Int=FineGrid) is the default in Gaussian 98. 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 optimization of certain kinds of systems). An alternate grid may be selected by including Int(Grid=N) in the route section (see the discussion of the Integral keyword for details).

DFT single point energy calculations involving basis sets which include diffuse functions should always use the SCF=Tight keyword to request tight SCF convergence criteria.

The Kohn-Sham equations for the optimal DFT density are similar to the Hartree-Fock equations, and for both cases it is possible to have SCF convergence problems and/or to converge to a saddle point in wavefunction space, rather than a minimum.

In general, the convergence properties of the hybrid methods are similar to Hartree-Fock: most cases converge readily with the standard methods. Pure DFT methods (i.e., without any Hartree-Fock exchange) tend to give small HOMO-LUMO gaps, and consequently convergence problems are much more frequent than for Hartree-Fock or hybrid methods. We recommend the following techniques for difficult DFT cases:

• Level shifting, using the SCF=VShift keyword, is useful for many problem cases. Note that you need to make sure that the final wavefunction respresents the desired electronic state, as this can change with level shifting.

• Use SCF=QC if the SCF procedure fails even with level shifting.

Using either method, you will want to use stability testing (Stable=Opt) to confirm that the solution is a local minima in wavefunction space [85-86]. See the discussion of the Stable keyword for details.

Energies, analytic gradients, and analytic frequencies.

IOp, Int=Grid, SCF=(VShift,QC), Stable

The energy is reported in DFT calculations in a form similar to that of Hartree-Fock calculations. Here is the energy output from a Becke3LYP 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

Return to TOC