Gaussian 03 Online Manual
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Density Functional (DFT) MethodsGaussian 03 offers a wide variety of Density Functional Theory (DFT) [75,76,448,449] models (see also [448,450,451,452,453,454,455,456,457,458,459,460,461] for discussions of DFT methods and applications). Energies [78], analytic gradients, and true analytic frequencies [197,198,199] are available for all DFT models. The same optimum memory sizes given by freqmem are recommended for the more general 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 here for details. The next subsection presents a very brief overview of the DFT approach. Following this, the specific functionals available in Gaussian 03 are given. The final subsection surveys considerations related to accuracy in DFT calculations. Note: Polarizability derivatives (Raman intensities) and hyperpolarizabilities are not computed by default during DFT frequency calculations. Use Freq=Raman to request them. BACKGROUNDIn 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: 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 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 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. KEYWORDS FOR DFT METHODSNames 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 03:
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 correlation functionals are available, listed by their corresponding keyword component:
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. 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. Three hybrid functionals, which include a mixture of Hartree-Fock exchange with DFT exchange-correlation, are available via keywords:
User-Defined Models. Gaussian 03 can use any model of the general form: P_{2}E_{X}^{HF} + P_{1}(P_{4}E_{X}^{Slater} + P_{3}ΔE_{x}^{non-local}) + P_{6}E_{C}^{local} + P_{5}ΔE_{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). You specify the values of the six parameters with various non-standard options to the program:
For example, IOp(3/76=1000005000) sets P_{1} to 1.0 and P_{2} 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: # BLYP IOp(3/76=1000002000) IOp(3/77=0720008000) IOp(3/78=0810010000) ACCURACY CONSIDERATIONSA 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 Integral=FineGrid) is the default in Gaussian 03. 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 Integral=(Grid=N) in the route section (see the discussion of the Integral keyword for details). Energies, analytic gradients, and analytic frequencies; ADMP calculations. IOp, Int=Grid, Stable, TD, DenFit 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 |