Basis Sets

Most methods require a basis set be specified; if no basis set keyword is included in the route section, then the STO-3G basis will be used. The exceptions consist of a few methods for which the basis set is defined as an integral part of the method; they are listed below:

  • All semi-empirical methods, including ZIndo for excited states.
  • All molecular mechanics methods.
  • Compound model chemistries: all Gn, CBS and W1 methods.

The following basis sets are stored internally in the Gaussian 09 program (see references cited for full descriptions), listed below by their corresponding Gaussian 09 keyword (with two exceptions):

  • STO-3G [Hehre69, Collins76]

  • 3-21G [Binkley80a, Gordon82, Pietro82, Dobbs86, Dobbs87, Dobbs87a]

  • 6-21G [Binkley80a, Gordon82]

  • 4-31G [Ditchfield71, Hehre72, Hariharan74, Gordon80]

  • 6-31G [Ditchfield71, Hehre72, Hariharan73, Hariharan74, Gordon80, Francl82, Binning90, Blaudeau97, Rassolov98, Rassolov01]

  • 6-31G†: Gaussian 09 also includes the 6-31G† and 6-31G‡ basis sets of George Petersson and coworkers, defined as part of the Complete Basis Set methods [Petersson88, Petersson91]. These are accessed via the 6-31G(d') and 6-31G(d',p') keywords, to which single or double diffuse functions may also be added; f functions may also be added: e.g., 6-31G(d'f), and so on.

  • 6-311G: Specifies the 6-311G basis for first-row atoms and the McLean-Chandler (12s,9p) → (621111,52111) basis sets for second-row atoms [McLean80, Raghavachari80b] (note that the basis sets for P, S, and Cl are those called negative ion basis sets by McLean and Chandler; these were deemed to give better results for neutral molecules as well), the basis set of Blaudeau and coworkers for Ca and K [Blaudeau97], the Wachters-Hay [Wachters70, Hay77] all electron basis set for the first transition row, using the scaling factors of Raghavachari and Trucks [Raghavachari89], and the 6-311G basis set of McGrath, Curtiss and coworkers for the other elements in the third row [Binning90, McGrath91, Curtiss95]. Note that Raghavachari and Trucks recommend both scaling and including diffuse functions when using the Wachters-Hay basis set for first transition row elements; the 6-311+G form must be specified to include the diffuse functions. MC-311G is a synonym for 6-311G.

  • D95V: Dunning/Huzinaga valence double-zeta [Dunning77].

  • D95: Dunning/Huzinaga full double zeta [Dunning77].

  • SHC: D95V on first row, Goddard/Smedley ECP on second row [Dunning77, Rappe81]. Also known as SEC.

  • CEP-4G: Stevens/Basch/Krauss ECP minimal basis [Stevens84, Stevens92, Cundari93].

  • CEP-31G: Stevens/Basch/Krauss ECP split valance [Stevens84, Stevens92, Cundari93].

  • CEP-121G: Stevens/Basch/Krauss ECP triple-split basis [Stevens84, Stevens92, Cundari93].

    Note that there is only one CEP basis set defined beyond the second row, and all three keywords are equivalent for these atoms.

  • LanL2MB: STO-3G [Hehre69, Collins76] on first row, Los Alamos ECP plus MBS on Na-La, Hf-Bi [Hay85, Wadt85, Hay85a].

  • LanL2DZ: D95V on first row [Dunning77], Los Alamos ECP plus DZ on Na-La, Hf-Bi [Hay85, Wadt85, Hay85a].

  • SDD: D95 up to Ar [Dunning77] and Stuttgart/Dresden ECPs on the remainder of the periodic table [Fuentealba82, Szentpaly82, Fuentealba83, Stoll84, Fuentealba85, Wedig86, Dolg87, Igel-Mann88, Dolg89, Schwerdtfeger89, Dolg89a, Andrae90, Dolg91, Kaupp91, Kuechle91, Dolg92, Bergner93, Dolg93, Haeussermann93, Dolg93a, Kuechle94, Nicklass95, Leininger96, Cao01, Cao02]. The SDD, SHF, SDF, MHF, MDF, MWB forms may be used to specify these basis sets/potentials within Gen basis input. Note that the number of core electrons must be specified following the form (e.g., MDF28 for the MDF potential replacing 28 core electrons). OldSDD requests the previous default.

  • SDDAll: Selects Stuttgart potentials for Z > 2.

  • cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pV5Z, cc-pV6Z: Dunning’s correlation consistent basis sets [Dunning89, Kendall92, Woon93, Peterson94, Wilson96] (double, triple, quadruple, quintuple-zeta and sextuple-zeta, respectively). These basis sets have had redundant functions removed and have been rotated [Davidson96] in order to increase computational efficiency.

    These basis sets include polarization functions by definition. The following table lists the valence polarization functions present for the various atoms included in these basis sets:

    Atoms    cc-pVDZ    cc-pVTZ    cc-pVQZ    cc-pV5Z    cc-pV6Z
    H    2s,1p    3s,2p,1d    4s,3p,2d,1f    5s,4p,3d,2f,1g    6s,5p,4d,3f,2g,1h
    He    2s,1p    3s,2p,1d    4s,3p,2d,1f    5s,4p,3d,2f,1g    not available
    Li-Be    3s,2p,1d    4s,3p,2d,1f    5s,4p,3d,2f,1g    6s,5p,4d,3f,2g,1h    not available
    B-Ne    3s,2p,1d    4s,3p,2d,1f    5s,4p,3d,2f,1g    6s,5p,4d,3f,2g,1h    7s,6p,5d,4f,3g,2h,1i
    Na-Ar    4s,3p,1d    5s,4p,2d,1f    6s,5p,3d,2f,1g    7s,6p,4d,3f,2g,1h    not available
    Ca    5s,4p,2d    6s,5p,3d,1f    7s,6p,4d,2f,1g    8s,7p,5d,3f,2g,1h    not available
    Sc-Zn    6s,5p,3d, 1f    7s,6p,4d,2f,1g    8s,7p,5d,3f,2g,1h    9s,8p,6d,4f,3g,2h,1i    not available
    Ga-Kr    5s,4p,2d    6s,5p,3d,1f    7s,6p,4d,2f,1g    8s,7p,5d,3f,2g,1h    not available

    These basis sets may be augmented with diffuse functions by adding the AUG- prefix to the basis set keyword (rather than using the + and ++ notation—see below).

  • Basis sets from of Ahlrichs and coworkers: the SV, SVP, TZV, TZVP keywords refer to the initial formations of the split valence and triple zeta basis sets from this group [Schaefer92, Schaefer94]. The keywords Def2SV, Def2SVP, Def2SVPP, Def2TZV, Def2TZVP, Def2TZVPP, Def2QZV, Def2QZVP, Def2QZVPP, and QZVP request the newer formulations [Weigend05, Weigend06].

  • MIDI! of Truhlar and coworkers [Easton96]. The MidiX keyword is used to request this basis set.

  • EPR-II and EPR-III: The basis sets of Barone [Barone96a] which are optimized for the computation of hyperfine coupling constants by DFT methods (particularly B3LYP). EPR-II is a double zeta basis set with a single set of polarization functions and an enhanced s part: (6,1)/[4,1] for H and (10,5,1)/[6,2,1] for B to F. EPR-III is a triple-zeta basis set including diffuse functions, double d-polarizations and a single set of f-polarization functions. Also in this case the s-part is improved to better describe the nuclear region: (6,2)/[4,2] for H and (11,7,2,1)/[7,4,2,1] for B to F.

  • UGBS: The universal Gaussian basis set of de Castro, Jorge and coworkers [Silver78, Silver78a, Mohallem86, Mohallem87, daCosta87, daSilva89, Jorge97, Jorge97a, deCastro98]. Additional polarization functions may be added by including a suffix to this keyword:


    where n is an integer indicating whether to add 1, 2 or 3 polarization functions for each function in the normal UGBS basis set. The second item is a code letter indicating which function should be augmented polarization functions: P adds them to all functions, V adds them to all valence functions, and O requests the scheme used in Gaussian 03 (see below). For example, the UGBS1P keyword requests this basis set with one additional polarization function to all orbitals, and UGBS2V adds two additional polarization function to all valence orbitals.

    The O suffix adds the same functions as the UGBSnP keywords in Gaussian 03. UGBS1O adds a p function for each s, a d function for each p, and so on; UGBS2O adds a p and d function for each s, a d and f function for each p, and UGBS3O adds a p, d and f for each s, etc.

    Diffuse functions may be added as usual with + or ++; the first of these may be specified as 2+ to add two diffuse functions for heavy atoms.

  • MTSmall of Martin and de Oliveira, defined as part of their W1 method (see the W1U keyword) [Martin99].

  • The DGDZVP, DGDZVP2 and DGTZVP basis sets used in DGauss [Godbout92, Sosa92].

  • CBSB7: Selects the 6-311G(2d,d,p) basis set used by CBS-QB3 high accuracy energy method [Montgomery99]. The notation specifies two additional d polarization functions on second rows atoms, one d function on first row atoms and a p function on hydrogens (note that this three-field polarization function syntax is not supported by Gaussian 09).

Adding Polarization and Diffuse Functions

Single first polarization functions can also be requested using the usual * or ** notation. Note that (d,p) and ** are synonymous—6-31G** is equivalent to 6-31G(d,p), for example—and that the 3-21G* basis set has polarization functions on second row atoms only. The + and ++ diffuse functions [Clark83] are available with some basis sets, as are multiple polarization functions [Frisch84]. The keyword syntax is best illustrated by example: 6-31+G(3df,2p) designates the 6-31G basis set supplemented by diffuse functions, 3 sets of d functions and one set of f functions on heavy atoms, and supplemented by 2 sets of p functions on hydrogens.

When the AUG- prefix is used to add diffuse functions to the cc-pV*Z basis sets, one diffuse function of each function type in use for a given atom is added [Kendall92, Woon93]. For example, the AUG-cc-pVTZ basis places one s, one d, and one p diffuse functions on hydrogen atoms, and one d, one p, one d, and one f diffuse functions on B through Ne and Al through Ar.

There are several options for augmenting the cc-pV*Z basis sets with diffuse functions:

  • spAug-cc-pV*Z augments with s and p functions only, including s functions on H and He.

  • dAug-cc-pV*Z augments with 2 shells of each angular momentum instead of one.

  • Truhlar’s “calendar” basis set variations [Papajak11] are available. The naming of this series of basis sets come from the fact that the cc-pV*Z basis sets with added polarization functions are known as Aug-cc-pV*Z. Truhlar noted that “Aug” is also an abbreviation for the month of August in English, so he proposed new augmentation schemes for the cc-pV*Z basis sets, also named after months of the year. They are constructed by removing diffuse functions from the Aug basis sets. For example, the Jul-cc-pV*Z basis sets remove the diffuse function from H and He from Aug-cc-pV*Z. Jun-cc-pV*Z also removes the highest angular momentum diffuse function from all other atoms, May-cc-pV*Z removes the two highest angular momentum functions, and Apr-cc-pV*Z removes the three highest angular momentum functions.

    Nevertheless, by default, at least s and p diffuse functions are always included in these basis sets. This serves to avoid some inherent inconsistencies, but it differs from Truhlar and coworkers’ original definitions. Use the forms TJul, TJun, and so on to specify the original versions where the limit is applied unconditionally: e.g., TMay-cc-pVDZ includes only a diffuse s function on Cl but both diffuse s and p functions on Fe and Br, while May-cc-pVDZ has diffuse s and p functions on all of these atoms.

Adding a single polarization function to 6-311G (i.e. 6-311G(d)) will result in one d function for first and second row atoms and one f function for first transition row atoms, since d functions are already present for the valence electrons in the latter. Similarly, adding a diffuse function to the 6-311G basis set will produce one s, one p, and one d diffuse functions for third-row atoms.

When a frozen core calculation is done using the D95 basis, both the occupied core orbitals and the corresponding virtual orbitals are frozen. Thus while a D95** calculation on water has 26 basis functions, and a 6-31G** calculation on the same system has 25 functions, there will be 24 orbitals used in a frozen core post-SCF calculation involving either basis set.

The following table lists polarization and diffuse function availability and the range of applicability for each built-in basis set in Gaussian 09:

Basis Set    Applies to    Polarization Functions    Diffuse Functions
3-21G    H-Xe         +
6-21G    H-Cl    * or **    
4-31G    H-Ne    * or **    
6-31G    H-Kr    through (3df,3pd)    +,++
6-311G    H-Kr    through (3df,3pd)    +,++
D95    H-Cl except Na and Mg    through (3df,3pd)    +,++
D95V    H-Ne    (d) or (d,p)    +,++
SHC    H-Cl    *    
CEP-4G    H-Rn    * (Li-Ar only)    
CEP-31G    H-Rn    * (Li-Ar only)    
CEP-121G    H-Rn    * (Li-Ar only)    
LanL2MB    H-La, Hf-Bi         
LanL2DZ    H, Li-La, Hf-Bi         
SDD, SDDAll    all but Fr and Ra         
cc-pVDZ    H-Ar, Ca-Kr    included in definition    added via AUG- prefix (H-Ar, Sc-Kr)
cc-pVTZ    H-Ar, Ca-Kr    included in definition    added via AUG- prefix (H-Ar, Sc-Kr)
cc-pVQZ    H-Ar, Ca-Kr    included in definition    added via AUG- prefix(H-Ar, Sc-Kr)
cc-pV5Z    H-Ar, Ca-Kr    included in definition    added via AUG- prefix (H-Na, Al-Ar Sc-Kr)
cc-pV6Z    H, B-Ne    included in definition    added via AUG- prefix (H, B-O)
SV    H-Kr         
SVP   H-Kr   included in definition  
TZV and TZVP    H-Kr    included in definition    
QZVP and Def2*     H-La, Hf-Rn    included in definition    
MidiX    H, C-F, S-Cl, I, Br    included in definition    
EPR-II, EPR-III    H, B, C, N, O, F    included in definition    
UGBS     H-Cn    UGBS(1,2,3)P    +,++,2+,2++
MTSmall    H-Ar         
DGDZVP    H-Xe         
DGDZVP2    H-F, Al-Ar, Sc-Zn         
DGTZVP    H, C-F, Al-Ar         
CBSB7    H-Kr    included in definition    +,++

STO-3G and 3-21G accept a * suffix, but this does not actually add any polarization functions.

Additional Basis Set-Related Keywords

The following additional keywords are useful in conjunction with these basis set keywords:

  • 5D and 6D: Use 5 or 6 d functions (pure vs. Cartesian d functions), respectively.

  • 7F and 10F: Use 7 or 10 f functions (pure vs. Cartesian f functions), respectively. These keywords also apply to all higher functions (g and beyond).

Other basis sets may also be input to the program using the ExtraBasis and Gen keywords. The ChkBasis keyword indicates that the basis set is to read from the checkpoint file (defined via the %Chk command). See the individual descriptions of these keywords later in this chapter for details.

Issues Arising from Pure vs. Cartesian Basis Functions

Gaussian users should be aware of the following points concerning pure vs. Cartesian basis functions:

  • All of the built-in basis sets use pure f functions. Most also use pure d functions; the exceptions are 3-21G, 6-21G, 4-31G, 6-31G, 6-31G†, 6-31G‡, CEP-31G, D95 and D95V. The preceding keywords may be used to override the default pure/Cartesian setting. Note that basis functions are generally converted to the other type automatically when necessary, for example, when a wavefunction is read from the checkpoint file for use in a calculation using a basis consisting of the other type [Schlegel95a].

  • Within a job, all d functions must be 5D or 6D, and all f and higher functions must be pure or Cartesian.

  • When using the ExtraBasis, Gen and GenECP keywords, the basis set explicitly specified in the route section always determines the default form of the basis functions (for Gen, these are 5D and 7F). For example, if you use a general basis set taking some functions from the 3-21G and 6-31G basis sets, pure functions will be used unless you explicitly specify 6D in the route section in addition to Gen. Similarly, if you add basis functions for a transition metal from the 6-311G(d) basis set via ExtraBasis to a job that specifies the 6-31G(d) basis set in the route section, Cartesian d functions will be used. Likewise, if you want to add basis functions for Xe from the 3-21G basis set to the 6-311 basis set via the ExtraBasis keyword, the Xe basis functions will be pure functions.

Density Fitting Basis Sets

Gaussian 09 provides the density fitting approximation for pure DFT calculations [Dunlap83, Dunlap00]. This approach expands the density in a set of atom-centered functions when computing the Coulomb interaction instead of computing all of the two-electron integrals. It provides significant performance gains for pure DFT calculations on medium sized systems too small to take advantage of the linear scaling algorithms without a significant degradation in the accuracy of predicted structures, relative energies and molecular properties. Gaussian 09 can generate an appropriate fitting basis automatically from the AO basis, or you may select one of the built-in fitting sets.

The desired fitting basis set is specified as a third component of the model chemistry, as in this example:


Note that slashes must be used as separator characters between the method, basis set, and fitting set when a density fitting basis set is specified.

The following fitting sets keywords are available in Gaussian 09:

  • DGA1 and DGA2 [Godbout92, Sosa92]. DGA1 is available for H through Xe, and DGA2 is available for H, He and B through Ne.
  • SVPFit [Eichkorn95, Eichkorn97] and Def2SV [Weigend05], corresponding to the SVP basis set.
  • TZVPFit [Eichkorn95, Eichkorn97] and Def2TZV [Weigend05], corresponding to the TZVP basis set.
  • The W06 fitting set of Ahlrichs and coworkers [Weigend05, Weigend06], designed for use with the Def2SVP, Def2TZVP and QZVP basis sets.
  • Fit: Select the fitting set corresponding to the specified basis set. If there is no such fitting set, an error results.
  • NoFit: Turn off fitting set use for this calculation. This keyword is used to override the DensityFit keyword with a Default.Route file.
  • Auto: Generate a fitting set automatically (see below).

Density fitting sets can be generated automatically from the AO primitives within the basis set. This is requested using the Auto fitting set keyword. The program automatically truncates the set at a reasonable angular momentum: the default is Max(MaxTyp+1,2*MaxVal), where MaxTyp is the highest angular momentum in the AO basis, and MaxVal is the highest valence angular momentum. You can request that all generated functions be used with Auto=All, or request those up to a certain level with Auto=N, where N is the maximum angular momentum retained in the fitting functions. Finally, the PAuto form generates all products of AO functions on one center instead of just squares of the AO primitives, but this is typically more functions than are needed.

By default, no fitting set is used. Density fitting basis sets may be augmented with the ExtraDensityBasis keyword, defined in full with the Gen keyword, and optionally retrieved from the checkpoint file (use ChkBasis to do so). The options to the DensityFit keyword can be used to control some aspects of the fitting set used within calculations.

Density fitting can be made the default for jobs using pure DFT functionals by adding the DenFit keyword to the route section (-#-) line in the Default.Route file. Fitting is faster than doing the Coulomb term exactly for systems up to several hundred atoms (depending on basis set), but is slower than exact Coulomb using linear scaling techniques (which are turned on automatically with exact Coulomb) for very large systems.


Last update: 24 February 2016