Polarization functions for double zeta basis sets


            To further increase the flexibility of the orbital description, double zeta basis sets are frequently augmented with basis functions of higher angular momentum. A typical first step consists of the addition of a set of d-type functions to the basis sets of those atoms, which have occupied s- and p-shells in their electronic ground states. For hydrogen, this corresponds to the addition of a set of p-type functions. Two different notations exist to specify the addition of polarization functions. The first notation adds one asterisk to the basis set to specify addition of polarization functions to non-hydrogen atoms, while two asterisks symbolize the addition of polarization functions to all atoms (including hydrogen). The 6-31G** basis set10 is thus constructed from the split valence 6-31G basis set through addition of one set of d-functions to all non-hydrogen atoms and one set of p-functions to all hydrogen atoms. In the second (preferable) notation the polarization functions are specified through their angular quantum number explicitly. The 6-31G** basis set would then be termed "6-31G(d,p)". This latter notation is much more flexible as multiple sets of polarization functions can be specified much more easily.


%Kjob L301
#P HF/6-31G(d,p) GFInput GFPrint

methanol basis set
O2  1  r2
H3  1  r3  2  a3
H4  1  r4  2  a4  3  d4


Kjob command kills the job after checking the input

The GFInput (“Gaussian Function Input”) output generation keyword causes the current basis set to be printed in a form suitable for use as general basis set input, and can thus be used in adding to or modifying standard basis sets.

GFPrint command: This output generation keyword prints the current basis set in tabular form.    


Further characteristics of polarization functions can be discussed after inspection of an example. The 6-31G(d) basis set for carbon is:


C 0

 S    6 1.00

   .3047524880D+04   .1834737130D-02

   .4573695180D+03   .1403732280D-01

   .1039486850D+03   .6884262220D-01

   .2921015530D+02   .2321844430D+00

   .9286662960D+01   .4679413480D+00

   .3163926960D+01   .3623119850D+00

 SP   3 1.00

   .7868272350D+01  -.1193324200D+00   .6899906660D-01

   .1881288540D+01  -.1608541520D+00   .3164239610D+00

   .5442492580D+00   .1143456440D+01   .7443082910D+00

 SP   1 1.00

   .1687144782D+00   .1000000000D+01   .1000000000D+01

 D    1 1.00

   .8000000000D+00   .1000000000D+01


            The polarization functions for the 6-31G basis appear in the basis set listing as a single set of uncontracted d-type Gaussians. There are six cartesian d-type Gaussians (x2, y2, z2, xy, yz, zx) exp (-ad r2), which are equivalent to five pure d-type functions (xy, yz, zx, x2-y2, 3z2-r2) exp (-ad r2) plus one additional s-type function. In some basis sets (such as 6-31G(d)) calculations are performed with all six polarization functions, in other cases only the pure d-type functions are used. Unfortunately, this is not reflected in the notation currently used for polarization functions. Great care must therefore be taken when theoretical results using polarized basis sets are compared.

            For the elements C, N, O, and F the orbital exponents of the d-type functions in the 6-31G(d) basis set have uniformly been set to a value of 0.8.10 The rational for this somewhat arbitrary assignment was that optimized orbital exponents of 0.73-0.92 were calculated for a small set of organic molecules and that the use of an uniform average value of 0.8 would be advantageous. For hydrogen, a standard orbital exponent of ap=1.1 has been adopted for the p-type Gaussian polarization functions of the 6-31G(d,p) basis set. The full 6-31G(d,p) basis set for hydrogen is:


H 0

 S    3 1.00

   .1873113696D+02   .3349460434D-01

   .2825394365D+01   .2347269535D+00

   .6401216923D+00   .8137573262D+00

 S    1 1.00

   .1612777588D+00   .1000000000D+01

 P    1 1.00

   .1100000000D+01   .1000000000D+01


As there is no inner core the 1s orbitals are described by two basis functions. The inner basis function is composed of three s-type Gaussians and the outer basis function is a single uncontracted s-type Gaussian with comparatively small exponent. The last two lines specify a single set of uncontracted p-type Gaussians with exponent 1.1.

            Returning to the example of methanol, one can readily see that the addition of polarization functions increases the number of basis functions and primitives significantly: The number of basis functions is: 26 [6-31G], 38 [6-31G(d)], and 50 [6-31G(d,p)] and the number of primitives is: 64 [6-31G], 72 [6-31G(d)], 84 [6-31G(d,p)]. Calculations involving polarized basis sets are therefore much more time consuming than those using the corresponding unpolarized double zeta basis sets. The results in Table 1 show, however, that already the addition of a single set of polarization functions on non-hydrogen atoms goes a long way towards the results obtained with much more sophisticated basis sets. The use of polarized basis sets is especially important for the proper description of bonds of strongly electronegative elements such as oxygen and fluorine and for theoretical studies using correlated methods.11



last changes: 01.04.2008, AS
questions & comments to: axel.schulz@uni-rostock.de