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 set^{10} 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 0,1 C1 O2 1 r2 H3 1 r3 2 a3 H4 1 r4 2 a4 3 d4 r2=1.20 r3=1.0 r4=1.0 a3=120. a4=120. d4=180. |
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 (x^{2}, y^{2}, z^{2}, xy, yz, zx) exp
(-a_{d }r^{2}), which are equivalent to
five pure d-type functions (xy, yz, zx, x^{2}-y^{2}, 3z^{2}-r^{2}) exp (-a_{d }r^{2}) 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 a_{p}=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