Split-valence basis sets I: the 3-21G basis set
The description of the valence
electrons can be significantly improved over that in the minimal STO-3G basis
set, if more
than one basis function is used per valence electron. Basis sets of
this type are called "split valence" basis sets as the description
of valence orbitals are split into two (or more) basis functions. A related
term is "double
zeta" in reminiscence of the greek symbol used for the orbital
exponents of STOs. The term "double zeta" (DZ) does not imply,
however, whether two basis sets are use for all of the orbitals or only for the
valence space.
One very economical, small split
valence basis set is the 3-21G basis set.2-4 The non-valence
electrons are described by single basis functions composed of a contraction of
three Gaussians. Each valence electron is described by two basis functions. The
first of these basis functions is composed of two Gaussian primitives while the
second consists of a single uncontracted Gaussian primitive.
For carbon the 3-21G basis set is2 (Gaussian 94
format):
C 0
S 3 1.00
.1722560000D+03 .6176690000D-01
.2591090000D+02 .3587940000D+00
.5533350000D+01 .7007130000D+00
SP 2 1.00
.3664980000D+01
-.3958970000D+00 .2364600000D+00
.7705450000D+00
.1215840000D+01 .8606190000D+00
SP 1 1.00
.1958570000D+00 .1000000000D+01 .1000000000D+01
%Kjob L301 #P HF/3-21G 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. |
The main
difference to the STO-3G listing before is that there are two sections now for
the 2s and 2p orbitals. Again, s and p orbitals share the same exponents a2,x in the first column, but
have different contraction coefficients listed in the second and third column.
The contraction coefficient of the outer SP shell is, of course, unity as the
outer SP shell consists of only one Gaussian function. Inspection of the
orbital exponents of the first SP shell (3.66 and 0.77) and the second SP shell
(0.19) shows that the first SP shell has larger exponents and therefore
describes electron density closer to the nucleus as compared to the second SP
shell. Consequently, the first and second
SP shells are sometimes referred to as the "inner" and
"outer" shells, respectively.
The 3-21G basis set has somewhat of
a curious developmental history. Straightforward variation of the orbital
exponents and expansion coefficients as practiced in the development of other
basis sets led to an "falling inward" of the inner valence shells.
This is due to the fact that there are only three Gaussians available for the
description of the core region and that addition of more primitives to the core
region lowers the overall energy more than an adequate description of the
valence shell. For most elements the basis set parameters were therefore first
optimized using a much larger core space of six Gaussians (6-21G). At this
stage all basis set parameters were varied to minimize the energy of the atoms
in their electronic ground states at the unrestricted Hartree-Fock (UHF) level
of theory. The large core was then replaced by a smaller one of only three
Gaussians and the basis set parameters of the new core region were reoptimized,
keeping the valence region constant. A similar strategy has also been used for
second row elements.3,4
How much more computational effort
is required for the 3-21G basis set as compared to the minimal STO-3G solution?
Each carbon and oxygen atom in methanol now needs 9 and each hydrogen atom
needs 2 basis functions. For methanol this equates to 26 instead of 14 basis
functions. This much larger number of basis functions and thus variable MO coefficients
is achieved, however, with the same number of Gaussian primitives as for the
STO-3G basis set: 42.
Split-valence basis sets II: the 6-31G
basis sets
Development of the larger 4-31G and
6-31G split valence basis sets5,6 predates that of
the 3-21G basis set considerably. The main difference between 3-21G and 6-31G
is that a much larger number of primitives is used in the latter in the core as
well as the inner most valence shell. The use of a contraction of six Gaussian primitives for each
core orbital improves the description of the core region significantly.
The valence region is again described by two basis functions per atomic
orbital. The inner shell is composed of a contraction of three Gaussians and
the outer shell consists of one single Gaussian primitive. As in other basis
sets developed by the Pople group, s and p functions share common
exponents.
The 6-31G basis 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
The orbital exponents and expansion
coefficients were optimized to yield the lowest possible UHF energies for the
respective atoms in their electronic ground states. The exponents of the
valence shell atoms have then been scaled uniformly using scale factors
developed for the 4-31G basis set in order to achieve the best possible results
in MO calculations on a set of small organic molecules.
For methanol, the
6-31G basis set includes 26 basis functions, which are composed of a total of
60 Gaussian primitives.
Double zeta basis sets: Dunnings D95 basis set
Dunnings D95 basis set has been
derived from an already existing large atomic basis set of nine uncontracted
Gaussian primitives of s- and five uncontracted Gaussian primitives of p-type.8 Six of the nine
s-type functions have then been grouped into a single contraction, while the
other three s-type functions have been left alone. Similarly, four of the five
p-type functions have been contracted into a single function, while one
function was left uncontracted. Overall, this yields a basis set of four s-type and two p-type
basis functions. In contrast to the split valence basis sets
discussed before, the D95 basis set is a full double zeta basis set in that it allocates
two basis functions for each atomic orbital of the core as well as the valence
region occupied in the electronic ground state.
The D95 basis set for carbon is:
C 0
S 6 1.00
.4232610000D+04 .2029000000D-02 |
.6348820000D+03 .1553500000D-01 |
.1460970000D+03 .7541100000D-01 |
.4249740000D+02 .2571210000D+00 |
.1418920000D+02 .5965550000D+00 | -- s1
+ s2 mainly describe
.1966600000D+01 .2425170000D+00 | the 1s core
of carbon
S 1 1.00 |
.5147700000D+01 .1000000000D+01 |
S 1 1.00 |
.4962000000D+00 .1000000000D+01 | -- s3 + s4
mainly describe
S 1 1.00 | the 2s
valence orbital of carbon
.1533000000D+00 .1000000000D+01 |
P 4 1.00
.1815570000D+02 .1853400000D-01
.3986400000D+01 .1154420000D+00
.1142900000D+01 .3862060000D+00
.3594000000D+00 .6400890000D+00
P 1 1.00
.1146000000D+00 .1000000000D+01
The format used here for the listing
of orbital exponents and expansion coefficients is different from those used
before for the Pople basis sets as s- and p-type functions of the D95 basis set
do not share the same orbital exponents.
Using an uncontracted atomic basis
set as the starting point for the development of contracted versions suitable
for the treatment of larger systems is common practice. The standard
nomenclature used specifies the uncontracted basis set in brackets and the
resulting contracted version in square brackets. Using the D95 basis set as an
example, the contraction can be described as (9s5p) -> [4s,2p]. This notation
does not specify, how many primitives are contained in each contraction. This
can be specified in more detail as (6111,41), listing first the s-type functions
(here distributed over four contractions) and then the p-type functions. Basis
sets, in which a given primitive appears in only one of the contractions, are
termed segmented.
Why
are contractions done? MO calculations using the uncontracted 9s5p atomic
basis set would need to handle 9*1+5*3=24 MO coefficients for each carbon atom
while only 4*1+2*3=10 MO coefficients are necessary for the [4s,2p]
contraction. If proper care is taken during the contraction process, calculations using
the contracted basis sets can be performed with similar accuracy but
dramatically reduced computational cost.
For methanol, the D95 basis set uses
10 basis functions for each carbon and oxygen atom, and two functions for
hydrogen, yielding a total of 28 basis functions. These basis functions are
constructed from a total of 64 Gaussian primitives.