The advent of optimization procedures that use analytical gradients led to the development of split-valence basis sets with fewer primitive Gaussians than 4-31G. 3-21G [Binkley et al., 1980, Gordon et al., 1982, Pietro et al., 1982, Dobbs and Hehre, 1986, 1987] became the basis set used most commonly for geometry optimization. It uses three primitive Gaussians for the core orbitals and a two/one split for the valence functions. Because the procedures used to calculate the atomic forces are very sensitive to the number of primitive Gaussians, a 3-21G optimization can be up to twice as fast as the same calculation with 4-31G, although the difference is not large for single point calculations.
            The next step in improving a basis set is usually the addition of d-orbitals for all heavy (nonhydrogen) atoms. For most organic compounds these do not function as d-orbitals in the normal sense of being involved in bond formation as in transition-metal compounds. Their purpose is far more to allow a shift off the center, for instance, a p-orbital away from the position of the nucleus. This polarization is illustrated in Fig. 1.2. Mixing the d-orbital, which has lower symmetry, with the p-orbital results in a deformation of the resulting orbital to one side of the atom. This adjustment is particularly important for small ring compounds and second-row elements. The most commonly used polarization basis set (i.e., one including d-orbitals) in the Gaussian03 program is 6-31G*. This basis set uses six primitive Gaussians for the core orbitals, a three/one split for the s- and p-valence orbitals, and a single set of six d-functions (indicated by the asterisk). Six d-functions (equivalent to five d- and one s-orbital) are used for computational convenience, although Gaussian programs can also handle basis sets with five real d-orbitals. A further development is the 6-31G** basis, in which a set of p-orbitals has been added to each hydrogen in the 6-31G* basis set.

Figure 1.3 Polarization of a p-orbital by mixing with a d-orbital

.............The diffuse function augmented bases, is intended for use in calculations on anions or molecules that require very good descriptions of nonbonding electron pairs for e.g Ionic Liquids. These basis sets are obtained by adding a single set of very diffuse s- and p-orbitals (with exponents ς between 0.1 and 0.01) to the heavy atoms in a standard basis such as 6-31G*. This basis is then designated 6-31+G*, or 6-31++G* if diffuse functions are to improve the basis set at large distances from the nucleus and thus describe the high-energy electron pairs associated with anions better.
            The number of basis functions rises rapidly with increasing sophistication of the basis set, however the number of basis functions the program can handle is limited. The computer time required is roughly proportional to the fourth power of the number of basis functions. The List of basis sets in the order of their applicability and evolution is given in Table 1.1.The basis sets are named as used in the commercially available GAUSSIAN03 package.

Basis Set	Applies to	Polarization Functions	Diffuse Functions
STO-3G	H-Xe	*
3-21G	H-Xe	* or **	+
6-21G	H-Cl	(d)
4-31G	H-Ne	(d) or (d,p)
6-31G	H-Kr	(3df,3pd)	++
6-311G	H-Kr	(3df,3pd)	++
D95	H-Cl except Na and Mg	(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-Ba, La-Bi
LanL2DZ	H, Li-Ba, La-Bi
SDD, SDDAll	all but Fr and Ra
cc-pV(DTQ5)Z	H-He, B-Ne, Al-Ar, Ga-Kr	included in definition	added via AUG- prefix
cc-pV6Z	H, B-Ne	included in definition	added via AUG- prefix
SV	H-Kr
SVP	H-Kr	included in definition
TZV and TZVP	H-Kr	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-Lr	UGBS(1,2,3)P
MTSmall	H-Ar
DGDZVP	H-Xe
DGDZVP2	H-F, Al-Ar, Sc-Zn
DGTZVP	H, C-F, Al-Ar

Table 1.1 : List of Basis sets built in Gaussian 03.