The Basis Set STO-3G, 6-31G, and All That

The requirement that the basis functions should describe as closely as possible the correct distribution of electrons in the vicinity of nuclei is easily satisfied by choosing hydrogen-like atom wave functions, //, the solutions to the Schrodinger equation for one-electron atoms for which exact solutions are available  

Unfortunately, the exponential radial dependence of the hydrogenic functions makes the evaluation of the necessary integrals exceedingly difficult and time consuming for general computation, and so another set of functions is now universally adopted. These are Cartesian Gaussian functions centered on nuclei. Thus, tj, l) is a function centered on atom /  

The superscripts, nx, ny, and nz, are simple positive integers or zero. Their values determine whether the function is s-type (nx = ny — nz — 0), p-type (nx + ny + nz = 1 in three 

Molecular orbitals will be very irregular three-dimensional functions with maxima near the nuclei since the electrons are most likely to be found there and falling off toward zero as the distance from the nuclei increases. There will also be many zeros defining nodal surfaces that separate phase changes. These requirements are satisfied by a linear combination of atom-centered basis functions. The basis functions we choose should describe as closely as possible the correct distribution of electrons in the vicinity of nuclei since, when the electron is close to one atom and far from the others, its distribution will resemble an AO of that atom. And yet they should be simple enough that mathematical operations required in the solution of the Fock equations can actually be carried out efficiently. The first requirement is easily satisfied by choosing hydrogenic AOs as a basis 

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Figure 1 shows the structure of all-trans-polymethineimine and Table 1 lists the optimized geometries for both HF and MBPT(2) methods with three basis sets, STO-3G, 6-31G, and 6-31G [68], We can see that both the size of the basis sets and electron correlation have a strong influence on the stable structure, especially on the difference between the two bond lengths, Tn=c and Yc-n-Table 2 presents the HF and MBPT (2) fundamental vibrational frequencies for all-trans-polymethineimine [68], The vibrational frequencies listed in each... 

A comparison of calculated and experimental anion geometries are provided in Table 5-16. Included are Hartree-Fock models with STO-3G, 3-21G, 6-31G and 6-311+G basis sets, local density models, BP, BLYP, EDFl and B3LYP density functional models and MP2 models, all with 6-31G and 6-311+G basis sets, and MNDO, AMI and PM3 semi-empirical models. Experimental bond lengths are given as ranges established from examination of distances in a selection of different systems, that is, different counterions, and mean absolute errors are relative to the closest experimental distance. 

Full ab initio optimizations of molecular geometries of enamines (and of any other kind of molecules) depend strongly on the kind of applied basis sets application of STO-3G 2 3, 3-21G 3-2lG 4-3lG 6-3lG 6-31G " and 6-31G basis sets leads to optimizations for the coplanar framework of all atoms of vinylamine, but it was not stated in these references whether coplanarity was assumed by input constraint or not. Contrary to that, the use of a double-zeta basis set with heavy atom polarization functions as well as 6-31 - -G ° based optimization yielded a non-planar amino group for 115. 

The above basis sets have been introduced by Pople and collaborators (see Hehre et aL in Further Reading at the end of this chapter) and have been used extensively, by a number of workers for calculations on a large variety of molecules. Apart from a few instances, all the calculations in this book use the STO-3G, 4-31G, 6-31G, and 6-31G " hierarchy of basis sets. By restricting our example calculations to a very limited set of molecules and the above basis sets, we are attempting to illustrate in a systematic way how specific attributes of a basis set affect calculated quantities. We are not attempting to provide a general review of current calculations. Such a review would be out of date very quickly. While the basis sets we use are not necessarily optimum, and may themselves be out of date shortly, they do have characteristics that can be used to illustrate all basis sets. 

In quantum mechanics it is always important to define a calculation in terms of the basis set used. The standard basis set used for a calculation has always been dependent on the time period in which the calculation was carried out. The earliest calculations were mostly carried out at the STO-3G level. Later calculations were usually at the 4-3IG level, followed still later by MP2/6-31G. This last is the lowest level that has been commonly used in recent years, and to which we often refer. The better level that is now usually feasible in almost all cases adds diffuse functions to the basis set, and our standard basis for a long time has been 6-31G++(2d,2p). We refer locally to... 

We now have two ways of inserting the correct parameters into the STO-2G calculation. We can write them out in a gen file like Input File 8-1 or we can use the stored parameters as in Input File 8-2. You may be wondering where all the parameters come from that are stored for use in the STO-xG types of calculation. They were determined a long time ago (Hehre et al, 1969) by curve fitting Gaussian sums to the STO. See Szabo and Ostlund (1989) for more detail. There are parameters for many basis sets in the literature, and many can be simply called up from the GAUSSIAN data base by keywords such as STO-3G, 3-21G, 6-31G, etc. 

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