Fundamental Concepts and General Description

The most widely used quantum chemical methods are based on the Idea of the expansion of atomic or molecular orbitals as a linear combination of basis functions 

The LCAO approximation (2.1) may be taken as the expansion of a function as a series. As for any method which contains such an expansion, it may be assumed that upon augmenting the expansion by some other functions one arrives at a superior description of atomic (or molecular) orbitals because the number of variational parameters is increased. Examine now a basis with a doubled number of functions, i.e. a basis set in which each atomic orbital is represented by two functions. The result of this basis set extension is presented in Table 2,1 for the atoms from the first to fourth rows of the periodic system. The results for the fifth-row atoms are presented in Table 

The entries in the first columns of the two tables are total energies of atoms given by the SCF calculations with the minimal basis set of Slater-type orbitals (STO, see Section 2.B.). The entries in the second columns are energies given by the calculations in which each STO was replaced by two Slater-type functions with the exponents so optimized to give the minimum total energy. From Tables 2.1 and 

it is seen that a twofold enlargement of the basis set gave 

As the number of functions is increased, description of molecular orbitals becomes better and better and the total SCF energy decreases. In the limiting case of an infinite expansion (2.1), the set of functions is complete and the function is expressed accurately  

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