The basis functions employed by the ab initio Hartree-Fock and density functional methods are contracted Gaussian-type functions

where the constants d_wy are called contraction coefficients, the g_w's are primitive Gaussians functions, and the values of b typically range from 1 to 7. The primitive Cartesian Gaussian function has the form



The quantities m, n, and o are integers; x, y, and z are Cartesian coordinates; and a is the orbital exponent. When m + n + o = 0, then g is said to be a s-type Gaussian function; when m + n + o = 1, then g is a p-type Gaussian; and when m + n + o = 2, then g is a d-type Gaussian. By combining primitive Gaussian functions with different values of a contracted Gaussian functions that approximate the radial part of Slater-type orbitals (STOs) can be constructed. The advantage of the Gaussian function is that less computer time is required to evaluate the integrals in the F_zy terms. The values of the contraction coefficients d_wy and the orbital exponent a are obtained by fitting a contracted Gaussian funciton to a STO or by finding the contracted Gaussian functions that minimize the self-consistent-field energies of atoms. These optimized d_wy and a values are stored in the software and treated as constants when a contracted Gaussian function is used in a calculation.^1,2,3

   The minimal basis set is the minimum number of basis functions c needed to describe the ground states of the component atoms in a molecule. For example, the minimal basis set for H₂O would be a 1s orbital for each hydrogen atom plus a 1s, 2s, 2p_x, 2p_y, and 2p_z orbital for the oxygen atom. Because the contraction coefficients d_wy and orbital exponents a are fixed, the size of a basis function on an atom can not vary as the nature of the surrounding atoms changes. This problem is solved by replacing each orbital in the minimal basis set with two basis functions that differ in size. During the self-consistent-field process the coefficients c_yi for each basis function on the atom are optimized and hence the size of the orbital that is now expressed as a linear combination of two basis functions



is optimized. This type of basis set is called a double-zeta basis set. The double-zeta basis set for H₂O includes a 1s' and 1s" orbital for each hydrogen atom plus a 1s', 1s", 2s', 2s", 2p_x', 2p_x", 2p_y', 2p_y", 2p_z', and 2p_z" orbital for the oxygen atom.

   In a split-valence basis set the inner-shell atomic orbitals are represented by one basis function and the valence orbitals are represented by two or more basis functions. For example, the 3-21G basis set has one contracted Gaussian function that is a linear combination of three primitive Gaussian functions for each inner-shell atomic orbital and two basis functions, one contracted Gaussian function that is a linear combination of two primitive Gaussians and one primitive Gaussian function, for each valence orbital. To allow for the displacement of charge density away from the nuclei and toward the bonding regions, orbitals for which the l quantum number is greater than maximum value of the valence orbitals in the ground state atom are added to the basis set. The 6-31G* basis set represents each inner-shell orbital with one contracted Gaussian function that is a linear combination of six primitive Gaussian functions and each valence orbital with two basis functions, one contracted Gaussian function that is a linear combination of three primitive Gaussians and one primitive Gaussian function. In addition, six d-type Gaussian functions for each nonhydrogen atom in the second or third period are included in the basis set. Basis sets such as 6-31G* are said to be polarized basis sets.^1,2

    1) Levine, Ira N. Quantum Chemistry, 5th ed.; Prentice Hall : Upper Saddle River, 2000; pp 486-494.
    2) Lowe, John P. Quantum Chemistry; Academic Press: New York, 1978; pp 315-319.
    3) Feller, D.; Davidson, E. R. In Reviews in Computational Chemistry; Lipkowitz, K. B.; Boyd, D. B. Ed; VCH: New York, 1990; Vol. 1; p 1.

Nutt 01