11.4: Orbital Polarization Terms in Basis Sets

Notation

In the Pople basis sets, polarization functions are indicated with “stars”: for instance, 6-31G\(^*\) or 6-311G\(^{**}\). A single star indicates that \(\ell+1\) functions are added to all second-row and higher atoms (e.g., carbon gets a set of \(d\) orbitals), and a double star indicates that hydrogen and helium also receive \(p\) functions. The polarization functions are each built just from a single Gaussian primitive. Most Pople style basis sets with polarization functions are optimized through HF/SCF calculations; however, the 6-311G\(^{**}\) basis set was optimized by second order Møller Plesset perturbation theory (MP2) calculations, which partially account for dynamic electron correlation. A summary of some common Pople-style basis sets with polarization functions is shown in Table \(\PageIndex{1}\). For more on these basis sets, see R. Krishnan et al., J. Chem. Phys. (1980), 72, 650 and the other papers in that series.

Notation

The presence of diffuse functions in a Pople basis set is denoted with a \(+\) symbol, such as 6-31+G. A single plus indicates that one extra \(n+1\) function, built from a single Gaussian primitive, is added for each orbital in the valence shell for all atoms except H and He. Two plus symbols (e.g., 6-311++G) indicate that H and He also receive \(n+1\) functions. To illustrate this, in a 6-311++G basis set, a hydrogen atom’s \(1s\) (valence) orbital is represented by 3 orbital functions (built from 3, 1, and 1 GTOs, respectively), and an extra orbital function consisting of just one Gaussian primitive is added to represent the \(2s\) orbital. The contraction, therefore, is (\(6s\))\(\to\)[\(4s\)]. A carbon atom in the same basis set receives diffuse functions to represent the \(3s\) and three \(3p\) orbitals. The coefficients of the Gaussian primitives in basis sets with diffuse functions are optimized by calculations of either atomic excited states or atomic anions. Diffuse and polarization functions are very often combined to effectively include the next shell after the valence shell; such basis sets are essential for calculating accurate properties of loosely-bound species. Additional examples are given in Table \(\PageIndex{2}\).

In the end, the accuracy of a calculation depends on the basis set chosen, and there are tradeoffs to be made. For the best answer, the basis set must be flexible enough to fit the shape of the “true” wavefunction, which can be accomplished by adding additional functions with different shape parameters or polarization properties to the set. The more functions that are added, though, the more time is required to complete the calculation. The farther down the periodic table the atoms are, the more functions need to be included in the basis, and the more difficult the calculations will be. In fact, for many 4th-row and higher atoms, it is common to use basis sets that include effective core potentials: mathematical functions that represent a classical potential that is added to the Hamiltonian in place of many of the core electrons. These basis sets reduce considerably the numbers of integrals that need to be calculated, and can make calculations of very heavy atoms with many electrons tractable. However, there are limitations, particularly for molecular properties that are sensitive to the electronic wavefunction near the nucleus (e.g., X-ray spectroscopy, NMR shielding coefficients).