Basis set convergence

The abscissa is the basis set limit in a graph of 1/n, being n the dzeta value

Quantum mechanical methods that rely on basis sets converge to a limit when using an infinite basis set (for instance, we can talk about HF limit). In variational methods using bigger basis sets necessarily mean lower energies.

It is sometimes possible to extrapolate from several calculations using two or more basis sets the limit of the method. Graphically it can be seen by plotting a certain property (like the energy) vs. the size of the basis. For this purpose some basis sets where developed, like cc-pVnZ (with n=D, T, Q, 5 or 6). For post-HF methods the correlation usually means that bigger basis sets are needed to reach the basis set limit. For instance, an approximation to the HF limit can be made with

$ E_{HF,\infty }=\frac{x^{5}E_{HF,x}-y^{5}E_{HF,y}}{x^{5}-y^{5}} $

where x and y are the highest angular momentum of two sequential basis sets (for instance x=2 for cc-pVDZ, y=3 for cc-pVTZ). The correlation energy in a CCSD uses the third power of the terms instead of the fifth, showing the harder to achieve the limit in correlated methods. DFT appears to be less sensitive to the size of the basis set.

References Edit

  • Cramer, C. J. (2004) Essentials of Computational Chemistry, 2nd Ed. John Wiley & Sons.
  • Parthiban, S.; Martin, J.M.L. J. Chem. Phys. 2001, 114, 6014.