Recent Developments and Computational Issues

Recent developments have attempted to combine localized and plane wave basis functions, i.e. describing the core region by radial polynomials or Gaussian functions, and the valence region by plane waves. This price of this approach is increased computational complexity, since new integrals involving different types of basis function are required. 

The computational problem is formally the same whether a Gaussian, plane wave or polynomial basis is used - calculate matrix elements of quantum mechanical operators over basis functions and solve the variational problem by an iterative procedure - but the nature of the functions results in some differences. With a GTO basis the matrix elements are calculated directly, while with a plane wave basis the matrix elements involving the potential energy can be generated by simple multiplication, as long 

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