Basis set shell structure

The definition of basis sets involves the selection of a set of functions for each angular momentum of the atom. In non-relativistic calculations, is a good quantum number, but in relativistic calculations it is j or k which is the good quantum number. However for the lighter elements where the effects of relativity are small, is an approximately good quantum number, and basis sets for the spin-orbit components of a non-relativistic subshell can share exponents. Basis sets that are optimized with the same exponents for the two spin-orbit components are called -optimized. Similarly, basis 

Optimization of larger basis sets such as those of triple-zeta quality encounter problems of balance for heavy elements if functions with the same t share exponents. This is shown by performing K-optimization for the basis set. For Rn, for example, the outermost two exponents of the 6pi/2 are very close to the second and third outermost functions of the 6 3/2 the 6p3/2 needs one more diffuse exponent than the 6pi/2- An -optimized basis set is a compromise between the two, and both will be closer to double zeta than triple zeta. Going to the next row, for the superheavy elements of the 7p block, the overlap between the spin-orbit components is even smaller, and the 7pi/2 is much closer to the 7s than to the 7p3/2- This suggests that j-optimization might produce better quality and smaller basis sets than -optimization, and this proves to be the case even for the 6p block. For the Rn case quoted above, the -optimized set gives an energy of 0.132 Eh, as compared to the 0.021 obtained with the -optimized set. 

There are some advantages of -optimization. The functions for the large 

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