Even-tempered atomic orbitals. III. Economic deployment of Gaussian primitives in expanding atomic SCF orbitals

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Abstract
Expansion of atomic self‐consistent‐field orbitals in terms of Gaussian‐type primitive AO's calls for the most economic deployment and efficient use of all available basis functions to avoid computational waste. In even‐tempered representations maximal use of all primitives in representing all SCFAO's is ensured. Here, the optimal number of primitives per individual SCFAO, consistent with a given total number of primitives is investigated. Maximal effectiveness is found to occur when the following conditions prevail: (1) The number of primitives per SCFAO is approximately the same for all SCFAO's, say x; (2) the number of primitives needed to express the set of SCFAO's with angular‐momentum quantum number l in atom A is approximately given by Nl(A) = x[a + byl(A)] where a and b are constants and yl(A) is the number of different SCFAO's with quantum number l in atom A. Thus the efficiency rapidly increases with yl(A). Explicit results and optimal expansions based on 3–14 primitives are discussed for all atoms up to krypton. The achieved systematization permits the unambiguous selection of economic even‐tempered expansions for molecular calculations.