Improving numerical integration through basis set expansion

Calculations are presented to assess a theorem presented by S.F. Boys [(1969) Proc. R. Soc. A. 309:195], regarding the accuracy of numerical integration in quantum chemical calculations. The theorem states that the error due to numerical integration can be made proportional to the error due to basis set truncation, and thus goes to zero in the limit of a complete basis. We test this theorem on the hydrogen atom, showing that with a solution-spanning basis, the numerically exact orbital energy can indeed be calculated with a small number of integration points. Moreover, tests for H and H-2(+) demonstrate that even when only a near-complete basis is employed, Boys' Theorem can significantly reduce integration error. However, for other systems, like the oxygen atom and the CO2 molecule, the theorem yields no advantage for some occupied orbitals. It is concluded that the theorem would be most useful for calculations that demand large basis sets.