A consistent method for optimizing Gaussian primitives for Rydberg and multiply excited helium states is designed. A novel series for the "exponentially tempered Gaussians" is introduced, which is markedly more efficient than the commonly used series of even tempered Gaussians. The optimization is made computationally feasible due to an approximate calculation of excited states using the effective one-electron Hamiltonian that is defined as Fockian from which the redundant Coulomb and exchange terms are dropped. Finally, ExTG5G and ExTG7F Gaussian basis sets are proposed. They enable calculations of the helium spectrum all the way from the ground state up to the (5, 4)_5 ~1S~e and (6, 5)_7 ~1S~e doubly excited resonances, respectively, mostly in the spectroscopic accuracy of 1 cm~(-1).
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