  We have developed an improved scheme of a three-qubit Toffoli gate based on
fine structure state changing three-body Stark-tuned Rydberg interaction. This
scheme is a substantial improvement of our previous proposal [I.I.Beterov et
al., Physical Review A 98, 042704 (2018)]. Due to the use of a different type
of three-body F\"orster resonance we substantially simplified the scheme of
laser excitation and phase dynamics of collective three-body states. This type
of F\"orster resonance exists only in systems with more than two atoms, while
the two-body resonance is absent. We reduced the sensitivity of the gate
fidelity to fluctuations of external electric field and eliminated the
necessity to use external magnetic field for fine tuning of the resonant
electric field value, compared to the previous scheme of Toffoli gate based on
Rydberg atoms. A gate fidelity of >99% was demonstrated in the calculations.
