ATOMS: ALMA Three-millimeter Observations of Massive Star-forming
  regions -- IV. Radio Recombination Lines and evolution of star formation
  efficiencies
  We report detection of radio recombination line (RRL) H$_{40\alpha}$ toward
75 sources, with data obtained from ACA observations in the ATOMS survey of 146
active Galactic star forming regions. We calculated ionized gas mass and star
formation rate with H40U line emission. The mass of ionized gas is
significantly smaller than molecular gas mass, indicating that ionized gas is
negligible in the star forming clumps of the ATOMS sample. The star formation
rate (SFR$_{{\rm H}_{40\alpha}}$) estimated with RRL H$_{40\alpha}$ agrees well
with that (SFR$_{\rm L_{bol}}$) calculated with the total bolometric luminosity
(L$_{\rm bol}$) when SFR $\gtrsim 5 {\rm M_\odot My}r^{-1}$, suggesting that
millimeter RRLs could well sample the upper part of the initial mass function
(IMF) and thus be good tracers for SFR. We also study the relationships between
L$_{\rm bol}$ and the molecular line luminosities (L0mol) of CS J=2-1 and
HC$_3$N J=11-10 for all the 146 ATOMS sources. The Lbol-L0mol correlations of
both the CS J=2-1 and HC3N J=11-10 lines appear approximately linear and these
transitions have success in predicting L$_{\rm bol}$ similar to that of more
commonly used transitions. The L$_{\rm bol}$-to-L$_{\rm mol}$ ratios or
SFR-to-mass ratios (star formation efficiency; SFE) do not change with
galactocentric distances (R$_{\rm GC}$). Sources with H$_{40\alpha}$ emission
(or H$_{\rm II}$ regions) show higher L$_{\rm bol}$-to-L$_{\rm mol}$ than those
without H$_{40\alpha}$ emission, which may be an evolutionary effect.
