In Gaussian the SCF convergence criteria is set to $10^{-N}$ by
SCF=(Conver=N)
Note that this criteria applies to the root mean square change in density matrix between two SCF cycles, while the maximum change convergence criteria is set to $10^{-N+2}$. For instance, for the default case of $N=8$ the output says
Requested convergence on RMS density matrix=1.00D-08 within 128 cycles. Requested convergence on MAX density matrix=1.00D-06. Requested convergence on energy=1.00D-06.
Now in GAMESS (US) the SCF convergence criteria is set to $10^{-N}$ by
$SCF CONV=1.0d-N $END
However, neither manual, nor the output do not explicitly specify does this criteria refers to the root mean square change in density matrix or to the maximum change in it? Judging from the default value (1.0d-05), it is the last one, the maximum change in density matrix, otherwise it would be too small for a reliable convergence. Thus, SCF=(Conver=N)
in Gaussian corresponds to $SCF CONV=1.0d-(N+2) $END
in GAMESS (US). But I'm not sure, so can anyone confirm that?
P.S. I tried to look at the source code, but it is an old FORTRAN mostly undocumented spaghetti mess with good-old six-lettered totally undescriptive variable names. :D