  The semi-exponential basis set of radial functions (A.M. Frolov, Physics
Letters A {\bf 374}, 2361 (2010)) is used for variational computations of bound
states in three-electron atomic systems. It appears that semi-exponential basis
set has a substantially greater potential for accurate variational computations
of bound states in three-electron atomic systems than it was originally
anticipated. In particular, the 40-term Larson's wave function improved with
the use of semi-exponential radial basis functions now produces the total
energy \linebreak -7.47805413551 $a.u.$ for the ground $1^2S-$state in the
${}^{\infty}$Li atom (only one spin function $\chi_1 = \alpha \beta \alpha -
\beta \alpha \alpha$ was used in these calculations). This variational energy
is very close to the exact ground state energy of the ${}^{\infty}$Li atom and
it substantially lower than the total energy obtained with the original
Larson's 40-term wave function (-7.477944869 $a.u.$).
