We perform LCAO (linear combination of atomic orbitals) calculations f
or the ground state of the Yukawa potential V(r) = -(e2/r)e(-qr) as a
function of the screening parameter q. We obtain the best variational
result so far for the ground-state energy E0 as a function of q. We al
so obtain the critical exponents of both the probability density at th
e origin and the ground-state energy as functions of (q-q(c)), where q
(c) is the critical q above which V(r) does not have a bound state. Th
e use of the critical exponents permits the so far most precise determ
ination of q(c), q(c) = 1.190 612 27+/-0.000 000 04. We also show that
it is possible to use the LCAO calculations as a tool to determine th
e analytical form of very precise variational wave functions. We obtai
n, in such a way, the wave function psi=(e(-ar)-e(-br))/r+e(-br)/(r+al
pha). This variational wave function has a bound-unbound transition at
q(c)=1.190 61074.