  We study a class of Landau-de Gennes energy functionals in the asymptotic
regime of small elastic constant $L>0$. We revisit and sharpen the results in
[18] on the convergence to the limit Oseen-Frank functional. We examine how the
Landau-de Gennes global minimizers are approximated by the Oseen-Frank ones by
determining the first order term in their asymptotic expansion as $L\to 0$. We
identify the appropriate functional setting in which the asymptotic expansion
holds, the sharp rate of convergence to the limit and determine the equation
for the first order term. We find that the equation has a ``normal component''
given by an algebraic relation and a ``tangential component'' given by a linear
system.
