The complementary variational principle is used to derive suitable expressions for the first modified bounds on the phonon thermal conductivity. The variational trial function is taken to depend on a series of variational parameters. It is shown that the relations needed to select one decomposition of the collision operator in preference to the others are independent of the number of variational parameters. The convergence of the method is investigated both analytically and numerically when the number of variational parameters is increased. All types of phonon interactions are considered and the results agree quantitatively with the experimental data for Ge and LiF over a wide temperature range.