The tensor perturbations in the rotational fine structure of spherical-top fundamental vibrational bands are analyzed in detail. The shift of a fine-structure sublevel ( v = 1, J, R, p) is the result of fourth-rank tensor interactions with sublevels belonging to other manifolds with the same J but different R′, due to the matrix elements 〈 v = 1, J, R, p| H| v = 1, J, R′, p′〉, which are proportional to the symmetry-adapted vector-coupling coefficients F A 1 pp′ (4, R, R′) of Moret-Bailly. Sum rules are derived for general binary products of these matrix elements, and for particular linear combinations of ternary and binary products arising in perturbation theory. These lead to analytical expressions for the second- and third-order perturbation corrections to the transition wavenumbers. To this order of approximation the formulas are expressed in terms of scalar functions of J and R, diagonal F (4) and F (6) tensor functions, and the g ̃ function of Ozier. For the third-order formulas, many tedious algebraic manipulations were avoided by computing the sums numerically for each value of ( R - J) and a range of values of R, and then obtaining the algebraic form of the coefficients, which are known to be rational functions of R, by generalization from the numerical results. This technique will be useful in other higher-order perturbation calculations. The cluster properties of the F (4) and F (6) functions in high- J manifolds explain the major qualitative features of the behavior of the perturbations. In the high- R limit, 3 j coefficients give the positions of the clusters. Explicit expressions for these coefficients give a partial check of the present general algebraic expressions. The formalism is tested for the ν 3 and ν 4 bands of the SiF 4 molecule, using as data energy levels computed from the known spectroscopic parameters. Because of the different ζ Coriolis coupling constants, these two cases differ markedly in the rate of convergence of the perturbation expansions. For each band, wavenumber data generated by computer diagonalization of the Hecht Hamiltonian operator are fitted using the perturbation formulas, and the results demonstrate the accuracy, reliability, and limitations in the use of this approximate treatment for the fitting of spectral lines.