We describe the implementation of a new grid-free density-functional technique for exchange-correlation potentials of rho(1/3) form (exchange-only local density-functional theory potentials). The potential is fitted to integrable functional forms by solving a set of nonlinear equations, rather than by fitting on a three-dimensional grid of points. This completely analytical method produces smooth energy surfaces and exact energy gradients. The method is found to be several times faster computationally in single-point calculations than a comparable grid-based method with a moderate number of grid points, and it is more than an order of magnitude faster for geometry optimizations. The analytical method is tested on the torsional energy surfaces of the classic isoelectronic series C2H6, N2H4, and H2O2, using the Hartree-Fock-Slater potential (alpha = 2/3). The locations and relative energies of energy extrema, and the structural variations across the potential surfaces, are in good agreement with experimental data and the results of high-quality ab initio studies.