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Rigorously continuous and smooth potential energy surfaces, as well as exact analytic gradients, are obtained for a conductorlike screening solvation model (CPCM, a variant of the general COSMO) with Hartree–Fock (RHF, ROHF, UHF, and MCSCF) and density functional theory (R-DFT, RO-DFT, and U-DFT) methods using a new tessellation scheme, fixed points with variable areas (FIXPVA). In FIXPVA, spheres centered at atoms are used to define the molecular cavity and surface. The surface of each sphere is divided into 60, 240, or 960 tesserae, which have positions fixed relative to the sphere center and areas scaled by switching functions of their distances to neighboring spheres. Analytic derivatives of the positions and areas of the surface tesserae with respect to atomic coordinates can be obtained and used to evaluate the solvation energy gradients. Due to the accurate analytic gradients and smooth potential energy surface, geometry optimization processes using these methods are stable and convergent.