The equivalent internal field model, originally developed for linear one-dimensional chromophores, is extended to three-dimensional multipolar chromophores. The extension requires two generalizations to the model. First, the equivalent internal field must be generalized to an equivalent internal potential. Second, all tensor components of the hyperpolarizability must be taken into account. A general formalism is developed for analyzing the hyperpolarizabilities induced by application of various internal potentials to a molecular skeleton, assuming the hyperpolarizabilities are linearly related to the internal potential. This formalism utilizes a symmetry analysis, along with quantum chemical calculations done here with the Huckel, Pariser-Parr-Pople and intermediate neglect of differential overlap (INDO) Hamiltonians and using single-configuration interaction theory. The formalism is applied to and tested on the benzene molecular skeleton. The hyperpolarizability is found to be linearly related to the internal potential provided the potential difference between carbon atoms is below a threshold of the order of 2 eV. Various predictions of the model are tested against explicit INDO and ab initio calculations on mono-substituted, ortho-disubstituted, meta-disubstituted, and tri-substituted benzenes. For the substituents F, CH3, OCH3, OH, and NH2, many of the predictions of the equivalent internal potential model apply, indicating that the internal potentials arising from these substituents are in the linear regime. For the NO2 and CN substituents, strong effects from nonlinear coupling are observed. The disagreements between the various quantum chemical treatments (INDO and various ab initio approaches) are largest for molecules exhibiting nonlinear coupling effects, indicating that the nonlinear effects are more difficult to calculate than linear effects. (C) 2003 American Institute of Physics.