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We address the question of an appropriate choice of basis functions for the self‐consistent field (SCF) method of simulation of the  N ‐body problem. Our criterion is based on a comparison of the orbits found in  N ‐body realizations of analytical potential–density models of triaxial galaxies, in which the potential is fitted by the SCF method using a variety of basis sets, with those of the original models. Our tests refer to maximally triaxial Dehnen γ‐models for values of γ in the range  0 ≤γ≤ 1 , i.e. from the harmonic core up to the weak cusp limit. When an  N ‐body realization of a model is fitted by the SCF method, the choice of radial basis functions affects significantly the way the potential, forces or derivatives of the forces are reproduced, especially in the central regions of the system. We find that this results in serious discrepancies in the relative amounts of chaotic versus regular orbits, or in the distributions of the Lyapunov characteristic exponents, as found by different basis sets. Numerical tests include the Clutton‐Brock and the Hernquist–Ostriker basis sets, as well as a family of numerical basis sets which are ‘close’ to the Hernquist–Ostriker basis set (according to a given definition of distance in the space of basis functions). The family of numerical basis sets is parametrized in terms of a quantity ɛ which appears in the kernel functions of the Sturm–Liouville equation defining each basis set. The Hernquist–Ostriker basis set is the ɛ= 0 member of the family. We demonstrate that grid solutions of the Sturm–Liouville equation yielding numerical basis sets introduce large errors in the variational equations of motion. We propose a quantum‐mechanical method of solution of the Sturm–Liouville equation which overcomes these errors. We finally give criteria for a choice of optimal value of ɛ and calculate the latter as a function of the value of γ, i.e. of the power‐law exponent of the radial density profile at the central regions of the galaxy.

Appropriate SCF basis sets for orbital studies of galaxies and a ‘quantum‐mechanical’ method to compute them