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We obtain analytical expressions for the total magnetic moment and the static spin-correlation functions of the classical Heisenberg model for ultrasmall systems of spins ͑unit vectors͒, that interact via isotropic, nearest-neighbor ͑n-n͒ exchange and that are subject to a uniform dc magnetic field of arbitrary strength. Explicit results are presented for the dimer, equilateral triangle, square, and regular tetrahedron arrays of spins. These systems provide a useful theoretical framework for calculating the magnetic properties of several recently synthesized molecular magnets. The tetrahedron as well as the equilateral triangle systems, each considered for n-n antiferromagnetic exchange, are of particular interest since they exhibit frustrated spin ordering for sufficiently low temperatures and weak magnetic fields. ͓S0163-1829͑99͒01538-6͔

We obtain the exact critical relaxation time L ( ), where is the bulk correlation length, for the Glauber kinetic Ising model of spins on a one-dimensional lattice of finite length L for both periodic and free boundary conditions ͑BC's͒. We show that, independent of the BC's, the dynamic critical exponent has the well-known value zϭ2, and we comment on a recent claim that zϭ1 for this model. The ratio L ( )/ ϱ ( ), in the double limit L, →ϱ for fixed xϭL/ , approaches a limiting functional form, f (L/ ), the finite-size scaling function.

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We present a highly accurate, ab initio recursive algorithm for evaluating the Wigner 3 j and 6 j symbols. Our method makes use of two-term, nonlinear recurrence relations that are obtained from the standard threeterm recurrence relations satisfied by these quantities. The use of two-term recurrence relations eliminates the need for rescaling of iterates to control numerical overflows and thereby simplifies the widely used recursive algorithm of Schulten and Gordon.