  A pair of magnetic atoms with canted spins Sa, Sb can give rise to an
electric dipole moment P. Several forms for the behavior of such a moment have
appeared in the theoretical literature, some of which have been invoked to
explain experimental results found in various multiferroic materials. The forms
specifically are P1 ~ R x (Sa x Sb); P2 ~ Sa x Sb, and P3 ~ Sa (R . Sa) - Sb (R
. Sb), where R is the relative position of the atoms and Sa, Sb are unit
vectors. To unify and generalize these various forms we consider P as the most
general quadratic function of the spin components that vanishes whenever Sa and
Sb are collinear, i.e. we consider the most general expressions that require
spin canting. The study reveals new forms. We generalize to the vector P,
Moriya's symmetry considerations regarding the (scalar) Dzyaloshinskii-Moriya
energy D. Sa x Sb (which led to restrictions on D). This provides a rigorous
symmetry argument which shows that P1 is allowed no matter how high the
symmetry of the atoms plus environment, and gives restrictions for all other
contributions. The analysis leads to the suggestion of terms omitted in the
existing microscopic models, suggests a new mechanism behind the
ferroelectricity found in the 'proper screw structure' of CuXO2, X=Fe, Cr, and
predicts an unusual antiferroelectric ordering in the antiferromagnetically and
ferroelectrically ordered phase of RbFe(MoO4)2.
