  There is recent interest in the inter and intra element interactions of
metamaterial unit cells. To calculate the effects of these interactions which
can be substantial, an ab initio general coupled mode equation, in the form of
an eigenvalue problem, is derived. The solution of the master equation gives
the coupled frequencies and fields in terms of the uncoupled modes. By doing
so, the problem size is limited to the number of modes rather than the, usually
large, discretized spatial and temporal domains obtained by fullwave solvers.
Therefore, the method can be considered as a numerical recipe which determines
the behavior of a complex system once its simpler ingredients are known.
Besides quantitative analysis, the coupled mode equation proposes a pictorial
view of the split rings hybridization. It can be regarded as the
electromagnetic analog of molecular orbital theory. The solution of the
eigenvalue problem for different configurations gives valued information and
insight about the coupling of metamaterials unit cells. For instance, it is
shown that the behavior of split rings as a function of the relative position
and orientation can be systematically explained. This is done by singling out
the effect of each relevant parameter such as the coupling coefficient and
coupled induced frequency shift coefficients.
