The resonant dipole-dipole interaction is calculated in an ideal 1 D waveguide and compared to the results expected in a 3D photonic band-gap structure . IntroductionSince Purcell's initial observation that altering the electromagnetic mode structure surrounding an atom can change its spontaneous emission rate [1], several basic quantum electrodynamic phenomena have been studied in this context . These include the inhibition of spontaneous emission by atoms, molecules, and electrons [2][3][4], as well as basic intermolecular interactions such as the potential between an atom and a wall [5, 6] . Most recently, it has been shown that a three-dimensional (3D) dielectric structure can also produce strong electromagnetic mode exclusion effects [7-10] . Such structures, constructed for microwave frequencies, have been shown to exhibit a full photonic band-gap (PBG) which excludes a set of electromagnetic modes propagating in all directions [11]. In addition, it has been experimentally verified that the 3D periodic structure can inhibit spontaneous emission in the visible region of the spectrum [12] .The concept of a PBG structure has also generated a number of theoretical calculations on a variety of molecular quantum electrodynamic effects such as molecular interaction potentials [13, 14] . The possibility of controlling the molecular interaction potentials in the same way as spontaneous emission has many ramifications for a number of chemical phenomena including the control of reaction rates and laser processes .In this paper, we will discuss the particular interaction between two identical two-level atoms, one of which is in the upper state . This process is important in determining molecular bonding energy surfaces and is an important line broadening mechanism in the absorption spectra of dipolar molecules . The process, referred to as resonant dipole-dipole interaction (RDDI), is calculated for two molecules in a PBG structure, where the atomic transition energy lies within the region of excluded modes, and in a rectangular waveguide below cut-off . The rectangular waveguide has the advantage that since the exact vector potential can be obtained, the interaction potential can be determined without any approximations . The calculation of the interaction potential can be determined without any approximations . The calculation of the interaction potential is performed using the multipolar Hamiltonian method in the electric dipole approximation . The interaction Hamiltonian is given by :

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