Abstract. Multi-dimensional potential energy surfaces are associated with optical binding. A detailed exploration of the available degrees of geometric freedom reveals unexpected turning points, producing intricate patterns of local force and torque. Although optical pair interactions outweigh Casimir-Polder coupling even over short distances, the forces are not always attractive. Numerous local potential minimum and maximum can be located, and mapped on contour diagrams. Islands of stability appear, and structures conducive to the formation of rings. The results, based on quantum electrodynamics, apply to optically trapped molecules, nanoparticles, microparticles and colloids.Keywords: Optical binding, optical matter, quantum electrodynamics, optical manipulation, nano-manipulation Electrically neutral molecules and larger nanoparticles, separated from each other beyond significant wavefunction overlap, generally experience weak forces of mutual attraction. Typically, such forces are associated with pair potentials characterized by an inverse sixth power dependence on inter-particle distance. In the case of molecules, the latter feature (the attractive component of a Lennard-Jones interaction) is interpreted as a dynamic coupling of fluctuating electric dipoles. Experimentally verifiable retardation effects modify the distance dependence at longer distances, exhibiting the deeply quantum electrodynamical character of the Casimir-Polder interaction [1-3] -ultimately attributable to virtual photon exchange. The additive effect of pairwise interactions between the constituents of larger particles conveys an inverse sixth power distance relationship into the Hamaker interaction [4], where it determines properties such as adhesion, wetting, multilayer adsorption and colloid flocculation [5,6].A suspicion that intense laser light might engage with virtual photon exchange, producing an optically modified potential energy surface, was first developed into theory by Thirunamachandran almost thirty years ago [7] -but the laser intensities that appeared necessary then represented a major deterrent. Within the decade, however, a landmark paper by Burns et al. [8] verified the effect experimentally, and introduced the term 'optical binding'; this work also drew attention to the long-range linearly inverse dependence on distance, tempered by an oscillatory factor. The study inspired increasingly adventurous theoretical and experimental investigations [9][10][11][12][13][14][15][16][17][18][19][20][21][22] and the development of quantum electrodynamical studies [23,24]. Mostly, attention has focused on the intensity and distance dependence of the pair forces. The pair separation is, however, only one of the geometric degrees of freedom. In this Letter, we present the first results to emerge from an investigation into the complete potential energy surfaces for optically induced interactions. Despite a relative simplicity in the field equations, a richly textured potential energy landscape emerges.For a pair of particles, B di...