Complexity and Geometry of Potential Energy Surfaces: A Structural NP-Hardness Framework

Potential energy surfaces (PESs) are a fundamental mathematical concept used to describe molecular systems. Despite the central role of searching for the global minimum in theoretical chemistry applications, there is still no general explanation for the associated computational difficulty. In this study, we present a theoretical framework linking the geometry of PES to computational complexity theory principles. We introduce formal mapping between continuous molecular configurations and discrete optimisation problems, constructing polynomial-time reductions from NP-hard problems, such as 3-SAT and frustrated Ising models, to the global optimisation of generic classes of PES. In this representation, bistable conformational degrees of freedom encode discrete variables, while suitable interaction terms reproduce the energy structure of the original combinatorial problems. This enables the intractability of global optimisation to be interpreted as a structural property of the geometry of the energy landscape. The proposed framework suggests that phenomena such as the exponential proliferation of local minima, energy frustration, fragmentation of the configuration space, and dynamics characterised by metastable trapping, emerge as geometric manifestations of the underlying computational complexity. This distinguishes between 'smooth' landscapes, which are associated with relatively tractable structures, and 'rugged' landscapes, in which the search for the global minimum is inherently difficult. This perspective unifies concepts from complexity theory, landscape theory, and the physics of disordered systems. It provides a possible basis for a 'complexity theory of molecular systems' and for developing exploration and optimisation strategies that consider the limits imposed by computational complexity.