On the origins of intersecting potential energy surfaces

Techniques developed for investigating nonadiabatic processes in molecular systems are adapted to study the structure and properties of holomorphic and meromorphic functions of a complex variable, \(f(z)=\mathfrak {R}(f)+i\,\mathfrak {I}(f)\). The connection is that \(\mathfrak {R}(f)\) and \(\mathfrak {I}(f)\) are correlated two-dimensional scalar functions, interrelated by the Cauchy–Riemann equations. Exploiting this fact, it is demonstrated that \(\mathfrak {R}(f)\) and \(\mathfrak {I}(f)\) of f can be envisaged in Euclidean \({\mathbb {R}}^{3}\) space as a two-state set of constrained, intersecting two-dimensional potential energy surfaces (PESs), called the graph of f. Importantly, the analytic and algebraic properties of f dictate the geometric structure evinced in the graph of f. This parallels multi-state sets of higher-dimensional, constrained, intersecting PESs linked with correlated electronic eigenstates of the parameterized molecular Hamiltonian operator. In view of this association, the language and mathematical infrastructure devised by chemists for discussing and analyzing intersections in higher-dimensional PESs are suitably modified for f. Notably, an algorithm capable of optimizing roots and poles of f through analysis of the real, two-dimensional \(\mathfrak {R}(f)\) and \(\mathfrak {I}(f)\) functions is derived, which is based on intersection-adapted coordinate and constrained Lagrangian methodologies. As constrained, intersecting PESs are indispensible for conceptualizing and characterizing the physics governing nonadiabatic phenomena, f represents a foundational bridge to these more abstract constructions.