Calculation of time autocorrelation and spectral functions using locally expanded potentials

We present theory for autocorrelation and spectral functions and for the description of molecular motions on a multidimensional potential energy surface. The basic idea is to express the autocorrelation functions by stepwise series expansions of the molecular surface using only local information. This approach thus presents one way to avoid the dimensionality catastrophe that is associated with dynamic studies using globally calculated and fitted polyatomic energy surfaces. The autocorrelation function is obtained by solving the quantum equations of motions in a set of locally defined internal coordinates uncoupled to rotation and translation. A Fourier transform of the autocorrelation function gives a time‐evolved interpretation of the formation of the vibronic spectrum. Locally expanded and quadratically approximated Hamiltonians are assumed throughout the reaction path. The formalism implements in a direct fashion the information obtained from molecular gradient and Hessian techniques, so that the buildup of the spectrum can be followed along with the reaction walks on the molecular hypersurface. The method applies to unbound or quasi‐bound as well as to bound potentials. In the latter case, the calculation of the autocorrelation function also provides a cost‐effective way to perform Franck–Condon analyses, since, in contrast to conventional Franck–Condon methods, it avoids sum‐over‐state and basisset procedures completely. A set of 3N – 6 Cartesian coordinates are analyzed for the purpose of translational and rotational‐free walks on a locally and quadratically expanded multidimensional potential energy surface without recoursing to the standard Eckart conditions and/or the necessity of performing supplementary transformations on the gradient vector and the Hessian matrix. Demonstrations of the formalism are given by calculations of time autocorrelation and spectral functions using locally expanded potentials corresponding to N₂²II_u and O₂²II_g photoionization.