Linear scaling geometry optimisation and transition state search in hybrid delocalised internal coordinates

An algorithm for linear scaling geometry optimisation and transition state search using hybrid delocalised internal coordinates (HDLC) has been developed and implemented in the context of a semiempirical quantum-chemistry program (MNDO) and a modular quantum-mechanical/molecular-mechanical (QM/MM) package (ChemShell). Linear scaling is achieved by a divide-and-conquer approach: the system is partitioned into user-defined fragments, and all coordinate manipulations are performed exclusively within these fragments. The optimiser employs a limited-memory quasi-Newton algorithm (L-BFGS) for energy minimisation, and a microiterative scheme for transition state search using a Hessian eigenmode-following algorithm (P-RFO) for the reaction core and the L-BFGS algorithm for the environment. There are automatic procedures for generating redundant sets of internal coordinates and non-redundant sets of HDLC from Cartesian coordinates. The input to the optimiser consists of the initial Cartesian geometry, the fragmentation of the system, the choice of the working coordinate system, and any constraints to be imposed in Cartesian and/or internal coordinates. The optimiser requires an external function that provides the energy and gradient at a given Cartesian geometry. Systems with thousands of atoms have been optimised, and transition states of a model enzymatic reaction have been determined.