Geometry optimization with QM/MM methods II: Explicit quadratic coupling

Geometry optimization of large QM/MM systems is usually carried out by alternating a second-order optimization of the QM region using internal coordinates ('macro- iterations'), and a first-order optimization of the MM region using Cartesian coordinates ('microiterations'), until self-consistency. However, the neglect of explicit coupling between the two regions (the Hessian elements that couple the QM coordinates with the MM coordinates) often interferes with a smooth convergence, while the Hessian update procedure can be unstable due to the presence of multiple minima in the MM region. A new geometry optimization scheme for QM/MM methods is proposed that addresses these problems. This scheme explicitly includes the coupling between the two regions in the QM optimization step, which makes it quadratic in the full space of coordinates. Analytical second derivatives are used for the MM contributions, with O(N) memory and CPU requirements (where N is the total number of atoms) by employing direct and fast multipole methods. The explicit coupling improves the convergence behaviour, while the Hessian update is stable since it no longer involves MM contributions. Examples show that the new procedure performs significantly better than the standard methods.