A fast linear scaling analytical Xalpha method & an adaptive importance function in Monte Carlo simulations

Abstract

The need for more accurate theoretical data pushes computational chemists to perform calculations at the quantum mechanical level on systems of ever increasing size and for comprehensive explorations of their potential energy surface. Fast and reliable ab initio methods must be developed in order to take advantage of the readily available and increasingly powerful computing resources. A grid-free analytical Xalpha functional was implemented, focusing on speed and scaling behaviour, within our own density functional theory software, DeFT. Appropriate basis sets for the requisite fits of rho⅓ and rho⅔ are constructed so as to have a method that is both fast and reliable. Our analytical Xalpha functional is tested against a conventional numerical approach that uses a grid to integrate the exchange-correlation contribution to the Kohn-Sham matrix. The analytical method reproduces the electronic energy and optimized geometrical parameters obtained with a sophisticated numerical grid. More importantly, compared to the numerical approach, it is 4 to 8 times faster for the evaluation of the Xalpha energy and 40 to 60 times faster for the evaluation of the Xalpha energy gradient on systems of medium to large sizes. The formal cubic scaling factor for the analytical Xalpha functional is improved to 1.31, ensuring that the method will remain the method of choice for larger systems. Finally, the utility of the analytical Xalpha functional as a low level method within hybrid schemes is demonstrated while taking advantage of its adaptive nature by treating alpha as an atom-dependent parameter. In a separate project, a novel adaptive semi-empirical importance function has been developed to better reproduce a quantum mechanical potential and improve the sampling efficiency of Monte Carlo simulations at the quantum mechanical level. The efficiency of our adaptive importance function is demonstrated in a proof of concept application focusing on the ring opening of cyclobutene. Using the adaptive importance function leads to a significant improvement in the sampling of all areas of the potential energy surface, especially near the transition state, when compared with a traditional single chain or when using a rigid importance function.