| Large molecules and clusters figure prominently in biophysics and nanoscience. With the advent of large computing platforms and novel algorithms, it is becoming feasible to simulate these systems at an accurate ab initio level. In this context, ab initio implies solving for the electronic wavefunction or density with a fixed configuration of nuclei, and perhaps updating the nuclear positions utilizing forces obtained from the electron density. In this way, highly accurate results can be obtained for systems with hundreds or even thousands of electrons. The predominant theoretical framework for such large calculations is currently density functional theory, since the Kohn-Sham method provides for efficient solution while including some degree of electron correlation. This dissertation is directed at the development of novel multiscale algorithms for making these electronic structure calculations more efficient.; Recently it has been shown that the higher-order real-space methods utilizing pseudopotentials can produce results in electronic structure calculations comparable to those of plane-wave methods. Multiscale methods provide efficient and robust algorithms for large scale electronic structure calculations. In this dissertation, I discuss multiscale methods to solve self-consistent eigenvalue problems for non-periodic systems such as molecules with pseudopotentials. The two most expensive operations on the fine grid are the Gram-Schmidt orthogonalization and the Ritz projection. It has been shown that, for systems with few wavefunctions or well defined cluster structures (degeneracies), these two operations can be brought to coarser levels. But the algorithm stalls in its original form when applied to realistic systems such as large molecules having tens of wavefunctions. I found a new method which is called Ritz projection performed on clusters along with GRBR to solve this problem. The main advantage of the new method is that it scales as N2e for modest-sized systems where Ne is the number of wavefunctions, compared to the Ritz projection method which scales as N3e . |
