The Linear Scaling Algorithms For Localized Molecular Orbitals In Large Systems

In the past decades, quantum chemistry calculations have become more and more important in the areas of chemistry, physics, biology, material science, and etc. Since the computational costs of the conventional quantum chemistry methods increase steeply with the system size, it is still difficult to extend the applications of ab initio quantum chemistry methods, especially electron correlation methods, to very large systems. In order to overcome this difficulty, many linear scaling electronic structure methods for large systems have been developed. Among these methods, local correlation methods have received much attention. In the local correlation methods, the equations for configuration amplitudes are reformulated in terms of localized molecular orbtials (LMOs)(or atomic orbitals), and these equations can be dramatically simplified by neglecting correlation between distant orbitals. However, most of present local correlation methods have not achieved a complete linear scaling for large systems, because the conventional localization algorithms, which scale cubically with the system size, are still employed to construct occupied and virtual LMOs for large systems. Although the conventional localization algorithms have small prefactors, their computational costs may dominate for sufficiently large molecules. In order to achieve the complete linear scaling for electron correlation calculations or extend excited state calculations to large systems, it is very desirable to develop linear scaling algorithms for computing LMOs in large systems.The main content of this thesis is to develop linear scaling algorithms to construct occupied and virtual LMOs in large systems. With the present algorithms for computing LMOs, one can achieve truly linear scaling post-HF calculations for sufficiently large systems or extend excited state calculations to large systems. These algorithms include a linear scaling algorithm for the construction of occupied LMOs based on the sparse matrix technique, an improved fragment-based algorithm to construct occupied LMOs, and a linear scaling algorithm to obtain virtual LMOs. The main contributions of the present work can be summarized as follows:1. In chapter2, we have developed a linear scaling algorithm for obtaining the Boys LMOs from the one-particle density matrix. The algorithm is made up of two steps:the Cholesky decomposition of the density matrix to obtain Cholesky molecular orbitals (MOs) and the subsequent Boys localization process. Linear scaling algorithms have been proposed to achieve linear scaling calculations of these two steps, based on the sparse matrix technique and the locality of the Cholesky MOs. The present algorithm has been applied to compute the Boys localized orbitals in a number of systems including a-helix peptides, water clusters, and proteins. Illustrative calculations demonstrate that the computational time of obtaining Boys localized orbitals with the present algorithm is asymptotically linear with increasing the system size.2. In chapter3, an improved localized molecular-orbital assembler (LMOA) approach is developed for constructing occupied LMOs of general large systems. Different from the original LMOA approach, each subsystem is placed into background point charges, which are used to model the electrostatic and polarization interactions between a given subsystem and all other atoms (beyond this subsystem). From separate calculations on such "embedded" subsystems, occupied LMOs from different subsystems are then assembled to construct the total HF density matrix of a large system, from which the ground-state HF energy or energy gradients can be approximately computed. For a wide range of neutral and charged systems, the present LMOA approach is demonstrated to offer a significant improvement on the original LMOA approach, providing satisfactory descriptions on their total HF energies. The present LMOA approach is also implemented for energy gradient calculations and geometry optimizations. The energy gradients and optimized structures obtained the LMOA approach agree well with those from the conventional HF calculations.3. In chapter4, we have developed a linear scaling algorithm for computing the virtual LMOs of large systems within the Boys localization scheme, which is based on the linear scaling algorithm for computing occupied LMOs (Chapter2). The basic idea of this algorithm is to divide a large system into many small fragments and assemble the virtual LMOs on the fragments into a set of approximate virtual LMOs of the whole system. The virtual LMOs on each fragment are constructed in terms of projected atomic basis functions within an extended fragment, which consists of a central fragment and its neighboring fragments. The present algorithm has been applied to compute the virtual LMOs of several systems including α-helix peptides, water clusters, and DNA double helices. Illustrative calculations demonstrate that the computational time of obtaining virtual LMOs with the present algorithm is asymptotically linear with increasing the system size. The quality of the resulting virtual LMOs is verified by excited state calculations using the configuration interaction singles (CIS) method.