Recent advances in the theory of polarization and the development of linear-scaling methods have revitalized interest in the use of Wannier functions for obtaining a localized orbital picture within a periodic supercell. To examine complex chemical systems it is imperative for the localization procedure to be efficient; on the other hand, the method should also be simple and general. Motivated to meet these requirements we derive in this paper a spread functional to be minimized as a starting point for obtaining maximally localized Wannier functions through a unitary transformation. The functional turns out to be equivalent to others discussed in the literature with the difference of being valid in simulation supercells of arbitrary symmetry in the ⌫-point approximation. To minimize the spread an iterative scheme is developed and very efficient optimization methods, such as conjugate gradient, direct inversion in the iterative subspace, and preconditioning are applied to accelerate the convergence. The iterative scheme is quite general and is shown to work also for methods first developed for finite systems ͑e.g., Pipek-Mezey, Boys-Foster͒. The applications discussed range from crystal structures such as Si to isolated complex molecules and are compared to previous investigations on this subject.

We present a modified version of the polarized atomic orbital (PAO) method [M. S. Lee and M. Head-Gordon, J. Chem. Phys. 107, 9085 (1997)] to construct minimal basis sets optimized in the molecular environment. The minimal basis set derives its flexibility from the fact that it is formed as a linear combination of a larger set of atomic orbitals. This approach significantly reduces the number of independent variables to be determined during a calculation, while retaining most of the essential chemistry resulting from the admixture of higher angular momentum functions. Furthermore, we combine the PAO method with linear scaling algorithms. We use the Chebyshev polynomial expansion method, the conjugate gradient density matrix search, and the canonical purification of the density matrix. The combined scheme overcomes one of the major drawbacks of standard approaches for large nonorthogonal basis sets, namely numerical instabilities resulting from ill-conditioned overlap matrices. We find that the condition number of the PAO overlap matrix is independent from the condition number of the underlying extended basis set, and consequently no numerical instabilities are encountered. Various applications are shown to confirm this conclusion and to compare the performance of the PAO method with extended basis-set calculations.

Following an approach to density functional theory calculations based on the matrix representation of operators, we implemented a scheme as an alternative to traditional grid-based methods. These techniques allow integrals over exchange-correlation operators to be evaluated through matrix manipulations. Both local and gradient-corrected functionals can be treated in a similar way. After deriving all the required expressions, selected examples with various functionals are given.

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