text: Spectral projectors of second order differential operators play an important role in quantum physics and other scientific and engineering applications. In order to resolve local features and to obtain converged results, typically the number of degrees of freedom needed is much larger than the rank of the spectral projector. This leads to significant cost in terms of both computation and storage. In this paper, we develop a method to construct a basis set that is adaptive to the given differential operator. The basis set is systematically improvable, and the local features of the projector is built into the basis set. As a result the required number of degrees of freedom is only a small constant times the rank of the projector. The construction of the basis set uses a randomized procedure and only requires applying the differential operator to a small number of vectors on the global domain, while each basis function itself is supported on strictly local domains and is discontinuous across the global domain...
  @cite_22: SUMMARY In this paper, classical and enriched flnite element formulations to impose Bloch-periodic boundary conditions are proposed. Bloch-periodic boundary conditions arise in the description of wave-like phenomena in periodic media. We consider the quantum-mechanical problem in a crystalline solid, and derive the weak formulation and matrix equations for the Schrodinger and Poisson equations in a parallelepiped unit cell under Bloch-periodic and periodic boundary conditions, respectively. For such second-order problems, these conditions consist of value- and derivative-periodic parts. The value-periodic part is enforced as an essential boundary condition by construction of a value-periodic basis, whereas the derivative-periodic part is enforced as a natural boundary condition in the weak formulation. We show that the resulting matrix equations can be obtained by suitably specifying the connectivity of element matrices in the assembly of the global matrices or by modifying the Neumann
  @cite_54: We have developed and implemented a selfconsistent density functional method using standard norm-conserving pseudopotentials and a flexible, numerical linear combination of atomic orbitals basis set, which includes multiple-zeta and polarization orbitals. Exchange and correlation are treated with the local spin density or generalized gradient approximations. The basis functions and the electron density are projected on a real-space grid, in order to calculate the Hartree and exchange-correlation potentials and matrix elements, with a number of operations that scales linearly with the size of the system. We use a modified energy functional, whose minimization produces orthogonal wavefunctions and the same energy and density as the Kohn-Sham energy functional, without the need for an explicit orthogonalization. Additionally, using localized Wannier-like electron wavefunctions allows the computation time and memory required to minimize the energy to also scale linearly with the size of the system. Forces and stresses are also calculated efficiently and accurately, thus allowing structural relaxation and molecular dynamics simulations.
  @cite_19: A simple and practical method for variationally optimizing numerical atomic orbitals used in density functional calculations is presented based on the force theorem. The derived equation provides the same procedure for the optimization of atomic orbitals as that for the geometry optimization. The optimized orbitals well reproduce convergent results calculated by a larger number of unoptimized orbitals. In addition, we demonstrate that the optimized orbitals significantly reduce the computational effort in the geometry optimization, while keeping a high degree of accuracy.
  @cite_43: We describe a complete set of algorithms for ab initio molecular simulations based on numerically tabulated atom-centered orbitals (NAOs) to capture a wide range of molecular and materials properties from quantum-mechanical first principles. The full algorithmic framework described here is embodied in the Fritz Haber Institute “ab initio molecular simulations” (FHI-aims) computer program package. Its comprehensive description should be relevant to any other first-principles implementation based on NAOs. The focus here is on density-functional theory (DFT) in the local and semilocal (generalized gradient) approximations, but an extension to hybrid functionals, Hartree–Fock theory, and MP2 GW electron self-energies for total energies and excited states is possible within the same underlying algorithms. An all-electron full-potential treatment that is both computationally efficient and accurate is achieved for periodic and cluster geometries on equal footing, including relaxation and ab initio molecular dynamics. We demonstrate the construction of transferable, hierarchical basis sets, allowing the calculation to range from qualitative tight-binding like accuracy to meV-level total energy convergence with the basis set. Since all basis functions are strictly localized, the otherwise computationally dominant grid-based operations scale as O(N) with system size N. Together with a scalar-relativistic treatment, the basis sets provide access to all elements from light to heavy. Both low-communication parallelization of all real-space grid based algorithms and a ScaLapack-based, customized handling of the linear algebra for all matrix operations are possible, guaranteeing efficient scaling (CPU time and memory) up to massively parallel computer systems with thousands of CPUs.
  @cite_13: In studies of the electronic structure of solids, the augmented plane wave (APW) method is the basis for the solution of the Kohn–Sham equations of density functional theory (DFT). The different versions and developing steps are discussed in terms of linearization, full potential, local orbitals, mixed basis sets, relativistic effects and computational aspects, as employed in the WIEN2k code.
summary: In the context of quantum physics, many tailored basis set have been designed to reduce the number of DOFs to represent spectral projectors (or density matrices in physics terminology). Notable examples include the Gaussian basis set and the atomic orbital basis set @cite_54 @cite_19 @cite_43 . Such basis sets are developed based on physical intuition, and can provide relatively accurate solution with much reduced number of degrees of freedom compared to more conventional basis sets. However, expert knowledge is often required to systematically converge the solution. These basis sets have also been used to enrich'' conventional basis sets to achieve a balance between the small number of DOFs and the systematic convergence property @cite_13 @cite_22 . However, the number of DOFs in the mixed basis representation is often much larger than those using Gaussian orbital or atomic orbital basis sets alone.