A full-band -method for solving the Kohn–Sham equation
Introduction
When performing first-principles calculations on crystals within density functional theory (DFT), the single-particle Kohn–Sham (KS) equation is solved for each k-point in the Brillouin zone (BZ) separately. The electron density is then obtained by an integration over the k-space. The same procedure, i.e., BZ integration on a discrete k-mesh, also applies to the calculation of many other properties of the crystal, such as the density-of-states (DOS) or the complex dielectric function , for which a dense grid usually is needed to obtain these quantities accurately. Given a complex crystal comprised of many atoms in the unit cell, or a low-dimensional system like a surface, this implies time-consuming calculations. A similar problem appears when describing the electronic band structure, where one has to calculate values at many k-points in certain directions of the BZ. In this work, we show, how to speed up the procedure by combining methods to solve the KS equation with the so-called -method [1], [2], [3], hence proposing a more suitable basis set for computing the eigenfunctions of solids.
The approach has been used since the nineteen-fifties in semiconductor physics to describe the electronic structure around the band gap. In this context, one usually expands the eigenfunctions at certain k-points into known eigenfunctions of a high-symmetry point (here denoted ), normally the s and p states of the Γ-point. This leads to typically a or Hamiltonian matrix, which is then solved numerically, or gives rise to an approximated energy expression involving the so-called Luttinger parameters [1]. In contrast, the Hamiltonian is in principle exact when taking into account a complete basis set of eigenfunctions, i.e. all bands of the given -point. In this context, one first calculates all these eigenstates, which then are used as the basis functions for the k-points around it.
There have been other attempts to use the method in combination with APW calculations. For example, convergence of the Al dielectric function with the number of k-points was achieved by a interpolation scheme based on six bands obtained in an APW calculation [4]. Much earlier, there was significant interest and effort at use of the scheme [5] with the APW basis. That effort ran into a difficulty: with only one reference point, , possible because of computer resource limitations, the basis set was inadequate. In this regard our approach differs substantially, as it also differs from the ELAPW-kp method [6] in which the scheme also is used with only one point to expand the APW basis set.
In this work, we show that the full-potential linearized augmented plane wave (FPLAPW) method for obtaining the eigenfunctions of some k-points in combination with the full-band approach yields a very accurate and efficient way of calculating electronic and optical properties of various solids. It turns out that one can even restrict the number of bands to a rather small size for accurately describing these quantities. We note that the matrix size is in particular much smaller than the number of basis functions in the underlying band structure method, e.g., the augmented plane waves within the FPLAPW method. Thus a matrix eigenvalue problem with a matrix size of typically only 10% of the original one has to be solved.
Section snippets
Theory
We follow the discussion by J. Callaway [7]. Let be a point in the BZ for which one knows the orthonormal Bloch functions and the corresponding eigenvalues of all the bands. One can make use of the Luttinger–Kohn functions [1] which obey the orthogonality relations to expand an arbitrary unknown eigenfunction at a point k Notice that the
Results
The present full-band -approach can in general be combined with any kind of computational scheme for solving the KS equation of periodic systems. It is independent of the choice of the external potential, and can be employed for spin-polarized systems and spin-spirals, and also in combination with various kinds of perturbations.
We have implemented this scheme into the FPLAPW program package WIEN2k [8]. To demonstrate the validity of the full-band -method and the convergence with respect
Conclusions and outlook
Having demonstrated the high accuracy of the method, we now can discuss the speed-up gained by our approach. For this purpose, the CPU-times are compiled in Table 1 for two examples, namely, the band structure of YBa2Cu3O7 and the dielectric function of PPP. The computing times per k point and point refer to the determination of the eigenstates, i.e. the solution of the KS equation (column 6) and setup and diagonalization of the Hamiltonian (column 8). Here, factors of 72 and 65,
Acknowledgements
This work was financially supported by the Swedish Research Council (VR), the Austrian Science Fund (project P16227), and the European Research and Training Network “EXCITING”, contract number HPRN-CT-2002-00317.
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