International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D, ch. 2.2, pp. 301-303
Section 2.2.11. Band-theory methods^{a}Institut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria |
There are several methods for calculating the electronic structure of solids. They have advantages and disadvantages, different accuracies and computational requirements (speed or memory), and are based on different approximations. Some of these aspects have been discussed in Section 2.2.9. This is a rapidly changing field and thus only the basic concepts of a few approaches in current use are outlined below.
For the description of crystalline wavefunctions (Bloch functions), one often starts with a simple concept of placing atomic orbitals (AOs) at each site in a crystal denoted by , from which one forms Bloch sums in order to have proper translational symmetry: Then Bloch functions can be constructed by taking a linear combination of such Bloch sums, where the linear-combination coefficients are determined by the variational principle in which a secular equation must be solved. The LCAO can be used in combination with both the Hartree–Fock method and DFT.
A simple version of the LCAO is found by parameterizing the matrix elements and in a way similar to the Hückel molecular orbital (HMO) method, where the only non-vanishing matrix elements are the on-site integrals and the nearest-neighbour interactions (hopping integrals). For a particular class of solids the parameters can be adjusted to fit experimental values. With these parameters, the electronic structures of rather complicated solids can be described and yield quite satisfactory results, but only for the class of materials for which such a parametrization is available. Chemical bonding and symmetry aspects can be well described with such schemes, as Hoffmann has illustrated in many applications (Hoffmann, 1988). In more complicated situations, however, such a simple scheme fails.
In many respects, core electrons are unimportant for determining the stability, structure and low-energy response properties of crystals. It is a well established practice to modify the one-electron part of the Hamiltonian by replacing the bare nuclear attraction with a pseudo-potential (PP) operator, which allows us to restrict our calculation to the valence electrons. The PP operator must reproduce screened nuclear attractions, but must also account for the Pauli exclusion principle, which requires that valence orbitals are orthogonal to core ones. The PPs are not uniquely defined and thus one seeks to satisfy the following characteristics as well as possible:
There are many versions of the PP method (norm-conserving, ultrasoft etc.) and the actual accuracy of a calculation is governed by which is used. For standard applications, PP techniques can be quite successful in solid-state calculations. However, there are cases that require higher accuracy, e.g. when core electrons are involved, as in high-pressure studies or electric field gradient calculations (see Section 2.2.15), where the polarization of the charge density close to the nucleus is crucial for describing the physical effects properly.
The partition of space (i.e. the unit cell) between (non-overlapping) atomic spheres and an interstitial region (see Fig. 2.2.12.1) is used in several schemes, one of which is the augmented plane wave (APW) method, originally proposed by Slater (Slater, 1937) and described by Loucks (1967), and its linearized version (the LAPW method), which is chosen as the one representative method that is described in detail in Section 2.2.12.
The basis set is constructed using the muffin-tin approximation (MTA) for the potential [see the discussion below in connection with (2.2.12.5)]. In the interstitial region the wavefunction is well described by plane waves, but inside the spheres atomic-like functions are used which are matched continuously (at the sphere boundary) to each plane wave.
In the KKR scheme (Korringa, 1947; Kohn & Rostocker, 1954), the solution of the KS equations (2.2.10.3) uses a Green-function technique and solves a Lippman–Schwinger integral equation. The basic concepts come from a multiple scattering approach which is conceptually different but mathematically equivalent to the APW method. The building blocks are spherical waves which are products of spherical harmonics and spherical Hankel, Bessel and Neumann functions. Like plane waves, they solve the KS equations for a constant potential. Augmenting the spherical waves with numerical solutions inside the atomic spheres as in the APW method yields the KKR basis set. Compared with methods based on plane waves, spherical waves require fewer basis functions and thus smaller secular equations.
The radial functions in the APW and KKR methods are energy-dependent and so are the corresponding basis functions. This leads to a nonlinear eigenvalue problem that is computationally demanding. Andersen (1975) modelled the weak energy dependence by a Taylor expansion where only the first term is kept and thereby arrived at the so-called linear methods LMTO and LAPW.
The LMTO method (Andersen, 1975; Skriver, 1984) is the linearized counterpart to the KKR method, in the same way as the LAPW method is the linearized counterpart to the APW method. This widely used method originally adopted the atomic sphere approximation (ASA) with overlapping atomic spheres in which the potential was assumed to be spherically symmetric. Although the ASA simplified the computation so that systems with many atoms could be studied, the accuracy was not high enough for application to certain questions in solid-state physics.
Following the ideas of Andersen, the augmented spherical wave (ASW) method was developed by Williams et al. (1979). The ASW method is quite similar to the LMTO scheme.
It should be noted that the MTA and the ASA are not really a restriction on the method. In particular, when employing the MTA only for the construction of the basis functions but including a generally shaped potential in the construction of the matrix elements, one arrives at a scheme of very high accuracy which allows, for instance, the evaluation of elastic properties. Methods using the unrestricted potential together with basis functions developed from the muffin-tin potential are called full-potential methods. Now for almost every method based on the MTA (or ASA) there exists a counterpart employing the full potential.
Conventional quantum-mechanical calculations are done using the Born–Oppenheimer approximation, in which one assumes (in most cases to a very good approximation) that the electrons are decoupled from the nuclear motion. Therefore the electronic structure is calculated for fixed atomic (nuclear) positions. Car & Parrinello (1985) suggested a new method in which they combined the motion of the nuclei (at finite temperature) with the electronic degrees of freedom. They started with a fictitious Lagrangian in which the wavefunctions follow a dynamics equation of motion. Therefore, the CP method combines the motion of the nuclei (following Newton's equation) with the electrons (described within DFT) into one formalism by solving equations of motion for both subsystems. This simplifies the computational effort and allows ab initio molecular dynamics calculations to be performed in which the forces acting on the atoms are calculated from the wavefunctions within DFT. The CP method has attracted much interest and is widely used, with a plane-wave basis, extended with pseudo-potentials and recently enhanced into an all-electron method using the projector augmented wave (PAW) method (Blöchl, 1994). Such CP schemes can also be used to find equilibrium structures and to explore the electronic structure.
The various techniques outlined so far have one thing in common, namely the scaling. In a system containing N atoms the computational effort scales as , since one must determine a number of orbitals that is proportional to N which requires diagonalization of matrices, where the prefactor k depends on the basis set and the method used. In recent years much work has been done to devise algorithms that vary linearly with N, at least for very large N (Ordejon et al., 1995). First results are already available and look promising. When such schemes become generally available, it will be possible to study very large systems with relatively little computational effort. This interesting development could drastically change the accessibility of electronic structure results for large systems.
References
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Car, R. & Parrinello, M. (1985). Unified approach for molecular dynamics and density-functional theory. Phys. Rev. Lett. 55, 2471–2474.
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