SUMMARY Ch. 12 "THE METHOD OF ORTHOGONALIZED PLANE WAVES (OPW)" 12.1 Introduction ================= C. Herring, Phys. Rev. 57, 1169 (1940) The method to be descibed in this capture has a significantly better convergence than the plane wave method explained in Ch. 11. However, this better performance has its price: a more complicated mathematical apparatus of OPW in comparison to PW. 12.2 Definition of core functions ================================= The main reason for the poor convergence of the PW method is caused by the fact that - in this method - all electrons in the crystal are assumed to be equally delocalized, i.e. no more bounded to a special atom. This ansatz is not justified: in most solid state materials, one has to distinct between strongly localized "core" electrons and delocalized "valence" electrons (see Fig. 12.1: Core and valence electrons in metallic sodium). A different theoretical treatment of these two types of electrons is the basic feature of the OPW bandstructure method. A core function |k,j> is defined in Eq. (12.1): k is a Bloch vector, R a vector of the crystal lattice, N is the number of electrons within the volume of the crystal (Omega), and w_j(r) means an atomic-like wave function of a localized core electron at space point R in the quantummechanical state |j> . One can easily verify that functions defined in Eq. (12.1) obay Bloch's theorem phi_{k,j}(r+R) = exp{ikR} phi_{k,j}(r) . Henceforth, another property of these core functions will be used, namely their periodicity in the reciprocal space (k space) - see Eq. (12.2). What are the atomic functions w_j(r) used in Eq. (12.1)? One could assume that they are represented by eigenfunctions of the corresponding isolated atoms, but this is not the case, as it is shown in Eqs. (12.3)-(12.5): The main point of this calculation is that the core functions phi_{k,j}(r) are to describe the physics of electrons in a crystal; consequently, these functions have to be eigenfunctions of the "crystalline Hamiltonian" H including the crystal potential V(r) [Eq. (12.3)] and not simply the atomic potential! In the following, we write the crystal potential V(r) as a sum potentials v(r) centered in the unit cells of the crystal: V(r) = \sum_{R'} v(r-R') , and we assume that the CONDITION holds that the potential function and an atomic function centered in different unit cells [i.e., v(r-R') and w_j(r-R) for R .ne. R'] "do not overlap". In this case, the functions w_j(r) must be eigenfunctions of the Schroedinger equation (12.5) to the eigenvalues E_j: E_j is atomic-like, i.e., it does not disperse with the Bloch vector k. REMARK 1: the above condition is fulfilled in many cases (alkali metals, aluminum, silicon, ...), but, e.g. not for transition metals (vanadium, iron, nickel, ...). REMARK 2: to present the OPW formulas as simple as possible, all further explanations are given in case of Bravais lattices (only one electron within the unit cell). In most OPW calculations, the "cell potential" v(vec r) is spherically approximated by v(vec r) = v(|vec r|) . In that case, the eigenfunctions of Eq. (12.5) can be written as Eq. (12.6), where |j> mean the well known atomic quantum numbers |n,l,m>. 12.3 ARE THE CORE FUNCTIONS ORTHOGONAL ? ========================================= In this section, it is shown that, provided that atomic functions centered around different unit cells [w(r-R) and w(r-R')] do not significantly overlap, one gets the relation Eq. (12.7). 12.4 THE ORTHOGONALIZED PLANE WAVE (OPW) ========================================= We are now going to define "new" basis functions to describe solutions of the one-particle Schroedinger equation (11.1). These basis functions shall have the following properties: (*) In order to improve the convergence of the PW method, the plane-wave ansatz is to be enlarged by a "core term" which enables a better description of the Bloch wavefunction psi_k(r) especially nearby the atomic nuclei. This goal can be reached by the definition of a function phi_k(r) according to Eq. (12.8): this function consists of a plane-wave and of a linear combination of core functions. (*) The function (12.8) is evidently a Bloch function, because both terms of it fulfills Bloch's condition. (*) Now to the CENTRAL POINT of the theory: the coefficients mu_{k,j} in Eq. (12.8) are chosen according to the orthogonality condition (12.9): "Each function phi_k(r) is orthogonal to all core functions |k,j>". If this condition is fulfilled, the function (12.8) is called an "orthogonalized plane wave (OPW)". *************************************************************************** What is the consequence of such choice of basis functions for the expansion of electron wave functions in a crystal? The most important consequence is that the (pre-calculated) wavefunctions of the core electrons are not part of the Hilbert space of the functional basis used in the calculation; that means that the Hamilton matrix obtained by a basis (12.8), (12.9) will only describe the valence part of the electronic system (i.e., its eigenvalues and eigenfunctions are only eigenvalues and eigenfunctions of the valence electrons of the system investigated). *************************************************************************** The calculation of the orthogonalization coefficients mu_{k,k} in Eq. (12.8) is described in Eqs. (12.10) to (12.14). At the end of Sec. 12.4, it is shown that the orthogonality condition (12.9) is also valid in the more general case (12.16). This result will be of some importance in the next section. 12.5 THE SECULAR MATRIX OF THE OPW =================================== Using the prevously explained OPW basis, the wavefunctions of the valence electrons can be expanded according to Eq. (12.17) psi_k(r) = \sum_{1}^{infin} a_t(k) phi_{k,t}(r) . The t^{th} OPW function is given by Eq. (12.18) [for this result, one uses the periodicity of the core functions in the reciprocal space, see Eq. (12.2)]. The following procedure is quite the same like it has been explained for the PW expansion: the ansatz for the wavefunction (12.17) is inserted into the Schroedinger equation, resulting a linear system of homogeneous equations of (theoretically) infinite order for the expansion coefficients a_t(k), see Eq. (12.19). The corresponding matrix in [....] is called the secular matrix for this problem. As for the PW case, this matrix consists of the Hamilton matrix H and the structure matrix S. On page 156 of the lecture notes, the evaluation of these matrices is presented in detail: these calculations are not difficult but tedious, and their results are given in Eqs. (12.20) and (12.21). These results are remarkable: each element of the S and the H matrix consists of two terms, where the first ones are exacty those of the PW result, and the second ones represent the "OPW coerrection". Especially, the result (12.20) shows that the OPW structure matrix is no more simply the identity matrix than in the PW case! Including the results (12.20), (12.21) into Eq. (12.19) gives the system of OPW equations: Eq. (12.22). REMARK 1: As it can be easily seen, both matrices S and H are self-adjoint. REMARK 2: We again emphasize here one important feature of the OPW method: the wavefunctions w_j(r) and the eigenenergies E_j of the core electrons in the crystal must have been pre-calculated before the OPW calculation starts! REMARK 3: The last remark notes an important DISADVANTAGE of the OPW method. it can only be applied to crystals where the bandstructure of the core electrons and the bandstructure of the valence electrons can (energetically) be clearly separated! The greatest ADVANTAGE of OPW is - as it will shown in the next section - its fast convergence which is much better than in case of the PW method. 12.6 APPLICATION OF THE OPW METHOD ON LITHIUM ============================================== Let's go back to the standard example of this lecture, the simplest metallic system lithium. The atoms that build up this metal have only one core state, the 1s state with n=1, l=m=0, and the corresponding atomic-like wavefunction is given as w_{100}(vec r) = R_{10}(r) Y_{00}(delta,phi) = R_{10}(r) / sqrt(4 pi). The radial function R_{10}(r) and the corresponding eigenvalue of the energy E_{10} = -3.064 Ry has been pre-calculated by numerically evaluating the differential equation (12.5). For this calculation, exactly the same crystal potential has been used as for the PW calculations in Ch. 11. Consequently, there is only one orthogonality coefficient [according to Eq. (12.14)], namely mu_{k,100} = sqrt(4 pi/omega0) int_0^{infin} dr r^2 R_{10}(r) sin(kr)/(kr) with omega0 as the volume of the unit cell. The integral over r in the above function is shown in Fig. 12.2. Fr the example Li, the OPW matrix elements (12.20) and (12.21) are given in Eqs. (12.23) and (12.24). Using these expressions in the eigenvalue problem (12.19), one obtains the bandstructure of the VALENCE electrons in metallic lithium. The efficiency of the OPW method for the example Li is demonstrated in the following figures: Fig. 12.3: The convergence of the OPW valence bandstructure, shown by comparing the bands resulting from a Hamilton matrix of the order of 55 (black points) and 87 (red points). Fig. 12.4: The convergence of the OPW method applied to Li, demonstrated by showing the band width of the s band as a function of the order of the Hamilton matrix [= the number of reciprocal-lattice vectors used for the ansatz (12.17)]. At the begin of this chapter, it has been emphasized that it is important the calculate the atomic-like(!) eigenenergies and wavefunctions of the core electrons with the use of the crystal potential and not the potential of a single atom. The importance of this statement shall now be demonstrated for the test case lithium: In Fig. 12.5, the left panel shows the difference between the potential within a lithium unit cell ("Li metal") and the potential of a lithium atom ("Li atom"). The right panel gives the corresponding curves for the function R_{10}(r) . As it can be seen, the differences between these functions are quite small! Taking this into account, the results of Table 12.1 are really surprising: in this table you find energy values of two different OPW calculations, namely (left column) by using core functions obtained with the "correct" crystal potential, and (right column) with the atomic potential. As you can see, the deviations between the energy values are partially considerable! In our times, the OPW method is very seldomly used for scientific calculations. Nevertheless, they indeed play a very important role in modern solid state physics, because they are the "starting point" for a bandstructure method which is very frequently used in the actual science, especially in semiconductor physics: I speak about the so-called "Method of Pseudopotentials" which is the topic of the following chapter.