SUMMARY Ch. 12 "THE METHOD OF ORTHOGONALIZED PLANE WAVES (OPW)"
12.1 Introduction
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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
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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 ?
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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)
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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)".
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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).
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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
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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
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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.