SUMMARY Ch. 15 "THE APW METHOD"
15.1 Introduction
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At the end of Ch. 14 ("Cell methods"), it has been demonstrated how
useful the so-called "muffin-tin" concept is. Approximating the
true crystal potential by a spherically symmetric potential inside
the muffin-tin sphere with radius r_MT, and by a CONSTANT potential
between the muffin-tin spheres (in the "interstitial region"),
the solution of the Schroedinger equation can be defined as follows:
(*) inside the MT-sphere, the solution is given by a linear expansion
of radial functions and spherical harmonics (see Eq. 14.6),
(*) within the interstitial region, the solution by given by a simple
plane wave with wavevector k (see Eq. 14.7).
These two parts can be put together to a so-called "augmented plane
wave" (APW) [(Eq. (14.9)] which is a solution of the Schroedinger
equation for the whole Wigner-Seitz cell; it is continuous in the
whole cell including the surface of the MT sphere
What concerns the first derivative of such an APW, it is also
continuous within the WSC EXCEPT THE SURFACE OF THE MUFFIN-TIN
SPHERE.
This last property means that one single APW cannot be considered
a quantum mechanically correct wavefunction. Only a linear expansion
of APW's can give such a solution [see Eqs. (14.10) and (15.1)].
Starting with Eq. (15.1), our problem is to find a set of coefficients
a_s(k) that yields a function which is continuous both with respect
to its values and first derivatives in the whole WSC including the
surface of the muffin-tin sphere.
In the following, will shall prove that such a function will also be
an Eigenfunction of the crystal Hamiltonian.
The method is the following one: we use Eq. (15.1) as a trial function
within a variational ansatz for non-continous functions.
15.2 The variational method
===========================
Mathematically, such an ansatz is provided by Eq. (15.2) in the lecture
notes. One can easily see that the usual formula is enlarged by 2
surface integrals: the first one is caused by the non-continuous trial
function (u_a .ne. u_i that means u_a-u_i .ne. 0 on the surface S,
and the second one appears from the fact that the gradients of the trial
function are not continous on S.
Note (1): Eq. (15.2) is more general as our situation in the WSC: the
surface S must not be a sphere but any closed surface within
the region where the differential equation shall be solved.
Note (2): what concerns the notation in Eqs. (15.2-15.9), we use the
functions u_i and u_a :
"i" means the trial function inside the surface S (in German:
Innen), "a" means the function outside S (in German: Aussen).
In order to prove that Eq. (15.2) is a proper expression for the
"condition of stationarity", we make the following ansatz:
Let's assume that u_t ("t" means "total", i.e., in the whole region
of interest) is an eigensolution of the Hamiltonian for the eigen-
energy E_t:
H u_t = E_t u_t .
An arbitrary trial function (not continuous on S) will differ from this
eigensolution by
u_i = u_t + delta u_i and u_a =u_t + delta u_a .
Calculating the expectation energy of such function will lead to a
somewhat different Energy
E = E_t + delta E .
Including Eqs. (15.3) and (15.4) into the variational ansatz (15.2)
will show that this leads to the fulfillment of the stationary condition
for the energy.
The variational principle (15.2) is a quite general one, and we have to
reduce it to the trial functions used in the APW case:
(1) the trial function is continuous on S, so we have u_i(S) = u_a(S) .
(2) S means the surface of the muffin-tin sphere with radius r_MT.
These two aspects lead to the somewhat simpler variational principle (15.7).
15.3 The APW eigenvalue problem
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Our job is now the calculation of stationary energy values, based on Eq. (15.7)
and the trial function
u = \sum_{i=1}^infin a_s \phi^APW_s
where the functions phi have the form of Eq. (14.8).
The parameters to be optimized are the coefficients a_s or (equivalently)
their cc values a^*_s . The condition for an energy extremal reads
\partial E / \partial a^*_s must be zero !!
Using Eq. (15.7), one gets the last equation on p. 182 of the textbook,
and - correspondingly - the linear homogenous system of equations written
on top of p. 183.
The corresponding secular matrix is given by Eq. (15.8).
NOTE: the further evaluation of Eq. (15.8) is not difficult but rather tedious.
therefore, this calcuation is shifted to the appendix 15.4 of the textbook.
One obtains the result Eq. (15.9):
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Omega_0 = volume of the unit cell (WSC),
P_l = Legendre polynomial,
j_l = spherical Bessel function,
R_l(r,E) = a particular solution of Eq. (14.2).
As it can be seen, the APW secular matrix is an symmetric one;
consequently, their eigenvalues (= the energies of the Bloch electrons)
are real numbers.
The APW method is extremely efficient, i.e. its convergence is much better than
that of the PW method. Additionally, in contrast to OPW and PP, no separation
between "core" and "valence" electrons is necessary: "all electrons in the
crystal are treated on equal footing!"
On effect, to get a sufficient convergence of the Bloch energies, a relatively
small number of APW basis functions is necessary; this is documented by the
following table (Table 15.1 in the textbook):
Number of APW functions, necessary to obtain d-bands converged to 0.001 Ry,
for different crystal structures:
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structure atoms/WSC number APW's
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fcc 1 27
bcc 1 43
simple cubic 1 57
hcp 2 67 ideal c/a ratio
CsCl 2 81 equal MT radii
NaCl 2 113 equal MT radii
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On the other hand, APW has a significant DISADVANTAGE: the eigenvalue problem
of the matrix (15.9) is NON-LINEAR with respect to E (due to the last term in
Eq. (15.9)! Therefore, the numerical determination of the eigenenergies is
much more time-comsuming that in case of PW, OPW, or PP.
(to be continued)