Construction of the Hamiltonian Matrix







 X

G

V_I^Ge^iGr interstitial region X

L

V_{M T}^L (r)YL(ˆr)muffin-tin (54) This method became possible with the development of a technique for obtaining the Coulomb potential for a general periodic charge density without shape-approximations and with the inclusion of the Hamiltonian matrix elements due to the warped interstitial and non-spherical terms of the potential. The charge densityn, is represented analogously to Eq.(54), just exchangingV byn. Details of the solution of the Poisson equation for an arbitrarily shaped periodic potential are described in Sect. 4.6.

4.2 Construction of the Hamiltonian Matrix

The FLAPW Hamiltonian and overlap matrices consist of two contributions from the two regions into which space is divided.

H = HI+ HM T and S = SI+ SM T (55) Both contributions have to be computed separately.

4.2.1 Contribution of the Muffin-Tins

Writing the product of the radial functionsuwith the spherical harmonics asφL =ulYL, the contribution of the muffin-tin to the Hamiltonian matrix and the overlap matrix is given

by:

HM T^G0G(k) =X

µ

Z

M T µ

X

L⁰

a^µ_L^G0 ⁰(k)φ^αL⁰(r) +b^µ_L^G0 ⁰(k) ˙φ^αL⁰(r)

!^∗ HˆM T α

X

L

a^µ_L^G(k)φ^α_L(r) +b^µ_L^G(k) ˙φ^α_L(r)

!

d³r (56)

(The overlap matrixS^G_{M T}^0G(k) is obtained by replacing HˆM T α by 1.) It is distinguish between the atom indexµand the atom type indexα(µ). In most application they are symmetry equivalent atoms in the unit cell, i.e. some atoms can be mapped onto each other by space group operations. Clearly, these atoms must possess the same physical properties, e.g. the potential has to be equal. As a consequence, the Hamiltonian and the basis functionsϕ^α_L(r)do not differ among the atoms of the same type. This fact is exploited in that the muffin-tin potential of an atom type is only stored once for the representative atom, and the matrices Eq.(58) is also calculated for the representative only. HˆM T^α is the scalar relativistic Hamiltonian operator. It can be split up into two parts, the spherical HamiltonianHˆsp(cf. Eq. (40)) and the nonspherical contributions to the potentialVns.

HˆM T α= ˆH_sp^α +V_ns^α (57) The above integrations contain the following type of matrix elements.

t^αφφ_L0L = Z

M T α

φ^α_L⁰(r) ˆHM T αφ^α_L(r)d³r (58) These matrix elements do not depend on theA^µ_L^G(k)andB_L^µG(k)coefficients. Thus, they are independent of the Bloch vector and need to be calculated only once per iteration. The functionsφ^α_L andφ˙^α_L have been constructed to diagonalize the spherical partHˆ_sp^α of the muffin-tin HamiltonianHˆM T^α:

Hˆ_sp^αφ^α_L=Elφ^α_L and Hˆ_sp^αφ˙^α_L+ =Elφ˙^α_L+φ^α_L. (59) Multiplying these equations withφ^α_L0(r)andφ˙^α_L0(r)respectively and integrating over the muffin-tins gives

hφ^α_L⁰|Hˆ_sp^αφ^α_LiM T α =δll⁰δmm⁰El ; hφ^α_L⁰|Hˆ_sp^αφ˙^α_LiM T α =δll⁰δmm⁰

hφ˙^α_L⁰|Hˆ_sp^αφ^α_Li^{M T α} = 0 ; hφ˙^α_L⁰|Hˆ_sp^αφ˙^α_Li^{M T α}=δll⁰δmm⁰Elhu˙^α_l|u˙^α_li^{M T α} (60) Where the normalization condition foru^α_l has been used. So, only the expectation values of the nonspherical part of the potential are left to be determined. Since the potential is also expanded into a product of radial functions and spherical harmonics,

V^α(r) =X

L⁰⁰

V_L^α00(r)YL⁰⁰(ˆr), (61) the corresponding integrals consist of product of a radial integrals and angular integrals over three spherical harmonics, the so-called Gaunt coefficients:

t^αφφ_L0L =X

l⁰⁰

I_l^αuu0_ll⁰⁰G^m_l⁰_ll^{0mm00 00} +δ_ll⁰δ_mm⁰El (62)

with as well as similar expressions forI_l^αu0ll^u00˙ and others. The I matrices contain the radial integrals. Finally, the Hamiltonian and overlap matrix elements become

H_{M T}^G0G(k) =X

The interstitial contributions to the Hamiltonian and overlap matrix have the following form.

The potential is also expanded into planewaves in the interstitial region.

V(r) =X

G⁰

VG⁰e^−iGr (68)

Without the existence of the muffin-tin spheres the integration would stretch over the entire unit cell and the integration becomes rather simple. The kinetic energy is diagonal in momentum space and the potential is local, diagonal is real space and of convolution form in momentum space.

However, these matrix elements are not as straightforward to calculate as they appear at first glance, because of the complicated structure of the interstitial region. The integrations have to be performed only in between the muffin-tins. Therefore, a step functionΘ(r)has to be introduced, that cuts out the muffin-tins.

Θ(r) =

1interstitial region

0muffin-tins (69)

Using the step function the matrix elements can be written:

Apparently these coefficients are needed up to a cut-off of2Gmax. The step function can be Fourier transformed analytically.

whereτ^µindicates the position of atomµ. The Fourier transform of the product ofV(r) andΘ(r)is given by a convolution in momentum space.

(VΘ)^G=X

G⁰

VG⁰Θ₍G−G⁰)

This convolution depends on both, G andG⁰, therefore the numerical effort increases like(Gmax)⁶. However,(VΘ)^Gcan be determined more efficiently, using Fast-Fourier-Transform (FFT). In Fig. 10 it is shown schematically how(VΘ)^Gcan be obtained using FFT. Using this scheme the numerical effort increases like(Gmax)³ln(Gmax)³withGmax.

(r)

Figure 10. Schematic representation of the calculation of(VΘ)G. FirstΘ(r)is Fourier transformed analytically with a cut-off of2GmaxyieldingΘ˜^G. ThenΘ˜^GandVGare fast Fourier transformed and multiplied on a real space mesh. Finally, the result(VΘ)(˜ r)is back-transformed to momentum space.

S S^g

S^α

µ

µ

pα p

Figure 11. Local coordinate frames inside each muffin-tin.

4.2.3 The Muffin-Tin a- and b-Coefficients

Within FLAPW the electron wavefunctions are expanded differently in the interstitial re-gion and the muffin-tins. Each basis function consists of a planewave in the interstitial, which is matched to the radial functions and spherical harmonics in the muffin-tins. The coefficients of the function inside the spheres are determined from the requirement, that the basis functions and their derivatives are continuous at the sphere boundaries. These coef-ficients play an important role. In this section we will therefore discuss how the matching conditions can be solved and what properties they induce.

In many systems that the FLAPW method can be applied to some atoms are symmetry equivalent, i.e. these atoms can be mapped onto each other by a space group operation {R|t}. Such a group of atoms is called an atom type, represented by one of the atoms. Let {R^µ|t^µ}the operation that maps the atomµonto its representative. This atom can now be assigned a local coordinate frameS^µ(cf. Fig. 11), where the origin ofS^µis at the atoms positionτ^µ. The local frame is chosen such that the unit vectors of the local frameS^µ are mapped onto those of the global frame byR^g(R^µS^µ =S^g). The local frame of the representative atomS^αis only translated with respect to the global frame, i.e. the same rotationR^µ mapsS^µontoS^α. The potential (and other quantities) inside the muffin-tins can now be written in terms of the local coordinate system. Due to the symmetry we find VM T^α(r^α) = VM T^µ(r^µ), wherer^αandr^µare expanded in terms of the local framesS^α andS^µrespectively. As a consequence the radial functionsul(r)and the t-matrices are the same for all atoms of the same type. This way symmetry is exploited to save memory and computer time (during the calculation of the t-matrices).

Any planewave can be expanded into spherical harmonics via the Rayleigh expansion.

e^iKr= 4πX

L

i^ljl(rK)Y_L^∗(Kˆ)YL(ˆr) (74) Wherer = |r|,K = |K|andKabbreviates(G+k). Looked at from the local frame Kandτ^µappear rotated, besides the origin of the local frame is shifted. Therefore, the

planewave has the following form in the local frame:

e^{i(RµK)(r+Rµ}τ^µ) (75) Thus, the Rayleigh expansion of the planewave in the local frame is given by:

e^iKτ^µ 4πX

L

i^ljl(rK)Y_L^∗(R^µKˆ)YL(ˆr) (76) The requirement of continuity of the wavefunctions at the sphere boundary leads to the equation:

X

L

a^µ_L^G(k)ul(RM T^α)YL(ˆr) +b^µ_L^G(k) ˙ul(RM T^α)YL(ˆr)

=e^iKτ^µ 4πX

L

i^ljl(rK)Y_L^∗(R^µKˆ)YL(ˆr), (77) whereRM T^αis the muffin-tin radius of the atom typeα. The second requirement is, that the derivative with respect tor, denoted by∂/∂r= ⁰, is also continuous.

X

L

a^µ_L^G(k)u⁰_l(RM T^α)YL(ˆr) +b^µ_L^G(k) ˙u⁰_l(RM T^α)YL(ˆr)

=e^iKτ^µ 4πX

L

i^lKj_l⁰(rK)Y_L^∗(R^µKˆ)YL(ˆr) (78) These conditions can only be satisfied, if the coefficients of each spherical harmonicYL(ˆr) are equal. Solving the resulting equations forA^µ_L^G(k)andB^µ_L^G(k)yields:

a^µ_L^G(k) =e^iKτ^µ 4π 1

Wi^lY_L^∗(R^µKˆ)

[ ˙ul(RM T^α)Kj_l⁰(RM T^αK)−u˙⁰_l(RM T^α)jl(RM T^αK)]

b^µ_L^G(k) =e^iKτ^µ 4π 1

Wi^lY_L^∗(R^µKˆ)

[u⁰_l(RM T^α)jl(RM T^αK)−ul(RM T^α)Kj_l⁰(RM T^αK)].

(79) The WronskianW is given by:

W = [ ˙ul(RM T^α)u⁰_l(RM T^α)−ul(RM T^α) ˙u⁰_l(RM T^α)] (80) 4.3 Brillouin-Zone Integration and Fermi Energy

In the current implementation of the FLAPW method the Fermi energy is determined in two steps. First the bands are occupied (at all k-points simultaneously), starting from the lowest energy, until the sum of their weights equals the total number of electrons per unit cell, i.e. the discretized equivalent of Eq.(27) is solved atT = 0. Then, the step function is replaced by the Fermi and the Fermi energy is determined from the requirement that:

N =X

k

X

ν

w(k, ν(k)−EF) (81)

Where the weights are given by:

w(k, ν(k)−EF) =w(k) 1

e^{(ν(k)−EF)/kBT} + 1 (82) The weightsw(k, ν(k)−EF)are stored to be used for later Brillouin zone integrations.

Im Dokument 4 The FLAPW Method (Seite 21-27)