Linear Dielectric Response of Systems with

Crystalline Order: Calculations on LiF

Richard J Mathar

Quantum Theory Project, Department of Physics,

Univ. of Florida, Gainesville, FL 32611-8435

30. September 1998

Abstract

The wave-vector and frequency-dependent

dielectric function (longitudinal-longitudinal

element of the dielectric tensor) becomes a

matrix in the case of bulk crystals. The stan-

dard approach is the use of first-principles,

all-electron Kohn-Sham states in the inte-

gral of the irreducible polarizability in the

Random Phase Approximation. From this

microscopic information on the length-scale

of the lattice constant (umklapp processes)

one may derive the macroscopic (long-wave-

length) element, the (0,0)-element, of the

inverse matrix (the energy loss function).

Supported by grant DAA-H04-95-1-0326 from the U.S. Army

Research Office

1 Cultural, Historical, Educational, Physi-

cal, Statistical, Linguistic, Religious and

Biological/Agricultural Background

The word “seminar” is derived from “semen” (Latin, English)

which means “seed(s).” It is not derived from “semi” (Greek:

half) and “Narr” (German: fool).

See also: “Seminit” (1Chronicles 15), “disseminate” and

“sembrar” (Span.).

In the classroom located in a country with English the

dominating language, we may easily associate “seed” with

“seat,” which everyone is usually offered one during our sem-

inars (the case of overpopulation excluded for the moment

and hardly probable in our case).

2 Dielectric Response in Crystals

2.1 Dielectric Matrix

General (inhomogeneous) media:

P (r, t) = ǫ0

∫

d3r′dt′χ(r, r′, t − t′)E(r′, t′)

Translational symmetry for infinite crystals

E(r’) P(r)

E(r’+R) P(r+R)

Figure 1: If the field configuration is shifted by a lattice vec-

tor R, the response (polarization) is shifted by the same

vector.

χ(r, r′, t − t′)!= χ(r + R, r′ + R, t − t′)

Definition of the Fourier transform (decompose the field into

plane waves):

χ(k,k′, ω) ≡

∫

d3rd3r′dτ χ(r, r′, τ ) exp(ik · r − ik′ · r′ − iωτ )

χ(k,k′, ω) = χ(k,k + G, ω)δk′,k+G

One momentum argument is coupled to the other by a re-

ciprocal lattice vector G. By convention the remaining con-

tinuous k is also decomposed into higher Brillouin zones,

k ≡ q + G′, which leads to the following notation of the di-

electric matrix

ǫG,G′(q, ω) = δG,G′ −e2

ǫ0|q + G|2Π(q + G,q + G′, ω).

In the context of the Dyson equation

W (12) = U(12) +

∫ ∫

U(13)Π(34)W (42) d3 d4

Π. . . irreducible polarizability (graphs that cannot be cut into

decoupled pieces by cutting one single photon line).

Notes:

• Thematrix notation is not enforced by quantummechan-

ics.

• The matrix is not symmetric/hermitian, but other def-

initions exist that use the equivalence between exter-

nal “probe” and internal “response” (ie the superposition

principle of electrodynamics) to define symmetric matri-

ces.

• The crystallinity may be in 3 dimensions (bulk), 2 dimen-

sions (slabs), or 1 dimension (rods). The Fourier trans-

forms is only economical for the subspaces with repeti-

tion.

ǫG,G′ is the information obtained from standard quantum

mechanical perturbation theories (“direct” methods left aside),

because it describes the reaction to the total (“self-consistent”)

field E.

Unfortunately, the application may mean that the exter-

nal “driving” field Eext is given (under control experimen-

tally) and the inverse matrix needed:

Eind(k + G, ω) =∑

G′

[

ǫ−1G,G′(k

′, ω) − 1]

Eext(k′ + G′, ω)

E ≡ Eext + Eind

Examples: polarization by ionizing particles; inelastic x-ray

scattering.

2.2 Macroscopic Dielectric Function

The “head” element ǫ0,0 of the dielectric matrix does not rep-

resent the macroscopic dielectric function (which would rep-

resent the long-wavelength response of the crystal to a long-

wavelength external field) best, because it cannot decouple

the higher harmonics of the spatial variation of the effective

potential W and the bare potential U (assume UG6=0 = 0):∑

G′

ǫG,G′W (q + G′, ω) ∝ U(q + G, ω).

The head element of the inverse dielectric matrix actually

represents the long-wavelength response as demanded

W (q + G, ω) ∝∑

G′

ǫ−1G,G′U(q + G′, ω).

The influence of the “wing” elements of the dielectric matrix

on the macroscopic dielectric function 1/ǫ−10,0 6= ǫ0,0 is named

local field effects.

3 Applications

• “background” dielectric function for exciton spectra

• energy loss function ℑǫ−1(q, ω) to obtain the (stopping)

force on massive particles (ions) [Mathar et al, (preprint)]

• (complex) index of refraction√

ǫ0,0(q → 0, ω); reflectivity

of (semi-infinite) crystals

• force constant that couples small displacements of atoms

s and s′ in the unit cell, moved into Cartesian directions

α and β, in the harmonic approximation

Cαβss′ (q) =

ZsZs′

ǫ0Ω

∑

G,G′

(q + G)α(q + G′)β

|q + G′|2ǫ−1G,G′(q)ei(G·Rs−G′·R

s′)

→ dynamic matrix and phonon spectra [Pick et al, Phys Rev B 1,

910 (1970)]

• re-formulation of the electron-electron interaction as an

effective, “screened” interaction, the W in Hedin’s GW

expansion of the vertex function:

W (1, 2) = U(1, 2) +

∫ ∫

W (1, 3)Π(3, 4)U(4, 2) d3 d4

Π(1, 2) = −i~G(1, 2)G(2, 1)

−(i~)2∫ ∫

G(1, 3)G(4, 1)W (3, 4)G(2, 4)G(3, 2) d3 d4 + . . .

The exchange-correlation functional of (static) density-

functional theory is equivalent to some ǫ−1G,G′(q, ω = 0)

4 Irreducible Polarizability

4.1 Random Phase Approximation

In the RandomPhase Approximation (RPA) to the irreducible

polarizability [Adler, Phys Rev 126, 413 (1962)]

Π(q + G,q + G′, ω) =∑

ν,ν′

∫

BZ

d3k

(2π)3M ∗

GMG′(fνk − fν′k+q)

~ω + iη + Eνk − Eν′k+q

.

Eνk . . . energy dispersion of band ν

fνk . . . occupancy (0 or 2 for spin degenerate bands)

η → 0 . . . no broadening by other mechanisms assumed

Decomposition of the denominator: contributions from “vir-

tual” and “real” transitions

1

~ω + iη + Eνk − Eν′k+q

= P1

~ω + Eνk − Eν′k+q

−iπδ(~ω+Eνk−Eν′k+q),

An economical implementation calculates only the real tran-

sitions, and obtains the virtual transitions via a Kramers-

Kronig transformation (de facto two Fourier transformations).

The “real” transitions are related to the Joint Density of

States

JDOS = VUC

∑

ν,ν′

∫

BZ

d3k

(2π)3(fνk − fν′k+q)δ(~ω + Eνk − Eν′k+q).

This quantity is faster to calculate and ignores transition

matrix elements.

4.2 Matrix Elements

Plane wave matrix elements

MG ≡ 〈ν ′k + q|ei(G+q)·r|νk〉.

Kohn-Sham states |νk〉 and energies Eνk of the GTOFF pro-

gram are functions of bands ν and crystal momentum k.

Matrix elements may be either evaluated in real space or

with an intermediate plane wave representation

MG =1

VUC

∑

K

u∗ν′,k+q+G,Kuν,k,K .

k +Kj

k +Kj + q +Gq +G

1

Figure 2: Bloch states (which are not eigenfunctions of

the momentum operator) are virtually decomposed into

plane waves, and scattered individually according to the

PW representation of MG.

〈r|νk〉 ≡∑

G

ei(k+G)·ruν,k,G

Risk: this artificially reduces MG for large ~q because the

uν,k,G are tabulated just to a max |G|.

k+K

q+G

k+q+G+L

|G| max

An optional evaluation of matrix elements MG in real space

is wishful to treat localized band states (core electrons) ac-

curately.

4.3 Electron Momentum Distribution (EMD)

Contributions to this sum of the correlation type (“shift” and

“add-multiply”) are related to the electron momentum dis-

tribution (EMD)

EMD =∑

G

fνG|uν,k+G|2

which gives an impression of how many plane waves must

be incorporated until uν,k,G ≈ 0 (|G| > Gmax).

Experimental probe: (e,2e) experiments are sensitive be-

cause the colliding partners are of equal mass.

5 Challenge

• Include a broad range of energies ~ω and typically dozens

of band pairs (ν, ν ′)

• Compute ǫG,G′ for virtually all row- and column indexes

and build the inverse (local field effects)

. . . with finite computing resources. Example: storing the

1291 plane wave expansion coefficients for 25 bands and a

12 × 12 × 12 mesh of points in the BZ with 8 bytes per value

(uν,k,G real-valued for lattices with inversion center) takes

7 × 12 × 12 × 1291 × 25 × 8 bytes = 260 Mbytes.

A

B

A’

B’

C’

IBZ

Figure 3: The concept of an

“irreducible” Brillouin zone

(IBZ) does not apply if q 6=

0. Let point A be equiva-

lent to A′ and B be equiva-

lent to B′ within the ground-

state calculation. When Π

is calculated, however, a

transition A → B contributes

to the integral together with

the (in-equivalent) transition

A′ → C ′, and it does not

help that the matrix ele-

ments for A′ → B′ are

known, because they refer

to a different q.

6 Bulk Lithium-Fluoride

Figure 4: LiF has the NaCl lattice structure (Symmorphic

space group with 48 point group elements, fcc lattice

with one formula unit in the primitive unit cell).

KS Hermite GTO basis (67 orbitals/UC, 67 bands):center at spd contraction pattern

F 6111111111/31111111/11

Li 51111111/11111

6.1 Kohn-Sham Bandstructure

-50

-40

-30

-20

-10

0

10

20

30

Γ X W K Γ L W

ener

gy (

eV)

A1g

A1g

T1u

A1g

T1u

T2g

Eg

EA1B2

E

B1

A1

A1

A1

A1

A1

E

B2

E

E

A1

A1

B1

A1g

A1g

A2u

Eu

A2u

B2g

A1g

Eu

A1g

B1g

A1

A1

B2

B1A1

B2B1

A1

A1

B1

A1

B2

A1

A1

B2

EA1

E

B2

A1

E

B2

A1

A’

A’

A’’A’A’

A’A’’A’A’

A’

A’’

A’A’

A1

A1

B1

A1B2

A1

B1

A1

B2

A1

B1A2A1

B2

A1

A2B1

A1B2

A1

A1

B1

B1

B2

A1

A1

B2

A1

A1

A1B2

A2B1

A1B2 B1

A1g

A1g

T1u

A1g

T1u

T2g

Eg

EA1

E

A1

A1

A1

E

A1

A1

E

A1

E

A1

E

A1g

A2u

A1g

Eg

A2u

A1g

Eu

A1g

Eg

A2u

Eu

AA

B

A

B

ABA

B

ABA

BA

B

BA

A1

B2

EA1

E

B2

A1

E

B2

A1

Figure 5: KS band-structure Eν,k of LiF for the Xα=2/3 form of

the XC potential using 16 × 16 × 16 k-points in the BZ (145

in the IBZ). Bottom of the conduction band at 0 eV. The

direct gap at Γ is 8.79 eV. The band with two F K-shell

electrons at −654 eV is not shown.

6.2 Electron Momentum Distribution

0 0.5 1 1.5 2 2.5 3px (100)

00.5

11.5

22.5

3

py (010)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7EMD

Figure 6: The electron momentum distribution of the three

top valence bands of LiF for q in the plane spanned by

(100) and (010) shows a p symmetry: The bands are formed

in essence by F 2p orbitals. A small directional depen-

dence exists. (Results with the Hedin-Lundqvist LDA to

the XC potential at a = 7.60841a0, the experimental lattice

constant. Band gap at Γ: 8.94 eV. Other values in the

plane follow from the C4z symmetry in reciprocal space.

q-components are given in atomic units.)

0 0.5 1 1.5 2 2.5 3px (100)

00.5

11.5

22.5

3

py (010)

0

0.2

0.4

0.6

0.8

1

1.2

1.4EMD

Figure 7: For the all-electron momentum distribution of the

12 electrons in the UC of LiF only a small dip at q = 0 re-

mains, as the additional bands have 1s and 2s character.

(Parameters as in Fig. 6.)

6.3 Joint Density of States

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80

JDO

S (

arbi

t. un

its)

eV

29 kpts.72 kpts.

Figure 8: The joint density of states is rather independent on

how dense the grid of the integration over the BZ is. Here:

8 × 8 × 8 versus 12 × 12 × 12 points in the BZ, ie, 29 versus

72 points in the IBZ. (q = 0.357/a0. q ‖ (111). Transitions

between lowest 16 bands included. Other parameters as

in Fig. 6).

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80

JDO

S (

arbi

t. un

its)

eV

qa0=0.12qa0=0.24qa0=0.36

Figure 9: The joint density of states has negligible dispersion.

(q ‖ (111). 72 points in the IBZ. Other parameters as in Fig.

8.)

6.4 Dielectric Response

6.4.1 Convergence Tests

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 10 20 30 40 50

Im ε

qa0=0.358

72 kpts145 kpts29 kpts

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50

Im ε

eV

qa0=0.715

72 kpts145 kpts29 kpts

Figure 10:

ℑǫ0,0(q, ω) with

q ‖ (111). A

comparison of

the cases with

72 and 145

points in the

IBZ shows that

the results are

hardly con-

verged with

respect to the

k-mesh den-

sity. (Integra-

tion including

transitions be-

tween the 16

lowest bands

and ~ω < 90

eV.)

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50

Im ε

qa0=0.358

1291 PWs645 PWs

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50

Im ε

eV

qa0=0.715

1291 PWs645 PWs

Figure 11:

ℑǫ0,0(q, ω) with

q ‖ (111).

Represent-

ing the Bloch

states with 645

plane waves

(|G|a0 < 7.0)

or 1291

plane waves

(|G|a0 < 8.85)

shows that

these plane

wave bases

are sufficient

to converge

the matrix el-

ements. (In-

tegration in-

cluding transi-

tions between

the 16 lowest

bands and

~ω < 90 eV. 72

points in the

IBZ.)

6.4.2 Dispersion

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 10 20 30 40 50

Im ε

eV

qa0=0.119qa0=0.238qa0=0.358

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50

Im ε

eV

qa0=0.476qa0=0.596qa0=0.715

Figure 12:

ℑǫ0,0(q, ω) with

q a multiple

of 0.119/a0

and q ‖ (111).

The dispersion

of the ma-

jor peaks is

rather weak,

as expected

from the very

flat valence

bands. (Inte-

gration with 72

points in the

IBZ, including

transitions be-

tween the 19

lowest bands

and ~ω < 90

eV. The sum

rule for the first

moment ωℑǫ

is fulfilled to

only 74% for 10

participating

electrons).

0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50

Im 1

/ε

qa0=0.119qa0=0.238qa0=0.358

0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50

Im 1

/ε

eV

qa0=0.476qa0=0.596qa0=0.715

Figure 13:

ℑ1/ǫ0,0(q, ω)

exhibits the

plasmon peak

and its disper-

sion defined

by ǫ(q, ω) ≈ 0.

Width and

heights de-

pend on the

number of

bands (avail-

able “virtual”

transitions)

and density of

the reciprocal

space mesh.

(Parameters

as in Fig. 12.)

6.5 Local Field Effects

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 10 20 30 40 50

Im ε

qa0=0.119

w/o LFEw LFE

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 10 20 30 40 50

Im ε

eV

qa0=0.238

w/o LFEw LFE

Figure 14: ℑǫ0,0

(open sym-

bols) versus

ℑ1/ǫ−10,0 (filled

symbols) char-

acterizes local

field effects.

(Only a 9×9 di-

electric matrix

of the 9 short-

est reciprocal

lattice vec-

tors is inverted.

Transitions be-

tween the 20

lowest band

are included.

Other param-

eters as in Fig.

12.)

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50

Im ε

qa0=0.358

w/o LFEw LFE

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50

Im ε

eV

qa0=0.477

w/o LFEw LFE

Figure 15: (Pa-

rameters as in

Fig. 14.)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60

Im 1

/ε

qa0=0.119

w/o LFEw LFE

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60

Im 1

/ε

eV

qa0=0.238

w/o LFEw LFE

Figure 16: ℑ1/ǫ0,0

(open sym-

bols) versus

ℑǫ−10,0 (filled

symbols) char-

acterizes local

field effects in

the energy loss

function. They

are smaller

than in the

model by

Michiels et

al [Phys Rev B 50,

11386 (1995)]. (Pa-

rameters as in

Fig. 14.)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60

Im 1

/ε

qa0=0.358

w/o LFEw LFE

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60

Im 1

/ε

eV

qa0=0.477

w/o LFEw LFE

Figure 17: (Pa-

rameters as in

Fig. 14.)

7 Outlook - Things To Do

The finite basis set of GTO’s (s, p, . . . functions) in GTOFF

implies that bands are missing in the region Eνk ≫ 0. Reme-

dies are

• Green’s function approach that replaces the sum over

conduction bands by the construction of “polarization”

functions [Quong et al, Phys Rev Lett 70, 3955 (1993)], or

• Completion of the GTO basis by one more diagonaliza-

tion of the SCF Hamiltonian with orthogonalized plane

waves.

The current implementation ignores the symmetry labels

of bands when it connects the discrete points in the BZ to use

a tetrahedron method to integrate over the BZ, equivalent

to a no-crossing rule for all bands. The number of erroneous

connectivities becomes negligible only for very dense meshes

in the BZ.

8 Summary

The first ab-initio calculation of local-field effects on themacro-

scopic dielectric function of the ionic insulator LiF has been

presented. They generally shift absorption peaks to higher

frequencies, but are less important than proposed by the

model by Michiels et al.