Real space calculation of optical constants from optical to x-ray frequencies
M. P. Prange,
1
J. J. Rehr,
1
G. Rivas,
2
J. J. Kas,
1
and John W. Lawson
3
1
Department of Physics, University of Washington, Seattle, Washington 98195, USA
2
Instituto de Ingeniería y Tecnología, Universidad Autónoma de Ciudad Juárez, Juárez 32310, Mexico
3
NASA Ames Research Center, Mail Stop 229-1, Moffett Field, California 94035, USA
Received 17 October 2008; revised manuscript received 20 July 2009; published 5 October 2009
We present a theory of linear optical constants based on the single-particle density operator and implemented
in an extension of the real space multiple scattering code known as FEFF. This approach avoids the need to
compute wave functions explicitly and yields efficient calculations for frequencies ranging from the IR to hard
x-rays, which is applicable to arbitrary aperiodic systems. The approach is illustrated with calculations of
optical properties and applications for several materials and compared with existing tabulations.
DOI: 10.1103/PhysRevB.80.155110 PACS numbers: 78.20.Bh, 78.70.Dm
I. INTRODUCTION
This work focuses on theoretical calculations of optical
constants, i.e., the long-wavelength limit q
→ 0 of the dielec-
tric function q
, . These quantities include the complex
dielectric constant , the complex index of refraction, the
energy-loss function, the photoabsorption coefficient, and the
optical reflectivity. Many other important physical properties
can be derived from the optical constants such as the photon
scattering amplitude per atom, inelastic mean-free paths, and
Hamaker constants. However, the ab initio calculation of
these optical properties for arbitrary materials has been a
long-standing problem in condensed-matter physics.
1–5
Thus
in practice, these properties are often approximated from
atomic calculations or taken from a variety of tabulated
compilations.
6–11
However, such tabulations are available
only for a small number of well-characterized materials over
limited spectral ranges and atomic calculations ignore solid-
state effects. Thus we aim to develop an efficient theoretical
method covering a broad range of frequencies and applicable
to aperiodic materials, thereby providing a practical alterna-
tive or complement to tabulated data, atomic models or first
principles methods for periodic systems.
The theory of dielectric response has been developed ex-
tensively over the past several decades, especially for peri-
odic systems,
1
following pioneering works of Nozières and
Pines,
2
Ehrenreich and Cohen,
3
Adler,
4
and Wiser.
5
These
authors developed the self-consistent field approach for the
dielectric function within the time-dependent Hartree ap-
proximation, also known as the random phase approximation
RPA. Subsequently the theory has been extended to include
exchange effects within the time-dependent density-
functional theory TDDFT.
12,13
More elaborate theories
have been developed that take into account quasiparticle ef-
fects and particle-hole interactions based on the Bethe-
Salpeter equation BSE and Hedin’s GW approximation to
the electron self-energy.
14–16
Several implementations of
GW-BSE Refs. 17–21 and TDDFT Refs. 20, 22, and 23
have recently become available. However, computational de-
mands generally restrict these implementations to limited
spectral ranges and relatively small or periodic systems.
In an effort to help remedy this situation we have devel-
oped an efficient, real space approach within the adiabatic
local-density approximation that can be applied to arbitrary
condensed systems from the visible to hard x-rays. Our ap-
proach is based on a density operator formalism within an
effective single-particle quasiparticle theory. Here the den-
sity operator E =Im GE is the imaginary part of the one-
particle Green’s operator GE = E - H + i
-1
, where see
Appendix H is the one-particle Hamiltonian including the
self-energy and is a positive infinitesimal. From E, both
the spectral function r
, r
' , E and the local density of states
r
, E can be obtained. This approach is a generalization of
the real space Green’s function method implemented in the
FEFF codes,
24
which includes both core- and valence-level
spectra and builds in inelastic losses and other solid-state
effects. Our work is intended to extend the capabilities and
ease-of-use of FEFF to enable full spectrum output with a
quality roughly comparable to that in currently available
tabulated data.
6–11
This effort was begun by one of us using
an atomic approximation for the initial core and semicore
states.
25
That approximation is often adequate at medium
high i.e., soft x-ray energies, but can be unsatisfactory for
optical and UV spectra.
The remainder of this paper is arranged as follows. Sec-
tion II describes the theoretical formalism behind our ap-
proach; Sec. III presents typical results for various optical
constants for a number of materials; Sec. IV discusses some
additional applications and diagnostics, and Sec. V presents a
brief summary and conclusions.
II. THEORY
A. Real space theory of dielectric response
We consider the macroscopic linear response of extended
systems to an external electromagnetic field of polarization ˆ
and frequency
V
ext
t = V
ext
e
t-it
+ cc , 1
where is a positive infinitesimal corresponding to adiabatic
turn on of the perturbing potential. Throughout this work we
use Hartree atomic units = m = e
2
= a
0
=1, unless otherwise
specified. This perturbation polarizes the material, inducing a
steady-state change nr
, e
-it
+ cc in the microscopic
electron density, which leads to a macroscopic polarization
P
e
-it
+ cc, representing the average screening dipole re-
PHYSICAL REVIEW B 80, 155110 2009
1098-0121/2009/8015/15511014 ©2009 The American Physical Society 155110-1