density matrix calculation of optical...
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Density Matrix Calculation of Optical Constants
Micah Prange
A dissertation submitted in partial fulfillment ofthe requirements for the degree of
Doctor of Philosophy
University of Washington
2009
Program Authorized to Offer Degree: University of Washington Physics
University of WashingtonGraduate School
This is to certify that I have examined this copy of a doctoral dissertation by
Micah Prange
and have found that it is complete and satisfactory in all respects,and that any and all revisions required by the final
examining committee have been made.
Chair of the Supervisory Committee:
John Rehr
Reading Committee:
John Rehr
Michael Schick
Larry Sorensen
Date:
In presenting this dissertation in partial fulfillment of the requirements for the doctoraldegree at the University of Washington, I agree that the Library shall make its copiesfreely available for inspection. I further agree that extensive copying of the dissertationis allowable only for scholarly purposes, consistent with fair use as prescribed in the U.S.Copyright Law. Requests for copying or reproduction of this dissertation may be referredto ProQuest Information and Learning, 300 North Zeeb Road, Ann Arbor, MI 48106-1346,1-800-521-0600, or to the author.
Signature
Date
University of Washington
Abstract
Density Matrix Calculation of Optical Constants
Micah Prange
Chair of the Supervisory Committee:
Professor John Rehr
Physics
The purpose of this work is to develop a quantitative theoretical understanding of the mi-
croscopic processes which give rise to the dielectric response of solids at all frequencies. It
contains the first ab initio theory to predict the long-wavelength dielectric constant of real
systems from optical through hard x-ray frequencies. The first calculations of dielectric
response based on quantum mechanics were published by Ehrenreich and Cohen in 1959. In
the intervening decades many theoretical treatments of various aspects of dielectric response
have been considered; the major developments are reviewed in this dissertation. An expo-
sition of the theory of dielectric response of a solid is given in terms of the density matrix.
In this historical context a theoretical framework for evaluating the density matrix and di-
electric constant using the techniques of real-space multiple scattering theory is developed.
I give the first applications of such a theory to dielectric response at visible frequencies.
My method relies on a novel expansion of the density matrix. This formulation, based on
a separable representation of the imaginary part of the Green’s function, combined with a
reasonable set of approximations leads to an efficient calculation of the imaginary part of the
dielectric constant. Since the density-matrix method can be applied over broad frequency
ranges, the full dielectric function and all linear optical constants can be recovered using the
analyticity of the dielectric function. I present extensions to the FEFF code and calculations
of the optical constants of a variety of materials completed using these extensions. Since
our work is based on real-space multiple scattering theory, the method can be applied to
most solids periodic or not containing any elements from the periodic table. Developer’s
notes are included for those who are interested in refining and extending these calculations.
The status of the code which implements our density matrix method is discussed. We give
recommendations for future refinements.
TABLE OF CONTENTS
Page
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Dielectric Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Optical regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 X-ray regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Goals and outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 2: Density Matrix Theory of Dielectric Response . . . . . . . . . . . . . . 5
2.1 Formal Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Chapter 3: Optical Constants in the Muffin-Tin Approximation . . . . . . . . . . 12
3.1 Separable Representation for the Density Matrix . . . . . . . . . . . . . . . . 12
3.2 Core state response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Valence response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Comparison of muffin-tin RSMS electronic structure with other methods . . . 27
Chapter 4: Calculations and Applications . . . . . . . . . . . . . . . . . . . . . . . 34
4.1 Theoretical Optical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Applications and diagnositcs . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Chapter 5: Conclusions and future directions . . . . . . . . . . . . . . . . . . . . . 61
5.1 Cconclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Appendix A: Developer’s Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A.1 Overview of full spectrum algorithm and codes . . . . . . . . . . . . . . . . . 69
A.2 Obtaining and installing FEFFOP . . . . . . . . . . . . . . . . . . . . . . . . . 74
i
A.3 Description of Routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A.4 Notes on Future Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Appendix A: Tables of optical constants . . . . . . . . . . . . . . . . . . . . . . . . . 85
ii
LIST OF FIGURES
Figure Number Page
3.1 The Voronoi cell in diamond with the first two shells of atoms and the planefrom which the data in Figures 3.2, 3.3, and 3.4 was drawn. . . . . . . . . . . 29
3.2 Valence charge density in diamond calculated by real-space multiple scatter-ing and the plane-wave pseudopotentials method. The scale for the horizontalplane is Bohr and the scale for the vertical axis is electrons per cubic Bohr. . 30
3.3 Valence charge density in diamond calculated by real-space multiple scatter-ing and the plane-wave pseudopotential method, weighted by the square ofthe distance from the atom at the origin. . . . . . . . . . . . . . . . . . . . . 31
3.4 Fractional difference in the valence charge density in diamond calculated byreal-space multiple scattering and the plane-wave pseudopotential methodfor two angular momentum cutoffs in the real-space calculation. . . . . . . . . 32
3.5 DOS in Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 Calculated ǫ2 for Cu plotted with the experimental data of Stahrenberg, et.al. [48], the DESY compilation [18], and K-edge data of Newville [31]. . . . . 35
4.2 Calculated and experimental ǫ2 for Au [18]. . . . . . . . . . . . . . . . . . . . 36
4.3 Calculated and experimental ǫ2 diamond [34] and amorphous C [53] (bottom).In the bottom panel the diamond curves have been shifted vertically for clarity. 36
4.4 Calculated ǫ1 for diamond compared to experiment [18, 34]. . . . . . . . . . . 38
4.5 Calculated ǫ1 for Cu compared to experiment [18]. . . . . . . . . . . . . . . . 38
4.6 Calculated ǫ1 for Al2O3 compared to experiment [18]. . . . . . . . . . . . . . 39
4.7 Calculated energy-loss function (Eq. 4.3) for Cu compared to experiment[18, 34]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.8 Calculated energy-loss function (Eq. 4.3) for Au compared to experiment [18]. 40
4.9 Calculated energy-loss function (Eq. 4.3) for Al2O3 compared to experimen-tal x-ray [18] and EELS [30] data. . . . . . . . . . . . . . . . . . . . . . . . . 41
4.10 Calculated real index of refraction for Cu compared to experiment [18]. . . . 42
4.11 Calculated real index of refraction for Al2O3 compared to experiment [18]. . . 43
4.12 Calculated real index of refraction for diamond compared to experiment [34]. 43
4.13 Calculated absorption coefficient µ in inverse cm for Cu compared to exper-iment [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
iii
4.14 Calculated absorption coefficient µ in inverse cm for Au compared to theexperimental data of Hageman, et. al. [18] and Windt, et. al. [55] whichinclude uncertainty estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.15 Calculated absorption coefficient µ in inverse cm for diamond compared toexperiment [34] and a reciprocal-space calculation. . . . . . . . . . . . . . . . 46
4.16 Calculated real part of the anomalous atomic scattering factor for diamond[34]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.17 Calculated real part of the anomalous atomic scattering factor for Au com-pared to experiment [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.18 Calculated real part of the anomalous atomic scattering factor for Cu com-pared to experiment [18, 11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.19 Calculated imaginary part of the atomic scattering factor for Cu comparedto experiment [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.20 Calculated imaginary part of the atomic scattering factor for Au comparedto experiment [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.21 Calculated imaginary part of the atomic scattering factor for Al2O3 comparedto experiment [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.22 L edge dynamic structure factor q2S(q, ω) ∼ −q2Im ǫ−1(q, ω) weighted by q2
for Al metal as computed by FEFFOP running on top of FEFF8q. The peakin the loss function moves to higher ω as q increases. . . . . . . . . . . . . . . 54
4.23 Calculated Hamaker constant ǫ(iω) compared to ǫ2(ω) for Pd metal. . . . . . 55
4.24 neff(ω) calculated from the ǫ2 sum rule for Mg using Eq. (4.9). . . . . . . . . 56
4.25 JDOS/ω2 for p → d (solid line) and p → s (dashed line) transitions and thecalculated ǫ2 of this work (dots) for diamond vs. photon frequency in eV. . . 58
A.1 Flowchart of full-spectrum calculation. . . . . . . . . . . . . . . . . . . . . . . 70
iv
ACKNOWLEDGMENTS
This work has been encouraged and facilitated by very many people to whom I am
indebted. I have listed those whose contributions were absolutely essential here: John
Rehr, Shelley Prange, Josh Kas, Gildardo Rivas, Alex Ankudinov, Yoshi Takimoto, Jerry
Seidler, Tim Fister, Hadley Lawler, Fernando Vila, Adam Sorini, Larry Sorensen, and
Michael Schick.
v
1
Chapter 1
INTRODUCTION
Quantum mechanics has been well tested experimentally and is believed to account
satisfactorily for all observed properties of bulk materials due to the motion of electrons.
This intellectual triumph has led to many technological advances and can be viewed as the
fundamental basis of the modern understanding of condensed matter as well as molecular
and atomic systems. However, the field of condensed matter physics now has more open
problems and puzzles than ever before. The key challenge is to use quantum mechanics
to describe quantitatively the properties of macroscopic condensed systems. The principle
difficulty lies in solving the Schrdinger equation for a system with more than one interacting
electron. Since the accurate numerical computation of the ground-state wave function of a
molecular system is already at the limit of current computational techniques, there is little
hope for a direct treatment of a macroscopic system containing 1023±3 electrons. Therefore
one must look to approximations that bypass the complications of the many-body wave
function. This dissertation is an addition to a great number of complimentary efforts to
treat the century-old problem of making ab initio calculations of the properties of bulk
systems.
1.1 Dielectric Response
The electron is a charged particle and hence interacts with the electric field. It is the electric
field of the nuclei which binds the electrons of a condensed system. If the system is neutral,
the average (over a volume containing many atoms) electric field vanishes when the system
is in its ground state in spite of the locally high fields that give rise to the coherence of
the system. The application of an electric field originating from a remote source will act
on the electrons just as the field of the nuclei. If one then considers the average field, two
2
nonvanishing contributions will be found: one from the external field and another from the
perturbation of the electrons from their ground state orbits. From a macroscopic point
of view it appears as though the electrons are screening the external field. It is precisely
this response which we endeavor to calculate in this work. In other words, this work is
primarily concerned with theoretical calculations of optical constants, which are obtained
from the long-wavelength limit ~q → 0 of the dielectric function ǫ(~q, ω). These include the
complex dielectric constant ǫ(ω), the complex index of refraction, the energy-loss function,
the photoabsorption coefficient, the photon scattering amplitude per atom, and the optical
reflectivity.
The ab initio calculation of these optical properties for arbitrary materials has been
a long-standing problem in condensed-matter physics [33, 32, 12, 1, 56]. Thus in prac-
tice, these properties are often estimated from atomic calculations or taken from tabulated
sources [34, 13, 22, 57]. However, such tabulations are available only for a small number of
materials over limited spectral ranges. The work presented here attempts to overcome some
of these problems. Thus we aim to develop an efficient and widely applicable method cov-
ering a broad range of frequencies, thereby providing a practical alternative or complement
to tabulated data.
1.2 Optical regime
The theory of dielectric response of the valence bands of crystalline systems has been devel-
oped extensively over the past several decades [33], following pioneering works of Nozieres
and Pines [32], Ehrenreich and Cohen [12], Adler [1], and Wiser [56]. These works devel-
oped the self-consistent field approach for the dielectric function within the time-dependent
Hartree approximation, also known as the random phase approximation (RPA). Subse-
quently the theory has been extended to include exchange effects within the time-dependent
density functional theory (TDDFT) [59, 43]. While ground-state DFT calculations are
now routine, theoretical methods for accurate calculations of optical spectra are still not
widely available. More elaborate theories have been developed that take into account quasi-
particle effects and particle-hole interactions based on the Bethe-Salpeter equation (BSE)
[44, 50, 41], but these are even more computationally demanding.
3
1.3 X-ray regime
The problem of developing a full-spectrum 1 theory has recently been examined by Rivas [40].
Prior to his work there were limiting forms available for very high frequencies, atomic models
[13, 22, 57] at intermediate energies, and good calculations available for many systems in the
optical [33]. In contrast to the optical region where the response of the system is dominated
by the weakly bound electrons whose binding energies are ∼ 10 eV, the x-ray response
of a material is due to polarization of core and semi-core electrons with binding energies
closer to the photon energy. The response of such core electrons has been the subject of
study by the x-ray absorption community for most of the last century [38] where the focus
has been primarily on the oscillatory part of the absorption coefficient. Far off resonance
(i.e. high above an absorption edge) atomic models are adequate and one can avoid the
theoretical difficulties of the solid state. Cromer and Liberman [10] developed a relativistic
theory for this region. Later Ankudinov and Rehr included solid-state corrections [5]. All
these works focused on the response of a single core state. Levine and coworkers were
the first to attempt to describe x-ray optical properties by adding the response of all the
electrons [28]. Following this attempt, Rivas developed and tested techniques described in
this dissertation and [40] based upon real-space multiple scattering methods developed for
x-ray absorption spectroscopy. Rivas was able to achieve semiquantitative agreement with
experimental spectra for far UV and higher photon energies. The techniques of Rivas lose
applicability in the optical; full spectrum capabilities require a theory that can handle both
core and valence initial states.
1.4 Goals and outline of this thesis
The goal of the present work is to summarize and extend the work of Rivas to the optical
regime, to implement such a full spectrum theory in an efficient computer code, to analyze
the resulting calculations, to use the results in important applications in electronic structure
1For the purposes of this dissertation ”full” spectrum refers to the complete electronic response; polariza-tion of the nuclei is not considered. Practically, this means our theory is valid above infrared frequencies.We also do not consider processes which change the particle number (i.e. pair production), so there is alsoan upper limit near the MeV scale where such QED processes dominate the total scattering cross-section.
4
calculations, and to directly compare with experimental results. As a result of this effort, we
have developed an efficient, real-space approach that can be applied to arbitrary condensed
systems over a broad range of frequencies from the visible to hard x-rays. Our approach is
based on a real-space density matrix formulation within an effective single-particle theory.
This approach is a generalization of the real-space Green’s function method implemented in
the FEFF codes that includes both core- and valence-level spectra. Our work is intended to
extend the capabilities and ease-of-use of FEFF to enable full spectrum output for general
aperiodic systems with a quality roughly comparable to that in currently available tabulated
data [34, 13, 22, 57]. This work has produced a new software package FEFFOP.
The remainder of this dissertation follows the following outline. Chapter 2 is devoted
to formulating a general theory for the dielectric response relevant to any Green’s function
method. Next, in Chapter 3 we specialize to our case of interest, which is real-space multiple
scattering with a muffin-tin type potential, showing how the density matrix can be computed
in a separable form that leads to an efficient evaluation of the optical constants in the optical
regime. This muffin-tin density matrix is then compared in some detail to a standard plane-
wave calculation. Some practical aspects of computing at all photon frequencies are explored
as the algorithms used in FEFFOP to construct a full spectrum are described. Chapter 4
contains graphs of our theoretically computed optical constants compared to experimental
data and other calculations. A few important applications are briefly described as well
as some diagnostic checks on the calculations. The thesis closes with a discussion of the
status of our optical constant calculation effort. New code developments are suggested.
A developer’s guide describing the computer code FEFFOP intended to facilitate further
development of these algorithms is included as an appendix. Directions for obtaining and
running the code as well as descriptions of all new and modified routines can be found here.
5
Chapter 2
DENSITY MATRIX THEORY OF DIELECTRIC RESPONSE
2.1 Formal Theory
In this chapter we discuss general aspects of the theory of the response to electromagnetic
waves but avoid specific discussions of the underlying computational issues which we will
return to in the next chapter. We write down our theory in terms the density matrix instead
of wave functions. This feature of our theory allows for high-energy photoelectron states to
be included easily, avoiding common basis-set limitations of other methods. Along the way
we revisit some of the history of this field.
2.1.1 Density matrix theory of dielectric response
We consider the macroscopic linear response of extended systems to an external electro-
magnetic field of polarization ǫ and frequency ω
Vext(~r, t) = Vext(ω)e(−iω+δ)t + cc, (2.1)
where δ is a positive infinitesimal corresponding to adiabatic turn-on of the perturbing
potential. Throughout this work we use Hartree atomic units (h = m = e2 = a0 = 1) un-
less otherwise specified. This perturbation polarizes the material, inducing a steady-state
change δn(~r, ω)e−iωt + cc in the microscopic electron density, which leads to a macroscopic
polarization ~P (ω)e−iωt + cc, representing the average screening dipole response of the elec-
trons to the applied field. For simplicity of discussion, we assume that ~P has no component
perpendicular to the applied electric field. This is the case for systems of cubic or higher
symmetry in the q → 0 limit, but relaxing this restriction poses no computational difficulty.
In this case one can define a scaler electric susceptibility χ, and the dielectric function is
6
[23]
ǫ(ω) = 1 + 4πχ(ω)
~P = χ~E,(2.2)
where ~E is the electric field. Recall that we are interested in the /it linear response, so χ
in Equation 2.2 is independent of the magnitude of ~E.
Our calculations here make use of an effective single-particle microscopic theory in which
the N -electron state of the system at time t is described by a Slater determinant of time-
dependent single-particle orbitals φi(t). Thus the state can be characterized by the single-
particle density matrix ρ which is simply the projector onto the orbitals:
ρ(t) =N∑
i=1
|φi(t)〉 〈φi(t)| . (2.3)
Their time evolution is governed by the time-dependent Schrdinger equation
id
dt|φi(t)〉 = H |φi(t)〉 (2.4)
for the time-dependent Kohn-Sham Hamiltonian
H = −1
2∇2 + Vnuc + VH + Vxc + Σd + Vext(t). (2.5)
The terms in Eq. (2.5) are respectively the kinetic energy, the electrostatic attraction to the
nuclei Vnuc, the Hartree potential VH , the ground-state exchange-correlation potential Vxc,
the dynamical contribution to the quasi-particle self-energy correction in the GW plasmon-
pole approximation [21] Σd, and the time-dependent external potential Vext(t) of Equation
(2.1). The time evolution in Eq. (2.4) implies the Liouville equation [12] for the density
matrix
idρ
dt= Hρ− ρH†. (2.6)
For a Hermitian Hamiltonian, the right-hand side of Eq. 2.6 reduces to the commutator
[H, ρ]; we have written the more general form to allow for losses introduced by the finite
quasiparticle lifetimes of the excited states. In order to obtain the optical constants, we first
linearize this equation with respect to the ground-state by decomposing the Hamiltonian
7
and density matrix into their values in the ground-state and parts induced by Vext
H = H0 +H1(t) = H0 + Vext(t) + Vind(t)
ρ = ρ0 + ρ1.(2.7)
H1 consists of the external field and a term Vind due to the response of the electrons.
Second order terms, i.e., the products ρ1H1 and H1ρ1 are discarded. We assume that the
induced potential Vind and hence H1 have the same time dependence as Vext. With these
assumptions, the time derivative in Eq. (2.6) becomes trivial and we can solve Eq. (2.6) for
the induced density matrix in terms of the Kohn-Sham (KS) orbitals∣
∣φ0i
⟩
and eigenvalues
Ei of the ground-state system,
ρ1(ω) =∑
i,j
(fi − fj)
∣
∣φ0i
⟩
⟨
φ0i H1 φ
0j
⟩⟨
φ0j
∣
∣
∣
ω − (Ej − Ei) + iδ, (2.8)
where fi = f(Ei) ≈ θ(µ − Ei) is the Fermi occupation number of state∣
∣φ0i
⟩
and µ is the
Fermi level. The KS orbitals obey the unperturbed Schrdinger equation
id
dt|φi(t)〉 = H0 |φi(t)〉 . (2.9)
The induced electron density δn(~r, ω) due to the perturbation Vext is then given by
δn(~r, ω) = 〈~r ρ1(ω) ~r〉 . (2.10)
At this point it is convenient to introduce the bare and full susceptibilities whose local
behavior is given by
δn(~r, ω) =⟨
~r χ0(ω)H1 ~r⟩
= 〈~r χ(ω)φext ~r〉 . (2.11)
Typically, the bare response χ0 to an external perturbation is first computed from a single-
particle (i.e. non-interacting) description of the ground state. The full response χ of the
system can be related to response χ0 of some non-interacting reference system. This pro-
cedure gives rise to the Dyson equation for χ with an interaction kernel K
χ = χ0 + χ0Kχ = χ0(1 −Kχ0)−1. (2.12)
8
Methods for computing optical response that start from a single-particle description of
the ground state can be classified by their approximations to the particle-hole interaction
kernel K. The accuracy of the calculated macroscopic properties reflect that of the non-
interacting response and the interaction kernel. Note, in particular, that one needs to find
the frequency-dependent response of the non-interacting system, which involves different
considerations than those for static, ground state properties (e.g., the ground state energy
and density).
In the crudest approximation K = 0: the resulting polarizability is that of the non-
interacting reference system and local fields are neglected. In this case there is no screening,
and the single-particle potential is the sum of the ground-state potential and V ext. An
obvious deficiency of the non-interacting response is that the Coulomb field of the induced
density is neglected. To address this deficiency Adler [1] and Wiser [56] developed formally
equivalent theories of the macroscopic dielectric response of periodic solids based on the
RPA in which K is taken to be the bare Coulomb interaction. These theories were originally
built on band structure calculations for periodic materials in the Hartree approximation, and
Hartree local fields were included through the Adler-Wiser formula. In this approach the
operator inversion of Eq. (2.12) is reformulated using the inverse of the microscopic dielectric
matrix ǫGG′(ω,~k), which is then spatially averaged to give the macroscopic response ǫ(ω) =
limk→0 1/[ǫ−1(~k, ω)]0,0. However, the Adler-Wiser dielectric function is that of the Hartree
system and has the deficiency that the underlying electronic wave function is not anti-
symmetric under particle interchange.
Going beyond RPA thus requires additional exchange-correlation effects in K. There
have been efforts along these lines of two types: those based on time-dependent formulations
of density-functional theory (TDDFT), and those based on many body perturbation theory
and the Bethe-Salpeter Equation (BSE). These approaches have been critically compared
by Onida et al. [33] Consideration of excited states from a quasi-particle viewpoint [41]
leads to the decomposition of the interaction kernel into a direct term KD which is the
Coulomb interaction between the quasi-particles and an exchange interaction KX ,
K = KX +KD. (2.13)
9
Eq. (2.12) can be expanded in singly-excited (one electron, one hole) states, and KD can
be taken to be the Coulomb interaction screened by an effective (microscopic) dielectric
function. Doing so yields a set of approximations referred to as the BSE. Various screening
models are used ranging from parametrized models (e.g. the Levine-Louie dielectric func-
tion) to independent-particle approximations such as the static RPA. BSE schemes can
become computationally demanding since the inverse in Eq. (2.12) must be dealt with in
a product basis which can be large. The differences between the independent-particle ex-
citation energies and optical spectra and their interacting counterparts are referred to as
excitonic effects. However, the non-locality of the exchange-correlation terms can be avoided
by including exchange-correlation effects in K in terms of a density-functional fxc. Then
the approach reduces to the TDDFT [59] where
K(ω) = v + fxc(ω), fxc(ω) =δVxc
δρ. (2.14)
Consequently a local approximation to vxc leads to a local kernel (i.e., K depends only on
the diagonal elements of the real-space single-particle density matrix). This locality implies
that Eq. (2.12) can be expanded in a single-particle basis, thus circumventing the need
for particle-hole states needed for the BSE. The cost of this simplification is that direct
information about the particle-hole interaction (e.g. exciton wave-functions) is only im-
plicit. This makes it difficult to systematically improve on the local density approximation
(LDA) [33]. Nevertheless, calculations in such TDLDA frameworks have been carried out
for a variety of systems [39, 52]. While the TDDFT has achieved good agreement with
experiment for optical spectra in many cases, quantitative agreement at higher frequencies
has been more elusive. Calculations with the BSE tend to be even more computationally
limited. In addition these methods are built on various ground-state KS calculations, de-
pending on the system. Each approach can work well for a specific class of materials, but
can lose accuracy or applicability for others. Also, the ground-state methods used were
originally developed to calculate static properties and calculations of frequency-dependent
(non-interacting) response can become cumbersome due to the need for large basis sets and
special exchange-correlation functionals to describe unoccupied and excited states.
The above difficulties have led us to consider a different approach with the goal of
10
developing a general method for calculations of optical response that can handle a variety
of systems and spectral ranges. Our approach is based on an extension of real-space multiple
scattering theory (RSMS) in terms of the one-particle density matrix. The RSMS approach
is well suited to treat arbitrary aperiodic condensed-matter systems over a very broad
frequency range (from the visible to hard x-rays). Indeed, this scattering-theoretic approach
provides a superior basis for very high energy spectra where scattering is weak and the
approach converges rapidly. Further the approach goes beyond the Born-Oppenheimer
approximation and can include nuclear motion effects in terms of correlated Debye-Waller
factors [35].
In this work, we present calculations within this RSMS approach using an independent
quasi-particle approximation for the single particle states. Comparison of Equations (2.10)
and (2.11) gives an expression for the bare response function or susceptibility
χ0(~r,~r ′, ω) =∑
i,j
(fi − fj)φ0
i (~r)φ0∗i (~r ′)φ0
j (~r′)φ0∗
j (~r)
ω − (Ej − Ei) + iδ. (2.15)
Formally the imaginary part of the dielectric function is related to the full susceptibility by
[59]
ǫ2(ω) =4π
VIm
∫
d~r d~r ′ Tr dχ(~r,~r ′, ω) d†, (2.16)
where V is the volume of the system, and d = ~α · ǫpei~k~r is the transition operator between
the incident photon of wave vector ~k and polarization ǫp. In practice the transition operator
is replaced by the truncation to rank-one of its expansion into tensors developed by Grant
[17], which is equivalent to the dipole approximation.
To evaluate Eq. (2.16) for both optical and x-ray frequencies, we must first compute
the response function χ0(~r,~r ′, ω). Formally Eq. (2.15) can be expressed in terms of the
single-particle Green’s function as
χ0(~r,~r ′, ω) =
∫ EF
ρ(~r,~r ′, E)G+(~r,~r ′, E + ω)
+ ρ(~r ′, ~r,E)G−(~r ′, ~r,E − ω) dE.
(2.17)
Using the symmetries ρ(~r,~r ′, E) = ρ(~r ′, ~r,E) and G−(~r,~r ′, E) = [G+(~r ′, ~r,E)]∗ on the real
11
E-axis we can express the results entirely in terms of the one-particle density matrices ρ(E)
− Imχ0
π=
∫ EF
EF−ωρ(~r ′, ~r,E)ρ(~r ′, ~r,E + ω) dE. (2.18)
In this work we calculate these density matrices for energies ranging from the lowest occupied
states to very high energies of order 100 KeV [38].
We have related ǫ2 to the density matrix ρ(E). Now we turn to the problem of evaluating
the density matrix in a manner suitable for the calculation of optical constants.
12
Chapter 3
OPTICAL CONSTANTS IN THE MUFFIN-TIN APPROXIMATION
This chapter discusses the technicalities of calculating optical constants in the muffin-tin
approximation. It contains the key results of this dissertation. We first develop expressions
for the nonrelativistic Green’s function in a muffin-tin multiple scattering context. We
repeat this exercise for the Green’s function for the relativistic system (which is the one
actually used in calculations), arriving at an expression for the density matrix that avoids
the irregular solutions usually needed to evaluate the multiple-scattering Green’s function.
We then describe the efficient evaluation of the integrals to give ǫ2 first for the core regime
and then the valence. This calculation takes the form of a matrix multiplication. Having
developed a multiple-scattering formalism for computing the electronic structure, we test
it by comparing the resulting electron density ρ(~r) with a more conventional calculation of
the same quantity in the diamond crystal.
3.1 Separable Representation for the Density Matrix
Our calculations use an independent electron model in which each electron moves in an
effective quasi-particle scattering potential V (~r) which implicitly includes a dynamic self-
energy correction Σd(E) to the ground state exchange and correlation potential. In this
work Σd(E) is calculated using the local GW plasmon-pole model of Hedin and Lundqvist
[21]. The potential V (~r) =∑
n vn(rn) + V0 is taken to be the self-consistent muffin-tin
potential for a cluster of atoms at fixed locations ~Rn. Here ~rn = ~r − ~Rn is the position
relative to the nth atom, and V0 is a constant interstitial potential.
3.1.1 Multiple scattering Green’s function
In this subsection we derive an expression for the non-relativistic, real-space Green’s function
that is valid for any points within muffin-tin spheres. In a subsequent section we will write
13
down a closely related, relativistic form that is actually used in our calculations. Formally
the Greens functions operator is given by
G+(E) = [E −H + iδ]−1, (3.1)
where δ is a positive infinitesimal. Expanding G+ in the scattering potentials and free
propagators G0 yields the multiple scattering (MS) expansion
G = G0 +G0V G = G0 +G0TG0 + · · ·
= [1 − G0T ]−1G0 (3.2)
Here we have introduced the local t-matrix tn = vn + vnG0t to sum implicitly over all
scatterings at a given site n, where 〈~r|tn|~r ′〉 = tn(~r,~r ′, E) vanishes outside a given cell n
where v(rn)=0.
Free propagator
In position space the free propagator G0(E) is given by the FT,
G0(~r,~r ′, E) =
∫
d3k
(2π)3ei
~k·(~r−~r ′)
E − k2
2 + iδ. (3.3)
Below we evaluate this expression in terms of site-angular momentum scattering states
|L,R〉 which diagonalize ti
jL(~rR) =〈~r|L,R〉 = iljl(krR)YL(rR)
jL(~rR) =〈L,R|~r〉 = i−ljl(krR)Y ∗L (rR),
(3.4)
where k =√
2(E − V0).
In terms of spherical Bessel functions the free propagator is given everywhere by
G0(~r,~r ′, E) = −2k∑
L
YL(r)gl(r, r′)Y ∗
L (r′) (3.5)
= −2k∑
L
h+L (~r>)jL(~r<), (3.6)
where gl(r, r′) = h+
l (kr>)jl(kr<) and h+L (~r) = ilh+
l (kr)YL(r). This result can be obtained,
e.g., from the FT using the identity exp(i~k · ~r) = 4πΣLjL(~r)Y ∗L (k) and carrying out the
14
radial integrals in the complex k-plane. Alternatively the same result follows from the
inhomogeneous radial differential equation, where the prefactor is obtained from the Wron-
skian 2/r2W (jl, h+l ) = −2k. Here, as in the treatment of Rehr and Albers, [37] we have
used the phase and normalization conventions of Messiah, with jl = (h+l − h−l )/2i and
ilhl(x) = eixcl(1/ix)/x, cl is a polynomial of degree l with cl(0) = 1. Also, for convenience,
we have included the phase factors il and i−l in h+L and jL respectively, which do not change
G0, but simplify the asymptotic behavior.
The expansion of the free propagator for points at different sites has the form of a matrix
product
G0(~r,~r ′, E) =∑
L,L′
jL(~rR)G0LR,L′R′ jL(~rR′)
=∑
L,L′
〈~r|LR〉〈LR|G0(E)|L′R′〉〈L′R′|~r ′〉.(3.7)
This follows directly from Eq. (3.6) and the translation formulae for the spherical Hankel
functions [37]
h+L′(~r
′R) =
∑
L
jL(~rR)G0LR,L′R′ . (3.8)
Note the implicit factors of il′
and il in jL(~rR) and jL′(~rR) in this representation. In some
works, e.g. that of Faulkner and Stocks [14], these phase factors are included in the definition
the propagator matrix elements. The above expression can be checked, e.g., by compar-
ing ilh+l (kr) =
∑
L′ j′l(krR)il′
G0+L′R,L0. Eq. (3.7) can be derived, e.g., by expanding the
exponential product ei~k·(~r−~r ′) = ei
~k·(~r−~R)e−i~k·(~r ′−~R ′)ei~k·(~R−~R ′) in spherical Bessel functions,
and then carrying out the integration over k. This procedure yields for the dimensionless
propagator matrix elements:
G0LR,L′R′ ≡
G0LR,L′R′
−2k= 4π
∑
L′′
〈YLYL′′ |YL′〉h+L′′(k ~R
′′). (3.9)
which depend explicitly on k ~R′′ = k(~R − ~R ′). The FEFF code uses dimensionless matrix
elements G0L,L′(k ~R) which have a separable representation
G0L,L′(k ~R) ≡ G0
LR,L′R′ =eikR
kR
∑
λ
ΓLλΓλ,L′ , (3.10)
→ 4πeikR
kRclc
′lY
∗L (R)YL′(R), (kR→ ∞), (3.11)
15
where Γλ,L(k ~R) are generalized spherical harmonics [37]. This can be obtained, for ex-
ample, by substituting the asymptotic form of ilhl and and the completeness relation∑
L Y∗L (k)YL(R) = δ(k − R).
Full propagator
Let us now evaluate the behavior of the full propagator G(~r,~r ′, E) for ~r and ~r ′ in different
cells n and n′ respectively. For this case the MS series can be viewed as a sequence of
scattering events consisting of all scatterings at site n followed by all sequences of scatterings
not scattering at site n first or site n′ last, followed by all scatterings at site n′,
Gnn′ = [1 +G0tn]Gnn′ [1 + tn′G0], (3.12)
where the notation Gnn′ refers to the propagator starting and ending in cells n and n′
respectively, while Gnn′ refers to those terms in the MS expansion with first scatterings
at sites other than n and last scatterings at sites other than n′. This can be evaluated
by substituting the representation of Eq. (3.7) into Eq. (3.12) and then re-expressing the
terms in the site-angular momentum basis. Then Gnn′ can be expressed in terms of the
dimensionless full multiple scattering matrix elements GLn,L′n′ where
G(~r,~r ′, E) =∑
L,L′
jL(~rn) GLn,L′n′ jL(~rn′)
GLn,L′n′ =[
1 − G0T]−1
G0∣
∣
Ln,L′n′
(3.13)
where
G0Ln,L′n′ = G0
Ln,L′n′(1 − δnn′).
The complementary delta-function in G0 ensures that G only includes initial scatterings
from sites other than n and and final scatterings from sites other than n′. Next the terms
on the left and the right sides of Eq. (3.12) can be expressed in terms of scattering states
RLn(~rn). To see this note that matrix elements of the dimensionless t-matrices can be
expressed in terms of phase shifts as
〈jL|tn|jL′〉 = tlnδL,L′
tln = eiδln sin δln.(3.14)
16
Then using the representation of G0 in terms of Bessel functions in Eq. (3.6), one obtains
〈~r|[1 + G0tn]|LR〉 ≡ RLn(~rn)eiδln
= il[jl(rn) + h+l (rn)tln]YL(rn), (rn > rmt
n ),(3.15)
where RL(~r) = ilRln(r)YL(r). Asymptotically Rln(r) = [h+l e
iδln − h−l e−iδln ]/2i → sin(kr −
lπ/2 + δln)/kr. For r < rmt, the radial states can be obtained from the regular solution to
the radial equation, matched to the above result. Similarly one obtains 〈LR|(1+ tnG0)|~r〉 =
RL(~rn) exp(iδln). Note that the radial functions Rln(r) in the scattering states are real for
real, nonnegative k, but are otherwise the analytic continuation to complex k. Combining
all these results in Eq. (3.12) then yields
G(~r,~r ′, E) = −2k
×∑
LL′
RLn(~rn)GLn,L′n′RL′n′(~rn′);
GLn,L′n′ = eiδlnGLn,L′n′eiδl′n′ . (3.16)
It is straightforward to show that this expression is equivalent to that of Faulkner and Stocks
[14].
For ~r and ~r ′ at the same site n, G = G0+G0tnG0+Gn,n, where G is given by Eq. (3.13).
This yields
G(~r,~r ′, E) = −2k[
∑
L
HLn(~r>)RL(~r<)
+∑
L,L′
RLn(~rn)GLn,L′nRL′n(~rn)]
, (3.17)
where HL(~r) is the outgoing scattering state at site R which matches to ileiδlnh+l (krn) for
rn > rmtn .
In summary, within real-space multiple scattering theory, the Green’s function for this
potential can be written as a double angular momentum expansion
G(~r,~r ′, E) = −2k[
∑
LL′
RLn(~rn)GLn,L′n′RL′n′(~r ′n′)
+ δn,n′
∑
L
HLn(~r>)RLn′(~r<)]
, (3.18)
17
where n and n′ are the sites nearest ~r and ~r ′ respectively, and ~r> (~r<) is the larger (smaller)
of the two position vectors. The terms in equation (3.18) are the right-hand-side regular and
irregular solutions RLn, HLn of the spherically symmetric single-site problems and their left-
side counterparts RLn, HLn, the partial-wave phase shifts δln, and the multiple scattering
(MS) matrix GLn,L′n′ . The wave functions are normalized so that in the interstitial region
RLn coincides with YL[h+l e
iδln − h−l e−iδln ]/2i, and HLn coincides with YLh
+l e
iδln . The bar
for the left-sided solutions indicates that all factors except the Bessel functions are to be
complex conjugated. All these ingredients except the MS matrix can be found from the
solution of a spherically symmetric single-particle quantum mechanics problem. The full
MS matrix G for the system is found by numerical matrix inversion (e.g., with the LU or
Lanczos algorithms in FEFF) with typical matrix dimensions of order 2 × 103 or using the
MS path expansion.
3.1.2 Relativistic real-space Green’s function for the muffin tin potential
We develop an expression for the single-particle propagator that can be evaluated at all
points and is written in terms of the non-relativistic scattering matrix. We use notation
similar to Tamura [51]. The Green’s function is defined as the resolvant of the Dirac Hamil-
tonian:
(E14 − Hdirac(~r))G(~r,~r ′;E) = δ(~r − ~r ′)14
G(~r,~r ′;E)(E14 − Hdirac(~r′)) = δ(~r − ~r ′)14
(3.19)
with outgoing wave boundary conditions. G and Hdirac are operators on the Hilbert space
of a single Dirac particle; they act on 4-component spinor wave functions, and the 14 on the
RHS of Eq. 3.19 is the identity in the spin-space, which subsequently will not be explicitly
written. The Hamiltonian is given in terms of two-component operators as
Hdirac = −ic ~α · ~∇ + c2β + V
~α =
0 ~σp
~σp 0
β =
12 0
0 −12
(3.20)
18
with ~σp denoting the vector of Pauli matrices. We take
V (~r) =∑
i
Vi(|~r − ~Ri|) (3.21)
to be a scaler (i.e. proportional to 14 in the space of spinors) muffin-tin potential with cells
centered at the sites Rr.
Separation of variables
For a spherically symmetric potential the eigenfunctions of Hdirac can be classified by eigen-
values of ~J 2, Jz, and parity. In such a basis K = β(1414 + ~J 2 + ~L2) is also diagonal and its
eigenvalues κ can be used in place of those of ~J 2 and parity. The eigenfunctions then take
the form [42, 17, 29]:
〈~r ψκµ〉 =1
r
Pκ(r)χµκ(r)
iQκ(r)χµ−κ(r)
(3.22)
which confines the dependence on the radial coordinate to the functions P/r, Q/r, separate
from the dependence on the spin and angular coordinates. The latter is expressed in terms
of the spin-orbit eigenfunctions [42]
χµκ(r) =
∑
σ=± 12
Y µ−σl (r)φσ
⟨
l,1
2, µ− σ, σ j, µ
⟩
. (3.23)
Here Y ml is a spherical harmonic, and φσ is a (two-component) Puali spinor. The integer
quantum number κ assumes all values except 0 and combines j and l such that
j = |κ| − 1
2
Sκ = κ/|κ|
l =
−κ− 1 κ < 0
κ κ > 0.
(3.24)
Note that 〈~r ψ〉 is a Dirac spinor. We call P the upper component and Q the lower compo-
nent. In a similar manner operators in the Dirac space will be written in terms of their action
on the upper and lower components as 2x2 matrices. Generally, the bra 〈ψ| corresponding
to |ψ〉 is given by
〈ψκµ ~r〉 =(−1)µ−1/2Sκ
r
(
Pκ(r)χ−µκ
†(r), −iQκ(r)χ−µ
−κ†(r)
)
. (3.25)
19
Free solutions
To construct the Green’s function for the case V = 0 we take the following spherical wave
solutions to the free Dirac equation
〈~r jκµ(E)〉 =
jl(kr)χµκ(r)
cκjl(kr)χµ−κ(r)
(3.26)
which indeed obey
H0dirac |jκµ〉 = E |jκµ〉 (3.27)
with the definitions
k =√
(E2 − c4)/c
cκ(E) =ickSκ
E + c2
l = l − Sκ.
(3.28)
We also adopt a similar definition for the states∣
∣h±κµ
⟩
with an upper component h±l . We
use Messiah’s [29] conventions for the Bessel functions. The relativistic Wronskian [42] [51]
between angular momentum states 〈ψκµ| (with upper and lower components Pκ, Qκ) and∣
∣
∣ψ′κ′µ′
⟩
(with upper and lower components P ′κ′ , Q′
κ′)
[〈ψκµ| ,∣
∣ψ′κ′µ′
⟩
] ≡ c(
PκQ′κ′ − P ′
κ′Qκ
)
δκµκ′µ′ (3.29)
is independent of r. For the free spherical waves we have
W ≡ [〈hκµ| , |jκµ〉] =−1
k
c2
E + c2≈ −1
2k(3.30)
for all κ, µ. The approximate equality in Eq. 3.30 holds in the non-relativistic limit where
E ≈ c2 and is included to aid comparisons with ref. [8] .
Single-site propagator
We can classify solutions to the single-site problem by their behavior in the interstitial
region where every solution is a linear combination of |jκµ〉 and |hκµ〉. In particular we take(
E − Hdirac
)
|Rκµ〉 =(
E − Hdirac
)
|Hκµ〉 = 0
|Rκµ〉 →1
2i
(
|hκµ〉 eiδκ +∣
∣h−κµ
⟩
e−iδκ
)
|Hκµ〉 → |hκµ〉 eiδκ
(3.31)
20
which, with the requirement that 〈~r Rκµ〉 be regular at the origin1, define |Rκµ〉, |Hκµ〉,and the phase shift δκ. These definitions and the result that the relativistic Wronskian for
solutions of the Dirac equation at the same energy does not depend on r [51] imply
[〈Hκµ| , |Rκµ〉] = W. (3.32)
With this notation, the Green’s function for the single-site problem is
Gc(~r,~r ′, E) =
1W
∑
κµ 〈~r Rκµ〉 〈Hκµ ~r′〉 r < r′
1W
∑
κµ 〈~r Hκµ〉 〈Rκµ ~r′〉 r > r′.
(3.33)
The free Green’s function is obtained as special case of Eq. 3.33 with free-particle so-
lutions |jκµ〉, |hκµ〉 for |Rκµ〉, |Hκµ〉. For ~r within the range of our single potential at the
origin and ~r ′ outside the range of this potential, we can re-expand the upper and lower
components of the outgoing wave about any other site R at position ~R using the addition
theorem for the Bessel functions [8]. Note that expanding the lower component yields the
same expansion coefficients as expanding the upper component. This can be seen by noting
that, in the interstitial region, the outgoing spherical wave |Hκµ〉 = eiδκ |hκµ〉 is a solution
of the free-particle Dirac equation with energy E and hence must be expressible as a linear
combination of the states |jκµ〉 at the same energy about R. A slightly more elegant proof
of this fact is given by Wang, et. al. [54] in their appendix. The expansion gives
Gc(~r,~r ′, E) =1
W
∑
κµκ′µ′
〈~r Rκµ〉 eiδκG0κµnκ′µ′n′
⟨
jκ′µ′ ~rn⟩
(3.34)
with ~rn = ~r − ~Rn the position relative to site n and
G0κµnκ′µ′n′ = il−l′
∑
σ
〈J Lσ〉G0LnL′n′
⟨
L′σ J ′⟩
= il−l′∑
σ
⟨
J Lσ⟩
G0LnL′n′
⟨
L′σ J ′⟩ .(3.35)
The matrix elements
〈J Lσ〉 = 〈j, µ l,m, σ〉 (3.36)
1As pointed out by Wang et. al. [54] the ’regular’ solution diverges at the origin for κ = ±1. Grant [17]discusses the near-origin behavior of the radial wave functions.
21
appearing in 3.35 are Clebsch-Gordon coefficients, J = (j, µ) (j is the total angular mo-
mentum quantum number corresponding to κ), and the orbital angular momentum states
are
L = Lκµσ = (lκ, µ− σ, σ)
L = Lκµσ =(
lκ, µ− σ, σ)
.(3.37)
Equation 3.34 is an exact representation of the single-site Green’s function.
Relativistic full propagator
We represent the full Green’s function for the muffin-tin potential in a form similar to
equations 3.34 and 3.35:
G(~r,~r ′, E) =Gc(~r,~r ′, E)δnn′+
1
W
∑
κµκ′µ′
〈~r Rκµ〉GFMSκµnκ′µ′n′
⟨
Rκ′µ′ ~rR⟩ (3.38)
GFMSκµnκ′µ′n′ =
∑
σ
〈J Lσ〉 ileiδκGFMSLnL′n′il
′
eiδ′
κ⟨
L′σ J ′⟩ (3.39)
Relativistic density matrix in angular momentum space
Once the propagator is obtained, the density matrix
ρ(~r,~r ′, E) = (−1/π) ImG+(~r,~r ′, E)
=∑
j
φj(~r)φ∗j (~r
′)δ(E − Ej) (3.40)
can be evaluated.
To simplify expressions for the density matrix, we use real scattering states 〈~r bκRκµ〉where bκ is a common normalization factor for the radial wave functions Pκ, Qκ defined by
the requirement
limr→0
bnκPκ = r√
κ2−Z2/c2 (3.41)
22
where c is the speed of light and Z is the atomic number.2 It is also necessary to use real
spherical harmonics Ylµ = alµmYlm. With these adjustments, the scattering states are real
for real energies, and Im 〈~r bκHκµ〉 = 〈~r bκRκµ〉, so
ρ(~r,~r ′, E) =∑
κµκ′µ′
〈~r bκRκµ〉 ρκµnκµ′n′
⟨
bκ′Rκ′µ′ ~r ′⟩ (3.42)
ρκµn,κ′µ′n′ = Im
[
1
Wπbκb′κ
(
iδn,n′δκµκ′µ′ + il−l′ei(δκ+δκ′ )Gκµnκ′µ′n′
)
]
. (3.43)
where G and G are related by the unitary coefficients alµm:
GL,L′ = δl,l1δl ′,l2∑
m1,m2
(alm,m1
)∗GL1R,L2R′alm′,m2
(3.44)
alµm =
1√2
(δµ,m + (−1)µδ−µ,m) µ > 0
δµ,0 µ = 0
i√2
(−δµ,m + (−1)µδ−µ,m) µ < 0.
(3.45)
Broadening
In finite systems ρ(E) has very sharp energy dependence near the eigenfrequencies of electron
system. To arrive at a numerically stable procedure for computing integrals involving ρ(E)
it is necessary employ some method that broadens the sharp energy dependence. Since
G(E) is analytic in the upper half-plane
G(E + iΓ) =
∮
dE′ ΓG(E′)/π
(E′ − E)2 + Γ2(3.46)
where Γ is a broadening parameter that can be set to match experimental broadening and
the contour integral has been closed in the upper half-plane. Taking the imaginary part of
equation 3.46 gives the Lorentzian-broadened density matrix
ρΓ(E) =∑
j
φj(~r)φ∗j (~r
′)δΓ(E − Ej) =−1
πImG(E + iΓ) (3.47)
where δΓ is a Lorentzian of width Γ. The replacement G(E) → G(E + iΓ) has two conse-
quences that require some care in the current context: off the real line the radial solution Rκ
2The exponent in equation 3.41 is a consequence of assuming a point nucleus of charge Z [17].
23
are no longer real and the irregular solution Hκ contributes to the density matrix. We use
a model density matrix that has does not have sharp energy dependence, can be expressed
in the separable form 3.43 in terms of real wave functions RK and matrix element ρKn,K ′n′ ,
and coincides with the broadened density matrix when ~r = ~r′. To get this model we use
the conventions:
Rκ = Re (bκRκ)
ρcκ(E) =
Im
−2kπ
∫ RNrm
0 r2Rκ(r)Hκ(r)dr
∫ RNrm
0 r2R2κ(r)dr
ρScKn,K ′n′(E) = Im
−2k
πil−l′ei(δκ+δκ′ )GKn,K ′n′
ρ(~r,~r ′, E) =∑
K,K ′
⟨
~r RK
⟩
[
ρcκ(E)δn,n′δK,K ′ + ρSc
Kn,K ′n′(E)]
⟨
RK ′ ~r ′⟩
(3.48)
where∣
∣
∣RK
⟩
is the ket constructed from the (real) radial wave function Rκ. With these
conventions, our density matrix ρ(~r,~r′, E + iΓ) differs from the true broadened density
matrix
−ImG(~r,~r′, E + iΓ)/π
but gives the same DOS and becomes exact as Γ → 0.
3.1.3 Complex scattering potential
The construction of the self-consistent muffin-tin scattering potential for the one-particle
states is described elsewhere [4], and we only briefly summarize the process here. First, a
Dirac-Fock solver is used to calculate free-atomic potentials and densities which are then
overlapped to obtain a starting point for the self-consistency loop. In this loop the one-
particle Green’s function for the full multiple scattering problem is calculated, from which
a new electron density is calculated. Finally a new ground state muffin-tin potential is
constructed within the LDA. The loop is iterated to self-consistency which typically takes
about 10-20 iterations. Self-energy corrections are subsequently added for unoccupied states
within the GW plasmon-pole approximation.
24
3.2 Core state response
At low energies (below the bottom of the valence band), the density matrix becomes sparse
in energy, taking non-zero values only at isolated eigenvalues. In this regime, it is more
computationally efficient to use orbitals to describe the electronic structure. Thus, we
separate the single particle density matrix into two energy regions: the core region in which
the atomic approximation is valid and the solid-state region where solid-state corrections
are important,
ρ(E) =
ρcore(E) E < Ecv,
ρval(E) E > Ecv.
(3.49)
The core-valence separation energy Ecv is chosen to be an energy away from all KS eigen-
values that separates the two regimes and is set by default to −40 eV, which is typically
about 30 eV below the Fermi level. Above this energy ρ(E) is derived from the single-
particle Green’s function as described below. Note that in general there are occupied and
unoccupied states above Ecv, but there are no unoccupied states below Ecv. Similarly, the
dielectric function ǫ2(ω)can be separated into contributions ǫcore2 (ω) and ǫval2 (ω) arising from
transitions with core and valence initial states respectively.
The core states are represented by single-particle atomic-like orbitals φν . Here the index
ν = (n, i) denotes both a site index n and atomic level index i for the particular bound
state at that site (e.g. 1s, 2s, 2p1/2, etc.). We replace ρ(E) in Eq. (2.18) for E < Ecv with
ρcore(E) =∑
ν
ρ(ν)at (E), (3.50)
ρ(ν)at (~r,~r ′, E) = φν(~r)φν(~r ′)δ(E − ǫν).
Thus we recover an expression equivalent to Fermi’s golden rule for the absorption of light
ǫcore2 (ω) =∑
ν
ǫν2(ω) (3.51)
=4π
ω
∑
ν
Im〈 i | d †G(ω + ǫi)d | i 〉θ(ω + ǫi − ǫF ).
The initial core states |φν〉 and their associated eigenvalues ǫν are described accurately by
Dirac-Fock atomic states for a single atomic configuration [7]. For energies below Ecv ≈
25
µ − 30 Ev < V0 the eigenfunctions of the central site problem are tightly bound to the
central atom; their wave-functions decay rapidly as a function of the distance from the
central site and can be taken to vanish in all cells except the central cell. This, along
with the selection rules, limits the elements GKnK′n′ (representing the final states) that
contribute to absorption. For core initial states, the final state energy includes the inverse
core-hole lifetime Γν which broadens ǫν . The calculation of the density matrix elements
appearing in equation (2.16) is handled differently depending on the photoelectron energy
E = ω + ǫν . For low-energy (less than ≈ 50 eV+V0) final states G is calculated by FMS
just as in the calculation of ǫval2 . At very high energies we again employ an atomic model
and neglect scattering contributions (i.e. GKnK ′n′ = 0 in Eq. (3.38)). At intermediate
energies (50 eV +V0 ≤ E ≤ 1000 eV) we use efficient path filters [58] developed to treat
EXAFS to find the dominant terms in the multiple scattering path expansion and sum these
contributions to obtain the necessary GKnK ′n′ elements,
G = G0 +G0TG0 + · · · . (3.52)
The calculation of ǫcore2 is accomplished by summing the edges corresponding to initial states
ν with eigenvalues below Ecv. For each such edge we calculate ǫν2 via FMS, path-expansion,
and the atomic approximation on appropriate energy grids. At this stage, correlated Debye-
Waller factors can be included as in conventional XAS calculations using FEFF.
3.3 Valence response
Using the formal relation between the density matrix and the one-particle Green’s function
ρ(E) = (−1/π)ImG(E) one obtains from Eq. (3.38)
ρval(~r,~r ′, E) =∑
K,K ′
RKn(~r)ρKn,K ′n′RK ′n′(~r ′n′), (3.53)
which is valid for ~r in cell n and ~r′ in cell n′, where RKn(~r) = χK(rn)Rκn(rn). For real
energies, the density matrix can be expressed entirely in terms of the regular solutions RKn,
and the irregular solutions do not enter. Below the Fermi level on the real energy axis, the
density matrix is a rapidly varying function of energy. Away from the real axis, however,
the behavior is much smoother. As mentioned above, to both retain the separable form of
26
Eq. (3.53) and the smoothness obtained by calculating the Green’s function away from the
real axis, we introduce a small broadening Γ and renormalize the regular solutions according
to Equation 3.48, so that the central atom density matrix gives the same density of states
(DOS) in each Norman sphere as the actual broadened density matrix. This result is a key
simplification in our approach. Here the Norman radius rNrmn is defined as the radius of a
neutral sphere centered on the nth atom in the charge distribution formed by overlapping
the charge distributions of the isolated atoms in their solid-state positions. The separable
representation of the density matrix in Eq. (3.53) permits a separation of the double spatial
integral in Eq. (2.16) into a product of two one-dimensional integrals. To complete the
spatial integral in Eq. (2.16), we make the approximation that the spherical Norman cells
n partition space and write the full integrals as sums of integrals over individual cells
∫
d~r −→∑
n
∫
~r∈nd~r =
∑
n
∫ r(n)N
0r2ndrn
∫
dΩn. (3.54)
The dipole matrix elements at each site n are defined as
MnK,K ′(E,E ′) =
∫
~r∈nd~r RKn(~r;E) d ˜RKn(~r;E ′)). (3.55)
In the dipole approximation the matrix elements vanish except for transitions with j ′ =
j ± 1. Left (right) circularly polarized light only induces transitions with mj′ = mj + 1
(mj′ = mj−1). Thus the transition matrix M is sparse. Relaxing the dipole approximation
is straightforward. Doing so introduces additional non-zero elements to M . With these
conventions, the contribution to the spectrum from the response of the valence states (i.e.
those occupied single-particle states with eigenvalues above Ecv) is given entirely in terms
of density matrices and matrix elements,
ǫval2 (ω) =4π
V
∫ EF
EF−ωdE
∑
n,n′
Tr ρnn′(E)Mn′(E,E + ω)
×ρn′n(E + ω)MTn (E + ω,E), (3.56)
where ρnn′ and Mn are matrices in a truncated relativistic angular momentum K = (κ, µ)-
space. By symmetry, the sum over sites n in Eq. (3.56) can be reduced to a sum over
inequivalent sites in the solid. To compute ǫval2 we first solve the Dirac equation at each
27
inequivalent site which yields T . Then GLnL′n′ is found by inverting the full multiple
scattering matrix, and matrix elements M are evaluated using the wave functions from the
calculation of T . Finally, Eq. (3.56) is evaluated using trapezoid rule integration for the
energy integrals.
3.3.1 Spectrum construction
With the response of both the valence band and the more tightly bound electrons calculated,
the contribution from each core edge is then interpolated onto a final output grid and
combined with the other core edges and with the valence contribution:
ǫ2(ω) =∑
ν
ǫ(ν)2 (ω) + ǫval2 (ω). (3.57)
3.4 Comparison of muffin-tin RSMS electronic structure with other methods
The venerable muffin-tin approximation is still the de facto standard for x-ray absorption
calculations and the biggest approximation after the single-particle approximation employed
in the current work.3 In the muffin-tin approximation, the single-particle potential is ap-
proximated as a sum V (~r) =∑
n vn(rn) + V0 of (possibly overlapping) potentials vn having
spherical symmetry about a cell center (always an atomic site in our case) n. There have
been some attempts to implement full potential schemes within multiple-scattering theory
[20, 54], but these efforts have encountered numerical difficulties and are still not widely
used, in contrast to the muffin-tin codes. On the other hand, optical spectra of crystals are
most often and easily calculated using the plane-wave pseudopotential method, which is a
full potential method. Since we are calculating optical properties, it makes sense to estimate
the magnitude of the effects of the muffin-tin approximation on the calculated electronic
structure, which we do presently.
Within methods that are based or rely on DFT, the electron density ρ(~r) is an important
quantity. From the density-matrix the electron density is simply the energy integral of the
3In fact, one could argue that the muffin-tin approximation is more severe than the single-particle one,since exact knowledge of the exchange-correlation functional would produce an exact Kohn-Sham (single-particle) theory.
28
~r-diagonal density matrix:
ρ(~r) =
∫ EF
ρ(~r,~r,E)dE. (3.58)
We have evaluated this expression numerically for the case of a cluster of carbon atoms
arranged on the diamond lattice. In the plane-wave method, the density is recovered as
a Brillouin-zone integral and sum over bands. Practically, we have used ABINIT [16] to
compute the band structure and wave functions and the included utility cut3d to print
out the density. We have used pseudopotentials to represent the two core electrons in the
plane-wave calculation, so we only integrate over the valence band when evaluating 3.58.
The diamond lattice has a Voronoi cell with 16 faces. For each method the density has been
computed on a grid spanning the intersection of the Voronoi cell and the plane shown in
Figure 3.1 which contains a “central” atom and 4 second neighbors. The density is plotted
in Figure 3.2 and (weighted by r2) in Figure 3.3. The agreement is generally good except
near the atomic site at the origin where the calculations differ. This discrepancy is an
artifact of the pseudopotentials and not of interest here. Some basic observations are clear
from these pictures. First, the muffin-tin approximation does not give a spherical density
and, in fact, has almost as much variation with angle as the full-potential calculation. This
can be seen most clearly in Figure 3.3. Secondly, the disagreement is greatest in portions
of the Voronoi cell that are far from the atomic site. The origin of this discrepancy is the
angular momentum expansion. The angular momentum arrays in the real space calculation
were kept through l = 4. To exhibit the effects of angular momentum truncation the real-
space calculation was repeated with the angular momentum arrays truncated at l = 2. The
fractional discrepancies between each of these two calculations and the reciprocal space
calculation are plotted in Figure 3.4, which demonstrates the convergence of the calculation
with angular momentum.
29
Figure 3.1: The Voronoi cell in diamond with the first two shells of atoms and the planefrom which the data in Figures 3.2, 3.3, and 3.4 was drawn.
30
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5-2
-1.5-1
-0.5 0
0.5 1
1.5 2
2.5
0 0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45 0.5
ABINIT w/ psuedopotentialsFEFF
Figure 3.2: Valence charge density in diamond calculated by real-space multiple scatteringand the plane-wave pseudopotentials method. The scale for the horizontal plane is Bohrand the scale for the vertical axis is electrons per cubic Bohr.
31
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.5 1 1.5 2 2.5
distance from atom (Bohr)
FEFFABINIT
Figure 3.3: Valence charge density in diamond calculated by real-space multiple scatteringand the plane-wave pseudopotential method, weighted by the square of the distance fromthe atom at the origin.
32
1e-05
0.0001
0.001
0.01
0.1
1
0 0.5 1 1.5 2 2.5
distance from atom (Bohr)
lmax=4lmax=2
Figure 3.4: Fractional difference in the valence charge density in diamond calculated byreal-space multiple scattering and the plane-wave pseudopotential method for two angularmomentum cutoffs in the real-space calculation.
33
0
0.1
0.2
0.3
0.4
0.5
0.6
-60 -40 -20 0 20 40 60 80 100 120 140
stat
es/e
V/a
tom
electron energy (eV)
abinit: total DOSabinit: sum of ldos through l=4abinit: sum of ldos through l=3FEFF: sum of ldos through l=4FEFF: sum of ldos through l=3
Figure 3.5: Density of one-electron states (DOS) in Diamond. ABINIT total DOS is shownas well as sums of the l-projected DOS though l = 3 and l = 4 for the FEFF Norman sphere.These are compared to the corresponding FEFF quantities. Note that, due to overlapping ofthe Norman spheres about neighboring atoms, the sum of the l-projections exceeds the totalBrillouin zone integral at some energies. Also note that the truncated angular momentumexpansion does not represent all states at high energies.
34
Chapter 4
CALCULATIONS AND APPLICATIONS
We now present our results. The chapter opens with plots for several optical constants,
then turns to diagnostics and applications. We describe a few important applications of our
optical constant calculations that have been developed to date.
4.1 Theoretical Optical Constants
The examples presented here are primarily monatomic crystals (metals and insulators)
with a single inequivalent site. However, the generalization to heterogeneous materials is
straightforward, and an example is also presented for Al2O3. Non-periodic materials can be
treated by averaging over sites. We have found that such a site average converges quickly.
The calculations presented in this section used FMS matrices truncated at l = 3 and 147
atoms for all materials except diamond. The diamond calculation used l = 2 and 450 atoms.
All spectra were obtained by summing the contributions from 70 atoms. The response for
the valence bands is obtained by calculating ρval on a regular energy grid of 200 points.
Then the dipole matrix elements M(E,E′) are calculated for all pairs (E,E′) with E below
the Fermi level and E′ above it. Eq. (3.56) is then evaluated by matrix multiplication and
simple numerical integration. To compute ǫval2 (ω) to high frequencies, we employ an atomic
model of the valence bands based on average band energies and occupations calculated
from ρval. The core state response is first calculated on a set of five 100 point frequency
grids for each core initial state κ in the embedded-atom approximation. The FMS and
path-expansion calculations are then carried out in cluster sizes of around 175 atoms on
frequency grids of approximately 120 points. The contribution to ǫ2 for each core initial
state and the valence bands are then interpolated onto a large (5 × 105 points) frequency
grid which spans the full spectrum (e.g. 10−3 through 106 eV) and serves as the final output
grid. This grid has a higher density of points at low frequencies and around each core edge.
35
0.0001
0.001
0.01
0.1
1
10
1 10 100 1000
TheoryHageman, et. al.
Stahrenberg, et. al.
9000 9500 10000
Figure 4.1: Calculated ǫ2 for Cu plotted with the experimental data of Stahrenberg, et. al.[48], the DESY compilation [18], and K-edge data of Newville [31].
4.1.1 Dielectric function: Imaginary part
The fundamental quantity needed in our calculations of optical response is the imaginary
part of the dielectric function ǫ2(ω) given by Eq. (3.57). All other optical constants
can be obtained in terms of ǫ2(ω) as described below. As illustrative examples our density
matrix calculations of ǫ2(ω) for Cu and Au are plotted in Fig.’s 4.1, 4.2, and 4.3 compared to
experiment. To demonstrate the effects of structural disorder on the dielectric response, we
compare the imaginary part of the dielectric function for Diamond and amorphous Carbon in
Fig. 3.57. Amorphous carbon structures were obtained with a “melt-and-quench” algorithm
[15] using first principles molecular dynamics as implemented in the VASP package [25]. A
periodic cell of 32 carbon atoms was melted at T=5000K, and then quenched down to
T=300K. The system was further evolved at T=300K, and the final structure was obtained
by optimizing the total energy via a conjugate gradient method [27]. These results, as well
as the results presented below and calculations for other materials, are currently available
36
0.0001
0.001
0.01
0.1
1
10
100
1 10 100 1000
TheoryExperiment
Figure 4.2: Calculated and experimental ǫ2 for Au [18].
0 5 10 15 20 25 30
ω (eV)
Diamond
aC
TheoryExperiment
Figure 4.3: Calculated and experimental ǫ2 diamond [34] and amorphous C [53] (bottom).In the bottom panel the diamond curves have been shifted vertically for clarity.
37
in both graphical and tabular form on the FEFF website [36].
4.1.2 Dielectric function: Real part
Owing to the analyticity of the dielectric response, the real and imaginary parts of the
dielectric function are related by the Kramers-Kronig relation [23]
ǫ(ω) = 1 +2
πP∫ ∞
0dω ′ω
′ǫ2(ω′)
ω2 − ω ′2 . (4.1)
Here P indicates the principal value of the integral. Since the denominator of the integrand
in Eq. (4.1) has a pole at ω ′ = ω care must be taken when evaluating the transform
numerically. To evaluate the integral appearing in Eq. (4.1) over the interval (ωi, ωi+1)
between the ith and (i+ 1)th grid points we find a linear approximation ǫ2(ω′) = mω ′ + b,
which allows us to rewrite the Kramers-Kronig integral as follows:
P∫ ωi+1
ωi
dω ′ω′ǫ2(ω
′)
ω2 − ω ′2 = m(ωi+1 − ωi) + (4.2)
b−mω
2ln
(
ωi+1 + ω
ωi + ω
)
+b+mω
2ln
(
ωi+1 − ω
ωi − ω
)
.
This expression is used to produce ǫ1 on the same output grid used for the imaginary part.
The results of this procedure for diamond, Cu and Al2O3 are plotted in Fig.’s 4.1.2, 4.1.2,
and 4.1.2. Even though the numerical transform Eq. (4.3) is stable and accurate and (along
with the calculated ǫ2) completely determines ǫ1 via Eq. (4.1), we find that the real part of
the dielectric function is more sensitive to errors and approximations than the imaginary
part.
4.1.3 Energy-loss
With both real and imaginary parts of ǫ(ω) one can easily obtain the energy loss function
−Im ǫ−1(ω) =ǫ2(ω)
ǫ22(ω) + ǫ21(ω). (4.3)
This is illustrated for Cu, Al2O3 , and Au in Fig.’s 4.7, 4.9, and 4.8. The loss function is
proportional to the long-wavelength limit of the dynamic structure factor S(~q, ω), which can
be measured by inelastic scattering of either electrons in electron energy loss spectroscopy
38
-8
-6
-4
-2
0
2
4
6
8
10
12
0 5 10 15 20 25 30 35 40
TheoryExperiment
Figure 4.4: Calculated ǫ1 for diamond compared to experiment [18, 34].
-80
-70
-60
-50
-40
-30
-20
-10
0
10
TheoryExperiment
Figure 4.5: Calculated ǫ1 for Cu compared to experiment [18].
39
0
1
2
3
4
5
6
1 10 100 1000
ω (eV)
TheoryExperiment
Figure 4.6: Calculated ǫ1 for Al2O3 compared to experiment [18].
(EELS) or photons in non-resonant inelastic x-ray scattering (NRIXS). Calculations of the
latter performed in a framework similar to ours have recently been reported by Soininen et.
al. who only address the response of core electrons, but at finite ~q [45]. In contrast to ǫ1
we find that the loss function is less sensitive to errors and approximations in the density
matrix than ǫ2. Onida, et. al. [33], in an illuminating discussion of the differences between
absorption and EELS experiments, have given an explanation of this observation in terms
of the long-range part of the coulomb interaction.
4.1.4 Index of refraction
Under the assumption that the (relative) permeability is 1, the complex index of refraction
is simply the square root of the complex dielectric function
n(ω) + iκ(ω) ≡ ǫ(ω)1/2. (4.4)
40
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9Theory
Experiment
Figure 4.7: Calculated energy-loss function (Eq. 4.3) for Cu compared to experiment [18,34].
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 10 100 1000
ω (eV)
TheoryExperiment
Figure 4.8: Calculated energy-loss function (Eq. 4.3) for Au compared to experiment [18].
41
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1 10 100 1000
ω (eV)
TheoryHageman, et. al.
French, et. al.
Figure 4.9: Calculated energy-loss function (Eq. 4.3) for Al2O3 compared to experimentalx-ray [18] and EELS [30] data.
42
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6Theory
Experiment
Figure 4.10: Calculated real index of refraction for Cu compared to experiment [18].
Typical results for the real part of the index of refraction are given in Fig.’s 4.10, 4.11, and
4.12.
4.1.5 Absorption coefficient
The photon absorption coefficient µ(ω) is defined as the (natural) logarithm of the ratio of
the incident and transmitted intensities for a photon beam across a thin sample, divided by
the thickness. Theoretically µ(ω) can be expressed in terms of the imaginary part of the
index of refraction κ(ω)
µ(ω) = 2ω
cκ(ω). (4.5)
Thus, µ(ω) is directly measurable with optical absorption experiments. Such experiments
are currently performed to high accuracy using synchrotron light sources. We compare
our calculated results with experiment for several materials and with a calculation based
on electronic structure calculated with ABINIT. This calculation was accomplished using
the AI2NBSE package developed by Lawler, et. al. [26] which employs a BSE solver
43
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
1 10 100 1000
TheoryExperiment
Figure 4.11: Calculated real index of refraction for Al2O3 compared to experiment [18].
0
0.5
1
1.5
2
2.5
3
3.5
4
5 10 15 20 25 30 35 40
ω (eV)
TheoryExperiment
Figure 4.12: Calculated real index of refraction for diamond compared to experiment [34].
44
0.0001
0.001
0.01
0.1
1
10
100Theory
Experiment
Figure 4.13: Calculated absorption coefficient µ in inverse cm for Cu compared to experi-ment [18].
developed at NIST to generate optical spectra. The calculation shown excludes both local
fields and excitonic effects and was generated using a regular grid of 83 k-points to sample
the Brillouin zone, 50 bands, and an energy cutoff of 30 Hartree for the plane wave basis.
The C 1s electrons were treated with a Troullier-Martins pseudopotential. For a sensible
comparison, no gap corrections were included in either calculation.
4.1.6 Reflectivity
An important optical experiment for materials that can be prepared by vapor deposition
methods is the measurement of the reflectivity R defined as the ratio of the power reflected
from a planar face of a sample to the incident power. This quantity can be related to
the dielectric response of the material by considering the boundary conditions satisfied
by Maxwell’s equations at the interface between the sample and vacuum. This procedure
produces the familiar Fresnel equations [23] relating the amplitudes of the transmitted
45
0.0001
0.001
0.01
0.1
1
10
100
1 10 100 1000 10000 100000
TheoryHageman, et. al.
Windt, et. al
Figure 4.14: Calculated absorption coefficient µ in inverse cm for Au compared to theexperimental data of Hageman, et. al. [18] and Windt, et. al. [55] which include uncertaintyestimates.
46
0
5
10
15
20
25
30
35
40
45
5 10 15 20 25 30
ω (eV)
Real-space theoryReciprical-space theory
Experiment
Figure 4.15: Calculated absorption coefficient µ in inverse cm for diamond compared toexperiment [34] and a reciprocal-space calculation.
47
(refracted) and reflected waves to the amplitude of the incident wave. As discussed by
Stratton, [49] R can be found by squaring the Fresnel equations. For example, for normal
incidence
R(ω) =[n(ω) − 1]2 + κ2(ω)
[n(ω) + 1]2 + κ2(ω). (4.6)
The general expression for a lossy material (ǫ2 6= 0) and arbitrary angle of incidence is com-
plicated. However it is interesting to note that off normal incidence R(ω) has polarization
dependence even for isotropic media.
4.1.7 Photon scattering amplitude
The Rayleigh forward scattering amplitude f(ω) for photons can also be computed from
the dielectric function [5]
f(ω) =ω
4πr0c2V
N[ǫ (ω) − 1] . (4.7)
Thus it is straightforward to calculate the x-ray scattering factors including anomalous
terms using our RSMS approach f(~q, ω) = g(q, ω) + f ss(~q, ω) + f1(ω) + if2(ω) in terms of
f . Typical calculations of the real and imaginary parts of f(ω) are illustrated in Fig.’s 4.16,
4.17, and 4.18 and 4.19, 4.20, and 4.21.
4.1.8 NRIXS
As described in Chapter A the full spectrum output is computed by the opcons code which
is an auxiliary program that can be trivially modified to work with any version of FEFF
that uses xmu.dat for output. This containment allows us to extend all the developments
in the FEFF project to full spectrum output. One such development is FEFF8q [45]. This
modification of the FEFF code due to J. A. Soininen extends the capabilities of FEFF to the
contribution to the dynamic structure S(q, ω) from core excitations. Figure 4.22 shows the
L edges of q2S(q, ω) for a variety of momentum transfers q in Aluminum metal as calculated
by FEFFOP running on top of FEFF8q.
48
-8
-6
-4
-2
0
2
4
6
8
10
12
5 10 15 20 25 30
TheoryExperiment
Figure 4.16: Calculated real part of the anomalous atomic scattering factor for diamond[34].
49
-10
0
10
20
30
40
50
60
1 10 100 1000 10000
ω (eV)
TheoryExperiment
Figure 4.17: Calculated real part of the anomalous atomic scattering factor for Au comparedto experiment [18].
50
-5
0
5
10
15
20
25
30
35
1 10 100 1000 10000
ω (eV)
TheoryExperiment
9000 9500 10000
Figure 4.18: Calculated real part of the anomalous atomic scattering factor for Cu comparedto experiment [18, 11].
51
0
2
4
6
8
10
12
14
16
18
20Theory
Experiment
Figure 4.19: Calculated imaginary part of the atomic scattering factor for Cu compared toexperiment [18].
52
0
5
10
15
20
25Theory
Experiment
Figure 4.20: Calculated imaginary part of the atomic scattering factor for Au compared toexperiment [18].
53
0
5
10
15
20
25
1 10 100 1000 10000
ω (eV)
TheoryExperiment
Figure 4.21: Calculated imaginary part of the atomic scattering factor for Al2O3 comparedto experiment [18].
54
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 200 400 600 800 1000 1200 1400 1600 1800 2000
ω(eV)
q=0.03 bohr-1
q=1 bohr-1
q=2 bohr-1
q=3 bohr-1
q=4 bohr-1
q=5 bohr-1
q=6 bohr-1
q=7 bohr-1
q=8 bohr-1
q=9 bohr-1
q=10 bohr-1
Figure 4.22: L edge dynamic structure factor q2S(q, ω) ∼ −q2Im ǫ−1(q, ω) weighted by q2
for Al metal as computed by FEFFOP running on top of FEFF8q. The peak in the lossfunction moves to higher ω as q increases.
55
0.001
0.01
0.1
1
10
100
0.1 1 10 100
ω (eV)
Hamaker constant: ε(i ω)ε2(ω)
Figure 4.23: Calculated Hamaker constant ǫ(iω) compared to ǫ2(ω) for Pd metal.
4.2 Applications and diagnostics
4.2.1 Hamaker constant
The Hamaker constant is the (real) function ǫ(iω) of a real frequency ω. For separation
distances beyond the tunneling regime, the interaction between the tip and sample in an
atomic force microscopy experiment is dominated by the van der Waals force, which can
be calculated given the tip-sample geometry and the Hamaker constants of the tip and
sample[19]. Using the analyticity of ǫ in the upper half-plane, one can derive the following
Kramers-Kronig type transform for the Hamaker constant
ǫ(iω) =1
π
∫ ∞
0dω ′ω
′ǫ2(ω′) + ωǫ1(ω
′)
ω2 + ω ′2 . (4.8)
We evaluate Eq. (4.8) numerically in the same way we evaluate the Kramers-Kronig trans-
form from ǫ2 to ǫ1, although away from ω = 0 the integrand is regular. The Hamaker
constant for Pd metal is plotted in Figure 4.23.
56
0
2
4
6
8
10
12
14
1 10 100 1000 10000 100000
ω (eV)
TheoryExperiment
Figure 4.24: neff(ω) calculated from the ǫ2 sum rule for Mg using Eq. (4.9).
4.2.2 Sum rules
Included in the output of our code are a few quantities useful for understanding the rela-
tionship between the underlying electronic structure and the frequency dependence of the
optical constants. The f -sum rules for the imaginary parts of the dielectric function and the
inverse dielectric function provide an important quantitative check of the calculation. We
define the effective number of electrons per atom participating in transitions at frequency
ω
neff(ω) =V
2π2N
∫ ω
0dω′ω′ǫ2(ω
′). (4.9)
This quantity has the limit [2]
limω→∞
neff(ω) = Z, (4.10)
where Z is the number of electrons in the subsystem whose number density is N/V . The
theory and calculations presented here are valid over a frequency range large enough to
quantitatively evaluate the limit (4.10); missing or extra oscillator strength implies invalid
57
approximations or unconverged calculations. Another check can be given by the index of
refraction sum rule.∫ ∞
0[n(ω) − 1] dω = 0. (4.11)
4.2.3 Joint Density of States
As stated above, the selection rules constrain the angular momentum of final and initial
states that can contribute to the absorption of light to a few channels (e.g. p → d, s → p,
etc.). The joint density of states (JDOS) corresponding to a certain dipole allowed channel
(l → l′) is defined in terms of the normal l-projected DOS ρl:
∫ Ef
Ef−ωρl(E)ρl′(E + ω) dE, (4.12)
where the l-projected DOS is given in terms of the density matrix by
ρl(E) =∑
m
∫
d~r|YL(r)|2ρ(~r,~r,E). (4.13)
Neglecting energy dependence of the dipole matrix elements in the calculation of ǫ2 gives a
spectrum which is a sum of terms proportional to the JDOS for the dipole allowed channels.
We show the JDOS/ω2 for transitions with initial p states compared to the calculated ǫ2
for Diamond in Fig. 4.25.
4.2.4 Inelastic losses for electrons
An important physical problem is the loss of energy by an electron passing through the
system. This classic problem has recently been reexamined by Sorini [47] who has applied
FEFFOP calculations to make the first quantitative ab initio calculations of such quantities
as the collision stopping power and inelastic mean free path. The seminal work on this
topic was done by Bethe [9] who developed the Bethe formula for the stopping power. This
formula depends on a parameter I defined by
ln(I) =
∫
dωω lnωImǫ−1
∫
dωωImǫ−1, (4.14)
58
0 5 10 15 20
Figure 4.25: JDOS/ω2 for p → d (solid line) and p → s (dashed line) transitions and thecalculated ǫ2 of this work (dots) for diamond vs. photon frequency in eV.
59
called the mean excitation energy in which Bethe placed much of the material dependence
which is difficult to calculate since it depends on the dielectric response of the material.
The mean excitation energy is an automatic output of the FEFFOP package. Sorini has
compared theoretical values of I with empirical values for a variety of materials finding
generally good agreement [46, 47]. While this is gratifying, a generally applicable theory of
inelastic losses needs to go beyond the mean excitation energy. Fortunately, one does not
need to go beyond linear response.
The dielectric function ǫ(~q, ω) contains all the information about the linear response of
the system (in fact, the Kramers-Kronig relations imply that complete knowledge of either
the real or imaginary part of ǫ or its inverse at all ~q and omega is sufficient to describe the
linear response). The work of this thesis gives a method to compute the q → 0 dielectric
function over a broad spectral range. By extending these calculations to finite ~q using
analytical models inspired by the electron gas we arrive at a method that gives an efficient
calculation of quantities that depend on the dielectric function or its inverse. One possible
way to extend our long-wavelength loss function to finite q is a “pole model:”
−Im[ǫ(q, ω)−1] = π∑
i
giω2i δ[ω
2 − ωi(q)2] (4.15)
where (ωi(q))2 = (ωi(0))
2 + aq2 + bq4 is the Hedin-Lundqvist [21] dispersion relation for
a plasmon with plasma frequency ωi(0). The strengths gi and locations ωi(0) for many
(typically 10-100) δ-functions can be found by fitting the calculated q = 0 loss function in
some suitable way [46, 24]. This type of pole model is especially well suited to quantities
which depend on the dielectric response through integrals over the loss function because the
δ-function of Equation 4.15 “does the integral for us.” Such quantities include the inelastic
mean free path (IMFP) for an electron traversing the sample [46] as well as the quasi-particle
self-energy in the GW approximation [24]. Our calculations of inelastic losses [46] based on
such a many-pole representation open new possibilities for computing excited state electronic
structure that are complimentary to existing many-body techniques. Our approach is to
use a preliminary, fast FEFFOP calculation employing a traditional Hedin-Lundqvist (single)
plasmon-pole model to compute an approximate loss function which is then used as the basis
for a more accurate, material specific many-pole self-energy model. Kas [46] has found that
60
it is often sufficient to include fine structure only for a few edges leading to quick self-energy
calculations. This should be contrasted with “full GW” calculations which can be very
computationally intensive and are typically only carried out for a few excited states (e.g. to
compute a band-gap correction). Our perspective is that most many-body effects relevant
for spectra are broadened and hence it is more important to capture the energy dependence
semiquantitatively than a highly precise band gap.
61
Chapter 5
CONCLUSIONS AND FUTURE DIRECTIONS
5.1 Conclusions
In this thesis we have contributed to the ab initio theory of dielectric response. This
work is a practical contribution: the foundations and many of the key approximations
utilized in the work presented here were previously known. What sets our work apart from
that of Ehrenreich and Cohen [12]? Formally, the theories are quite similar, but ours is
much more applicable: it can be applied to a wide range a materials with little effort. It
works over a vast range of frequencies. We have contributed, in one field, to the grand
challenge of condensed matter physics: the quantitative description of condensed systems
from microscopic principles.
In particular, we have built on the work of Rivas to implement the first full-spectrum
ab initio calculations of optical constants in a fast computer program that can be run on
a typical desktop computer by any reasonably proficient scientist who knows the struc-
ture of the system of interest. We have tested this implementation by comparison with
experimental data and other theoretical calculations. This method is based on a real-space
density matrix calculated within an effective single-particle theory. Our particular effective
theory was originally developed for x-ray absorption calculations and uses the muffin-tin
approximation. We have shown that the muffin-tin model can be used to calculate spectra
as well as more direct descriptions of the electronic structure, like the electron density and,
more generally, the real-space density matrix. Thus real-space multiple scattering repre-
sents a complimentary alternative to basis-set based electronic structure methods. We have
tested muffin-tin multiple scattering in the severe case of diamond and found satisfactory
agreement with plane-wave methods.
We have demonstrated that our ab initio optical constants can be used in lieu of measured
ones when the latter are unavailable in important applications including the calculation of
62
Hamaker constants, stopping powers for fast electrons, and inelastic mean free paths for
the same. We have also shown that full-spectrum capability is useful for improving the
treatment of quasiparticle and other many-body effects, an important problem in excited
state electronic structure.
5.2 Future work
As alluded to above, the theory published here is already sufficient to describe much of di-
electric response. But there are key physical processes which have been neglected or coarsely
approximated in our approach. Also, like most scientific codes, the FEFFOP package could
use some basic code maintenance and improvements to basic algorithms. We list improve-
ments that change the physics of our calculations here, leaving more practical improvements
for the appendix.
• The most important missing physics is that, ironically, screening has been neglected
in the calculation of the electron dynamics. Currently, FEFFOP calculates valence
spectra in the independent particle approximation which neglects local fields. Local
field effects have been treated for core edges by Ankudinov [3], but these routines have
only been tested in a few materials. This treatment of screening is based on time-
dependent density functional theory (TDDFT); it’s extension to valence electronic
response is theoretically straightforward, but some practical difficulties exist. Also,
as pointed out by Ankudinov [6], the real-space TDDFT routines use approximations
that are inadequate for situations in which excitonic effects are important.
• The atomic sphere approximation of Equation 3.54 should be removed. The matrix
elements should be computed by full integrals over the Voronoi cell. Note that this
improvement is independent of and, in the author’s estimation, more important than
relaxing the muffin-tin approximation.
• The valence response should be generalized to finite ~q so that the wave-vector depen-
dence of the dielectric function can be calculated. The only ingredient that is currently
missing are the matrix elements of Equation 3.55. Once these have been evaluated
63
beyond the dipole approximation, as Soininen [45] has done for core edges, the rest of
the calculation remains unchanged.
• It is time to move beyond the muffin-tin approximation. We have shown that this
useful approximation performs surprisingly well, and the spherical symmetry greatly
simplifies the mathematical structure of the resulting equations. Nonetheless, states
close to (either side of) the Fermi level are substantially affected by non-spherical
parts of the potential. There has been important work [51, 54] done along these lines
but full potential calculations are still the exception in real-space multiple scattering.
64
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69
Appendix A
DEVELOPER’S GUIDE
A.1 Overview of full spectrum algorithm and codes
The calculations described in this dissertation have been accomplished with standard release
versions of FEFF (8.2 and 8.4) and a new package FEFFOP, which consists of two compo-
nents: opcons and op abs. To describe all these codes, which by now are becoming quite
a complicated software package with many options and auxiliary calculations, we begin
with an outline of a full spectrum calculation making only limited reference to the specific
routines. We then indicate which routines are responsible for completing each part of the
calculation.
A.1.1 Algorithm
Given the structure (i.e. the Cartesian coordinates ~Ri of the nuclei and their charges
Zi) for a cluster of N atoms we proceed as follows to compute the dielectric function ǫ(ω)
for a specified polarization. This algorithm is depicted in a flowchart in Figure A.1.
1. Choose a number nph of model potentials and sites Ri to serve as representative sites.
Each atom of type iph in the cluster will have the same muffin-tin radius and potential
inside that muffin tin. If we are simulating a crystal, each inequivalent site gets its own
potential. Often, in practice, atoms are considered equivalent if they are of the same
species, regardless of environment. In a system without symmetry, in principle, every
atom should have its own potential, but the convergence of the SCF loop becomes less
and less likely as the number of distinct potentials in increased.
2. For each model atom ~Ri(iph) compute the atomic wave functions and potentials using
the Dirac-Fock atomic routines [7].
70
Figure A.1: Flowchart of full-spectrum calculation.
71
3. Create a starting muffin-tin model of the cluster by overlapping the atomic potentials
and symmetrizing the potential about each site representative site ~Ri(iph). Muffin-
tin radii are chosen at this time. The FEFF conventions for this step are described
in reference [8]. This step uses the LDA evaluated on the symmetrized overlapped
density.
4. Compute the density of this potential using FMS. Briefly, the integral
−1
πIm
∫
CdEd~rG(~r,~r,E) (A.1)
is evaluated along a contour C starting at the core-valence separation energy ECV
on the real axis, continuing into the upper half plane, traversing toward higher real
part. Of course the integral is evaluated by computing the Green’s function at discrete
points along the contour. At each step, a single-energy FMS inversion is done, and the
integral A.1 is updated. Once the integral equals the total number of valence electrons
in the cluster, the contour makes its way back to the real line moving to larger or
smaller real part as needed to keep A.1 equal to the total number of electrons. The
point at which C returns to the real axis is then the Fermi level associated with the
starting potential.
5. The potential is now updated by evaluating the Hartree potential and LDA exchange-
correlation potential for the charge density given by the previous step, symmetrizing
this potential and adding it to the potential from the ion cores which is taken from
the overlapped atom calculation (step 3).
6. The new Fermi level is compared to the previous estimate, if it exists. If the Fermi
level has changed by less than a small tolerance, the SCF loop is exited. Otherwise,
control is directed back to step 4 with the starting potential replaced by the potential
calculated in step 5.
7. With the potentials computed, an energy grid covering the valence bands and the final
states that are best described by FMS is chosen. At each energy point and about each
72
model site iph selected in step 1 the wave functions Riphκ , phase shifts δiph
κ , and dipole
matrix elements M iphK,K ′(E,E ′) are determined.
8. Using the phase shifts from the previous step, the FMS inversion is run for each point
in this grid. The results are saved to disk for later use.
9. The results of the FMS inversion are read from disk and the density matrix elements
are constructed using the second line of Equation 3.43.
10. ǫval2 is computed according to Equation 3.56. The atomic model is used to compute
the tails of this contribution. The results are saved to disk.
11. A new core edge is selected. The path expansion and FPRIME routines are run for
this edge at this site. An FMS calculation is also carried out using the results of step
8. All these results are saved.
12. If there are core edges which have not yet been simulated, return to step 11. Here we
must loop over all the core states at each distinct potential iph.
13. Select a frequency grid on which to output the optical constants. Initialize an array
to hold ǫ2 on this grid.
14. Select a model potential iph the contribution to ǫ2 of which will be added to our
spectrum. The number density of this type of site is calculated from the structure or
specified stoichiometry.
15. Loop over core edges ν combining the path expansion, FMS, and FPRIME calculations
into a single spectrum tabulated on the final output grid. Compute neff via the ǫ2
sum rule (Equation 4.9). Add ǫν2 to our final results.
16. Test for model potentials that have not yet been included in the final output. If one
exists, return to step 14.
73
FEFF 1, 2, 3, 4, 5, 6
op abs 7, 8, 9, 10
opcons 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Table A.1: Table showing which steps in the full spectrum algorithm are completed by eachcode in the FEFFOP package.
17. Perform Kramers-Kronig transform on the full-spectrum ǫ2 produced by the above
loops.
18. From the complex dielectric function compute the other optical constants.
19. Evaluate the sum rules, writing out the values of the integrals as a function of the
upper integration limit.
20. Evaluate the Hamaker constants by evaluating Equation 4.8.
A.1.2 opcons
The first of the new codes is an automated implementation of the prescription of Rivas
[40] for calculating core-level response. Written in bash and FORTRAN, this code uses a
master input file appropriate for FEFF and completes many runs using FEFF in a variety
of modes and keeping track of all necessary input and output files. Once these runs are
finished, the information from them is collected and assembled into a full spectrum. The
opcons package is an add-on to FEFF: it is not incorporated into FEFF proper, but rather
acts as a controller by preparing, executing, collecting, and combining many individual FEFF
calculations. There are two main parts of the opcons package: a controller for FEFF and a
”spectrum utility”. The controller can parse and write input files and run FEFF using them
as input. The spectrum utility has the capability to combine several FEFF calculations into
a single photoabsorption cross-section spectrum using a suitable numerical grid, convert
this spectrum to ǫ2, perform Kramers-Kronig transformations, evaluate various sum rules,
74
and calculate other optical constants. Features include automatic grid generation based on
a table of binding energies, evaluation of the f−sum on a large grid for each shell (i.e. each
core edge and the valence bands), and calculation of the mean excitation energy.
A.1.3 op abs
The primary work of this thesis is the development of the density-matrix formulation for
computing, within real-space multiple scattering theory, the dielectric response of electrons
for which the atomic approximation is not adequate. This formulation has been implemented
in op abs, a new extension to FEFF. Previous versions of FEFF simulated a single x-ray
absorption edge, i.e. the response of a single initial state which, due to its high binding
energy, can be unambiguously classified by its atomic quantum numbers and described by
the wave-function for this state in the atomic system. This new module provides a ’valence
edge’ capability for FEFF. Just like FF2X of the FEFF8 series, op abs produces xmu.dat
which tabulates photoabsorption cross-section as a function of photon frequency. Hence, it
can be used with opcons when creating a full spectrum. Roughly, op abs takes the place of
all modules except the potentials module. To accomplish an optical calculation, one should
first run the RDINP and POT modules from FEFF8. Once this is accomplished, the op abs
code can be run, producing xmu.dat.
A.2 Obtaining and installing FEFFOP
The FEFFOP codes reside in the FEFF project CVS repository. They can be obtained by
$cvs checkout opcons
or
$cvs checkout op abs
Alternatively, a tar’d archive can be obtained from the FEFF project upon request.
A.2.1 Installation of opcons
To compile opcons use the included scripts and makefiles. The makefiles assume the Intel
fortran compiler can be called as ifort for serial compilations and mpif77 for parallel
75
compilations. If this is not the case you will need to adjust two files: src/feff comp and
src/opcons/Makefile. The only obvious changes that need to be made are to define F77
in each of these files to be the appropriate local compiler. There is a script to compile all
fortran source at once:
$cd src
$sh install.sh
The scripts currently assume one is running LAM so that FEFF can be called with lamboot
and mpirun. If this is not the case bin/optical runmods will need to be modified. The
scripts will run FEFF under LAM started with the default boot schema (i.e. the scripts will
simply call lamboot with no arguments).
To test that everything is working, try doing the Aluminum calculation:
$cd runs/Al
$sh opcons
This should produce a file called opcons.dat which contains the optical constants. If
gnuplot is installed, you can check your result against the sample:
$gnuplot epsilon2.gnuplot
which will plot ǫ2 against the sample calculation and the DESY experiment.
A.2.2 opcons input and output
The opcons code looks for a file called core.inp in your run directory. This is an input file
for the full spectrum calculation. There are several documented cards that one can use in
core.inp.
COMPONENT compname iz [numden] [EDGES | DETAIL] Include absorption from compo-
nent compname in the final spectrum. For each component card the code requires a
FEFF input file named feff.compname that has as absorbing atom the model atom
for component compname. In addition, the user can specify the number density of this
absorber in inverse cubic angstroms. If the fourth argument is EDGES, the code will
76
interpret successive lines as the names of edges to be calculated. This allows the user
to produce spectra with just some of the edges included. If, instead of EDGES, the
fourth argument is DETAIL, the code will only compute the atomic background to ǫ2
for edges that are not in the list. Fine structure will be computed for those edges
that are listed (i.e. potentials will be computed using the SCF loop, xanes will be
calculated via FMS, and exafs will be calculated via path expansion with parameters
set in the original input file).
EGRID e max npts Use an output grid with npts points going up to e max eV. Defaults
(if the card is omitted) are npts=50000, e max=25000.
VALENCE Include valence response from op abs. This card causes opcons to include the
file valence/xmu.val from each component directory in addition to any core edges.
Currently, these files must be placed in the correct place by hand (i.e. run op abs for
each absorber, run opcons for the core edges using the EDGE card, move the op abs
output into the proper directories, then run opcons step4).
DETAIL If present fine structure is computed for all edges, otherwise only atomic back-
ground is computed unless the DETAIL option of the COMPONENT card is used.
The main output is the file opconsKK.dat which has eight columns containing ω (eV),
n, k, ǫ1, ǫ2, µ 105 (cm−1), −ǫ−1, R. The file osc str.dat lists the contribution from each
edge to the ǫ2 f-sum rule. The file sumrules.dat contains the ǫ2 partial sum rule as a
function of frequency for all the edges included in the calculation.
A.2.3 Installation of op abs
The code is installed by make, which should be run in the directory op abs/OPTICAL. The
Makefile has targets defined for serial (seq) and parallel (par) versions. Only the FMS
matrix inversion is currently parallelized. There are also several auxiliary codes that can be
compiled with the Makefile. They are described below. The resulting executable should
be in op abs/bin. It will be named op abs new bin or op abs new bin.
77
A.2.4 op abs input and output
To produce xmu.dat for the valence response using op abs follow the following instructions.
1. Create a normal feff.inp. Include FPRIME 0 w max where w max is the highest fre-
quency (in eV) photoelectron state to be calculated for the atomic tail (500 is a good
start). Also include an LDOS card whose energy range specifies the photoelectron states
(both final and initial) to consider in the calculation.
2. Calculate the potentials by running rdinp and ffmod1 of FEFF 8.4.
3. Run the actual op abs executable.
The script op abs automates the last two steps of this procedure.
In addition to the main output in xmu.dat, there are also files jdosl.dat for 0 ≤ l ≤ lx
which contain the joint density of states for l → l± 1 transitions as a function of frequency
for each initial state l. The file kappados.dat lists the DOS evaluated from the density
matrix used in the optical calculation. This should, of course, be quite similar to the normal
FEFF LDOS output. One notable difference is that the op abs DOS will show the effects of
the self energy unless ground-state exchange is used.
A.3 Description of Routines
We now describe the new parts of the code at the subroutine level. Partial calling trees for
op abs ( figure ??) and opcons ( figure ??) are provided for reference. Below we give a
brief description of some of the new and existing routines that might be of interest to FEFF
developers.
A.3.1 op abs
ang fac: computes the angular part of the integrals for the dipole matrix elements M of
equation 3.55 such that the matrix element between two states is given by the product
of this factor and the radial matrix element rkk computed by dipole. This routine
is called by tracind during the spectrum construction phase of mat mult.
78
atomicxs: computes a ’tail’ for the optical spectrum given by equation 3.56 based on
occupation numbers for the various angular momentum channels computed from the
density matrix. This routine is basically a loop over the FEFF8.4 FPRIME calculation,
which is accomplished by calling ffmod2. The cross-section for the absorbing site is
computed as a sum of partial cross-sections obtained by using the atomic-like orbitals
from pot.bin as initial states weighted by the occupation numbers derived from the
density matrix during the main phase of op abs. The results are appended to xmu.dat
if it exists. There is a simple program addtail which is a wrapper for this routine
that can be used to add this tail to an existing xmu.dat.
changelittle and changebig: routines to change basis from complex to real YL’s. Uses
cmplx2realY.
chhole: rewrites the hole index in pot.bin. Used by atomic xs to loop over the ini-
tial state angular momenta (hole indices) when constructing the tail for the optical
spectrum.
chei: simple interactive program that changes the orbital energies of the initial state
orbitals by rewriting pot.bin.
chxmu: routine to change the Fermi level written in pot.bin.
dfovrg: solves for radial wave functions given potential. This routine is briefly described
by Ankudinov [8].
dipole: computes the radial matrix elements rkk from the radial wave functions. Along
with the angular factors computed by ang fac, rkk can be used to construct the dipole
matrix elements of equation 3.55.
egrid: sets the energy grid used for the op abs calculation. The parameters are currently
read from the LDOS card and passed to egrid.
79
eps2: this is the main routine of op abs. It executes three basic steps:
1. wave functions, phase shifts, and matrix elements (wave funct, dipole, etc.);
2. FMS matrix (fmsnbr);
3. spectrum construction (mat mult).
ffmod2: a copy of the second module (XSPH) of FEFF8.4 converted to a subroutine. This
inelegant hack is used in the calculation of the tails of the optical spectrum atomicxs.
fmsnbr: FMS routine modified to write out gg00.bin which is similar to fms.bin except
that one FULL column (all sites and all L’s) of the FMS matrix GFMSLL′ of equation
3.39 is written to disk instead of gtr. Saving more of the FMS matrix allows reuse of
the FMS inversion, which is typically the bottleneck of a FEFF calculation. Of course
saving the extra information costs disk space: large clusters will produce gg00.bin’s
occupying 100’s of MB.
getrkk: integral for the radial matrix elements rkk.
getxs: gets the FPRIME calculation for a specified edge. Used in the construction of the
tail for the valence response.
jdos: auxiliary routine to print out the joint dipole-allowed density of states of equation
4.13 for all sensible l,l′.
l2kap: effects the change of basis described by equation 3.45.
mat mult: evaluates equation 3.56 by doing the indicated matrix multiplication, summa-
tion over remote site n′, and energy integration. This routine is currently slow; it’s
performance can be easily improved.
occptn: returns the occupations of each angular momentum channel; the occupations form
the basis of the atomic model for the tail of the valence response.
80
op abs: main program. Reads input and calls eps2 which is responsible for the meat of
the calculation.
rd atomic state: reads in the radial wave function written in atomic state.dat as part
of the ’atomic’ test which uses the embedded-atomic orbital specified by the EDGE
or HOLE card instead of the Green’s function as the basis for the initial state density
matrix.
rdgg: reads the FMS matrix GFMSLL′ of equation 3.39.
rdrho: constructs the matrix ρKn,K ′n′ of equation 3.53.
traceind: forms the full matrix element MnK,K ′(E,E ′) defined by equation 3.55 for given
n, K, K ′, E, E′ by looping over the dipole allowed channels and summing the appro-
priate products of radial matrix elements and angular factors.
wave fnct: calculates the radial wave functions bκPκ, bκQκ of equations 3.22 and 3.43.
The normalization constants bκ (amp) and the phase shifts δκ (ph) needed to construct
ρKn,K ′n′ are also output, as is the central atom renormalization factor ρcκ(E) from
equation 3.48 which is passed in the variable rhoc.
wr atomic state: writes the radial wave function written in atomic state.dat as part
of the ’atomic’ test which uses the embedded-atomic orbital specified by the EDGE
or HOLE card instead of the Green’s function as the basis for the initial state density
matrix. The file atomic state.dat contains the upper and lower components dgc0,
dpc0 of the initial state orbital written in pot.bin as well as the energy of this orbital,
its hole index, and the quantum number κ associated with this state. This output is
produced automatically by op abs, but the atomic test is only run if the logical flag
atomic is set to .true. in the header of op abs.f.
wrg0: neutered copy of fmsnbr that skips the actual matrix inversion and writes out the
G0L,L′ matrix instead of the normal scattering matrix GFMS
LL′ . Output is g0 00.bin
81
in the same format as gg00.bin so it can be read by rdgg. Intended to assist with
testing.
A.3.2 opcons
The opcons package consists of three FORTRAN programs:
writedirs– reads in the master input file core.inp and one FEFF 8.4 format input file
for each inequivalent site in the calculation, produces a set of directories to hold all
the FEFF calculations needed for the spectrum, and populates these directories with
FEFF input files;
step2– for each edge, copies pot.bin from each FMS/XANES calculation to directories
for all other calculations for that edge;
step4– produces a composite spectrum for each edge at each site, sums these edges, com-
putes all optical constants and sumrules.
In addition there is a bash script optical runmods which is run after writedirs and step2
that actually runs FEFF on the relevant input files. This structure is summarized in figure
??. We now describe the important subroutines and scripts involved in this package.
addedg: combines several FEFF runs for a single edge into a single contribution suitable
for summing with the other edges. A smooth transition is made from XANES/FMS
to EXAFS/RPATH and then later to FPRIME as the photoelectron energy increases.
Output is in the form of scattering factor per atom (like FPRIME); two arrays are
produced: one containing the atomic background (column 5 of xmu.dat) and one
containing the full signal including fine structure (column 4 of xmu.dat. The integrated
oscillator strength neff of equation 4.9 is evaluated (at the highest frequency for which
an FPRIME calculation is available) and passed back. This output is eventually
written in osc str.dat.
82
drdtrm: computes ǫ− 1 for a Drude model. Input is a lifetime τ in seconds and a number
density n of electrons in atomic units which sets the plasma frequency ωp of the Drude
model through the relation
ω2p = 4πn.
The output is
ǫDrude(ω;ωp, γ) − 1 =
(
ω2p
ω2 + γ2
)
(
iγ
ω− 1
)
where γ = 1/(hτ) is the Drude width (in atomic units). The output is written to the
file drude.dat as well as passed back in the arguments.
egrid: constructs the energy grid for the full spectrum calculation (column 1 of opconsKK.dat).
The grid is regular in k for the lowest energy electron that contributes to absorption
at the grid point frequency. The size of the k-step is set by a parameter xkstep in
params.h. The highest and lowest frequencies (Hartree) for the grid are input and the
number of points in the grid is output. The routine requires the atomic numbers of
all constituent atoms in order to look up the edge energies so that the photoelectron
momenta can be estimated. Note that op abs contains a distinct routine with the
same name.
eps2opt: from ǫ(ω) − 1 computes N − 1 = (n − 1) + iκ, R, µ (105 cm−1), and −Imǫ−1.
Output is via arguments.
getedg: encoding of the table of x-ray edge frequencies by Elam, et. al. [13]. Given the
atomic number and hole index for a given initial state, returns the frequency of the
onset of transitions from that initial state listed on the table if it exists.
gtedgs: uses the getorb table to list the occupied initial states. Used to guess which edges
to run if the user does not specify them in core.inp.
hamaker: computes the integral in equation 4.8 for each ω in the input grid.
83
kk: Kramers-Kronig integral. Takes ǫ2(ω) as input and computes the integral of equation
4.1 numerically by employing equation 4.3.
opcons: takes an array containing ǫ − 1 and writes an opcons-format file of optical con-
stants.
qsum: evaluates the ǫ2 sum rule.
rdbkg: combines a collection of FPRIME calculations with cascading energy grids into a
single spectrum. Output is the atomic background of the contribution to the scattering
factor from a given initial state. The oscillator strength for the edge is also computed.
rddens: uses the cluster from a FEFF input file to estimate the number density of given
species of atoms in the system described by the input file.
rdop: parser that reads core.inp, the input file for opcons.
rdopcn: reads dielectric function from an opcons-format file. Useful for processing exper-
imental data or other calculations with opcons technology (e.g. the sum rule routines
or KK transforms).
rdst: combines XANES and EXAFS runs into a single spectrum on a grid passed as input.
Output is scattering factor per atom. Also reports the frequency range spanned by
the XANES and EXAFS calculations.
rdval: reads the output of op abs, converting the output from cross-section per atom to
ωǫ2(ω) for inclusion in the final output.
rdxmu: reads photoabsorption cross-section from a FEFF output file (xmu.dat). Output is
σ(ω) in A2.
rdxmunorm: reads normalized photoabsorption cross-section from the FEFF output file
xmu.dat.
84
rwfeff: rewrites a master FEFF input file including or excluding various cards based on
logical arguments.
rwinputs: main routine of writedirs. Prepares all needed FEFF input files for an opcons
calculation.
sumrules: reads an opcons-format file of optical constants and writes the partial sum
rules neff(ω) (c.f. equation 4.9) as a function of ω in a file.
A.3.3 Auxiliary codes
In addition to op abs and opcons there are a few small but useful programs that FEFF
developers may be interested in. They are described here.
A.4 Notes on Future Development
85
Appendix A
TABLES OF OPTICAL CONSTANTS
Here we present our calculated results in a tabular format inspired by [18].
86Table A.1: Optical constants of Pd
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.200000E+00 .944874E+01 .494334E+00 .223335E+01 .764369E-01 .154330E-01 .278556E+00 .451637E-02
.300000E+00 .935208E+01 .372458E+00 .221799E+01 .578709E-01 .175488E-01 .276645E+00 .347093E-02
.400000E+00 .936655E+01 .295474E+00 .222004E+01 .458816E-01 .185485E-01 .276836E+00 .274750E-02
.500000E+00 .943582E+01 .244008E+00 .223067E+01 .377661E-01 .190934E-01 .278063E+00 .223977E-02
.100000E+01 .103236E+02 .203089E+00 .236516E+01 .301777E-01 .305669E-01 .293607E+00 .158413E-02
.150000E+01 .128466E+02 .374214E+00 .272113E+01 .502280E-01 .765437E-01 .332247E+00 .194457E-02
.200000E+01 .251175E+02 .139784E+02 .427923E+01 .132391E+01 .268369E+01 .487223E+00 .159294E-01
.250000E+01 .328620E+01 .135957E+02 .204456E+01 .223261E+01 .565685E+01 .429402E+00 .669360E-01
.300000E+01 .100900E+01 .890597E+01 .135987E+01 .188683E+01 .573680E+01 .364286E+00 .106878E+00
.325000E+01 .759386E+00 .750418E+01 .117556E+01 .172441E+01 .567934E+01 .333535E+00 .126380E+00
.350000E+01 .721873E+00 .647699E+01 .105226E+01 .157788E+01 .559664E+01 .304657E+00 .144236E+00
.360000E+01 .738972E+00 .614788E+01 .101592E+01 .152470E+01 .556233E+01 .293912E+00 .150640E+00
.370000E+01 .765831E+00 .586632E+01 .986458E+00 .147645E+01 .553589E+01 .284070E+00 .156331E+00
.377000E+01 .785875E+00 .568714E+01 .968098E+00 .144480E+01 .552039E+01 .277563E+00 .160059E+00
.380000E+01 .796806E+00 .561523E+01 .961185E+00 .143147E+01 .551223E+01 .274812E+00 .161568E+00
.390000E+01 .833401E+00 .539521E+01 .940576E+00 .139003E+01 .549365E+01 .266258E+00 .166170E+00
.400000E+01 .867251E+00 .521294E+01 .924130E+00 .135455E+01 .549076E+01 .258900E+00 .170025E+00
Continued on next page
87
Table optical constants of Pd – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.410000E+01 .895615E+00 .504585E+01 .908644E+00 .132179E+01 .549201E+01 .252046E+00 .173676E+00
.420000E+01 .922287E+00 .490037E+01 .895553E+00 .129257E+01 .550183E+01 .245921E+00 .176854E+00
.430000E+01 .941988E+00 .477893E+01 .884201E+00 .126814E+01 .552657E+01 .240760E+00 .179597E+00
.450000E+01 .960197E+00 .456682E+01 .861440E+00 .122667E+01 .559450E+01 .231805E+00 .184907E+00
.475000E+01 .939983E+00 .435520E+01 .831352E+00 .118906E+01 .572447E+01 .223215E+00 .191597E+00
.500000E+01 .879432E+00 .417060E+01 .796373E+00 .116084E+01 .588271E+01 .216177E+00 .199304E+00
.550000E+01 .687196E+00 .377696E+01 .706394E+00 .110665E+01 .616854E+01 .201616E+00 .220766E+00
.600000E+01 .521412E+00 .331482E+01 .607558E+00 .103093E+01 .626871E+01 .182129E+00 .249231E+00
.650000E+01 .465822E+00 .282885E+01 .525102E+00 .927312E+00 .610777E+01 .156942E+00 .278693E+00
.700000E+01 .539452E+00 .237065E+01 .477597E+00 .802142E+00 .568938E+01 .128530E+00 .296553E+00
.750000E+01 .728190E+00 .200096E+01 .478665E+00 .676669E+00 .514204E+01 .104118E+00 .286012E+00
.800000E+01 .988194E+00 .175186E+01 .522888E+00 .575278E+00 .466336E+01 .903068E-01 .249480E+00
.900000E+01 .154827E+01 .165111E+01 .670978E+00 .494051E+00 .450672E+01 .941085E-01 .179156E+00
.100000E+02 .177458E+01 .202134E+01 .761758E+00 .573669E+00 .581544E+01 .114304E+00 .171495E+00
.110000E+02 .160272E+01 .215116E+01 .729065E+00 .622059E+00 .693537E+01 .117234E+00 .188674E+00
.120000E+02 .158773E+01 .216123E+01 .726162E+00 .626019E+00 .761424E+01 .117488E+00 .190126E+00
.130000E+02 .155886E+01 .230550E+01 .732508E+00 .665366E+00 .876715E+01 .123815E+00 .194338E+00
.140000E+02 .148753E+01 .244491E+01 .728506E+00 .707232E+00 .100357E+02 .129758E+00 .200965E+00
Continued on next page
88Table optical constants of Pd – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.145000E+02 .140654E+01 .254306E+01 .718696E+00 .739830E+00 .108734E+02 .134016E+00 .207442E+00
.150000E+02 .129424E+01 .261919E+01 .699445E+00 .770621E+00 .117167E+02 .137441E+00 .216030E+00
.160000E+02 .112628E+01 .267800E+01 .665207E+00 .804111E+00 .130409E+02 .140534E+00 .229017E+00
.170000E+02 .954951E+00 .295806E+01 .658410E+00 .891837E+00 .153666E+02 .156296E+00 .235290E+00
.180000E+02 .338806E+00 .308529E+01 .533237E+00 .100631E+01 .183603E+02 .174680E+00 .272798E+00
.190000E+02 -.184157E+00 .269591E+01 .347492E+00 .100018E+01 .192597E+02 .172228E+00 .340122E+00
.200000E+02 -.211687E+00 .223895E+01 .257381E+00 .890322E+00 .180474E+02 .145866E+00 .397369E+00
.210000E+02 -.126341E+00 .220606E+01 .274083E+00 .865772E+00 .184286E+02 .139294E+00 .391774E+00
.215000E+02 -.247978E+00 .222117E+01 .244375E+00 .892532E+00 .194500E+02 .146831E+00 .403918E+00
.220000E+02 -.399425E+00 .216190E+01 .192464E+00 .906510E+00 .202136E+02 .152654E+00 .429520E+00
.225000E+02 -.544231E+00 .198620E+01 .116348E+00 .889341E+00 .202795E+02 .152736E+00 .478968E+00
.230000E+02 -.571119E+00 .176181E+01 .587999E-01 .831836E+00 .193888E+02 .141050E+00 .536055E+00
.235000E+02 -.526037E+00 .159788E+01 .346160E-01 .772115E+00 .183876E+02 .126154E+00 .575217E+00
.240000E+02 -.444656E+00 .146514E+01 .301430E-01 .711177E+00 .172982E+02 .109526E+00 .596593E+00
.245000E+02 -.360369E+00 .141059E+01 .461168E-01 .674251E+00 .167409E+02 .984419E-01 .587760E+00
.250000E+02 -.289573E+00 .136942E+01 .614041E-01 .645099E+00 .163458E+02 .900067E-01 .575380E+00
.255000E+02 -.234104E+00 .135843E+01 .782843E-01 .629924E+00 .162798E+02 .854515E-01 .558546E+00
.260000E+02 -.191848E+00 .135581E+01 .923692E-01 .620588E+00 .163536E+02 .826531E-01 .544218E+00
Continued on next page
89
Table optical constants of Pd – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.265000E+02 -.165374E+00 .136334E+01 .102985E+00 .618026E+00 .165995E+02 .817081E-01 .533540E+00
.270000E+02 -.151989E+00 .137292E+01 .109438E+00 .618745E+00 .169326E+02 .817007E-01 .527234E+00
.275000E+02 -.150102E+00 .138453E+01 .112311E+00 .622365E+00 .173470E+02 .824779E-01 .524593E+00
.280000E+02 -.159124E+00 .139238E+01 .110737E+00 .626786E+00 .177879E+02 .835647E-01 .526258E+00
.285000E+02 -.175538E+00 .139689E+01 .106010E+00 .631504E+00 .182421E+02 .848243E-01 .530922E+00
.290000E+02 -.197899E+00 .139513E+01 .980369E-01 .635286E+00 .186730E+02 .859892E-01 .538714E+00
.300000E+02 -.250941E+00 .137422E+01 .756676E-01 .638773E+00 .194227E+02 .877328E-01 .561050E+00
.310000E+02 -.300988E+00 .132786E+01 .487192E-01 .633088E+00 .198916E+02 .876836E-01 .589681E+00
.320000E+02 -.331378E+00 .126483E+01 .245179E-01 .617266E+00 .200193E+02 .851920E-01 .617980E+00
.330000E+02 -.338484E+00 .120327E+01 .862307E-02 .596491E+00 .199506E+02 .810587E-01 .638182E+00
.340000E+02 -.331126E+00 .116292E+01 .260571E-02 .579946E+00 .199848E+02 .773787E-01 .646138E+00
.350000E+02 -.324934E+00 .114263E+01 .553067E-03 .570998E+00 .202554E+02 .753284E-01 .648735E+00
.360000E+02 -.330601E+00 .113787E+01 -.261065E-02 .570427E+00 .208135E+02 .754110E-01 .652881E+00
.370000E+02 -.353363E+00 .113612E+01 -.115953E-01 .574728E+00 .215533E+02 .771370E-01 .664824E+00
.380000E+02 -.390439E+00 .112807E+01 -.274331E-01 .579947E+00 .223367E+02 .797456E-01 .686148E+00
.390000E+02 -.436240E+00 .110885E+01 -.492951E-01 .583173E+00 .230518E+02 .826309E-01 .716622E+00
.400000E+02 -.481833E+00 .107809E+01 -.741783E-01 .582232E+00 .236046E+02 .851125E-01 .753545E+00
.420000E+02 -.562121E+00 .988541E+00 -.128512E+00 .567140E+00 .241420E+02 .884421E-01 .845820E+00
Continued on next page
90Table optical constants of Pd – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.440000E+02 -.615893E+00 .872021E+00 -.182407E+00 .533252E+00 .237798E+02 .885376E-01 .960604E+00
.460000E+02 -.621153E+00 .747853E+00 -.219872E+00 .479298E+00 .223453E+02 .818254E-01 .106405E+01
.470000E+02 -.592259E+00 .692911E+00 -.221609E+00 .445099E+00 .212020E+02 .735712E-01 .107175E+01
.480000E+02 -.589449E+00 .696209E+00 -.219366E+00 .445921E+00 .216939E+02 .732979E-01 .106580E+01
.490000E+02 -.601281E+00 .668668E+00 -.232788E+00 .435771E+00 .216417E+02 .736799E-01 .110328E+01
.500000E+02 -.614977E+00 .640311E+00 -.247617E+00 .425515E+00 .215636E+02 .745387E-01 .114710E+01
.510000E+02 -.635275E+00 .595911E+00 -.270826E+00 .408620E+00 .211219E+02 .761217E-01 .122079E+01
.520000E+02 -.628222E+00 .552604E+00 -.279653E+00 .383568E+00 .202157E+02 .725303E-01 .124576E+01
.530000E+02 -.633150E+00 .501095E+00 -.297191E+00 .356494E+00 .191501E+02 .711720E-01 .129925E+01
.540000E+02 -.619548E+00 .454521E+00 -.302436E+00 .325792E+00 .178310E+02 .661377E-01 .129367E+01
.560000E+02 -.581065E+00 .390490E+00 -.295855E+00 .277280E+00 .157380E+02 .551543E-01 .119055E+01
.580000E+02 -.548254E+00 .349707E+00 -.284796E+00 .244481E+00 .143720E+02 .469334E-01 .107150E+01
.600000E+02 -.533195E+00 .314751E+00 -.282431E+00 .219317E+00 .133372E+02 .426490E-01 .992973E+00
.620000E+02 -.519987E+00 .270340E+00 -.282045E+00 .188271E+00 .118309E+02 .385012E-01 .890750E+00
.640000E+02 -.497029E+00 .227557E+00 -.273699E+00 .156655E+00 .101617E+02 .330998E-01 .746669E+00
.660000E+02 -.472810E+00 .190757E+00 -.262494E+00 .129326E+00 .865111E+01 .282075E-01 .606892E+00
.680000E+02 -.446935E+00 .157977E+00 -.248917E+00 .105167E+00 .724816E+01 .237285E-01 .477513E+00
.700000E+02 -.417802E+00 .130708E+00 -.232247E+00 .851254E-01 .603933E+01 .195354E-01 .367143E+00
Continued on next page
91
Table optical constants of Pd – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.720000E+02 -.391574E+00 .109736E+00 -.216843E+00 .700613E-01 .511259E+01 .163077E-01 .287121E+00
.740000E+02 -.366275E+00 .914071E-01 -.201874E+00 .572637E-01 .429490E+01 .136050E-01 .222967E+00
.760000E+02 -.340697E+00 .766895E-01 -.186658E+00 .471452E-01 .363152E+01 .112646E-01 .174080E+00
.780000E+02 -.318497E+00 .676146E-01 -.173456E+00 .409020E-01 .323357E+01 .951487E-02 .144163E+00
.800000E+02 -.300011E+00 .591681E-01 -.162601E+00 .353290E-01 .286452E+01 .819848E-02 .119906E+00
.850000E+02 -.258803E+00 .406331E-01 -.138749E+00 .235899E-01 .203224E+01 .571707E-02 .737449E-01
.900000E+02 -.223261E+00 .262345E-01 -.118546E+00 .148814E-01 .135746E+01 .403232E-02 .434340E-01
.950000E+02 -.192672E+00 .185913E-01 -.101426E+00 .103449E-01 .996072E+00 .288356E-02 .285090E-01
.100000E+03 -.167173E+00 .144760E-01 -.873720E-01 .793097E-02 .803834E+00 .210400E-02 .208647E-01
.105000E+03 -.146952E+00 .120745E-01 -.763710E-01 .653645E-02 .695617E+00 .158777E-02 .165897E-01
.110000E+03 -.130244E+00 .109901E-01 -.673740E-01 .589205E-02 .656900E+00 .122462E-02 .145258E-01
.120000E+03 -.104913E+00 .989755E-02 -.538953E-01 .523069E-02 .636182E+00 .774191E-03 .123522E-01
.130000E+03 -.872624E-01 .932444E-02 -.446145E-01 .487994E-02 .642982E+00 .526816E-03 .111915E-01
.140000E+03 -.732381E-01 .813902E-02 -.373060E-01 .422721E-02 .599826E+00 .365929E-03 .947551E-02
.150000E+03 -.622445E-01 .815608E-02 -.316131E-01 .421117E-02 .640233E+00 .262513E-03 .927405E-02
.160000E+03 -.539856E-01 .814699E-02 -.273582E-01 .418808E-02 .679169E+00 .196855E-03 .910268E-02
.170000E+03 -.475386E-01 .784882E-02 -.240505E-01 .402112E-02 .692849E+00 .152290E-03 .865128E-02
.180000E+03 -.420650E-01 .733592E-02 -.212513E-01 .374760E-02 .683703E+00 .118931E-03 .799387E-02
Continued on next page
92Table optical constants of Pd – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.200000E+03 -.334853E-01 .660199E-02 -.168795E-01 .335767E-02 .680630E+00 .753138E-04 .706704E-02
.220000E+03 -.274706E-01 .596646E-02 -.138262E-01 .302506E-02 .674524E+00 .507795E-04 .630806E-02
.240000E+03 -.225190E-01 .507770E-02 -.113203E-01 .256792E-02 .624646E+00 .340709E-04 .531421E-02
.260000E+03 -.185846E-01 .453656E-02 -.933324E-02 .228965E-02 .603369E+00 .233054E-04 .470990E-02
.280000E+03 -.153558E-01 .397885E-02 -.770558E-02 .200488E-02 .568964E+00 .159721E-04 .410386E-02
.300000E+03 -.123857E-01 .345911E-02 -.621063E-02 .174037E-02 .529177E+00 .104654E-04 .354637E-02
.350000E+03 -.431419E-02 .335109E-02 -.215802E-02 .167917E-02 .595672E+00 .187323E-05 .338016E-02
.370000E+03 .848490E-03 .590061E-02 .428556E-03 .294904E-02 .110593E+01 .221959E-05 .589040E-02
.400000E+03 -.192799E-02 .139113E-01 -.940180E-03 .696218E-02 .282259E+01 .123505E-04 .139623E-01
93
Table A.2: Optical constants of Au
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.250000E-01 -.113083E+05 .405336E+05 .121419E+03 .162623E+03 .402212E+01 .988200E+00 .241031E-04
.500000E-01 -.913169E+04 .157850E+05 .658585E+02 .116706E+03 .584191E+01 .985373E+00 .483321E-04
.100000E+00 -.507057E+04 .434142E+04 .271361E+02 .765058E+02 .771625E+01 .983268E+00 .977342E-04
.200000E+00 -.182539E+04 .813551E+03 .824052E+01 .436077E+02 .878836E+01 .981647E+00 .204758E-03
.300000E+00 -.869632E+03 .274340E+03 .358948E+01 .298041E+02 .904479E+01 .980048E+00 .331489E-03
.400000E+00 -.502925E+03 .131682E+03 .190491E+01 .225570E+02 .911522E+01 .977831E+00 .492184E-03
.500000E+00 -.323472E+03 .765044E+02 .111099E+01 .180576E+02 .912650E+01 .974826E+00 .701837E-03
.100000E+01 -.780155E+02 .198576E+02 .122396E+00 .884103E+01 .894851E+01 .945537E+00 .316770E-02
.150000E+01 -.323073E+02 .118928E+02 .453478E-01 .568885E+01 .864007E+01 .885281E+00 .106939E-01
.200000E+01 -.162552E+02 .953605E+01 .170468E+00 .407580E+01 .825607E+01 .780062E+00 .296345E-01
.250000E+01 -.995664E+01 .871162E+01 .330175E+00 .327468E+01 .829549E+01 .670527E+00 .558670E-01
.300000E+01 -.693959E+01 .749463E+01 .345983E+00 .278376E+01 .846187E+01 .593681E+00 .820484E-01
.325000E+01 -.590453E+01 .669689E+01 .303191E+00 .256921E+01 .846080E+01 .562068E+00 .973165E-01
.350000E+01 -.480861E+01 .598377E+01 .281505E+00 .233468E+01 .828190E+01 .518947E+00 .118945E+00
.360000E+01 -.434516E+01 .578608E+01 .292434E+00 .223876E+01 .816494E+01 .496334E+00 .129767E+00
.370000E+01 -.391823E+01 .568477E+01 .318170E+00 .215726E+01 .808635E+01 .474195E+00 .139235E+00
.377000E+01 -.363518E+01 .565557E+01 .342595E+00 .210646E+01 .804795E+01 .458904E+00 .145278E+00
Continued on next page
94Table optical constants of Au – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.380000E+01 -.353305E+01 .564380E+01 .351699E+00 .208795E+01 .804031E+01 .453322E+00 .147467E+00
.390000E+01 -.322598E+01 .560589E+01 .379675E+00 .203194E+01 .803019E+01 .436402E+00 .154069E+00
.400000E+01 -.296296E+01 .557049E+01 .404447E+00 .198358E+01 .803927E+01 .421831E+00 .159642E+00
.410000E+01 -.272661E+01 .553668E+01 .427199E+00 .193988E+01 .805996E+01 .408707E+00 .164577E+00
.420000E+01 -.252183E+01 .550489E+01 .447446E+00 .190174E+01 .809438E+01 .397330E+00 .168732E+00
.430000E+01 -.234411E+01 .547428E+01 .465216E+00 .186829E+01 .814059E+01 .387469E+00 .172240E+00
.450000E+01 -.204211E+01 .541680E+01 .495741E+00 .181084E+01 .825810E+01 .370750E+00 .177996E+00
.475000E+01 -.177219E+01 .534819E+01 .521800E+00 .175724E+01 .845882E+01 .355678E+00 .183148E+00
.500000E+01 -.158124E+01 .524495E+01 .532289E+00 .171147E+01 .867329E+01 .343888E+00 .188347E+00
.550000E+01 -.130356E+01 .503456E+01 .539507E+00 .163513E+01 .911494E+01 .324980E+00 .197908E+00
.600000E+01 -.119078E+01 .484948E+01 .526835E+00 .158809E+01 .965751E+01 .314313E+00 .205890E+00
.650000E+01 -.114633E+01 .455764E+01 .485538E+00 .153400E+01 .101059E+02 .303466E+00 .219190E+00
.700000E+01 -.110509E+01 .430282E+01 .448962E+00 .148478E+01 .105341E+02 .293360E+00 .232274E+00
.750000E+01 -.111135E+01 .404775E+01 .403194E+00 .144233E+01 .109638E+02 .285511E+00 .246870E+00
.800000E+01 -.113586E+01 .376685E+01 .347852E+00 .139735E+01 .113301E+02 .277773E+00 .265135E+00
.900000E+01 -.107156E+01 .324588E+01 .259966E+00 .128805E+01 .117491E+02 .255177E+00 .307965E+00
.100000E+02 -.967210E+00 .277060E+01 .183968E+00 .117001E+01 .118579E+02 .228514E+00 .360922E+00
.110000E+02 -.863806E+00 .242419E+01 .132295E+00 .107044E+01 .119336E+02 .204364E+00 .411259E+00
Continued on next page
95
Table optical constants of Au – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.120000E+02 -.792150E+00 .210379E+01 .774607E-01 .976231E+00 .118727E+02 .182019E+00 .470805E+00
.130000E+02 -.691002E+00 .174358E+01 .197638E-01 .854831E+00 .112619E+02 .152001E+00 .556139E+00
.140000E+02 -.430236E+00 .139990E+01 .202278E-01 .686184E+00 .973487E+01 .103594E+00 .612231E+00
.145000E+02 -.235028E+00 .129800E+01 .657978E-01 .609107E+00 .895022E+01 .810285E-01 .571581E+00
.150000E+02 -.440417E-01 .127588E+01 .129202E+00 .565036E+00 .858957E+01 .692921E-01 .502076E+00
.160000E+02 .202827E+00 .134713E+01 .226535E+00 .549166E+00 .890561E+01 .671091E-01 .413061E+00
.170000E+02 .444902E+00 .144049E+01 .320047E+00 .545606E+00 .940109E+01 .704452E-01 .346104E+00
.180000E+02 .614598E+00 .175876E+01 .414584E+00 .621581E+00 .113415E+02 .898071E-01 .308495E+00
.190000E+02 .464448E+00 .231643E+01 .450500E+00 .798763E+00 .153876E+02 .126797E+00 .307540E+00
.200000E+02 -.247472E+00 .233668E+01 .266373E+00 .922591E+00 .187017E+02 .154007E+00 .387743E+00
.210000E+02 -.589389E+00 .199969E+01 .106963E+00 .903022E+00 .192190E+02 .157494E+00 .480521E+00
.215000E+02 -.662885E+00 .171870E+01 .219081E-01 .840830E+00 .183211E+02 .147549E+00 .560434E+00
.220000E+02 -.598197E+00 .154559E+01 -.260291E-03 .772963E+00 .172337E+02 .129999E+00 .605981E+00
.225000E+02 -.541990E+00 .145031E+01 -.525457E-02 .728974E+00 .166225E+02 .117832E+00 .626907E+00
.230000E+02 -.507507E+00 .137464E+01 -.118976E-01 .695589E+00 .162152E+02 .109095E+00 .644708E+00
.235000E+02 -.474040E+00 .129576E+01 -.190487E-01 .660439E+00 .157288E+02 .100129E+00 .662461E+00
.240000E+02 -.425263E+00 .121981E+01 -.193632E-01 .621967E+00 .151284E+02 .898623E-01 .670697E+00
.245000E+02 -.359298E+00 .116476E+01 -.746181E-02 .586806E+00 .145699E+02 .798528E-01 .658832E+00
Continued on next page
96Table optical constants of Au – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.250000E+02 -.289716E+00 .114669E+01 .146790E-01 .565073E+00 .143176E+02 .729978E-01 .630246E+00
.255000E+02 -.235074E+00 .115217E+01 .363030E-01 .555910E+00 .143676E+02 .696667E-01 .602472E+00
.260000E+02 -.201968E+00 .116928E+01 .520596E-01 .555704E+00 .146441E+02 .689257E-01 .583475E+00
.265000E+02 -.183527E+00 .118465E+01 .618886E-01 .557800E+00 .149820E+02 .690357E-01 .572302E+00
.270000E+02 -.166760E+00 .119738E+01 .705132E-01 .559252E+00 .153046E+02 .690774E-01 .562690E+00
.275000E+02 -.155286E+00 .122786E+01 .805250E-01 .568167E+00 .158369E+02 .707975E-01 .552800E+00
.280000E+02 -.165397E+00 .127788E+01 .864832E-01 .588081E+00 .166894E+02 .751871E-01 .548546E+00
.285000E+02 -.216658E+00 .131412E+01 .754831E-01 .610989E+00 .176504E+02 .809946E-01 .561361E+00
.290000E+02 -.282338E+00 .131467E+01 .524771E-01 .624576E+00 .183584E+02 .853684E-01 .586059E+00
.300000E+02 -.375622E+00 .126037E+01 .768260E-02 .625377E+00 .190154E+02 .884729E-01 .637158E+00
.310000E+02 -.479200E+00 .119629E+01 -.446796E-01 .626089E+00 .196712E+02 .935103E-01 .703092E+00
.320000E+02 -.515360E+00 .979030E+00 -.111961E+00 .551164E+00 .178743E+02 .817944E-01 .820146E+00
.330000E+02 -.372378E+00 .941946E+00 -.620621E-01 .502117E+00 .167950E+02 .638773E-01 .735312E+00
.340000E+02 -.393079E+00 .949083E+00 -.690142E-01 .509719E+00 .175652E+02 .663347E-01 .747835E+00
.350000E+02 -.391188E+00 .879779E+00 -.838157E-01 .480127E+00 .170304E+02 .608842E-01 .768408E+00
.360000E+02 -.335984E+00 .826475E+00 -.715035E-01 .445067E+00 .162391E+02 .518778E-01 .735286E+00
.370000E+02 -.269174E+00 .842939E+00 -.391688E-01 .438638E+00 .164499E+02 .480466E-01 .677365E+00
.380000E+02 -.246676E+00 .906927E+00 -.170610E-01 .461321E+00 .177687E+02 .514200E-01 .652393E+00
Continued on next page
97
Table optical constants of Au – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.390000E+02 -.284000E+00 .968242E+00 -.201421E-01 .494081E+00 .195306E+02 .587280E-01 .667655E+00
.400000E+02 -.352941E+00 .996223E+00 -.421370E-01 .520052E+00 .210849E+02 .663578E-01 .705934E+00
.420000E+02 -.536560E+00 .998221E+00 -.115695E+00 .564435E+00 .240278E+02 .858217E-01 .824134E+00
.440000E+02 -.727758E+00 .754421E+00 -.267156E+00 .514477E+00 .229401E+02 .102932E+00 .117400E+01
.460000E+02 -.631475E+00 .534428E+00 -.286614E+00 .374644E+00 .174650E+02 .723839E-01 .126748E+01
.470000E+02 -.588133E+00 .502535E+00 -.271411E+00 .344886E+00 .164279E+02 .620063E-01 .119003E+01
.480000E+02 -.561154E+00 .471971E+00 -.264011E+00 .320645E+00 .155986E+02 .553636E-01 .113614E+01
.490000E+02 -.539369E+00 .442656E+00 -.258546E+00 .298511E+00 .148245E+02 .499624E-01 .108447E+01
.500000E+02 -.519676E+00 .414514E+00 -.253407E+00 .277608E+00 .140678E+02 .451760E-01 .102966E+01
.510000E+02 -.500552E+00 .387481E+00 -.247803E+00 .257571E+00 .133135E+02 .407333E-01 .969617E+00
.520000E+02 -.480904E+00 .361487E+00 -.241159E+00 .238189E+00 .125530E+02 .364750E-01 .903351E+00
.530000E+02 -.459469E+00 .336477E+00 -.232778E+00 .219291E+00 .117792E+02 .322574E-01 .829971E+00
.540000E+02 -.434523E+00 .312900E+00 -.221611E+00 .201006E+00 .110004E+02 .279559E-01 .749191E+00
.560000E+02 -.361692E+00 .277856E+00 -.183172E+00 .170119E+00 .965469E+01 .187998E-01 .573908E+00
.580000E+02 -.372931E+00 .369037E+00 -.176996E+00 .224201E+00 .131798E+02 .241862E-01 .697073E+00
.600000E+02 -.353229E+00 .365218E+00 -.166470E+00 .219082E+00 .133231E+02 .222055E-01 .661973E+00
.620000E+02 -.421789E+00 .282243E+00 -.218452E+00 .180568E+00 .113467E+02 .250518E-01 .681761E+00
.640000E+02 -.393261E+00 .233966E+00 -.207212E+00 .147560E+00 .957168E+01 .199985E-01 .553286E+00
Continued on next page
98Table optical constants of Au – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.660000E+02 -.379960E+00 .202209E+00 -.202434E+00 .126767E+00 .847979E+01 .175690E-01 .475408E+00
.680000E+02 -.352835E+00 .169247E+00 -.188798E+00 .104319E+00 .718976E+01 .141366E-01 .378240E+00
.700000E+02 -.327930E+00 .148119E+00 -.175297E+00 .898018E-01 .637128E+01 .116232E-01 .312741E+00
.720000E+02 -.299810E+00 .125475E+00 -.159901E+00 .746801E-01 .544970E+01 .918419E-02 .247988E+00
.740000E+02 -.253578E+00 .122220E+00 -.133169E+00 .704914E-01 .528728E+01 .650882E-02 .213552E+00
.760000E+02 -.260470E+00 .122613E+00 -.137111E+00 .710481E-01 .547278E+01 .686179E-02 .218198E+00
.780000E+02 -.262328E+00 .124911E+00 -.138070E+00 .724598E-01 .572839E+01 .700288E-02 .223147E+00
.800000E+02 -.246228E+00 .107188E+00 -.129619E+00 .615752E-01 .499272E+01 .588009E-02 .184916E+00
.850000E+02 -.225602E+00 .908163E-01 -.118494E+00 .515121E-01 .443784E+01 .471235E-02 .149384E+00
.900000E+02 -.198607E+00 .661278E-01 -.104034E+00 .369031E-01 .336627E+01 .338843E-02 .102269E+00
.950000E+02 -.173410E+00 .530960E-01 -.903615E-01 .291852E-01 .281015E+01 .247207E-02 .773915E-01
.100000E+03 -.156737E+00 .438395E-01 -.813965E-01 .238621E-01 .241851E+01 .195429E-02 .614853E-01
.105000E+03 -.138134E+00 .357838E-01 -.714324E-01 .192683E-01 .205058E+01 .147157E-02 .480905E-01
.110000E+03 -.124078E+00 .300540E-01 -.639553E-01 .160537E-01 .178983E+01 .115993E-02 .391256E-01
.120000E+03 -.979260E-01 .215124E-01 -.501567E-01 .113242E-01 .137731E+01 .695405E-03 .264216E-01
.130000E+03 -.775224E-01 .173781E-01 -.395005E-01 .904639E-02 .119195E+01 .427238E-03 .204144E-01
.140000E+03 -.620430E-01 .151341E-01 -.314867E-01 .781305E-02 .110864E+01 .271600E-03 .171980E-01
.150000E+03 -.491664E-01 .145273E-01 -.248646E-01 .744885E-02 .113247E+01 .172703E-03 .160648E-01
Continued on next page
99
Table optical constants of Au – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.160000E+03 -.394559E-01 .151073E-01 -.198962E-01 .770700E-02 .124983E+01 .116111E-03 .163698E-01
.170000E+03 -.314923E-01 .161837E-01 -.158378E-01 .822205E-02 .141669E+01 .808855E-04 .172485E-01
.180000E+03 -.257630E-01 .180146E-01 -.129234E-01 .912522E-02 .166480E+01 .633874E-04 .189735E-01
.200000E+03 -.189189E-01 .209333E-01 -.944825E-02 .105665E-01 .214194E+01 .507070E-04 .217386E-01
.220000E+03 -.149416E-01 .227993E-01 -.743245E-02 .114850E-01 .256092E+01 .471348E-04 .234836E-01
.240000E+03 -.138463E-01 .239947E-01 -.687384E-02 .120804E-01 .293858E+01 .486284E-04 .246587E-01
.260000E+03 -.136524E-01 .236150E-01 -.677849E-02 .118881E-01 .313276E+01 .471358E-04 .242593E-01
.280000E+03 -.134534E-01 .226897E-01 -.668380E-02 .114212E-01 .324125E+01 .440721E-04 .233005E-01
.300000E+03 -.133508E-01 .216634E-01 -.663799E-02 .109041E-01 .331552E+01 .410111E-04 .222429E-01
.350000E+03 -.128582E-01 .182320E-01 -.640756E-02 .917479E-02 .325468E+01 .315094E-04 .187037E-01
.370000E+03 -.130349E-01 .168991E-01 -.650245E-02 .850487E-02 .318942E+01 .288404E-04 .173434E-01
.400000E+03 -.126133E-01 .145349E-01 -.629974E-02 .731353E-02 .296504E+01 .234408E-04 .149054E-01
.500000E+03 -.105139E-01 .941972E-02 -.525960E-02 .473476E-02 .239944E+01 .125864E-04 .962009E-02
.600000E+03 -.860681E-02 .647394E-02 -.430741E-02 .325097E-02 .197701E+01 .731210E-05 .658655E-02
.740000E+03 -.650914E-02 .385717E-02 -.325801E-02 .193489E-02 .145121E+01 .360132E-05 .390782E-02
.800000E+03 -.582057E-02 .321306E-02 -.291322E-02 .161123E-02 .130643E+01 .277882E-05 .325076E-02
.100000E+04 -.406096E-02 .173056E-02 -.203215E-02 .867045E-03 .878790E+00 .122283E-05 .174470E-02
.150000E+04 -.188633E-02 .510182E-03 -.943559E-03 .255332E-03 .388186E+00 .239101E-06 .512112E-03
Continued on next page
100Table optical constants of Au – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.200000E+04 -.950318E-03 .204846E-03 -.475280E-03 .102472E-03 .207720E+00 .591263E-07 .205236E-03
.250000E+04 -.649807E-03 .426877E-03 -.324961E-03 .213509E-03 .541001E+00 .378086E-07 .427433E-03
.300000E+04 -.554796E-03 .261801E-03 -.277445E-03 .130937E-03 .398132E+00 .235366E-07 .262091E-03
.350000E+04 -.443205E-03 .170866E-03 -.221623E-03 .854518E-04 .303132E+00 .141078E-07 .171017E-03
.360000E+04 -.425782E-03 .155514E-03 -.212891E-03 .777737E-04 .283778E+00 .128456E-07 .155647E-03
.400000E+04 -.358053E-03 .108805E-03 -.179042E-03 .544121E-04 .220597E+00 .875577E-08 .108883E-03
.450000E+04 -.289495E-03 .726425E-04 -.144743E-03 .363264E-04 .165683E+00 .556835E-08 .726845E-04
.500000E+04 -.237246E-03 .504360E-04 -.118624E-03 .252210E-04 .127813E+00 .367741E-08 .504599E-04
.600000E+04 -.166041E-03 .265403E-04 -.830476E-04 .132712E-04 .807062E-01 .176840E-08 .265491E-04
.800000E+04 -.927821E-04 .948932E-05 -.463905E-04 .474488E-05 .384732E-01 .543673E-09 .949108E-05
.100000E+05 -.581133E-04 .422328E-05 -.290714E-04 .211170E-05 .214030E-01 .212408E-09 .422377E-05
.200000E+05 -.155251E-04 .142246E-05 -.774860E-05 .711236E-06 .144174E-01 .151368E-10 .142250E-05
.250000E+05 -.999590E-05 .617354E-06 -.500679E-05 .308678E-06 .782151E-02 .629084E-11 .617366E-06
.260000E+05 -.924310E-05 .532175E-06 -.464916E-05 .266088E-06 .701202E-02 .542140E-11 .532184E-06
.300000E+05 -.693761E-05 .308812E-06 -.345707E-05 .154407E-06 .469496E-02 .299381E-11 .308817E-06
.400000E+05 -.388471E-05 .101535E-06 -.196695E-05 .507674E-07 .205820E-02 .967872E-12 .101536E-06
.500000E+05 -.247484E-05 .421443E-07 -.125170E-05 .210722E-07 .106788E-02 .391798E-12 .421444E-07
.750000E+05 -.108395E-05 .819321E-08 -.536442E-06 .409661E-08 .311407E-03 .719467E-13 .819322E-08
Continued on next page
101
Table optical constants of Au – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.100000E+06 -.660415E-06 .153942E-08 -.357628E-06 .769711E-09 .780063E-04 .319759E-13 .153942E-08
.200000E+06 -.153365E-06 .872813E-15 -.119209E-06 .436406E-15 .884635E-10 .355271E-14 .872813E-15
.500000E+06 .000000E+00 .000000E+00 .000000E+00 .000000E+00 .000000E+00 .000000E+00 .000000E+00
102Table A.3: Optical constants of Cu
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.250000E-01 -.659809E+04 .278710E+05 .102919E+03 .132187E+03 .327405E+01 .985329E+00 .356456E-04
.500000E-01 -.561386E+04 .114499E+05 .584444E+02 .956968E+02 .481116E+01 .981464E+00 .713176E-04
.100000E+00 -.352034E+04 .360180E+04 .262951E+02 .652701E+02 .656659E+01 .978528E+00 .142847E-03
.200000E+00 -.140711E+04 .713050E+03 .818435E+01 .385535E+02 .778102E+01 .976977E+00 .287384E-03
.300000E+00 -.694209E+03 .232753E+03 .335955E+01 .266855E+02 .811287E+01 .976465E+00 .435281E-03
.400000E+00 -.408518E+03 .104048E+03 .154787E+01 .203238E+02 .821802E+01 .976109E+00 .588747E-03
.500000E+00 -.264763E+03 .543202E+02 .658603E+00 .163081E+02 .824577E+01 .975745E+00 .749779E-03
.100000E+01 -.627297E+02 .694342E+01 -.559423E+00 .786422E+01 .796162E+01 .972438E+00 .180337E-02
.150000E+01 -.215404E+02 .634879E+01 -.307219E+00 .458356E+01 .696386E+01 .883738E+00 .138064E-01
.200000E+01 -.119844E+02 .596776E+01 -.128942E+00 .342563E+01 .693930E+01 .771056E+00 .383656E-01
.250000E+01 -.720463E+01 .485854E+01 -.845859E-01 .265375E+01 .672412E+01 .658137E+00 .782469E-01
.300000E+01 -.408791E+01 .418493E+01 .282331E-01 .203586E+01 .618898E+01 .502033E+00 .154695E+00
.325000E+01 -.303163E+01 .438097E+01 .182710E+00 .185215E+01 .610088E+01 .422706E+00 .187843E+00
.350000E+01 -.240938E+01 .455636E+01 .296225E+00 .175777E+01 .623482E+01 .380049E+00 .200210E+00
.360000E+01 -.224426E+01 .462209E+01 .330919E+00 .173665E+01 .633571E+01 .370012E+00 .201613E+00
.370000E+01 -.211631E+01 .468556E+01 .360217E+00 .172242E+01 .645893E+01 .362736E+00 .201919E+00
.377000E+01 -.205246E+01 .472836E+01 .376880E+00 .171710E+01 .656089E+01 .359473E+00 .201474E+00
Continued on next page
103
Table optical constants of Cu – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.380000E+01 -.202510E+01 .474670E+01 .384021E+00 .171483E+01 .660458E+01 .358075E+00 .201283E+00
.390000E+01 -.201869E+01 .480416E+01 .395026E+00 .172186E+01 .680656E+01 .358683E+00 .199205E+00
.400000E+01 -.202240E+01 .477394E+01 .389144E+00 .171820E+01 .696525E+01 .358394E+00 .200317E+00
.410000E+01 -.203118E+01 .466786E+01 .369161E+00 .170462E+01 .708347E+01 .357100E+00 .204272E+00
.420000E+01 -.199360E+01 .456579E+01 .356264E+00 .168312E+01 .716384E+01 .352988E+00 .209172E+00
.430000E+01 -.194508E+01 .446782E+01 .345661E+00 .166000E+01 .723335E+01 .348154E+00 .214299E+00
.450000E+01 -.182640E+01 .429947E+01 .332644E+00 .161309E+01 .735600E+01 .337254E+00 .224342E+00
.475000E+01 -.169553E+01 .413891E+01 .323149E+00 .156401E+01 .752890E+01 .325190E+00 .234998E+00
.500000E+01 -.160816E+01 .399123E+01 .309415E+00 .152405E+01 .772347E+01 .315886E+00 .244864E+00
.550000E+01 -.148180E+01 .368070E+01 .270885E+00 .144808E+01 .807226E+01 .299195E+00 .267114E+00
.600000E+01 -.137678E+01 .337537E+01 .228727E+00 .137351E+01 .835265E+01 .282890E+00 .292623E+00
.650000E+01 -.127489E+01 .310712E+01 .192554E+00 .130271E+01 .858215E+01 .266611E+00 .319350E+00
.700000E+01 -.122687E+01 .289854E+01 .157686E+00 .125185E+01 .888134E+01 .255834E+00 .342927E+00
.750000E+01 -.119954E+01 .265199E+01 .109034E+00 .119561E+01 .908838E+01 .245237E+00 .374973E+00
.800000E+01 -.115028E+01 .240784E+01 .635405E-01 .113198E+01 .917844E+01 .232043E+00 .413710E+00
.900000E+01 -.989189E+00 .199555E+01 .160298E-02 .996166E+00 .908643E+01 .198518E+00 .501130E+00
.100000E+02 -.826832E+00 .172261E+01 -.241668E-01 .882622E+00 .894511E+01 .166480E+00 .574725E+00
.110000E+02 -.695871E+00 .149897E+01 -.424681E-01 .782729E+00 .872589E+01 .138257E+00 .640691E+00
Continued on next page
104Table optical constants of Cu – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.120000E+02 -.555138E+00 .135973E+01 -.316090E-01 .702061E+00 .853838E+01 .113090E+00 .664275E+00
.130000E+02 -.471938E+00 .128696E+01 -.204177E-01 .656893E+00 .865497E+01 .992890E-01 .665037E+00
.140000E+02 -.444156E+00 .122424E+01 -.252294E-01 .627955E+00 .891015E+01 .919804E-01 .677231E+00
.145000E+02 -.443960E+00 .118921E+01 -.333515E-01 .615115E+00 .903961E+01 .893729E-01 .690050E+00
.150000E+02 -.449628E+00 .114856E+01 -.450188E-01 .601344E+00 .914202E+01 .869233E-01 .708091E+00
.160000E+02 -.461983E+00 .104522E+01 -.743707E-01 .564596E+00 .915570E+01 .805364E-01 .756346E+00
.170000E+02 -.451139E+00 .923362E+00 -.991543E-01 .512491E+00 .883006E+01 .703038E-01 .800222E+00
.180000E+02 -.407692E+00 .809698E+00 -.106804E+00 .453260E+00 .826840E+01 .572313E-01 .804343E+00
.190000E+02 -.342529E+00 .722864E+00 -.959286E-01 .399801E+00 .769822E+01 .446711E-01 .756931E+00
.200000E+02 -.269932E+00 .668169E+00 -.727083E-01 .360280E+00 .730318E+01 .351404E-01 .682188E+00
.210000E+02 -.204173E+00 .647973E+00 -.455174E-01 .339443E+00 .722462E+01 .298120E-01 .615290E+00
.215000E+02 -.177962E+00 .648162E+00 -.333400E-01 .335259E+00 .730569E+01 .285204E-01 .591481E+00
.220000E+02 -.159179E+00 .655347E+00 -.235626E-01 .335580E+00 .748279E+01 .281603E-01 .576672E+00
.225000E+02 -.149446E+00 .665798E+00 -.174765E-01 .338819E+00 .772691E+01 .284549E-01 .570649E+00
.230000E+02 -.149330E+00 .677161E+00 -.156341E-01 .343957E+00 .801822E+01 .292286E-01 .572804E+00
.235000E+02 -.160032E+00 .684480E+00 -.193090E-01 .348981E+00 .831236E+01 .302025E-01 .583010E+00
.240000E+02 -.178379E+00 .685589E+00 -.274526E-01 .352470E+00 .857387E+01 .311313E-01 .598746E+00
.245000E+02 -.199976E+00 .674232E+00 -.392078E-01 .350870E+00 .871263E+01 .314178E-01 .615986E+00
Continued on next page
105
Table optical constants of Cu – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.250000E+02 -.220214E+00 .654618E+00 -.518717E-01 .345206E+00 .874663E+01 .311348E-01 .631518E+00
.255000E+02 -.235259E+00 .626407E+00 -.637114E-01 .334505E+00 .864492E+01 .300349E-01 .640989E+00
.260000E+02 -.243506E+00 .594630E+00 -.729802E-01 .320711E+00 .845078E+01 .283521E-01 .642153E+00
.265000E+02 -.244474E+00 .561412E+00 -.789036E-01 .304744E+00 .818432E+01 .261970E-01 .633526E+00
.270000E+02 -.239090E+00 .529674E+00 -.812910E-01 .288271E+00 .788823E+01 .238337E-01 .616161E+00
.275000E+02 -.228863E+00 .501759E+00 -.804358E-01 .272828E+00 .760351E+01 .215290E-01 .592705E+00
.280000E+02 -.215928E+00 .476934E+00 -.775556E-01 .258517E+00 .733646E+01 .193609E-01 .566270E+00
.285000E+02 -.201107E+00 .456576E+00 -.728854E-01 .246242E+00 .711226E+01 .174782E-01 .539229E+00
.290000E+02 -.185815E+00 .439082E+00 -.674673E-01 .235430E+00 .691952E+01 .158285E-01 .513133E+00
.300000E+02 -.155187E+00 .412565E+00 -.552803E-01 .218360E+00 .663902E+01 .132525E-01 .466772E+00
.310000E+02 -.126394E+00 .393714E+00 -.429645E-01 .205697E+00 .646273E+01 .114055E-01 .428811E+00
.320000E+02 -.993115E-01 .380313E+00 -.308825E-01 .196218E+00 .636389E+01 .100767E-01 .397881E+00
.330000E+02 -.706981E-01 .370213E+00 -.177522E-01 .188456E+00 .630301E+01 .904048E-02 .370012E+00
.340000E+02 -.376156E-01 .376526E+00 -.104970E-02 .188461E+00 .649446E+01 .881071E-02 .352568E+00
.350000E+02 -.229817E-01 .394286E+00 .762019E-02 .195650E+00 .694082E+01 .942287E-02 .355193E+00
.360000E+02 -.197337E-01 .411094E+00 .107533E-01 .203361E+00 .742022E+01 .101535E-01 .363825E+00
.370000E+02 -.293399E-01 .426692E+00 .771215E-02 .211716E+00 .793994E+01 .110139E-01 .379549E+00
.380000E+02 -.473735E-01 .423475E+00 -.121718E-02 .211995E+00 .816483E+01 .111249E-01 .389644E+00
Continued on next page
106Table optical constants of Cu – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.390000E+02 -.555703E-01 .414537E+00 -.606068E-02 .208532E+00 .824288E+01 .108285E-01 .389679E+00
.400000E+02 -.604584E-01 .406057E+00 -.927422E-02 .204929E+00 .830802E+01 .105075E-01 .387597E+00
.420000E+02 -.668483E-01 .390293E+00 -.139381E-01 .197905E+00 .842452E+01 .988068E-02 .381479E+00
.440000E+02 -.717797E-01 .375972E+00 -.177339E-01 .191379E+00 .853462E+01 .931444E-02 .374865E+00
.460000E+02 -.770607E-01 .361317E+00 -.217147E-01 .184668E+00 .860960E+01 .875812E-02 .367799E+00
.470000E+02 -.788968E-01 .353858E+00 -.233119E-01 .181152E+00 .862930E+01 .846677E-02 .363433E+00
.480000E+02 -.804565E-01 .346710E+00 -.247361E-01 .177752E+00 .864753E+01 .818863E-02 .358997E+00
.490000E+02 -.818656E-01 .339855E+00 -.260517E-01 .174473E+00 .866485E+01 .792473E-02 .354578E+00
.500000E+02 -.832047E-01 .333274E+00 -.273007E-01 .171313E+00 .868164E+01 .767531E-02 .350229E+00
.510000E+02 -.845510E-01 .326952E+00 -.285238E-01 .168276E+00 .869823E+01 .744070E-02 .346000E+00
.520000E+02 -.860497E-01 .320874E+00 -.297960E-01 .165364E+00 .871531E+01 .722254E-02 .341985E+00
.530000E+02 -.878583E-01 .314534E+00 -.312402E-01 .162338E+00 .872035E+01 .700344E-02 .337867E+00
.540000E+02 -.891893E-01 .307932E+00 -.324599E-01 .159131E+00 .870934E+01 .676931E-02 .333114E+00
.560000E+02 -.910785E-01 .295437E+00 -.344294E-01 .152986E+00 .868303E+01 .632655E-02 .323437E+00
.580000E+02 -.925702E-01 .283801E+00 -.361003E-01 .147215E+00 .865394E+01 .592383E-02 .313947E+00
.600000E+02 -.942257E-01 .272920E+00 -.377702E-01 .141817E+00 .862420E+01 .556487E-02 .304968E+00
.620000E+02 -.956076E-01 .261335E+00 -.393266E-01 .136017E+00 .854707E+01 .519000E-02 .294885E+00
.640000E+02 -.959140E-01 .250345E+00 -.402621E-01 .130424E+00 .846003E+01 .482992E-02 .284468E+00
Continued on next page
107
Table optical constants of Cu – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.660000E+02 -.956289E-01 .240023E+00 -.408200E-01 .125119E+00 .836954E+01 .449435E-02 .274155E+00
.680000E+02 -.949154E-01 .230241E+00 -.410955E-01 .120054E+00 .827412E+01 .418051E-02 .263979E+00
.700000E+02 -.930237E-01 .220507E+00 -.407362E-01 .114936E+00 .815426E+01 .386047E-02 .253099E+00
.720000E+02 -.896477E-01 .212933E+00 -.394596E-01 .110840E+00 .808852E+01 .358992E-02 .243607E+00
.740000E+02 -.841760E-01 .207378E+00 -.369752E-01 .107670E+00 .807547E+01 .335316E-02 .235193E+00
.760000E+02 -.827319E-01 .209014E+00 -.361410E-01 .108426E+00 .835196E+01 .337657E-02 .236156E+00
.780000E+02 -.841942E-01 .209306E+00 -.368735E-01 .108660E+00 .859027E+01 .340602E-02 .237172E+00
.800000E+02 -.944760E-01 .208453E+00 -.422079E-01 .108820E+00 .882348E+01 .354331E-02 .241425E+00
.850000E+02 -.110893E+00 .185480E+00 -.520137E-01 .978288E-01 .842806E+01 .322693E-02 .224848E+00
.900000E+02 -.110027E+00 .155905E+00 -.530318E-01 .823181E-01 .750894E+01 .252504E-02 .190977E+00
.950000E+02 -.103305E+00 .139587E+00 -.502135E-01 .734835E-01 .707549E+01 .208066E-02 .169496E+00
.100000E+03 -.988607E-01 .124345E+00 -.484704E-01 .653395E-01 .662243E+01 .173593E-02 .150263E+00
.105000E+03 -.933410E-01 .113411E+00 -.459602E-01 .594372E-01 .632543E+01 .147710E-02 .135839E+00
.110000E+03 -.895241E-01 .103885E+00 -.442649E-01 .543480E-01 .605924E+01 .128351E-02 .123708E+00
.120000E+03 -.813276E-01 .864800E-01 -.404673E-01 .450636E-01 .548087E+01 .954854E-03 .101569E+00
.130000E+03 -.739371E-01 .755198E-01 -.368801E-01 .392058E-01 .516581E+01 .751479E-03 .874784E-01
.140000E+03 -.697299E-01 .651312E-01 -.349048E-01 .337434E-01 .478806E+01 .610182E-03 .748941E-01
.150000E+03 -.647143E-01 .557021E-01 -.324700E-01 .287857E-01 .437635E+01 .486292E-03 .634520E-01
Continued on next page
108Table optical constants of Cu – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.160000E+03 -.596461E-01 .486075E-01 -.299580E-01 .250544E-01 .406299E+01 .392921E-03 .548229E-01
.170000E+03 -.558285E-01 .425843E-01 -.280682E-01 .219071E-01 .377463E+01 .325983E-03 .476722E-01
.180000E+03 -.520383E-01 .368066E-01 -.261834E-01 .188982E-01 .344773E+01 .267617E-03 .408970E-01
.200000E+03 -.448319E-01 .282302E-01 -.225663E-01 .144410E-01 .292730E+01 .183556E-03 .309154E-01
.220000E+03 -.387016E-01 .223530E-01 -.194755E-01 .113985E-01 .254162E+01 .129817E-03 .241760E-01
.240000E+03 -.341704E-01 .180652E-01 -.171907E-01 .919056E-02 .223561E+01 .966490E-04 .193593E-01
.260000E+03 -.300010E-01 .140002E-01 -.150891E-01 .710734E-02 .187293E+01 .706096E-04 .148765E-01
.280000E+03 -.264012E-01 .117475E-01 -.132710E-01 .595276E-02 .168934E+01 .535971E-04 .123914E-01
.300000E+03 -.233834E-01 .946364E-02 -.117493E-01 .478808E-02 .145587E+01 .407199E-04 .992133E-02
.350000E+03 -.177608E-01 .609183E-02 -.891545E-02 .307332E-02 .109023E+01 .224322E-04 .631388E-02
.370000E+03 -.161291E-01 .517229E-02 -.809395E-02 .260726E-02 .977741E+00 .182249E-04 .534313E-02
.400000E+03 -.138797E-01 .406995E-02 -.696198E-02 .204925E-02 .830794E+00 .132594E-04 .418526E-02
.500000E+03 -.885248E-02 .208503E-02 -.443553E-02 .104716E-02 .530672E+00 .521577E-05 .212244E-02
.600000E+03 -.605446E-02 .115365E-02 -.303166E-02 .578579E-03 .351848E+00 .238870E-05 .116775E-02
.740000E+03 -.367120E-02 .612971E-03 -.183722E-02 .307049E-03 .230294E+00 .869018E-06 .617497E-03
.800000E+03 -.297270E-02 .497891E-03 -.148740E-02 .249316E-03 .202154E+00 .569482E-06 .500864E-03
.100000E+04 -.165671E-02 .178474E-02 -.828326E-03 .893111E-03 .905209E+00 .371251E-06 .179066E-02
.150000E+04 -.135006E-02 .510200E-03 -.675233E-03 .255272E-03 .388094E+00 .130364E-06 .511580E-03
Continued on next page
109
Table optical constants of Cu – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.200000E+04 -.812510E-03 .179897E-03 -.406324E-03 .899848E-04 .182407E+00 .433168E-07 .180189E-03
.250000E+04 -.523838E-03 .800599E-04 -.261981E-03 .400404E-04 .101457E+00 .175640E-07 .801438E-04
.300000E+04 -.362718E-03 .411558E-04 -.181369E-03 .205816E-04 .625811E-01 .833112E-08 .411856E-04
.350000E+04 -.265102E-03 .232990E-04 -.132552E-03 .116510E-04 .413311E-01 .442704E-08 .233114E-04
.360000E+04 -.250292E-03 .209865E-04 -.125169E-03 .104945E-04 .382922E-01 .394482E-08 .209970E-04
.400000E+04 -.201796E-03 .141830E-04 -.100909E-03 .709220E-05 .287531E-01 .255848E-08 .141887E-04
.450000E+04 -.158523E-03 .913254E-05 -.792685E-04 .456663E-05 .208282E-01 .157621E-08 .913544E-05
.500000E+04 -.127674E-03 .614865E-05 -.638669E-04 .307452E-05 .155808E-01 .102217E-08 .615022E-05
.600000E+04 -.876441E-04 .308879E-05 -.438482E-04 .154446E-05 .939229E-02 .481284E-09 .308933E-05
.800000E+04 -.475577E-04 .103174E-05 -.238136E-04 .515881E-06 .418295E-02 .141842E-09 .103184E-05
.100000E+05 -.311708E-04 .363861E-05 -.156164E-04 .181934E-05 .184398E-01 .617965E-10 .363884E-05
.200000E+05 -.824286E-05 .268415E-06 -.411272E-05 .134208E-06 .272052E-02 .423314E-11 .268420E-06
.250000E+05 -.526719E-05 .111663E-06 -.262260E-05 .558314E-07 .141469E-02 .172030E-11 .111664E-06
.260000E+05 -.486784E-05 .955241E-07 -.244379E-05 .477622E-07 .125864E-02 .149360E-11 .955250E-07
.300000E+05 -.365048E-05 .538040E-07 -.184774E-05 .269021E-07 .817995E-03 .853722E-12 .538044E-07
.400000E+05 -.204748E-05 .165997E-07 -.101328E-05 .829985E-08 .336491E-03 .256701E-12 .165998E-07
.500000E+05 -.132210E-05 .118996E-07 -.655651E-06 .594979E-08 .301519E-03 .107479E-12 .118996E-07
.750000E+05 .629009E+02 .106933E-03 .629009E+02 .629009E+02 .629009E+02 .629009E+02 .629009E+02
Continued on next page
110Table optical constants of Cu – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.100000E+06 .838679E+02 .142577E-03 .838679E+02 .838679E+02 .838679E+02 .838679E+02 .838679E+02
.200000E+06 .167736E+03 .285153E-03 .167736E+03 .167736E+03 .167736E+03 .167736E+03 .167736E+03
.500000E+06 .419339E+03 .712883E-03 .419339E+03 .419339E+03 .419339E+03 .419339E+03 .419339E+03
111
Table A.4: Optical constants of Diamond
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.200000E+00 .478553E+01 .248929E+00 .140587E+01 .517292E-01 .104576E-01 .170577E+00 .742080E-02
.300000E+00 .467810E+01 .200927E+00 .138325E+01 .421540E-01 .128166E-01 .167290E+00 .622427E-02
.400000E+00 .463779E+01 .177452E+00 .137470E+01 .373627E-01 .151317E-01 .166040E+00 .557717E-02
.500000E+00 .461200E+01 .154883E+00 .136919E+01 .326867E-01 .165374E-01 .165228E+00 .491383E-02
.100000E+01 .461358E+01 .997396E-01 .136939E+01 .210475E-01 .213240E-01 .165211E+00 .316414E-02
.150000E+01 .468201E+01 .979608E-01 .138378E+01 .205472E-01 .312469E-01 .167267E+00 .303325E-02
.200000E+01 .477883E+01 .156426E+00 .140414E+01 .325310E-01 .660636E-01 .170216E+00 .467982E-02
.250000E+01 .481669E+01 .217100E+00 .141220E+01 .450004E-01 .114024E+00 .171431E+00 .640773E-02
.300000E+01 .491597E+01 .204429E+00 .143264E+01 .420179E-01 .127762E+00 .174311E+00 .583410E-02
.325000E+01 .499739E+01 .209437E+00 .144933E+01 .427538E-01 .140834E+00 .176675E+00 .581562E-02
.350000E+01 .509177E+01 .224774E+00 .146857E+01 .455261E-01 .161532E+00 .179403E+00 .604835E-02
.360000E+01 .513245E+01 .233238E+00 .147682E+01 .470816E-01 .171866E+00 .180573E+00 .619196E-02
.370000E+01 .517422E+01 .243390E+00 .148527E+01 .489632E-01 .183715E+00 .181771E+00 .637329E-02
.377000E+01 .520411E+01 .252022E+00 .149131E+01 .505774E-01 .193341E+00 .182628E+00 .653555E-02
.380000E+01 .521692E+01 .255722E+00 .149390E+01 .512692E-01 .197467E+00 .182995E+00 .660509E-02
.390000E+01 .526089E+01 .267950E+00 .150274E+01 .535276E-01 .211690E+00 .184247E+00 .682161E-02
.400000E+01 .530570E+01 .281189E+00 .151173E+01 .559703E-01 .227047E+00 .185520E+00 .705574E-02
Continued on next page
112Table optical constants of Diamond – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.410000E+01 .535133E+01 .295225E+00 .152086E+01 .585556E-01 .243349E+00 .186813E+00 .730247E-02
.420000E+01 .539871E+01 .307902E+00 .153029E+01 .608393E-01 .259096E+00 .188141E+00 .750120E-02
.430000E+01 .544842E+01 .319445E+00 .154014E+01 .628753E-01 .274127E+00 .189524E+00 .766204E-02
.450000E+01 .555833E+01 .339343E+00 .156177E+01 .662283E-01 .302151E+00 .192544E+00 .786736E-02
.475000E+01 .572690E+01 .360608E+00 .159454E+01 .694895E-01 .334620E+00 .197080E+00 .794523E-02
.500000E+01 .594700E+01 .389180E+00 .163672E+01 .737939E-01 .374058E+00 .202874E+00 .803767E-02
.550000E+01 .657414E+01 .612177E+00 .175430E+01 .111048E+00 .619904E+00 .219030E+00 .105736E-01
.600000E+01 .723922E+01 .131897E+01 .187960E+01 .228822E+00 .139429E+01 .237409E+00 .188719E-01
.650000E+01 .744510E+01 .242215E+01 .193535E+01 .412589E+00 .272069E+01 .250138E+00 .313565E-01
.700000E+01 .707038E+01 .332962E+01 .189838E+01 .574444E+00 .407673E+01 .253367E+00 .436884E-01
.750000E+01 .656209E+01 .362108E+01 .182369E+01 .641196E+00 .487411E+01 .248606E+00 .515110E-01
.800000E+01 .671489E+01 .356822E+01 .184735E+01 .626578E+00 .508060E+01 .250436E+00 .493865E-01
.900000E+01 .758994E+01 .536584E+01 .205929E+01 .876943E+00 .800024E+01 .290476E+00 .522963E-01
.100000E+02 .701146E+01 .830323E+01 .212661E+01 .132784E+01 .134602E+02 .334508E+00 .623390E-01
.110000E+02 .373972E+01 .113088E+02 .191576E+01 .193949E+01 .216250E+02 .389258E+00 .751841E-01
.120000E+02 -.890656E+00 .106471E+02 .131906E+01 .229585E+01 .279245E+02 .430588E+00 .938689E-01
.130000E+02 -.326051E+01 .793325E+01 .730119E+00 .229242E+01 .302043E+02 .455548E+00 .116587E+00
.140000E+02 -.366920E+01 .560074E+01 .329245E+00 .210630E+01 .298857E+02 .460910E+00 .145546E+00
Continued on next page
113
Table optical constants of Diamond – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.145000E+02 -.354901E+01 .482523E+01 .205739E+00 .200072E+01 .294020E+02 .456174E+00 .162060E+00
.150000E+02 -.346194E+01 .424498E+01 .105615E+00 .191950E+01 .291812E+02 .455275E+00 .176301E+00
.160000E+02 -.325049E+01 .321958E+01 -.842464E-01 .175756E+01 .284998E+02 .458099E+00 .208729E+00
.170000E+02 -.295268E+01 .249934E+01 -.219345E+00 .160064E+01 .275785E+02 .455326E+00 .248494E+00
.180000E+02 -.265611E+01 .195365E+01 -.327306E+00 .145211E+01 .264921E+02 .451593E+00 .297837E+00
.190000E+02 -.238846E+01 .156900E+01 -.405582E+00 .131977E+01 .254151E+02 .444984E+00 .357443E+00
.200000E+02 -.210324E+01 .120729E+01 -.484143E+00 .117016E+01 .237197E+02 .437309E+00 .451492E+00
.210000E+02 -.183018E+01 .103299E+01 -.502474E+00 .103812E+01 .220958E+02 .400622E+00 .588198E+00
.215000E+02 -.172401E+01 .953783E+00 -.513458E+00 .980145E+00 .213582E+02 .386162E+00 .665317E+00
.220000E+02 -.161505E+01 .879847E+00 -.521204E+00 .918824E+00 .204875E+02 .368149E+00 .763650E+00
.225000E+02 -.151515E+01 .838453E+00 -.515774E+00 .865776E+00 .197432E+02 .343977E+00 .866013E+00
.230000E+02 -.143568E+01 .800466E+00 -.512299E+00 .820660E+00 .191303E+02 .324219E+00 .963940E+00
.235000E+02 -.136606E+01 .764100E+00 -.509474E+00 .778865E+00 .185507E+02 .306262E+00 .106461E+01
.240000E+02 -.130106E+01 .730055E+00 -.505693E+00 .738479E+00 .179630E+02 .288337E+00 .117081E+01
.245000E+02 -.124304E+01 .704928E+00 -.498691E+00 .703094E+00 .174588E+02 .270366E+00 .126790E+01
.250000E+02 -.119511E+01 .681600E+00 -.493100E+00 .672331E+00 .170355E+02 .255323E+00 .135608E+01
.255000E+02 -.115446E+01 .659183E+00 -.488826E+00 .644778E+00 .166642E+02 .242535E+00 .143811E+01
.260000E+02 -.112235E+01 .636672E+00 -.487173E+00 .620746E+00 .163576E+02 .232863E+00 .151481E+01
Continued on next page
114Table optical constants of Diamond – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.265000E+02 -.109359E+01 .606524E+00 -.490043E+00 .594681E+00 .159724E+02 .225466E+00 .161042E+01
.270000E+02 -.106354E+01 .576604E+00 -.491778E+00 .567272E+00 .155233E+02 .217080E+00 .171360E+01
.275000E+02 -.103475E+01 .547756E+00 -.492984E+00 .540172E+00 .150552E+02 .208684E+00 .181841E+01
.280000E+02 -.100892E+01 .519344E+00 -.494764E+00 .513957E+00 .145851E+02 .201177E+00 .192504E+01
.285000E+02 -.983393E+00 .486810E+00 -.498143E+00 .485003E+00 .140094E+02 .194068E+00 .205185E+01
.290000E+02 -.955478E+00 .454839E+00 -.499223E+00 .454133E+00 .133480E+02 .185258E+00 .217766E+01
.300000E+02 -.895386E+00 .394260E+00 -.493764E+00 .389410E+00 .118402E+02 .163386E+00 .236916E+01
.310000E+02 -.834476E+00 .344927E+00 -.476457E+00 .329445E+00 .103504E+02 .138121E+00 .235513E+01
.320000E+02 -.772928E+00 .299665E+00 -.450843E+00 .272899E+00 .885010E+01 .112285E+00 .211862E+01
.330000E+02 -.713486E+00 .267556E+00 -.417536E+00 .229676E+00 .768197E+01 .888119E-01 .174103E+01
.340000E+02 -.658586E+00 .238677E+00 -.384370E+00 .193886E+00 .668074E+01 .700246E-01 .137592E+01
.350000E+02 -.606839E+00 .217242E+00 -.351021E+00 .167391E+00 .593764E+01 .550676E-01 .107705E+01
.360000E+02 -.557723E+00 .197526E+00 -.319318E+00 .145096E+00 .529417E+01 .432299E-01 .841921E+00
.370000E+02 -.511979E+00 .186564E+00 -.289200E+00 .131243E+00 .492158E+01 .342668E-01 .683651E+00
.380000E+02 -.470827E+00 .176412E+00 -.262790E+00 .119655E+00 .460828E+01 .275049E-01 .567163E+00
.390000E+02 -.432212E+00 .169414E+00 -.238326E+00 .111217E+00 .439604E+01 .222051E-01 .482677E+00
.400000E+02 -.394056E+00 .163343E+00 -.214665E+00 .103997E+00 .421619E+01 .177947E-01 .414801E+00
.420000E+02 -.323937E+00 .168147E+00 -.171537E+00 .101481E+00 .431995E+01 .118490E-01 .346491E+00
Continued on next page
115
Table optical constants of Diamond – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.440000E+02 -.276684E+00 .198229E+00 -.141716E+00 .115478E+00 .515003E+01 .964147E-02 .352397E+00
.460000E+02 -.264981E+00 .227553E+00 -.132689E+00 .131183E+00 .611636E+01 .993607E-02 .384360E+00
.470000E+02 -.268412E+00 .230308E+00 -.134387E+00 .133032E+00 .633720E+01 .102217E-01 .391506E+00
.480000E+02 -.273688E+00 .230215E+00 -.137378E+00 .133438E+00 .649167E+01 .105186E-01 .396558E+00
.490000E+02 -.277211E+00 .218411E+00 -.140390E+00 .127041E+00 .630924E+01 .103184E-01 .383089E+00
.500000E+02 -.273130E+00 .205653E+00 -.139106E+00 .119442E+00 .605270E+01 .966864E-02 .360396E+00
.510000E+02 -.265120E+00 .193508E+00 -.135473E+00 .111917E+00 .578481E+01 .885112E-02 .335091E+00
.520000E+02 -.251964E+00 .182121E+00 -.128816E+00 .104526E+00 .550892E+01 .783566E-02 .307267E+00
.530000E+02 -.237457E+00 .179300E+00 -.120830E+00 .101972E+00 .547762E+01 .705923E-02 .292214E+00
.540000E+02 -.223704E+00 .177640E+00 -.113250E+00 .100164E+00 .548209E+01 .640422E-02 .280116E+00
.560000E+02 -.204181E+00 .187746E+00 -.101812E+00 .104513E+00 .593211E+01 .589072E-02 .280811E+00
.580000E+02 -.201405E+00 .202502E+00 -.993157E-01 .112416E+00 .660859E+01 .620694E-02 .298343E+00
.600000E+02 -.211832E+00 .207919E+00 -.104652E+00 .116111E+00 .706108E+01 .677651E-02 .312929E+00
.620000E+02 -.225931E+00 .200823E+00 -.112936E+00 .113195E+00 .711306E+01 .715440E-02 .314022E+00
.640000E+02 -.236395E+00 .183341E+00 -.119968E+00 .104167E+00 .675681E+01 .712023E-02 .297284E+00
.660000E+02 -.238773E+00 .160229E+00 -.122749E+00 .913244E-01 .610882E+01 .662692E-02 .264775E+00
.680000E+02 -.232041E+00 .138559E+00 -.120136E+00 .787398E-01 .542650E+01 .582899E-02 .227541E+00
.700000E+02 -.221607E+00 .124980E+00 -.114913E+00 .706041E-01 .500900E+01 .511207E-02 .201096E+00
Continued on next page
116Table optical constants of Diamond – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.720000E+02 -.214477E+00 .116361E+00 -.111288E+00 .654660E-01 .477725E+01 .466784E-02 .184528E+00
.740000E+02 -.210373E+00 .105812E+00 -.109407E+00 .594053E-01 .445545E+01 .433198E-02 .166711E+00
.760000E+02 -.205078E+00 .928103E-01 -.106903E+00 .519604E-01 .400218E+01 .393962E-02 .144905E+00
.780000E+02 -.197052E+00 .801432E-01 -.102813E+00 .446636E-01 .353096E+01 .348915E-02 .123080E+00
.800000E+02 -.187264E+00 .700639E-01 -.976456E-01 .388235E-01 .314773E+01 .305025E-02 .105295E+00
.850000E+02 -.163521E+00 .533862E-01 -.849431E-01 .291711E-01 .251309E+01 .219900E-02 .759909E-01
.900000E+02 -.143020E+00 .437311E-01 -.739667E-01 .236121E-01 .215386E+01 .162492E-02 .593912E-01
.950000E+02 -.126442E+00 .398741E-01 -.651134E-01 .213256E-01 .205337E+01 .125383E-02 .521440E-01
.100000E+03 -.118207E+00 .369503E-01 -.607557E-01 .196703E-01 .199360E+01 .108434E-02 .474381E-01
.105000E+03 -.110521E+00 .296211E-01 -.567475E-01 .157016E-01 .167095E+01 .918028E-03 .373984E-01
.110000E+03 -.100809E+00 .233182E-01 -.516631E-01 .122943E-01 .137068E+01 .742923E-03 .288204E-01
.120000E+03 -.848670E-01 .166351E-01 -.433347E-01 .869435E-02 .105743E+01 .510247E-03 .198573E-01
.130000E+03 -.724026E-01 .107267E-01 -.368653E-01 .556866E-02 .733715E+00 .360694E-03 .124650E-01
.140000E+03 -.605705E-01 .703432E-02 -.307516E-01 .362876E-02 .514894E+00 .247263E-03 .797027E-02
.150000E+03 -.521705E-01 .567216E-02 -.264302E-01 .291308E-02 .442869E+00 .181533E-03 .631358E-02
.160000E+03 -.470967E-01 .301617E-02 -.238311E-01 .154491E-02 .250515E+00 .146040E-03 .332172E-02
.170000E+03 -.389551E-01 .141490E-02 -.196708E-01 .721648E-03 .124339E+00 .988028E-04 .153194E-02
.180000E+03 -.338549E-01 .111563E-02 -.170730E-01 .567505E-03 .103533E+00 .742146E-04 .119519E-02
Continued on next page
117
Table optical constants of Diamond – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.300000E+03 -.209607E-01 .121280E-01 -.105169E-01 .612857E-02 .186344E+01 .374693E-04 .126516E-01
.350000E+03 -.101529E-01 .710617E-02 -.508300E-02 .357124E-02 .126686E+01 .969688E-05 .725233E-02
.370000E+03 -.994470E-02 .507926E-02 -.498152E-02 .255235E-02 .957161E+00 .787168E-05 .518168E-02
.400000E+03 -.916446E-02 .396298E-02 -.459078E-02 .199063E-02 .807036E+00 .628832E-05 .403657E-02
.500000E+03 -.621647E-02 .172382E-02 -.311272E-02 .864602E-03 .438156E+00 .261730E-05 .174545E-02
.600000E+03 -.440438E-02 .931206E-03 -.220451E-02 .466632E-03 .283771E+00 .127221E-05 .939463E-03
.740000E+03 -.292514E-02 .423348E-03 -.146364E-02 .211984E-03 .158993E+00 .547601E-06 .425835E-03
.800000E+03 -.250191E-02 .319689E-03 -.125172E-02 .160045E-03 .129770E+00 .398607E-06 .321294E-03
.100000E+04 -.159308E-02 .138033E-03 -.796851E-03 .690718E-04 .700071E-01 .160064E-06 .138475E-03
.150000E+04 -.696905E-03 .288569E-04 -.348519E-03 .144335E-04 .219434E-01 .304292E-07 .288972E-04
.200000E+04 -.387855E-03 .900432E-05 -.193916E-03 .450303E-05 .912807E-02 .940769E-08 .901131E-05
.250000E+04 -.246541E-03 .371683E-05 -.123272E-03 .185864E-05 .470954E-02 .380036E-08 .371867E-05
.300000E+04 -.170470E-03 .178953E-05 -.852700E-04 .894842E-06 .272089E-02 .181810E-08 .179014E-05
.350000E+04 -.124876E-03 .959995E-06 -.624603E-04 .480027E-06 .170285E-02 .975441E-09 .960234E-06
.360000E+04 -.117977E-03 .856266E-06 -.589985E-04 .428158E-06 .156225E-02 .870303E-09 .856468E-06
.400000E+04 -.954061E-04 .558275E-06 -.477356E-04 .279151E-06 .113173E-02 .569721E-09 .558382E-06
.450000E+04 -.752647E-04 .344638E-06 -.376558E-04 .172326E-06 .785968E-03 .354512E-09 .344690E-06
.500000E+04 -.608915E-04 .223623E-06 -.304759E-04 .111815E-06 .566648E-03 .232205E-09 .223651E-06
Continued on next page
118Table optical constants of Diamond – continued from previous page
ω (eV) ǫ1 ǫ2 n κ µ (cm−1) R −Imǫ−1
.600000E+04 -.422145E-04 .105159E-06 -.211235E-04 .525809E-07 .319759E-03 .111555E-09 .105168E-06
.800000E+04 -.237016E-04 .316185E-07 -.118613E-04 .158094E-07 .128189E-03 .351732E-10 .316200E-07
.100000E+05 -.151544E-04 .123106E-07 -.756979E-05 .615532E-08 .623869E-04 .143256E-10 .123110E-07
.200000E+05 -.378302E-05 .619427E-09 -.190735E-05 .309714E-09 .627818E-05 .909496E-12 .619432E-09
.250000E+05 -.242065E-05 .232199E-09 -.125170E-05 .116100E-09 .294182E-05 .391687E-12 .232201E-09
.260000E+05 -.223798E-05 .195119E-09 -.113249E-05 .975594E-10 .257090E-05 .320633E-12 .195120E-09
.300000E+05 -.168185E-05 .103385E-09 -.834465E-06 .516926E-10 .157178E-05 .174083E-12 .103385E-09
119
VITA
Micah Prange received a B.S. in Physics and Math from Pacific University in 1996 and
an M.S. in Mathematics from Portland State University in 2000. Micah then returned to
Pacific, joining the faculty of the Physics department there for two years before returning
to school at the University of Washington in 2002. He received his Ph.D. in Physics in 2009.