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\documentstyle[preprint,aps]{revtex}
\begin{document}
% \draft command makes pacs numbers print
\draft
%\preprint{}
\preprint{WFU preprint for Phys. Rev. B -- 2/2/00}

%Title of paper
\title{Electronic Structure
and Optical Properties of  CdMoO$_4$ and CdWO$_4$}

% repeat the \author\address pair as needed
\author{Y. Abraham, N. A. W. Holzwarth, and R. T. Williams}
%\email[]{abray00g@wfu.edu} \homepage[]{www.wfu.edu/~abray00g}
%\author{N. A. W. Holzwarth}
%\email[]{natalie@wfu.edu} \homepage[]{www.wfu.edu/~natalie}
%\author{R. T. Williams}
%\email[]{williams@wfu.edu}\homepage[]{www.wfu.edu/~williams}
\address{Department of Physics,
Wake Forest University, Winston-Salem, NC, 21709}

\date{\today}
\maketitle
\begin{abstract}
% insert abstract here
In previous work, we  studied the electronic structure of several
molybdate and
 tungstate crystals having the scheelite structure.
  The wolframite structure is another common form for molybdate and tungstate materials.
   In this paper, we compare two chemically
 similar materials which naturally form in the scheelite and
 wolframite structures --
 CdMoO$_4$ and CdWO$_4$, respectively.
We compare the
 electronic structure and approximate optical properties of these
 materials within  the framework of density functional
 theory.  One major difference between the two materials is in
 their lower conduction bands which are dominated by the crystal-field-split
 $d$ bands of the transition metal.  For
 CdMoO$_4$  the Mo ions have an
 approximately tetrahedral coordination causing the lower conduction band to be two-fold degenerate,
  while for CdWO$_4$ the W ions have an approximately octahedral
 coordination causing the lower conduction band to be three-fold dengenerate.

\end{abstract}
% insert suggested PACS numbers in braces on next line
\pacs{71.20.Ps,78.20.Ci,77.84.Bw,71.15.Mb}


% body of paper here
%introduction
\section{Introduction}
Cadmium tungstate (CdWO$_4$) and cadmium molybdate (CdMoO$_4$) are
interesting materials from several points of view, including their
 optical, chemical, and structural properties. Optical properties of CdWO$_4$
  are of interest because this material
has been used as a scintillator for detecting $X$-rays and
$\gamma$-rays in medical
applications\cite{ishii:1991,chernov:1997}. It has advantages over
other scintillator materials due to its relatively large $X$-ray
absorption coefficient and scintillation output, although its
relatively long response time is  disadvantageous for some
applications.

 Chemical interest is due to possible catalytic
activity of mixed CdMo$_x$W$_{1-x}$O$_4$ crystals, which have been
recently studied by Daturi and
co-workers\cite{daturi:1996,daturi:1997}.  These authors found
increased catalytic activity for values of $x$ in the range
$\frac{1}{8} < x < \frac{1}{4}$.


 Structural interest in these materials is due to the
fact that CdWO$_4$ crystallizes in wolframite structure while
CdMoO$_4$ crystallizes in scheelite structure. These two crystal
structures have many examples in ternary molybdate, tungstate, and
other ABO$_4$ materials. According  to a compilation by
Sleight\cite{sleight:1972}, the wolframite structure is a
generally more closely-packed structure, naturally forming in
materials with small A$^{2+}$ ions while the scheelite structure
naturally forms in materials with larger A$^{2+}$ ions.  For
example, ZnMoO$_4$ and ZnWO$_4$ have the wolframite structure,
while CaMoO$_4$ and CaWO$_4$ have the scheelite structure. In
contrast, CdWO$_4$ and CdMoO$_4$ have different crystal structures
-- the wolframite material (CdWO$_4$)having a slightly larger unit
cell volume than that of the scheelite material (CdMoO$_4$).
Interestingly, under pressure there seems to be a tendency for the
scheelite form of CdMoO$_4$  to transform to the wolframite
form\cite{shieh:1996} and for the wolframite form of CdWO$_4$
 to transform to the scheelite form\cite{macavei:1993}.
In their mixed crystal work, Daturi and
co-workers\cite{daturi:1996,daturi:1997} found that the
$50\%-50\%$ alloy ($x=\frac{1}{2}$) takes the scheelite structure,
while alloys with smaller Mo concentrations have the wolframite
form.

 In order to build a basis for understanding some of these unusual properties,
 we have carried out detailed  electronic
 structure calculations
for  CdWO$_4$ and CdMoO$_4$
 within the framework of density functional
theory.  In addition to determining the ground state electronic
density and distribution of one-electron eigenstates, we have also
estimated the optical dielectric functions and reflectivity for
these materials.  In Sec. \ref{crystalstruc}, the crystal
structure is discussed.  The calculational methods are presented
in Sec. \ref{calcmeth}. Density functional results are presented
in Sec. \ref{electronicstructure} and compared with results for
related materials, especially with our recent study of several
tungstate and molybdate scheelite materials\cite{zhang:1998}. The
calculated optical properties are presented in Sec. \ref{optical}
and discussed with reference to recent reflectivity measurements
of Nagirnyi and coworkers\cite{nagirnyi:1999,nagirnyi:private} . A
summary and conclusions are presented in Sec. \ref{conclusions}.

%crystalstruc


\section{Crystal Structures} \label{crystalstruc}

\subsubsection{CdMoO$_4$ -- {S}cheelite {S}tructure}



Cadmium molybdate  has the same crystal structure as CaMoO$_4$,
CaWO$_4$, PbMoO$_4$, and PbWO$_4$ whose electronic structures were
reported in our previous
 paper [\onlinecite{zhang:1998}]. It belongs to the scheelite-type structure with a space group
 of $I4_1/a \equiv C_{4h}^6$ which is  No. 88 in {\em{The International Tables}}\cite{internationaltable}.
 This structure has 8 symmetry elements and a body-centered
 orthorhombic primitive cell which includes  two formula units of
 CdMoO$_4$.  Each Mo site is surrounded by 4 equivalent O sites at
 a bond length of 1.75~\AA  $\;$    in approximately tetrahedral symmetry about that site.
 Each Cd site is surrounded by 8
 O sites at bond lengths of 2.40~\AA $\;$  and 2.44~\AA    $\;$    in approximately
 octahedral symmetry.  The crystal parameters
 used in the present work were taken from the recent measurements of Daturi
 and co-workers\cite{daturi:1997}
 which are listed in the first column of Table
 \ref{structtable2}. For comparison, earlier results are also
 listed in this table.

\subsubsection{CdWO$_4$  -- {W}olframite {S}tructure}




Cadmium tungstate has the wolframite-type structure which  is in
the monoclinic class, with  one axis not orthogonal to the other
two \cite{daturi:1996,daturi:1997}. This wolframite structure has
a space group of {\it P}2/c $\equiv$ $C_{2h}^4$ which corresponds
to No. 13 in the {\em{International
Tables}}\cite{internationaltable}. In this structure, each W is
surrounded by  6 near O sites in approximately octahedral
coordination. There are two distinct oxygen atoms present. Type
one (O$_1$) forms
  a bond to one tungsten atom with a very short bond length (1.78~\AA) and to two cadmium
   atoms with longer bond lengths (2.41 and 2.27~\AA). Type two (O$_2$) connects to
   two different
  W atoms with  longer bond lengths (1.91 and 2.15~\AA) and to Cd with a  bond
  length of  2.20~\AA. The unit cell  contains two formula units of CdWO$_4$. Table \ref{st} lists the
  measured lattice constants $a$, $b$, and $c$, as well as the   nonorthogonal angle $\beta$.  In this work,
  we used the parameters of Daturi and co-workers\cite{daturi:1996}.  Table \ref{st} also lists
  other crystallography data for comparison.  The
crystal volume of  CdWO$_4$  is apparently less than 1\% larger
than that of CdMoO$_4$, and there is considerable variation in the
experimental results.

%calcmeth

\section{Calculational Methods} \label{calcmeth}

The computations were performed using the Linearized Augmented
Plane Wave (LAPW) method using the WIEN97 code \cite{wien}, using
the same  calculational parameters (muffin tin radii, reciprocal
space truncations, and  convergence tolerances) as
 in our previous calculations on the Ca and Pb molybdates and tungstates in the
 scheelite structure \cite{zhang:1998}. Specifically, the muffin tin radius was
 chosen to be 1.65~bohr for O, Mo, and W, and 1.90~bohr for Cd. The Rkmax parameter
 was taken to be 10 and the Gmax parameter was taken to be 14 bohr$^{-1}$. All relativistic effects
 including the spin-orbit interaction were taken into account.
The exchange-correlation functional was taken within the local
density approximation using the form developed by Perdew and Wang
\cite{perdew:1992}.  Brillouin zone integrations within the
self-consistency cycles were performed using a tetrahedron
method\cite{blochl:1994a} using $12$ and $16$ inequivalent {\bf
k}- points  corresponding to $64$ and $48$ {\bf k}- points
throughout the Brillouin zone for the   scheelite and wolframite
structures, respectively. For calculating the density of states
and the imaginary part of the dielectric tensor, a denser sampling
of the Brillouin zone was needed.  For these, we used $35$ and
$63$ inequivalent {\bf k}- points  corresponding to $216$ and
$252$ {\bf k}- points throughout the Brillouin zone for the
scheelite and wolframite structures, respectively. The larger
number of inequivalent {\bf k}-points for the wolframite structure
are necessary because  of its lower symmetry.  We checked the
convergence of the k-point sampling and found nearly identical
results with variations in the number of k-points of about 15\% .

In Sec. \ref{dos}, results for the total and partial one-electron
density of states are presented.  In order to avoid the sharp
spikes of the tetrahedron method\cite{blochl:1994a}, these  are
calculated using a Gaussian smearing function\cite{fu:1983} of the
form:
\begin{equation}\label{doseq}
  N^x(E) = \sum_{n{\bf k}} f^x_{n{\bf k}} w_{{\bf k}}
  \frac{e^{-(E-E_{n{\bf k}})^2/{\sigma^2}}}{\sqrt{\pi}\sigma} .
\end{equation}
In this equation, $w_{{\bf k}}$ denotes the Brillouin zone
sampling weight factor while $f^x_{n{\bf k}}$ denotes the partial
density of states weight factor.  For the total density of states
$N^T$, the $f^T_{n{\bf k}}$ factor is equal to the degeneracy of
the state, while for  atomic partial density of states the
$f^a_{n{\bf k}}$ factor is the degeneracy of the state times the
charge within the muffin tin sphere for atom $a$, averaged over
all  spheres of that type. The Gaussian smearing parameter was
chosen to be $\sigma = 0.05$~eV.   In all of the density of states
plots and also in presentation of the band dispersions (Sec.
\ref{bands}), the zero of energy is taken at the top of the
valence band.

In Sec. \ref{optical}, results for components of the dielectric
tensor and for the reflectivity are presented.   In fact,
calculation of the optical properties of materials is beyond the
scope of density functional theory, which is rigorously a
ground-state formalism. However, there has recently been
considerable progress toward developing methods for calculating
optical properties from first-principles, using density functional
results as the starting point.\cite{delsole:1993,rohlfing:1998} As
a first step toward investigating the optical properties of
CdMoO$_4$ and CdWO$_4$, we have calculated the imaginary part of
the dielectric tensor from the LAPW one-electron eigenvalues
$E_{n{\bf k}}$ and corresponding wavefunctions $\Psi_{n{\bf
k}}({\bf r})$ using the approach of Del Sole and
Girlanda\cite{delsole:1993} and the code developed by
Ambrosch-Draxl and Abt\cite{ambrosch:1997}. In this approximation,
the imaginary part of the   dielectric tensor is given by a sum of
all transitions between occupied ``initial" bands $n_i {\bf k}$
and unoccupied ``final" bands $n_f {\bf k}$, integrated over the
Brillouin zone:
%\begin{widetext}
\begin{equation}\label{diel}
   \varepsilon_2^{\alpha \beta}(h \nu) =
  \left(\frac{2 \pi \hbar e}{m (h \nu - \Delta)} \right)^2
  \sum_{n_i n_f} \int \frac{ 2 d^3 k}{(2\pi)^3} \; M^{\alpha
  \beta}_{{n_i} {n_f}}({\bf k}) \;
\delta(E_{n_f{\bf k}} - E_{n_i{\bf k}} + \Delta - h \nu).
\end{equation}
%\end{widetext}
This expression depends upon  momentum matrix elements in the
$\alpha$ and $\beta$ directions between occupied and unoccupied
states according to
\begin{equation}  \label{matrixel}
M^{\alpha  \beta}_{{n_i} {n_f}}({\bf k})  \equiv  {\rm Re} \left[
\langle \Psi_{n_i{\bf k}}|p_{\alpha}|\Psi_{n_f{\bf k}}\rangle
\langle \Psi_{n_f{\bf k}}|p_{\beta}|\Psi_{n_i{\bf k}}\rangle
\right].
\end{equation}
  The so called "self-energy" parameter
$\Delta$ is a type of ``scissors operator"
\cite{levine:1989,delsole:1993,ambrosch:1997} which approximates
the leading correction to the infamous band gap underestimation
problem of density functional theory\cite{godby:1988}. In the
present work, the value of $\Delta$ is estimated from experiment.
The Brillouin zone integration in Eq. (\ref{diel}) is approximated
using a modified tetrahedron
method\cite{blochl:1994a,ambrosch:1997}.  The real part of the
dielectric tensor is determined by numerically evaluating the
Kramers-Kronig relations
\begin{equation}\label{kram}
  \varepsilon_1^{\alpha \beta}(h \nu) \approx \delta_{\alpha \beta} +
  \frac{2}{\pi}\mathcal{P}\int_{\nu_{\rm{min}}}^{\nu_{\rm{max}}} \;
  \frac{\nu^{\prime} \; {\ \varepsilon_2^{\alpha
  \beta}(h \nu^{\prime})}}{{\nu^{\prime}}^2 -
  {\nu}^2} \; d \nu^{\prime},
\end{equation}
where $\delta_{\alpha \beta}$ indicates the Kronecker $\delta$
function in the directional indices $\alpha$ and $\beta$.

Strictly speaking, the equality of Eq. (\ref{kram}) holds only
when the integration includes the entire frequency range. However,
the integration must exclude part of the low frequency range
because we have only included electronic  (no vibrational)
contributions to $\varepsilon_2$, and it must exclude part of the
high frequency range because we have only calculated electronic
transitions over a $30$~eV range.
  From the form of the Kramers-Kronig transform
(\ref{kram}), for $h \nu_{\rm{min}} \lesssim h \nu \lesssim h
\nu_{\rm{max}}$, one can reasonably argue that
 the neglected
 integration in the low frequency range will contribute
 a slowly varying negative value, while the neglected integration in the high
  frequency range  will contribute a slowly varying positive value. These corrections affect the
  calculation of the reflectivity and will be discussed further in section \ref{optical}.
  When the electric field direction, $\alpha$, coincides with an optic axis of the crystal, the
  corresponding normal incidence reflectivity  can then be estimated
  from the complex components of the dielectric tensor:
\begin{equation}\label{ref}
  R_N^{\alpha}= \left| \frac{(\varepsilon^{\alpha \alpha})^{1/2}-1}{(\varepsilon^{\alpha \alpha})^{1/2}+1}
  \right|^2.
\end{equation}
This can be conveniently evaluated using the following form:
\begin{equation}\label{refn}
  R_N^{\alpha}=
  \frac{U^{\alpha}+1 - V^{\alpha}}{U^{\alpha} +1 + V^{\alpha}},
\end{equation}
where,
\begin{equation}\label{refdef}
  U^{\alpha} \equiv \sqrt{(\varepsilon^{\alpha\alpha}_1)^2 + (\varepsilon^{\alpha\alpha}_2)^2}  \;\;\;
  {\mbox{\rm{and}}} \;\;\; V^{\alpha} \equiv \sqrt{2 \left( \sqrt{(\varepsilon^{\alpha\alpha}_1)^2 +
  (\varepsilon^{\alpha\alpha}_2)^2} + \varepsilon^{\alpha\alpha}_1 \right) }.
\end{equation}
In section  \ref{optical} we will also discuss the calculation of
the reflectivity for other geometries in order to compare with
experimental data.

%Electronicstructure
\section{Electronic Structure Results} \label{electronicstructure}

\subsection{Density of States} \label{dos}



Figure \ref{fdos} compares the total density of states for
CaMoO$_4$ and CdWO$_4$ ranging between $-80$~eV below the top of
the valence band to $+12$~eV above the conduction band.  The
distribution of the Mo and W core states for these materials is
very similar to their distribution in the scheelite molybdates and
tungstates studied in our earlier work\cite{zhang:1998}.  The Cd
$4p$ core states are located close to the Mo $4s$ states in
CaMoO$_4$ and $10$~eV above the W $5s$ states in CaWO$_4$.   For
both materials the Cd $4d$ states are concentrated near the bottom
of the valence band which, as we will see below, has largely O
$2p$ character.



Figure \ref{fpdos} shows the partial density of states for
CdMoO$_4$ and CdWO$_4$ in a $20$~eV energy range starting at  $8$
~eV below the top of the valence bands, comparing the spectral
contributions from the Cd, W, Mo, and O muffin-tin spheres.
Comparing the partial density of states plots of these two
materials, we see that the main features are similar.   One finds
the density of states features for CdWO$_4$ are generally not as
highly peaked as for CdMoO$_4$,  due to the lower local symmetry
and wider distribution of ligand bond lengths  in the wolframite
CdWO$_4$ compared with that of the scheelite CdMoO$_4$.

The contributions of the Cd $4d$ states are concentrated in double
peaks (due to spin-orbit splitting) of approximate width  $1$~eV
at the bottom of the valence bands of both materials and hybridize
to a small extent throughout the valence bands.

The O $2p$ states dominate the upper part of the valence bands and
have significant contributions throughout the main portion of the
valence bands.  From analysis of the electron density contours,
the O $2p$ contributions to  the lower and upper valence band
states are primarily $\sigma$ and $\pi$ states, respectively,
relative to their nearest neighbor Mo or W bonds. For CdWO$_4$,
the separation of the $\sigma$ and $\pi$ contributions of the O
$2p$ states is less well-defined because of the different W--O
bond lengths in this structure.

The Mo $4d$ states contribute to a small extent near the bottom of
the main valence band in CdMoO$_4$, and dominate the lower
  conduction bands. As in other scheelite materials,
the Mo $4d$ states are split by the tetrahedral crystal field into
two groups of states which will be discussed further below. For
CdWO$_4$, the W $5d$ states contribute to a small extent to the
valence bands, with a larger contribution near the bottom of the
bands. The dominant contributions of the  W $5d$ states occur in
the lower portion of the conduction bands where they are split by
the approximately octahedral crystal field into two main groups of
states.

For the lower portion of the conduction band, contributions
associated with the W $5d$ states span more than 7~eV in CdWO$_4$,
while those  associated with the Mo $4d$ states span less than
6~eV in CdMoO$_4$. For both of these materials the crystal field
splitting of the $d$ states by the approximate octahedral or
tetrahedral ligand geometries can be described by 2-fold
degenerate $e$ and 3-fold degenerate $t_2$ states.  For CdWO$_4$,
the lower peak in the conduction band partial density of states is
associated with the  $t_2$ states, while the upper peaks are
associated with
 $e$ states which are actually split into two
peaks due to deviations from octahedral symmetry at the W site.
For CdMoO$_4$, the lower peak in the conduction band partial
density of states is associated with the  $e$ states while the
upper peak is associated with the  $t_2$ states. Figure
(\ref{fpdos}) shows a very small contribution to the partial
density of states in the upper conduction band energy range near
$12$~eV, although the total density of states at these energies is
significant. This implies that the states in this energy range are
spatially diffuse so that they have a very small contributions
inside the muffin tin spheres used for the partial density of
states analysis.





\subsection{Energy band dispersions} \label{bands}



The Brillouin zones for the scheelite and wolframite structures
are shown in Fig. (\ref{bz}). The energy dispersion curves
calculated for CdMoO$_4$ and CdWO$_4$ are shown in Fig.\ref{band}.
In both cases the dispersion curves are plotted along three
different symmetry directions.  Both materials show a group of
narrow bands between $-6.5$ and $-5.5$~eV corresponding to the Cd
$4d$-like states, although other features of the band dispersions
are quite different from each other.

 For
 CdMoO$_4$, the minimum band gap of $2.4$~eV is located at the
$\Gamma$ point  at the center of the Brillouin zone. The valence
band dispersion is similar to that of the scheelite materials
CaMoO$_4$ and CaWO$_4$ studied previously.\cite{zhang:1998} The
conduction bands of  CdMoO$_4$ are much more dispersive than the
corresponding conduction bands of both CaMoO$_4$ and CaWO$_4$. In
particular, the Ca materials are characterized by a small band gap
between the lower and upper conduction bands which correspond to
the crystal field split states of Mo $4d$ and W $5d$ character,
while for CdMoO$_4$ the crystal field split $4d$ Mo states are not
separated by a well-defined band gap.

By contrast, the minimum band gap of $2.9$~eV for CdWO$_4$ does
not occur at the $\Gamma$ point, but at Y, which is located at the
center of the Brillouin zone boundary plane perpendicular to the
$b$ crystal axis. The valence band width of $5.3$~eV is slightly
larger (by $0.3$~eV)  than the valence band width of CdMoO$_4$.
The conduction band of CdWO$_4$ is generally less dispersive than
that of CdMoO$_4$.

\subsection{Electron density maps} \label{density}



In order to further understand the electronic states in greater
detail, we have constructed contour plots of the electron density
associated with selected states.  The results for CdMoO$_4$ in the
scheelite structure  were very similar to those of CaMoO$_4$
presented in our earlier paper\cite{zhang:1998} and therefore are
not presented here.

For the electron density of CdWO$_4$, we chose to construct
contour plots in a plane which passes through a W site and its
nearest-neighbor O sites. Figure \ref{unit} shows the drawing of
the atoms contained in a unit cell of CdWO$_4$ and of the plane
used for plotting the contours.   From this figure, we can see the
approximate octahedral symmetry about the W site.   For perfect
octahedral symmetry, the W site would be surrounded by $6$
equidistant O atoms, with $4$ in a plane and $2$ along an axis
perpendicular to that plane. In CdWO$_4$, we see that only two O
atoms are in the plane, while another two are close to that plane
at a larger bond distance. Bonds from the W to the remaining two O
atoms have an intermediate  length and make an angle with the
plane that is only approximately  $90^o$.




Figure \ref{2dplots}(a-f) show contour plots of the electron
density for various energy ranges of states of CdWO$_4$ in the
plane defined in Fig. (\ref{unit}).  The contours in Fig.
\ref{2dplots}(a) correspond to the energy range dominated by the
Cd $4d$ states and thus shows a very small contribution to the
electron density in the W--O plane shown. Plots \ref{2dplots}(b)
and (c)  show O $2p \sigma$ orbitals hybridizing with the W $5d$
orbitals, representing the middle portion of the valence band.
Plot \ref{2dplots}(d) shows the O $2p \pi$ states that comprise
the upper portion of the valence band, with very little
contribution from the W states.



Plots \ref{2dplots}(e) and (f) shows the electron density contours
for the lower  and upper conduction band states, respectively.
These plots clearly show the crystal field split $5d$ states of W.
If one compares the shapes of these contours with those of the
conduction bands of the scheelite materials plotted in a similar
way\cite{zhang:1998}, one finds that they look very similar.  In
both structures, the bottom conduction band state has W $5d$
orbitals which have peak densities in the non-bonding directions
and which hybridize with the $\pi$ states of the nearest neighbor
O atoms. The upper conduction band state has W $5d$ orbitals which
have peak densities along the bond directions, making anti-bonding
hybrid states with the $\sigma$ states of the nearest neighbor O
atoms.    There is one important difference between the wolframite
and scheelite conduction bands, however. In the wolframite
structure, the local symmetry about the W site is approximately
octahedral and the lower band is 3-fold degenerate with $t_2$
symmetry while the upper band has $e$ symmetry. As shown in Fig.
(\ref{fpdos}),  the 2-fold degenerate $e$ states are split by
approximately 2~eV by  deviations from octahedral symmetry. In the
scheelite structure, the local symmetry about the Mo site is
approximately tetrahedral and the lower band is 2-fold degenerate
having $e$ symmetry, while the upper band is 3-fold degenerate
having $t_2$ symmetry.

%optical

\section{Optical Properties} \label{optical}




As described in Sec.(\ref{calcmeth}), in order to calculate the
imaginary part of the dielectric tensor from the density
functional results, it is necessary to estimate a self-energy
correction $\Delta$.  A rough estimate of $\Delta$ is the
difference between the experimental band gap and the value
determined from our results calculated within the local density
approximation (LDA). Unfortunately, the experimental values of
E$_{\rm{gap}}$ have not been precisely measured. Table \ref{gap}
lists the LDA values of E$_{\rm{gap}}$ compared with several
literature results. For CdMoO$_4$, if we take the reference band
gap to be 3.8~eV, the average of the two experimental values, the
self-energy correction can be taken to be $\Delta = 1.4$~eV.
Measurements of the absorption edge in CdWO$_4$ lead to estimates
of E$_{\rm{gap}}=3.8$ to
$4.09$~eV\cite{kroger:1948,tanaka:1995,gillette:1950}. Yochum and
Williams\cite{yochum:1999} have measured the two-photon absorption
spectrum of CdWO$_4$ using a 200 femtosecond pulse of 420~nm light
and white-light continuum of similar duration.   The spectrum
covers final state energies from 4.5 to 5.5~eV.   The absorption
rises from a finite value at 4.5~eV to a plateau at approximately
4.7~eV. Taking the published linear absorption edge in bulk
crystals as a definite lower limit for the band gap and the
two-photon absorption plateau at 4.7~eV as a definite upper bound,
and finally influenced by measured reflectivity
spectra\cite{nagirnyi:1999,nagirnyi:private}, we estimate an
interband edge of $4.4 \pm 0.3$~eV for CdWO$_4$. The value of the
self-energy correction, $\Delta = 1.4$~eV, used in the present
work corresponds to the choice of $E_{\rm{gap}}=4.3$~eV.





Since CdMoO$_4$ has orthorhombic symmetry, it has two distinct
optic axes.   The corresponding dielectric tensor components are
$\varepsilon^{xx} \equiv \varepsilon^{yy}$, corresponding to the
electric field oriented along one of the a-axes and
$\varepsilon^{zz}$, corresponding to the electric field oriented
along the c-axis.  The calculated values of $\varepsilon^{xx}_2$
and $\varepsilon^{zz}_2$ and of $\varepsilon^{xx}_1$ and
$\varepsilon^{zz}_1$ are shown in figures \ref{cdmoe2} and
\ref{cdmoe1} respectively.  The $\varepsilon_2$ results indicate
that the dominant transitions occur within approximately 8~eV
above the band gap with two main peaks due to transitions between
the O valence band and the crystal-field split Mo $4d$ conduction
bands. A small peak near 10~eV is like to be due to transitions
from the Cd $4d$ states. The average of the calculated values of
$\varepsilon_1$ is close the experimental measurements in the
transparent region\cite{bakhshieva:1977}, although the calculated
birefringence is considerably larger than experiment. As discussed
in Section \ref{calcmeth} there are some systematic errors  due to
truncation error in the Kramer's-Kronig integral (\ref{kram}) used
to calculate $\varepsilon^{\alpha \alpha}_1$. Since
$\varepsilon^{\alpha \alpha}_2$ is not zero at $\nu_{\rm{max}}$,
we expect our calculated $\varepsilon^{\alpha \alpha}_1(\nu)$ to
be underestimated. The fact that $\varepsilon^{xx}_1$ is larger
than experiment in the visible range suggests that either the
neglected vibrational contributions of $\varepsilon^{xx}_2$ in the
Kramer's-Kronig integral are appreciable, or that our calculations
generally over-estimate the value of  $\varepsilon^{xx}_2$.

Using the calculated values of the dielectric tensor components
shown for CaMoO$_4$ in Figs. \ref{cdmoe2} and \ref{cdmoe1} and
Eq.~(\ref{ref}), results for the normal incidence reflectivity are
shown in Fig. (\ref{cdmor}).  The calculated reflectivity spectrum
shows two main peaks, similar to the spectral form of
$\varepsilon_2$. Unfortunately, we do not know of any reflectivity
data available for comparison.


Although the crystallographic $b$ and $c$ axes of CdWO$_4$ are
mutually orthogonal, the $a$ axis makes an angle of $91.519^o$
with respect to the $c$ axis in the $a-c$-plane.  The principal
axis system of the dielectric tensor has been determined
experimentally in the transparency region.  This axis system
includes the crystallographic $b$-axis and two perpendicular axes
in the $a-c$-plane.  One of those axes is oriented at an angle of
$13.5^o$\cite{lotem:1987} or $16^o$\cite{burshtein:1988} relative
to the crystallographic $c$-axis  at $\lambda =633$~nm and does
not rotate appreciably within the transparency region. However,
throughout the 30~eV range addressed in the present work, the
principal axes of the dielectric tensor could vary considerably.
To facilitate comparisons between theory and experiment referenced
approximately to the crystallographic axes, we have adopted a
fixed orthogonal coordinate system with the $y$ axis along the
crystallographic $b$ axis and the $z$ axis along the
crystallographic $c$ axis.  The $x$ axis then lies in the $a-c$
plane and makes an angle of $1.519^o$ with the crystallographic
$a$ axis.   The dielectric tensor expressed in this coordinate
system thus has 4 nontrivial components: $\varepsilon^{xx},
\varepsilon^{yy}, \varepsilon^{zz}$, and $\varepsilon^{xz} \equiv
\varepsilon^{zx}$. The remaining elements are zero.
 These are shown in figures \ref{cdwoe2} and
\ref{cdwoe1} for the imaginary and real parts, respectively. The
 $\varepsilon^{\alpha \alpha}_2$ spectra for CdWO$_4$ are similar to those
 of CdMoO$_4$, although the upper energy peak is smaller and broader than the lower energy peak.
   The off-diagonal component
 $\varepsilon^{xz}_2$, is much smaller than the diagonal components.
 The real parts of the dielectric tensor presented in figure
 \ref{cdwoe1} are compared with experimental points\cite{lotem:1987,burshtein:1988}  in the
 transparent region.    As for CdMoO$_4$, we find that average calculated
 real part of the dielectric tensor is close to the experimental average in the transparent
 range, although the dispersion and birefringence  differ.
  In addition,  one of the principal axes of the dielectric tensor calculated in our results
  makes a much smaller angle with the $z$-axis than $13.5^o$ or $16^o$
  measured by experiment\cite{lotem:1987,burshtein:1988}.
    This suggests that our calculated diagonal components of $\varepsilon^{\alpha\alpha}_1$
    are comparable or larger than experiment  while
  the calculated off-diagonal $\varepsilon^{xz}_1$ is smaller than experiment in the
  visible spectral region.  These discrepancies are partly due to
  the truncation error in the Kramer's-Kronig transform discussed
  above and
  in section \ref{calcmeth}.

  Normal incidence reflectivity, calculated with the electric field
  oriented along the $y$-axis can be calculated from Eq. (\ref{ref}), while other orientations
  are more complicated.   Since the magnitude of the off-diagonal matrix
   element $\varepsilon^{xz}$
  is generally less than 10\% of that of the corresponding diagonal components it is also reasonable
  to use  Eq. (\ref{ref}) to approximate the normal incidence reflectivity for the electric field
  oriented along the $x$- and $z$- axes. Figure \ref{cdwor} shows the
 normal incidence reflectivity spectra calculated in this way from
 the uncorrected dielectric tensor components.

 Nagirnyi and co-workers\cite{nagirnyi:1999,nagirnyi:private} have measured the
 reflectivity spectrum of a single-crystal sample  of
 CdWO$_4$, cleaved in the $x-z$ ( or $a-c$)  plane.   The incident photon beam
 was oriented with $p-$ polarization at an angle of $\theta=45^o$ relative to the normal
 direction.  Assuming that the off-diagonal dielectric
 contributions can again be neglected, the corresponding
 calculated $p$- polarized
 reflectivity for an incidence angle of $\theta$ and for the
 electric field in the $y-\alpha$ plane (where $\alpha$ represents either $x$ or
 $z$)
  can be determined from the complex dielectric tensor components
  according to the
 equation:
 \begin{equation} \label{pref}
 R_P^{\alpha} = \left| \frac{(\varepsilon^{yy}-\sin^2
 \theta)^{1/2} - (\varepsilon^{yy}\varepsilon^{\alpha
 \alpha})^{1/2} \cos \theta}{(\varepsilon^{yy}-\sin^2
 \theta)^{1/2} + (\varepsilon^{yy}\varepsilon^{\alpha
 \alpha})^{1/2} \cos \theta} \right|^2.
 \end{equation}

Results for $\theta=45^o$ are plotted in Fig. \ref{cdwop}.   The
shape of the $p-$polarized reflectivity spectrum is similar to
that at normal incidence, although its magnitude is reduced.
Recent preliminary reflectivity measurements of Nagirnyi and
co-workers\cite{nagirnyi:1999,nagirnyi:private} demonstrate that
the spectra change with crystal orientation. Their data show a
two-peak structure below 12~eV which is qualitatively similar to
the
 results  shown in Fig. \ref{cdwop},
although features in the spectra at higher energy differ from our
calculations. Detailed comparisons will be done in the future.

%conclusions
\section{Summary and Conclusions} \label{conclusions}

We have carried out detailed electronic structure calculations for
CdMoO$_4$ and CdWO$_4$.  The electronic structure of  CdMoO$_4$ is
 similar to that of other scheelite materials, especially to
that of  CaMoO$_4$.  Differences are due to the fact that the
CdMoO$_4$ crystal is more compact than other scheelite materials,
resulting in larger band widths. In addition, the Cd ions
introduce a narrow band of $3d$ character at the bottom of the
valence band.

Despite the fact that CdWO$_4$ has the lower symmetry wolframite
structure, there are more similarities than differences of its
electronic structure compared with that of  CdMoO$_4$.  In both
materials, the lower conduction bands are controlled by the
crystal field-split $d$ bands of the transition metal. In the case
of CdMoO$_4$, the Mo ions have an approximately tetrahedral
environment which causes the lowest conduction band to be composed
of doubly degenerate $4d-e$ states.  In the case of CdWO$_4$, the
W ions have an approximately octahedral environment which caused
the lowest conduction band to be composed of triply degenerate
$5d-t_2$ states. The optical spectra of the two materials are
effected by these differences.  The calculated imaginary
components of the dielectric tensor for CdMoO$_4$ show two peaks
of similar strength, while those for CdWO$_4$ show a larger peak
near threshold and a smaller peak at higher energy.  Detailed
comparison of the present results with experiment will be possible
as soon as single crystal optical measurements are
completed.\cite{nagirnyi:private}


\begin{acknowledgments}
% put your acknowledgments here.
This work was supported by NSF grants DMR-9403009, DMR-9706575,
and DMR-9732023.  We would also like to thank Y. C. Zhang, P.
Blaha, and C. Ambrosch-Draxl for  help with the WIEN97 code, Z.
Burshtein for helpful comments on the refractive index of
CdWO$_4$, and V. Nagirnyi for sharing his preliminary reflectivity
data on CdWO$_4$.
\end{acknowledgments}


% now the references. delete or change fake bibitem. delete next three
%   lines and directly read in your .bbl file if you use bibtex.
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\expandafter\ifx\csname bibnamefont\endcsname\relax
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\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\expandafter\ifx\csname bibinfo\endcsname\relax \def\bibinfo#1#2{#2}\fi
\expandafter\ifx\csname eprint\endcsname\relax \def\eprint#1{#1}\fi

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%\end{thebibliography}

\end{references}

% figures follow here
%
% Here is an example of the general form of a figure:
% Fill in the caption in the braces of the \caption{} command. Put the label
% that you will use with \ref{} command in the braces of the \label{} command.
%
% \begin{figure}
% \caption{}
% \label{}
% \end{figure}
%figures.tex




\begin{figure}
%\includegraphics*[width=6in]{../epsfiles/totaldos.eps}
\caption{Total density of states of  CdMoO$_4$ and CdWO$_4$
including upper core, valence, conduction band states. The zero of
energy is placed at the top of the valence band. Labels represent
dominant atomic character of spectral peaks.} \label{fdos}
\end{figure}

\begin{figure}
%\includegraphics*[width=6in]{../paper_original/cdwo-cdmo-fpdos.eps}
\caption{Partial density of states for valence and conduction
bands of CdMoO$_4$ and CdWO$_4$, indicating Cd, Mo or W, and O
contributions with dashed, solid, and dotted lines respectively. }
\label{fpdos}
\end{figure}

\begin{figure}
%\includegraphics*[width=6in]{../epsfiles/bz.eps}
\caption{Diagram of the Brillouin zones for CdMoO$_4$ and CdWO$_4$.}
\label{bz}
\end{figure}

\begin{figure}
%\includegraphics*[width=6in]{../epsfiles/bandstructure.eps}
\caption{Energy band dispersions for CdMoO$_4$ and CdWO$_4$ plotting along
Brillouin zone directions indicated in Fig. (\ref{bz})}
\label{band}
\end{figure}

\begin{figure}
%\includegraphics[width=100mm]{../paper_original/cdwo_unit.eps}
\caption{A perspective drawing of the crystal structure of
CdWO$_4$. The spheres represent  O, W, and Cd positions using
large, medium, small sizes, respectively. The bonds represent the
near-neighbor W--O interactions. The shaded plane is drawn to pass
through a W atom and its nearest-neighbor O atoms.  This is the
plane that is used to construct the two-dimensional electron
contour plots described below. The contours superposed on this
diagram correspond the upper conduction band states presented in
Fig. (\ref{2dplots}(f)).} \label{unit}
\end{figure}

\begin{figure}
%\includegraphics*[width=6in]{../epsfiles/2dplot.eps}
\caption{Contours of  electronic charge density for various states
of CdWO$_4$ plotted in the plane shown in Fig. (\ref{unit}) with
contour levels uniformly spaced at intervals of $0.5\:~e/$\AA$^3$,
starting at $0.5\:~e/$\AA$^3$. The one-electron energy ranges for
these plots are $-7.2$~eV to $-5.6$~eV for (a), and $-5.6$~eV to
$-4.0$~eV for (b), $-4.0$~eV to $-2.0$~eV for (c), $-1.8$~eV to
$0$~eV for (d), $2.5$~eV to $5.1$~eV  for (e), and $5.1$~eV to
$11.9$~eV  for (f). For plots (e) and (f), the  electron density
was calculated supposing full occupancy of these unoccupied
bands.} \label{2dplots}
\end{figure}



\begin{figure}
%\includegraphics*[width=6in]{../epsfiles/cdmoe2.eps}
\caption{Calculated values of $\varepsilon^{xx}_2$ (solid line)
and $\varepsilon^{zz}_2$ (dotted line) for CdMoO$_4$. }
\label{cdmoe2}
\end{figure}

\begin{figure}
%\includegraphics*[width=6in]{../epsfiles/cdmoe1.eps}
\caption{Calculated values of $\varepsilon_1^{xx}$ (solid line)
and $\varepsilon_1^{zz}$ (dotted line)  for CdMoO$_4$.
Experimental values from Ref. \onlinecite{bakhshieva:1977} are
plotted with filled and open triangles corresponding to
$\varepsilon_1^{xx}$ and $\varepsilon_1^{zz}$, respectively. }
\label{cdmoe1}
\end{figure}

\begin{figure}
%\includegraphics*[width=6in]{../epsfiles/cdmor.eps}
\caption{Calculated values of normal incidence $R_N^x$ (solid
line) and $R_N^z$ (dotted line) for CdMoO$_4$. } \label{cdmor}
\end{figure}


\begin{figure}
%\includegraphics*[width=6in]{../epsfiles/cdwoe2.eps}
\caption{Calculated values of $\varepsilon^{yy}_2$ (solid black
line), $\varepsilon^{zz}_2$ (dotted line), and
$\varepsilon^{xx}_2$ (grey line) for CdWO$_4$. Inset shows
 off-diagonal element $\varepsilon^{xz}_2$. } \label{cdwoe2}
\end{figure}

\begin{figure}
%\includegraphics*[width=6in]{../epsfiles/cdwoe1.eps}
\caption{Calculated values of $\varepsilon^{yy}_1$ (solid black
line), $\varepsilon_1^{zz}$ (dotted line), and
$\varepsilon_1^{xx}$ (grey line) for CdWO$_4$. Experimental values
from Ref. \onlinecite{lotem:1987} are plotted with filled
triangles and open circles corresponding to the three eigenvalues
of $\varepsilon^{\alpha \beta}_1$ in the transparent region. Inset
shows
 off-diagonal element $\varepsilon_1^{xz}$.} \label{cdwoe1}
\end{figure}

\begin{figure}
%\includegraphics*[width=6in]{../epsfiles/cdwor.eps}
\caption{Values of normal incidence reflectivity for CdWO$_4$;
$R_N^y$ (solid black line), $R_N^z$ (grey line) and $R_N^x$
(dotted line), calculated from Eq. (\ref{ref}), neglecting
off-diagonal dielectric contributions. } \label{cdwor}
\end{figure}

\begin{figure}
%\includegraphics*[width=6in]{../epsfiles/cdwop.eps}
\caption{Values of  reflectivity for the $z-x$ ($c-a$) face of
CdWO$_4$ for $p-$ polarization at incidence angle $\theta=45^o$;
$R_P^z$ (solid black line) and $R_P^x$  (dotted line), calculated
from Eq. (\ref{pref}), neglecting off-diagonal dielectric
contributions. } \label{cdwop}
\end{figure}

% tables follow here
%
% Here is an example of the general form of a table:
% Fill in the caption in the braces of the \caption{} command. Put the label
% that you will use with \ref{} command in the braces of the \label{} command.
% Insert the column specifiers (l, r, c, d, etc.) in the empty braces of the
% \begin{tabular}{} command.
%
% \begin{table}
% \caption{}
% \label{}
% \begin{tabular}{}
% \end{tabular}
% \end{table}
\begin{table}
\caption{Compilation of experimental  crystal parameters for
CdMoO$_4$. Atom positions are specified using the same conventions
as in Ref. \onlinecite{zhang:1998}.}
\begin{tabular}{||c||c|c|c||c|c|c||} \hline
 &\multicolumn{3}{c||}{Ref. [\onlinecite{daturi:1997}]}&\multicolumn{3}{c||}
 {Ref. [\onlinecite{chichagov:1966a}]}\\ \hline

a(\AA)&\multicolumn{3}{c||}{5.156}&\multicolumn{3}{c||}{5.17}
\\

c(\AA)&\multicolumn{3}{c||}{11.196}&\multicolumn{3}{c||}{11.19}\\

V(\AA$^3$)&\multicolumn{3}{c||}{148.82}&\multicolumn{3}{c||}{149.55}\\  \hline

 &x&y&z&x&y&z\\ \hline

Cd&0.0&0.25&0.625&0.0&0.25&0.625\\

Mo&0.0&0.25&0.125&0.0&0.25&0.125\\

O&0.24&0.091&0.0421&0.25&0.11&0.03\\
\end{tabular}
\label{structtable2}
\end{table}

\begin{table}
\caption{Compilation of experimental  crystal parameters for
CdWO$_4$. Cd sites are located at the fractional positions
$\pm(\frac{1}{2},y_{Cd},\frac{3}{4})$, W sites are located at the
fractional positions $\pm(0,y_{W},\frac{1}{4})$, and O$_1$ and
O$_2$ sites are located at the fractional positions
$\pm(x_i,y_{i},z_i)$ and $\pm(x_i,-y_{i},z_i+\frac{1}{2})$, where
$i=1$ or $2$. }
\begin{tabular}{||c||c|c|c||c|c|c||c|c|c||c|c|c||}\hline
&\multicolumn{3}{|c||}{Ref.
[\onlinecite{daturi:1996}]}&\multicolumn{3}{|c||}{Ref.
[\onlinecite{daturi:1997}]}&\multicolumn{3}{|c||}{Ref.
[\onlinecite{macavei:1993}]}&\multicolumn{3}{|c||}{Ref.
[\onlinecite{chichagov:1966}]}\\\hline
a(\AA)&\multicolumn{3}{|c||}{5.0289}&\multicolumn{3}{|c||}{5.026}&\multicolumn{3}{|c||}{5.028}&\multicolumn{3}{|c||}{5.02}\\
b(\AA)&\multicolumn{3}{|c||}{5.8596}&\multicolumn{3}{|c||}{5.867}&\multicolumn{3}{|c||}{5.862}&\multicolumn{3}{|c||}{5.85}\\
c(\AA)&\multicolumn{3}{|c||}{5.0715}&\multicolumn{3}{|c||}{5.078}&\multicolumn{3}{|c||}{5.067}&\multicolumn{3}{|c||}{5.07}\\
$\beta
(\deg)$&\multicolumn{3}{|c||}{91.519}&\multicolumn{3}{|c||}{91.47}&\multicolumn{3}{|c||}{91.50}&\multicolumn{3}{|c||}{91.5}\\
V(\AA$^3$)&\multicolumn{3}{|c||}{149.39}&\multicolumn{3}{|c||}{149.69}&\multicolumn{3}{|c||}{149.3}&\multicolumn{3}{|c||}{148.84}\\\hline
&x&y&z&x&y&z&x&y&z&x&y&z\\\hline

Cd&0.5&0.3027&0.75&0.5&0.3020&0.75&0.5&0.3023&0.75&0.5&0.308&0.75\\

W&0.0&0.17847&0.25&0.0&0.1784&0.25&0.0&0.1785&0.25&0.0&0.180&0.25\\

O$_1$&0.242&0.372&0.384&0.250&0.360&0.393&0.242&0.372&0.383&0.240&0.380&0.380\\
O$_2$&0.202&0.096&0.951&0.189&0.099&0.954&0.203&0.098&0.949&0.210&0.10&0.960\\\hline
\end{tabular}
\label{st}
\end{table}

\begin{table}
\caption{Estimates of energy band gaps.} \label{gap}
\begin{tabular}{|l|l|l|}
\hline
 Material&E$_{\rm{gap}}$ (eV)&Reference \\ \hline
  CdMoO$_4$& 2.43 & LDA (present work)  \\
 & 3.96 &Excitation energy [\onlinecite{zhang:1993}] \\
& 3.72 &Absorption edge [\onlinecite{kroger:1948}] \\ \hline
 CdWO$_4$& 2.94 & LDA (present work) \\
& 4.5 $\pm$ 0.2 & Two-photon absorption [\onlinecite{yochum:1999}]
 \\
  & 3.8& Absorption edge [\onlinecite{tanaka:1995,gillette:1950}]\\
 & 4.09 &Absorption edge [\onlinecite{kroger:1948}] \\
  \hline
\end{tabular}
\end{table}


\end{document}
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% ****** End of file template.aps ******
