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\documentstyle[prb,preprint,aps]{revtex}
\begin{document}
% \draft command makes pacs numbers print
\preprint{Phys. Rev. B preprint}
\draft
\title{Electronic band structures of the scheelite materials -- CaMoO$_4$, 
CaWO$_4$, PbMoO$_4$, and PbWO$_4$}

% repeat the \author\address pair as needed
\author{Y. Zhang, N. A. W. Holzwarth, and R. T. Williams}
\address{Department of Physics, Wake Forest University, Winston-Salem
, NC 27109}

\date{\today}
\maketitle

\begin{abstract}
% insert abstract here
%\input{mainabs}
Density functional calculations  using the linearized-augmented-plane-wave 
(LAPW) method were carried out for the scheelite materials  CaMoO$_4$, CaWO$_4$, 
PbMoO$_4$, and PbWO$_4$ in order to determine their ground state electronic 
properties.  The results indicate that CaMoO$_4$ and CaWO$_4$ have  direct band 
gaps at the center of the Brillouin zone, while PbMoO$_4$ and PbWO$_4$ have  
band  extrema at wave vectors away from the zone center with possibly indirect 
band gaps. The magnitudes of the band gaps increase in the order: PbMoO$_4$ $<$  
PbWO$_4$ $<$ CaMoO$_4$ $<$ CaWO$_4$. The valence and conduction bands near the 
band  gap are dominated by molecular orbitals associated with the MoO$_4^{-
\alpha}$ and WO$_4^{-\alpha}$ ions, where $\alpha \approx 2$. The  valence band 
widths are 5~eV and 5.5~eV, for the Ca and Pb materials respectively. In the Pb 
materials, the Pb $6s$ states form  narrow bands  1~eV below the bottom of the 
valence bands  and also hybridize with states throughout the valence bands, 
while the Pb $6p$ states hybridize with states throughout the conduction bands.  
In the Ca materials, the Ca $3d$ states contribute to a high density of states  
3-4~eV above the bottom of the conduction bands. 
\end{abstract}

% insert suggested PACS numbers in braces on next line
\pacs{71.20.-b,71.15.Ap,71.15.Mb} 

% body of paper here

%\input{intro}
\section{Introduction}
\label{intro}

The calcium and lead molybdates and tungstates  are naturally occuring minerals 
which can also be made synthetically and which have very interesting 
luminescence and structural properties.  The mineral names associated with these 
materials are powellite (CaMoO$_4$),  scheelite (CaWO$_4$), wulfenite 
(PbMoO$_4$), and stolzite (PbWO$_4$). The name ``scheelite" is also used to 
describe the  common crystal structure of these materials.  These materials 
have been very extensively studied during the past century.  For example, a 1948 
monograph by Kr\"{o}ger\cite{kroger:1948} presents a thorough summary of the 
luminescence properties of these and related materials and their solid 
solutions.  Recently, PbWO$_4$ has attracted special interest because of plans 
to use it as a scintillator in detectors at the Large Hadron Collider in 
CERN.\cite{lecoq:1996} There are still some very fundamental questions 
concerning the electronic properties of these materials, especially concerning 
the nature of the states of the ideal crystal in the vicinity of the band gap.   
In order to address some of these questions, we have calculated the self-
consistent electronic structure of the four materials in the framework of 
density functional theory\cite{hohenberg:1964,kohn:1965} using a relativistic 
linearized-augmented-plane-wave (LAPW)\cite{singh:1994,wien} technique.  

The paper is organized as follows.   In Section \ref{strct} the crystal 
structure is described.  The calculational method is discussed in Section 
\ref{calc}.  The results for the densities of states, the band structures, and 
the electron density distributions are presented in Section \ref{results}.  In 
Section \ref{conclude} the work is analyzed in terms of earlier theoretical and 
experimental work and the main results are summarized.

%\input{struct}
\section{Crystal structure}
\label{strct}

The scheelite crystal structure is characterized by the tetragonal space group  
$I4_1/a$ or C$^6_{4h}$ (listed as No. 88 in the {\em{International 
Tables}}\cite{international}). In this structure, the primitive unit cell has 
two  ABO$_4$ units (where A = Ca or Pb and B = Mo or W). The A and B sites have 
S$_4$ point symmetry and the crystal has an inversion center.  The O sites have 
only a trivial point symmetry, and are arranged in approximately tetrahedral 
coordination about each B site.  The crystal has three crystal parameters 
$\{x,y,z\}$ which describe the location of the O sites. There are a number of 
equivalent ways to describe the structure which have appeared in the 
literature.\cite{sillen:1943} Table \ref{strctpar} lists some representative 
values of the crystal parameters that have been determined by X-ray and neutron 
diffraction with the $\{x,y,z\}$ defined relative to the origin of the crystal  
at the inversion center and confined to the sector $0 < \{x,y\} < \frac{1}{2}$ 
and $0 < z < \frac{1}{8}$. The starred references indicate the values used in 
the present electronic structure calculations.

The crystal structure has been described\cite{sillen:1943} as highly ionic with 
A$^{+\alpha}$ cations and tetrahedral BO$_4^{-\alpha}$ anions, where $\alpha 
\approx 2$.  The BO$_4^{-\alpha}$ anions are themselves highly ionic with formal 
charges of B$^{+\beta}$ and O$^{-\gamma}$, where $4\gamma - \beta \equiv 
\alpha$. If there were no covalent bonding, $\beta$ and  $\gamma$ would take the 
values 6 and 2, respectively. The tetrahedral BO$_4^{-\alpha}$ anions have  
short B--O bond lengths ranging from 1.77 \AA \ for CaMoO$_4$ to 1.79 \AA \ for 
PbWO$_4$ which are quite rigid even under pressure.\cite{hazen:1985} 
 Figure \ref{cryst} shows a perspective drawing of the  crystal structure of 
PbWO$_4$ in the conventional unit cell, indicating the a-- and c-- axes and the  
nearly tetrahedral bonds between O and W.   Also included in this figure is a 
plane which passes through two W--O bonds and one of the Pb sites.  A contour 
map of the valence electron density which is indicated on this plane will be 
discussed further in Section \ref{results}.

%\input{method}
\section{Calculational methods}
\label{calc}

The calculations reported in this paper were performed with the LAPW technique 
using the {\bf{WIEN97}}\cite{wien} code.   The exchange-correlation functional 
was taken within the local density approximation using the form developed by 
Perdew and Wang\cite{perdew:1992}. The calculational and convergence parameters 
used for all four materials are listed in Table \ref{lapwpar}. The muffin tin 
radii for O, Mo, and W were chosen to be slightly less than half the O--Mo and 
O--W bond lengths. For Pb, the muffin tin radius  of 1.9~Bohr was chosen in 
order to avoid the numerical instabilities of ``ghost" states. The plane wave 
expansion cut-offs were chosen to be 6~Bohr$^{-1}$ for expanding the 
wavefunctions and 14~Bohr$^{-1}$ for expanding the densities and potentials.   
The {\bf{k}}-point sampling was chosen to be 6 irreducible points within the 
Brillouin zone.  For this choice of parameters, the bottleneck of the 
calculation is  diagonalization of the  plane wave Hamiltonian matrices which 
have a dimension of roughly 5000.  We estimate that the convergence error of the 
calculation is dominated by the plane wave cut-off and the total energy error is 
at most 0.2~eV.   The {\bf{k}}-point sampling error for the total energy is 
estimated to be at most 0.01~eV.   Although, for structural optimization 
studies, the convergence parameters would have to be increased, the present 
choice of parameters ensures very good convergence of the one-electron energies, 
with estimated errors of at most 0.002~eV.  In fact, for this choice of plane-
wave cut-off parameters,  non-relativistic LAPW density of states results for 
CaMoO$_4$  were compared with the density of states  determined from a 
completely independent calculational method --- namely the PAW 
technique.\cite{holzwarth:1997,blochl:1994}   The two density of states plots 
were {\em{virtually indistinguishable}}  over a 66~eV range of upper core, 
valence, and conduction band states, providing further evidence of the 
convergence of the one-electron energies.



Both scalar-relativistic and spin-orbit contributions were calculated self-
consistently using the {\bf{WIEN97}}\cite{wien} code.  In each self-consistency 
iteration, the  scalar relativistic Hamiltonian was diagonalized in the usual 
LAPW representation. The full relativistic Hamiltonian was then evaluated and 
diagonalized in the basis of the scalar relativistic eigenstates, with the spin-
orbit interaction approximated as a spherically symmetric contribution confined 
within each muffin-tin sphere. This approximation has been well studied and has 
been shown to work well for elements in the 5th row of the periodic 
table.\cite{singh:1994} It may, however cause greater errors for W and Pb which 
are in the 6th row of the periodic table.


%\input{results}
\section{Results}
\label{results}

Density functional theory\cite{hohenberg:1964,kohn:1965}  is designed to 
accurately determine the ground state total energy and electron density of many-
electron materials. The one-electron energies $E_{n{\bf{k}}}$ and their 
corresponding wavefunctions $\Psi_{n{\bf{k}}}({\bf{r}})$ are  variational 
parameters which are only qualitatively related to the measurable spectrum of 
the material.  For example, although it is tempting to interpret the energy gap 
as the energy difference between the bottom of the first unoccupied band and the 
top of the last occupied band, density functional theory is notorious for 
underestimating the measurable band gap of most materials.  Nevertheless, by 
focusing on  the comparison of the results for the four materials of this study 
we can obtain useful qualitative information about their ground state 
properties. 

\subsection{Densities of states}

The  densities  of states were calculated by approximating the delta function in 
energy with a Gaussian smearing function\cite{fu:1983} of the form:
\begin{equation}
N^x(E) \equiv \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}.  \label{gauss}
\end{equation}
The Gaussian smearing parameter was chosen to be $\sigma = 0.1$~eV. In Eq. 
(\ref{gauss}),
 $w_{{\bf{k}}}$ is the Brillouin zone sampling weight factor and  
$f^x_{n{\bf{k}}}$ is a weight factor associated with the state $n{\bf{k}}$.  
The 6 ${\bf{k}}$-point sampling of the Brillouin zone did a good job of 
approximating the occupied densities of states. However, for states having 
larger dispersion such as in the upper portion of the conduction band, the 
approximation deteriorated somewhat.
For the total density of states $N^T$, $f^T_{n{\bf{k}}}$ is equal to the 
degeneracy  (including spin degeneracy) of the state. For the partial density of 
states $N^a$, $f^a_{n{\bf{k}}}$ is equal to the degeneracy multiplied by the 
fractional charge  within the muffin tin sphere $a$ for the state.   In addition 
to the total density of states, there are 4 interesting choices for partial 
densities of states that will be presented below.

Figure \ref{tdos} compares the total densities of states ($N^T$) for the 4 
compounds within a 110~eV range, with the zero of energy taken at the top of the 
last occupied band.  The upper core states are labeled according to their 
dominant atomic character.   The similarities in the electronic structure of the 
four materials is quite apparent from this figure.  The one-electron energies of 
the
upper core states including the Pb $5p_{1/2,3/2}$, Pb $5d_{3/2,5/2}$, W $5s$, W 
$5p_{1/2,3/2}$, W $4f_{5/2,7/2}$, Mo $4s$, Mo $4p_{1/2,3/2}$, Ca $3s$, Ca $3p$, 
and O $2s$ states have very small relative chemical shifts in these materials.  
The O $2s$ states are among the upper core states forming a narrow band having a 
width of about 2~eV.
  In the Pb materials, this O $2s$ band peaks at -17~eV  and is  degenerate with 
the Pb $5d_{3/2}$ states but separated from the very narrow Pb $5d_{3/2}$ band 
at -14~eV. In the Ca materials, the O $2s$ band peaks at -16~eV and is well 
separated from other core states of the system.

Partial densities are presented on expanded scales in Figs. \ref{pdos}, 
\ref{sigdos}, \ref{cryfdos}, and \ref{ados}. The muffin-tin sphere radii used to 
calculate the weight factors for the partial densities of states are listed in 
Table \ref{lapwpar}.  In general, these radii are much smaller than typical 
atomic radii for these atoms and the total muffin-tin volume accounts for only 
10--12\% of the volume of the crystal.   Nevertheless, one can obtain 
qualitative information about the atomic origins of the valence and conduction 
band states of these materials.  

The partial densities of states based on charge within each muffin tin sphere 
($N^a$ for $a$ = Ca, Pb, Mo, W, or O) are presented for the valence and 
conduction bands in Fig. \ref{pdos}. 
In PbMoO$_4$ and PbWO$_4$ there is a narrow Pb contribution to the density of 
states, having a width of 0.5~eV and centered at 7.1~eV and 7.2~eV respectively, 
separated by more that 1~eV from the bottom of the main part of the valence 
bands.  These bands, due to the Pb $6s$ states, accommodate 4 electrons per unit 
cell.   Since the  Pb $6s$ wavefunctions are so diffuse, there is significant 
mixing with O and Mo or W states as will be discussed below.
 For all four materials, the main part of the valence bands accommodates 48 
electrons per unit cell with a band width of 5~eV for the calcium materials and 
5.5~eV for the tungsten materials. The shape of the density of states for these 
valence bands  has two main features. The lower portion of the bands has roughly 
equal contributions from O  and Mo  or W states per atom, while the upper 
portion contains states of primarily O  character.   If we had plotted the 
partial density of states per unit cell, the O contributions would be 4 times 
larger than the contributions from Ca, Pb, Mo, and W presented in Fig. 
\ref{pdos}. From this point of view, it is clear that O states dominate the 
character of these valence bands. In fact, these valence bands  accommodate the 
same number of electrons per unit cell (48) as if they were filled with  pure  O 
$2p$ states. The bottom of the conduction bands of these materials is dominated 
by Mo  and W  states. The calcium materials have additional conduction band 
contributions from the Ca  states at approximately 3-4~eV above the bottom of 
the conduction band. 


In order to investigate the nature of the valence and conduction bands  further, 
we analyzed the partial densities in three additional ways --- in terms of the 
$2p \sigma$ and $2p \pi$ contributions on the O sites (Fig. \ref{sigdos}), the 
crystal field split $4d$ or $5d$ contributions on the Mo  or W sites (Fig. 
\ref{cryfdos}), and in terms of the partial wave composition of the Ca and Pb 
contributions (Fig. \ref{ados}). 

On the oxygen sites, the valence band states are almost entirely described by 
atomic-like $2p$ wavefunctions.  The strong crystal field due to the nearby Mo 
or W ions splits the atomic $2p$ states into $\sigma$ and $\pi$ contributions as 
shown in the partial density of states plot of  Fig. \ref{sigdos}. From this 
figure, it is apparent that the $\sigma$-like contributions are weighted toward 
the bottom of the valence band and the top of the conduction band, while the 
$\pi$-like contributions are stronger at the top of the valence band and bottom 
of the conduction band.

The geometry in the vicinity of the Mo or W sites is approximately tetrahedral 
and the $4d$ or $5d$ states split into  $e$-like and $t_2$-like states.  The 
corresponding partial densities of states are plotted in  Fig. \ref{cryfdos}.     
From this result it is apparent that the bottom of the valence band has roughly 
equal contributions from both the  $e$-like and $t_2$-like states, while the top 
of the valence band has very little contribution from either symmetry.  The 
bottom of the conduction band, however, is dominated by $e$-like contributions 
while the upper conduction band is dominated by $t_2$-like  contributions.  Not 
included in this plot are the  Mo $5s$ and $5p$ and W $6s$ and $6p$ 
contributions, since these were found to be negligible.

Finally, it is  interesting to investigate the partial wave distribution of the 
Ca and Pb contributions to the valence and conduction bands.  Since these sites 
are 2.5 {\AA} or 2.6 {\AA}  away from the nearest neighbor O sites, the crystal 
field splittings for these states are very small.   The Ca $4s$, $4p$, and $3d$ 
and Pb $6s$, $6p$, and $d-$like contributions are shown in Fig.\ref{ados}. The 
vertical  scale has been expanded relative to that used in the previous figures 
in order to compensate for the small muffin tin used in the calculations. In 
fact, since the atomic or ionic wavefunctions corresponding to  Ca $4s$ and $4p$ 
states and Pb $6s$ and $6p$ states are very diffuse, even a larger muffin tin 
radius could not contain the majority of the charge associated with these 
states.    This figure shows that there is very little Ca $4s$ and $4p$ 
contributions to the valence and conduction band states between -6~eV and +8~eV.   
However, there is a significant contribution of the Ca $3d$ states above the Mo 
$4d$ and W $5d$ conduction band states.  In the Pb materials, the corresponding 
Pb $5d$ states are filled as discussed above so that there is no appreciable 
$d$-like density in the valence and conduction band region.   Although the Pb 
$6s$ states form a well-defined narrow band below the bottom of the valence 
band, there is some additional Pb $6s$ character throughout the valence bands of 
PbMoO$_4$ and PbWO$_4$.  Figure \ref{ados} also shows significant contribution 
of Pb $6p$ states throughout the conduction bands of these materials.

\subsection{Electronic band structure diagrams}

Band dispersions for the four materials are plotted along 3 symmetry directions 
within the body-centered tetragonal Brillouin zone  (Fig. \ref{BZ}) and 
presented in Figs. \ref{Cabands} and \ref{Pbbands}.   The shapes of the bands 
for CaMoO$_4$ and CaWO$_4$ are very similar to each other.  The valence band 
maxima and the conduction band minima are located at the $\Gamma$ point so that 
these are direct gap materials. Although the values of the band gaps calculated 
within density functional theory are known to be 
underestimated\cite{hybertsen:1987}, it is interesting to compare the gaps 
calculated for the two materials as listed in Table \ref{valband}. The 
dispersion of the valence bands is relatively small, with comparable dispersions 
along both the a- and c- directions. The lower part of the conduction band, 
which is composed primarily of e states associated with the Mo $4d$ or W $5d$ 
states, is separated by approximately 0.5~eV from the upper part of the 
conduction band formed from the t$_2$ states  of Mo or W and the $3d$ states of 
Ca.   

The shapes of  the bands for PbMoO$_4$ and PbWO$_4$ are very similar to each 
other, but are somewhat different from those of the Ca materials.   For the Pb 
materials, the band extrema are located away from the $\Gamma$ point. Within the 
limited region of the Brillouin zone studied for the dispersion plot shown in 
Fig. \ref{Pbbands}, we can make the following statements about the band extrema.  
The valence band has maxima in the $\Delta$ directions and slightly lower 
maximum in the $\Sigma$ directions.   The conduction band minima are located in 
the $\Sigma$ directions. The values of these indirect gaps are listed in Table 
\ref{valband}.   The direct gaps for both materials appear to lie in the 
$\Sigma$ direction and are approximately 0.1~eV larger than the minimum 
(indirect) band gaps listed in  Table  \ref{valband}.

The calculated  ordering of the  band gaps is given by PbWO$_4$ $<$ PbMoO$_4$ 
$<$ CaMoO$_4$ $<$ CaWO$_4$.  Experimental data for exciton reflectance peak, 
luminescence excitation threshold, and free-carrier threshold summarized in 
Table \ref{exptable} are consistent with this ordering.

\subsection{Electron density distributions}

In order to further understand the electronic states of these materials, we have 
constructed contour plots for their occupied and unoccupied states. Fig. 
\ref{cryst} shows a perspective diagram of a  PbWO$_4$ crystal together with a 
plane which contains two W--O bonds and also contains a Pb site. A contour plot 
of the valence electron density (including states within 5.5~eV of the valence 
band maximum, but not the narrow Pb $6s$ band) is also indicated on this plane.   
Most of the contours shown in this diagram are associated with the 4 atoms in 
the plane, although there are significant contributions from two O's at the top 
of the diagram which lie in front or in back of the plane.  

We have used the plane indicated in Fig. \ref{cryst} to construct a series of 
contour diagrams for various partitions of the  electronic charge densities of 
the four materials.   Since the plots for CaMoO$_4$ and CaWO$_4$ and for 
PbMoO$_4$ and PbWO$_4$  look so similar to each other, we present only the plots 
for CaMoO$_4$ and PbWO$_4$. These contour plots all have the same uniform 
contour level spacing for all of the plots, namely the contour levels are spaced 
at intervals of  0.2 e/\AA$^3$,  with the first contour level chosen to be  0.1 
e/\AA$^3$.

Fig. \ref{pwoden1} shows the electron density corresponding to the valence band 
states of PbWO$_4$.   The plot labeled   I   shows the contours corresponding to 
the narrow Pb $6s$ states below the valence band.   With the exception of the 
small region of high density near the Pb site, most of the contours shown in the 
diagram correspond to oscillations about the 0.1  
e/\AA$^3$ level around the Pb site and near the O site.  Plots labeled   II   
and   III   correspond to the lower and upper portions of the valence band, 
divided at the minimum in the density of states located 3.2~eV below the  
valence band maximum.  From these plots, it is evident that the lower bands (II) 
are dominated by $\sigma$ bonding states formed from the O $2p$ and the W  $5d$ 
states, while the upper bands (III) are dominated by $\pi$ bonding states formed 
from the O $2p$ with somewhat less density associated with the W $5d$ states. 
This result is consistent with the density of states analysis discussed above. 
The sum of the densities   II   and   III   correspond to the density  shown in 
the 3-dimensional plot  shown in Fig. \ref{cryst}. 


Fig. \ref{cmoden1} shows the electron density corresponding to the valence band 
states of CaMoO$_4$. Plots labeled   II   and   III   correspond to the lower 
and upper portions of the valence band, divided at the minimum in the density of 
states located 3.0~eV below the top of the valence band maximum.  These plots 
show very similar distributions to those of PbWO$_4$, showing the dominance of 
the O $2p\sigma$ states in the lower portion  and  the O $2p\pi$ states in the 
upper portion of the valence band.    

Figures \ref{pwoden2} and \ref{cmoden2} show the electron density distributions 
in the unoccupied bands of PbWO$_4$ and CaMoO$_4$, respectively.  The bottom 
portion of the conduction bands (labeled   IV) is dominated by states of $e$ 
symmetry about the tetrahedral site formed from the W $5d$ or Mo $4d$ states 
with a small antibonding contribution from the   O $2p\pi$ states.  The upper 
portion of the conduction bands (labeled   V) is dominated by states of $t_2$ 
symmetry formed from the W $5d$ or Mo $4d$ states with a small antibonding 
contribution from the   O $2p\pi$  and $2p\sigma$ states. For CaMoO$_4$, the 
upper most portion of the conduction band ($E > 6.7$~eV) is plotted separately 
(labeled   VI) showing the contributions from the Ca $3d$ states.

%\input{conclude}
\section{Discussion and conclusions}
\label{conclude}

The results presented here represent an accurate self-consistent evaluation of 
the Schr\"{o}dinger equation for the ground state of the system within the 
density functional formalism. The total energy of the system and the 
corresponding electron density are determined variationally.  However, the band 
structure and the corresponding densities of states  are only qualitatively 
related to the ``quasiparticle" states\cite{hybertsen:1987} which are accessible 
experimentally. Within this caveat, it is useful to relate the present results 
to some of the experimental and theoretical results in the literature.    For 
the purpose of the present  qualitative discussion, we first relate our results 
to the ligand field model used extensively in the literature and then present a 
tentative comparison with some spectroscopic results. 

\subsection{Ligand field model}


For the purpose of this analysis, it is helpful to visualize the formation of 
these materials in the following way.  Imagine that one could place the neutral 
atoms at the correct atomic positions in the crystal.  First allow the neutral 
spherical atoms to self-consistently transfer charge to produce spherical 
A$^{+\alpha}$, B$^{+\beta}$, and O$^{-\gamma}$ ions as described in Sec. 
\ref{strct}. If the valence bands were formed only from  O $2p$ states, the band 
filling would indicate that $\gamma = 2$, $\beta = 6$, and     $\alpha = 2$. Now 
apply degenerate perturbation theory to describe the splittings of the valence 
states of the spherical ions.\cite{ballhausen:1962}  The  splitting of the O 
$2p$ states is dominated by the Coulomb attraction of the  electron to the 
nearest neighbor  B$^{+\beta}$ ion. The  first order $\sigma$ and $\pi$ energy 
shifts are given, respectively,  by:

\begin{equation}
E^1_{p\sigma} = - \frac{2 \beta e^2 <r_{p}^2>}{5R_{BO}^3} 
\:\:\:{\rm{and}}\:\:\:E^1_{p\pi} = + \frac{ \beta e^2 <r_{p}^2>}{5R_{BO}^3}.
\end{equation}
Here, $R_{BO}$ is the Mo--O or W--O bond length  and $<r_{p}^2>$ is the 
expectation value of the squared radius of the O $2p$ orbital. Using the 
estimated values of the parameters $\beta = 6$, $R_{BO} = 3.4$~Bohr, and 
$<r_{p}^2> = 2$~Bohr$^2$ (estimated from numerical integration using neutral O 
$2p$ wavefunctions), we find $E^1_{p\pi}-E^1_{p\sigma} \approx 5$~eV.   


The splitting of the $4d$ or $5d$ states of the B ion is dominated by the 
Coulomb repulsion of the electron from the four tetrahedral nearest neighbor 
O$^{-\gamma}$ ions.  The first order energy shift for states of   ``t$_2$" and 
``e" symmetries are given, respectively, by:

\begin{equation}
E^1_{t_2} = - \frac{8 \gamma e^2 
<r_d^4>}{27R_{BO}^5}\:\:\:{\rm{and}}\:\:\:E^1_{e} =  \frac{4 \gamma e^2 
<r_d^4>}{9R_{BO}^5}.
\end{equation}
Here $<r_d^4>$ is the expectation value of the fourth power of the radius of the 
$d$ orbitals of the B ions. Using the estimated values of the parameters $\gamma 
= 2$, $R_{BO} = 3.4$~Bohr, and $<r_{d}^4> = 9$~Bohr$^4$ (the average value from 
numerical integration using Mo$^{+6}$ $4d$ or W$^{+6}$  $5d$ wavefunctions), we 
estimate $E^1_{e}-E^1_{t_2} \approx 1$~eV.  

Finally, following the approach of Ballhausen and Liehr\cite{ballhausen:1958} we 
allow the t$_2$ and e  orbitals on the B sites to hybridize with   linear 
combinations of $p \sigma$ and $p \pi$ ligand orbitals, symmetrized for the 
tetrahedral geometry, forming bonding and anti-bonding combinations which 
approximate the eigenstates of the BO$_4^{-\alpha}$ ions.  The 4 ligand $p 
\sigma$ orbitals are compatible with the tetrahedral representation of symmetry 
a$_1$ and t$_2$, while the 8 ligand $p \pi$ orbitals are compatible with the 
tetrahedral representation of symmetry t$_1$, t$_2$, and e.  From our partial 
densities of states analyses, we can roughly guess the ordering which would 
correspond to the molecular orbitals for the BO$_4^{-\alpha}$ clusters.    These 
are shown in Fig. \ref{diagram}, where we have also indicated in the same 
diagram the  positions of the atomic-like bands that we have identified in our 
calculations --- Pb $6s$ states and the Ca $3d$ states.   This qualitative 
picture (apart from the ordering of the lower molecular orbital states) is 
consistent with the early semi-empirical  molecular orbital calculations for 
WO$_4^{-2}$ and  MoO$_4^{-2}$ by several 
authors.\cite{ballhausen:1958,dahl:1968,kebabcioglu:1971} In particular, the 
early calculations recognized that the top occupied state has t$_1$ symmetry 
formed from the O $2p\pi$ states and the lowest unoccupied state has e symmetry 
formed from an antibonding combination of Mo $4d$ e or W $5d$ e state with O 
$2p\pi$ states. In fact, this model has been used more generally to describe the  
class of tetrahedral BO$_4^{-\alpha}$ anions.\cite{mcglynn:1981}  

The fact that there is hybridization between the $d$ orbitals of the 
B$^{+\beta}$ ions and the O $2p$ orbitals means that there is some covalent 
character to the bonding.   From the ratio of occupied  to unoccupied charge 
density within the muffin tin spheres of the B$^{+\beta}$ ions, we estimate 
$\beta \approx$ 3 for all four materials.   

\subsection{Comparison with optical spectra}


A  large portion of the interest in these materials is due to their intrinsic 
luminescence spectra.\cite{kroger:1948,lecoq:1996}   While a detailed 
understanding of the luminescence processes goes far beyond the scope of the 
present study, we can make some qualitative comments.   For each of the four 
materials, the intrinsic luminescence spectrum has been interpreted in terms of 
a molecular diagram such as  Fig. \ref{diagram}, similar to that given by 
Ballhausen and Liehr\cite{ballhausen:1958}.  The ground state of the system 
corresponds to filling all   one-electron states below the band gap, resulting 
in a many-electron state of $^1A_1$ symmetry. The lowest excited  states involve 
one hole in the $t_1$ (primarily O 2p$\pi$) states and one electron in the $e$ 
(primarily Mo 4d or W 5d) states, corresponding to the many-electron states 
$^1T_1$, $^3T_1$, $^1T_2$, and $^3T_2$.  Of these states only the transition 
$^1A_1$ $\leftrightarrow$ $^1T_2$ is a dipole allowed transition. 
However, it is the lower $^3T_1$ or $^3T_2$ states which were shown to  account 
for the intrinsic luminescence by a spin-forbidden transition to the ground 
$^1A_1$ state in optically detected electron paramagnetic resonance experiments 
on CaMoO$_4$ and PbMoO$_4$.\cite{barendswaard:1986,vantol:1996}


%\input{richard}

	While the molecular orbital model describes the general features of the 
excitation and luminescence of these materials, a more accurate  description 
corresponds to a Frenkel-type 
exciton\cite{grasser:1975,belsky:1995,kolobanov:1996,nedelko:1996,murk:1997}  
having energies within the band gap of the one-electron states.\cite{knox:1963}    
Since the present study has calculated the one-electron states only, we have no 
detailed information about the excitons.  However, having information about the 
band states, we can comment on the qualitative differences between the excitons 
in the Ca and Pb materials.  In particular, several 
authors\cite{belsky:1995,nedelko:1996,murk:1997}  have suggested that the Pb 
$6s$ and $6p$ states contribute in a significant way to the excitation and 
luminescence spectra of PbWO$_4$.   One experimental impetus for this suggestion 
is that an exciton peak is found at considerably lower energy in PbWO$_4$  
(4.2~eV) than in CaWO$_4$ (5.9~eV), despite the fact that both crystals have the 
same WO$_4^{-\alpha}$ complex anion and the same  scheelite structure. 

Thus, Belsky {\em{et al.}} suggested  that the first sharp reflectance peak in 
PbWO$_4$  at 4.17~eV is associated with Pb $6s \rightarrow 6p$ 
transitions.\cite{belsky:1995}    In advance of an available band structure 
calculation, they speculated that a narrow Pb$^{+2}$ $6p$ sub-band at the bottom 
of the conduction band should be expected.   However, the partial density of 
states ($N^a(E)$)  calculations presented in figures \ref{pdos} and \ref{ados} 
do not exhibit a narrow Pb  peak at the bottom of the conduction band nor at the 
top of the valence band.   Rather, a nearly uniform density of Pb  states is 
distributed throughout the conduction and valence bands.  The only distinct 
narrow Pb $6s$ band is located {\em{below}} the valence bands.   We will return 
later to the discussion of partial Pb character in the lowest exciton.     
Comparing second ionization potentials of Ca and Pb,  M\"{u}rk {\em{et al.}} 
suggested another exciton model for PbWO$_4$ in which   a hole residing mainly 
on the tungstate group is paired with an electron described as being in  the 
Pb$^+$ state.\cite{murk:1997}   

	The peak energies of intrinsic recombination luminescence in the four 
scheelite materials under consideration are listed in Table \ref{exptable}.  The 
intrinsic luminescence energy $E_{lum}$ is about 2.9~eV in both tungstates and 
about 2.3~eV in both molybdates, consistent with the view that the intrinsic 
luminescence mainly involves the tungstate (molybdate) complex anion.    We have 
already seen that the exciton energy depends strongly on whether Ca or Pb is the 
cation. In contrast, the luminescence energy is almost independent of this 
factor.\cite{ODEPR}     The luminescence peaks are broad, having full width at 
half maximum (FWHM) around 0.7~eV at temperature T = 295~K.  The intrinsic 
luminescence energy is Stokes-shifted about 3~eV below the exciton reflectance  
peak in CaWO$_4$, and about 1.3~eV below the exciton in PbWO$_4$.    Since the 
2.9~eV luminescence in tungstates has been satisfactorily identified as 
intrinsic recombination luminescence, the large Stokes shift and broad band 
imply that excitons self-trap before emitting luminescence 
photons.\cite{nedelko:1996,nagirnyi:1997,williams:1996}    This conclusion  is 
further supported by the Urbach slope parameter $\sigma_0$ =  0.9 measured for 
PbMoO$_4$ by van Loo.\cite{vanloo:1975}   According to Toyozawa's 
theory,\cite{toyozawa:1964,toyozawa:1982} $\sigma_0  < 1$ implies an exciton-
phonon coupling constant strong enough to induce self-trapping.  This has been 
confirmed by observations reported for a wide variety of 
crystals.\cite{williams:1996}   Because of the significant local lattice 
distortion that accompanies self-trapping, the states giving luminescence can be 
described reasonably well as molecular orbitals of the single distorted atomic 
group on which the exciton localizes.   Van der Waals and co-workers have used 
optically detected electron paramagnetic resonance  to study the   excited state  
responsible for the intrinsic green luminescence in 
CaMoO$_4$\cite{barendswaard:1986}   and PbMoO$_4$.\cite{vantol:1996}  The 
luminescence was found to originate from a spin triplet excited state localized 
on a MoO$_4^{-2}$ ion distorted by a Jahn-Teller instability.    In PbWO$_4$, 
the complex shape of the spectrum of the blue luminescence band was attribted by 
Polak {\em{et al.}}\cite{polak:1997} to Jahn-Teller splitting of the nearly 
degenerate localized excited states of a WO$_4^{-2}$ complex anion.   In 
contrast, it is the  delocalized exciton band states which determine the near-
edge excitation spectrum.
	
The band gaps of PbWO$_4$ and CaWO$_4$ can be assigned on the basis of 
excitation spectra of thermoluminescence\cite{murk:1997,nagirnyi:1997} or 
phosphorescence.\cite{gurvich:1971}  Excitation of thermoluminescence or trap-
mediated recombination phosphorescence implies charge separation followed by 
capture in traps.   Such excitation spectra may be considered to yield 
information similar to photoconductivity spectra, without the need to apply an 
external field.   We have chosen the threshold for production of free carriers, 
$E_{fc}$ , as the beginning of the steep rising edge of the thermoluminescence 
excitation spectrum in PbWO$_4$ single crystals measured at 80~K by M\"{u}rk 
{\rm{et al.}}\cite{murk:1997}  It is listed as $E_{fc}$ = 4.7~eV in Table 
\ref{exptable}.     Estimating the free-carrier threshold for a single crystal 
of CaWO$_4$ at a roughly corresponding point on its thermoluminescence 
excitation spectrum\cite{murk:1997},  we find $E_{fc} \approx  7$~eV.   Gurvich 
{\em{et al.}}\cite{gurvich:1971} assigned $E_{fc} \approx 6.5$~eV for calcined 
powder samples of CaWO$_4$ at 295~K and Nagirnyi {\em{et 
al.}}\cite{nagirnyi:1997} recently found  $E_{fc} \approx 6.8$~eV for single 
crystals of CaWO$_4$ at 8~K.   We  will  adopt $E_{fc} \approx 6.8$~eV as the  
band gap for data comparisons below.           
  	
We have made use of $E_{fc}$ in order to overlay the valence band density of 
states from the present calculation with spectra of reflectivity and 
luminescence excitation near the interband edge in PbWO$_4$ (Fig. \ref{pwoexp}) 
and in CaWO$_4$ (Fig. \ref{cwoexp}).    Although it is not expected that optical 
spectra should be directly comparable to the valence band density of states in 
general,  our rationale for the present comparison is that the main differences 
in band structure which we want to compare between CaWO$_4$ and PbWO$_4$  occur 
near the top of the valence band, whereas the conduction band edge is similar in 
both materials.  Furthermore, as long as we confine attention to transitions 
near the interband edge, overlapping optical transitions should not complicate 
the spectra so much as to obscure basic valence structure.  

	With these caveats in mind, we have aligned the top edge of the valence 
band density of states, $N^T(E)$, with the photon energy ($h \nu$) corresponding 
to the experimental band gap $E_{fc}$ , such that deeper valence states 
correspond to the higher photon energies needed to excite them.    The implied 
assumption is that the important spectroscopic features near the interband edge 
represent mainly transitions from successively deeper valence states to just the 
conduction band edge, and that a rough comparison to the density of initial 
states may be made without  treating transition matrix elements explicitly.  In 
future work, we will calculate optical constants from the band structure for a 
more accurate comparison over a larger photon energy range.   
     
We focus now on the PbWO$_4$ reflectance and luminescence excitation spectra  
measured at liquid helium temperature by Kolabanov {\em{et 
al.}}\cite{kolobanov:1996}  and reproduced in Fig. \ref{pwoexp} with the aligned 
and overlaid valence band density of states.\cite{apology}       It is obvious 
that the lowest-energy peak in reflectivity lies within the band gap, in  
agreement with the previous identification of the 4.2~eV peak as an  
exciton.\cite{murk:1997,kolobanov:1996,belsky:1995,grasser:1975,nedelko:1996}     
The exciton binding energy is 0.5~eV if the band gap is $E_{fc}$ = 4.7~eV.   The 
rise of reflectivity on the high energy side of the exciton line coincides with 
the onset of interband transitions out of the highest valence states in the 
overlaid $N^T(E)$, starting at 4.7~eV on the photon energy scale.  The 
luminescence excitation spectrum, which should have some correspondence to the 
absorption spectrum in this region, exhibits a shoulder roughly matching the 
shoulder at the top of the calculated valence $N^T(E)$.  The strong O $2p$ peaks 
in the valence $N^T(E)$ spectrum rise sharply at about 6~eV on the aligned 
photon-energy scale, just where the experimental excitation spectrum turns 
sharply upward.   The deeper group of valence bands accessible by photons above 
8~eV  seem a little mis-aligned with the valley at 8.6~eV and the peak at 10~eV 
in the excitation spectrum.   This may indicate that we are getting too far from 
the interband edge for this simplistic comparison to valence structure to be 
faithful.  Another reason for the apparent mis-alignment with the 10-eV peak in 
the excitation spectrum is that at least parts of the peaks at 10~eV and 15~eV 
probably correspond to excitation 
multiplication\cite{luschik:1996,nagirnyi:1997}  at photon energies 2 and 3 
times the band gap. We note that the isolated narrow Pb $6s$ peak in the 
calculated $N^T(E)$ at $h \nu$ = 12~eV ($E$ = -7.2~eV) aligns well with the bump 
in the excitation spectrum at the corresponding energy.  

Similar comparisons can be made for CaWO$_4$  reflectivity and the overlaid 
valence $N^T(E)$  in Fig. 16.    With the band gap taken to be $E_{fc} \approx 
6.8$~eV as discussed above, the exciton reflectance peak at 5.9~eV implies  a 
0.9~eV binding energy of the exciton.    Furthermore, we find reasonable 
qualitative correspondence  between both the energy location and width of the 
valence $N^T(E)$ structure aligned in this way and the region of main 
reflectivity structure above the exciton, measured by Grasser {\em{et 
al.}}.\cite{grasser:1975} 

The shape of the optical spectra near the interband edge in PbWO$_4$  is 
suggestive of a weaker absorption threshold followed at higher energy by a 
stronger threshold in stair-step fashion, whereas in CaWO$_4$ there appears to 
be  a  main absorption threshold with which the exciton is associated.   The 
``strong" absorption thresholds in both materials involve direct transitions at 
$\Gamma$ starting from valence states with overwhelming O $2p$ character.   The 
weaker threshold and associated  exciton in PbWO$_4$  involve the topmost 
valence band which disperses upward away from $\Gamma$, extending about 1~eV 
higher than the dominant O $2p$ bands at $\Gamma$.  According to Fig. 
\ref{Pbbands} and Table \ref{valband}, there is a minimum-energy direct 
transition at about 0.5 along $\Sigma$ and another direct transition 0.19~eV 
higher at about 0.5 along $\Delta$. In this regard, it is interesting to note 
that the 4.2~eV reflectance peak in PbWO$_4$ was resolved as a doublet with a 
0.1~eV splitting at 4.2~K by Kolobanov {\em{et al.}}\cite{kolobanov:1996}.   An 
indirect gap from $\Sigma$ to $\Delta$ lies about 0.1~eV lower in energy 
according to the present calculations.  Since the exciton luminescence is 
emitted after self-trapping with 1.3 eV relaxation, luminescence spectra are 
unlikely to indicate whether an indirect edge lies 0.1~eV below the minimum 
direct edge.  We know of no data yet available to support or rule out a minimum-
energy indirect  edge slightly below the direct edge.  

We have seen in Fig. \ref{pdos} that there is no peak in the partial density of 
states corresponding to Pb $6s$ orbitals ($N^{Pb}(E)$) at the energy of the top-
most valence band.  However, because  $N^O(E)$ dips sharply above the energy of 
the $\Gamma$ critical point, the relative importance  of the nearly constant  
$N^{Pb}(E)$  increases significantly in the top-most band.  With respect to  
some of the earlier discussions on the involvement of Pb 
states,\cite{murk:1997,belsky:1995,nedelko:1996}  the top-most valence band may 
be described as having Pb $6s$ character with a strong admixture of O $2p$ 
character.  Even though the distinct Pb $6s$ band actually lies below the 
valence bands at about -7.2~eV, we  conclude that the presence of Pb (rather 
than Ca) in the PbWO$_4$ scheelite-type material also causes a band with hybrid 
Pb $6s$ and O $2p$ character to disperse upward from $\Gamma$ and establish the 
minimum gap.   However,  a ``Pb $6s \rightarrow 6p$ exciton" is not supported 
because no Pb$^{+2}$  $6p$ discrete band was found at the bottom of the 
conduction band.   The minimum-energy direct exciton in PbWO$_4$ should involve 
charge transfer from both lead and oxygen to mainly tungsten.  In CaWO$_4$ and 
also in PbWO$_4$ at its second absorption threshold, the  direct exciton at 
$\Gamma$ involves charge transfer mainly from oxygen to tungsten. 


 

\subsection{Summary}

For the four scheelite materials studied in this work,
the results of the partial densities of states analyses (Figs. \ref{pdos}, 
\ref{sigdos}, \ref{cryfdos}, and \ref{ados}) and electron density contour plots 
(Figs. \ref{pwoden1}, \ref{cmoden1}, \ref{pwoden2}, and \ref{cmoden2}) together 
with the band structure plots (Figs. \ref{Cabands} and \ref{Pbbands}) are in 
qualitative agreement with the energy level diagram presented in Fig. 
\ref{diagram}. Namely, the main features of the valence and conduction band 
states near the energy band gap are well approximated by the molecular 
orbitals\cite{ballhausen:1958} of the tetrahedral molybdate or tungstate ions.  
The valence band accommodates 48 electrons per unit cell in one-to-one 
correspondence with the number of O $2p$ states.  The shape of the valence band 
density of states has two main peaks, with the upper peak primarily composed of 
O $2p \pi$ states and the lower peak composed of a combination of O $2p$ 
$\sigma$ and $\pi$ states together with contributions from the $d$ states of Mo 
or W. The lowest conduction band is largely formed from $e$- symmetry states 
derived from $d$ states of Mo or W and, in the Ca materials, is separated from 
the upper conduction band by a small gap. The upper conduction band is largely 
formed from $t_2$- symmetry states derived from $d$ states of Mo or W. In the Pb 
materials,  a narrow band of filled states   is formed 1~eV below the valence 
bands as a result of the Pb $6s$ states.  In the Ca materials, the $3d$ orbitals 
form a narrow band above the molybdate or tungstate conduction bands. 

In addition to these general features, there is significant hybridization of the 
Pb $6s$ and $6p$ states throughout the valence and conduction bands of PbMoO$_4$ 
and PbWO$_4$, so that the band dispersions and splittings in the Pb materials 
are different from those of the  Ca materials. In particular, the top-most 
valence band acquires significant Pb $6s$ character along with O $2p$ character 
and disperses upward away from the $\Gamma$ point of the Brillouin zone. We have 
analyzed some of the optical properties of these materials as summarized in 
Table \ref{exptable} and have made a rough comparison of our densities of states 
with reflectance and excitation yield data in Figs. \ref{pwoexp} and 
\ref{cwoexp}.  In the future, we hope to use the present results as a basis for 
a more thorough analysis of the optical properties of these materials.


\acknowledgements
This project was supported by NSF grants No. DMR-9403009, DMR-9706575, and DMR-
9510297.  We would like to thank  Dimitri Alov for introducing us to the 
scheelite materials, providing samples for the experimental work associated with 
this project, and for helpful discussions. We thank Martin Nikl and Vitali 
Nagirnyi for helpful comments on the manuscript, and also  thank  Eric Carlson, 
Abdessadek Lachgar, and Ekaterina Anokhina for symmetry and crystallographic 
advice and Peter Blaha for help with the WIEN97 computer program.



% now the references. delete or change fake bibitem. delete next three
%   lines and directly read in your .bbl file if you use bibtex.
%\begin{references}
%\bibitem{tag} Fake bibitem.
%\end{references}

%\input{bibtexbib}
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\bibitem{suryanarayana:1972}
S.~V. Suryanarayana and V.~T. Deshpande, Current Science {\bf 41},  837
  (1972).

\bibitem{moreau:1996}
J.~M. Moreau, Ph. Galez, J.~P. Peigneux, and M.~V. Korzhik, Journal of Alloys
  and Compounds {\bf 238},  46  (1996).

\bibitem{alov:unpublished}
Dimitri Alov, private communication.

\bibitem{saito:1996a}
Nobuhiro Saito, Noriyuki Sonoyama, and Tadayoshi Sakata, Bull. Chem. Soc. Jpn.
  {\bf 69},  2191  (1996).

\bibitem{kobayashi:1993}
Masaaki Kobayashi, Mitsuru Ishii, Yoshiyuki Usuki, and Hiroshi Yahagi, Nuclear
  Instruments and Methods in Physics Research A {\bf 333},  429  (1993).

\bibitem{nikl:1996}
M. Nikl, K. Nitsch, K. Polak, E. Mihokova, I. Dafinei, E. Auffray, P. Lecoq, P.
  Reiche, R. Uecker, and G.~P. Pazzi, Phys. Stat. Sol. (b) {\bf 195},  311
  (1996).

\bibitem{groenink:1980}
J.~A. Groenink and G. Blasse, Journal of Solid State Chemistry {\bf 32},  9
  (1980).

\bibitem{hemphill:1988}
William~R. Hemphill, R.~Michael Tyson, and Arnold~F. Theisen, Economic Geology
  {\bf 83},  637  (1988).

\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}
%\input{figures}
\begin{figure}
\caption{Perspective drawing of the  structure of PbWO$_4$ indicating the a and 
c axes.  The atomic positions are indicated by spheres. The largest spheres 
(with no bonds) are the Pb sites; the middle size spheres at the W sites and the 
smallest spheres at the O sites are connected along nearest neighbor bonds.   
The shaded plane passes through two W--O bonds and one of the Pb sites.  Contour 
levels corresponding to the occupied electron density in the valence band are 
indicated on this plane with lowest contour level at 0.1~e/\AA$^3$ and with a 
spacing of 0.25~e/\AA$^3$ between contours.}
\label{cryst}
\end{figure}



\begin{figure}
\caption{Plot of total densities of states per unit cell for the four ABO$_4$ 
scheelite materials, evaluated using Eq. (\ref{gauss}) with $\sigma=0.1$~eV 
including upper core, valence, and conduction band states.   The zero of energy 
is taken at the top of the last occupied states.   The upper core states are 
labeled according to their dominant atomic behavior.}
\label{tdos}
\end{figure}

\begin{figure}
\caption{Atomic partial densities of states per muffin tin sphere for the four 
ABO$_4$ scheelite materials, evaluated as described in Fig. \ref{tdos}. The 
partial densities were weighted by the charges within a muffin tin sphere for Ca 
or Pb (solid line), Mo or W (dashed line) and O  (dotted line).}
\label{pdos}
\end{figure}

\begin{figure}
\caption{Crystal field split O $2p$ partial densities of states per muffin tin 
sphere for the four ABO$_4$ scheelite materials, evaluated as described in Fig. 
\ref{tdos}.  The partial densities were weighted by the  $\sigma$-like (full 
line) and $\pi$-like (dotted line) charges within each O muffin tin sphere.}
\label{sigdos}
\end{figure}



\begin{figure}
\caption{Crystal field split $4d$ and $5d$ partial densities of states per 
muffin tin sphere for the four ABO$_4$ scheelite materials, evaluated as 
described in Fig. \ref{tdos}. The partial densities were weighted by the  $e$-
like (full line) and $t_2$-like (dotted line) charges within a Mo or W muffin 
tin sphere.}
\label{cryfdos}
\end{figure}

\begin{figure}
\caption{Ca and Pb atomic orbital  partial densities of states for the four 
ABO$_4$ scheelite materials, evaluated as described in Fig. \ref{tdos}. The 
partial densities were weighted by the  $s$-like (full line), $p$-like (dashed 
line), and $d$-like (dotted line) charges within each Ca or Pb muffin tin 
sphere.}
\label{ados}
\end{figure}

\begin{figure}
\caption[]{Diagram of the Brillouin zone for a body-centered tetragonal crystal 
structure, based on Ref. \onlinecite{yamagami:1992}.}
\label{BZ}
\end{figure}


\begin{figure}
\caption{Band structure diagram for CaMoO$_4$ and CaWO$_4$ plotted in the range 
of 6~eV below and 10~eV above the valence band maximum (taken as the 0 of 
energy). The bands are plotted along the ${\bf{\hat{z}}}$ (c) axis from $\Gamma$ 
to Z and within the a plane from its boundary at $\frac{\pi}{a}(1 + \left( 
\frac{a}{c} \right)^2)$  along the $\Sigma$ direction to the $\Gamma$ point and 
from the $\Gamma$ point along the $\Delta$ direction to the X point, where the 
labels correspond to Fig. \ref{BZ}.}
\label{Cabands}
\end{figure}

\begin{figure}
\caption{Band structure diagram for PbMoO$_4$ and PbWO$_4$ plotted in the  same 
energy range and along the same Brillouin zone directions as
in Fig. \ref{Cabands}.}
\label{Pbbands}
\end{figure}

\begin{figure}
\caption{Contour plot of the electronic charge density associated with the 
valence band of PbWO$_4$, showing separately the narrow 0.5~eV band associated 
with the Pb $6s$ states which is separated from the valence band by 1.5~eV 
(labeled  I), the bottom 2.3~eV of the valence bands (labeled   II), and the top 
3.2~eV of the valence bands (labeled   III).  The plane of the contour plot is 
the same plane  as shown in Fig. \ref{cryst}, and passes through four atomic 
sites: 2 O sites, 1~W site, and 1~Pb site.
 The contour levels are uniformly spaced starting at 0.1~e/\AA$^3$ with  a 
spacing of 0.2~e/\AA$^3$ between contours.}
\label{pwoden1}
\end{figure}




\begin{figure}
\caption{Contour plot of the electronic charge density associated with the 
valence band of CaMoO$_4$, showing separately the bottom 1.9~eV (labeled   II) 
and the top 3.0~eV (labeled   III) portions.  The plane of the contour plot and 
the contour levels are the same as in Fig. \ref{pwoden1}.}
\label{cmoden1}
\end{figure}

\begin{figure}
\caption{Contour plot of the electronic charge density associated with the 
conduction bands of PbWO$_4$, showing separately the lower 2.1~eV portion (IV)  
and the next higher 3.4~eV portion (V) of  the  conduction band.  The plane of 
the contour plot and the contour levels are the same as in Fig. \ref{pwoden1}, 
with the density calculated from full occupancy of these states.}
\label{pwoden2}
\end{figure}



\begin{figure}
\caption{Contour plot of the electronic charge density associated with the 
conduction bands of CaMoO$_4$, showing separately the lower  conduction band 
having a width of 1.2~eV (IV), the bottom 1.6 eV of the upper conduction band 
(which is 0.9~eV above the lower  conduction band) (V), and the next 2.8~eV of 
the upper conduction band (VI).  The plane of the contour plot and the contour 
levels are the same as in Fig. \ref{pwoden1}, with the density calculated from 
full occupancy of these states.}
\label{cmoden2}
\end{figure}

\begin{figure}
\caption[]{Schematic diagram of the crystal field splitting and hybridization of 
the molecular orbitals of a tetrahedral WO$_4^{- \alpha}$ 
(or MoO$_4^{- \alpha}$) cluster using the notation of Ref. 
\onlinecite{ballhausen:1958}, with   ``*" indicating anti-bonding (unoccupied) 
states.  The numbers in parentheses indicate the degeneracy (including spin 
degeneracy) of each cluster state.  The ordering of the molecular orbitals in 
the diagram approximates  the partial density of states analyses. The shaded 
boxes are included to emphasize the fact that  the discrete states are broadened 
by neighboring cluster interactions in the solid material. The relative 
positions of the Pb $6s$ states (for PbWO$_4$ and PbMoO$_4$) and Ca $3d$ states 
(for CaWO$_4$ and CaMoO$_4$) are also indicated on this diagram.}
\label{diagram}
\end{figure}


\begin{figure}
\caption[]{Experimental spectra of reflectance {[}R($h \nu$){]} and excitation 
yield of intrinsic luminescence {[}Y($h \nu$){]} measured for PbWO$_4$ at liquid 
helium temperature in Ref. \onlinecite{kolobanov:1996} are overlaid with  the 
calculated total density of  states {[}$N^T(E)${]}.   The experimental free 
carrier excitation threshold $E_{fc} =$ 4.7~eV, as discussed in the text, is 
used to align the top ($E = 0$~eV) of the valence density of states with the 
photon energy scale such that $h \nu$ is the energy from each plotted valence 
state to the conduction band edge.}
\label{pwoexp}
\end{figure}

\begin{figure}
\caption[]{Reflectance spectra {[}R($h \nu$){]} measured for CaWO$_4$  in Ref. 
\onlinecite{grasser:1975} are overlaid with  the calculated total density of  
states {[}$N^T(E)${]}.   The broken and solid reflectance spectra were measured 
at 77~K and 295~K, respectively.   The experimental free carrier excitation 
threshold $E_{fc} \approx$ 6.8~eV, as discussed in the text, is used to align 
the top ($E = 0$~eV ) of the valence density of states with the photon energy 
scale such that $h \nu$ is the energy from each plotted valence state to the 
conduction band edge. }
\label{cwoexp}
\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}

%\input{strctable}
\begin{table}
 \caption{Some representative experimental data for ABO$_4$ scheelite crystals 
at room temperature. Quoted values refer to the conventional unit cell structure 
with A sites at $\pm(0,\frac{1}{4},\frac{5}{8})$; B sites at  
$\pm(0,\frac{1}{4},\frac{1}{8})$; and O sites at $\pm(x,y,z)$, $\pm(\frac{1}{2}-
x,-y,\frac{1}{2}+z)$, $\pm(-\frac{1}{4}-y,\frac{1}{4}+x,\frac{1}{4}+z)$, and 
$\pm(-\frac{1}{4}+y,-\frac{1}{4}-x,-\frac{1}{4}+z)$ , where 12 additional 
lattice sites are generated by adding $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$. 
Some references use the alternative O site parameters $x \rightarrow 
\frac{1}{4}-y$, $y \rightarrow \frac{1}{4}-x$, and $z \rightarrow \frac{1}{4}-
z$. Still other references choose the origin of the unit cell at the site of a B 
atom with the corresponding O positions described with  parameters $x 
\rightarrow x$, $y \rightarrow \frac{1}{4}-y$, and $z \rightarrow \frac{1}{8}-
z$. The experimental methods are abbreviated with ``XP" for X-ray powder, ``XC" 
for X-ray single crystal, and ``N" for neutron diffraction. The starred rows 
indicate the parameters used in the present calculations. }
 \label{strctpar}
 \begin{tabular}{||c|c|c|c|l|l|l|l|l|l||}
Material & Year& Ref. &Method&$\: a$ (\AA) &$\: c$ (\AA) &$\:\: c/a$ &$\:\:\: x$ 
&$\:\:\: y$ &$\:\:\: z$ \\ \hline
CaMoO$_4$& 1943&[\onlinecite{sillen:1943,sillen} ]&XP & 
5.213&11.426&2.192&0.25&0.10&0.05\\
&1971&[\onlinecite{gurmen:1971}]&N&5.226&11.43&2.187&0.2428&0.1035&0.0424\\
&1972&[\onlinecite{sleight:1972}] &XP& 5.2256 &11.434&2.188&&&\\
&$^*$1985&[\onlinecite{hazen:1985} ]&XC& 
5.222&11.425&2.188&0.2431&0.1010&0.0411\\ \hline

CaWO$_4$& 
1943&[\onlinecite{sillen:1943,sillen}]&XP&5.230&11.348&2.170&0.25&0.10&0.05\\
&1964&[\onlinecite{zalkin:1964}]&XC& 5.234&11.376&2.170&0.2415&0.0996&0.0389\\
&1964&[\onlinecite{kay:1964}]&N& &&&0.2413&0.0989&0.0389\\
&1972&[\onlinecite{deshpande:1972}]&XP&5.2437&11.3754&2.169&&&\\
&1972&[\onlinecite{sleight:1972} ]&XP& 5.2419&11.376&2.170&&&\\
&$^*$1985&[\onlinecite{hazen:1985}]&XC & 
5.2429&11.3737&2.169&0.2414&0.0993&0.0394\\
&1989&[\onlinecite{blanchard:1989} ]&XP&5.24294&11.373&2.169&&&\\ \hline

PbMoO$_4$&1943&[\onlinecite{sillen:1943,sillen}]&XP&5.424&12.076&2.226&0.25&0.12
&0.05\\
&$^*$1965&[\onlinecite{leciejewicz:1965}]&N&5.4312&12.1065&2.229&0.2352&0.1143&0
.0439\\
&1972&[\onlinecite{suryanarayana:1972}]&XP&5.4360&12.1107&2.228&&&\\
&1972&[\onlinecite{sleight:1972}]&XP& 5.4355 &12.108&2.228&&&\\
&1985&[\onlinecite{hazen:1985}] &XC & 5.4351&12.1056&2.227&&&\\ \hline

PbWO$_4$&1943&[\onlinecite{sillen:1943,sillen}]&XP&5.448&12.016&2.206&0.25&0.12&
0.05\\
&1972&[\onlinecite{sleight:1972} ]&XP& 5.4622 &12.048&2.206&&&\\
&1985&[\onlinecite{hazen:1985} ] &XC & 5.4595&12.0432&2.206&&&\\
&$^*$1996&[\onlinecite{moreau:1996}]&XC 
&5.456&12.020&2.203&0.2388&0.1141&0.0429\\
&1996&[\onlinecite{alov:unpublished}] 
&XC&5.4612&12.0452&2.206&0.2524&0.1364&0.0446\\ 
 \end{tabular}
\end{table}


%\input{lapwtable}
\narrowtext
\begin{table}
 \caption{List of calculational and convergence parameters used for LAPW 
calculations}
 \label{lapwpar}
 \begin{tabular}{||cr|c||}
Muffin tin radii: & Ca, Pb & 1.9 Bohr \\
			& Mo, W  & 1.65 Bohr \\
			& O      & 1.65 Bohr \\ \hline 
\multicolumn{2}{|l|}{ Maximum ${\bf{|k + G|}}$ for wavefunctions } & 6 Bohr $^{-
1}$ \\ 
\multicolumn{2}{|l|}{ Maximum ${\bf{|G|}}$ for charge density}  & 14 Bohr $^{-
1}$ \\ 
\multicolumn{2}{|l|} {Inequivalent ${\bf{k}}$-points for BZ integrals}  & 6  \\ 
\multicolumn{2}{|l|} {Convergence tolerance of total energy}  & 0.001 eV  \\ 
 \end{tabular}
 \end{table}
\widetext

%\input{valtable}

\begin{table}
 \caption{Band gaps determined from the band dispersion calculations shown in 
Figs. \ref{Cabands} and \ref{Pbbands} are listed for various values of {\bf{k}}  
for the valence and conduction bands ($k_v$ and $k_c$, respectively).}
 \label{valband}
 \begin{tabular}{||cccc||}
Material & $E_{gap}$ (eV) & $k_v$ $\left( \frac{2\pi}{a} \right)$& $k_c$ $\left( 
\frac{2\pi}{a} \right)$\\ \hline 
CaWO$_4$  & 4.09 & $\Gamma$ & $\Gamma$ \\
&&&\\ 
CaMoO$_4$ &  3.41 & $\Gamma$ & $\Gamma$ \\
&&&\\
PbWO$_4$ &  2.96 & 0.51 ($\Delta$) &0.47 ($\Sigma$)\\
         &  3.06 & 0.47 ($\Sigma$) &0.47 ($\Sigma$)\\
         &  3.25 & 0.47 ($\Delta$)& 0.47 ($\Delta$)\\
         &  4.13 & $\Gamma$ & $\Gamma$ \\
&&&\\
PbMoO$_4$ & 2.59 & 0.57 ($\Delta$) & 0.47 ($\Sigma$)\\
        &  2.71 & 0.47 ($\Sigma$) &0.47 ($\Sigma$)\\
         & 2.93 & 0.51 ($\Delta$) & 0.51 ($\Delta$)\\
         &  3.62 & $\Gamma$ & $\Gamma$ \\
\end{tabular}

 \end{table}


%\input{exptable}
\begin{table}
\caption{Optical parameters (in units of eV) measured and deduced  from 
experimental spectra.   In a few cases,    multiple entries are listed in order 
to illustrate the variability of these parameters, depending on temperature (T), 
sample preparation,  and other factors.}
\label{exptable}
\begin{tabular}{||l|c|c|c|c||} 
&CaWO4&PbWO4&CaMoO4&PbMoO4\\ \hline
Luminescence peak ($E_{lum}$)& 2.9\tablenotemark[1]&	  2.9\tablenotemark[2]	& 
2.3\tablenotemark[1]		&   2.2\tablenotemark[3] \\
&&&&\\
Luminescence width (FWHM) &  0.8\tablenotemark[1] &		  
0.8\tablenotemark[2]	&	   0.6\tablenotemark[1]	&	   
0.6\tablenotemark[3]\\
&&  0.7\tablenotemark[2]$^,$\tablenotemark[4] &&				   
0.5\tablenotemark[5] \\
&&&&\\
Exciton reflectance peak ($E_{exc}$) & 5.9\tablenotemark[6]	&  
4.2\tablenotemark[7]$^,$\tablenotemark[8]	&	   5.0\tablenotemark[6]	&\\	   
&&&&\\
Luminescence Stokes shift ($\Delta E_S$)&	  3.0&   1.3&   2.7& \\		   
&&&&\\
Free-carrier threshold ($E_{fc}$)&	   7.0\tablenotemark[9]	& 
4.7\tablenotemark[9] &&\\
	&   6.5\tablenotemark[10] &&&\\
      &   6.8\tablenotemark[11] &&&\\
&&&&\\
Exciton binding energy ($E_b$)&   0.9	&   0.5 &&\\
&&&&\\
Luminescence excitation threshold ($Y_{th}$) &   5.1\tablenotemark[12] &   
4.1\tablenotemark[7]&	   4.6\tablenotemark[12]	&	   
3.4\tablenotemark[3]$^,$\tablenotemark[5]\\
&				   4.7\tablenotemark[13]	&	   
4.0\tablenotemark[13] &&\\
\end{tabular}		   

\tablenotetext[1]{Ref. [\onlinecite{saito:1996a}], T = 295 K.}
\tablenotetext[2]{Ref. [\onlinecite{kobayashi:1993}],  T = 295 K.}
\tablenotetext[3]{Ref. [\onlinecite{vanloo:1975}], 1.6 K $<$ T $<$ 144 K.}
\tablenotetext[4]{Ref. [\onlinecite{nikl:1996}],  T = 4.2 K.}
\tablenotetext[5]{Ref. [\onlinecite{groenink:1980}], at T = 4.2 K 
(luminescence spectrum); at T = 77 K (excitation spectrum). }
\tablenotetext[6]{Ref. [\onlinecite{grasser:1975}], T = 295 K.}
\tablenotetext[7]{Ref. [\onlinecite{kolobanov:1996}],  T = 4.2 K, 295 K.}
\tablenotetext[8]{Ref. [\onlinecite{belsky:1995}],  T = 295 K.}
\tablenotetext[9]{Deduced from the thermoluminescence excitation spectra of
 Ref. [\onlinecite{murk:1997}],  T = 80 K.}
\tablenotetext[10]{Ref. [\onlinecite{gurvich:1971}], calcined powder sample, T = 
295 K.}
\tablenotetext[11]{Ref. [\onlinecite{nagirnyi:1997}], single crystal, T = 8 K.}
\tablenotetext[12]{Ref. [\onlinecite{hemphill:1988}], T = 295 K.}
\tablenotetext[13]{Ref. [\onlinecite{murk:1997}],  T = 80 K.}
	
\end{table}




 

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