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\input amssym.def
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\title{Transition-metal interactions in aluminum-rich intermetallics}

\author{Ibrahim Al-Lehyani$^{a,b}$ and Mike Widom$^{a}$}

\address{$^{a}$Department of Physics, Carnegie Mellon University,
Pittsburgh, Pennsylvania 15213}
\address{$^{b}$Department of Physics, King AbdulAziz University, Jeddah, Saudi Arabia}

\author{Yang Wang}

\address{Pittsburgh Supercomputer Center, Pittsburgh, Pennsylvania 15213}

\author{Nassrin Moghadam and G.~Malcolm Stocks}

\address{Oak Ridge National Laboratory, Oak Ridge, TN 37831-6114}

\author{John~A. Moriarty}

\address{Lawrence Livermore National Laboratory,
University of California, Livermore, California 94551}

\date{\today}

\maketitle

\begin{abstract}
The extension of the Generalized Pseudopotential Theory (GPT) to
transition-metal (TM) aluminides produces pair and many-body
interactions that allow efficient calculations of total energies.  In
aluminum-rich systems treated at the pair potential level, one
practical limitation is a transition-metal over-binding that creates
an unrealistic TM-TM attraction at short separations.  Even with this
limitation, the GPT pair potentials have been used effectively in
total energy calculations for systems with TM atoms at separations
greater than 4~\AA.  An additional term may be added for systems with
shorter TM atom separations, formally folding repulsive contributions
of the 3- and 4-body interactions into the pair potentials, resulting
in structure-specific interatomic potentials.  We have performed
numerical ab-initio total-energy calculations using VASP for an AlCoNi compound in a particular quasicrystalline approximant structure. The results allow us to fit a correction of the form $a/r^{b}$ to the GPT pair potentials
.
\end{abstract}

\newpage

\section{Introduction}
\label{sec:intro}

Total energy calculations are an important tool in theoretical
condensed matter physics, giving insight into structures and mechanical
properties of solids~\cite{hafner87,pettifor95}.  Accurate
calculations of total energy are notoriously difficult.
Theoretically, one must solve the Schroedinger equation simultaneously
for all electrons in the presence of fixed atomic nuclei.  Density
functional theory simplifies this problem by reducing it to the
self-consistent solution of Schroedinger's equation for a single
electron in a potential that depends upon the electron density.  Even
with this simplification, such full ab-initio methods are
computationally demanding~\cite{abInitio}, usually limited to systems
of less than a hundred atoms, and may not yield immediate physical insight
once an answer is obtained.

Instead, one may expand the energy in terms of pair and many-body
interatomic potentials~\cite{hafner87,pettifor95,torrens,interatomic}
so that the total energy appears as an explicit function of atomic
separations. Depending on the physical system under study and the type
of information sought, the expansion may often be truncated after a
small number of terms.  Such a truncated expansion trades off a degree
of accuracy in favor of computational simplicity and potentially
greater physical insight as compared with a full ab-initio approach.

Many physical systems have been studied using interatomic
potentials~\cite{moriartymcmahan82,krajcihafner92,phillipszou94,windischhafner94,mihalkzhu96,moriarty88}. These potentials are especially simple in the case of non-transition
metals.  There, the d-electron shells are either empty or else are deeply
buried under the Fermi energy level, allowing rapidly convergent
expansions of the total energy.  The presence of partially filled
d-bands in transition metals complicates the analysis.  The
d-band electronic states are highly localized in the vicinity of the
atoms and have strong angle-dependence.  In contrast to non-transition
metals, transition metal d-bands are at or near the Fermi level.
Total energy expansions will not converge as quickly as for
non-transitions metals, and 3- and 4-body interactions may contribute
significantly~\cite{finnispaxton}.

Moriarty~\cite{moriarty88} developed a rigorous treatment for
transition metals in the context of the Generalized Pseudopotential
Theory (GPT).  The treatment was later extended to binary and ternary
alloys of aluminum with first row transition
metals~\cite{moriartywidom97}.  These studies found that 3- and 4-body
interactions could be important in determining energetic and
mechanical stability of structures with large TM concentrations.  The
treatment of d-electron interactions created strong attractive
interactions at unphysically short distances in the pair potentials,
balanced by repulsive forces contained in 3- and 4-body interactions.
We wish to modify the pair potentials to remove this unphysical
attraction so that a truncation of the total energy expansion at the
level of pair potentials will be more accurate when transition metal
atoms are near neighbors.

One motivation for this study is the need for fast total energy
calculations in systems with short TM separations to enable structural
relaxation, molecular dynamics and Monte Carlo simulations. We focus
our attention on Al-Co-Ni compounds in decagonal quasicrystalline
structures~\cite{tsai89}.  The precise modification required depends
on the particular structure studied, but should be at least
approximately valid for many similar structures.  Furthermore, the
modifications obtained may allow us to treat Al-Co-Cu and Al-Cu-Ni
decagonal phases~\cite{otherdecag} because the Cu-Cu interactions do not
appear to require modification~\cite{moriartywidom97}.  Limited
numbers of full ab-initio calculations are sufficient to determine the
required modifications.

We intend to apply these potentials to predict the structures of
decagonal quasicrystals~\cite{allehyani_to}. A great deal of
experimental data is available that identifies the positions of most
atoms and identifies the chemical identity of many of those.  However,
in order to determine the quasicrystal structures from X-ray
diffraction one faces degenerate structures because elements near each
other in a row of the periodic table (such as Co, Ni and Cu) have
similar X-ray form factors.  A common approach to this problem is to
supplement the experimental data with total energy calculations.  This
approach is well established in crystallography~\cite{XRAY}.

The newly modified pair potentials can be applied to total energy
calculations in quasicrystals and related structures with a great
reduction in computational times compared with the full ab-initio
calculations. The time savings results from two features of the
potentials.  First, the potentials themselves depend on composition
and atomic volume but not the structure to which they will be applied,
so they may be precalculated and then applied repeatedly with a simple
lookup and interpolation. Second, to calculate the change in energy
when a single atom is moved, only interactions affecting that atom are
needed.  If the interactions are cut off at a certain spatial
separation, the time required to calculate the change in total energy
becomes independent of the number of atoms in the complete
structure. In contrast, full ab-initio methods must recalculate the
entire system when a single atom is moved.


In section~\ref{sec:problem}, GPT potentials are briefly described and
the difficulty of truncating the total energy expansion at pair
potentials is discussed.  Section ~\ref{sec:mod} gives details
about the methods we employ to determine the needed modifications.  In
section~\ref{sec:results}, we present the results of our full
ab-initio calculations and the modified pair potentials are
introduced.

\section{Interatomic Potentials}
\label{sec:problem}

JOHN: PLEASE REVISE OR ENHANCE THE FOLLOWING DISCUSSION OF GPT.

The Generalized Pseudopotential Theory starts with a full ab-initio
representation of the total energy.  Only valence electrons are
treated, and the interactions of valence electrons with ionic cores
are treated via pseudopotentials. A mixed basis is employed allowing
$sp$ electrons to be represented as superpositions of plane waves,
while $d$ electrons are represented as localized states.  The energy
and the electron density are expanded in terms of weak $sp$
pseudopotential matrix elements $W_{kk'}$, $sp-d$ hybridization
$\Delta_{kd}$, and $d-d$ tight-binding matrix elements $\Delta_{dd'}$.
The interatomic potentials can be calculated as functions of these
matrix elements.  For an alloy, the GPT expands the total energy in
the form:
\begin{equation}
\label{eq:etot}
E({\vec{R}_{i}^{\alpha}})=E_{vol}+
\frac{1}{2}\sum_{\alpha \beta}\sum_{ij}\mbox{}^\prime \ 
v^{\alpha \beta}_{2}(R_{ij})+
\frac{1}{6}\sum_{\alpha \beta \gamma}\sum_{ijk}\mbox{}^\prime \ 
v^{\alpha \beta \gamma}_{3}(R_{ij}R_{jk}R_{ki})+...
\end{equation}
where $\; \vec{R}_i \;$ is the set of all positions of $ N $ ions in
the metal, $E_{vol}$ is a volume term which includes all single-ion
contributions, and $v_{2}$, $v_{3}$,... are the two-, three-, and
many-ion interatomic potentials.  The primes on sums over ion
positions exclude all self-interaction terms.  Indices $\alpha, \beta,
\gamma, \cdot\cdot\cdot$ run over all chemical species, and indices
$i, j, k, \cdot\cdot\cdot = 1,\cdot\cdot\cdot, N$ run over the
individual ions.  All the interatomic potentials are atomic volume and
composition-dependent, but structure-independent.  They are functions
of the relative positions of small subsets of atoms, independent of
the positions of all other atoms in a structure.  The entire
dependence on the structure comes analytically through the summations
over all $ N $ ions.  This makes these potentials transferable among
different structures at fixed atomic volume and composition.  Detailed
treatments of GPT are given in
references~\cite{moriarty88,moriarty90,moriartywidom97}.

The separation of total energy between the 2- and higher-body terms is
not unique in principle, since we can add contributions to $v_{2}$
provided we make suitable subtractions from $v_{3}$ or higher-body
interactions. In the GPT potentials, the total energy is calculated to
second order in the weak pseudopotential $W_{kk'}$, so
their contribution to the total energy enters only in the volume term
and the pair potentials $v_2$.  The $d-d$ tight-binding contributions
are carried to all orders in the matrix element $\Delta_{dd'}$.  Terms
are allocated to pair- and many-body potentials according to how many
distinct ionic position explicitly enter.  Thus the pair potentials
$v_2$ contain contributions that are even powers of $\Delta_{dd'}$
associated with repeated hopping of $d$ electrons back and forth
between a pair of ions.  The three-body interactions contain a term of
third order order proportional to
$\Delta_{dd'}\Delta_{d'd''}\Delta_{d''d}$ and terms of fourth order
proportional to $\Delta_{dd'}^2\Delta_{d'd''}^2$, as well as higher-order
terms. The four-body interaction starts at fourth order.

The overbinding in $v_2$ at short distances results from the
contribution of second-order in $\Delta_{dd'}$.  This term is
attractive because it relates to $d$-band broadening resulting from
$d$ electron hopping. The attraction is strong at short distances
because the matrix element for atoms separated by distance $r$ varies
roughly as $r^{-5}$.  Contributions of higher order in $\Delta_{dd'}$
should balance this attractive contribution of the second order
term. However, as discussed above, only certain higher order terms are
explicitly included in $v_2$. The others belong to the many-body
potentials.

JOHN: DOES THE ABOVE DISCUSSION IMPLY THAT THE TM ATTRACTION MIGHT
BE CORRECT FOR AN ISOLATED PAIR OF TM ATOMS SURROUNDED BY ALUMINUM? DO
WE ALSO NEED TO DISCUSS MANY-BODY TERMS CONNECTING A TM PAIR WITH
SURROUNDING ALUMINUM ATOMS?

Fig~\ref{fig:alal} shows the Al-Al and Al-TM pair potentials for
Al-Co-Ni~\cite{moriartywidom97}.  These are calculated in the
aluminum-rich limit, but in practice they do not depend strongly on
composition.  The first minima of the Al-TM pair potentials occur near
2.3~\AA~with depths of about 0.2 eV (Al-Ni) and 0.3 eV
(Al-Co). Rather than a potential minimum, the Al-Al potential exhibits
a shoulder near 3~\AA.  The TM-TM pair potentials are shown in
Fig.~\ref{fig:tmtm}. The TM overbinding is most evident for Co.  The
Co-Co potential has a depth of 2.1 eV at 1.7~\AA.  The NiNi potential
depth of 0.1 eV at 2.2~\AA~ is not obviously unreasonable, but in the
following we will find it requires some modification. The Co-Ni pair
potential $v_2^{CoNi}$ is defined as an average of the Co-Co and Ni-Ni
potentials,
\begin{equation}
\label{vconi}
v_2^{CoNi} \equiv (v_2^{CoCo}+v_2^{NiNi})/2.
\end{equation}
This amounts to a perturbative expansion of $v_2^{\alpha\beta}$ in the
difference in atomic number $Z^{\alpha}-Z^{\beta}$. Clearly
$v_2^{CoNi}$ so-defined suffers overbinding due to the overbinding of
$v_2^{CoCo}$.

We wish to devise potentials for Al-Co-Cu and Al-Cu-Ni as well as
Al-Co-Ni. Previously the Al-Cu potentials were found to be well
behaved up to large Cu composition~\cite{moriartywidom97} so perhaps no
modification of $v_2^{CuCu}$ is needed. Our modification to
$v_2^{CoCo}$ obtained for Al-Co-Ni compounds may be approximately
valid for these other compounds. We previously
defined~\cite{moriartywidom97} $v_2^{CoCu}$ as equal to $v_2^{NiNi}$
because Ni lies between Co and Cu in the periodic table. Thus our
modified Ni-Ni potential should serve as an approximate modified Co-Cu
potential. For the modified Cu-Ni potential we may take
$(v_2^{CuCu}+v_2^{NiNi})/2$, using the modified Ni-Ni potential. The so-obtained Al-Co-Cu and Al-Co-Ni potentials should still be validated using full ab-initio calculations.

\section{Modification of Pair Potentials}
\label{sec:mod}

As discussed above, the unphysical short-ranged attraction in the
TM-TM pair potentials is balanced by repulsive terms contained in the
three- and higher-body potentials.  If one chooses to truncate the GPT
expansion at the pair potential level, these repulsive many-body
contributions must be ``folded in'' to effective pair potentials.
Formally, we may define an effective pair potential by averaging over
atomic positions, holding a single pair of ions fixed~\cite{moriarty90}
\begin{equation}
\label{veff}
v_2^{eff} \equiv v_2 + <v_3> + <v_4> + \cdot\cdot\cdot.
\end{equation}
The redefined three- and four-body interactions then must act on
deviations of a particular structure from isotropy and homogeneity.
Within the simplified model GPT~\cite{moriarty88}, the four-body
interaction oscillates with respect to angles between atoms, with a
nearly zero mean, so it does not contribute to $v^{eff}$.  The
third-order contribution to $v_3$ also averages away, but the fourth
order contributions to $v_2$ survive, yielding a short-ranged
repulsive term proportional to $\Delta_{dd'}^2$ balancing against the
attractive second order term in $v_2$.

Inspired by the short-ranged repulsion found in equation.~(\ref{veff}) and
the power law variation of $\Delta_{dd'}$ within the model
GPT~\cite{moriarty90}, we propose to modify the full GPT pair
potentials $v_2^{\alpha\beta}$ by adding terms of the form
\begin{equation}
\label{Umod}
U^{\alpha\beta}(r)=a(r_{0}/r)^b
\end{equation}
where $a$ and $b$ are positive and depend upon the elements $\alpha$
and $\beta$ of pair potential modified. The value $r_{0}$ is a common atomic separation in quasicrystals of 2.55 \AA. Then the effective pair potential is written as
\begin{equation}
\label{effpair}
V^{\alpha\beta}(r)=v_{2}^{\alpha\beta}(r)+U^{\alpha\beta}(r).
\end{equation}
We determine the unknowns $a$ and $b$ by matching binding energies and forces obtained from full
ab-initio calculations on a quasicrystal approximant.  Cockayne and
Widom~\cite{cockaynewidom98,widomallehyani00} suggested a structure
for decagonal AlCoCu. An approximant of that structure is shown in
fig~\ref{fig:str} with Ni atoms replacing Cu. The orthorhombic unit
cell (a=23.3~\AA, b=7.57~\AA, c=4.09~\AA) contains 50 atoms
(Al$_{34}$Co$_{10}$Ni$_{6}$).  Most atoms occupy either
z=0.25 or z=0.75 layers.  Al atoms at the centers of hexagons occupy
the z=0.5 layer.  Two Co atoms occupy symmetric positions around these
central Al atoms. In AlCoCu, alternation of Co and Cu on tile edges is
thought to be energetically advantageous~\cite{cockaynewidom98}.  We
find that alternation of Co and Ni shown in Fig.~\ref{fig:str} is
slightly {\it dis}advantageous in AlCoNi.

To investigate TM bonding energetics, we alter the basic structure
shown in Fig.~\ref{fig:str} by swapping a Co atom on a horizontal tile
edge (atom b in fig~\ref{fig:str}) with the Ni atom on the other
horizontal tile edge (atom c ). Focusing on near-neighbor
interactions, we find this swap of atoms replaces 4 CoNi bonds with 2
CoCo and 2 NiNi bonds, all of length = 2.55~\AA. These numbers are
twice as large as is apparent by inspection of Fig.~\ref{fig:str}, the
extra factor of 2 coming from periodic boundary conditions in the
direction perpendicular to the plane.

Now consider the energy change evaluated using pair potentials. Atoms
b and c occupy nearly equivalent sites. An exact symmetry in the Al
atom positions guarantees that no bond involving an Al atom is
affected by the swap. We already noted the change in TM near-neighbor
interactions. At further neighbors, with separations of 4.6~\AA~ or
greater, we also find interchanges between CoCo and NiNi bonds for
pairs of CoNi bonds.  If the approximate form~(\ref{vconi}) of
$v_2^{CoNi}$ as the average of $v_2^{CoCo}$ and $v_2^{NiNi}$ were valid, all
changes in bonding would exactly cancel each other, resulting in a
vanishing energy change. We presume that approximation~(\ref{vconi})
is more accurate at large separations than small separations. Thus
we attribute the entire energy change of the bc swap to near neighbor
binding energy differences
\begin{equation}
\label{E1}
\Delta E_{1}=2V^{CoCo}+2V^{NiNi}-4V^{CoNi}
\end{equation}
where $V^{\alpha\beta}$ denotes the strength of the pair potential evaluated
at the near-neighbor distance 2.55~\AA.

Next we swap one of the Co atoms inside the tiles (atom e ) with one of
the Ni on a horizontal tile edge (atom a ).  Two CoNi bonds are broken
and two CoCo bonds are produced after this swap.  All other
interactions that are affected are Al-TM interactions, which we
presume to be described accurately by the GPT pair potentials.  This
swap energy can be written as:
\begin{equation}
\label{E2}
\Delta E_{2}=2V^{CoCo}-2V^{CoNi}+V^{AlTM}
\end{equation}
where $V^{AlTM}$ represents a calculable collection of interactions
between Al atoms and TM atoms at many separations.  $V^{AlTM}$ should
be described accurately by the unmodified GPT pair potentials.

Lastly, we replace the Co-Ni
pair on one horizontal tile edge (atoms c and d) with Al atoms.  Then we
swap one of the newly introduced Al
(at position c) with a Ni atom on the other horizontal tile edge (atom a).
This breaks two CoNi bonds.  All other interactions are either Al-TM
or Al-Al interactions, and again those are described well within the
GPT.  The energy change of this swap is
\begin{equation}
\label{E3}
\Delta E_{3}=-2V^{CoNi}+V^{AlTM}+V^{AlAl}
\end{equation}
where $V^{AlAl}$ and $V^{AlTM}$ represent collections of interactions
involving Al atoms that, as before, we presume to be accurately
calculable within the unmodified GPT.

Full ab-initio values for the energy changes $\Delta E_1$, $\Delta
E_2$ and $\Delta E_3$ were calculated using VASP~\cite{VASP}.  VASP
calculates total energies within the local density approximation using
pseudopotentials to treat valence-core electron interactions.  We performed calculations using a 4x4x4 k-space grid and also using a 4x4x8 k-space grid to observe the convergence as k-points are added. All calculations were done using medium precision which expected to be sufficient for out needs. We iterate the self-consistent calculation until an accuracy of $10^{-6}$ eV is achieved.

By comparing the energy differences $\Delta E_1$, $\Delta E_2$ and
$\Delta E_3$ calculated by VASP with the same quantities calculated
with the unmodified GPT potentials, we can obtain the values of
$U^{\alpha\beta}$ evaluated at the near neighbor separation
2.55~\AA. Specifically, when energy changes calculated by unmodified GPT are
subtracted from energy changes calculated by VASP, assuming that the contributions $V^{AlAl}$ and $V^{AlTM}$ are accurately calculated with the
unmodified GPT, we find
\begin{eqnarray}
\label{simul}
\Delta E_1^{VASP}-\Delta E_1^{GPT} = 2U^{CoCo}+2U^{NiNi}-4U^{CoNi}\nonumber \\
\Delta E_2^{VASP}-\Delta E_2^{GPT} = 2U^{CoCo}-2U^{NiNi} \\
\Delta E_3^{VASP}-\Delta E_3^{GPT} = -2U^{CoNi}. \nonumber \\ \nonumber
\end{eqnarray}
Since each correction $U^{\alpha\beta}(r)$ involves two unknowns, $a$
and $b$, equation~(\ref{simul}) consists of three equations in six
unknowns. Additional information is obtained from the forces on atoms calculated by VASP. By examining the forces on the Co-Ni pair (atoms c and d) in
Fig.~\ref{fig:str}, and on the Co-Co and Ni-Ni pairs created by the bc swap,
we obtain three additional equations governing the derivatives of
$U^{\alpha\beta}$ at the near-neighbor separation. This additional information
allows closure of the equations and determination of the unknowns.

\section{Results}
\label{sec:results}

Table~\ref{tab:diff} shows the energy differences $\Delta E_{i}$ in
equations~\ref{E1}-\ref{E3} calculated using GPT pair
potentials and VASP.  Comparing the VASP data for the two grid sizes, we note that the signs and approximate magnitudes of $\Delta E_{i}$ are consistent with each other.  One immediate result from table~\ref{tab:diff} is that
mixed Co-Ni bonds are disfavored over pure Co-Co and Ni-Ni bonds.  The
energy difference $\Delta E_{1}$ results from breaking 4 CoNi bonds
and producing 2 CoCo and 2 NiNi bonds.  $\Delta E_{1}$ calculated by
VASP is negative, showing that the swap lowers the system energy.
This means that for AlCoNi, similar TM atoms prefer to reside near
each other on the tile edges.  Cockayne and Widom found the opposite
for the case of AlCuCo using mock ternary
potentials~\cite{cockaynewidom98}, and this was confirmed later using
a full ab-initio technique~\cite{widomallehyani00}.

Also concerning the calculated values of $\Delta E_1$, we see that
the averaged potential approximation~(\ref{vconi}) is fairly
accurate. GPT yields $\Delta E_1 = 0$ because it employs this approximation.
The small value of $\Delta E_1$ obtained by VASP confirms that this
approximation is not far off the mark.

Fig.~\ref{fig:forces} shows the x-component of the total force on
certain TM atoms. Our (4x4x4) and (4x4x8) VASP calculations yield forces that agree to 0.06 eV/\AA~or better. We examine the horizontal bonds ab and cd in
Fig.~\ref{fig:str} in both the original and swapped configurations. As
expected, at 2.55~\AA, GPT pair potentials predict attractive forces
between TM pairs while the actual forces obtained from VASP are
repulsive. The small force asymmetry on atoms in the CoNi pair is due
to the different ways Co and Ni atoms interact with their surrounding
environments. The difference between the forces calculated by our two
methods is greatest for Co-Co bonds and smallest for Ni-Ni bonds,
consistent with our expectation that overbinding is more severe for Co
than for Ni.

Calculated corrections to the GPT pair potentials are given in table~\ref{tab:para}. Examining the magnitude of $U^{\alpha\beta}$ at
$r$=2.55~\AA~(i.e. the value of $a$), we note that $U^{NiNi}$ is smaller than $U^{CoCo}$, as is
expected since Ni is closer to a noble transition metal structure,
with its d-band almost full. It should be noted that $r$=2.55~\AA~is not the potential minimum. It is the nearest neighbor distance that the calculations were performed at. $V^{\alpha\beta}$ and $F^{\alpha\beta}$ are, respectively, the binding energy and force of the modified GPT (eq.(~\ref{effpair})) at the near neighbor distance. The large powers of inverse length we obtain show that our modifications of the GPT pair potentials fall off rapidly beyond the near-neighbor separation. The modified potentials are illustrated in fig.~\ref{fig:mod_tmtm}. The (4x4x4) and (4x4x8) VASP calculations agree in positions of the potential minima to about 0.05~\AA~ and agree in the values at the minima to about 0.02 eV.

\acknowledgments
%
IA wishes to thank King Abdulaziz University (Saudi Arabia) for
supporting his study and MW acknowledge support by the National
Science Foundation under grant DMR-9732567.  The work of the JAM was
performed under the auspices of the U. S. Department of Energy at the
Lawrence Livermore National Laboratory under contract number
W-7405-ENG-48. VASP calculations were performed at the Pittsburgh Supercomputer
Center and at NERSC.

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\bibitem{pettifor95}
{\it Bonding and Structure of Molecules and Solids}, David Pettifor
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\bibitem{abInitio}
Ab-initio studies of Al-rich intermetallics include:
%AlCuLi ab-initio
M. Windisch, M. Krajci and J. Hafner, 
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G. Trambly de Laissardiere and T. Fujiwara, 
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%

\bibitem{torrens}
%
{\it Interatomic Potentials.}, I.~McC. Torrens (Academic Press, New
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%interatomic potentials in condensed matter physics
\bibitem{interatomic}
A large collection of interatomic forces applications in condensed matter physics can be found in Phil. Mag. A {\bf 58} No. 1 (1988)

%Early Al GPT results
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J.~A. Moriarty and A.~K. McMahan, Phys. Rev. Lett. {\bf 48}, 809 (1982); 
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%AlMgZn pair potentials
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%AlCo potentials
\bibitem{phillipszou94}
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%AlCuLi pair potentials
\bibitem{windischhafner94} 
M. Windisch, J. Hafner, M. Krajci and M. Mihalkovic,
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%AlMn potentials
\bibitem{mihalkzhu96}
M. Mihalkovic, W.-J. Zhu, C.~L. Henley, and R. Phillips, 
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%
%
\bibitem{moriarty88}
J.~A. Moriarty, Phys. Rev. B {\bf 38}, 3199 (1988).
%

\bibitem{finnispaxton}
%
Simple methods for transition metals are reviewed in
M. W. Finnis, A.~T. Paxton, D.~G. Pettifor, A.~P. Sutton and
Y. Ohta, Philos. Mag. A {\bf 58}, 143 (1988)
%


\bibitem{moriartywidom97}
J.~A. Moriarty and M. Widom, Phys. Rev. B {\bf 56}, 7905 (1997);
M. Widom and J.~A. Moriarty, Phys. Rev. B {\bf 58}, 8967 (1998);
M. Widom, I. Al-lehyani and J.~A. Moriarty, in preparation.
%

%Early report of stable AlCoNi
\bibitem{tsai89}
A.-P. Tsai, A. Inoue and T. Masumoto, Mat. Trans. JIM, {\bf 30} (1989) 463.

%Al-Co-Cu and Al-Cu-Ni decagonal phases.
\bibitem{otherdecag}
L.~X. He, Y.~K. Wu and K.~H. Kuo, J. Mat. Sci. Lett. {\bf 7} (1988) 1284;
A.-P. Tsai, A. Inoue and T. Masumoto, Mat. Trans. JIM, {\bf 30} (1989) 300.
%
\bibitem{allehyani_to}
I. Al-Lehyani, Y. Wang, M. Widom, M. Mihalkovic, N. Moghadam, 
J.~A. Moriarty, C.~L. Henley, and E. Cockayne, in preparation.

\bibitem{XRAY}
{\it X-ray structure determination; a practical guide },
G.~H. Stout, L.~H. Jensen. p 370 (Wiley, New York, 1989).

\bibitem{moriarty90}
J.~A. Moriarty, Phys. Rev. B {\bf 42}, 1609 (1990).
%
\bibitem{cockaynewidom98} 
E. Cockayne and M. Widom, Phys. Rev. Lett. {\bf 81}, 598 (1998)

\bibitem{widomallehyani00} M. Widom, I. Al-Lehyani, Y. Wang and E. Cockayne,
Mat. Sci. Eng. A, to appear (2000).
%
%

%

% VASP references
\bibitem{VASP}
G. Kresse and J. Hafner, Phys. Rev. B {\bf 47}, RC558 (1993);
G. Kresse and J. Furthm\"uller, Phys. Rev. B {\bf54}, 11169 (1996).

\end{references}



\begin{table}
\caption{Total energy differences defined by equations~(\ref{E1})-(\ref{E3})
as calculated by VASP and GPT. Units are eV/cell.}
\vspace{10pt}
\begin{tabular}{|l|r|r|r|}
 Energy  & GPT   & VASP (4x4x4) & VASP (4x4x8) \\
\hline
$\Delta E_{1} $       &  0.000	& -0.020 & -0.031 \\ 
$\Delta E_{2} $       &  0.116	&  0.298 &  0.279  \\  
$\Delta E_{3} $       & -0.945	& -1.384 & -1.419  \\    
\end{tabular}
\label{tab:diff}
\end{table}




\begin{center}
\begin{table}
\caption{Modifications for GPT potentials, $U^{\alpha\beta}(r)=a(\frac{r_0}{r})^b$ where $r_0$=2.55~\AA. Units of $a$ and $V^{\alpha\beta}$ are eV while $b$ is dimensionless and $F^{\alpha\beta}$ has units of ev/\AA.}
\vspace{10pt}
\begin{tabular}{|l|c|c|c|c|}
$\alpha\beta$ &  $a$   & $b$ & $V^{\alpha\beta}$ & $F^{\alpha\beta}$  \\
\hline
CoCo        & 0.319 & 16.6 &0.0946   & 0.978  \\ 
CoNi        & 0.237 & 19.3 &0.0941   & 0.994 \\  
NiNi        & 0.140 & 21.3 &0.0779   & 0.674  \\    
\end{tabular}
\label{tab:para}
\end{table}
\end{center}



\newpage

\begin{figure}
\epsfig{figure=alal.eps}
\vspace{100pt}
\caption{GPT interatomic pair potentials for Al-Al, Al-Co and  Al-Ni.}
\label{fig:alal}
\end{figure}
%
%
%
\begin{figure}
\epsfig{figure=tmtm.eps}
\vspace{100pt}
\caption{GPT interatomic pair potentials for Co-Co, Co-Ni and Ni-Ni.}
\label{fig:tmtm}
\end{figure}
%
%
%

%\begin{center}
\begin{figure}
\epsfig{figure=d0c0s0_3.eps}
\vspace{50pt}
\caption{The initial structure used in our calculations. Labeled atoms
participate in swaps.}
\label{fig:str}
\end{figure}
%\end{center}
%
%
%
\begin{figure}
\epsfig{figure=forces.eps}
\vspace{30pt}
\caption{Horizontal components of forces on transition metal atom pairs calculated from the GPT and VASP.}
\label{fig:forces}
\end{figure}
%
\begin{figure}
\epsfig{figure=mod_tmtm.eps}
\vspace{100pt}
\caption{Modified transition metal pair potentials using the parameters in
table~\ref{tab:para}.}
\label{fig:mod_tmtm}
\end{figure}
%
%
%
%
\end{document}        
