Materials Science and Engineering A365 (2004) 2–13
Analytic bond-order potentials for multicomponent systems
D.G. Pettifor a,∗ , M.W. Finnis a,1 , D. Nguyen-Manh a ,
D.A. Murdick b , X.W. Zhou b , H.N.G. Wadley b
a
b
Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK
School of Engineering and Applied Science, University of Virginia, Charlottesville, VA, USA
Abstract
Classical interatomic bond-order potentials (BOPs) have previously been obtained by coarse-graining the quantum-mechanical electronic
structure within the chemically intuitive reduced tight-binding (TB) framework. This paper generalizes the reduced tight-binding approximation to the case of multicomponent sp-valent systems, thereby allowing a rigorous derivation of expressions for the ␴ and ␲ bond orders within
chemically heterogeneous situations. The close link between the different bond-order potential parameters and different physical properties
is illustrated for the particular choice of Goodwin–Skinner–Pettifor (GSP) radial dependences for the repulsive pair potential and two-centre
bond integrals.
© 2003 Elsevier B.V. All rights reserved.
Keywords: Classical interatomic bond-order potential; Multicomponent; Tight-binding
1. Introduction
Materials modeling is often contingent upon having reliable knowledge about the key mechanisms operating at the
atomistic level. Unfortunately, however, for many materials processes of technological importance the results of the
atomistic simulations are questionable due to the unsatisfactory nature of the classical interatomic potentials used.
For example, the growth of films by molecular beam epitaxy (MBE) or chemical vapor deposition (CVD) involves
the breaking and re-making of chemical bonds. The classical Stillinger and Weber [1], Tersoff [2], or EDIP [3,4] interatomic potentials cannot provide a reliable description of
the underlying growth mechanisms since they are intrinsically unable to model the creation and destruction of dangling bonds. Although Brenner [5] and Brenner et al. [6]
have modified the Tersoff potential to include explicitly radical formation and conjugacy, this scheme has been applied
only to the hydrocarbon system where there is a large experimental database with which to fit the many new parameters
introduced. In general, these classical interatomic potentials
∗ Corresponding author. Tel.: +44-1865-273751;
fax: +44-1865-273783.
E-mail address: [email protected] (D.G. Pettifor).
1 Present address: School of Mathematics and Physics, Queen’s
University, Belfast, UK.
0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.msea.2003.09.001
have not been successfully extended to multicomponent systems such as, for example, GaAs [7].
In this paper, following from our earlier work [8–15],
we show that it is possible to derive classical interatomic
bond-order potentials (BOPs) for multicomponent systems by coarse-graining the quantum-mechanical electronic
structure within the chemically intuitive tight-binding (TB)
framework. This BOP theory provides a direct bridge
between the electronic modeling hierarchy (with its full
treatment of the electronic degrees of freedom) and the
atomistic modeling hierarchy (where the electronic degrees
of freedom have been removed by imagining the atoms are
held together by some sort of glue or interatomic potential).
We will see that this theoretical link between the electronic
and atomistic modeling hierarchies helps address the fundamental problem with empirical potentials, namely how best
to choose their analytic form and how best to decide the
significance of the many fitting parameters that are required
for multicomponent systems.
The plan of the paper is as follows. Section 2 focuses
on the electronic modeling hierarchy. In particular, in
Section 2.1 we give a brief introduction to the two-centre,
orthogonal TB approximation [16], stressing that in general
the bond integrals are environmentally dependent [17,18].
In Section 2.2, we demonstrate that this TB approximation
is the simplest quantum-mechanical description of the covalent bond which contains the necessary ingredients for
D.G. Pettifor et al. / Materials Science and Engineering A365 (2004) 2–13
predicting the well-known structural trends within the periodic table for the elements [19] and within the structure
map for binary AB compounds [20]. In Section 2.3, we
generalize for the first time our earlier reduced TB approximation [9,11,12] to the case of multicomponent sp-valent
systems where spσ AB = psσ AB . This approximation, which
reduces the number of independent two-centre bond integrals by one, was introduced in order to recover directly
the chemically powerful concept of ␴ and ␲ bond orders
within a TB framework [13].
Section 3 focuses on the atomistic modeling hierarchy.
In particular, in Section 3.1 we show how the TB electronic
structure may be coarse-grained in terms of its moments.
Cyrot–Lackmann [4] had proved in 1967 that the nth moment of the local density of states associated with a given
atom could be written as a sum over all the self-returning
hopping or bonding paths of length n about that site. This
provides the direct link between the electronic structure
(usually calculated by diagonalizing a Hamiltonian) and
the interatomic potential (often expressed as a sum over
many-body contributions). We demonstrate that the concept
of moments is a very powerful tool for understanding the
origin of the observed structural trends within elements
and binary compounds [16]. Critically, the second-moment
approximation, which is implicit in the form of the Tersoff
potential, fails to distinguish between the three-dimensional
structure-types of diamond, simple cubic and face-centred
cubic [22]. Their relative stability is determined by higher
moments such as, for example, the fourth which characterizes the unimodal versus bimodal shape of the electronic
density of states.
In Section 3.2, BOP theory is used to go beyond the simplest second-moment approximation and provide analytic
expressions for the bond orders of sp-valent systems [13,14].
We indicate that these allow the quantification of the ubiquitous concept of single, double, triple, conjugate and radical bonds in hydrocarbon systems [13]. In Section 3.3, we
present analytic BOPs for multicomponent sp-valent systems
in which the Goodwin–Skinner–Pettifor (GSP) functional
form [23] is used to model the distance dependences of the
repulsive potential and two-centre bond integrals. We derive
for the first time a set of equations, which link the BOP parameters to physical quantities. In Section 4 we conclude.
2. The electronic modeling hierarchy
2.1. The tight-binding approximation
The two-centre, orthogonal TB model [16,24] approximates the total energy of a multicomponent sp-valent system
as follows:
U = Urep + Uprom + Ubond ,
(1)
where we have assumed that each atom is locally charge neutral. This LCN constraint is achieved within the Oxford Or-
3
der N (OXON) code [25] by adjusting self-consistently the
on-site atomic energy levels. It is an excellent constraint for
multicomponent metallic systems [26,27] but clearly starts
to break down as the degree of ionicity in the bond increases.
This constraint can be lifted within the TB model if required
[28].
The first term contains the overlap repulsion and may be
written in the form:
1 µν
(2)
φij
Urep =
2
i=j
µν
where φij is the repulsive interaction between a µ atomic
species at site i and a ν atomic species at site j. In the simplest
µν
approximation it is assumed to be pairwise so that φij =
µν
φ (Rij ) where the two sites are a distance Rij apart. Recently, however, more quantitative orthogonal TB schemes
have assumed that this pairwise repulsion is also dependent
on the local environment about the bond [17,18,29–32].
The second term represents the promotion energy for spvalent systems, which arises from the change in the hybridization state of the sp orbitals as the atoms are brought
together from infinity. It is given by
µ
µ
µ
Uprom =
(3)
(Ep − Es )(Np )i
i
µ
µ
where (Ep − Es ) is the splitting between the valence s and
µ
µ
p atomic energy levels of the µth species. δµ = (Ep − Es )
is assumed to be a constant and thus independent of the
µ
environment [17]. (Np )i is the change in the number of p
electrons on the µth atom at site i compared to the free atom
value. Note that due to local charge neutrality Ns +Np =
0 so that Ns = −Np .
The third term is the attractive covalent bond energy,
which may be written in the form:
1
µν
(Ubond )ij
(4)
Ubond =
2
i=j
The individual bond energies are given by
µν
µν
νµ
(Ubond )ij = 2
HiL,jL ΘjL ,iL
(5)
L,L
in terms of the Hamiltonian and bond-order matrix elements with respect to the valence orbitals |iL and |jL on
sites i and j, respectively, where L = (l, m) and L =
(l , m ) represent the appropriate orbital and magnetic quantum numbers. The prefactor 2 accounts for the assumed spin
degeneracy.
The fundamental two-centre ␴ and ␲ bond integrals
µν
µν
(βll ␴ )ij and (βll ␲ )ij , which determine the intersite Hamiltonian matrix elements, can be obtained from first principles
density functional theory, either indirectly by fitting the
band structure [17,30,31] or directly by using screened TBLMTO theory [33]. For example, the open squares in Fig. 1
shows the predicted TB-LMTO values for elemental Si and
4
D.G. Pettifor et al. / Materials Science and Engineering A365 (2004) 2–13
to provide a computationally fast but reliable method for
modeling point defects, dislocations, and surfaces in the elemental systems C [34,35], Si [36,37], and Mo [30,31,38].
2.2. Structural prediction
The TB model provides the simplest quantum-mechanical
description of the covalent bond that contains the necessary
ingredients for predicting structural trends within the periodic table for the elements or the structure maps for binary
compounds. The relative stability of two competing structures in equilibrium under a binding energy law of the type
given by Eq. (1) can be computed directly using the structural energy difference theorem [39]. This states that the energy difference U between two structures is given in first
order in U/U by:
U = [Uprom + Ubond ]Urep=0 ·
Fig. 1. The ␴ and ␲ bond integrals within elemental bcc Mo and bcc
Si and within binary Cllb MoSi2 . The analytic unscreened (solid curves)
and screened results for the bcc structure (dotted curves) and the C11b
structure (dashed curves) are plotted. The numerical screened TB-LMTO
values are presented by the squares and circles for the bcc and Cllb
structures, respectively, (1) and (2) label the first and second nearest
neighbour curves, respectively [18].
Mo with respect to the bcc structure, whereas the open circles show the values for binary MoSi2 with respect to the
bcc-related C11b structure [18]. (The clusters of three open
points correspond to values obtained at (0.9, 1.0, 1.1) Ω0 ,
where Ω0 is the equilibrium volume). We see at once that
these two-centre bond integrals are indeed environmentally
dependent.
This environmental dependence is well represented by the
expression:
µν
µν
µν
(βll τ )ij = βll τ (Rij )[1 − (Sll τ )ij ]
(6)
where τ = σ, π or δ. Sij is a screening function whose
analytic dependence on the neighbouring atoms k has recently been derived by using BOP theory to invert the nonorthogonality matrix [18]. Comparing the unscreened (solid)
and screened (dashed) curves in Fig. 1, we observe that
Eq. (6) not only accounts for the discontinuities in the values
of the TB-LMTO bond integrals between first and second
nearest neighbour shells, but also provides a robust, transferable TB representation from one environment (say elemental
Mo) to another (say binary MoSi2 ). These environmentallydependent orthogonal TB schemes have recently been shown
(7)
This theorem generalizes the usual procedure for studying
the structural stability of ionic compounds by first packing
together hard spheres until they touch and then comparing
their electrostatic or Madelung energies in order to see which
is most stable. Eq. (7) extends this two-stage process to the
case of realistic atoms or ions which do not exhibit hard-core
behaviour. In the first step (analogous to packing together
hard spheres), the bond lengths of the competing structure
types are adjusted to guarantee the same repulsive energy. In
the second step (analogous to evaluating the ionic Madelung
energies), the bond and promotion energies are computed
and compared.
The structural trends across the sp-valent elements within
the periodic table may thus be investigated directly [19] by
taking Harrison’s [40] canonical parameterization for the
bond integrals ssσ, spσ, ppσ and ppπ. Fig. 2 shows the result
of comparing the bond energies for the particular choice of
δ = Ep − Es = 0, corresponding to zero promotion energy.
We see that this simple nearest-neighbour TB model predicts
the structural trend correctly from close-packed fcc and hcp
metallic structures for less than half-full bands through the
open four-fold coordinated diamond structure for Group IV
to the arsenic puckered sheets for the Group V pnictides, the
helical linear chains for the Group VI chalcogenides and the
dimers for the Group VII halogens. A similarly successful
prediction of the structural trends across the transition metal
and rare earth series is also provided by the TB model [16].
The power of the locally charge neutral, orthogonal TB
approximation for modeling metallic and covalently bonded
multicomponent systems is illustrated in Fig. 3 for the pdbonded AB compounds [20,26]. The upper panel shows the
experimental structure map (χp , χd ) for 169 binary compounds which comprise a p-valent element from Groups III
to VII and a d-valent element from the transition metal series. The chemical scale χ is a phenomenological coordinate
that was chosen to arrange all the elements in sequential order by providing the best structural separation within a twodimensional map [41]. We see that the compounds fall within
D.G. Pettifor et al. / Materials Science and Engineering A365 (2004) 2–13
5
Fig. 2. A comparison of the structural energy curves (in units of |ssσ| for
the simple cubic lattice) as a function of sp-band filling for 10 different
structures, ranging from a coordination of 1 for the dimer to a coordination
of 12 for the fcc and hcp close-packed lattices. They were evaluated
under the assumption that the repulsive interaction falls off as the square
of the bond integral [19].
well-defined domains that characterize the seven most frequently occurring AB structure types, namely NaCl, CsCl,
NiAs, MnP, FeB, CrB and FeSi. The lower panel shows the
predictions of the structural energy difference theorem using Andersen’s [42] canonical parameterization for the bond
integrals pp(σ, π), pd(σ, π) and dd(σ, π, δ).
We see that broad agreement is obtained between the
topological features of the experimental and theoretical
structure maps. In particular, NaCl in the top left-hand corner adjoins NiAs running across to the right and boride stability down to the bottom. MnP stability is found in the middle of the NiAs domain and towards the bottom right-hand
corner where it adjoins CsCl towards the bottom. The main
failure of this simple pd TB model is its inability to predict
the FeSi stability of the transition metal silicides, which is
probably due to the total neglect of the valence s electrons
within the bonding. Thus, the observed stability of the
NaCl, CsCl, NiAs, MnP and boride domains is determined
solely by the covalent bond energy from Eq. (7), once the
bond lengths have been adjusted to provide the same repulsion. This gives us confidence that the quantum-mechanical
TB description is sufficiently accurate at the electronic
level to provide the springboard from which to derive
classical interatomic bond-order potentials at the atomistic
level.
2.3. The reduced tight-binding approximation
Fig. 3. A comparison of the experimental (upper panel) and theoretical
(lower panel) domains of structural stability for the pd-bonded AB compounds. χp and χd in the upper panel give the values of a phenomological
chemical scale for the p- and d-valent elements, respectively. Np and Nd
in the lower panel give the values of the number of electrons on p- and
d-valent sites, respectively, which are predicted by the TB calculations
[20,26].
It follows from Eq. (5) that for sp-valent systems the ijth
bond energy can be decomposed into explicit ␴ and ␲ bond
contributions, namely
µν
µν
(8)
Choosing the z-axis along the direction of the bond from
i to j, we have
µν
µν νµ
νµ spσij
Θjs,is Θjs,iz
ssσij
µν
(U␴ bond )ij = 2Tr
µν
µν
νµ
νµ
psσij
ppσij
Θjz,is Θjz,iz
(9)
and
µν
(U␲ bond )ij
= 2Tr
µν
ppπij
0
0
ppπij
µν
Θjx,ix
νµ
Θjx,iy
νµ
Θjy,iy
Θjy,ix
νµ
.
νµ
(10)
Thus, the ␴ bond energy contributon can be written as:
µν
There is one further approximation that we need to make
before we can derive analytic expressions for the bond order.
µν
(Ubond )ij = (U␴ bond )ij + (U␲ bond )ij ·
µν
νµ
µν
νµ
µν
νµ
(U␴ bond )ij = 2(ssσij Θjs,is + spσij Θjz,is + psσij Θjs,iz
µν
νµ
+ ppσij Θjz,iz )
(11)
6
D.G. Pettifor et al. / Materials Science and Engineering A365 (2004) 2–13
and the ␲ bond energy contribution can be written as:
µν
µν
νµ
νµ
µν
νµ
(U␲ bond )ij = 2ppπij (Θjx,ix + Θjy,iy ) = 2(β␲ )ij (Θ␲ )ji ,
(12)
where β␲ ≡ppπ and Θ␲ is the sum of the two contributions
in parantheses.
It, therefore, follows that within the conventional TB approximation there is not a single, scalar bond order which
characterizes the ␴ bond but the four separate matrix quantities appearing in Eq. (11). In order to make contact with
the powerful chemical concept of a single ␴ bond order, the
first author suggested constraining the sp␴ bond integral to
be the geometric mean of |ssσ| and ppσ for the case of single component systems [9]. In this sub-section, we generalize for the first time this reduced TB approximation to the
µν
µν
case of multicomponent systems where |psσij | = spσij for
µ = ν. (Note that it follows from our choice of z-axis that
µν
µν
spσij > 0 but psσij < 0).
We begin by choosing ␴ hybrids on sites i and j, namely
the bonding hybrids pointing into the bond

1
µ

[|iµs
+
p
|iµz]
|iµσ = 
␴

µ

1 + p␴
(13)

1

ν

|jνσ = [|jνs − p␴ |jνz] 
1 + pν␴
and the non-bonding hybrids pointing away from the bond

1
µ

[
p
|iµs
−
|iµs]
|iµσ ∗ = 
␴

µ

1 + p␴
.
(14)

1


|jνσ ∗ = [ pν␴ |jνs − |jνs] 
1 + pν␴
We then reduce the number of independent ␴ bond integrals from four (ssσ, spσ, psσ and ppσ) to three by constraining
µν
µν
µν
µν
spσij psσij = ssσij ppσij
(15)
For the case µ = ν we recover our earlier constraint
equation [9], namely
µµ
µµ
µµ
(16)
spσij = |ssσij | ppσij ·
With respect to this new hybrid basis the intersite Hamiltonian matrix can be written as:
µν
0
(β␴ )ij
µν
(17)
Hij =
0
0
provided we choose
µν
ppσij
µ ν
p␴ p␴ =
µν
|ssσij |
and
µν
spσij
pν␴
µ =
µν ·
p␴
|psσij |
(18)
The single ␴ bond integral in Eq. (17) takes the value
µν
µ
µν
(20)
(β␴ )ij = (1 + p␴ )(1 + pν␴ ) ssσij ·
It follows that the ␴ bond energy can now be written in
terms of a single bond order, namely
µν
µν
νµ
(U␴ bond )ij = 2(β␴ )ij (Θ␴ )ji ·
(21)
Thus, the reduced TB approximation allows us to recover
the powerful concept of a single bond order for the ␴ bond.
The four independent TB ␴ bond integrals have been exµν
pressed in terms of the three reduced TB parameters (β␴ )ij ,
µ
p␴ and pν␴ as:

µν 
−1 
ssσij 





ν


µν 

µν
p␴ 
spσij 


|(β␴ )ij |
µ =
·
(22)
µν
µ
− p␴ 
ν)


psσij 
(1
+
p
)(1
+
p


␴
␴






µ ν 
µν 


p␴ p␴
ppσij
Substituting the constraint Eq. (15) into Eqs. (18) and (19)
we have
µν 2
ppσij
µ
p␴ =
(23)
µν
spσij
and
ν
p␴ =
µν 2
ppσij
µν
psσij
=
νµ 2
ppσij
νµ
spσij
·
(24)
In general, the three reduced TB parameters will depend
not only on the bond distance Rij but also on the local environment through the screening functions in Eq. (6). However, if we assume that the bond integrals between the orbitals of different species (say |µl and |νl with µ = ν) are
given by the geometric mean of the bond integrals between
the same orbitals of the species (i.e. |µl with |µl and |νl with |νl ), then from Eq. (23)
µµ
ppσij ppσijνν
ppσ µµ
=
=
(25)
pµ
µµ
␴
|ssσ| ij
|ssσij |ppσijνν
and similarly for pν␴ . Thus, if the ssσ and ppσ bond integrals
are assumed to be simply pairwise with identical distance
µ
dependences, then p␴ will be a constant that is characteristic
of the species µ being considered. It determines how much p
character is present in the original bonding hybrid, Eq. (13).
We will see in the following sections that it plays a critical
role in determining the angular character of the BOPs.
3. The atomistic modeling hierarchy
3.1. Coarse-graining by moments
(19)
The structural trend, which is displayed in Fig. 2 as a
function of the sp-band filling, can be understood by coarse-
D.G. Pettifor et al. / Materials Science and Engineering A365 (2004) 2–13
graining the electronic structure in terms of its moments.
Mathematically, the nth moment of the local density of states
(DOS) ρi (ε), associated with atom i, is defined to be
µn = εn ρi (ε) dε,
(26)
where the integral runs over the entire energy range of the
band. Thus, as is well known, µ0 , µ1 , µ2 and µ3 give the
area under the DOS (normalized to unity per orbital per
spin), the center of gravity, the mean square width, and the
skewness, respectively. The fourth moment µ4 determines
the unimodal versus bimodal character of the DOS through
the dimensionless shape parameter
s=
µ23
µ4
−
− 1·
µ22
µ32
(27)
If s < 1 the DOS is said to show bimodal behaviour,
whereas if s > 1 it shows unimodal behaviour.
Physically, within the TB approximation, the nth moment
of the local DOS ρi (ε) can be related to all self-returning
bonding or hopping paths of length n that start and finish on
atom i [21]. That is,
µn =
Hii1 Hi1 i2 . . . Hin−1 i
(28)
i1 ,i2 ,...in−1
This is a key result since it links the moments of the
eigenspectrum to the local environment about the atoms.
It follows that the local DOS of all the structures, which
are considered in Fig. 2, have the same first moment since
µ1 = Hii = 0, taking the on-site atomic energy levels as the
reference zero (i.e. Ep =
Es = 0). They also have identical
second moments µ2 = j=i Hij2 , since the structural energy
difference theorem has prepared their bond lengths to have
the same repulsive energy per atom. If we make the common
assumption that the overlap repulsion φij is proportional to
Hij2 , then Urep = 0 implies that
φij = 0 ⇒ Hij2 = 0 ⇒ µ2 = 0.
(29)
j=i
j=i
Thus, the structural trends in Fig. 2 are not driven by
differences in the second moments, because the local DOS
of all the structures have identical variance or mean square
width.
We are now in a position to understand the oscillatory
behaviour of the structural energy curves shown in Fig. 2.
In order to display the very small differences in energy between one structure and the next, these structural energy
curves were obtained by comparing the bond energy corresponding to the tight-binding DOS for a given structure
with that resulting from a constant, rectangular DOS with
the same second moment or variance. In 1971, Ducastelle
and Cyrot-Lackmann [43] had proved a very important moments theorem which states that if two DOS have moments that are identical up to some level n0 (i.e. µn =
0 for n ≤ n0 ), then their bond energy difference curve
7
must cross zero at least (n0 − 1) times as a function of
band filling. Thus, since the close-packed structures fcc
and hcp have very skew DOS with many three-membered
rings contributing to the third moment, their structural energy curves in Fig. 2, will cross zero once since n0 = 2
with µ3 = 0. However, for the open structures where
there are no three-membered rings so that n0 = 3 since
µ3 = 0, their structural energy curves will cross zero
at least twice, as illustrated graphically by the diamond
curve.
Therefore, the occurrence of close-packed stability for
less than half-full bands in Fig. 2 is due to the presence
of the three-member rings which skew the DOS to lower,
more bonding energies. However, as is seen from the
fcc and hcp structural energy curves, this skewing destabilizes the close-packed structures with respect to more
open structures for more than half-full bands. The trend
amongst the open structures from diamond at half-full
through As-type and linear chains to dimers for the nearlyfull sp-shell of Group VII is driven by the fourth moment
through the shape parameter s in Eq. (27). Diamond has
the smallest value of s, since its DOS is most bimodal
with its well-known hybridization gap. The dimer has the
largest value of s, since its electronic structure is most
unimodal with its non-bonding ␴ states. In conclusion, we
see that any interatomic potential, which claims to predict
rather than fit the structural stability of covalently bonded
systems, must go beyond the second-moment approximation, including at least the very important fourth-moment
contribution.
3.2. Beyond the second-moment approximation
During the past decade BOP theory has been developed
[9–15] to provide analytic expressions for the ␴ and ␲ bond
orders that are required for evaluating the bond energy. In
particular, bond orders have been derived for sp-valent systems with half-full shells such as C or Si within the so-called
symmetric four-level approximation. The ␴-bond order can
be written [14] in the form as:
µv
(Θσ )ij
1
=
ij
j
1+((2Φ2σ +R4σ +Φ̃i2σ Φ̃2σ (2+Φ̃4σ ))/(1+Φ̃4σ )2 )
(30)
where
j
Φ̃i2σ Φ̃2σ = Φ̃i2σ = j
Φi2σ Φ2σ
j
Φ4σ + Φi2σ Φ2σ
Φ4σ
j
Φ4σ + Φi2σ Φ2σ
(31)
(32)
8
D.G. Pettifor et al. / Materials Science and Engineering A365 (2004) 2–13
Fig. 4. Self-returning hopping paths of length 2 (panel (a)) and length
4 (panel (c)) which contribute to the potential functions Φi2␴ and Φi4␴ ,
respectively. The interference hopping path of length 3 (panel (b)) which
ij
contributes to the four-member ring function R4␴ [14].
with
Φ4␴ =
Φ4␴ − Φ22␴
·
Φ2 ␴
(33)
j
'n␴ and Φ22␴ are defined by Φn␴ = 1/2(Φin␴ + Φn␴ ) and
j
Φ22␴ = 1/2[(Φi2␴ )2 + (Φ2␴ )2 ], respectively. Note that in
order to avoid indeterminancies in Eqs. (31) and (32) for
i(j)
i(j)
the dimer and trimer due to Φ2␴ and Φ4␴ vanishing, we
add in practice an extremely small positive number γ to
their definitions, in order that the bond order approaches
numerically the correct dimer and trimer limit.
Thus, the ␴ bond order depends not only on the expected
self-returning second-moment two-hop and fourth-moment
i(j)
i(j)
four-hop contributions Φ2␴ and Φ4␴ , respectively, but also
ij
on the interference three-hop contribution R4␴ which couples atom i with atom j [44]. These terms are illustrated diagrammatically in Fig. 4 from which analytic expressions
follow straight-forwardly [13,14]. For example, the two-hop
contribution is given by
µk
Φi2␴ =
[g␴µ (θjik )]2 [(β̂␴ )ik ]2 ,
(34)
Fig. 5. The angular functions [g␴ (θ)]2 for p␴ = 0, 2, 3 and ∞. They are
compared with the empirical Tersoff [47] curve for Si [46].
for a pure p␴ bond corresponds to p␴ = ∞ and vanishes at
90◦ . The curve for p␴ = 3 corresponds to the usual sp3 hybrid, and vanishes at the tetrahedral bond angle of 109◦ . The
curve for p␴ = 2, which falls between the Chadi [47] and
Harrison [40] values of 1.57 and 2.31 for p␴ = ppσ/|ssσ|,
is close to that for the empirical Tersoff angular function for
silicon [46].
The ␲ bond order is the sum of two contributions as
expected from Eq. (12). Within the symmetric four-level
approximation for sp-valent systems it can be written [13]
in the form:
It reflects the directional dependence of the bonding hybrid, Eq. (13), and is illustrated in Fig. 5 for different values of p␴ [45]. The curve for a pure s bond corresponds to
p␴ = 0 and displays no angular character, whereas the curve
1 + Φ2 ␲ −
√
Φ4 ␲
+
1
1 + Φ2 ␲ +
√
Φ4 ␲
·
(36)
The two-hop contribution is given by:
Φ2␲ =
1
µκ
{(1 + cos2 θjik )[(β̂␲ )ik ]2
2
k=i,j
µκ
+ [g␲µ (θjik )]2 [(β̂␴ )ik ]2 + (iµ ↔ jν)}
k=i,j
where the weak second-order contribution from ␲ bonding
with the neighbouring atoms k has been neglected [9]. The
µκ
µκ
µν
normalized bond integral, (β̂␴ )ik = (β␴ )ik /(β␴ )ij , measures the relative strength of a neighbouring ␴ bond to that
of the ␴ bond of interest. The angular function is given by:
µ p␴
µ −1
g␴µ (θjik ) =
+ cos θjlk ]·
(35)
µ [(p␴ )
1 + p␴
1
µν
(Θ␲ )ij = (37)
and the four-hop contribution is given by:
Φ4 ␲ =
1 µκ
µκ
{[g␲µ (θjik )]2 [g␲µ (θjik )]2 [(β̂␴ )ik ]2 [(β̂␴ )ik ]2
4 k,k =i,j
µκ
2
+ [g␲µ (θjik )]2 [g␲ν (θijk )]2 [(β̂␴ )ik ]2 [(β̂␴ )νκ
jk ]
+ (iµ ↔ jν)}cos 2(φk − φk )·
(38)
The capped bond integrals have been normalized by the
µν
␲ bond integral (β␲ )ij rather than by the ␴ bond integral
as in Eq. (34). The term (iµ ↔ jν) implies an additional
contribution obtained by interchanging (iµ) and (jν) in the
preceding terms. Note that Eq. (38) differs from that in [13]
D.G. Pettifor et al. / Materials Science and Engineering A365 (2004) 2–13
in that we have made the valid assumption that (β␲ /β␴ )2 µ
µ
p␴ /(1 + p␴ ). The angular function is given by:
µ
p␴
g␲µ (θjik ) =
(39)
µ sin θjlk ·
1 + p␴
It reflects the angular dependence of projecting a ␲ orbital
on atom i through the bond angle φjik onto the appropriate
␴ bonding hybrid along ik. We see that the ␲ bond order
depends not only on the bond angles φijk and φjik , but also
on the dihedral angles (φk − φk ).
These analytic expressions for the bond orders provide
the first ‘classical’ potentials that have been derived directly
from a ‘quantum-mechanical’ description of the ␴ and ␲
bonds. They account naturally for the occurrence of single,
double, triple, conjugate and radical bonds in covalent systems. For example, consider the hydrocarbons where the ␴
bond order is predicted close to unity due to the open nature
of the structures [13]. The CC ␲ bond orders in going from
C2 H2 → C2 H4 → C2 H5 → C2 H6 can then be evaluated
from Eq. (36) assuming pC
␴ = 1 (which is close to the TB
value of ppσ/|ssσ| = 1.1 [48]). We find
(Θ␲ )CC

1+1









1
+
1/
1 + (3/2)β̂␴2

=


2 + 1/ 1 + (17/12)β̂2

1/
1
+
(2/3)
β̂
␴
␴







1/ 1 + (4/3)β̂␴2 + 1/ 1 + (4/3)β̂␴2
for C2 H2
where z is the local coordination within a first nearest neighbour model. Thus, the bond order of these directionally
bonded sp-valent systems varies inversely with the square
root of the coordination just as for the non-directional svalent systems considered by Abell [49].
This has important consequences on the nature of the
binding energy curves. Neglecting the promotion energy and
the ␲ bond contribution in Eq. (1), the binding energy per
atom can be written as:
1
Uz (R) = z[φ(R) − 2Θ␴(2) β␴ (R)]
(43)
2
where R is the nearest neighbour distance. Assuming the
overlap repulsion φ(R) is proportional to [β␴ (R)]2 (just as
we had done in computing the structural energy curves in
Fig. 2), we have
√
Uz (R) = z A[β␴ (R)]2 − zB β␴ (R)
(44)
for C2 H4
for C2 H5
β␴ (Rz ) =
for C2 H6
Substituting this into Eq. (44), we find the cohesive energy
Uz given by:
CC
Taking β̂␴ = (β␴ )CH
ik /(β␲ )ij = 6 [48], we see that the ␲
bond order decreases from 2 → 1.135 → 0.339 → 0.286
on going from C2 H2 → C2 H4 → C2 H5 → C2 H6 , where
the small non-integer contributions reflect the unsaturated
components of the ␲ bond within a TB or Molecular Orbital rather than a Valence Bond framework. Thus, including the single ␴ bond order, we have recovered the expected
behaviour from a triple bond for C2 H2 , a double bond for
C2 H4 , and a single bond for C2 H6 . Importantly, the radical C2 H5 remains singly bonded when a hydrogen atom is
abstracted from C2 H6 . These BOPs should, therefore, be
invaluable for modeling the growth of covalently bonded
films.
Finally, before ending this subsection, we must reiterate
that the second-moment approximation is unable to make
structural prediction within covalently bonded systems. If
we had made the symmetric two-level approximation within
BOP theory, then we would have predicted the ␴ bond order
1
,
1 + Φ2σ
dimensional lattices diamond, simple cubic and face-centred
cubic, then remarkably summing over the angular function
in Eq. (34) leads to the expression [22]
µ
µ
[(1 + p␴ )/ 1 + ((1/3)(p␴ )2 )]
(2)
(42)
Θ␴ =
z1/2
where A and B are constants. Setting Uz (Rz ) = 0 gives us
the implicit condition for the equilibrium bond length Rz ,
namely
(40)
Θ␴(2) = √
9
(41)
which appears not too dissimilar from the empirical Tersoff form [2]. However, if we consider the simple three-
Uz =
B
√
(2 z A)
1 B2
1 B2
1 B2
−
=−
,
4 A
2 A
4 A
(45)
(46)
which is independent of coordination.
Thus, although we recover the empirical Pauling relation
between the bond length and coordination (or bond number)
by assuming an exponential form for the bond integral in
Eq. (45), the second-moment approximation cannot provide
structural differentiation. This is consistent with our earlier
conclusions using the structural energy difference theorem.
(It follows from substituting Eq. (45) into Eq. (28) that the
second moment of the DOS of the three lattices are all identical; in addition, it follows from substituting Eq. (45) into
Eq. (44) that they have the same equilibrium repulsive energy per atom.) The behaviour of the binding energy curves
within the second-moment approximation is illustrated by
the left-hand panel in Fig. 6 [11], where we see that the diamond, sc and fcc lattices have identical cohesive energies.
Turning on the fourth-moment contributions in the righthand panel leads to their structural differentiation, the closepacked lattice being destabilized by the presence of many
four-member rings [14]. Structural prediction requires information about the shape of the electronic density of states,
not just its mean square width or variance [16].
10
D.G. Pettifor et al. / Materials Science and Engineering A365 (2004) 2–13
Fig. 6. Model binding energy curves within 2-level (left-hand panel) and
4-level (right-hand panel) BOP theory, which has been simplified by
neglecting the promotion energy and the ␲ bond contribution. The overlap
repulsion has been assumed to fall off with distance as the square of the
␴ bond integral. The curves correspond to the dimer (z = 1), graphite
(z = 3), diamond (z = 4), simple cubic (z = 6) and fcc (z = 12) [11].
3.3. Analytic BOPs with GSP radial functions
These analytic bond-order potentials might at first appear
very complicated. It is, therefore, important that we identify
the different physical roles played by the various contributions and how to find the numerous fitting parameters. We
will investigate this behaviour for the first time by assuming
that the distance dependences of the repulsive potential and
the bond integrals between species µ and ν are determined
by the same GSP [23] functional form, namely



µν nµν
c


R
R
R
0
µν 

f µν
=
−
1
,
exp
−λ
µν
µν


R
R0
R0
(47)
µν
where R0 is the equilibrium nearest neighbour distance
for the µν ground state structure (or a nearby metastable
phase with a simple structure such as in Fig. 6 if the true
µν
ground state is distorted). λµν and nc are shape parameters
determining how fast the function cuts off with respect to
µν
R0 . (The original GSP paper used the two independent
shape parameters Rc and nc where λ = (R0 /Rc )nc ). The GSP
functional form has the desirable property that f µν (1) = 1.
The total binding energy is then determined by Eq. (1).
The repulsive interactions and bond integrals are written in
the form:
µν 
µν
µν
µν
φij = φ0 [f µν (Rij /R0 )]m 


µν
µν
µν nµν
µν
(β␴ )ij = β␴,0 [f (Rij /R0 )]
(48)


µν

µν
µν
µν
(β␲ )ij = β␲,0 [f µν (Rij /R0 )]n
where the exponents m and n depend on the species pair µν.
We will assume here that the prefactors φ0 , β␴,0 and β␲,0
are constant that are environment independent. However,
this constraints could be relaxed by assuming the screening
functions, such as in Eq. (6) are structure dependent but
volume independent [17] without changing the logic of the
argument presented in this subsection. The bond energy is
Fig. 7. The normalized promotion energy, Uprom /δ, vs. the argument y,
which is a measure of the normalized bond integral, β␴ (R)/δ. For carbon
systems a typical value of y is around 5, where the promotion energy is
saturating.
completely determined by the ␴ and ␲ bond integrals above
µ
and the angular coefficients p␴ that enter the bond order
through Eqs. (35) and (39). (Note that in general the bond
order also depends on the occupancy of the band [8,11,15]
and weakly on the sp atomic energy level separation δµ [13]).
The promotion energy cannot be neglected in Eq. (1). For
the case of C and Si the ␴ and ␲ atomic energy level separation, δ = (Ep − Es ), is approximately 7 eV, so that there
is an energy penalty of about 7 eV per atom to promote the
free atom s2 p2 configuration to the hybrid sp3 configuration
which is more appropriate for the bulk diamond structure,
for example. Following Eq. (108) of [11], we approximate
the promotion energy associated with the atom µ at site i by
the expression:


1
µ
,
(Uprom )i = δµ 1 − (49)
µ 2
1 + (yi )
where
µ
(yi )2
=
µν
Aij
µν 2
(β␴ )ij
δµ
j=i
µν
·
(50)
The prefactor Aij is environment dependent. For bulk
CH
carbon and the hydrocarbons, we found that ACC
ij = Aij =
10/zi resulted in excellent values of the promotion energy
compared to TB (c.f. Fig. 2 of [12]).
The behaviour of the normalized promotion energy,
Uprom /δ, is displayed in Fig. 7 as a function of y or the normalized bond integral, β␴ (R)/δ. We see that as the atoms
are brought together from infinity (corresponding to y = 0)
the promotion energy increases quadratically as y2 before
saturating and tending asymptotically to unity as (1 − 1/y).
Carbon atoms in graphite, diamond and the hydrocarbons
take values of y around 5 where we see from the curve the
promotion energy has nearly saturated at a value of about
80% [12]. In this subsection, we will, therefore, neglect
for algebraic simplicity the influence of the derivatives of
D.G. Pettifor et al. / Materials Science and Engineering A365 (2004) 2–13
the promotion energy on the binding energy curves around
equilibrium.
We are now in a position to derive equations linking
the parameters to physical quantities in multicomponent spvalent systems, such as Ga–As. We begin by considering the
elemental systems with respect to simple lattices with equivalent nearest neighbour bonds such as z = 1 (the dimer),
z = 2 (the linear chain), z = 3 (a graphene sheet), z = 4
(diamond), z = 6 (sc), and z = 12 (fcc). Choose those
two simple lattices with the lowest cohesive energies (which
may be evaluated by first principles density functional theory if they are unavailable experimentally). For example, we
could take fcc and sc Ga the former being only 0.07 eV per
atom higher in energy than the seven-fold coordinated ␣-Ga
ground state [50]; and sc and a single graphene sheet of As,
the former being 0.06 eV per atom higher in energy than the
(3 + 3)-fold coordinated ␣-As ground state [51].
Elemental systems are characterized by 10 parameters:
the GSP shape parameters, λ and nc ; the repulsion parameters, φ0 and m; the bond integral parameters, β␴,0 , β␲,0 and
n; the angular dependence parameter, p␴ ; the promotion energy parameters, δ and A. We will assume that β␴,0 and β␲,0
have already been predicted by screened TB-LMTO theory. In this paper, for algebraic simplicity, we will implicitly
µ
choose A by taking the value of (yi )2 in the expression for
the promotion energy to be its carbon value with respect to
the diamond lattice, namely 24 (see Fig. 2 of [12]), and neglect its change with bond length for small changes about
equilibrium. This leaves seven unknown parameters which
will be fitted to the six equations resulting from the cohesive energy, equilibrium bond length, and curvature of the
two lowest binding energy curves corresponding to simple
first-nearest neighbour structure-types. A seventh equation
follows from the assumption that the model is first nearest neighbour only so that the GSP shape parameters must
guarantee small values out at second nearest neighbours and
beyond.
We start by assuming physical values of m/n and p␴ ,
namely m/n = 2 (c.f. this was used to evaluate the structural energy curves in Fig. 2) and p␴ = ppσ/|ssσ| from TBLMTO. We then iterate as follows:
(i) The equilibrium bond length of the ground state simple
lattice: U0 (R0 ) = 0 implies the repulsive interaction
can be evaluated from:
φ0 =
2β0
m/n
(51)
Θ4␲ vanishes identically (c.f. Eq. (99) of [11]). Thus,
summing over all the nearest neighbours in Θ2␲ using
Eq. (13) of [22], we find
Θ␲
=
2
,
{(4/3)z[1+(1/2)(p␴ /(1+p␴ ))(β␴,0 /β␲,0 )2 ]−1}1/2
(53)
where z = 4, 6 and 12 for diamond, sc and fcc, respectively.
(ii) The cohesive energy constraint of the ground state simple lattice: the cohesive energy implies the sp atomic
energy level separation δ through
1
0.8δ = U0 − z0 (φ0 − 2β0 )
2
(54)
where z0 is the coordination of the most stable simple
lattice. This fitted value of δ can be checked against
what is expected from the known value of (Ep − Es ).
(iii) The curvature of the ground state simple lattice: the
curvature U0 about equilibrium implies values for the
effective exponents m̃ and ñ, since
m
!
U0 = z0 β0
(55)
− 1 ñ2
n
where
m̃ = (1 + λnc )m
(56)
and
ñ = (1 + λnc )n.
(57)
It follows that m̃/ñ = m/n. We would expect the
fitted value of ñ to be close to 2 to reflect Harrison’s
first nearest neighbour canonical TB parameters [40].
However, we see that we are unable to unravel the
values of m and n explicitly, only their effective values
m̃ and ñ at the equilibrium distance R0 . In order to
obtain m and n separately we need to consider the
nearby metastable structure with coordination z, bond
length Rz , and curvature Uz .
(iv) The equilibrium bond length of the metastable phase:
Uz (Rz ) = 0 implies that the GSP function at the equilibrium bond length Rz takes the value
1/(m−n)
βz
Rz
fz = f
=
(58)
R0
β0
where
where
β0 = (|β␴,0 |Θ␴,0 + |β␲,0 |Θ␲,0 )·
11
(52)
For a nearest neighbour model Θ␲,0 is a function
only of p␴ (and the structure) as the normalized bond
integrals are unity. Θ␲,0 also depends on the ratio
(β␲,0 /β␴,0 ). For example, for the cubic lattices diamond, simple cubic and fcc the four-hop contribution
βz = (|β␴,0 |Θ␴,z + |β␲,0 |Θ␲,z )
(59)
can be evaluated from the analytic BOP expressions.
(v) The curvature of the metastable phase: using the above
Eq. (58) the curvature Uz implies values for the effective exponents m̃z and ñz , since
m
!
Uz = zβz
(60)
− 1 ñ2z fzn
n
12
D.G. Pettifor et al. / Materials Science and Engineering A365 (2004) 2–13
where
m̃z =
and
ñz =
R0
Rz
R0
Rz
+ λnc
+ λnc
Rz
R0
Rz
R0
nc −1 nc −1 m
(61)
n.
(62)
It follows that m̃z /ñz = m̃/ñ = m/n.
(vi) The first nearest neighbour approximation: the GSP
shape parameters λ and nc must be chosen so that they
are small by the second nearest neighbours, say at R =
1.5R0 . We, therefore, have the equation:
f(1.5) =
1
exp[−λ(1.5nc − 1)] = 0.1
1.5
(63)
where the right-hand side has been chosen somewhat
arbitrarily as one-tenth. In practice, the tail of the GSP
function is connected to a cubic spline to guarantee it
vanishes identically before the second nearest neighbour distance. From Eqs. (56), (57), (61) and (62) we
have a second equation involving λ and nc , namely
m̃z
ñz
(R0 /Rz ) + λnc (Rz /R0 )nc
=
=
·
m̃
ñ
1 + λnc
(64)
Eqs. (63) and (64) can be solved numerically for
values of λ and nc , obtaining physical values by adjusting the constraint on the right-hand side of Eq. (63)
if necessary. This allows Eq. (58) to be solved for the
individual values of m and n. Hopefully, the new ratio
of m/n is not too far from the expected value of 2 for
sp-valent elements.
(vii) The energy difference between the two structures: the
structural energy difference theorem allows us to express U as
1−(n/m) βz
z
U = 1 −
z 0 β0 ·
(65)
β0
z0
We find the key result, as noted earlier by Goodwin
et al. [23], that the structural energy differences are
functions only of the ratio m/n, not of the GSP shape
parameters λ and nc . Assuming the m/n value previously fitted, the above equation can be used to adjust
p␴ through the ratio βz /β0 .
Finally, steps (i)–(vii) would be iterated to try and find
a sensible physical set of parameters for the elemental systems, which could be further tested against the shear moduli [45], defect energies and predicting the true (distorted)
ground state, if different from that of a simple lattice. The
binary AB system would then be fitted in a similar fashion
with the initial values of β␴,0 and β␲,0 again being provided
by first principles screened TB-LMTO calculations. Hopefully, the on-site parameters δA and δB (influencing the proB
motion energies) and pA
␴ and p␴ (influencing the angular
dependences) will be found to be transferable from the elemental situation to the binary systems. However, the latter
parameters which are key to the shear moduli and structural
energy differences, may need to be refined through the introduction of some environmental dependence (c.f. Eqs. (23)
and (24)). A fitting procedure to the above equations is currently being automated by using a genetic algorithm, the results of which will be presented elsewhere for the case of
GaAs.
4. Conclusions
We have shown that analytic bond-order potentials can
be derived for sp-valent multicomponent systems within a
reduced tight-binding framework using BOP theory. The
expression for the ␴ bond order includes the critical fourthmoment contribution, which we have argued is key for
structural differentiation as it controls the unimodal versus
bimodal character of the local density of states. The expression for the ␲ bond order depends explicitly on the dihedral
angles, so that any given bond experiences a torsional stiffness. Together the ␴ and ␲ bond orders give a good account
of single, double, triple, conjugate and radical bond behaviour in hydrocarbon systems. The close link between the
different BOP parameters and different physical properties
was illustrated for the particular choice of a GSP-type radial
dependence for the repulsive interaction and bond integrals.
Currently, these analytic sp-valent BOPs are being developed in order to model the growth of C and GaAs films.
Acknowledgements
We thank Sergei Dudarev, Nigel Marks, Sanghamitra
Mukhopadhyay, Ivan Oleinik, Dave Pankhurst, and Vasek
Vitek for helpful discussions. This research on the development of analytic BOPs is funded by DARPA under contract
No. MDA972-00-1-0016.
References
[1] F.H. Stillinger, T.A. Weber, Phys. Rev. B 31 (1985) 5262.
[2] J. Tersoff, Phys. Rev. Lett. 56 (1986) 632.
[3] J.F. Justo, M.Z. Bazant, E. Kaxiras, V.V. Bulatov, S. Yip, Phys. Rev.
B 58 (1998) 2539.
[4] N. Marks, J. Phys.: Condens. Matter 14 (2002) 2901.
[5] D.W. Brenner, Phys. Rev. B 42 (1990) 9458.
[6] D.W. Brenner, O.A. Shenderova, J.A. Harrison, S.J. Stuart, B. Ni,
S.B. Sinnott, J. Phys.: Condens. Matter 14 (2002) 783.
[7] K. Nordlund, A. Kuronen, Nucl. Instrum. Meth. Phys. Res. B 159
(1999) 183.
[8] D.G. Pettifor, Phys. Rev. Lett. 63 (1989) 2480.
[9] D.G. Pettifor, Springer Proc. Phys. 48 (1990) 64.
[10] M. Aoki, D.G. Pettifor, in: P.M. Oppeneer, J. Kübler (Eds.), Physics
of Transition Metals, World Scientific, Singapore, 1993, p. 299.
[11] D.G. Pettifor, I.I. Oleinik, Phys. Rev. B 59 (1999) 8487.
D.G. Pettifor et al. / Materials Science and Engineering A365 (2004) 2–13
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
I.I. Oleinik, D.G. Pettifor, Phys. Rev. B 59 (1999) 8500.
D.G. Pettifor, I.I. Oleinik, Phys. Rev. Lett. 84 (2000) 4124.
D.G. Pettifor, I.I. Oleinik, Phys. Rev. B 65 (2002) 172103.
D.G. Pettifor, I.I. Oleinik, Prog. Mat. Sci., in press.
D.G. Pettifor, Bonding and Structure of Molecules and Solids, Clarendon Press, Oxford, 1995.
M.S. Tang, C.Z. Wang, C.T. Chan, K.M. Ho, Phys. Rev. B 53 (1996)
979.
D. Ngyuen-Manh, D.G. Pettifor, V. Vitek, Phys. Rev. Lett. 85 (2000)
4136.
J.C. Cressoni, D.G. Pettifor, J. Phys.: Condens. Matter 3 (1991) 495.
D.G. Pettifor, R. Podloucky, Phys. Rev. Lett. 53 (1984) 1080.
F. Cyrot-Lackmann, Adv. Phys. 16 (1967) 393.
S.R. Nishitani, P. Alinaghian, C. Hausleitner, D.G. Pettifor, Philos.
Mag. Lett. 69 (1994) 177.
L. Goodwin, A.J. Skinner, D.G. Pettifor, Europhys. Lett. 9 (1989)
701.
A.P. Sutton, M.W. Finnis, D.G. Pettifor, Y. Ohta, J. Phys. C 21
(1988) 35.
http://www.materials.ox.ac.uk/mml/software/detail.html.
D.G. Pettifor, R. Podloucky, J. Phys. C 19 (1986) 315.
D.G. Pettifor, Solid State Phys. 40 (1987) 43.
M.W. Finnis, A.T. Paxton, M. Methfessel, M. van Schilfgaarde, Mat.
Res. Soc. Symp. Proc. 491 (1998) 265.
A.J. Skinner, D.G. Pettifor, J. Phys.: Condens. Matter 3 (1991)
2029.
H. Haas, C.Z. Wang, M. Fähnle, C. Elsässer, K.M. Ho, Mat. Res.
Soc. Symp. Proc. 491 (1998) 327.
H. Haas, C.Z. Wang, M. Fähnle, C. Elsässer, K.M. Ho, Phys. Rev.
B 57 (1998) 1461.
13
[32] D. Nguyen-Manh, D.G. Pettifor, S. Znam, V. Vitek, Mat. Res. Soc.
Symp. Proc. 491 (1998) 353.
[33] O.K. Andersen, O. Jepsen, Phys. Rev. Lett. 53 (1984) 2571.
[34] C.Z. Wang, K.M. Ho, M.D. Shirk, P.A. Molian, Phys. Rev. Lett. 85
(2000) 4092.
[35] D. Nguyen-Manh, D.G. Pettifor, D.J.H. Cockayne, M. Mrovec, S.
Znam, V. Vitek, Bull. Mat. Sci. 26 (2003) 43.
[36] G.D. Lee, C.Z. Wang, Z.Y. Lu, K.M. Ho, Phys. Rev. Lett. 81 (1998)
5872.
[37] M. Pei, W. Wang, B.C. Pan, Y.P. Li, Chin. Phys. Lett. 17 (2000) 215.
[38] M. Mrovec, D. Nguyen-Manh, D.G. Pettifor, V. Vitek, Mat. Res.
Soc. Symp. Proc. 653 (2001) Z6.3.
[39] D.G. Pettifor, J. Phys. C 19 (1986) 285.
[40] W.A. Harrison, Electronic Structure and Properties of Solids, Freeman, San Francisco, 1980.
[41] D.G. Pettifor, Solid State Commun. 51 (1984) 31.
[42] O.K. Andersen, Phys. Rev. B 12 (1975) 3060.
[43] F. Ducastelle, F. Cryot-Lackmann, J. Phys. Chem. Solids 32 (1971)
285.
[44] D.G. Pettifor, M. Aoki, Philos. Trans. Roy. Soc. London A334 (1991)
429.
[45] P. Alinaghian, S.R. Nishitani, D.G. Pettifor, Philos. Mag. B 69 (1994)
889.
[46] J. Tersoff, Phys. Rev. B 38 (1988) 9902.
[47] D.J. Chadi, Phys. Rev. B 29 (1984) 785.
[48] C.H. Xu, C.Z. Wang, C.T. Chan, K.M. Ho, J. Phys.: Condens. Matter
4 (1992) 6047.
[49] G.C. Abell, Phys. Rev. B 31 (1985) 6184.
[50] M. Bernasconi, G. Chiarotti, E. Tosatti, Phys. Rev. B 52 (1995) 998.
[51] R. Needs, R. Martin, O. Nielsen, Phys. Rev. B 35 (1987) 9851.