Calculation of total energies in multicomponent oxides

A. F. Kohan and G. Ceder

Department of Materials Science and Engineering

Massachusetts Institute of Technology

77 Massachusetts Ave. Room 13-4061

Cambridge, MA 02139

U.S.A.

phone #: (617)252-1507

Fax #: (617)258-6534

(October 4, 1996)

1

Abstract

The accuracy of di�erent total{energy methods to compute the properties

of multicomponent oxides is studied. These materials have typically large

unit cells and consequently, computer{running time considerations become

important. Many approximations are introduced in order to speed up calcu-

lations but at the expense of loosing accuracy. We show that while highly

sophisticated quantum{mechanics techniques such as pseudopotentials or the

full{potential linearized{augmented{plane{wave method can be used to ac-

curately compute materials properties, they may require prohibitively long

computer runs in oxides. On the other hand, simple potential models, or

even fast quantum{mechanics methods such as the spherical self{consistent

atomic deformation or the linear mu�n{tin orbital method (in the atomic

sphere approximation), are not always reliable to study oxides. Charge trans-

fer, breathing of the oxygen ions, and non{spherical charge relaxations are

some of the factors that can make any of these schemes fail. However, it is

not necessary to always use sophisticated techniques. We show that the self{

consistent semiempirical tight{binding formalism can be used as an interpo-

lation tool to extend the results of accurate calculations for a few compounds

in a system to the rest of them. This opens new possibilities for the use of ab

initio methods to study technologically{relevant materials properties, such as

the temperature behavior of oxides, since formation energies of many di�erent

compounds at 0 oK are a crucial input to these models.

I. INTRODUCTION

In the last decades, computer technology has been evolving at a very fast pace making

possible the use of powerful combinations of quantum{ and statistical{mechanics techniques

to study materials properties. The long sought goal of predicting materials properties before

2

they are even synthesized is �nally coming to a reality. Computational experiments, as

these calculations are usually called, have the advantage that they o�er full control of any

experimental condition and that they can be performed even under extreme values of such

conditions.

At the core of most of these methods lies a model that describes the energetics of the

system being studied. Nearly all physical properties are related to total energies (equations

of state) or their di�erences. By just computing total energies, valuable information such

as elastic properties, lattice constants, defect arrangements, and structural stability can

already be obtained. However, in most cases of interest for materials science, total{energy

calculations by themselves do not su�ce but need to be used in combination with statistical{

mechanics [1] or molecular{dynamics methods [2,3], as the temperature dependence and

time evolution of the system are usually essential to understand the properties of materials.

This adds a new dimension to the complexity of computing total energies since not only do

the methods have to be accurate but also fast. In all of the ab initio schemes developed

to study temperature{dependence behavior for real materials, total{energy methods are

usually the limiting step in the calculations. For example, temperature{composition phase

diagrams are one of the most important tools in designing and processing materials. Their

ab initio calculation usually requires the computation of formation energies ofmany di�erent

compounds at 0 oK with high accuracy [4{6] (of the order of a few hundredths of an eV).

Even for very simple systems, these calculations are very computer{time consuming if one

is looking for quantitative predictions.

Oxide materials present a particularly di�cult challenge to total{energy methods. They

usually have large unit cells, low symmetry, and a mixed ionic{covalent bonding. Under these

conditions, accurate quantum{mechanical techniques require extremely long computer runs.

Consequently, modeling in oxides has been mainly done using simple empirical potential

models [7]. These models have been successful in predicting defect energies, lattice con-

stants, and some elastic properties [8,9]. However, they are inadequate to make quantitative

predictions when computing phase diagrams [10]. It is an open question which of the more

3

elaborate, but more time consuming, quantum{mechanical models will improve this pic-

ture since some of them use approximations, such as spherically averaged electronic charge

densities or frozen electronic cores, that may not be su�ciently accurate in oxide systems.

The objective of this work is to evaluate the accuracy of di�erent total{energy tech-

niques for the calculation of formation energies. Several methods will be compared on two

problems: formation energies in the CaO{MgO system and polymorphic transition energies

between the phases of ZrO2 (zirconia). The former system presents a simple miscibility gap

in the temperature{composition phase diagram and consequently the di�erent CaO{MgO

compounds are expected to have positive formation energies. Zirconia is stable at room

temperature in a monoclinic structure (baddeleyite) and transforms into a tetragonal phase

(space group P42=nmc) at approximately 1180 oC [11]. Before melting, it transforms again,

but now into a cubic phase ( uorite{type structure) at approximately 2350 oC [12].

We will start in Section II by brie y introducing the di�erent techniques used in this

work. Then, in Section III, we apply these methods to compute formation energies and

cell parameters for compounds in the CaO{MgO system and to compute the di�erence in

energies between the observed ZrO2 phases. Finally, in Section IV, we discuss the e�ect

of the approximations used in the di�erent total{energy techniques and we show to what

degree they are acceptable in oxides.

II. TOTAL{ENERGY METHODS

The �rst total{energy calculations date back to the 1920s, when the main concern was

trying to understand cohesion in periodic solids [13,14]. Since then, many di�erent schemes,

from classical empirical models to highly sophisticated quantum{mechanical techniques have

been proposed and changed during the years. A major step was the introduction of density

functional theory [15] for the treatment of the many{body electronic problem. Currently,

most ab initio quantum{mechanical models are based on a local density approximation [16]

(LDA) to this theory or on improvements over the LDA by using gradient corrections [17,18].

4

Other methods, based on the Hartree{Fock approximation are also used [19].

Two di�erent approaches can be taken to compute total energies: We can assume a model

for the electronic density and solve for the energy, or we can directly solve Schr�odinger's

equation self consistently. Empirical potentials correspond to the �rst category and represent

one of the simplest energy models that can be used in oxides. In this case, the electronic

density and ionic charges are replaced by point charges centered at the ions. The energy

is computed as the sum of the electrostatic interaction and a short{range repulsive term

that represents the overlap of the electronic clouds. The value of the point charges can be

obtained from the ideal chemical valence or can be considered as a parameter of the potential.

On the other hand, in self{consistent quantum{mechanical methods no a priori assumption

is made about the electronic density (though in some cases spherical symmetry is assumed).

The main di�erence between the methods in the last category is in the basis they use to

expand the electronic density. A fast total{energy technique is fundamental for any attempt

of describing oxide's properties ab initio. Usually, fast methods imply using a large number

of approximations. Since accuracy is important to produce quantitative predictions, we will

evaluate di�erent total{energy techniques with di�erent degrees of approximations.

A two{body potential was used in this work to describe the short{range interaction

between ions with the following form:

V (rij) = Aije

��

rij

�ij

��

Cij

r6ij; (1)

where Aij, �ij, and Cij are parameters that depend on the identity of the ions i and j, and

rij is the distance between them. The polarizability of each ion was treated with a shell

model [20,21], where the total charge of the ion is split between a massless shell of charge

Qs and a core with charge Qc connected with an harmonic spring with constant ks. Thus,

the interaction between a core and a shell separated a distance d is given by,

V (ri) =1

2ksd

2: (2)

The ewald summation method [13] was used to compute the electrostatic energy of the point

charges. This is a completely classical approach where no contribution to the total energy

5

coming from the kinetic energy or exchange and correlation of the electrons is taken into

account explicitly. Point charges are �xed and do not depend on the environment.

Quantum{mechanical methods involve more complex, and consequently more time con-

suming calculations than potential models. One technique particularly fast in oxides is the

spherical self{consistent atomic deformation (SSCAD) model [22,23]. The SSCAD is a mod-

i�ed version of the Gordon and Kim electron gas theory [24]. The electronic charge density

is assumed to be localized around each ion, resulting in single{particle Schr�odinger's equa-

tion per ion, which is solved self consistently in the �eld of the other ions. The potential

energy term is spherically averaged. Consequently, its use is restricted in its present form, to

systems where the non-spherical components of the electronic charge density are negligible.

This condition is not necessarily satis�ed in all oxide materials and may lead to incorrect

predictions. In Ref. [25] it is demonstrated that a non{self{consistent version of the SSCAD

incorrectly predicts zirconia to be stable in the cubic phase (see Section III).

Other quantum{mechanical methods introduce less approximations, at the expense of

time performance. In this work we make use of two of these more accurate techniques: the

linear mu�n{tin orbital method in the atomic{sphere approximation [26,27] (LMTO{ASA)

and the pseudopotential (PP) method [28]. We will also compare all these techniques against

published results obtained with the linearized{augmented{plane{wave in its full potential

version [29,30] (FLAPW).

It is not our intention to review all these total{energy methods, instead we will brie y

mention the main approximations they use. In both LMTO and LAPW, space is divided

into spheres (generally centered around atoms) and a mu�n tin potential is used. Basis

functions are de�ned both in the interstitial region and within the spheres (MTO's in the

former method and APW's in the latter, but in both cases the energy dependence is linearized

around a given energy for the di�erent angular momenta). The basis sets used in these

methods are relatively small compared to the ones used in �xed basis methods such us

the pseudopotential plane wave technique. Usually, an order of magnitude less MTO's than

APW's are needed to accurately solve Schr�odinger's equation, making the LMTO a very fast

6

scheme. To make the LMTO less computational intensive it is frequently used in the atomic

sphere approximation [26,27] (ASA). In the ASA, all integrals over space are substituted by

integrals over space �lling spheres (Wigner{Seitz overlapping spheres) generally centered at

atoms, and the electronic density is spherically averaged within each sphere. The potential

is a superposition of spherical contributions from each sphere. In close{packed solids the

LMTO{ASA approximations are very accurate. Some corrections are sometimes used to

compensate for the sphere overlap (usually called \combined corrections" [27,31,32]) and

the symmetrization of the potential (usually called \mu�n tin corrections" [33]). In more

open structures the method is less accurate and empty spheres (not centered at atoms) are

usually used. A full{potential version of the LMTO, where no shape approximations are

made, has been developed [34] but we will not discuss it here. The full potential version

of the the LAPW also does not have shape approximations for the potential. In this case,

when the calculations are carefully performed, the only approximation within DFT is the

LDA.

Both LMTO and LAPW (or their full{potential versions) are used as all{electron meth-

ods with energy-linearized basis functions. On the other hand, in the pseudopotential

method only the wave functions of the valence electrons are computed (frozen core ap-

proximation) and they are usually expressed as a combination of plane waves. The frozen

core approximation is based on the fact that, in general, only valence electrons take part in

bonding while the core electrons, which are tightly bound to the nucleus, can be regarded

as frozen. The e�ect of the core electrons on the valence states is taken into account as a

potential that is added to the nuclear potential. This pseudopotential is chosen so that its

valence pseudo wave functions are equal to the actual all{electron wave functions beyond a

given core radius and so that they do not have nodes in the core region. Since the electrons

that participate in bonding usually reside beyond the core, using pseudopotentials is an

excellent approximation in most cases.

The use of the pseudopotential approximation not only allows for a reduction in the

number of plane{wave basis functions but also in the number of orbitals that need to be

7

computed. All{electron techniques need to accurately treat core energies that constitute a

large part of the total energy, while generally their e�ects are canceled out when energy dif-

ferences are taken. Consequently, the pseudopotential energy is much smaller and requires

a smaller relative accuracy than all-electron methods when computing formation energies.

The use of plane waves greatly simpli�es the calculations: Shape approximations are not

necessary and Pulay forces [35] are zero. The computational speed of the method depends

on the chemical identity of the elements that form the material. Sharply peaked valence

states, as in �rst row nonmetals such as oxygen, or in transition metals, require a large num-

ber of plane waves to be expanded. However, in the last years, new developments such as

Car{Parrinello molecular dynamics [3], conjugate{gradients [36], and optimized pseudopo-

tentials [37] have shown that pseudopotential calculations can be e�ciently performed for

any element in the periodic table [36].

In this work we will also use the semiempirical tight{binding method [38] in the self{

consistent form suggested in Ref. [39] and [40]. Contrary to empirical potential models, an

explicit treatment of the electronic structure is made and all the relevant terms for computing

total{energy di�erences in systems with charge transfer (such as oxides) are incorporated.

The diagonal terms of the Hamiltonian are not �xed but written as

"i;� = "oi� + uintrai� + uinter

i� + Vi� + gi�; (3)

where "oi� represents the kinetic energy and the interaction with its own ionic core (at site

i) of an electron at orbital �, and uintrai� and uinter

i� are the short{ and long{range interaction

of the electron in the orbital i� with the other electrons. Vi� represents the crystal �eld at

site i and gi� is a shift introduced by the orthogonalization of the basis functions [40,41] and

corresponds to

gi� = �X

i0�0 6=i�

si�;i0�0ti�;i0�0: (4)

si�;i0�0 are the overlap matrix elements of the original non{orthogonal basis functions and

ti�;i0�0 are o�{diagonal hopping terms. These terms are assumed to depend on the distances

between ions i and j.

8

The interaction between electrons, ui�, is written as

uintrai� =

X

�0

Ui�;i�0Qi�0 ; (5)

uinteri� =

X

i0 6=i;�0

Ui�;i0�0Qi0�0 ; (6)

where Ui�;i0�0 represents electron{electron two center integrals and Ui�;i�0 are intra{atomic

repulsion parameters [42]. Qi� is the occupation of the orbital i�

Qi� =occX

n;~k

j< 'i� j n~k >j2 : (7)

The sum over all occupied orbitals is associated with the number of electrons at site i.

Since Qi� depends on the solution eigenvectors, j n~k >, these eigenvectors need to be found

self consistently ('i� are the atomic{like basis functions). In this way, charge transfer is

incorporated into the formulation through the ui� terms, the e�ect of the crystal �eld (which

shifts the atomic eigenvalues when the solid is formed), and the hopping terms. The gi�,

on the other hand, is a repulsive term that represents an increase in the kinetic energy of

the electrons when they are brought together. Finally, the hopping integrals are essential

to determine the band structure of a solid and take into account the e�ect of lowering the

electronic energy when an electron is attracted not only by its own core but also by the

neighboring ones. Most of these terms are not explicitly accounted for in potential models,

and the charge on the ions is usually �xed, and independent of the environment of the

atom. The tight{binding model is not limited to a given kind of bonding and no shape

approximations for the ionic potentials need to be made.

This formulation of tight{binding is particularly fast since the time{consuming hopping

and overlap integrals are �tted to ab initio results. In this sense, this method represents a

much more sophisticated and accurate interpolation tool than potential models and is still

orders of magnitude faster than ab initio quantum{mechanical techniques.

We have reviewed the main approximations and we will now describe how they a�ect

the calculation of total{energy di�erences and cell parameters in oxide systems.

9

III. RESULTS

A. Ab initio Pseudopotential Results

In previous work [40] we reported formation energies obtained for 10 compounds in the

CaO{MgO system using the pseudopotential method (they are reproduced in Fig. 1). The

energy of some of those structures was also computed in Ref. [10] using the FLAPW. Table I

compares the FLAPW and PP results for the formation energies and lattice constants, the

largest absolute di�erence being 9 meV/atom. Since the FLAPW and pseudopotential cal-

culations were converged to within 3 meV, the results seem to agree well. These compounds

can not be observed experimentally since the formation energies are all positive. However

an indirect test of the accuracy of the formation values is obtained from the solubility limits

that can be predicted with these data. In the CaO{MgO phase diagram the computed sol-

ubility limits agree nicely with the experimental data as shown in Ref. [10]. (These values

were actually computed using the SSCAD but as we will show, they are close to the PP

results.)

We also computed the cubic to tetragonal transition in ZrO2. To our knowledge, this

is the �rst pseudopotential calculation of this sort. We did not compute the monoclinic

phase since it is very complex, with 12 atoms in the unit cell and 13 variables to optimize.

Performing this calculations with pseudopotentials requires a large number of supercomputer

hours. A single point calculation may take about 20 Cray90 cpu hours and the monoclinic

structure has 13 relaxation parameters. On the other hand the tetragonal phase takes about

40 minutes and has 3 relaxation parameters, and the cubic phase has just one relaxation

parameter and takes about 15 minutes.

We used non{local, optimized [37], Kleinman{Bylander [43] type pseudopotentials. We

considered the 4s, 4p and 4d orbitals in Zr as the valence orbitals, and we generated the

pseudopotential for a Zr+2 ionic con�guration. The Zr core radii used to generate the

pseudopotential were 1.3 a.u., 1.4 a.u., and 1.6 a.u. for the s, p, and d components. For

10

the oxygen pseudopotential the core radius was 0.8 a.u. for both the s and p components.

The energy functional was minimized employing the conjugate gradients technique and we

used the Perdew and Zunger [44] parameterization of the exchange{correlation energy. Ten

special Chadi{Cohen k points were chosen for the cubic phase and an equivalent set for the

tetragonal phase (12 k points). The energy cuto� was 700 eV.

The pseudopotential results are presented in Table II. The tetragonal phase is predicted

to have a lower energy than the cubic one, as one would expect from the experimental

information mentioned in Section I. The di�erence in energy between these two phases

agrees well with the estimated enthalpy of the transition measured at 2377 oC in Ref. [45].

The cell parameters for the cubic and tetragonal phases also agree well with experiment:

The reported numbers for the experiment are linear extrapolations to 0 oK using the data

from Ref. [12] for the tetragonal a and c, and the cubic a parameter (since the two phases

are very similar we assumed the same temperature dependence in both cases).

B. LMTO{ASA

The von Barth{Hedin [46] form for the exchange{correlation potential was used in all the

LMTO{ASA calculations in this work. Integrations in k{space where done in a 173 uniform

grid (considering crystal symmetry) and a s{p{d basis was used in the CaO{MgO system.

We found an important dependence of the total energies on the size of the atomic spheres

even when the \combined" and the \mu�n tin" corrections were used. For example, in

Fig. 2 we show the total energy versus sphere radius for the CaO structure. It is clear that

a criterion for choosing the relative sizes of the atomic spheres in the ASA is needed. By

choosing di�erent points in the curve, the formation energy of any compound in the system

can be switched from positive to negative.

For monoatomic solids the sphere radius is �xed (space{�lling condition). For all other

cases, where more than one atom is present in the unit cell, a criterion for choosing the

radii is needed. Four criteria are usually used. The �rst criterion is to just take equal{size

11

spheres. The second one is to make the atomic spheres charge neutral (in this way the

madelung energy error in the ASA vanishes). The third one is to minimize the formation

energy of the compound, though there is no variational principle that justi�es this procedure.

Finally, the last criterion, proposed by Andersen [31], gives an explicit formula for the sphere

radii in terms of Wigner{Seitz radii, bulk moduli, and elemental volumes. In metallic alloys

these four criteria usually agree and formation energy values are comparable to FLAPW

calculations [47].

In oxides, it does not make sense to make the spheres charge neutral. In general, we

found that the consistent use of any of these criteria along many compounds in the same

system does not provide accurate formation energies when compared to FLAPW or pseu-

dopotentials. For example, we found that both the Ca{ and Mg{rich L12 structures have

negative formation energies when using equal size spheres while all the other total{energy

methods predict them to be positive. Using a smaller radii for the cations than for the anions

makes the L12 structure formation energies positive but we could not �nd a consistent way

of setting the sphere sizes. We observed the same behavior in other oxide systems such us

ZrO2{CaO, Na2O, and Li{transition metal oxides.

C. SSCAD

In Ref. [10] we presented di�erent formation energies computed with the SSCAD for the

CaO{MgO system. Some of these values are reproduced in Fig. 1. The SSCAD values are

systematically lower than the pseudopotential and FLAPW results. The average error is of

the order of 17 %.

The phase transitions in ZrO2 were studied by Cohen, Mehl, and Boyer [25] using a

non{self{consistent version of the SSCAD (called the potential induced breathing (PIB)

method [48]). Their results are reproduced in Table II for comparison. Neither the tetragonal

phase nor the monoclinic phase were metastable at zero pressure. Cubic zirconia was found

to be the stable phase when compared to the rutile, the orthorhombic, and the cotunnite

12

structures.

D. Semiempirical tight-binding model

We showed in a previous work [40] that the semiempirical tight{binding method can

accurately reproduce the pseudopotential formation energies in the CaO{MgO system (see

Fig. 1). We will now apply the same scheme to study the phase transitions in zirconia.

The elements of the overlap matrix and the hopping integrals were �tted to the valence

bands of pseudopotential calculations for tetragonal ZrO2. Their distance dependence was

assumed to be the one proposed by Harrison [41,49]. The conduction bands were included

with a smaller weight (10 times) than the valence bands. Signi�cant departures from the

experimental lattice constants were penalized during the �t. In all the calculations, a s{

p{d and a s{p basis was used for zirconium and oxygen respectively. The results of this

calculations are shown in Table II. The di�erence in energy between the tetragonal and the

cubic phases are the same as in the pseudopotential method. This is a prediction of the

tight{binding scheme since only the tetragonal phase was included in the �t.

The monoclinic phase is predicted to have a lower energy than the tetragonal in agree-

ment with the fact that this is the observed phase at 0 oK. Not all the 13 crystallographic

parameters were relaxed for the monoclinic phase. The angle not �xed by the symmetry was

set to 90o and consequently, 12 (instead of 13) parameters were minimized. We found the

monoclinic phase to be unstable under variations of this angle. We think this is due to the

fact that we are evaluating the tight{binding parameters beyond the range in which they

were �tted. We do not discard also the possibility that more complex distance dependencies

of the overlap and hopping terms need to be considered. We are currently testing these

hypothesis. We expect that this will bring the energy di�erence between the tetragonal and

the monoclinic phase closer to the experimental values shown in Table II. Note that the

experimental monoclinic cell parameters are already well reproduced by the tight{binding

parameters we used.

13

E. Potential models

The potential results for the CaO{MgO system were computed in Ref. [10] and are

shown for comparison reasons in Fig. 1. Published potentials for this system overestimate

the formation energies by as much as 100%.

A Buckingham potential combined with a shell model were �tted to reproduce structural

parameters and dielectric constants of the tetragonal zirconia phase [50] (the O{O parame-

ters were taken from Ref. [21]). The predictions for the zirconia phase transitions are shown

in Table II and were taken from Ref. [50] and [51]. The monoclinic phase is the stable one

in this model.

IV. DISCUSSION

In Section III we presented total{energy studies of the CaO-MgO system and the poly-

morphic transitions in zirconia using di�erent total{energy techniques. They not only pro-

vide information regarding fundamental properties of materials but also new insights to

understand and improve them. We basically reported three approaches to perform these

calculations: highly accurate methods (in which the major approximation is the LDA), em-

pirical classical models, and a full range of techniques in between, from semiempirical to

fully ab initio schemes (that introduce di�erent approximations with the �nal objective of

improving the computational speed).

Full{potential methods, such as FLAPW or pseudopotentials, are clearly in the �rst

category. Their results usually compare well with experiments (see Table II). In our case,

the use of the frozen core and pseudopotential approximations simpli�ed considerably the

calculations without a large loss in accuracy. The di�erences between an all-electron method,

such as FLAPW, and the pseudopotential results were very small in the CaO{MgO system

as shown in Table I. On the other hand, for the tetragonal to cubic transition energy,

the agreement is not good (see Table II). However, we do not think this di�erence comes

14

from the pseudopotential approximations. The pseudopotential predictions are not only

much closer to the experimental values but also the FLAPW calculations [52] might have

not been fully converged.(In Ref. [52], a limited number of basis functions was used, which

introduced relative errors of the order of 1mRy/cell, compared to 0.1mRy/cell in our case,

and the relaxation of the tetragonal phase was not thoroughly performed.)

When there is an appreciable overlap between the valence and core orbitals the frozen{

core approximation can become an important source for error. In these cases some of the core

orbitals should be taken as valence orbitals reducing the size of the core. If a pseudopotential

is being used, this will also result in an deeper potential, considerably increasing the number

of plane waves and consequently, the computational demands of the method.

It is not uncommon for oxides to have structures more complex than the monoclinic

zirconia one. As we already mentioned, this structure requires prohibitively long PP runs.

Consequently, only the properties of simple oxides can be compute with highly accurate

techniques.

The LMTO{ASA method on the other hand, is orders of magnitude faster than the above

techniques but at the expense of loosing accuracy. The errors introduced by an incorrect

treatment of the overlap region and the spherical approximation can be very large in oxides.

As we showed in Section III B the computed formation energies can shift from being positive

to being negative according to how the atomic sphere sizes were chosen. We found that none

of the criteria usually used to set the sphere radii in metallic alloys gave consistently correct

formation energies in the CaO{MgO system. We could not �nd an equivalent criterion that

worked well in oxides. In metals, the error introduced by the ASA is less critical and it is

usually very small when using neutral spheres.It is expected that a full{potential non{ASA

LMTO method will overcome these di�culties in oxides and work is currently being done

in our group to test this.

The SSCAD is another fast quantum{mechanical technique. In Section III C we showed

that it underestimates the formation energies in the CaO{MgO, the larger di�erences being

of the order of 20%. The approximations introduced in the method clearly break down for the

15

zirconia phase transitions. The non{self consistent SSCAD predicts the cubic structure to be

the stable phase in contradiction with the experimental �ndings. Cohen et al. [25] suggested

that these discrepancies were related to non{spherical charge relaxations not included in the

model. Since many oxides have covalent bonding, signi�cant errors can be expected due to

this approximation. It is possible to avoid the spherical approximation in the SSCAD but

at the expense of an increase in the computational burden.

We already discussed in detail the accuracy of potential models in Ref. [10]. Although

they have been successfully used to study many oxide properties [8,9] they just render

qualitative agreement for phase diagrams. Most published potentials have been �tted to

energies on the scale of eV, while temperature e�ects are determined in the scale of several

meV. This is not the only source of error. Many body e�ects, charge transfer, oxygen

breathing, and other quantum{mechanical e�ects are di�cult to take into account within

the framework of potentials. However, the use of potentials is widespread since they are

simple and very fast. In the case of the CaO{MgO system, the ionic charge does not change

considerably from one structure to the other and a potential could be found that reproduces

the formation energies relatively well, but poorly predicts other properties such as elastic

constants [10]. We do not expect the same to happen in doped zirconias since the charge will

change when dopants and vacancies are introduced in the system [51]. This will seriously

limit the use of potentials in these materials.

A natural way of overcoming the shortfalls of potential models without going into the

complexities of full{potential ab initiomethods is the semiempirical tight{binding technique.

As we showed in Section IIID, this method provided, both in CaO-MgO and in ZrO2,

accurate energy di�erences comparable to the best quantum{mechanics techniques. Even

for monoclinic zirconia, the tight{binding cell parameters are in very good agreement with

the experimental values.

Although the need to use a self{consistent formalism (due to charge transfer e�ects)

slows down the tight{binding method compared to potentials, it is still orders of magnitude

faster than fully ab initio techniques.

16

V. CONCLUSIONS

We showed that full{potential quantum{mechanical methods are reliable tools to com-

pute total{energy di�erences on a scale relevant to the study of phase transitions. When

taken to their full capabilities their predictions compare well with experiments. Unfortu-

nately, these techniques are also the most computer{time demanding of all.

Oxides present a particularly di�cult case for any total{energy method due to the large

unit cells and low symmetry. Consequently, there are signi�cant bene�ts to using sim-

pler schemes. Approximations such as spherical averages of potentials, mapping space into

overlapping spheres, or replacing complex ion-ion interactions by simple pair potentials can

dramatically speed up total{energy calculations. However, we showed that even for the

relatively simple CaO{MgO system, most of these approximations break down.

Consequently, it is very important to accurately incorporate all the relevant e�ects in

any total{energy model. Our results in CaO{MgO and ZrO2 seem to indicate that a self{

consistent tight{binding model is capable of capturing the relevant physics of the problem

while retaining a low computational cost. The results are promising and more tests are

underway.

The scaling of the methods with the number of ions is another factor to be taken into

account. As computers are becoming more powerful and larger systems are being studied,

those methods that can be formulated to scale linearly in N will eventually succeed. Oxide

materials present a good case to test this new developments, since many di�erent mechanisms

are crucial to determine total{energy di�erences.

ACKNOWLEDGMENTS

This work was sponsored in part by the National Science Foundation under contract No.

DMR9501856, the National Institute for Health under contract No. 2-P30-ESO2109-16 and

the Petroleum Research Fund under contract No. ***********. Professor Joannopoulos

17

is gratefully acknowledged for providing us with the pseudopotential codes. We thank the

Pittsburgh Supercomputer Center for the possibility of using their C90 computer. We thank

Mark van Schilfgaarde for his permission to use the LMTO-ASA code.

18

FIGURES

FIG. 1. Formation energies for di�erent ordered structures in the CaO{MgO system obtained

with di�erent total{energy techniques. The number that identi�es each structure corresponds to

the one used in Ref. [40]. The tight{binding parameters were �tted only to structure number 1 in

the plot (and both CaO and MgO). The potential parameters were obtained from Ref. [21].

FIG. 2. Total energy for pure CaO as a function of the ratio between the Ca and the O sphere

radius in the LMTO{ASA. The volume of the cell was kept constant and corresponds to the

experimental value. Empty spheres, and mu�n tin and combined corrections were used to reduce

the errors introduced by the ASA. The total energy changes in a scale much larger than the scale

of the temperature e�ects in the system and a criterion is needed to choose the sphere size.

19

TABLES

TABLE I. Formation energies and cell parameters for the L12 structure in the CaO-MgO system

computed with the pseudopotential (PP) and the full{potential linearized{augmented{plane{wave

(FLAPW) methods. The CaO-MgO system orders on two interpenetrating fcc lattices. The L12

nomenclature identi�es the distribution of the Ca and Mg ions on the fcc cation lattice (that

corresponds to the minority species at the corners of the conventional fcc cell) while the oxygen fcc

sublattice remains fully occupied. Energy values are expressed in eV/ion and the cell parameters

are in �A. The values correspond to fully relaxed structures.

TABLE II. Cell parameters and structural energy di�erences for the experimentally observed

phases in zirconia at zero pressure. The energies are in eV per ZrO2 formula unit and the lattice

constants are in �A. All cell parameters were fully relaxed, including the internal positions, xi, yi,

and zi, that are shown here according to the Wycko� notation in reference [53]. The experimental

z value for the tetragonal structure was measured at 1295 oC. The tight{binding � angle was not

relaxed. PIB calculations for the tetragonal and monoclinic phases of zirconia are not reported in

the table since these structures are not predicted to be stable by this method.

20

Ca3MgO4 CaMg3O4

PP Formation energy 0.092 0.131

FLAPW Formation energy 0.101 0.137

PP Lattice constant 4.698 4.423

FLAPW Lattice constant 4.620 4.350

Table I: Kohan{Ceder

21

Expt. [45,12,54] Pot. [51,50] TB PIB [25] PP FLAPW [52]

Cell Parameter

Cubic ZrO2

a=b=c 5.092 5.075 5.075 5.10 5.050 5.050

Tetragonal ZrO2

a=b 3.571 3.588 3.607 3.575 3.568

c 5.20 5.216 5.203 Not 5.153 5.084

z(O) 0.303 0.31 0.296 stable 0.295 0.279

Monoclinic ZrO2

a 5.1505 5.241 5.03

b 5.2116 4.898 5.22

c 5.3173 5.578 5.39

� 99.230 90.0 90.0

x1(Zr) 0.2754 0.25 0.278

y1(Zr) 0.0395 0.0 0.036 Not

z1(Zr) 0.2083 0.1899 0.203 stable

x2(O) 0.0700 0.0753 0.073

y2(O) 0.3317 0.2818 0.344

z2(O) 0.3447 0.3958 0.343

x3(O) 0.4496 0.4247 0.445

y3(O) 0.7569 0.7182 0.764

z3(O) 0.4792 0.3958 0.474

Energy di�erences:

Ecubic-Etetragonal 0.058 0.019 0.045 0.045 0.009

Etetragonal-Emonoclinic 0.061 0.166 0.278

Table II: Kohan{Ceder

22

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25

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26

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10

For

mat

ion

Ene

rgy

(eV

/ato

m)

Structure Number

"Pseudopotentials""SSCAD"

"Tight-binding""Potentials"

Figure 1 Kohan{Ceder

Tot

al E

nerg

y (R

y/ce

ll)

r /r Ca O

-1507.31

-1507.3

-1507.29

-1507.28

-1507.27

-1507.26

-1507.25

-1507.24

-1507.23

-1507.22

0.8 0.85 0.9 0.95 1 1.05

Fig 2: Kohan-Ceder