THE EFFECT OF BAND STRUCTURE AND ELECTRON-ELECTRON INTERACTIONS ON THE METAL-INSULATOR TRANSITIONS IN Ti2O3 AND V2O3

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THE EFFECT OF BAND STRUCTURE AND

ELECTRON-ELECTRON INTERACTIONS ON THE

METAL-INSULATOR TRANSITIONS IN Ti2O3 AND

V2O3

J. Ashkenazi, M. Weger

To cite this version:

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JOURNAL D E PHYSIQUE Colloque C4, supplement au no 10, Tome 37, Octobre 1976, page C4-189

THE EFFECT OF BAND STRUCTURE

AND ELECTRON-ELECTRON INTERACTIONS

ON THE METAL-INSULATOR TRANSITIONS IN Ti203 AND

V 2 0 3

J. ASHKENAZI

DCpartement de physique de la mati2re condenste, Universitt de Gen2ve, Suisse

and

M. WEGER

Racah Institute of physics, the hebrew University of Jerusalem, Israel

Resume.

-

On modifie ici un calcul antMeur pour la bande 8 un Bectron du Ti203 et VzO3, en incluant de manihre autoconsistante l'effet des interactions intra-atomiques des electrons entre eux, a partir d'un Hamiltonien gknkralise de Hubbard. On montre, pour TizO3, que l'hypothbe des taux d'occupations diK6rents pour 1es.bandes alp et ex conduit

B

Ia crkation d'une barrikre d'energie qui n'est pas associBe aux moments magnktiques. Ce mkcanisme de crkation d'une barrikre concorde avec les effets expkrimentaux de la temperature et du dopage au V sur la transition et sur le rapport

c/a. On montre aussi, pour Ie V2Q3, qu'une barribre d'energie est crkke dans Ia phase aatiferroma- gnbtique oh les diffkrents taux d'occupation sont pris pour les diffkrents Btats de spin. Cette barribre est du type Hubbard avec, en plus la complexitk du probl6me qui pourrait constituer l'explication de la nature de la transition qui est du premier ordre.

Abstract. - A previous single-electron band calculation for Tiz03 and V203 is modified inclu- ding self consistently the effect of intra-atomic electron-electron interactions by means of a generalized Hubbard Hamiltonian. For Ti203 it is shown that the assumption of different occupations for the alp and the en bands results in a creation of a gap which is not associated with magnetic moments. This mechanism of gap creation is consistent with the experimental effects of temperature and doping V on the transition and the cla ratio. For V203 it is shown that a gap is created in the antiferromagnetic phase where different occupations are assumed for the different spin states. This gap is of the Hubbard type but has the complexity of the problem which might be the explanation for the first order nature of the transition.

1, Introduction.

-

Three general approaches have .

been applied to the study of the metal-to-insulator

-

[ ' ( a ) ' ' ' ' '

1

transition (MIT) in transition metal oxides :

2'1

o

A n

v2°3

J

(1) The rigid band picture.

(2) The full band picture in the unrestricted Hartree- Fock approximation.

(3) The correlated electron gas picture for a n extre- mely idealised band.

(1) is good when U

<

W ( W : electronic band- width ; U : Coulomb intra-atomic interaction), i. e. for ordinary metals. (3) is good when U B W i. e. for insulators like NiO ; (2) is the most difficult approach, since both the band structure and the correlations (to some extent) have to be taken into account. Still, for materials undergoing an MIT, with U

--

W, (2) is necessary, since we can neither assume that U is small, nor thkt W is small.

In figure 1 we show the band structure of V20,, calculated by the LCAO method [l] using Bloch sums of the metal 3d and the oxygen 2s and 2p atomic functions as a basis.

We see that the bonding al, band is at the bottom

ENERGY

.

eV

FIG. 1.

-

The density of states of V203 obtained neglecting

electron-electron interactions (ref. [l]) : (a) separated to the a1 component (dashed line), the en-eo component (full line) and the oxygen 2s-2p component (dotted-dashed line) ; (b) the total

density of states (full line) and its integral (dashed line).

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C4-190 J. ASHKENAZI AND M. WEGER of the 3d band, and about 1 eV wide ; the e, band

(including bonding and antibonding components) is about 3 eV wide ; the a:, band is about 2.5 eV above the bonding a,, band and roughly 1.5 eV wide. The e, band starts at the top of the e, band, and is about 6 eV wide. This large width is due to covalency with the oxygens. Thee, band contains several large and narrow (-- 0.25 eV wide) peaks in the density of states, close to the Fermi level. The band structure of Ti203 is similar ; it is somewhat wider (the t,, band is about 4 eV wide, instead of 3 in Vz03). The e, band is slightly higher with respect to the a,, band, but still the two bands overlap considerably (Fig. 2). The agreement of those band results with experiment has been discussed in ref. [l].

E N E R G Y

.

eV

FIG. 2.

-

The same as in figure 1 for Tiz03.

In this band calculation, the electron-electron interactions within the 3d band are largely ignored, an assumption which is not a pr'iori justified, since

U

E 2 eV and some strong structure of the 3d band is just 0.25 eV wide. Thus, the natural next step, is to include the effect of U.

In the present work we start presenting a generalized Hubbard Hamiltonian adequate for treating tightly bound electrons in degenerate bands, and we proceed applying unrestricted Hartree-Fock solutions of this Hamiltonian for the band results obtained in ref. [l]. The results are discussed with respect to the nature of the metal-insulator transitions (MIT) in those materials.

2. Presentation of the problem.

-

The Hamiltonian of tightly bound electrons in a degenerate band (such as the 3d electron) can be expressed as :

t X =

C

tzir C, C,*

+

aa'

Where a = (imo) is a collective index for the lattice site i, the orbit m and the spin o, c, are the annihilation operators, tau the LCAO band integrals and

<

a, a,

1

V ] a; a;

>

the matrix elements of the Cou- lomb interaction between the electrons.

Considering only the intra-atomic direct integrals U and exchange integrals J, the Coulomb interaction matrix elements are approximated by :

<

a1 a2 l V

I

a3 a4

>

g 8ilizi3i4

aUlu3

'02Uq X

X CBmlrn3 6rn2rn4 Umlrnz f (1

-

8mlm3)

X (arnlrnq Jm2m3

+

8mlm2 'mjrnq) Jm1rn31. (2)

In the Hartree-Fock approximation, the Hamiltonian is given by :

" t J ~ H F =

C

tau, c, c,,

.

aa' (3)

Where t,,, are defined self-consistently by the e q u a t i o ~ :

t,,, = t,,,

+

[<

aa,

I

V

I

a' a,

>

-

alaz

- <am,

I

V1 a2 a ' > ]

<

,:c ,,c

> .

(4) Angular brackets here denote the thermal average. (Here we make a zero temperature calculation, the finite temperature case will be worked out in a further publication.)

Turning to V203 and Ti203 we use the band structure results of ref. [l]. We concentrate on the t,, sub- band, and neglect the wide e, sub-band which lies fairly above the Fermi level. This sub-band is expressed in terms of three basis functions per atom [2] :

where the

+

signs stand for the a-b or the c-d atoms respectively (Fig. 3). More precisely those functions contain also admixture from oxygen and other metal orbitals. The

f,

and f2 functions form the doubly degenerate e, band (with the e, and e, states) and the f3

function forms the non-degenerate a, band. The band structure is constructed in terms of effective transfer integrals between a metal atom and its neighbouring metal atoms up to the fourth shell (Fig. 3). The values of those integrals are determined by making a least squares analysis to get best fit with the band structure. In this way the effect of the oxygens on the t,, band is included even though one has to diagonalize 12 X 12

instead of 44 X 44 matrices. Since in that band cal-

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THE EFFECT OF BAND STRUCTURE AND ELECTRON-ELECTRON INTERACTIONS C4-191

hexagonal vector

Fls. 3. - V z 0 3 and Tiz03 in the trigonal phase (only metal atoms are shown).

the a,-e, transfer integrals and the crystal field integrals are not changed. We see that the Ti203 e,-e, transfer integrals have to be decreased by more than a factor of two, and this might reflect correlation effects. The results are shown in table I.

Next we consider the effect of intra-atomic electron- electron interactions (determining YE,,). Let us express them in terms of three parameters (see eq. (2)) U(= U,, = J,,), U ' l,,, m # m') and

J(

= J,,., m # m'). For a t2, band in a cubic crystal it can be shown that U' = U

-

2 J. The values of U and J obtained for V203 and Ti203 using atomic functions are about 25 eV and 2 eV respectively, but screening effects [3], are expected to reduce their values up to an order of magnitude, so we assume that

U = 2.5 eV, J = 0.25 eV, and U' = 2 eV.

Before turning to the present HF solutions we would like to discuss the results of a model proposed [4] for V203 some year ago, considering only the e, band, that model would be called (using modern terminology) a

saddle point model. Since the e, band is about 3 eV

wide, but has peaks about 0.25 eV wide near E,, it was divided into two components, a heavy-electron one, responsible for the sharp peak, and a light-electron one, responsible for the broad structure.

For each type of carriers, the density of states was assumed constant. This model was treated in the HF approximation in some detail. It was found, that there

are instabilities of various kinds, namely, charge density waves, for which the trigonal symmetry is broken ; orbital magnetic states ; spin magnetic states, and

exotic states, for which the spin distribution of diffe-

rent orbitals on the same atom is different. These instabilities produce a gap (at 0 K). An interesting feature of this model is, that all these instabilities are nearly degenerate. The reason for this near-degeneracy is as follows : In the Bloch state, there is an average of ) of an electron per atom in each of the states e,?, ell, e2?, e21, due to symmetry. Thus, the probability to find 4, 3, 2, 1, 0 electrons on one atom is given by 0.004, 0.047,0.211, 0.422, 0.316 respectively, and the average

3

Coulomb energy is

-

U. In an ordered state, with both 8

electrons in the same orbital, the probabilities are 0.0, 0.25, 0.5, 0.25 respectively, and the average Coulomb energy is only

3

U. Because only the states near EF

(i. e. near the saddle-point) participate, this has to be reduced somewhat (yielding 0.1-0.2 eV, finally). Howe- ver, the main point of the argument is, that for any pair of states of the four, the gain in Coulomb energy is the same or nearly the same ; e,? e,, will give a CDW (breakdown of trigonal symmetry) ; elf e2.T give a ferromagnetic state ; e l t ezl an exotic spin state ; (elT.

+

ieZT) (e, l

+

ie,,) give orbital magnetism ;

sim~lar states result when the unit cell is doubled (as observed experimentally). The difference in energy between the different ordered states is due to several causes ; intra-atomic exchange favours the spin- ordered state ; electron-phonon coupling the CDW ; imperfect matching of the CDW to the lattice, slightly disfavours the CDW, etc.

Thus, the existence of such a diverse range of nearly- degenerate instabilities is due to orbital degeneracy. Localization of states of a degenerate band requires a breakdown in symmetry. This argument has been expressed in a more formal form by W. Kohn [5]. Because of the near degeneracy, the actual state is a mixture of instabilities ; also, if we use Landau theory, the various order parameters are coupled non-linearly, and this accounts for first-order-transitions, etc. [6].

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THE EFFECT OF BAND STRUCTURE AND ELECTRON-ELECTRON INTERACTIONS C4-193

role) or Slater (type where integrals like J play the major role). We feel that only a proper unrestricted- orbit H F calculation (i. e. without spherical averaging inside MT spheres) can provide an answer to this question.

Moreover, from the discussion of the preceding paragraph we see that even a Mott Hubbard transition originating from U, may have its energies reduced due to band structure effects ; thus a small effective splitting parameter (gap) is no proof that its origin is J (i. e. F,,

F, Slater integrals) rather than U (the

I;,

Slater integral).

In addition to spin-unrestriction and orbital- unrestriction, a proper H F calculation requires also k-unrestriction, i. e. Bloch functions with different values of k, see different H F potentials. k-unrestricted HF calculations have been performed a generation ago on the free electron gas (see review article by Hedin [8] for example). They gave an infinite value for m* at the

Fermi level, due to the long-range Coulomb interaction. In order to overcome this divergence, screening has to be taken into account (screening is some kind of correlation that goes beyond HF, naturally). Thus, the problem with H F is NOT that it does not give the mass enhancement at E,, but that the mass enhancement it gives, is too large. Also, H F calculations being as complicated as they are, k-unrestriction is usually not attempted.

Beside this technical point, there is also the funda- mental difficulty that for a uniform electron gas, there is no anomaly at E, of the standard exchange potentials (Kohn-Sham, Slater, Hedin-Lundquist), while for atoms, H F is a better approximation than the local- exchange ones. Thus it is by no means clear what should be done in the intermediate region U

=

W. There is the well-known work of Hubbard [9] and Brinkman- Rice [10], but apparently these formalisms have not been applied to real bands to see whether they can account for the effective mass at the Fermi level.

3. Non-Magnetic Solutions. - In the trigonal phase V,O, and Ti203 are non-magnetic, so making the calculation for this phase we neglect magnetic effects, but still assume different occupation for the e, and a, bands (unrestricted, orbit). In this case the effect of intra-atomic interactions in eq. (4) turns out to be an increase of 4(2 U'

-

J

-

U ) . (n,,

-

all) in the splitting between those bands.

Here we denote

<

c:cm

>

by n,, ; the index l

refers to the basis function &(nil = n,,, since

f,

and

f,

form the degenerate e, gand) ; in the non- magnetic case, the non-diagonal terms of n , vanish by

symmetry consideration. Since Zn,, = n, the number of 3d electrons per metal atom (which is 1 for Ti,O, and 2 for V203), we have to determine self-consistently only one independent parameter.

For a rapid convergence of the self-consistency procedure we use the Newton method. We have to solve vector equation of the form X = f(x). It can be

shown that if X, is close to X, than E = X

-

xo is given by :

solving this linear set of equations, self consistency is achieved in about three steps. (The derivatives 6f/6x

are calculated rapidly using a perturbation expansion.) The calculation has been done using the Gilat interpolation method, taking 240 points for Ti203 and 265 points for Vz03 in the irreducible Brillouin zone. The results obtained for the band structure and the density of states curves are shown in figures 4, 5, 6, 7.

FIG. 4. -The structure of the tzg band in insulating TizO,.

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C4- 1 94 J. ASHKENAZI AND M. WEGER c---o total 0 'g

-

en a1 integral m =. m m m

FIG. 6. -The density of states of the tzg band in insulating T i z 0 3 (the various components and the energy integral are also

shown). Vz O3 t r ~ g o n a l a +---o total m *i--w

-

elt al

-

integral m W m

-

0 0 D

FIG. 7.

-

The same as in figure 6 for metallic VzO3.

It turns out that in Ti203 the a l band is shifted down relatively ti the e, band, and there is gap of about 0.1 eV at the Fermi level at zero temperature in agreement with experiment. The values of the self- consistent occupation parameters for Ti,O, are n,, = nZ2 = 0.089, n,, = 0.822. In V203, on the other hand, the a, band is shifted up relatively to e, band and this band is almost empty (rill =n2, =0.964,

n,, = 0.073). There is no gap at the Fermi level in this phase of V203 in agreement with experiment. The anomalously high (in Ti203) and low (in V203) occupations of the a, band are in agreement with their anomalously high (in V203) and low (in Ti203) cla ratios [l11 (see section 5.1).

In figures 8, 9 the joint density states curves are shown. The peaks and slopes in those curves are in agreement with polarized light reflectance measure- ments for Vz03 [l21 and Ti203 [13], provided that

T i 2 0 3

o---o total al-e,

-

e r e x

a!-a,

FIG. 8. -The joint density of states within the tzg band of insulating Ti203 (the various components are also shown).

Vz 0, trigonal

c--o

-

total al-e, %-% al-a1

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THE EFFECT OF BAND STRUCTURE AND ELECTRON-ELECTRON INTERACTIONS C4-195

anomalies of light polarized parallel to the c-axis results mainly from a,-a, and e,-e, transition, while anomalies of light polarized perpendicular to it results mainly from a,-e, transitions.

The Ti,O, band structure is consistent with the experimental result that the holes are lighter than the electrons, as discussed in ref. [l]. It should be noted that if covalency had been treated more rigorously (not just by determining effective d-d integrals), then the pushing up of the a, band in V203 due to covalency would be smaller (the effective transfer integrals are determined for the band results of ref. [l] where the a, band is relatiirely low), resulting in a higher occupation for the a, band than obtained here. Besides, one should not rule out without further experimental results the possibility that for a certain range of the parameters, the stable solution is the one where the al band is shifted down also for V203. Such- a solution has been treated in previous works [2], [6].

4. Antiferromagnetic solution for V,O,.

-

In the antiferromagnetic monoclinic phase of V203, the e, band splits into two non-degenerate el and e, bands (after the basis functions f, and f,). The, interaction parameters U, U' and J, are assumed to be the same as in the trigonal phase. The variation in the effective transfer integrals is estimated by fitting transfer integrals to the band structure calculated for (V0.96Cr0.04)203 (using the inter-atomic distances of Dernier [14]), and extrapolating them to the inter- atomic distances in the monoclinic phase [15]. The a,

-

b, distance (Fig. 3) is taken as the one which increases considerably, while the a,

-

b, distance is the one which slightly decreases (ao

-

b2 almost does not change). The modified integrals are shown in table I.

The following assumptions are made : Different occupations are assumed for different orbits and spins (unrestricted orbit and spin), but those self consistent parameters are assumed to be diagonal in the spin index (denoted by nk), and to be real (neglecting orbital magnetism). Hence we have to diagonalize 24 X 24

matrices (the unit-cell is doubled due to antiferro- magnetism), and to determine self-consistently 12 real parameter n?,,, out of which 11 are independent. The Newton method (Cq. 5) is used again for a rapid convergence of the self-consistency procedure. The calculation is done using again the Gilat interpolation method taking 210 points in the irreducible B. Z. (which is & of the whole B. Z. in this phase). The results obtained for the band structure and the density of states curve are shown in figures 10, 11.

The values obtained for the self-consistent occupation parameter (in a spin-up site) are :

FIG. 10. -The structure of the tz, band jn monoclinic V203

(the arrows indicate the direction of the magnetic moments on sites where a spin up state of the band is concentrated).

Vz O3 monocl~nlc m

-

total "7

-

_a1 m

-

Integral m

-

0

FIG. 11. -The same as in figure 6 for monoclinic V203.

Again, if covalency had been treated more rigorously, higher occupation might be expected for the a l orbital which makes it closer to the stable solution found by Ranninger and co-workers [16]. It turns out that a gap of about 0.2 eV (which is some kind of a Hubbard gap) is created a t the Fermi level in agreement with experiment. It is found that ittineracy reduces the magnetic moments from 2 p, to l .85 p,.

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C4-196 J. ASHKENAZI AND M. WEGER covalency, which contributes an additional multi-

plication factor of l

-

A;

to the magnetic moments [16]. Using estimates for the covalency based on ref. [l] and on NMR results [l71 a value close to the experimental 1.2 p, is obtained.

The magnetic ordering in the monoclinic phase of V 2 0 3 1111 can be interpreted as composed of zig-zag antiferromagnetic chains in the Y direction and zig-zag ferromagnetic chains in the (101) direction. This is reflected in the bands (Fig. 7) being flatter in the Y direction than in the X and Z directions, and in the strong hybridization between the e, and a, bands (nT3 being relatively large) which have large transfer integrals in the X and Z directions respectively.

A simplified interpretation of the band structure in that phase is of coupled e2 band directed along the antiferromagnetic chains and e,-a, band directed along

FIG. 12. - The interpretation of the monoclinic phase of V203 in terms of antiferromagnetic and ferromagnetic chains.

the ferromagnetic chains. The first being half-filled, and the second quarter filled, both having Hubbard gaps at E,. This interpretation is similar to the band scheme given by Goodenough Ill], and also to the band model given by Ashkenazi and Weger [6] replac- ing the e, and the a, bands there by the e, and the e,-a, bands here. A sketch of this picture is given in figure 12. A diagram for the role of band structure electron-electron interactions in V203 is given in table 11.

5. Discussion.

-

5.1 Ti203.

-

It has been shown in section 3 that the inclusion of electron-electron interaction introduces an additional term

3(2

U' - J

-

U) (n,,

-

n,,)

in the splitting between the a, and the e, bands. In Ti203, this term might be responsible for the gap between the bands at zero temperature. As temperature rises, electrons are excited from the a, to the e, band, resulting in the decrease of n,,

-

n,,, and of the gap. This would reduce n,,

-

n,, further, especially as the gap shrinks, and cause a considerable band crossing at high temperatures, satisfying roughly

which would predict a band structure close to the one obtained without considering the electron-electron interactions (Fig. 2).

(In this case n,, S

3,

n,, = n,, S

B,

but for the high temperature state, the overlap between the a, and the e, bands might be even larger.) The thermo- dynamics for such a transition has been worked out for a simple band model by Zeiger [18], and a gradual insulator to metal transition is expected in agreement with experiment. The transition is followed by an abnormal increase in the c/a ratio, and this is also a consequence of the band crossing, since in a non-rigid lattice, an increase in the population of the e, band at the expense of that of the a, band, would affect the inter-atomic distances. A study of such an effect for the metal to insulator transition in Vz03 has been done in a previous work [ 6 ] . An analogous situation occurs in (Ti, ~,V,),O,. Experimentally, the effect of a few

Schematic representation of various factors implicit in the Hartree-Fock calculation Increased atomic separation t, Localization of +

along [l011 chains in the mono- al, el orbitals

clinic phase

I

J.

Ferromagnetic arrangement t Spin polarization

(band nearly

t

full)

I

Intra-atomic

exchange J [6]

due to orbital degeneracy and U [4]

4,

Antiferromagnetic arrangement t Spin polarization +

4

e2 electrons

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THE EFFECT OF BAND STRUCTURE AND ELECTRON-ELECTRON INTERACTIONS C%-197 percents of vanadium on the electric properties and the

c/a ratio is similar to that of temperature. Magnetic and specific heat properties of this substance tend to indicate the existence of an impurity e, band below the Fermi level [19]. The increase in the population of such a band with the vanadium concentration would result in the decrease of n3, - n,, and of the gap, resulting in an insulator to metal transition similar to the one achieved by increasing temperature.

The experimental observation that there remains an anomaly in about 400 K up to X Z' 6

%

might be connected with the randomness of the impurities which leaves spaces without V atoms.

Regarding the c/a ratio and the band structure in Ti203 and V203, we do not argue as was often done by others, that thi; ratio and its ihanges, are the cause i f the position of the bands and their changes, but rather that it is a result of them, although some degree of feedback does exist.

However, it seems that the electrons entropy is not sufficient for the transition in the observed temperature [18]. The missing entropy probably results from softphonons, and from polaron mass enhancement due to the change in the c/a ratio. If this is the case, then the question whether c/a determines the bands or vice versa

is a chicken-egg problem.

Mott argues (private communication) that correlation effects would prevent a large overlap between the bands in the metallic phase as obtained here. This problem is being studied at present.

5.2 V20,.

-

It has been shown in section 4 that the band structure in the monoclinic phase is analogous to that of coupled antiferromagnetic and ferromagnetic chains. This is similar to a previous model by the authors [6], where it has been shown that for such a coupled system, the insulator-to-metal transition would probably be of the first order, even when only intra- atomic interactions are considered. It has also been shown in that work that the increase in the distance between ferromagnetically coupled neighbours, is considerably larger than that between those coupled antiferromagnetically, in agreement with experiment [IS]. Those results might be applied by analogy also from the present calculation.

Considerable amount of work has been done on the extra (lower c/a trigonal paramagnetic phase in (Vl-xCrx)20f,), and there seems to be contreversy whether this phase is metallic or insulating. Goode- nough [l11 and Zeiger [l81 explain this phase within the framework of the H-F approximation as resulting from a change in the positions of the a, and the e, bands. While McWhan and Remeika [20], Mott [3] and others interpret it as an insulating one obtained by magnetic disorder of the antiferromagnetic phase (and therefore cannot be treated in the H-F approximation), relying also on the fact that the inter-atomic distances in that phase are close to those of the monoclinic one (on the average). We cannot say as yet definitely which explanation is better. However if Mott's explanation

which seems more probable is the true one, it is still hard to tell without a further calculation whether it is metallic or insulating since the small gap in the monoclinic phase might shrink due to the magnetic disorder and the cancellation of the monoclinic distortion. Beyond the HF approximation, we should also consider the short-range order. Because of the orbital degeneracy, we have even just for the e, band, 4 states :

l : eIT ; 2 : e2? ; 3 : e14 ; 4 : ezl: Thus, all correlations

where p and q denote neighbouring atoms, have to be considered, as well as the correlations with al, states. Thus, another contested question is the relative impor- tance of the H F parameters vis a vis the short range order ones.

6. Conclusion. - V,03 and the oxides similar to it have been a subject of intensive investigation for about a decade. The MIT in V203 is probably the most complicated one, and understanding it implies that we understand MIT's in transition metal oxides (and sulphides, etc.) in general. There'fore, the large amount of effort on a single compound is warranted. The cardinal questions are : (i) Is the MIT a Mott transition ? (ii) Can the H F approximation describe the MIT ? (iii) What are the most important points we should be aware of ?As for (i), we feel the answer is yes, but the transition is from partly localized molecular orbitals to somewhat more localized molecular orbitals, rather than between delocalized orbitals to ones localized on the atoms ; thus the energy changes involved are only a few tenths of an eV, rather than eV's, and, moreover, orbital degeneracy causes changes in symmetry. As for (ii), we feel the answer is positive, though considerable additional work has to be done to verify (or refute) this. As for (iii), we feel that the most important theoretical aspect is the question of unrestriction ; the HF calculation should be as unrestricted as possible, and if this is impractical (because of the amount of labour involved), we should be fully aware of what is restricted and what the implications of this restriction are.

Clearly these conclusions are controversial and thus this paper does not attempt to provide an objective review of the extensive litterature in this field.

Still, the band formalism appears to be a most fruitful approach, as demonstrated for instance by Ranninger and co-workers [16], obtaining the rather specialised magnetic order in the A F state from the band-structure parameters.

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C4-198 J. ASHKENAZI AND M. WEGER

References

[l] ASHKENAZI, J. and CHCHEM, T., Phil. Mug. 32 (1975) 763. [2] NEBENZAHL, I. and WEGER, M., Phil. Mag. 24 (1 971) 11 19.

[3] MOTT, N. F., Metal Insulator Transitions (London, Taylor

& Francis Ltd.) 1974.

[4] WEGER, M., Phil. Mug. 24 (1971) 1095. [5] KOHN, W., Phys. Rev. B 7 (1972) 4388.

[6] ASHKENAZI, J. and WEGER, M., Adv. Phys. 22 (1973) 207. [7] MATTHBISS, L. F., Phys. Rev. B 10 (1974) 995.

[8] HEDIN, L. and LUNDQUIST, S., Solid State Phys. 23 (1959)

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