The equivalent charge concept and its application to the electrostatic energy of charges and multipoles

HAL Id: jpa-00208876

https://hal.archives-ouvertes.fr/jpa-00208876

Submitted on 1 Jan 1978

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The equivalent charge concept and its application to the electrostatic energy of charges and multipoles

E.F. Bertaut

To cite this version:

E.F. Bertaut. The equivalent charge concept and its application to the electrostatic en- ergy of charges and multipoles. Journal de Physique, 1978, 39 (12), pp.1331-1348.

�10.1051/jphys:0197800390120133100�. �jpa-00208876�

THE EQUIVALENT ^CHARGE CONCEPT AND ITS APPLICATION

TO THE ELECTROSTATIC ENERGY OF CHARGES AND MULTIPOLES

E. F. BERTAUT

Laboratoire de Cristallographie, C.N.R.S., ¹⁶⁶ X, 38042 Grenoble Cedex, ^{France and} Laboratoire de Diffraction Neutronique, D.R.F., C.E.N.G., ^{38041 Grenoble} Cedex, ^France

(Reçu ^{le 24} avril 1978, révisé le 21 juillet 1978, accepté le 23 août 1978)

Résumé.

Le concept ^de charges ^sans empiètement, développé ^par l’auteur, s’est montré très utile dans l’évaluation de l’énergie Coulombienne, représentée par la différence de deux quantités positives dont l’une est un développement ^en série dans l’espace réciproque, ^{l’autre une constante.}

Nous montrons ici _que des dipôles ^et plus généralement ^des multipôles, peuvent ^être remplacés

par des distributions équivalentes ^de charges ^sans empiètement ^{de sorte que} même dans le cas le

plus général ^où charges ^et multipôles ^sont présents, l’énergie d’interaction électrostatique peut être représentée par la différence de deux quantités positives ^{avec de bonnes} propriétés de convergence et dans le cadre d’un formalisme simple. La méthode est applicable ^{à toute} symétrie. ^{A titre de}

vérification nous avons calculé l’énergie magnétostatique dipolaire, ^connue de MnO. Pour TiO2

et CdCl2 ^nous évaluons les énergies, potentiels ^et champs coulombiens et dipolaires ^ainsi que les moments dipolaires des anions d’une manière self-consistante.

Ce qui ^est remarquable dans la méthode est le très grand choix dont on dispose pour les fonctions de densité de charges ^sans empiètement..

Abstract.

The concept ^{of non} overlapping charges developed by the author has proven very useful for the evaluation of the Coulomb energy represented by the difference of two positive quan- tities of which one is an expansion ⁱⁿ reciprocal space, the other is ^{a constant.} We show here that

dipoles ^{and more} generally multipoles ^{can be} replaced by equivalent ^non overlapping charge ^dis-

tributions. Thus even in the most general ^{cases of} charges ^and multipoles ^the electrostatic interaction energy ^can be represented by the difference of two positive quantities ^with good convergence pro-

perties and in the frame of a simple formalism. The method is applicable ^{to any} symmetry. ^{As a}

test we calculate the known dipolar magnetostatic energy of MnO. For TiO2 ^and CdCl2 ^{we evaluate}

the Coulomb energies, potentials and fields as well as the dipolar moments, energies and fields self

consistently. What is remarkable in the method is still the great arbitrariness left in the choice of the

non overlapping charge density ^functions.

Classification Physics ^Abstracts

41.10

Introduction.

For the evaluation of the Coulomb energy in ionic lattices the author has generalized

the concept ^of charges, distinguishing overlapping

and non overlapping charges [1]. ^{He could} identify

the famous Ewald method [2] ^{with the case of} charge overlap ^{where two} expansions ^are needed, ^{one in} reciprocal space, representing ^the approximate ^lattice potential, the second one in direct _space representing

the overlap correction. The author could show that there was a very large choice of charges ^available (for instance that of the Ewald method is Gaus- sian [1,3]) ^{and, if} charges ^were chosen without overlap, only ^one expansion ^{is needed in} reciprocal space.

More particularly, ⁱⁿ this last event the Coulomb

energy could be expressed ^{in a} compact ^form by ^the

difference of two positive quantities, ^{where the first}

one is a structure dependent ^{summation and the}

second one a non structure dependent ^{constant. Thus}

no oscillations can occur in the evaluation of the Coulomb _energy. Also quantities ^{such as} potentials,

fields and field gradients, ^{derived from the} energy

expression ^{show a} simple ^{form and} good convergence

properties [4].

The question arises if such a procedure ^{can be}

extended to dipolar ^{and more} generally ^{to multi-} polar lattices. This is indeed shown to be the case

in the present paper by introducing ^the concept ^of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197800390120133100

equivalent charges, ^{i.e. of} charge distributions which

are equivalent ^to multipoles [5].

In the first part ^we recall the fundamentals for the case of non overlapping charges only. ^{The case}

of a dipole ^lattice is considered in the second part.

Simply replacing ^the charge operator qj ^{in the Coulomb}

energy expression by ^the equivalent charge operator

m;. grad r, ^{of the} dipole mj ^{we obtain the} energy of a dipole,lattice ^{which, as} expected, ^{is the difference}

of two positive quantities. Also another expression

of the dipolar energy is given ^where the Lorentz

polarization appears. Dipolar potentials ^{and fields}

are derived from the _energy expression in the usual way.

The case of induced dipoles is examined in the third part ^{where the} polarizability relates the dipole

moments self consistently ^to the local electric field.

The total interaction energy when charges ^and dipoles ^are present ^{is the sum of the Coulomb} energy,

quadratic ^{in the} charges qj, ^{of the} dipole energy,

quadratic ^{in the} dipoles _mj, ^{and of a} polarization ^term

bilinear in qi _mj for which the expansion ⁱⁿ reciprocal

space is given.

Part four generalizes ^the procedure ^to multipoles

with special emphasis ^on quadrupoles. ^{It is} conjectur-

ed that the charge-quadrupole interaction _energy

might ^{be of the same order of} magnitude ^{as the} dipole- dipole interaction _energy.

Part five is devoted to applications. ^{A check of}

the method is provided by ^the evaluation of the

magnetostatic dipolar energy of MnO which is found to be in good agreement with the Cohen-Keffer value. The Coulomb, dipole ^and polarization energies

as well as the potentials ^{and fields are} calculated for

CdC12 ^and Ti02. ^{The sum of} polarization ^and dipole

energy ^{amounts to} about 22 % ^for CdCl2 ^and only

4 % ^for Ti02 ^of the Coulomb _energy. The dipolar

energy is positive ⁱⁿ CDCI, ^and negative ⁱⁿ Ti02

which results in a moment depression in the former and a moment enhancement in the latter by the respective factors of 0.664 and 1.67. The induced moments are found to be m(Cl) ^{= 0.66} debye ⁱⁿ CdC’2 ^and m(O)

^0.79 debye ⁱⁿ Ti02. ^It may be estimated that the corresponding displacement ^bet-

ween the nucleus and the centre of gravity ^{of the} charge cloud around the anion can be detected by combined X-ray ^{and. neutron} diffraction measure-

ments at least ⁱⁿ Ti02.

Special topics ^{as non} overlapping density functions, lower limits of summations, self-energy computa- tions and overlap corrections with special emphasis

of the Ewald method are dealt with in appendices.

Part 1 : Fundamentals about charges.

^{1 . 1 THE}

COULOMB ENERGY.

The method of non over-

lapping charges [1] combines three main ideas. The

first one is that the total Coulomb _energy Wtot ^{of a} charge distribution p(r) generally ^{written as}

can be reexpressed by ^{means of the} self correlation function of the charge distribution p(r), say P(u) (the ^Patterson function ^{of the} crystallographers)

so that by ^the substitution of r’

r + u (1.1) ^becomes

The second idea makes use of the fact that the Fourier transform of a self-convolution as in (1.2)

is a positive quantity, ^a square. Thus the Fourier transform of P(u) is 1 F(h) 12, ^{where F(h), the Fourier}

transform of p(r) (1.4) is called the structure factor (1 . 5). ^In (1.4) V ^{is the volume of the unit cell} and 1

h

is a summation over the whole reciprocal space. In

(1.5) ^the integration ^{and the} summation 03A3 ^{are over}

j

the unit cell.

When P(u) is written as a Fourier expansion (1.6)

and the integration ⁱⁿ (1.3) ^is performed ^{over the}

whole _space, one gets ^a positive quantity (1.3’)

for Wtot.

Here il ^{means that h}

⁰ is omitted in the summation.

Note that F(0)

⁰ (neutrality condition). ^However Wtot( 1 3’) diverges ^when p(r) ^{is a sum of} point charges.

The reason is that Wtat also contains the self energy Ws,

i.e. the action of the charge ^on itself which in the Coulomb case is infinite. To _escape this difficulty,

one must remember that the physically interesting quantity is neither the total nor the self _energy, but their difference, ^the interaction _energy W; per unit cell

The third idea uses the theorem, well known in

electrostatics, that point charges of density qj ^{03B4(r -} r j)

can be replaced by ^non overlapping sphero-symmetri-

cal continuous charges

and vice versa without changing ^the interaction energy. Thus considering ^such continuous charges

it becomes possible ^to compute the Coulomb inter- action _energy Wi (1.7) ^{as the} difference of two positive and finite quantities Wtot (1.8) ^and Ws (1.9), ^both

evaluated in reciprocal space.

cp(h) ^{is the} Fourier transform of f (r). ^{The function} f (r) is chosen such as to be non zero in a sphere ^of

radius R where 2 R ru, ^ri¡ being ^{the smallest}

distance between atoms, and zero for r > R. Examples

of choices for f(r) ^are found in the literature

(f(r) ^{= constant} [1] ^and /(r) ^oc (R - r)" ^with

n ⁼ 1 [5] [7], 2, ^{3 and 4} [6]) ^{and in} appendix ^{A. For}

instance in the parabolic ^{case one has}

with

and

I .2 POTENTIALS, FIELDS AND FIELD GRADIENTS.

From the interaction _energy (1. 7 ^to 1. 9) ^one may derive the potential V(rj) ^{(1 . 14),} the electric field ( 1.15),

and the field gradient (1 .15) [4] ^at the lattice point rj,

with

In (1.14) ^to (1.16) ^{we have} introduced the abbreviation

1. 3 THE SELFCORRELATION FUNCTION.

It is worth- while to summarize here some properties of p(u) (1.18)

the selfcorrelation function of f(r) which will be used in the following :

One has

If f(r) ^{has the reach R,} p(u) has the reach 2 R.

(By ^{reach is meant the non zero} region ^{of the} function).

The Fourier transform of p(u) is ] ç(h) 12. ^{One has}

Thus when f(r) ^is normalized, p(u) ^{is also norma-}

lized. The self _energy of ^a unit charge »,, ^{can be}

evaluated either in direct or reciprocal space (1.21).

where 0(h) ^is given by (1.17). ^We finally ^mention

the important ^{sum rule} (1.22) which, proven in

appendix B, is valid without _any approximation ^and provides ^a good check for the correctness of calcula- tions and for convergence ^{tests :}

Throughout this paper the ^vector

is always ^{a vector of} reciprocal space. It is defined

as being ^discrete (hi integer, j ^{= 1, 2,} 3) ^{under the}

summation 1 ^{which is over all vectors of the reci-}

h

procal lattice, ^{and as} being continuous in hi ^{under the} integral which extends over the whole reciprocal

J oc

space. Note that h is ^not restricted to the k-space

of ^a Brillouin zone and that the ^zone concept ^{is not} needed here.

Part 2 : Dipole ^lattices.

^{2.1 1 THE EQUIVALENT}

CHARGE CONCEPT.

2.1.1 1 Point dipoles.

^A point dipole ^{m is defined as the limit of the} product ql when q - ^{00 and 1 - 0.} By analogy ^{we define the} equivalent ^electric density ô(’)(r - rj) ^{of a dipole m}

at point _ri ^{as the} limiting ^{value of the} difference of

two point charge ^densities

It is easily ^{checked that the Coulomb} potential produced by ^the equivalent ^electric density ô(1)(r - rj)

reproduces ^{the classical} potential ^{of a} dipole ^m

In the same way the Coulomb _{energy Wkl} produced by ^{the densities} £5(1 )(r - rk) ^and ô(1)(r’ - r,) equals

the interaction _energy of two point dipoles ^{mk and}

m, ^at points rk and _r,.

The demonstration is based on the fact that

Thus the operator - mk8 or ^{or + mk8} Vrk ^replaces

the charge operator qk ^{as well in the} potential ^{as in}

the energy formulae.

2 .1.2 Continuous charges.

^A continuous charge density f(O)(r) equivalent ^{to a} point charge density ô(r) ^is obviously ^obtained by the convolution of a

spherically symmetric ^function f(O)(x) ^with ô(x)

We apply ^{the same} procedure ^{to find a continuous}

electric density f(1)(r) equivalent ^{to that of a} point dipole by convoluting 03B4(1)(x) ^with f(O)(x). ^Thus

in agreement ^{with the} operator ^rule given ^above.

Here f(O)(r) obeys ^{the non} overlap conditions outlined in part ¹ (f(O)(r)

^{0 for r} > R). ^{The same condition}

also imposed upon f(l)(r) implies ^{however an addi-}

tional constraint on f(O)(r), ^the vanishing ^of V f(O)(r)

for r

R. Thus the choice of the parabolic charge density f(O)(r) (1 . 0) ^is convenient for charges ^and dipoles.

It is instructive to prove that the continuous electric

density f(1)(r) ^is equivalent ^to the action of a dipole by checking the relations

Here the first integral (2. 7a) ^is easily transformed

to a surface integral ^with f(O)(r)

^{0 on the surface}

r

R. The second formula is _proven by partial integration. ^The third relation (2. 7c) is demonstrated

by ^{the same} steps ^{as in} (2.2). ^We only ^{mention that}

here the last step ^{of the} proof ^is the evaluation of

where the integral ^can be rewritten as

This last relation just expresses the theorem,

alluded to in part 1, ^{of the} equivalence ^{of a} point charge ^{and a} continuous charge density, ^{non over-} lapping ^{on r.}

2.1.3 The operator formalism in Fourier _space.

The dipole operator - m. ~r corresponds in Fourier space ^to the scalar product 2 II im. h. One has indeed

This Fourier formalism is appropriate ^for many

calculations. For instance the self _energy ws(1) ^{of a}

dipole which expresses the action of the dipole ^on

itself is easily ^{evaluated as} (see appendix C)

2.1.4 The importance of ^the equivalent cllarge

concept.

^{If the} substitution qj ^- mj. ~rj ^{is made}

in the expression ^of p(r) ( 1. 4), ^{one is led to the} equi-

valent electric density

Here ^we have introduced a dipole ^structure jàctor

The importance ^{of the} equivalency concept ^deve- loped ^{here resides in} the scalar nature of the electric

density. Thus the total electric density will be additive

where p(O)(r) ^{is the Coulomb} charge density (1.4)

and p(1)(r) ^is given by (2 .11 ). The consequences of this additive relation will be examined later (Part 3).

For the time being ^we restrict ourselves to the study

of p(l)(r) ^{alone, i.e. of} dipolar ^lattices separately.

2. 2 DIPOLAR ENERGIES AND FIELDS.

2. 2. 1 Ener- gy evpressions.

The energy of a dipole ^lattice Wi( 1)

can be found directly ^{from the Coulomb} energy formulae ((1.7) ^to (1.9)) by applying ^{the above} operator formalism, say replacing ^{in the structure}

factor F(h) each gk exp 2 nlb.rk by

and in the self-energy ^term

(cf. Appendix C). The result is, ^{as stated} in the intro-

duction, ^the difference of two positive quantities

We leave it ^{as an} exercise to the reader to arrive at the same relation by inserting o(l)(r) (2 .11 ) ^into

the total energy expression (1.1) ^or (1.3) ^and by substracting ^{the self} energy expression (2 .10) ⁱⁿ complete analogy ^with (1.7) ^to (1.9).

A straightforward application ^of (2.14) ^is given

in our discussion of CdCl2 ^{and of} Ti02 ^{and the}

moments induced there. We shall describe still

another form of the dipolar energy which makes the total polarization appear. Consider a periodic

lattice of point dipoles _mi, ^The dipolar density M(r)

which is periodic ^{is described} by

where D(h) ^is given by (2 ,12). ^If M(r) ^{is convoluted}

with a continuous function f °(r) ^as specified ⁱⁿ part 1,

one obtains a continuous dipolar density, ^denoted M(r)

The convolution of M(r) ^{with itself} produces ^the following ^relation

Now put ^{u =} 0. In the left hand member peri - r,) ^is identically ^zero except ^for ri

rk. On the right ^hand side separate the h = 0 term which is

where

is the total polarization. ^{One has} finally ^the identity

which inserted into (2.14) gives ^{rise to}

This expression will be discussed in the paragraph

on dipolar ^fields.

Finally ^{we shall} separate ⁱⁿ Wil 1) ^{(2.14) terms of}

the same species ^{and of différent} species :

where

The dyadic Ajk ^is given by

It can be shown that due to thé sum rule (1 .22) in the cubic case, Wjj(1) ^{(2.23) is} identically ^{zéro for}

each species j (see appendix B) ^{so that} Wi( 1) strictly

reduces to

We have used the property expressed by Wjj(1) ^{= 0}

and by (2.26) ^{in the} computation ^{of the} dipolar magnetic energy of MnO.

2.2.2 Dipolar fields and potentials.

^{The field E(1) )}

created by ^the dipole mk ^at the point ^{where the} dipole m, is located is given by derivation of the _energy (2. 3)

This gradient ^{relation also holds for a set of} dipoles,

Thus in a periodic dipole lattice the field at point ri where the dipole _mj is located will be

Multiplication with ’ mj and sommation over the unit cell reconstitutes (2.14) ⁱⁿ agreement ^with

Remark.

Of course the dipolar ^field E(l)(rj) ^(2.28)

is also obtained by ^the application ^{of the} operator rule to the expression (1.15) of thé Coulomb field E(O).

In thé same way, the potential V(l)(rj) ^as derived from

(1.14) isgiven by

The version (2.21) ^{of the} dipolar energy is parti- cularly interesting because it gives ^{rise to}

Here the first term on the right ^side represents ^the classical Lorentz polarization whilst the second term

represents ^{the structure} dependent ^{deviation from}

the Lorentz field. This formulation might ^{be useful}

in polar crystals.

Tlm cuhic lattice case.

One has the relation (2. 32),

proven in the appendix B,

In other words in a cubic cell the dipoles ^{on the}

Bravais lattice.j ^{do not} contribute to the field E( 1)(ri),

which reduces to

The equivalent of the Ewald version for the _energy of dipolar ^{lattioes is described in} appendix ^E.

Put 3 : Induced moments. - 3 . 11 SELF CONSISTENT DETERMINATION OF THE INDUCED MOMENT. - The total electric field at rj, ^sum of the Coulomb field

El’o)(ri) ^{and the dipolar field} E(’)(rj) ^(2.28), is related

to the moment induced at r, by ^the polarizability ocj,

a symmetric ^tensor which for simplicity ^{will be}

written here as isotropic :

When the dipolar ^field E(’)(rj) ^{(2.28) which is a}

linear function of the moments is inserted in (3.1),

one obtains a linear set of équations of the form

where

Here 1 is the dyadic identity ^and Ajk ^{is given by} (2.25). (3.2) ^can be solved with respect ^{to the unknown} quantities ^{which are the} components of the moments, all other parameters being known. We shall not discuss hère (3. ) ^{in its} generality, but consider the

simple ^{case where} only ^one species ^{of n} équivalent polarizable ions with m, ^{colincar to} E)1) contributes to the dipolar energy (for instance n = 6 for Cl- in the hexagonal ^{cell of} CdC!2, 7ï

⁴ for 0 in the unit cell of Ti02).

One has then the simple ^relation

Here Wi(’)(1) ^{is the} dipolar energy for the unit dipole.

The solution or (3.2) ^is simply

One will have to distinguish ^{two cases} according

to the sign ^{of the} quantity W¡(1)(l).

i) ^The dipolar energy is positive ^{in which case the}

factor 2 ^{+ aj} Wi(1)(I) . 2) -1 depresses the action of

(2 + aj W ^{i (1) n} n)-1 dep

the Coulomb field.

ii) ^The dipolar energy is negative ^{in which case}

the action of the Coulomb field is enhanced.

Both cases will be encountered in our discussion of CdCl2 ^{and of} Ti02.

3.2 THE TOTAL INTERACTION ENERGY.

The total interaction energy is not the sum of the Coulomb energy ((1.7) ^to (1.9)), quadratic ^{in the} charges q, and the dipolar energy (2.14), quadratic ^{in the}

moments m. Cross terms in _{q. m} arise from the fact that the induced dipoles produce ^a supplementary potential acting ^{on the} charges, ^{whilst the} charges produce ^a Coulomb field acting ^on the induced

moments. The total interaction _energy must have the form (appendix D)

where Vj ^and Ej ^are respectively ^the potentials ^and

fields of both charges ^and dipoles.

The simplest way for constructing W; (3.6) ^is again ^the operator formalism which replaces qi ^by

qj ⁺ mj. Vrj ^{and thus F(h) by G(h) in the energy}

expression (1. 7) :

The result is

where W;(g, m) ^is given by

Remark.

In centrosymmetric crystals F(h) ^is

real and D(h) purely imaginary. Putting

one obtains a polarization contribution

which generally ^exceeds by far the contribution of

W.(l) ⁱ

Physical meaning of Wi(q, m).

Substituting ^the

definition of D(h) (2.12) ⁱⁿ (3.9), ^the comparison

with the Coulomb field E(’)(rj) ^{(1.15) leads to}

On the other hand comparing ^the potential V(l)(rj)

(2. 30) ^to (3. 9) ^{one finds}

Thus one has certainly

This relation joined ^{to the} defining ^relations

completes ^the proof ^{of the} equivalence ^of (3. ) ^and (3 . 8).

Remark 1.

The total interaction _energy (3.8) obeys ^{the conditions}

and

where Vj ^and Ej ^are respectively ^the potentials ^and

fields of both charges ^and dipoles. ^{The condi-}

tion (3.16b) ^is easily ^{shown to} hold from the identity

Remark 2.

The interaction _energy (3. ) ^can

also be found by evaluating first the total electro- static _energy according ^{to the} classical Coulomb

expression (1.1), ^{where the total} density p(r) ^is given by p(O)(r) ⁺ p(l)(r), ^{and then} subtracting ^{the corres-} ponding ^self energy ^terms.

Part 4 : Generalization to multipoles.

4 .1 CHAR-

GE OPERA TORS.

Our procedure consists in replacing

charges qj ^{by charge operators} Qj ^so that the inter- action energy Wi ^{continues to be} given by the Coulomb like expression

For instance if only charges ^and dipoles ^are present

Qj ^{is given by (4.2) and W;} will contain thé charge- charge, charge-dipole ^and dipole-dipole interactions

of (3 . 8)

For the sake of uniformity ⁱⁿ notation, ^{we write} for a charge, ^a dipole, ^a quadrupole ^the following :

Here the subscript j ^{stands for the} point r ¡, ^the

vectors n have unit length, ^and q(P) has the dimension of a charge. ^{We define the} charge operators ^at point j

and more generally ^{for a} 2P-pole, depending ^on 2 p ^{+ 1} parameters

so that

4. 1 . 1 Equivalent charge densities.

We have

already ^{found the} equivalent ^electric density ^{of a} dipole (2. . 1)

The equivalent ^electric density ^{of a} point quadrupole

at ri is obtained as the limiting ^{value of the} equi-

valent electric densities of two dipoles, displaced by ^{1 when} q(1) ^{tends to} infinity ^{and 1 to zero with the}

result (4.7)

More generally, ^the equivalent ^electric density

of ^a point multipole ^{of order 2P will be}

In complete analogy ^with (2.6) ^{the continuous}

electric charge density f(P)(r - rj) equivalent ^{to a} point multipole ^{is obtained} by ^the folding operation (4.9) ^of ô(P)(r - r;) ^{with a} continuous function

For instance,

and the condition of non overlap imposes ^upon f(O)(r) ^the constraint of having ^its second derivative

equal ^{to zero for r}

^{R. The} following ^function

satisfies this condition (cf. (A. 2)) :

for r R and zero for r > R.

Remark.

When the gradient operation Vj ^at point rj ^is replaced by ^{V at} point ^{r, one must} multiply o p(p) by ( - 1 )p (cf. (2. 4)).

4.2 INTERACTION ENERGIES.

The substitution qj --> Qj ^{(4. 5) in the} expression ^of the total energy

(1 .3’) ^{has the effect of} replacing F(h) by ^a genera- lised structure factor Y(h)

where

New terms will have to be added to the total inter- action energy (3.8) corresponding ^to charge-quadru- pole, dipole-quadrupole ^and quadrupole-quadrupole

interactions :

where the new self _energy terms ws(2) ^and ws(q(O) q(2»)

are given ⁱⁿ appendix ^C (1). ^{It is} quite remarkable that

a cross term of self-energy charge-quadrupole ^exists.

It may also be conjectured ^{that the} dipole-dipole

interaction energy Wi(’) ^{is of the same order of} magni-

tude as that of charge-quadrupole W¡(q(O) q(2»). ^How

far should one go in such multipole expansions ?

From the point ^{of view of} physics ^{it is} quite ^clear

that the static Jahn-Teller effect is equivalent ^to multipole ordering. ^{One must} question ^{however the}

usefulness of lattice sums for multipoles ^{of order} higher ^{than 22 where a near} neighbour ^summation

in direct space will probably turn out to be sufficient.

In the case of overlapping ^densities (for ^instance

in the Ewald method) correction terms have to be added to the expressions ((4.14)-(4.I6)). They ^can

be obtained according ^{to the rules} specified in appen- dix E.

(’) ^{In formulae} (4.14) ^to (4.16) ^the quantities 0(h), F(h), D(h)

and Q(h) ^{have been written without the} argument ^h.

Part 5 : Applications.

^{5. 1 THE} MAGNETOSTATIC ENERGY OF MNO. - As a test we have checked our

method on the magnetic ^structure of MnO where

good evaluations of the dipolar ^{energy are} already

available [8]. In MnO where ferromagnetic (111)

planes alternate in the [111] direction, ^the spin

direction being ^{in the} plane, ^the dipolar ^structure

factor can be written in the magnetic ^{cell a}

² ao

as follows

Here the first bracket corresponds ^{to a face centered} lattice, the second one to a body anticentered lattice and the last one to the spins ^{at the} origin ^{and to its}

three nearest neighbours ^at 1/4 1/4 0 ^and permutations.

for h, k, 1 odd and such that

Otherwise D(h) ^is identically ^{zero. The shortest}

distance between two Mn-atoms is a /2-/4 ^{so that}

R = a l18 is chosen with the continuous parabolic charges ((1.10)-(1.11 )).

Noting

we form the average of (8.h)2 ^{over the} equivalent

vectors hkl, lhk and klh with the result

where

a, f3, ^y

direction cosines.

The dipolar energy per magnetic ^{unit cell of}

volume V is then conveniently ^written

Here 1 ^{is the summation under the} conditions (5.2),

h

p is the multiplicity ^{and d =} 1 h ,- ^{1 the} equidis-

tance of the lattice planes. The first term in the

right ^{hand member of} (5.6) ^{turns out to be zero ;}

it expresses the ^sum rule, given ⁱⁿ appendix ^{B. The}

second term has been summed over the 14 vector

manifold 111, 311, 331, 333, 511, 531, 533, 551,

711, 553, 713, 733, 555, 715 with the result

Recalling ^{that N}

4la’ ^is the number of spins

per cm3 and that V

8 a30 one ^{finds for the} dipolar

energy per cm3

Cohen and Keffer [8] ^obtain nearly ^{the same result,}

their numerical coefficient being 3.615. Our precision

can be estimated from the degree ^{to which our sum-}

mation 32 L pcp2

^123.100 approaches ^{the theoreti-}

h

cal value

For the general ^{case of error} estimation we refer the reader to appendix ^A. The difference is about 0.3 % ^{so that we estimate the} precision ^{to be}

3.615 + 0.011.

T

yp ⁺ {3y + ya has two extrema : a maximum

ï

1 for a = {3

y, say the direction [111] ^which corresponds ^to (N.S.)2 ^3.615

^{1.640 x 10’} erg/cm3, representing ^the magnetic ^out of plane anisotropy (with ^N

^{4.59 x} 1022 Mn atoms/cm3 ^and

S

5 uR

^{4.64 x 10- 20} erg/gauss), ^{and a minimum}

T = - 2 ^{for a} + {3 ⁺ y

0, say for _any vector in the (111) plane ^which corresponds ^{to a} dipolar

energy of - 0.820 ^x 10’ erg/cm 3.

Note that the Cohen-Keffer result was obtained

by sampling ^{about 30} points ^{in the} k-space ^{of the}

first Brillouin zone. Their method is restricted to cubic lattices whilst ours is perfectly general. ^le

also compare ^our method to the Ewald version of

dipolar energy in appendix ^E.

5.2 ENERGIES, POTENTIALS AND FIELDS IN CDCL2.

As first example ^{of an ionic structure} with induced

dipoles ^{we choose the} rhombohedral structure of

CdCl2 ^{which is} described here in the hexagonal

system ^with parameters ^a

^3.85 Á and c

17.46 Á.

There are three formulae units in the unit cell of volume V

224.128 Â’. The atomic positions ^{are :}

Cd at 000 and Cl at ± (00z) plus translations 000,

333, 3- S3- ^In thé idealized structure z

1/4 ^{and the struc-}

ture factor is

It differs from zero only for - h + k + 1 = 3 n.

With z = 1/4 and q (Cd) ⁺ 2 q (Cl)

⁰ (neutrality condition), it takes the form

The smallest distance d (Cd-0), that between Cd (0, 0, 0) ^{and Cl} (-L, -Z, - 1/12), ^{is 2.657 A so that}

R ⁼ 1.329 A.

We have performed the summation in Wi(’) (1.4)

with the choice (1.10)-(1.11)) of f(r) ^and g(h), using ^{30 terms} (11 ^{with 1 even and 19 with 1} odd) ⁱⁿ

the order of increasing reciprocal ^{lattice vectors} until | 1 h | ^{= 1.06} Å-1, the last h used being 0, 0, 0, 18.

The explicit ^{form is}

The result is

Johnson et al. [7a] ^{find a} slightly ^{different va-}

lue ^{= -} 1.685 5 e2 that is Madelung constant/shor-

test distance due to different cell parameters ^a

^3.86 and c ⁼ 17.50 Á.

The Coulomb potentials ^at the lattice points ^are

found to be

and

One checks that -1 E qj Vi,01 ^{= wi(0).}

j

The electric field at the Cd nucleus is £(0) (Cd) ^{= 0}

which is evident from the point symmetry. ^{The field}

at the CI position ^{00z has a non zero} component

only along ^{Oz. Its} explicit ^{form is}

The quantity under 1 ^{turns out to be negative so}

h

that the field is along - ^z (q (Cl) ^is negative). Using

the conversion factor

the absolute value of E(0) is 4.82 V/Â ^{in close} agree- ment with the value of 4.81 calculated by ^Weenk

and Harwig ^{with a} very large ^{number of terms in}

their summation [9].

The Coulomb fields acting ^on 00z and 00E are

opposite ^{and the induced moments are noted + m}

and - m respectively ^{so that the} dipolar ^structure

factor is purely imaginary

The explicit form of the dipolar energy is _per unit cell

We have calculated Wi(’) ^first with the function (p(h) already ^{used for} Wi(0) and based ^on the shortest Cd-Cl distance (2 R = ^2.657 Â) obtaining

Wi(1) ~ ^{0.498 m2 Å -3 .}

Actually ^{Cd does not} intervene in the dipolar energy

so that one may ^{use as} shortest distance that of the nearest Cl-Cl atoms which is 2 R’ ⁼ 3.664 A. The convergence is then greatly improved ^{and one obtains}

with only 15 terms, the last ^one (0, 0, 0, 15) ^contri- buting ^{6 x 10- 5 of the total,}

It is remarkable that the dipolar energy of the induced

dipoles ^is positive ⁱⁿ CdCI2.

The dipolar ^{field is} explicitly (cf. (2.28)),

Thus E(1) is opposite ^{to m} and will reduce the action

of E(Ol. Using the value x ⁼ 2.97 x 10-24 CM3 for

the polarizability of chlorine [10], the reduction factor

(1 ^{1 +} aWi(1)(I)/3)-l ^{in relation (3.5) is 0.664, so that}

from (5.11) ^{we obtain}

Substituting this value in Wi(1) ^{one finds}

say about 4.4 % ^{of the Coulomb} energy, but of _oppo- site sign.

The dipolar ^{field is} numerically

The polarization ^{term, linear} in q ^{and m, evaluated} according ^to (3 . 9)-(3 .11 ) ^is explicitly

It is also equal ^{to - 6 E(O).m} according ^to (3.12)

as it should be. The interaction _energy due to the induced dipoles (Wi(q,m) ⁺ Wi( 1)) ^is quite ^conside- rable, ^{at 21.6} % of the Coulomb _energy.

Assuming q (Cl) = - ^{e, the} energies per molecule

CdC’2 ^{are :} Wi(0) ^{= - 1.695} Wi(1) ^{= 0.076 and}

Wi(q, m) = - ^{0.442 in units e2} Å-1, say

The conventional polarization ^energy (see appen- dix D) is 2 1 W¡(q, m) = - ^73.5 kcal/mol, ^{still an} appreciable ^{amount of 13} % of the Coulomb _energy.

5.3 ENERGIES, POTENTIALS AND FIELDS IN T’02,

RUTILE.

Ti02, ^rutile crystallizes ^{in the} tetragonal

space group P42/mnm. The lattice parameters ^are [9a]

a ⁼ 4.592 9 Å ; ^{c = 2.959 1. Ti is in the} positions ^{2 a :} 000, 222 0 is in the positions 4 g : ^xx0 (1), ^Ri0 (2), + x t - x 2 (3) ; 1/2 - x 1/2 + x 1/2 ^{(4) with}

x

0.305 3 [9a]. ^{For reasons of symmetry the}

moments induced on these four atoms have the

following ^form

(x ^and y ^are orthogonal ^unit vectors) ^{which is} imposed by ^the glide plane n. Furthermore the site symmetry of the oxygen site is mm and the intersection of these mirrors (one ⁱⁿ xyO ^and xy1/2, ^{the other one in xxz and}

xxz) ^{is a twofold} diagonal ^{axis which must be the}

direction of all polar vectors ; this implies

The central Ti atom at -1/2-1/2-1/2 ^{has four nearest} oxygen

neighbours ^at d,

^{1.946 4 À} (in ^{xxO and} homologous points) ^and two next nearest ones at d2 ^{= 1.983 0 A} (in points (3) ^and (4)). Consequently ^{the radius of}

non overlap ^{is chosen R} = 2 d,

^{0.973 2.}

The structure factors F(h) ^and D(h) ^are

Here we have abbreviated

Note that the dipoles (5.16) form four sublattices with m,

M2, m3 ^{= -} m4, and _ml orthogonal

to m3, and that consequently D(h) ^is purely imaginary

as expected.

Charges.

^{The Coulomb} energy (cf. (1.7) ^to (1.9)) amounts to - 4.911 q2 (0) Å -1 ¹ per unit cell.

If we _suppose that the _oxygen charge is q (0) = - 2(e)

we obtain

We have checked independently ^the potentials ^accord- ing ^to (1. 14) ^{and find}

The Coulomb field acting ^on the oxygen ^atom in xxo has a value of

in good agreement with the value of Weenk and

Harwig [9] ^{of 2.82} V/Å. The direction of the field is’ along l- 1, -, l, 0] (in ^{tact we have calculated}

Ex(0) ^{and found} Ex(O) ⁼ Ey(O) ^{by a} simple interchange

of the variates h and k under the summation in

(1.15)).

Dipoles.

Disregarding first the symmetry ^condi- tion (5.17) ^we have found the following ^numerical expression ^{for the} dipolar energy

Here the first term is the différence between the isotro-

pic part of the total energy and of the self energy ; the second term does not depend ^{on the self energy}

and represents ^a dipolar anisotropy. This result

has been checked with several charge distributions, respecting the conditions f(r)

^{0 and} Vf (r)

⁰

for r

R. One may wonder why Wi(1) ^{is not minimal}

for _mx

my. ^In fact one has to minimize

As

it turns out that the symmetry conditions (5.17) minimize the m-dependent part ^of the interaction energy for _mx negative. Finally

wherefrom the dipolar ^field

Adopting the value a(0)

^{2.4 x 10-24 cm3 for}

the _oxygen polarizability [10], ^{the factor}

becomes here an enhancement factor (cf. (3.5)) equal ^to 1.672 ^{so that}

The total field acting ^{on the} oxygen ^atom is thus

Finally the different contributions to the electro- static interaction _energy are given here per mole- cule Ti02 ^{in e2 A -1 1 and in} kcal/mol

Their _sum, 135.8 kcal/mol amounts to only 4 %

of the Coulomb _energy and the conventional pola-

rization _energy contribution to the cohesive _energy

(2 Wi(q, m)) ^is only ^1.6 % ^{of the Coulomb} energy.

Still the dipolar energy per molecule is about 57 times kT for T

300 K so that no transition _may be expected unless the charges ^and polarization ^are greatly overestimated.

Comparison ^with CdCI2.

CdCl2 ^{is a} typical layer ^structure and it is certain that the dipolar polarization stabilizes this structure type (’). ^This

does not seem to be the case for Ti02. ^{On the other}

(2) ^{If one considers as a measure of ionic} strength ^the quantity Coulomb _energy x smallest cation anion distance/q2 (cation)

one finds 375 for CdC12 ^{and 397 for} Ti02 (in ^{kcal x Á x mol-’} e-2).

hand, ^it is remarkable that the dipole ^{moment of}

the anion induced in Ti02, ⁱⁿ spite ^{of its smaller} polarizability and smaller Coulomb field is greater than in CdCl2. ^{If we make the} rough ^estimation

that the moment value of nearly ^0.8 debye ^corres- ponds ^{to a} charge ^{of 10 electrons} multiplied by ^a

shift s between the nucleus and the centroid of the

charge cloud, ^{we find s}

^0.08 Á. This corresponds

to a variation of 0.012 3 of the parameter-value x

between the neutron and the X-ray observation and should be observable. We hope ^{that such an} experi-

ment will be undertaken on Ti02 ^{in the near future.}

(From ^{a similar} reasoning the shift of the Cl nucleus in CdCl2 ^is expected ^{to be} equal ^{to 0.037 Á}

^Az.c

where from Az

0.002 only.)

Appendix ^{A :} Problems of self energy and ^conver- gence.

We first question ^the problem ^{of lower}

bounds of Coulomb and dipolar energy which is equivalent ^to the search of the smallest possible ^self-

energy.

COULOMB CHARGES.

Polynomials.

^{We have}

considered the simple polynomial

The normalizing ^constant Cn ^{is found to be}

By tedious, ^but straightforward procedures ^one gets for the self _energy

ws tends to infinity ^{with n} (at ^{the same time one tends}

toward the point case); ^{its smallest value is} ws(n

0)

3/(5 R).

Dirac Function.

The smallest self _energy is obtained for charges ^{at maximum} distance, ^i.e.

when all the charges ^are concentrated on the surface of a sphere ^{of radius} R, say for

The Fourier transform is formally ^{identical to the}

well known interference function of a linear grating :

The self _energy is readily ^{evaluated in} reciprocal

space

Thus ^a suitable lower bound for the Coulomb energy (with ^all f(r) identical) ^is

For instance in NaCI where R

a/4 ^and

£q2j

^{8 e2 this lower bound is}

^{16 e2/a whilst}

wil" J

^{13.980 8} e2/a.

Dipoles.

The self energy being proportional

to p(0), ^{one has to} look for the smallest possible

value of the self correlation function at the origin,

say of f 2(r) ^{d3r. We have} investigated polynomials fm(r) having ^their first m derivatives equal ^{to zero}

at the origin ^and having f ’(r)

^{0 for r}

^{R. These} polynomials ^{have the} simple ^form

where

The case m

0 reduces to (1. 0). ^{When m tends}

to infinity, y(x) ^{tends to a} rectangular shape ^{(= 1}

for 0 x 1 and = 0 for x = 1). One finds :

wherefrom (cf. (2. 10) ^and Appendix C)

Thus the lower bound for dipolar energies ^is

This result _agrees precisely ^{with an estimate} by

Griffiths [11], ^obtained by ^another method, CONVERGENCE. - Smallest self _energy is not syno- nymous with best convergence. For instance Wi(O)

converges ^as oc-6 for the uniform density (w°

3/(5 R)) ^{and as a-4 for} the Dirac density (A. 4) where ws

1/(2 R ).

Some special ^{cases :}

The infinite product

is such that cp(0)

^{1. In direct} space it corresponds

to the successive folding ^{of p} functions of type f(r)

R R R

and of reach R/2 , R/4 , ... R/2p , the reach of the resulting

2 ’ 4 ’... 2p ^g

convolution product being

The infinite product ^function

has been recommended by ^Tournarie [12]. ^{In direct}

space it corresponds ^to the infinite convolution _pro- duct of Dirac functions of type (A. 4).

The function fm(r) ^{with m}

¹ (A. 8) ^{has been}

found _very convenient [13]. We have used it for

Wi(1) ⁱⁿ Ti02

TERMINATION ERRORS.

Templeton [14] ^{was the}

first to suggest ^a significant improvement by ^an

order of magnitude. ^{When the summation is} stopped

at hmax, ^he replaces ^{the rest} by ^an integration ^between hmax ^and infinity. ^{In the Coulomb case} this correction to be added is

where

AWi(0) ^is tabulated for uniform [14] and linear densi-

ties [7].

In the dipole ^{case, we find with} obvious notations

where

Due to the sum rule of appendix ^{B one has} approxi-

mately

Appendix ^{B : The sum rule.}

The Fourier trans- form of the correlation function p(u) ^is

el

Conversely

Thus

Consider the special ^{case of} points ^{of mass one,}

distributed ^{on a} lattice. Their density ^{is a} lattice delta function

Summing ^relation (B .1) ^{over all h and} inverting

summation and integration ^on the r.h.s. one obtains, using (B. 4)

Relation (B. 3) ^and (B. 5) establish the sum rule

(1.22) ^{of the text.}

The sum L 1 ^(p(h) 12 has remarkable properties

h

of homogeneity for selected sets of indices h, k, ¹ which we list here without detailed proof. ^{One has}

where the indicated parities apply ^{to h, k and 1 res-} pectively.

Thus for instance

We have _proven and checked this relation (cf. ^rela-

tion (3.15") ^{in reference} [4]). Furthermore for h, ^k and 1 odd, ^{the sums} h + k, k + 1, 1 + h ^can only

have one of the following forms with equal weight

Finally, ^one has for the summation performed ^under

the condition (5.2) ^{of the text}

The cubic ^case.

Consider the dyadic

and evaluate the quantity m.R.m where m is an

arbitrary, but fixed vector. The _average of

over the symmetry equivalent ^{vectors in the cubic} system is 1 h 12 m2 . 1/3. Consequently

wherefrom

m being arbitrary ^{the vector} quantity ^{in brackets}

must be zero.

Remark.

We have derived the sum rule (1.22)

in the form needed for the text. In fact if qJ(h) ^{is the}

Fourier transform of _any function f(r)

one easily ^shows by ^{the same method as above} ((B 1)

to (B. 5)) ^{that the following sum rule holds}

privided f (0) ^is finite. However one has not

because :

i) the Fourier transform of 0(h) (1.17) ^is p(u)/u ^and

lim p(u)/u ^{is not finite and}

u- o

ii) W(0) ^{is infinite even when the} integral is finite.

The sum rules (1.22) ^and (B. 13) ^{do not} depend

on overlap (cf. appendix E).