E. 1. Burrows ond S. F. A. Kewle University of East Anglio Norwich NOR 88C. England
Madelung Constants and Other Lattice Sums
The ionic model is sometimes regarded as one of the cornerstones of inorganic chemistw. Bv its use a wide range of phenomena m a y be either correiated or predicted with remarkable accuracy. Although i t is not the purpose of the present article t o ~ a t t e m p t critical assessment of the ionic model, we intmduce it because it provides the best known example of the application of lattice sums to chemistry. In the ionic model i t becomes necessary to sum the electrostatic forces acting on any particular ion because of its interaction with all other ions in the lattice. So, for the case of sodium chloride, each ion is surrounded by six ions of opposite sign at a distance R(R being the Na+CI- ionic separation), by a further 12 ions of the same sign a t a distance of fl R, by eight ions of opposite signs a t a distance of d R , and so on (tacitly, to infinity). The sum of all these terms may be written as
a
The value of the Madelung constant, M, for this lattice is usually given ( 1 ) as 1.74756. Analogous sums occur for all ionic lattices and tables of Madelung constants are widely quoted. Rather similar lattice sums occur for molecular crystals. Firstly, one approach to understanding the cohesive forces of molecular crystals is $0 postulate that these forces are essentially electrostatic. That is, each molecule is regarded as having, superimposed, a dipole, a quadropole, an octapole, . . . etc. The crystal cohesion then results from the interaction between these multipoles extended throughout the crystal lattice (tacitly, assumed to extend to infinity). In that a dipole, quadropole, etc. can be regarded as a sum of suitably arranged (fractional) point changes, i t is evident that the resulting lattice sums will be of the same form as the one leading to the Madelung constant in ionic crystals. These sums are, however, much more difficult to perform and a variety of subtle mathematical ~rocedureshave been devised 12). Secondly, sums arise for molecular crystals whenever the inremrtion of the crvstal with radiation is considered. Thus. the energy of an electronic dipole allowed transition will, in a molecular lattice. be determined not solelv . bv . the enerav ... ot the correspondinq transition wtthin nn isolated molecule but will alsc, be ~nfluencedby the mapiturle ot the Intermolecular transition-dipole-transition-dipole interactions within the crystal; i t is this interaction which gives rise to Davydov (factor group) splittings in solid state spectra I
The Convergence of Lattice Sums
The lattice sums which appear in the theory of crystal properties have the general form
for infinite crystals. Numerous well-known rules to test for the convergence of this series are available although it is not possible to frame a general rule which will give a decisive test for an arbitrary series (4). If the series can be shown to converge then the limit, B, to which eqn. (2) converges is called the sum of the series. However, the properties of B are not necessarily the same as those of a sum of a finite number of terms. For the sums with which we are concerned the signs of the terms in the series alternate. Let us, as a specific example, consider an infinite one dimensional ionic chain (Fig. 1).The Madelung constant for this chain then arises in the energy equation as
where, again, M is the Madelung constant and R is the distance between neighboring ions. Clearly the sum to be evaluated is
which converges (5) to a value of 2 (In 2). Let us consider in more detail the sum in eqn. (4). The terms in this summation may be rearranged in an infinite number of ways. Consider the following rearrangement
Reference Ion
~. Figure I.One dimensional ionic chain.
58
(3). Again, in attempting to understand the magnitude of such sp1ittings.a lattice sum is involved. We should perhaps also note that the relative intensities of the Davydov components also depend, theoretically, on such lattice sums. Evidently our ability to understand the properties of crystals based on these models depends on our ability to carry out lattice sums. It is the purpose of the present communication to make more widely known the fact that in the evaluation of such lattice sums one is faced with a fundamental difficulty; the feature of the problem which enables the difficulty to be overcome is very seldom stated.
/ Journal of Chemical Education
In this rearrangement we have neither lost nor gained terms because 1/(2n - 1) includes all odd terms and those in 1/(4n - 2) and 1/4n, between them, include all even
terms. Evaluating the differences in round brackets we have 1
1
1
1
I
1
(6)
I
1
.
which now has the sum l/z In 2. We have reached the unexpected conclusion that a series which has a sum of In 2 may be rearranged to give a series with the sum ' 1 2 In 2. Further, this is a quite general problem. Thus, yet anotherrearrangement of eqn. (4) is
1
1
=
41
81
1
[ 121
11 6
1
+
I
...
thus yielding a sum of '14 In 2. The general term in eqn. (7) is obtained by using that in eqn. (6) twice; by repeating this procedure n times the sum is found to equal l / Z n In 2-scarcely a unique value. To understand this result consider the sum corresponding to eqn. (4) where, however, the absolute value of each term is summed, i.e.
When summed to infinitv eqn. (8) diverees to + m . Such a series (as eqn. (4)) is called conditionafiy convergent series. A series for which both the series itself and the series analogous to eqn. @-that of the moduli of the term&both converge is called absolutely convergent. The basic difference between conditionally convergent series and absolutely convergent series is that for the former the commutative law a + b = b + a does not hold. Note that this statement is true for infinite series only, not for finite ones. This lack of commutivity means that the order in which the terms in a conditionally convergent series are summed is important. The only difference between the sums in eqns. (4), (6), and (7) is the order in which the terms appear. This conclusion is summarized in the following theorem due to Riemann: By a suitable rearrangement of the terms in a conditionally convergent series, it can be made to converge to any given number, to alternate, or even to diverge (6). Evidently, all the sums of the type we discussed earlier as appearing in lattice problems are conditionally convergent. It follows therefore, that mathematically there is no unique answer to the lattice sum problem and so, for instance, no such thing as the Madelung constant of any particular infinite crystal lattice.
a
Physical Consequences of Conditional Convergence The different answers which may be obtained for an infinite lattice sum arise because there is an infinite number of ways of ordering the individual terms within the sum. The most natural way in which a difference in the order of terms mav arise in a lattice sum is with the ohilosophy of the cal&ation. In the usual Madelung c a l k l a tion, terms appear in the order of increasine distance from the reference-ion. As an alternate procedure we could select some crystal plane containing the reference ion, sum all the contributions from this plane (to infinity), move to a parallel plane and sum its contribution, and so on until all of the infinity of planes have been included. This alternate procedure would lead to a rearranged series and thus
.
.
* (a
.
Mn'-.,+[L n . Lm. ~r+aLm. . . ]
- . @I
.
- . . . ,._...-,
-
(0
,, + *&
~ n - . [a I a, ~r -r L ~ ) . ~ ,H ..-G '..] [I-
*.]
Figure 2. Same two dimensional lattices and their Madelung sum;. a different value for the Madelung constant (or any other lattice sum). As a further example consider Figure 2 showing a two dimension lattice. In fa) we assume a circular shape and measure our distances in terms of R whereas in (b) a triangular shape is assumed while R remains the same. A comparison of the series generated shows that assumption of a different shape to fill all space leads to a rearrangement in the terms of the series. Throughout the above discussion we have taken the infinite sums literally-as sums to infinity. In practice, of course, these sums are used to describe finite crystals. Now, the rearrangement that we carried out on the onedimensional lattice sum problem had the effect of including terms arising from distant ions while omitting some arising from closer ones. For a finite lattice this is clearly not permissible for it corresponds, in the limit, to including terms from non-existent ions while omitting those from real ones! This, then, is the justification for the numeralogical ordering of the terms in a Madelung or similar series; with this order the infinite sum can approximate the finite one. The problem of the crystal shape remains. The normal Madelung expansion for a crystal such as NaCl effectively assumes that the finite crystal is spherical in shape. The rearrangement of the terms in the expansion needed to make it appropriate to a cubic box crystal would involve a rearrangement of terms in the infinite sum and, consequently, a different Madelung constant. In practice, summations over all atoms up to -100 A from a reference ion give results which are similar to those obtained bv summine to infinity, so rhar it is not surprising that it appears chat thr difference berween the snherirnl and cubic \ladel!~n~. constants is negligible (7). However, in some cases the shape problem remains (Craig and Walsh (8) have discussed this in some detail for the case of the crystal spectrum of anthracene) but its importance may be more apparent than realCraig and Walsh have suggested that, for some phenomena, the shape problem is of no importance because of other. moresi&ificant, assumptions (for instance, that the wave: length of the light used in a particular study is infinite). Ultimately,-the solution to the lattice sum problem must lie in the recognition that crystals are both finite and delineated so that the summations involved are not infinite. Unfortunately, for the present, direct summation seems to be out of the question-to evaluate the sum for a 0.1 mm3 crystal of sodium chloride would involve -10l8 terms, a task which would take a fast computer years to complete. Acknowledgment One of us (E. L. B.) is indebted to the Science Research Council for financial support. Literature Cited
.
(11 Wsddingfon.T.C..Adu. In"?#. Chem, ondRodiochem 1.157, 119591. (21 Burrowa. E. L.. andKettle, S.F.A.. iobepublirhed. 131 Sw for example. Robinron. G. W i k Annual. XFU Ph.vs. ('hem. 429. (19701 and references therein. 141 Bmmwich. T. .I. I.. '"An Intmduction t o the Theoru of Infinite Srrier" . T ~ ~~-~ mar^ P millan Co.. New York. 1959, p. ifi. ( 5 ) Jolly, L.B. W., "Summalion of &ri%"Dover. New York. IS(.!, p . IS. (61 Knopp. K.. "Theory and Applicatian of Inflnife Series." Biarkie and Sons Lfd, London. 1965. p.318. 171 Jenkins. H. B. D.. penonai communication. 181 C r a i ~D.V.. and Walsh, J. R.. J. C h m S o r . 16i3.11953l. ~~~~
Volume52, Number 1, January 1975 / 59
