Volume 55, number 3 CHEhfICAL PHYSICS LET-I-ERS 1 May 1978

ON CO~PA~BiLi~ OF STRUCTURES

Ames Laboratory-USDOE, Iowa State University, Ames, Iowa 5OilZZ. USA

Received 3 January I978

The problem of comparability of structures and functions defined on structures is considered foliowing the early work of Muirhead (confined to intsnl parametrization) and more modern genercdization of Karamata. The basis of such com-

parabilities is the set of inequalities which when satisfied permit one to est,blish J relative order (importance) of the struc- tures or associated functions. Chemical illustrations are given for both, Muirhead and Karamata inequalities. It is outlined how to go beyond the partial ordering in such comparisons when additfom ’ information on the trends rtmong the param- eters involved is available.

1. introduction

In examining theoretical or experiiental results one frequently makes comparisons without prior proper clarification of the criteria used. For example, we speak of a more or a less important valence struc- ture, of a more or a less branched molecular skeleton (isomer), or of a system having a higher or a lower symmetry. In order to consider such questions more rigorously one has first to recognize the relevant crit- icaf variable. For the above examples the variables involve an estimate of the contribution of the valence structures to the moIecuIar resonance energy, they represent the formal vaiencies of vertices in the molec- ular graphs for molecules, and they signify the num- ber of symmetry operations of the pertinent moIec- ular point group respectiveIy.

A mathematical formulation of certain conditions which can help in defining such comparisons rigorous- ly have been suggested already in 1901 by Muirhead [ 1,2] who confined his study to sequences in which ordy integers appear. Since s&z4ctir~! codes for moIe- cules frequently can be expressed by sequences of integers the above restriction still permits useful chemical application. An illustration is provided in a more recent study of chirality of Ruth [3] whose rules for ordering Young diagrams become identical to those of Muirhead. Such and similar aIgebraic

manipulations have found application in a number of mathematical problems related to graphs* and are important in several chemical topics: chiraiity [IS], molecular branching is], as we11 as in the field of non-equilibrium thermodynamics f6].

The restriction of the parameters to integers has been found subsequently not to be essential [2] _ and the scheme has been generalized, in particular by Karamata [7,8] who derived a theorem valid for functions defined in the given sequences_ In this usu- al modern point of view the approach of comparabif- ity is very weIJ known to scientists working in prob- abilistic, statistical and infommtion theoretical prob- lems, It appears that it is not generally well known among chemists, although, as will be illustrated here. it can be of considerable help and interest.

[Ye will outline the theorem of Karamata (the in- equalities of Muirhead can be viewed then as a less relevant special case) and illustrate the basic inequali- ties on selected chemical exampies. Then we will point to limitations of such co~lparabi~ity tests and suggest extensions, which however necessitate ava% ability of addiimal information on the sequences investigatea. As will be seen the comparability fre-

* For example, the conditions far realirabiiity of B valence sequence that it corresponds to .S graph, call for a comptri- son oi sequences of integers. See ref. $4 1.

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Volume 55, number 3 CHEiCiICAL PHYSKS LETI-ERS 1 May 1978

quently leads only to a partial ordering (which in some applications is not a deficiency). However ad- ditional infomation on the magnitudes of the struc- tural parameters involved may resolve the relative ordering even though the current use of the inequali- ties does not allow a condusion to be reached_

2. Basic inequalities

Following Karamata [7,8] let k, > k2 3 ._. 3 k, > 0 and k; 3 k; > ___ > k; > 0 be sequences of real numbers, We say that (kl , . . . . k,) majorizes (k; , . . . . kk) if

kl +k2+ . . . tk,>kk; +k;+...+kk.

m = 1,2,...,n-1

and

k,+kz+_._+k,=k;+k;+...ik’. n

if the above conditions are not met the correspond- ing structures are considered as not comparable. if all parameters are fhe same, the structures cannot be differentiated. When the above inequalities are satis- fied then also hoids for any continuous, convex func- tion ql(k):

@(kl) + #(k$ + --- + Hk,)

2 #(k;) + @(k;) + a.. f t#(k;)_

3. ilhzstrations

Consider benzanthracene and its 7 KekuE valence structures (fig. 1). We wish to order these 7 valence structures according to their relative contribution to the molecular resonance energy. Such contributions can be estimated by enumeration of the con&f&d citwifs in the individual KekulC structure [9]. Con- jugated circuits are defmed as those circuits in a mole- cule for which a regular alternation of the formal CC single and double bond occurs. In benzenoid systems they are necessarily of a (4x2 f 2) size and are design- ated by R, (table I), The number of conjugated cir- cuits of size (482 t- 2) plays the role of the parameters km _ The smaller conjugated circuits make the larger

548

Et

& & 0

2 & E F

Fig. 1. The KekuIi valence structures of benzanthracene.

contribution to the overall molecular stability. With these preliminaries we can now attempt to establish the relative importance of the KekulE structures A to G of benzanthracene. First, we see that the structures A and B have an identical decomposition of the con- jugation, and hence cannot be differentiated using the basic inequalities_ Structures A and C can be com- pared, and A majorizes C. The pair of structures C, D as well as the pair of structures F, G provide exam- ples of structures which cannot be compared from the set of the basic inequalities_ The co~espond~g partial sums (table 1) do not satisfy afl the inequali- ties: in the first case the relative magnitude of the partial sums is reversed after the first term, while in the second case the relative order of the partial sums is reversed after the third term. The ambiguity in the

relative ordering here originates from a lack of infor- mation on the relative magnitudes of (R 1 + R3) as compared to 2R, and (RI + R4) as compared to fR2 + R3) respectively. However, when such infor- fiation is available the basic inequalities do not pro- vide a way to establish the relative ordering of the structures. We will outline in the next section a pro- cedure which elevates this particular deficiency in the cornFamb~ty tests.

Volume 55, number 3 CHEMICAL PHYSICS LE-lTERS 1 May 1978

Table 1 An ilhrstration of the Muhhcad inequalities: Tha decomposition of the conjugation in the individual Kekulé valence structures of benzanthracene into contributions of conjugated circuits of (4n f 2) size. The fust and the second partial sums are aíso given

Vatente structure

Conjugated circuits

Code Partial sums Iterated sums

A 3Rt fR2 3100 3, 4, 4, 4 3, 7, 11, 15

B 3Rl +R2 31uo 3, 4, 4, 4 3, 7, 11, 15 c 3RE+R3 3010 3. 3, 4, 4 3, 6, 10, 14

D 2R1+2Rz 2200 2, 4, 4, 4 2, 6, 10, 14 E 2Rl+R2+R3 2110 2, 3, 4, 4 2, 5; 9, 13

F 2R1+R2fR4 2101 2, 3, 3, 4 2, 5, 8, 12 G RI+2R2+-R3 1210 1, 3, 4, 4 1, 4, 8, 12

As an illustration of Karamata’s theorem consider contributions. The bond contributions can now serve hexane isomers: for comparability of additive functions of compounds

X /I,L

such as alkanes, alcohols, fatty acids, etc. The bonds of the isomers to be compared are sequenced accord- ïng to the relative magnitudes of the bond indices.

(al (bf (cl For hexane isomers (c) and (d) we have:

W (11% (1,3), (1,3), (2,2f, (2,3);

(dl (f,% (1521, (1,3), (2,3), (2,3).

(dl fel

In studies of bond additivity regularities one may proceed first to classify edges (bonds) of the corre- sponding molecular graphs. A particularly useful scheme divides bonds in various (m. n)-bond types, where m, n represent the forma1 valenties of vertices in the mobcular graph in which hydrogens are sup- pressed [IO] _ To each bond type one subsequently assigns a bond index wlüch numerïcally determines the relatïve additive importante for the bonds. For a discussion of bond additive properties along a se- ries of homoIogous compounds a simple assignment of a value (mn)-l/2 to the bond type (m, n) seems to provide an adequate differentiation among bond

The corresponding partial sums from table 2 show that isomer (d) majorizes isomer (c). It then follows from Karamata’s theorem that rnz~~ continuous convex function of the bond indices will preserve the indi- cated majorization. For instance, this wil1 be the case for the boilïng points of the two isomers 110 ] , chro- matographic retention values [! i 1, and several other physico-chemical functions”, at least within the ec- curacy of the empirïcal eorrelations found_ A sïmïlar definite answer however cannot be stated for ali pairs

* A Iinear function can bc considered as a limit to a convex function. thus in practica1 applications extending the use 0fKaramats’s theorem if the scatter of e‘tprrimental points in the correlation used is not excessive.

Table 2 An ilhrstration of Karamata’s theorem: The bond types for hexane isomers and the corresponding partial sums leading to the par- tial ordering of bond additive functions

2,2dhnethylbutane 2,3_dimcthylbutane 2-metbylpentane 3-methylpentane n-hexane

(12) 0.7071 (1,3) 0.5774 (1,2) 0.7071 (1,2) 0.7071 (1.2) 0.7071 (1,4) 1.2071 (1.3) Ll.548 (L3) 1.2845 Cl,21 1.4142 cl,-) X.4142 (1.4) 1.7071 (1,3) 1-7322 (1,3) 1.8639 CL31 19916 (2,2) 19 142 (1,4) 2.2071 <1,3) 2.3096 (2,2) 2.3639 (2.3) 2.3999 <2,2) 2.4142 (2,4) 2.5607 (3,3) 2.6129 (2,3) 2.7722 <22,3) 2.8082 <2,2) 29142

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Volume 5.5, number 3 CHEM fCAL PHYSICS LETTERS 1 May 1978

of isomers of hexane. As can be seen from table 2 the partiai sums for the pairs (a), (b) and then again (d), (e), indicate a reversal of the magnitudes of some partial sums, thus cause some ambiguity in the order- ing of isomers. The fmal sums, which represent the molecular connectivity indices defme an ordering which wih parallel many molecular properties, but in view of Karamata’s theorem, the possible deviations for selected molecular features should not be over- looked.

4. Refmement of the comparability test

The indicated limitations of the comparability tests are due to hmited information employed in such comparïsons. The only requirement used is that the variables form an ascendkzg sequence. It is in a way remarkable how much can be deduced from such a limited information. From a very plausible assump- tion that the contribution of conjugated rings de- pends on their size we have been able to order many Kekulé valence structures of benzanthracene- But we may have addïtional information, such as the rate of the change of conjugated ring contributions with the change of the size. It seems worthwhile then to con- sider extensions of the comparabihty tests that could mcorporate the additional information. We wïll out- hne one such refmement in a continuation of the dis- cussion of the relative importante of the Kekuié va- Ience structures of benzanthracene. Observe in the following that we do not employ any particular set of numerical vahres for the partial contributions of different conjugated circuits to the molecular reso- nance energy. The available (approximate) numerical vahres only provide a guidance on the rate of the change of the parameters. Ambiguities in ordering valence structures of benzanthracene originate from the presence of the terms (R1 +R3) and X2. How- ever, since the curvature in the plot of R, against n is positive we have deftitely that (R1 + R3) is her

than 2R2. The same is true for other combinations of the parameters which can cause ambiguity. Hence a knowledge of the shape of the function can resolve the ambiguities previously encountered. Now we have

to select a way of implementing thïs kind of addition- al information in the comparabïlity test. One way of accomplishing this consists in taküzg the already de-

550

rïved partkzl sums as an ìnput ùtfonnation and then

to repeat the process of cons~ction of new partkl

szuns. One can verify that the earher ambiguities which caused incomparabïhties among some struc- tures are now resolved: At the criticai step instead of comparing the partiai sums S, and Sn, we are comparing (S, + Sm+l) with (Sm + Sk,l) bypassing the problem of comparing individual contributions. The same maneuver wiU resolve ambiguities arising when non-adjacent entries in the sequence cause the probIem.

We used here, roughly speaking, ïnformation on the first derivative of the function dependence de- fmed on the sequence. If no further details (such as the trend of the second derivative) on the function are known the anaiysis ends. If however stih more in- formation is available one can proceed further by re- peating the process of the construction of partial sums from those previously derived. Thus in fact a series of the comparability tests of different hierar- chy is possible. In table 1 the iterated partial sums for benzanthracene valence structures are shown. Generally the iterated partial sums may not satïsfy the last of the basic conditions (equahty of the total sums), but this is not a serious obstacle, which can be easily formally removed by inclusion of an addi- tional entry at the end of the sequences. We con- sidered the special case of positive curvature along the sequence (when the differences of adjacent en- tries in the sequence decrease). If the differences in- crease, rather than decrease, one proceeds similarly using the partial sums as an input for construction of new partial sums, but in making the second partial sums the order of the partial sums is inverted, Le, the entries are read from right to left, rather than from left to right. In this way the sense of increase and de-

crease aiong the sequences are exchanged, and the case is converted formally to the prevïous state. In a stih more genera1 situation when a point of inflection can be identifïed, one proceeds as in the above two situa- tions, i.e., starts with the partial sums as a new input, and then dependïng on the position of the inflection point inverts a part of the sequence appropriately. Ah the three steps can also bz combined when suffï- cient infonnation warrants going beyond the second

partial sums, resulting in an increased orderïng of structures.

Without additional ïnformation on the sequences

Volume 55, number 3 CHEhffCAL PHYSICS LETTERS 1 hfay f978

ene canuot resolve the ambiguities of a partiaUy or- [Zj G.H. Hardy, J.E. Littfewood and G. PoIya, Inequ&ties dered set of structures. Since frequentiy trends along (Cambridge Univ. Press, London, 1934) p. 44.

seqttences are known and are more reIiabfe tban ac- [3] E. Ruch, Accounts Chem. Res. 5 (1972) 49;

tuai numerical values we anticipate tbat the basic in- E. Ruch and A. Schönhofer, Theoret. ChÏm. Acts 19

equaiities of Muirhead and Karamata and the pro- (1970) 225;

posed extension wiH find useful applïcation in struc- C.A. hfead, Topics Current Chem. 49 (1974).

[4f D.R. Fufkerson, Pacific J. Ma&, 10 (f960) 83i: tural chemistry. W.K. Chen, hfatrix Tensor Quart. (June 1974) p_ I23;

(Sept. 1974) p. 1. 151 L Gutman and M. RartdiC, Chem. Phys. Letters 47

Acknowledgement <1977) Is_

IS] E. Rucb, Theoret. Chim. Acts 38 (1975) 167;

~o~spondence with Professor A. Mead (Min- E. Ruch and C.A. Mead, Theoret. Chim. Acts 41(1976) 95.

neapolìs)-&d Professor H. Primas (Zurich) r&ulted in an ~provement of the work and the presentation- This work was supported by the U.S. Department of Energy, Division of Basic Energy Sciences.

17 1 J. Karam&a, PubI, Math. Univ. Belstade L (1932) 145. 181 E.F. Beckenbach and R. Beilman, Inequalities, Ergeb-

nissc der ~fat~cmatik tmd ihre Grenzgebicte, Neue FoIge, Heft 30 (Springer, Berlin, 1961).

[9] M. Rand& Chem. Phys. Letters 38 (1976) 68; J. Am. Chem. Sec. 99 (1977) 444; Tetmbedron 33 (1977) 1905; Mof. Phys. 34 (1977) 849.

References [lol M. Rand% J- Am. Ckem. Sec. 97 (1975) 6609. ff Il hi. Rand& to be published.

[lf R.F. Nuirhead, Proc. Edinburgh hfath. Sec. 19 (19Ult 36; 2iG903) 144; 24 (1906) 45.