Cyclopentadienyl anion analogues can be semi-embedded [9] by the complementary eigenvalues} eigenvectors of butadiene given in ®gure 5. Since furan (X ¯.
M o l e c u l a r P h y s i c s , 1996, V o l . 88, N o . 2, 407± 417
Properties and relationships of right-hand mirror-plane fragments and their eigenvectors : the concept of complementarity of molecular graphs M olecular orbital functional groupsÐ Part 2 By JERRY RAY D IAS D epartment of Chemistry, University of M issouri, Kansas City, M O 64110- 2499, USA (Recei Š ed 24 July 1995 ; re Š ised Š ersion accepted 5 December 1995) The correlations of properties and relationships of McClelland right-hand mirror-plane fragments (subgraphs) are presented. A complementary relationship is de® ned which has a strong analogy to the pairing relationship. Righthand mirror-plane fragments are molecular orbital functional groups and molecular graphs having the same mirror-plane fragments (functional groups) have a greater degree of similarity. Collections of subspectral structures are tabulated.
1.
Introduction
One of the guiding principles of chemistry is that a molecule’ s structure determines its properties. This leads one to analyse a molecule in term s of its elem entary substructures (atoms, bonds, functional groups, em bedding subgraphs [1], excised internal structures [2], etc.) in order to describe its potential chem istry. The more elem entary substructures that two molecules have in common, the more they are similar [3]. Thus, if the properties of one of these two molecules are known, then this comparison allows one to deduce the properties of the other. In this paper, we describe the study of right-hand mirror-plane fragm ents, which represent one class of functional group (or structural invarian t). Throughout this paper we will represent conjugated polyene hydrocarbons by m olecular graphs which only show the C > C r bond skeleton. W ithin the H u$ ckel approxim ation, an eigenvalue corresponds to an energy level with a ¯ 0 and an eigenvector is the set of coe cients belonging to the associated wave function. Tw o m olecular graphs having one or more eigenvalues in common are said to be subspectral. 2.
Similarity
Sim ilarity is the degree of overlap between two or more structures and has been the subject of numerous studies [3]. Sim ilarity is related to analogy, both of which have been used extensively in science. The more substructures (fragm ents, subgraphs, or functional groups) two molecules have in common, the more they are similar. Similarity plays an im portant role in molecular modelling, which involves the prediction of properties of unknown structures from a give n set of known structures. M olecular modelling involves the generation, manipulation and analysis of m olecular structures, model development, and prediction of structural, chem ical, physical, and biological properties. Similarity is a com ponent of the analysis of molecular structures. R ight-hand m irror-plane fragm ents represent a class of quantum -based structures that can be used in similarity comparisons. For reviews on the integration of graph 0026± 8976 } 96 $12± 00 ’
1996 Taylor & Francis Ltd
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theory and quantum chemistry for structure± activity relationships, the reader should consult the work of Balasubram anian [4]. The search for and study of structural invarian ts is a vital undertaking in similarity studies [5]. Characteristic polynomials, eigenvectors, the number of spanning trees [6], recurring eigenvalues [7], and right-hand mirror-plane fragm ents [8, 9] are just some exam ples of structurally dependent invarian ts. Similarity is most frequently used in a conceptual, qualitative sense. Organic chemists exploit similarity in this fashion when they state that two m olecules with the sam e functional groups are more similar than two with diŒerent functional groups. In fact, the practice of organic chemistry relies heavily on the functional group concept and its qualitative application. Likewise, it is our contention that the m ore eigenvalues two subspectral molecular graphs have in common the more they are similar, other things being equal. This similarity is strengthened further if the frontier molecular orbitals are included in the common eigenvalues. The Hu$ ckel molecular orbital (HM O) m odel considers only the p electron system and is devoid of strain related components which must be determined separately. For exam ple, both biphenylene and pyrene have twelve identical eigenvalues, including the frontier orbitals, but the ring strain associated with the cyclobutadiene substructure in biphenylene has not been accounted for by the HM O m odel. This strain should cause more mixing of the r electrons with the p electrons. N evertheless, both biphenylene (IP ¯ 7± 53 eV) and pyrene (IP ¯ 7± 43 eV) have ® rst " ionization potentials within 0± 1 eV" of each other, with the slightly higher IP of " biphenylene being consistent with greater r electron interaction with the p electronic system. 3.
Right-hand mirror-plane fragm ents
M cClelland’ s rules [8] have been reviewed several tim es [9]. Since we are concerned only with right-hand mirror-plane fragm ents, we will need to rem ember only the following. W hen an internal mirror-plane of sym metry divides a molecular graph into two parts, the vertices on the mirror-plane remain with the left-hand fragm ent and vertices in the right-hand fragm ent originally connected by a bisected edge have weights of ® 1. Thus, the vertices in the right-hand fragm ent are either normal or have weights of ® 1 ; the latter will be indicated on the fragm ent graphs by open circles. The M cClelland mirror-plane de® nes an antisymm etric relationship for the eigenvectors corresponding to the eigenvalues belonging to the right-hand mirror-plane fragm ents ; in this case, the vertices on the mirror-plane have zero eigenvector coe cients. 4.
Paired eigenvalues
If the two eigenvalues X within a single molecular graph or two related mirrorplane fragm ent graphs sum to zero (X X ¯ 0), they are said to be paired. The well " # eigenvalues in a conjugated alternant known pairing theorem [10] states that all hydrocarbon (AH) are either zero (non-bonding) or paired (bonding and antibonding) [11]. AHs have no odd size rings and every other carbon vertex can be starred so that no two starred and no two unstarred positions are adjacent. Consider a maxim ally starred A H. The eigenvector coe cients for the starred positions of the AH are unchanged in going from one eigenvalue (X ) to its paired partner (X ), and for the " changes in going from one # eigenvalue unstarred positions the sign (but not m agnitude) to its paired partner ; if an eigenvalue has no paired partner (i.e., X ¯ 0), then the coe cients of the unstarred positions are zero.
Properties of rig ht-hand mirror-plane fra g ments
Figure 1.
409
Complementary pairs of vertex-weighted mirror-plane fragment graphs.
Figure 2.
Self-complementary vertex-weighted graphs.
5.
Com plem entary eigenvalues
If two eigenvalues in a single molecular graph, two related molecular graphs or right-hand mirror-plane fragm ents sum to minus one (X X ¯ ® 1), they are said to " # fragm ents or m olecular be `complementary ’ . Two equal-size right-hand mirror-plane
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Figure 3.
Complementary pairs of vertex-weighted mirror-plane fragment graphs that are subspectral.
Figure 4.
Complementary pairs of vertex-weighted mirror-plane fragment graphs that are subspectral.
Properties of rig ht-hand mirror-plane fra g ments
Figure 5.
411
Molecular graphs containing complementary eigenvalue pairs.
graphs are complementary if all their eigenvalues are mutually com plem entary. The normal vertices of one of the complementary right-hand fragm ents correspond to ® 1 weighted vertices in the other, and both have the sam e sets of normalized eigenvector coe cients, except, perhaps, for sign. Figure 1 presents nine com plem entary pairs of right-hand mirror-plane fragm ents with up to four vertices, and ® gure 2 give s all the self-complementary right-hand m irror-plane fragm ents with up to six vertices. Selfcomplementary m irror-plane fragm ents necessarily have an even number of carbon vertices and S pseudosym metry. Two sets of subspectrally related complem entary # right-hand mirror-plane fragm ents are presented in ® gures 3 and 4. Except for sign, it is seen that the norm alized eigenvectors of non-degenerate com plementary eigenvalues in complem entary m olecular graphs are identical and antisymmetrical in regard to the relevant mirror-plane. Alm ost all the molecular graphs in ® gures 5 and 6 have their right-hand m irror-plane fragm ents listed somewhere in ® gures 1± 4. If the ® 1 weighted vertices of two com plementary right-hand m irror-plane fragm ents are changed to 1 weighted vertices, then the corresponding left-hand m irror-plane fragm ents so generated will be `paired complementary ’ with eigenvalues
412
Figure 6.
J. R. Dias
Two sets of subspectral molecular graphs that have mutually complementary eigenvalues.
that sum to plus one (X X ¯ 1). For exam ple, consider the 5th and 6th m irror-plane " 1.# Changing their ® 1 vertices to 1 vertices leads to the fragm ent graphs in ® gure corresponding left-hand mirror-plane fragm ent graphs belonging to 3,4-dimethylenylcyclobutene and 1,3,5-h exatriene, respectively (see ® gure 7). The corresponding eigenvalues are 2± 2470, 0± 5550, and ® 0± 8019 for the ® rst one and ® 1± 2470, 0± 4450, and 1± 8019 for the second one. If the diŒerence between two eigenvalues is minus one (X ® X ¯ ® 1), they are " # complem entary said to be `negatively complementary ’ . The occurrence of negatively eigenvalues is less frequent and less understood. All known exam ples have involved non-A Hs. Exam ples of molecular graphs having negatively complem entary eigenvalues can be found in ® gures 5 (11th molecular graph) and 6 (4th m olecular graph versus the 9th and 10th ones).
6.
Eigenvectors corresponding to com plem entary eigenvalue s
Except for sign, the eigenvector coe cients for com plementary eigenvalues have a one-to-one correspondence in complementary molecular graphs. This is illustrated in ® gures 7± 10. To discern this one-to-one correspondence, one needs to com pare the
Properties of rig ht-hand mirror-plane fra g ments
Figure 7.
413
Corresponding eigenvectors for com plementary eigenvalues belonging to complementary molecular graphs.
associated com plem entary right-hand mirror-plane fragm ents which head ® gures 7± 10. Note that the normal vertices of the mirror-plane fragm ents in one column correspond to the ® 1 weighted vertices in the other column in ® gures 7± 10. To determine the signs of the eigenvector coe cients, keep in mind that the mirror plane de® nes an antisymmetric relationship. In addition, we have detailed an algorithm which used the following equation in determining the unnormalized eigenvector coe cients for a given eigenvalue X :
i
®
XC
i iu
C
ir
C
is
C ¯
it
0,
where C is the eigenvector coe cient for some central (carbon) vertex u with adjacent iu vertices of r, s, and t [12]. Clearly this equation can be used to obtain the signs if we already know the absolute values of the coe cients from the above complem entary relationship. To elaborate further, consider the complementary molecular graphs for 1,3,5hexatriene and 3,4-dimethylenylcyclobutene in ® gure 7. Their corresponding com plementary right-hand m irror-plane fragm ents are listed at the head of ® gure 7 ; the vertex weighted ® 1 in one corresponds to the normal vertex in the other. For each pair
414
Figure 8.
J. R. Dias
Corresponding eigenvectors for com plementary eigenvalues belonging to complementary molecular graphs.
of com plementary eigenvalues listed in ® gure 7, a one-to-one correspondence of the absolute value of the eigenvalue coe cients is observed. Since the molecular graphs in ® gures 7± 10 are AHs, one can use the pairing theorem to obtain the remaining eigenvalues and associated eigenvectors not shown which are paired-complem entary eigenvalues. All the AHs in ® gures 7± 10 can be converted to non-AHs retaining the give n eigenvalues by connecting two degree-1 vertices which have corresponding eigenvectors with coe cients of the sam e m agnitude but opposite sign through a zero coe cient vertex (in this regard refer to the above equation). For exam ple, this procedure converts o-quinodimethane to indenyl, shown in ® gure 5. From ® gure 10, one should note that m olecular graphs having self-complem entary right-hand mirror-plane fragm ents possess a special type of `hidden ’ sym metry. The pairing relationship has contributed broadly but indirectly to the rapid evaluation of eigenvalues and eigenvectors and our understanding of molecular graphs. For exam ple, Dewar used the pairing relationship in formulating numerous derivations and theorems about m olecular orbital characteristics and in his perturbation m olecular orbital approach [11]. W e believe that the complementary relationship has a sim ilar potential utility. The indirect facility of this relationship in aiding our understanding of chem ical properties is illustrated in the next section.
Properties of rig ht-hand mirror-plane fra g ments
Figure 9.
415
Corresponding eigenvectors for com plementary eigenvalues belonging to complementary molecular graphs.
7.
Som e chemical exam ples
M any of the m olecules represented by the molecular graphs given in ® gures 5± 10 have been the subject of recent studies [13± 17]. Complementary eigenvalue pairs that occur together within a given molecular graph are presented in ® gure 5 (also see ® gure 10). As can be seen, the molecular graphs in ® gure 5 have self-com plem entary righthand m irror-plane fragm ents (® gure 2). 1,3-Butadiene is the smallest exam ple. Cyclopentadienyl anion analogues can be semi-embedded [9] by the complem entary eigenvalues } eigenvectors of butadiene given in ® gure 5. Since furan (X ¯ O), thiophene (X ¯ S), and selenole (X ¯ Se) have the sam e right-hand m irror-plane fragm ent [9] (® gure 1) which contains the HO M O for all three m olecules, these heterocyclic m olecules have the sam e ® rst ionization potential (IP ¯ 8± 9 eV) [18]. " o-Quinodimethane (o-xylylene) can be generated therm ally from benzocyclobutane, which functions as a reactive diene in D iels± A lder reactions ; the HOM O value
416
Figure 10.
J. R. Dias
Corresponding eigenvectors of complementary eigenvalues for o-quinodimethane.
changes approxim ately from 1± 0 to 0± 295 in this thermal transformation, resulting in a more electron-donating diene. This facile reaction has been exploited as a synthetic strategy [19]. The 2H -isoindene analogue of o-quinodimethane has been prepared and studied at low temperatures [20], and the photoelectron spectrum of o-quinodimethane, itself, has been reported [21]. 8.
Conc lusion
The structural origin of certain eigenvalues has been delineated. This has led to the identi® cation of a complementary relationship that is analogous to the well known pairing property of AH s. Thus, M cClelland’ s right-hand mirror-plane fragm ents [8] provides a powerful conceptual tool for analysing the topological properties of conjugated polyenes and related species [22]. Since the pairing theorem is applicable to higher level calculations [11], this com plementary relationship m ay also be applicable to higher level calculations. Sim ple HM O continues to be of practical utility. It provides a viable qualitative, sem iquantitative, and param eterized tool for interpreting a wide variety of phenomena. The description part of m ost ab initio reports are framed within H M O supplemental discussions because of its ease of conceptualization, and it predicts accurately the relative signs of the eigenvector coe cients. Like the pairing theorem, the complementary relationship demonstrated herein should facilitate our qualitative understanding of relevant chemical systems. References [1] H a l l , G. G., 1957, Trans Faraday Soc., 53, 573 ; 1981, Bull. Inst. Math. Applic., 17, 70. [2] D i a s , J. R., 1986, Handbook of Polycyclic Hydrocarbons, Part A (Amsterdam : Elsevier) ; 1990, J. chem. Inf. Comput. Sci., 30, 61 ; Theoret. Chim. Acta, 77, 143. [3] J o h n s o n , M., and M a g g i o r a , G. M., 1991, Similarity in Chemistry (New York : Wiley) ; T r i n a j s t i c , N., 1992, Chemical Graph Theory (Boca Raton, FL : CRC Press) ; Randic, M., 1992, J. chem. Inf. Comput. Sci., 32, 686. [4] B a l a s u b r a m a n i a n , K., 1994, SAR QSAR En Š iron. Res., 2, 59 ; 1985, Chem. Re Š ., 85, 599. [5] R a n d i c , M., 1992, J. math. Chem. 9, 97.
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[6] J o h n , P., and S a c h s , H., 1990, J. chem. Soc. Faraday Trans, 86, 1033. [7] J i a n g , Y., Y u , W., and K i r b y , E. C., 1991, J. chem. Soc. Faraday Trans, 87, 3631. [8] M c C l e l l a n d , B. J., 1974, J. chem. Soc. Faraday Trans ii, 70, 1453 ; 1982, J. chem. Soc. Faraday Trans, 78, 911 ; Molec. Phys., 45, 189. [9] D i a s , J. R., 1989, J. chem. Educ., 66, 1012 ; 1993, Molecular Orbital Calculations Usin g Chemical Graph Theory (New York : Springer-Verlag). [10] M a l l i o n , R. B., and R o u v r a y , D. H., 1990, J. math. Chem., 5, 1 ; 1991, J. math. Chem., 8, 399. [11] D e w a r , M. J. S., 1969, The Molecular Orbital Theory of Or ganic Chemistry (New York : McGraw-Hill). [12] D i a s , J. R., 1995, Molec. Phys., 85, 1043. [13] H o p f , H., and M a s s , G., 1992, An gew. Chem. Int. Edn En gl., 31, 931. [14] T o d a , F., and G a r r a t t , P., 1992, Chem. Re Š ., 92, 1685. [15] G a l a s s o , V., 1993, J. molec. Struct. Theoche m. 281, 253. [16] Z a h r a d n i k , R., H o b z a , P., B u r c l , R., H e s s , B. A., and R a d z i s z e w s k i , J. G., 1994, J. molec. Struct. Theochem., 313, 335. [17] G r a n u c c i , G., E l l i n g e r , Y., and B o i s s e l , P., 1995, Chem. Phys., 191, 165. [18] L e v i n , R. D., and L i a s , S. G., 1982, Ionization Potential and Appearance Potential Measurements, 1971± 1981, NSRDS-NBS 71 (Washington, D.C. : U.S. Governm ent Printing O ce). [19] N e m o t o , H., S a t o h , A., F u k u m o t o , K., and K a b u t o , C., 1995, J. or g. Chem., 60, 594 ; Kametani, T., Suzuki, K., and N e m o t o , H., 1981, J. Amer. chem. Soc., 103, 2890. [20] A l l a n , M., A s m i s , K. R., E l h o u a r , S., H a s e l b a c h , E., C a p p o n i , M., U r w y l e r , B., and W i r z , J., 1994, HelŠ . Chim. Acta, 77, 1541. [21] K r e i l e , J., M u n z e l , N., S c h u l z , R., and S c h w e i g , A., 1984, Chem. Phys. Lett., 108, 609. [22] D i a s , J. R., 1996, J. chem. Inf. Comput. Sci., in press.
