Correlation effects and electron delocalization in alternant

Chem. , 1992, 96 (2), pp 604–614. DOI: 10.1021/j100181a019. Publication Date: January 1992. ACS Legacy Archive. Cite this:J. Phys. Chem. 1992, 96, 2...
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J. Phys. Chem. 1992,96, 604-614

604

Correlation Effects and Electron Delocalization In Alternant Hydrocarbon ?r Compounds Johannes Schiitt and Michael C. Bohm* Institut fur Physikalische Chemie, Physikalische Chemie III, Technische Hochschule Darmstadt. 0-6100 Darmstadt, Federal Republic of Germany (Received: May 20, 1991)

The .rr electron correlations in altemant hydrocarbons are investigated by the local approach (LA); the many-particle procedure has been combined with simple model Hamiltonians of the tight-binding Hackel and extended Hubbard type. The localization properties of the ‘17 electrons are quantified by the charge fluctuations in the correlated ground state ((An;))con.and the associated correlation-strengthparameter 4measuring the reduction of the charge fluctuations due to interatomic T correlations. It is shown that the .rr electrons in linear polyenes and monocyclic 4n + 2 or 4n annulenes are sizably correlated. Bond length alternation, 6, in the latter compounds is accompanied by an enhancement of the charge fluctuations; the delocalization of the respective .rr electronsis enhanced correspondingly. The variation of the 7r correlation energy as a function of the dimerization coordinate depends on the size and geometry of the .rr system. Electronic correlations are one driving force for the dimerization in all 4n 2 annulenes. The &dependent variation of the many-body energy changes sign in CnHn+2polyenes with n I 8 and 4n annulenes with n I3. For larger n the correlation energy is again enhanced with increasing bond length alternation. The many-particle indices and Ai are highly site specific. They depend often on the nearest-neighbor sphere of the selected “reference”atom. In many alternant hydrocarbons an extensive transferability of the above parameters ( ( A ~ I , ~ ) ) ~ and Ai between similar compounds is realized. Depending on the nature of the .rr center, the molecular size, and geometry, correlation-strength parameters Ai are derived in alternant hydrocarbons that approach both the “free-electron” limit and the boundary of perfect interatomic correlations (localized limit). Transparent ordering principles realized in alternant a systems in the many-particle level are formulated.

+

I. Introduction Hydrocarbon compounds with extended ?r systems have absorbed the interest of physicists and chemists for more than a half century. It has been assumed throughout for many years that the electronic structure of conjugated linear polyenes and cyclic systems with 4n + 2 (n = 0, 1, 2, ...) ?F centers can be described adequately by theoretical models based on the independent-particle approximation. Two of the most popular classical one-electron approaches, e.g., are the Hiickel molecular orbital (HMO) model’ and the free-electron (FE) approximation of Kuhn.2 The zero differential overlap (ZDO)nature of the latter model Hamiltonians in combination with the underlying tight-binding approximation suggested a subdivision of ?r systems into alternant and nonaltemant hydrocarbon^.^ The one-electron energies t k in altemant monocyclic A rings are given in eq I. 1 in the framework of the simple tight-binding approximation.

~ ( 6 )= f12tol(cos2k + 62 sin2 k)’I2

(1.1)

to stands for the conventional tight-binding hopping integral and 6 abbreviates the bond length alternation. In chemistry there exists an almost universal trend to identify systems without bond alternation (6 = 0) as more “delocalii” configurations, while bond alternation is equated with increasing localization of the electrons. In this picture the static charge density p is adopted as measure for the localization properties of the ( T ) electrons. The momentum vector k in (1.1) labels the symmetry of the one-electron states. In monocyclic systems of the “aromatic” Hiickel-type with 4n 2 atoms the allowed k values in the first Brillouin zone are given in (1.2); for “antiaromatic” non-Hiickel rings of the 4n type, they are expressed in (1.3). * f- 2* ..., f- 4* N = 4n 2 (1.2) k=O,f2 n + 1’ 2 n + 1’ 2n+ 1 * 2* (2n - 1 ) ~?r , - N = 4n (1.3) k = O , f - 2n’ *%’ 2n 2 The Jahn-Teller theorem ensures the instability of the 4n rings as a result of an orbitally degenerate electronic ground state at

+

+

...9

(1) Salem, L. Molecular orbital theory of conjugated systems; Benjamin: New York, 1966. (2) Kuhn, H. Helu. Chim. Acra 1948, 31, 1441. (3) Coulson, C. A. Proc. R. SOC.London, Ser. A 1939, 164, 413.

0022-3654/92/2096-604$03.00/0

6 = 0. The one-electron energies collected in (1.2) and (1.3), lead to a widely accepted “historical” subdivision of cyclic alternant .rr rings into 4n systems on one side and 4n + 2 compounds, on the other. As a consequence of the Jahn-Teller instability the former molecules are intrinsically distortive. For the latter monocyclic rings it has been assumed for a long time that their ?r network should be nondistortive. Previous electronic structure investigations, however, have demonstrated that even the ‘K frameworks of the “aromatic” 4n + 2 Hiickel systems are distortive; the nondimerized structure is forced by the superimposed u frame.4J From this background it is clear that the electronic structure of hydrocarbon u compounds deserves new attention. References 4 and 5 are typical contributions, where the validity of popular one-electron descriptions for u systems have been analyzed critically. There exist many theoretical studies that have shown definitively that the independent particle approximation is an oversimplification in hydrocarbon u systems.+” The interatomic correlations between the .rr electrons are sizable; they are often of decisive influence for the respective physical properties. Also the optical gap in polyenes has been explained by electronic correlations.’ * In the present work we investigate several aspects of ?r electron correlations in altemant hydrocarbons. The most important ones are concisely summarized. Already the allowed k vectors in (1.2) and (1.3) for the one-electron energies tk in monocyclic systems indicate that 4n 2 and orbitally degenerate 4n materials must differ in their dependence of the u correlation energy on the bond dimerization at least in the limit of smaller n values. The principal

+

(4) Shaik, S.S.;Hierty, P. C.; Ohanessian, G.; Lefour, J. M. Noun J . Chim. 19115.9, 385. Shaik, S.S.; Lefour, J. M.; Ohanessian, G. J. Org. Chem. 1985, 50, 4657. (5) Shaik, S.S.;Hiberty, P. C.; Lefour, J. M.; Ohanessian, G. J. Am. Chem. Soc. 1987, 109, 363. (6) Harris, R. A.; Falicov, L. M. J . Chem. Phys. 1969, 51, 5034. (7) Schulten, K.; Ohmine, I.; Karplus, M. J. Chem. Phys. 1976,644422. Ohmine, I.; Karplus, M.; Schulten, K. J . Chem. Phys. 1978. 68, 2298. (8) Horsch, P. Phys. Rev. E.: Condens. Matter 1981, 24, 7351. (9) Paldus, J.; Chin, E. Int. J. Quantum Chem. 1983,24, 373. Paldus, J.; Chin, E.; Grey, M. Int. J. Quantum Chem. 1984, 26, 395. (10) Takahashi, M.; Paldus, J. Inr. J . Quantum Chem. 1985, 28, 459. (11) Kuwajima, S.; Soos. Z. G. J . Am. Chem. SOC.1987, 109, 107. (12) Ovchinnikov, A. A.; Ukrainskii, I. I.; Kventsel, G. V. Sou. Phys.Usp. (Engl. Transl.) 1973, 15, 575.

0 1992 American Chemical Society

The Journal of Physical Chemistry, Vol. 96, No.2, 1992 605

Electron Delocalization in Alternant u Systems conclusions derived in ref 8 (Le. u correlation as one driving force for the bond altemation) correspond probably only to one possible limit. It seems to be necessary to compare the u correlation energy in 4n 2 and 4n monocycles, on one side, and in linear polyenes, on the other hand, as a function of the dimerization coordinate. In addition to the u correlation energy we study the localizatidh properties of the u electrons in the correlated ground state I+o) together with the associated correlation strength. The localization of the u electrons is quantified by the mean-square deviation of the electronic charge ((An;)) at the ith u center. The fluctuations in the independent-particle ground state I+=F) correspond thereby to the ‘delocalized” boundary; correlations tend to reduce the charge fluctuations, thereby enhancing the localization properties of the electrons. In (1.4) and (1.5) the charge fluctuations are defined in the two levels of sophistication, i.e in I+scF) and I+o).

+

((~?))SCF= ((An?))corr

(n?) - ( n i ) 2

= (+olni?+~)- ( + O h l + ~ ) *

(1.4) (1.5)

The number operator ni for the u electrons at center i reads ni = C s i Uwith niu = aiu+ai,;u labels the electron spin t or 1. The air+ and ai, are creation and destruction operators for the u electrons at the ith carbon center. In (1.4) the abbreviation (...) has been used to symbolize expectation values which are calculated with respect to the independent-particle ground state I+scF); i.e. (...) = (+sCFl ...I+ScF). In eq (1.6) we combine expressions 1.4 and 1.5, respectively, to define the correlation-strengthparameter Ai, which is a quantitative measure for the strength of manyparticle interactions in chemical bonds and atomic regions i. In

computational capabilities of the conventional correlation schemes in quantum chemistry. Therefore we have selected the LA to investigate the electronic correlations in altemant hydrocarbons. The LA has been combined with a simple model Hamiltonian of the tight-binding type which is in line with the spirit of the aforementioned popular u electron models of the ZDO type. We have employed a theoretical framework, where all integrals are calculated parameter-free, i.e. ab initio, in a minimal basis of Slater-type orbitals. The organization of the present model study is as follows. In section I1 we give a formulation of the LA, which is most convenipnt for alternant u systems with only one atomic orbital per carbon center. In section I11 we analyze many-particle effects in linear hydrocarbon compounds; in the next one monocyclic u rings are studied. In section V polycyclic systems are considered. Some ordering principles on the many-particle level are also formulated in this part of our work. In VI we present transferable ‘group elements”, ((An;)).-, and Ai which are characteristic for different types of a centers. A fmal outlook is then given in section VII.

XI. Theoretical Basis The theoretical background of the LA is well documented in the The capability of the variational method in combination with model Hamiltonians of the ZDO type has been quantified in several investigation~.’~J~J~J’*~@~~-~~ Therefore it suffices to give a brief outline of the many-particle procedure. The correlated ground state is derived from the SCF wave function I+scF) by using the exponential ansatz (11.1) where S symbolizes a certain set of local projection operators. Ob!

) = exp(S) 1+SCF )

(11.1)

The two-particle operator S consists of local density operators 0, and Oij;S is of the form alternant hydrocarbons and homopolar two-center two-electron bonds, Ai has allowed values between 0 and 1; the first boundary symbolizes the independent-particlelimit, the second one perfect interatomic ( u ) correlations. As one of the interesting details evaluated in the present work we discuss some type of ‘correlation paradox”, i.e. an enhancement of the correlation energy lEEorrl coupled to a decreasing correlation strength Ai, a combination which is not expected a priori. In previous contributions of one of us it has been shown that localized chemical bonds can be divided into characteristicclasses according to their Ai numbers.13-17 Most of the Ai elements are found in an interval between 0.1 and 0.8. The lower limit is realized in u bonds formed by main-group atoms, the upper one in transition-metal compounds. u interactions between main-group atoms lead to Ai elements between the two boundaries. Subsequently it is demonstrated that the above Ai spectrum produced by many element combinations in different types of bonds is also covered by the u electrons in alternant hydrocarbons. The aforementioned Ai datal3-I7 have been derived by using the method of the local approach (LA) for the interatomic correlations; the local nature of the correlation hole is here explicitly taken into The latter many-particle model is easy to implement also for complicated molecules beyond the current ~~

OW, A.M.; Winch, F.; Fulde, P.; Mhm, M. C. 2.Phys. B Condens. 1987, 66, 359. OM,A. M.; Fulde, P.; Bbhm, M. C. Chem. Phys. 1987, 117, 385. Bijhm, M. C. Chem. Phys. 1988, 128, 457. Bbhm, M. C.; Bubeck, G.; Ole& A. M. Chem. Phys. 1989,135, 27. Bubeck, G.; Old A. M.; Mhm, M. C. Z . Phys. B Condens. Matter 1989, 143. (17) Old, A. M.; Fulde, P.; Whm, M. C. 2.Phys. B Condens. Matter 1989, 76, 238. (18) Stollhoff, G.; Fulde, P. Z . Phys. B Condens. Mutter 1977,26,257; 2.Phys. B Condens. Matter 1977, 29, 23 1. (19) Stollhoff, G.; Fulde, P. J . Chem. Phys. 1980, 73, 4548. (20) Pfirsch, F.; Whm, M. C.; Fulde, P. Z . Physik E Condens. Mutter 1985, 60, 171. (21) Old, A. M.; Wirsch, F.; Fulde, P.; Mhm, M. C. J . Chem. Phys. 1986, 85, 5183. (13) Matter (14) (15) (16)

s = -clIijoij

(11.2)

ij

with variational parameters vi) The i, j summation is over the respective u AOs. The local projectors 0,describe densityaensity correlations in terms of two-particle excitations. In the present model study based on a model operator defined in terms of Hartree-Fock hopping integrals, we have restricted the 0, to the leading intraorbital contributions Oii. We have used the term ‘interatomic correlations” to label those many-body contributions which are feasible in a minimal (i.5. valence) basis. In (11.3) intraorbital projectors Oii are defined with the aim to express the 0,in (II.Z)-as their irreducible parts beyond the single-particle operators Oii

Oii = 2nitni4 (11.3) The density operators ni, (u = t, 1) are those already given in section I. The interrelation between the two-particle operators 0,and the single-particle ones is expressed in (11.4)

ai,

oii

= 6ij - [Oij]SCF

On the right-hand side of the above equation we have used to abbreviate

[O~~ISCF = nit(nil)

+ (nit)nii + (nit)(niJ)

(11.4) [O&F

(11.5)

The variational parameters nii in (11.2) are determined by minimization of the functional for the ground-state energy E(7)

E ( d = (+olwl+o)

=

(eS(q)H@))c

(11.6)

H denotes the r-electron Hamiltonian. The index c symbolizes (22) Borrmann, W.; Old, A. M.; Pfirsch, F.; Fulde, P.; Bijhm, M. C. Chem. Phys. 1986, 106, 11. Ole&A. M.; Pfirsch, F.; Borrmann, W.; Fulde. P.; Bbhm, M. C. Chem. Phys. 1986, 106, 27. (23) Kbnig, G.; Stollhoff, G. J . Chem. Phys. 1989, 91, 2993. (24) Rbciszewski, K.; Chaumet, M.; Fulde, P. Chem. Phys. 1990,143, 47.

606 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992

that only connected diagrams are taken into account. As a consequence the expansion (11.6) is correctly normalized.2s To derive the functional (11.6) for the ground-state energy, we make use of a linearized expansion where $(?) is replaced by (1 S(q)). The validity of this expansion has been studied recently.26 The linearization is valid for not too strong electronic correlations; they are overestimated when approaching the perfectly correlated limit. The above approximation leads to eq 11.7 for the variational parameters. The simple structure of this equation is caused by the restriction to the intraorbital projectors Oii

+

CVii(OiiHOkk)c =

(0kkH)c

(11.7)

i

In this degree of sophistication the interatomic correlation energy can be written in the form of Ecorr

= -COii(OiiH)c

(11.8)

i

The variational parameters qii are those derived from (11.7). In the above formulation it is possible to decompose the interatomic A correlation energy as a sum of individual contributions Ecorr,i. We have Ecorr

= CEcorr,i

(11.9)

I

with = qii(Oi,H),. As can be seen, the variational parameter qii is of simple analytic form in systems with only one degree of freedom, i.e. in monocyclic A systems described by the above intraorbital approximation. qii depends in this case on the ratio (okkH)c/Zj( OjjHokk)c. The correlation energy, on the other side, depends on ( OkkH), square divided by the sum of all elements of the type ( OjjHOkk)c. The h e a r versus square dependence of the ( OkkH)c terms causes an interesting interrelation in many hydrocarbons between qii, Ai on one side, and E,,,,, on the other. To evaluate the above matrix elements (OkkH),and (OiiHOkk)c, respectively, by using Wick’s theorem, we need the first-order density matrix P and its pendant D from the independent-particle calculation. The elements P j k and Dik are defined in (11.10) and (11.1 l), respectively. The latter numbers are the only information from the independent-particle precursor entering the many-body part of the problem. Pik

= (ai,+ak,)

Dik = (arcuko’)

(11.10) (11.1 1)

The matrix elements ( OiiH), and (OjiHOkk), are defined in the subsequent formulas (11.12) and (11.13). The former ones are exclusively determined by the residual interaction of the A Hamiltonian and contain therefore the electron repulsion beyond the Hartree-Fock approximation. The respective Hamiltonian in the ( oii~okk)c expressions is approximated by the mean-field (Fock) operator F. The matrix elements describe essentially the average mean-field energy of the virtual excitations from the filled one-particle space to the virtual one. In our model approach they are measured in terms of the so-called Hartree-Fock (HF) hopping Tij;see eq 11.27. (OjiH), =

(0iiHres)c

= 2CVmnPjmDjmPjfljn (11.12) m.n

( OiiHOkk ) c = ( OiiFOkk ) c =

4PikDik~Fmn[PikDimDkn- PimPkflik] (11.13) m.n

Equation 11.13 is an approximation based on the neglect of the residual interaction when going from the first expectation value defined by the A Hamiltonian H to the second one formulated by the mean-field operator F. This substitution corresponds to a second-order perturbation expansion in the framework of the LA. For the present comparative discussion a simplified inter( 2 5 ) Horsch, P.;Fulde, P.2. Phys. E : Condens. Mazter 1979, 36, 23. (26) Whm, M. C.;Bubeck, G.;Old,A. M. Z . Naturforsch., A: Phys. Sri. 1989, 44, 117.

Schiitt and B6hm polation of the type (11.13) seems to be sufficient. A formula corresponding to (11.13) has been employed in a previous investigation of one of us on simple A systemsI4 where it has been of sufficient accuracy. The V,, on the right-hand side of (11.12) symbolize on-site (m = n ) and intersite (m # n ) Coulomb repdlsions between the A electrons. F,, on the right-hand side of (11.13) is the m,nth matrix element of the Fock operator. The maximum allowed variational parameter qiimaxin the many-body problem amounts to 1.0 for alternant A systems in the limit of perfect interatomic A correlations. qii = 0 indicates the validity of an independent-particle wave function to describe the ground state of the A electron system. In the introduction it had been emphasized already that our theoretical analysis is not restricted to A correlation energies Em,.,. Insight into the electronic-structure properties of alternant hydrocarbons is also derived by investigating the charge fluctuations ((An?)),,, in the correlated ground state and the correlation-strength parameter Ai, respectively. The charge fluctuations ( (An:))SCF in the independent-particle approximation I+SCF) are simply defined in (11.4); they amount to 0.5 in alternant hydrocarbons with Pii = Dii = 0.5 ((An:))SCF = 2PiiDii = 0.5

(11.14)

Electronic correlations lead to a suppression of the charge fluctuations according to eq 11.15 ((An?))corr = 2PiiDii(l - 4qiiPiiDii) = OS(1 - qii)

(11.15)

The reduction of the charge fluctuations is directly proportional to the variational parameter qii which depends on the ratio between the electron repulsion and the HF hopping; see the above equations. The above formula (11.15) corresponds to the most simple approximation for the charge fluctuations based on the neglect of crossterms which allow for a mutual coupling of correlation processes. A general expression for ((An?))COrr,which is valid for any strength of electronic correlations, is not easy to formulate and contains a number of mutually compensating elements.16 A particular situation arises in diatomic homopolar bonds with coupling elements that are remarkably large; but also here one has a strong compensation. Guided by this boundary and to restrict the computational expenditure we decided to adopt the “decoupled” interpolation (11.15). We expect minor errors in absolute magnitude of the ((An?)), numbers. Nevertheless the subsequent results for A systems are close to previous data of one of us14 where the couplings have been taken into account. Therefore we believe that (11.1 5) is sufficient for comparative semiquantitative discussions on classes of similar A systems. The last formula visualizes a remarkable result for alternant hydrocarbons. Perfect A correlations with qii = qiiW = 1.0 cause a complete suppression of the respective charge fluctuations. The A electrons are then perfectly localized at the different carbon centers. This situation characterizes of course the complete suppression of the probability of double and empty occupancy Pi(2) and Pi(0), respectively, at the ith A center. For the probability of single occupancy Pi(l) we have then Pi(l) = 1.0. To derive eq 11.15 we have adopted a linearization in the evaluation of the charge fluctuations in I+o);quadratic terms in the variational parameters qii have been neglected. In eq 1.6 of section I we have already defined the correlation-strength parameter A,. Insertion of the eqs 11.14 and 11.15, respectively, for the charge fluctuations in the independent-particle ground state and the ”exact” one leads to (11.16) Ai = 4q..p.,Dii I1 II = qii (11.16) The above formula indicates that the correlation-strength parameter Ai coincides with the variational parameter qii in altemant A systems in the adopted level of sophistication. The reduction of the charge fluctuations due to the A correlations changes the probabilities Pi(n) of double, single, and empty occupancy (n = 2, 1, 0) in the ith A AO. We have already mentioned that perfect correlations lead to Pi(l) = 1.0 and Pi(2) = Pi(0) = 0; the identity Pi(2) = Pi(0) is again a consequence of the particlehole symmetry of the alternant A systems. In eqs

The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 607

Electron Delocalization in Alternant u Systems 11.17 and 11.18, respectively, we have related the probabilities Pi(n) to the average number ( n i ) and number square ( n i ) 2of the A electrons in the ith A 0

10

11

12

The normalization condition for the Pi(n) reads 2

C P i ( n ) = 1.0

(11.19)

n=O

in the independent-particle The respective probabilities approximation I$scF) are given in the eqs 11.20 to 11.22. The particle-hole symmetry in alternant hydrocarbons leads to Pi( 1 ) = 0.5 and Pi(2) = Pi(0) = 0.25 Pj(2)sCF = 0.25(ni)2 = 0.25

(I1* 20)

Pi(1)SCF = (ni)(l - 0 . 5 ( n j ) ) = 0.50

(11.21)

Pj(0)ScF = 1.0 - ( n j ) - 0.5(ni)' = 0.25

(11.22)

Subsequently we have expressed the probabilities Pi(n) in the correlated ground state as a function of the SCF probabilities and the variational parameters qip As already mentioned Pi( 1) is enhanced by the A electron correlations on the cost of Pi(2) and Pi(0). The final Pi(n) formulas are extremely simple in alternant hydrocarbons Pj(2) = Pj(2)sCF - 4qji(PjiDji)2 = 0.25( 1 - qij) (11.23) Pi(1) = Pj(1)scF

+ 8qji(PiiDiJ2 = 0.5(1 + qji)

(11.24)

Pi(0) = Pi(0)ScF - 4qij(PjjDji)' = 0.25( 1 - qij) (11.25) After having presented the many-body procedure for alternant hydrocarbons, we have to define the underlying A Hamiltonians. To derive the maximum output from minimum computational expenditure, we have adopted a simple oneelectron operator HscF of the Hiickel type to calculate the P and D matrices. This setup is furthermore justified by the fact that the latter matrices are completely defined by the alternancy symmetry of the considered systems. A strict tight-binding approximation has been adopted in the latter step HscF = -Etijai,+aj,

(11.26)

iJ.0

The tij are the conventional tight-binding hopping integrals; we have tij # 0 only for bonded A centers. To calculate matrix elements of the type (OiiHOkk)c = ( OiiFOkk)c in the LA we have employed some kind of an extended Hubbard-Hamiltonian with off-diagonal elements defined by the so-called Hartree-Fock hopping Tit2*27928 Fjk

= Tjj = tij - 0.5PikVik

(11.27)

The integrals tij and vk are calculated parameter-free, i.e. ab initio, in a basis of Slater-type orbitals29with a screening coefficient [ derived amrding to Burns rules.3o The above design of the model operators and the adaptation of the LA for the correlation problem is very inexpensive in its numerical realization. The most timeconsuming step is the diagonalization of the mean-field Hamiltonian HscF This leads to the situation that ea. 80% of the total computer time is required for the independent-particle problem providing the above matrices P and D, respectively; less than 10% (27) Borrmann, W.; Fulde, P. Phys. Rev. E Condens. Mutter 1985, 31, 7800; Phys. Rev. B: Condens. Mutter 1987, 835, 9569. (28) Ole&A. M.; Pfirsch, F.;BBhm, M. C. Chem. Phys. 1988, 120,65. (29) Roothaan, C. C. J. J . Chem. Phys. 1991, 19, 1445. (30) Burns, G. J. Chem. Phys. 1964, 41, 1521.

Figure 1. Molecular graphs for CaHn+2polyenes ( n = 6) I, C,Hh2 isomethylidenes I1 (n = 12), and the CI2HIZ radialene 111 together with the adopted atomic numbering scheme. The same labeling is used for the other polyenes and isomethylidenes. TABLE I: Charge Fluctuations ((An:) )- and Correlation-Strength I (II = 18, Parameters Ai at the T Centers in Linear Polyenes 14, 10, 6 ) O ((An?) ) C1 = C18 C2 C17 C3 = C16 C4 = C15 Cs = C14 c 6 = clj C7 = C12 Cg = CII C9 = Clo ((An,:)) Ai

C1 = C18 C2 C17 C3 C16 C4 = CI5 C5 = C14 c 6 = clj C7 = C12 Cs = CII C9 = Clo

4"

%HZO

C14H16

0.124 0.418 0.289 0.324 0.309 0.316 0.313 0.315 0.314 0.304 0.752 0.165 0.422 0.353 0.382 0.367 0.374 0.370 0.372 0.395

CIOHIZ C6H8

C2H4

0.124 0.418 0.289 0.324 0.309 0.316 0.313

0.122 0.420 0.290 0.322 0.311

0.120 0.422 0.300

0.281

0.299

0.293

0.281

0.281

0.752 0.164 0.422 0.353 0.381 0.368 0.374

0.756 0.161 0.421 0.356 0.378

0.760 0.156 0.401

0.438

0.402

0.414

0.438

0.438

For convenience also the C2H4 data for a CC bond length of 1.4 A are given. ((Ana.))m, and Aav denote the mean values of the charge fluctuations and correlation strength. For the atomic numbering scheme, see Figure 1. Nondimerized structures have been assumed throughout.

of the CPU time is necessary to determine the interatomic u correlations. Even larger u systems with more than 50 carbon centers can be handled without difficulties on conventional Ws. Finally we should mention that we have selected the present simple theoretical formalism with intent to demonstrate that already this level of sophistication is sufficient to evaluate transparent ordering principles in alternant hydrocarbons on the many-particle level. 111. Linear Polyenes and Isometbylidenes of CnHn+* Stoichiometry We begin the analysis of the u electron localization and u correlation energy E, with a series of linear polyenes I and isomethylidenes I1 of CnHn+2stoichiometry. Molecular graphs together with the employed atomic numbering scheme in the class I and I1 compounds are shown schematically in Figure 1. Additionally we have studied the ((An;)), and Ai of the radialene molecule CI2Hl2I11 which is also displayed in Figure 1. The mean value of the charge fluctuations ((Ana:)) and correlation-strength parameter AB" of the radialene system show remarkable similarities with the many-body indices derived for linear isomethylidene arrangements. In Table I we have collected the ((An;)),,, and Ai numbers at the A centers in CnHn+Zpolyenes I (n = 18, 14, 10, 6); the C2H4data have been added. A common bond length of 1.4 A

Schiitt and Bijhm

608 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992

8 an increasing functionof the bond length alternation 6. In C&, reduction corresponds prevailingly to a dilution of the AnderC6H8,and C8H10, respectively, lE,,,l is reduced with increasing son-type localization at the end atoms. The ((Ant)),, and Ai 6. The energy gain is an increasing function of the chain length. elements at the inner carbon sites depend only weakly on the chain But Figure 4 indicates also a saturation in IEmrlwith increasing length. Below we will discuss model systems where the above “dilution” is not possible. The data in Table I indioate that the

1,

-

(32) Longuet-Higgins, H. C.; Salem, L. Proc. R. Soc. London, Ser. A

(31) Anderson, P. W. Phys. Rev. 1958, 109, 1402.

1959, 251. 172.

0.325

The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 609

Electron Delocalization in Alternant u Systems AE,,,,

n

[ k l /mole1 12.0

ICnHn+>

1

polyenes

/I

8.0

A4

-65,

-;,, -===---- ..5:: , _ _ - r r

0.300

,_,*’

*2

-8.01 0

0.02 0.04 0.06

0.275 0

0.08 6 [A]

H4

0.02 0.0~0.06 008 6[Al

Figure 4. Variation of the interatomic T correlation energy AE,,, in

Ln.2 : - n z 1.2.3

CmH,2 polyenes I (n = 4,6,8, 10, 12, 14, 16, 18,40 from top to bottom)

L”

as a function of the bond length alternation 6. The correlation energies for the nondimerized polyene structures have been used as common origin. The AE- values are always normalized to two T electrons.

TABLE Ik Charge Fluctuations ((An:))and Correlation-Strength Parameter A, in Isomethvlidenes C a d , II (II = 12,10,8.6)“ ~~

C12H14

CIOH12

C8H10

cl = c6

C2 = C s C3 = C4 C, = C12 C8 C I I C9’CI0 ((Anav))-

AI

C1 = C6 Cz Cs C3 = C4 C I Cl2 C8 C I I C9 = Clo

4”

CIZH12

C6H8

( (An?) )-

((An?)),rr 0.408 0.417 0.419 0.143 0.109

0.407 0.417 0.143 0.112

0.268

0.408 0.417 0.419 0.143 0.109 0.111 0.269

0.185 0.166 0.162 0.714 0.782 0.777 0.464

0.185 0.166 0.162 0.714 0.781 0.777 0.463

0.185 0.166 0.714 0.776

0.111

0.408 0.416 0.143 0.113

Cinnm

C,,,,,

0.270 0.272 ((Ana:))

0.425 0.122

0.274

Ai 0.185 0.169 0.714 0.775

0.460 0.457

CiMCr COu,,,

4”

0.151 0.756

0.454

“Additionally we have given the many-body parameters ( (An:))EOll and Ai for the Cl2HI2 radialene 111; see Figure 1. ((Ana:)) and 4. denote the corresponding mean values. The atomic numbering scheme employed for the isomethylidenes is shown in Fi ure 1. Nondimerized structures with a C C bond length of 1.40 have been adopted throughout.

x

n. For CaHd2, e.g., we derive a normalized gain in lE,,,l of ca. 7.75 kJ/mol for 6 = 0.08 A. For longer polyenes we have thus derived some type of “correlation paradox”, i.e. a reduction of the correlation-strength parameter A,, as a result of bond length alternation which is nevertheless coupled to a gain in the respective u correlation energy IE,,l. The physical origin of this “out-ofphase” modulation is quantified in the next section in connection with u dimerization in monocyclic u rings. To summarize, the present results coincide with the principal findings of ref 8 insofar that u electron correlations are one driving mechanism for the bond alternation in extended polyenes. We should mention, however, that the limitations of our model approach prevent the evaluation of exact E,, numbers. Thus it may be possible that less simplified computational procedures lead to another value of n where the gradient of lE,,,l changes its sign as a function of 6. Nevertheless we have visualized for the first time the size dependence of the correlation influence on the dimerization of polyenes and the above “correlation paradox”. The computational results collected in Table I for the CnHn+2 polyenes have shown an attenuation of end-atom effects with increasing chain length; the net correlation strength measured via 4, is a decreasing function of the number of a centers. In Table and Ai for CnH,,+2 isoI1 many-body indices ((An:)),,, methylidenes of class 11, see Figure 1, are collected together with the corresponding data for the ClzHlzradialene 111. In the C,,HH2 series I1 we predict some type of “negative saturation”. The exocyclic u bonds cause here an enhancement of the averaged

4

: _ _ _ n_ = 5,L,3,2,1 4

in 4n + 2 Hiickel annulenes (solid curves) and 4n non-Hiickel rings (broken curves) as a function of the T dimerization 6. At the left bottom we have added ((An:))- of ethylene at a C C bond length of 1.4 A. For the 4n + 2 annulenes we have considered n = 1,2,3 (from top to bottom) and for the 4n, n = 1, 2, 3,4,5 (from bottom to top). The “start geometry” in the 4n annulenes corresponds to 6 = 0.005 A.

Figure 5. Charge fluctuations ((An:))-

correlation-strength parameter 4, with increasing chain length. From n = 6 to n = 12 4, is raised from 0.457 to 0.464.The rather high correlation-strength parameter A,, indicate$ that isomethylidenes should be unstable and highly reactive hydrocarbon compounds, a theoretical prediction that is in line with chemical experience. A convenient classification of the electronic structure in isomethylidenes I1 and the C12Hlzradialene I11 on the many-body level is feasible by dividing their u network into two subsets: (i) The highly correlated end atoms with A, elements between 0.714 and 0.777, respectively. The second number is characteristic for exocyclic u centers in isomethylidenes at the chain end, while the first one is realized at the exocyclic u A O s in the chain. (ii) The “inner” carbon sites with Ai numbers between 0.164 and 0.185, respectively. In the CI2Hl2radialene I11 we derive Ai elements of 0.756 and 0.151 for the uouter” and “inner” carbon centers. We have the net situation that the T electrons are strongly localized at the terminal sites of the above hydrocarbons. Almost perfect u electron delocalization is possible in the inner carbon networks. The above discussion has shown that the linear u network in CnHn+2polyenes is far from being “free electron like”. In structures as realized in the class I1 and I11 compounds it makes sense, however, to speak of a highly delocalized “electron gas” with sizable charge fluctuations.

+

IV. 4a 2 and 40 AMulenes The calculation of ((An?) ,,) Ai, and E,,, numbers in 4n + 2 annulenes of the Hiickel type and 4n monocycles allows for the evaluation of rather surprising theoretical results. The principal difference between the two families of annulene rings in the one-electron description has been already mentioned in section I. In Figure 5 we have collected the charge fluctuations ((An:)), in a series of 4n 2 and 4n hydrocarbon rings as a function of the dimerization coordinate 6. We have considered Hiickel rings with n = 1, 2, and 3 and “antiaromatic” 4n cycles with n = 1, 2, 3,4, and 5, respectively. The ((An?)),, curves show immediately that the 4n 2 and 4n rings differ fundamentally for small n. A mutual assimilation is observed with increasing ring size. The charge fluctuations in the Hiickel rings are only weakly 6 dependent. In the 4n rings ((An:) ,,,) is sizably enhanced with increasing bond alternation. The 0.08-Adimerization reduces the calculated width in the ((An:))- numbers by more than 605% in the latter compounds. The gradient of the ((An:)),, curves is largest in cyclobutadiene; it is reduced with increasing size of the 4n rings. Figure 5 indicates that the r electrons in CsHs are far from being delocalized. Bond alternation is not accompanied by an increasing localization of the u electrons; even the opposite var-

+

+

Schutt and Bahm

610 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 AE ,

[kJ/molel

n

n

I

;I

2.4

-.-

2.L 2.3

yi /

002

0.04

0.0~3 0.00 6

I A1

Figure 6. Normalized r correlation energy AEm, in 4n + 2 monocycles (left diagram) and 4n non-Hiickel rings (right diagram) as a function of the CC alternation 6. For the 4n + 2 Hiickel systems (n = 1, 2, 3, 4,7) the equidistant Dd structure has been employed to fix the energy origin. For the 4n non-Hiickel rings (n = 2, 3,4,5 , 6 , 10)a slightly dimerized (6 = 0.01 A) arrangement has been used to define the origin of E,,,. AE,,, of C4H4is displayed on the top left for 6 values 10.03 A.

iation is found. For the D6,, structure we predict ((An;))mr, = 0.323 (Ai = 0.354). 6 = 0.08 8,leads to ((An;)),, = 0.328 (Ai = 0.344). The enhancement of the charge fluctuation with increasing bond alternation is even larger in the 4n + 2 rings with n = 2 and 3, respectively. In comparison to ethylene the charge fluctuations in C6H6 are enhanced by ca. 15%. This is on one side a consequence of a normalized hopping energy of -2.67ti, in benzene in comparison to -2tij in the localized two-center limit. This effect is supported by the Coulomb interaction which is reduced with increasing ring size; see below. We have identified several peculiarities of the a bonding in C6H6, which is frequently adopted as prototype of a delocalized a perimeter. (i) In comparison to localized twocenter a bonds the I electron delocalization in 4n + 2 Hiickel rings is enhanced due to the cyclic boundary conditions allowing for an increasing hopping contribution and decreasing interaction. (ii) Relative to all u bonds formed by main-group elements the a electrons in benzene and higher 4n 2 representatives are however more localized. (iii) Bond alternation allows for an enhancement in the a electron delocalization. The charge fluctuations in the neighborhood of the D4h structure of cyclobutadiene indicate a coincidence with the a electron delocalization in C2H4 (Le. localized limit). The a delocalization in C4H4is enhanced with increasing dimerization. Our model approach leads to ((An;)),,, = 0.31 for 6 = 0.08 8,. The discussion in the last section has shown that this number is also characteristic for the inner carbon centers in polyenes. It is the opinion of the present authors that the ((An;)), curves in Figure 5 for monocyclic hydrocarbons as well as the numerical results collected in the Figures 2 and 3 for C,H,,+* polyenes point toward the probable microscopic origin for the distortive nature of the ?r network in almost all molecular hydrocarbon arrangements. This seems to be the enhancement of the charge fluctuations ( (An;))mr, (i.e. increasing electronic delocalization) coupled to an energy gain on the many-particle level; see section I1 and below. In Figure 6 we have displayed the variation of the interatomic a correlation energy AE,,, in 4n + 2 annulenes (left diagram) and 4n rings (right one) as a function of the dimerization. 6 = 0 has been adopted as E,, origin in the Hiickel rings and 6 = 0.01 8,as common reference in the orbitally degenerate (6 = 0) 4n perimeters. The correlation energy lEcor,lis an increasing function of 6 in all Hiickel systems. The energy gain is thereby an increasing function of the ring size. In comparison to the CnHn+2polyenes the enhancement of lE,J as a function of 6 is enlarged. For ClsHzowe have predicted a many-particle stabilization of 4.4 kJ/mol in the dimerized structure. In the 18

+

2.2

0

06-

'

I

0.OL 0.08 6 I&

-C14H14 _ _ _ _

--

0

OOL 0 0 8 6 [ A ]

CH,:, CH ,.,

Figure 7. Matrix elements (OiiH),, (OiiH)?, and Ck(OiiHOkk)o,respectively, in C6H6, CgHg,and CI4Hl4as a function of the bond length elements are given in eV, dimerization 6. The ( OiiH),and Ck(OiiHOkk)e ( OiiH): in (eV)*. Full curves, C,H,; broken curves, C14H14; dot and dash CUNeS, CgHg

perimeter this contribution is more than doubled; AE,, amounts here to 9.1 kJ/mol. The above numbers have been always normalized to two a electrons. The many-particle energies derived for the 4n annulenes as a function of the dimerization coordinate 6 (right diagram in Figure 6) indicate a competition between two different physical effects. (i) A retarding component caused by the increasing average kinetic energy for the virtual excitations with increasing 6. 6 # 0 opens a gap between the occupied and virtual one-particle space; see section I. (ii) The general pushing "size-effect" allowing for increasing lE,J numbers as a function of the bond length alternation with increasing size of the a network. This holds for linear as well as for cyclic hydrocarbons. Our calculations show that the correlation energy in cyclobutadiene is strongly reduced with increasing bond alternation. The steep gradient of the corresponding AE,, curve allows only for the presentation of data points with 6 I0.03 8,in the adopted energy window. The experimentally verified dimerized structure3' of cyclobutadiene derivatives must be a consequence of the Jahn-Teller instability of the orbitally degenerate D4h structure. The Jahn-Teller splitting is attenuated by the a correlations. 6 = 0.08 8, in C4H4costs 13.6 kJ/mol correlation energy. Decreasing IEmrrlnumbers with increasing dimerization are also predicted for the 8 and 12 perimeters. The loss in the I correlation energy amounts here to ca. 2.5 and 1.5 kJ/mol (n = 2, 3). The polyene data collected in Figure 4 indicate that lE,,,l hampers the dimerization in chains with less than 10 carbon centers. The sign of A,!?,,, is reversed in CloH12. In the 4n rings "sign conversion" takes place in the 16 perimeter where the a correlation is again one driving force for the dimerization. The AE,, curves in the Figures 6 and 4, respectively, show some interesting interrelations between 4n 2 Hilckel rings, 4n 'antiaromatic" polyenes. One series with roughly comperimeters, and CnHn+2 parable A&,,, e.g., is C6H6 Cl6Hl6 C14H16. For all 4n 2 Htickel systems we have evaluated some type of "correlation paradox", i.e. an out-of-phase modulation of the correlationstrength parameter and lE,,,l. Ai is here a decreasing function of 6 and lE,,,l an increasing one. Some kind of IEco,,l and Ai conformity is only obeyed in the 4n rings with n = 1, 2, and 3; decreasing Ai is here coupled to decreasing a correlation energy.

+

- -

+

(33) Masamune, S.;Souto-Bachiller, F. A.; Machiguchi, T.; Bertie, J. E. J . Am. Chem. SOC.1978, 100. 4889.

The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 611

Electron Delocalization in Alternant u Systems

8

TABLE 111: Charge Fluctuations ((An:)), and Correlation-Strength Parameter Ai in Benzene, Naphthalene (IV), Biphenyl (V), and 1,fL)iphenylene (VI)” compound atom i ((An?))” Ai C6H6

IV, naphthalene

CI = C4 = C5 = C8 c 2 = c3 c 6=c 7 c 9 = ClO V, biphenyl C1 = Cs = C8 = C12 C2 = C4 C9 = CI1 c3 = ClO c 6=c 7 VI, 1,2-diphenylene CI = C9 C2 = C6 = Clo = C14 C3 = C5 = C11 = CI3 c4 = c 1 2

c, = c*

0.323 0.303 0.327 0.360 0.308 0.328 0.319 0.357 0.364 0.304 0.330 0.317 0.304

0.354 0.395 0.345 0.28 1 0.384 0.343 0.362 0.286 0.272 0.393 0.340 0.366 0.392

“The atomic numbering scheme is indicated in Figure 8. To understand the 6 dependence of the correlation energy AE,, and the correlation-strength parameter Ai in the above annulene systems, we have collected in Figure 7 the matrix elements ( Oi,H),, (OiiH); as Well as the Sum x k ( OiiHOkk), O f C6H6, C8H8, and C14H14, respectively, as a function of 6. In section I1 it has been demonstrated that the Ai numbers in systems with one degree of freedom depend on ( OiiH),/&( OiiHOkk),, while ( OiiH),enters E,,, as quadratic term. Most easy to explain in Figure 7 is the steep gradient of the kinetic x k ( OiiHOkk)ccontribution in CsH8 with increasing 6. This is simply caused by the increasing kinetic energy of the virtual excitations when the one-particle gap is opened for 6 # 0. The 6 dependence of the kinetic energy part is rather shallow in the two 4n 2 rings for not too large bond length alternation. All matrix elements in Figure 7 are a decreasing function of the ring size (6 = 0); the gradient of the ( OiiH),,( OiiH):, and &( 0iiHOkk),curves is however enlarged with increasing n. This reflects just that the number of nonvanishing matrix e k ” t S Of the types (OiiHOkk)cand (OiiH), are enlarged for 6 # 0 with an increasing number of u centers. The different curves collected in Figure 7 indicate that the 6 variation of Ai and E,, in C8Hs is always determined by the kinetic energy part &( OifiOkk),. The enhancement of the kinetic energy for the virtual excitations is neither compensated by ( OiiH), nor by ( OiiH),square. ((An?)),, and E, change “in-phase”. The &( OiiHOkk), gradient in the 4n 2 rings is remarkably reduced in comparison to the corresponding gradient in the orbitally degenerate (6 = 0) 4n perimeters. The correlation-strength parameter Ai changes here only weakly with the dimerization. We have already mentioned the shallow x k ( OiiHOkk),variation for smaller 6 values. For large dimerizations we predict nevertheless a Ai reduction (corresponding to a ((An;)), enhancement) which is determined by the x k ( OiiHokk), denominator. This influence is an increasing function of the ring size. The xk(OiiHOkk)c gradient becomes gradually steeper; see Figure 7. The latter effect is however not strong enough to control the variation of the correlation energy. The ( OiiH),square term is now the determining factor which allows for increasing lE,,l with increasing 6. The ”correlation paradox” encountered in the Figures 5 and 6 has thus found an intelligible explanation. Figure 7 gives also straightforward insight into the size effects of E,,, in annulenes and polyenes.

+

+

V. Miscellaneous Compounds and “Memory Effects” Already the discussion in the last two sections has shown that the Ai and ((An?)),,, elements are highly site-specific and therefore also transferable from one u system to the other. Several ordering principles in u systems on the many-body level have been derived so far: (i) Anderson-type localization of the u electrons at terminal atoms, (ii) “alternancy-symmetry” in the Ai and ( (An;)),,, numbers, and (iii) size and topology dependence of the above quantities (Le. discrimination between cyclic and linear and Ai structures, etc.). The local nature of the ((An;)),,r numbers suggests also the possibility of finding “memory effects”

7

2-7

y

3m 10

4

=

1 13

5

8

9

14 2 7

6

4 W 5

Figure 8. Molecular graphs of naphthalene (IV), biphenyl (V), and 1,2-diphenylene (VI) together with the atomic numbering scheme.

in extended u systems with more or less isolated electronic s u b structures. We have used the latter denotation to label the restoration or destruction of smaller electronic substructures. Such phenomena are of course also possible in the one-electron picture. In Table I11 we give the ((An;))w,, and Ai elements of naphthalene IV,biphenyl V, and 1,Zdiphenylene VI, respectively; for convenience the C6H6 data have been added. The molecular graphs together with the adopted atomic numbering scheme of the cycles IV to VI are displayed in Figure 8. The naphthalene data IV indicate immediately the aforementioned alternancy (oscillations) in the ( ( A n ; ) ) , and Ai elements between symmetry-inequivalent atoms. The charge fluctuations at the centers 9 and 10 of CloHBIV are sizably larger than the ((An;)), numbers at the other u atoms with only two bonded neighbors. This leads to a remarkable competition effect in naphthalene. On one side we find decreasing charge fluctuations at all ?r centers bonded to two neighbors relative to ((An;)), in the underlying C6H6 monocycle (size effect). But this is compensated by the increase of ( at C9 = Clo acting as triple junctions in the bicycle (i.e. atoms with three neighbors). The mean values ( (Anay2))w,in C6H6 and CloH8 are therefore of comparable magnitude; see also Table IV and Figure 10, respectively. The latter coincidence between C6H6and CloH8has been recognized phenomenologically by chemists insofar that the 4n + 2 Hackel approach for annulenes is often transferred to polycycles containing 4n 2 ?r electrons. In a simple topologic assignment all higher polycycles remember their C6H6 precursor. Also in biphenyl V the atoms c6 = C7 differ from the other u positions in their ?r localization properties. In VI it is without larger importance whether u centers bonded to two neighbors are part of the six-membered rings or belong to the exocyclic CC bridge. On the other side we find large similarities between C9, cloin IV, C6, c7in v, and cl, c9in VI, respectively. The corresponding ((An?)),, and Ai numbers depend only on the direct neighbor relations but not on the details of the molecular structure. Generally we have shown that the ((An;)),, are enlarged with an increasing number of bonded neighbors; they are minimal at terminal u centers. To estimate ((An?)),, and Ai in a typical two-dimensional (2D) r network with most of the u centers bonded to three neighbors, we have modeled a graphite layer by a 2D array built by 92 carbon atoms. For the “inner” ?r A O s we predict a ( (An:))wm number of 0.345 corresponding to A,, = 0.311. In comparison to benzene this is a ca. 12% reduction in the correlation-strength parameter Ab In Table IV elements of “conservative” we have collected ((An?)), and u systems. The latter label has been used to identify r systems, where the extension of molecular subfragments is not coupled to stronger modifications in the Ai or ((An;)),,, numbers. In the polyphenyls V and polyphenylidenes VI, respectively, we have extended the number m of six-membered rings. For convenience we have also given ((Ana:)), and A, of naphthalene IV and C&. The computational results indicate that the kV and ( elements are principally determined by the sixmembered rings as central structural building unit. Any further connection and condensation of the latter unit is without influence on the Aav and ((Ana?)),r, elements.

+

Schutt and BBhm

612 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992

TABLE I V Mean Values of the Cbarge Fluctuations and Correlation-StrengthParameter Aa,in Napbthalene IV as Well as

037 0 35

a Series of Polypbenyls (V) a d Polypbeoylidenes (VI) with an Increasing Number m of Six-MemberedT Rings

system

4"

m

IV, naphthalene V, polyphenyls

2 2 3 4 5 2 3 4 5

VI, polyphenylidenes

0 33 0 31

benzene

029 0 27

0.352 0.350 0.350 0.350 0.350 0.356 0.356 0.356 0.356 0.354

0.324 0.325 0.326 0.326 0.326 0.322 0.322 0.322 0.322 0.323

'2n.2

'4

0 25 C#4

Figure 11. Charge fluctuations ((An:)), at the T centers i of the polycycles VI11 formed by four-membered rings as key units. The lines elements. are only given to facilitate the observation of the ((An:))"

H4

n * 2 H2n

4

' 6 , +ZH4n+4

Figure 9. Molecular graphs for a one-dimensional (1D)arrangement of four-membered rings (VII), six-membered rings (VIII), and eight-membered rings (IX). -30.0 ))corrand ((An;))” pattern is the electronic outcome of different memory effects in the quasi-lD chains; they are symbolized in Figure 13. C6H4restores a 4n 2 like “delocalized Structure” caused by one triple junction. We find a pair of ((An;))mrr numbers characteristic for A sites bonded to three neighbors (Le. enhancement), second diagram in Figure 13. Such a “stable” 6~ electron structure is not possible in C8H4. The ( (Anma$))corr values are lowered; the amplitude function A indicated in Figure 12 is reduced correspondingly. Figure 13 visualizes that the respective electronic structure is best described in terms of a distorted 4n perimeter and two localized terminal A bonds. The ( ( A ~ I , , , ~numbers ~ ~ ) ) ~ in~ CsH4 are close to the charge fluctuations predicted for dimerized cyclobutadiene; see Figure 5. The minimum fluctuation coincide with ((An,2)), of ethylene. CI0H4shows memory effects to a 6 r monocycle and two terminal u bonds. Note that neither C1&4 nor C8H4allows for extensive triple junctions which would enhance the u electron delocalization; i.e., the A centers with three neighbors behave formally as atomic members bonded to two neighbors. Triple junctions are again possible in C12H4; see the maxima in Figure 1 1. The realized memory effect is shown in Figure 13. The respective electronic structure can be described by two decoupled 6u subunits with triple junctions. The remaining members in series VI1 follow throughout this building principle. The numerical results derived for the model compounds VI1 to VI11 can be finally summarized as follows. Extended bridged r networks developed from stable (i.e. 4n 2) monocyclic precursors and elongated by fragments adding 4n A electrons to the net balance are “conservative”; remarkable changes in the localization properties of the A electrons are suppressed. An example has been series VIII. Two types of memory effects have been observed in series VII. In the initial step from C4H4to C6H4 the unstable perimeter structure is finally bleached out. The extension of the chain length in series VI1 allows alternatively for the formation of n isolated (4n + 2) structures with extensive delocalization via triple junctions and the formation of ”pseudomonocyclic” 4n 2 subunits which are decoupled from 2 X 2 subdomains forming terminal z bonds. Experimental research has shown that the present topologic ordering principle is far from being of only academic interest. The material properties of many polycyclic hydrocarbon u systems seem to be consistent with the above ‘memory effects” realized in extended u domains. As a

+

+

+

0

0.6

Figure 14. Charge fluctuations ((Ann,:) , , ,) correlation-strength parameters &,,, and probabilities of double, single, empty ( n = 2, l, 0) occupancy PaV(n)characteristic for the I centers A to H shown schematically at the left margin. The I centers A to H are always discriminated on the basis of their nearest-neighbor structures. In the ((b:)),,, 4”:and P,,,(n) diagrams we have given the independentparticle (SCF)limit and the case of perfect interatomic correlations (localized boundary) at the right and left margin. Bottom diagram: Pa,,(2) = P,v(0), filled bars; Pav(l),empty ones. The given width of the different bars in the three diagrams corresponds to the theoretically calculated distribution in ((Ana:) )wm, &,,, and PaV(n),respectively.

+

principal result we have shown that electronically isolated 4n 2 subunits are preferred elements in alternant hydrocarbons.

VI. Transferable Elements for the Charge Fluctuations and CorrehtiOnstrength Parameters Aav On the basis of the numerical results collected in the latter three sections and previous calculation^,^^*^^ it is straightforward to summarize highly transferable many-particle indices ( (4:))” and 4, characteristic for the possible interconnections between A centers in alternant hydrocarbons. In Figure 14 mean values for the charge fluctuations ( (Ana:))corr and the correlationstrength parameter h,,are collected together with the occupation probabilities P,,(n). The underlying local structure elements A and H are indicated at the left-hand side of the figure. As already emphasized the presently derived width in the ((Ana>))mrrand $, elements covers almost the total allowed spectrum of manyparticle indices from the independent-electron (SCF) limit to the localized boundary of perfect interatomic correlations. The charge fluctuations predicted for the type A and B A centers are probably slightly underestimated in the adopted linearization of our correlation model. An “exception” in Figure 14 are the u functions of type H with a terminal site as neighbor and bonded to an “inner” carbon center. The charge fluctuations are here larger than at centers bonded to three neighbors; they approach the ‘free-electron” limit ((An;))SCF. The collection in Figure 14 is based on electronic structures which are exclusively determined by the local surrounding of a selected “reference” site. Superior ordering principles, i.e. certain “memory effects”as discussed in series VII,have (34) (35)

Mhm, M.C.; Schltt, J. Mol. Phys. 1991, 72, 1159. Schiltt, J. diploma thesis, Technischc Hochschule Darmstadt, 1989.

614

J. Phys. Chem. 1992, 96, 614-623

been neglected. For these compounds it has been shown that the latter “memory effects” may cancel the nearest neighbor control. Such processes have to be expected in compounds with “unstable” local electronic configurations in the absence of “memory effects”.

VII. Final Outlook The many-particle nature of a electron bonding in alternant hydrocarbon systems has been investigated by the method of the local approach supplemented by rather simple model Hamiltonians. The present authors believe that it is most convenient for the reader to review the principal findings in the form of characteristic catchwords. (i) The A electrons in annulenes and linear polyenes are sizably localized. It seems to be necessary to substitute widely accepted traditional models for a compounds. (ii) Bond alternation leads throughout to an increase in the A electron delocalization. (iii) The respective enhancement of the charge fluctuations is probably the microscopic origin for the distortive nature of A networks. (iv) An enhancement of the charge fluctuations (decreasing electronic correlation-strength Ai)and increasing interatomic A correlation energies caused by bond dimerization are not mutually exclusive in larger hydrocarbons. The out-of-phase modulation is obviously the rule. (v) Cyclic 6 7 electron structures allow for the optimum possible a delocalization

Radical Cations in Mixtures of ClsP and Me#. Chemical Study

in alternant hydrocarbons; but even this takes place far from the “free-electron” limit. The latter process may lead to the situation that extended a systems prefer to form more or less decoupled spatially localized 6 r subunits. (vi) Extension of linear monocyclic or polycyclic A systems is frequently not coupled to an increasing delocalization of the A electrons (effects due to the end atoms neglected); their fluctuations saturate quite early. Exceptions have been discussed in the above sections. (vii) The interatomic A correlation energy is one driving force for the bond length alternation in linear and monocyclic hydrocarbons exceeding a certain threshold dimension. (viii) Strong electronic correlations tend to attenuate Jahn-Teller or Peierls instabilities. The attenuation is maximized with increasing correlation strength; see the series C4H4,CsHs, CI2Hl2.(ix) The latter effect may be of some influence in “metastable” solids, i.e. solids where the instability is scarcely suppressed. In these compounds superconductivity under strong coupling conditions may become possible.

Acknowledgment. This work has been supported by the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie. M.C.B. acknowledges support due to a Heisenberg Fellowship. The drawings have been kindly prepared by Mr. H. A. Schmaltz.

A Combined ESR and Quantum

Olav M. Aagaard,* Bas F. M. de Waal, Marcoen J. T. F. Cabbolet, and Ren6 A. J. Janssen Laboratory of Organic Chemistry, Eindhouen University of Technology, P.O. Box 51 3, 5600 MB Eindhouen, The Netherlands (Received: June 10, 1991)

Exposure of phosphorus trichloride (C13P)and dimethyl sulfide (Me#) dissolved in halocarbons (CFC13,CF3CC13,CF2C1CFCl2, and CH2Clz)to X rays at 77 K results in the corresponding parent cations and several cationsubstrate adducts. The radicals are detected and identified by ESR spectroscopy. In dilute solution exclusive formation of the parent C13P.+and Me2S’+ radical cations is observed. In CFC13, Me2S’+ exhibits superhyperfine interactions due to chlorine and fluorine nuclei of the matrix molecule(s). At increased concentration, or on warming the sample, the parent radical cations readily react with dissolved C13Por Me2Smolecules to form homodimeric C13PLPC13+and Me+SMe2+ and heterodimeric C13P4Mq+ radical cations with a two-center three-electron bond. The heterodimer is formed in spite of a significant difference between the ionization potentials of the two constituents in reduced form. On further annealing, the CI3PLPCl3+cation rearranges to the well-known trigonal-bipyramidal C1,P radical and an as yet unidentified configuration. Candidates for the latter are proposed. In concentrated frozen solutions an unexpected reaction of Me2SLSMe2+and C13Pis observed, resulting in the heterotrimer C13P(SMe2)2’+with an octahedral configuration, exhibiting a very large 31Phyperfine interaction (Ai, = 51 15 MHz). Extensive ab initio calculations at the HF/3-21G* level, including calculation of isotropic and dipolar electron-nuclear hyperfine interactions, confirm the assignments and provide detailed insight into the molecular geometry, electronic configuration, and stability of the radical products.

Introduction The use of halocarbon matrices in combination with ESR spectroscopy has eminently contributed to the knowledge of radiogenic formation and reactivity of radical cations at low temperature. In general these experiments are conducted at low substrate concentration to ensure the detection of the parent radical cations and unimolecular decomposition reactions. However, at elevated concentration and depending on the temperature and mobility in the frozen halocarbon matrix, radical cations can react with free-electron pairs of other substrate molecules. Such ionmolecule reactions usually afford two-center three-electron bonds with a a2a*’ configuration (a* bond). Recently an increased interest has been noted in the literature on the nature and stability of these three-electron bonds. On the basis of high-level quantum chemical calculations, Clark postulates that, in vacuo, the stability of a a* bond will diminish exponentially with increasing difference in ionization potential of the two reduced substrates

Therefore, formation of an ion-molecule adduct consisting of two equivalent molecular parts (homodiier) should be preferred over ion-molecule adducts comprised of two different molecular fragments (heterodimer). Accordingly, a large number of ESR studies describe homodiieric radical cations, whereas reports on heterodimen are few. However, this reflects the lack of systematic study rather than the alledged intrinsic instability of the heterodimers. Recently we reported the results of radiogenic radical formation in a mixture of trimethylphosphine (Me3P) and dimethyl sulfide (Me$) in Freon? We demonstrated that besides the well-known (1) Clark, T.; Hasegawa, A.; Symons, M. C. R. Chem. Phys. Lett. 1985,

116, 79. (c)

(2) (a) Clark, T. J . Compur. Chem. 1981,2, 261. (b) Ibid. 1982,3, 112. Ibid. 1983, 4,404. (d) Clark, T. J. Am. Chem. SOC.1988, 110, 1672. (3) Janssen, R. A. J.; Aagaard, 0. M.; van der Woerd, M. J.; Buck, H.

M. Chem. Phys. Lett. 1990, 171, 127.

0022-365419212096-614%03.00/0 0 1992 American Chemical Society