3674
J . Phys. Chem. 1992,96, 3674-3683
expected to be underestimated somewhat for the same reasons, and the hydrogens are also not coplanar with the carbon skeletons. Ab initio force fields of both molecules were scaled by scale factors optimized with the al, b2, and bl fundamentals of Dewar benzene. On the basis of the scaled force field, the assignments of fundamental frequencies of Dewar benzene given by Griffith et a1.6 are essentially confirmed with the following modifications: (1) The band at 863 cm-l is a bl fundamental, while the two strong bl bands at 1281 and 1267 cm-I, both of which were assigned as fundamentals by Griffith et al., are believed to be a Fermi doublet arising from one fundamental at about 1267 cm-I and an a l bl combination. (2) The bands of a2 symmetry at 723,923, and 1178 cm-I are a2 fundamentals instead of those at 779,801, and 1028 cm-*, which were assigned by Griffith et aL6 For Dewar
+
pyridine, the scaled force field successfully predicts vibrational frequencies in good agreement with available experimental results, and assignments of all the observed FTIR frequencies are given for the first time. This study demonstrates again the practical value of the scaled quantum mechanical force field method for the interpretation of experimental spectra and derivation of reliable force fields.
Acknowledgment. This research was supported by US.National Science Foundation Grant CHE-8814143. We thank Professor G. Fogarasi for helpful discussions on the definition of internal coordinates and the geometry corrections. We also thank Prof. N. C. Handy for the use of the CADPAC program package. Registry No. Dewar benzene, 5649-95-6; Dewar pyridine, 6566-96-7.
Correlatlon Effects and Electron Delocalization in Nonalternant Hydrocarbon ?r Compounds Michael C. Bohm* and Johannes Schutt Institut f u r Physikalische Chemie, Physikalische Chemie III, Technische Hochschule Darmstadt, 0-6100 Darmstadt, Germany (Received: September 30, 1991)
Many-particle contributions to the *-electronic structure of nonalternant hydrocarbons are investigated by a local correlation approach. Numerical results are derived by adopting a simple tight-binding model as well as a model operator in the zero-differential overlap (ZDO) approximation definded in terms of the Hartree-Fock hopping element. The different integrals are calculated ab initio in a Slater-type basis. The atomic 7-electron density localization is quantified by the charge fluctuations ((An;)),,, in the correlated ground state I$o). By means of two correlation-strength parameters, A, and Zi, the reduction of the charge fluctuations ( (An?)),r, due to the n-electronic correlations relative to the free-electron value are measured. The ((An?))" and Ai numbers depend sensitively on the topology of the considered 7 center. Therefore it is possible to formulate topologicordering principles in the many-particle picture. The 7 electrons in nonaltemant hydrocarbons are throughout sizeably correlated. The mean values of the charge fluctuations ( (An?)),rr,av and correlation-strength parameters A,, are roughly comparable in all studied model systems. Furthermore the net charge fluctuations in several unstable nonalternant polycycles are rather close to the ((An?)),,, numbers calculated for aromatic systems. On the basis of the A, numbers an instability index (IS) for nonalternant hydrocarbons is suggested. The theoretical many-particle data coincide with experimental fmdings. It is demonstratedthat bond-length alternation in pentalene is accompanied by increasing charge fluctuations ( (An;))m, The 7 correlation energy is thereby a driving force for the bond alternation. Inter- and intraatomic contributions to the total 7 correlation are given.
I. Introduction Hydrocarbon a systems have been an object of theoretical investigation for more than ca. 60 years. One of the early fundamental contributions to the electronic theory of these compounds has been the subdivision into altemant and nonalternant systems due to Coulson. In altemant species the A centers can be divided into two atomic subsets, starred and unstarred, such that no members of one subset have bonded neighbors of the same one.' This altemancy symmetry has characteristic consequences on the splitting of the one-electron energies t, and one-electron densities Pi,. In the present work the label Pij stands for the i j t h element of the oneparticle density matrix per spin direction. For the on-site A density at center i ( n , ) = 2P,does hold. In alternant a systems one has ( n , ) = 1.0 a t any center. Concerning the canonical molecular orbital (CMO) energies 6, alternancy symmetry leads to a pairwise symmetric t, distribution of bonding and antibonding C M O S around the atomic basis energy. In nonalternant hydrocarbons the above symmetries are broken. The (ni) elements differ from site to site. Also the pair symmetry in the e, distribution is violated. For many years it has been generally accepted in the chemical community that the n-electronic structure of hydrocarbon com(1) Coulson, C. A. Proc. R. SOC.London Ser. A 1938, 164, 383.
pounds can be described adequately in terms of one-electron, Le., independent-particle, models. Two popular approaches, e.g., are the Htickel M O (HMO) theory2 or the free-electron approximation ( E A ) of K ~ h n .The ~ shortcomings of these one-electron techniques have been clarified in theoretical contributions of the past two decades where it has been shown that electronic correlation effects are sizeable in the a subspace of hydrocarbons.+I0 Strong correlations are more or less equivalent with remarkable localization of the corresponding A electrons. Also in the independent-particle picture, which coincides with the SCF description in alternant systems, it has been verified that the 7r electrons are sizeably l o c a l i ~ e d . ' ~ -Recently ~~ it has been shown that a frames (2) Salem, L. Molecular Orbital Theory of Conjugated Systems; Benjamin: New York, 1966. (3) Kuhn, H. Helv. Chim. Acro 1948, 31, 1441. (4) Harris, R. A.; Falicov, L. M. J . Chem. Phys. 1969, 51, 5034. (5) Schulten, K.; Ohmine I.; Karplus, M.J. Chem. Phys. 1976,644422. Ohmine, I.; Karplus, M.; Schulten, K. Ibid. 1978, 68, 2298. (6) Horsch, P. Phys. Reu. B 1981, 24, 7351. (7) Paldus, J.; Chin, E. Int. J. Quantum Chem. 1983,24, 373. Takahashi, M.; Paldus, J. Int. J . Quantum Chem. 1985, 28, 459. (8) Pfirsch, F.; Fulde, P.; BBhm, M. C. Z . Phys. 1985, 860, 171. (9) Kuwajima, S.; Soos, Z . G . J . Am. Chem. SOC.1987, 109, 107. (IO) Soos, Z. G.; Hayden, G. W.In Electroresponsive Molecular and Polymeric Systems; Skotheim, T., Ed.; Marcel Dekker: New York, 1988.
0022-3654/92/2096-3674$03.00/00 1992 American Chemical Society
Nonalternant Hydrocarbon
A
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3675
Compounds
are generally distortive.I2 The tendency of bond-length alternation increases thereby with increasing electronic corre1ati0ns.l~ Molecular geometries without bond-length alternation are forced by the superimposed u frame. The 17 electronic energy is lowered in structures with bond alternation. For a recent quite comprehensive discussion on advances in the theory of *-electronic hydrocarbons see ref 14. For polyacetylene (PA) models, e.g., it has been verified that the interatomic ?r correlation energy E,, is the important key element determining finally the direction (Le., alternating vs nonalternating structures) of the above competition.6JS'' ?r correlations always favor the bond-length alternation. With increasing correlations this effect is enhanced. u correlations, on the other side, are only a smooth function of the distortion amplitude. In chemistry bond-length alternation (Le., dimerization in the solid state) is throughout equated with increasing localization of the * electrons. The underlying property causing this assignment is the static one-electron density p of the many-electron system. In this step, however, a fundamental principle of quantum mechanics is violated, Le., the indistinguishability of electrons. In previous contributions on altemant hydrocarbons1s-20we have demonstrated that the latter one-to-one correspondence, i.e., equation of bond-length alternation with increasing electronic localization, cannot be justified in terms of an unambiguously defined quantitative theory. In the above references we have employed the charge fluctuations ((An?)), at the T centers i as localization parameter. Large atomic electron density localization is equivalent with small ((An?)) numbers. It is of course necessary to evaluate the ((An?)) numbers in the correlated ground state I+o). In the independent-particle (SCF) approximation I+=-) one has a constant ( (AnJ2)SFwhich is determined only by the I density. In alternant hydrocarbons, e.g., a value of 0.5 is found. The mean-square deviations of the *-electronic density in the above levels of sophistication are expressed in the eqs 1.1 and 1.2.
= ( n ? ) - (ni)2
(I. 1)
(+0ln?1+0)- (+olniI+o)2
(1.2)
((b?))SCF
((An?))mrr r:
In the first equation the symbols (...) have been used to abbreviate expectation values which refer to the independent-particle ground state I+SCF), i.e., (...) = (+SCFI ...(+SCF). ni in the expectation values stands for the *-electron number operator which reads ni = aia+aiu.u labels the electron spin t or 1, and uiu+and uisare the conventional *-electron creation and destruction operators, respectively. The charge fluctuations are of maximal size in the independent-particle wave function they are suppressed with increasing electronic correlations. ((An?))" = 0 is possible only in alternant systems with a common ( n i ) = 1.0. In the perfectly correlated limit one finds complete suppression of the probability of double occupancy at the ith center Pi(2) = 0. Alternancy symmetry forces here also suppression of the probability of empty occupancy; Pi(0) = 0. ((An?),,,) = 0 is therefore equivalent with Pi( 1) = 1.O; i.e., only "covalent" configurationswith one ?r electron per center are part of the correlated ground state. As a result of on-site densities ( ni) # 1.O, residual (1 1) Shaik, S. S.;Hiberty, P. C.; Ohanessian, G.; Lefour, J. M. Nouu. J . Chim. 9, 1985, 385. Shaik, S. S.; Lefour, J. M.; Ohanessian, G.J . Org. Chem. 1985. 50. 4657. Hibertv. P. C. Too. Curr. Chem. 1990. 153. 27. (12) Shaik, S. S.; Hi'derty, P. 6 . ; Lefour, J. M.; Ohanessian, G. J . Am. Chem. Soc. 1987, 109, 363. (13) Jug, K.; Kiwter, A. M. J. Am. Chem. SOC.1990, 112, 6772. (14) Gutman, I.; Cyvin, S. J., Eds. Top. Curr. Chem. (Springer: Berlin, 1990; Vol. 153. (15) Mazumdar,S.; Dixit, S. N. Phys. Rev.Lett. 1983,51,292. Baeriswyl, D.; Maki, K. Phys. B 1985, 31, 6633. Carmelo, J.; Baeriswyl, D. fbid. 1988, 37, 7541. (16) Hayden, G.W.; Soos, 2.G. Phys. Rev. B 1988, 38, 6075. (17) Kuprievich, V. A. Phys. Rev. B 1989, 40, 3882. (18) Bahm, M. C.; Schiitt, J. Mol. Phys. 1991, 72, 1159. (19) Schiitt, J.; BGhm, M. C. J . Phys. Chem. 1992, 96, 604. (20) Bbhm, M. C.; Schiitt, J. Mol. Phys., in press.
fluctuations ((An?)), # 0 are possible in nonalternant molecules also in the perfectly correlated limit. A convenient measure for the strength of the *-electron correlations is given in eq 1.3, where we have related the charge fluctuations in I+o)to the maximum fluctuations in I+scF) normalized by the fluctuations in I+scF). Ai
=
((Ani?
)SCF
- ((An?) )mrr
((An?))sm
(1.3)
In recent contributions of one of us it has been shown that chemical bonds can be divided into characteristic subclasses according to their many-body parameters ((An;) and Electrons are only weakly correlated in u bonds formed by main-group elements. Increasing correlations are found in localized T bonds. The correlation strength Ai in bonds of transition-metal atoms is generally larger than in main-group elements. and Ai Rather detailed numerical investigation of ( indexes in alternant hydrocarbons leads to a rather surprising result.1s-20 The accessible ((An?))wrr and Ai width in these materials covers the same width as realized in different types of bonds (a, ?r symmetry, homopolar vs heteropolar ones, etc.) formed by main-group elements as well as transition metals. The ((An?))mrr and Ai elements in alternant hydrocarbons are furthermore characteristic for the type of the considered ?r center; i.e., they reflect the topology of the corresponding carbon site. As a result of the new ordering principles evaluated for alternant hydrocarbons we found it of interest to extend our work to nonalternant hydrocarbons. In Figure 1 the considered ?r systems are collected. As can be seen many of the studied examples have focused remarkable research activities in the past two decades. Stable systems, e.g., are cyclopropylidenecyclopentadiene (3), azulene (a), or the polycycles 15 and 17. Pentalene (5) and s-indacene (8) are rather unstable; their characterization has been possible only by ring s u b s t i t u t i ~ n . ~Also ~ . ~ ~pentafulvene (1) is a highly reactive A system. 6 and 6' derivatives are important key compounds in the synthesis of many nonalternant polycycles. The method of the local approach (LA)27928has been adopted as theoretical tool to derive the many-body indices in the nonalternant r networks. The local nature of the correlation-hole is explicitly taken into account in the LA. To define the underlying independent-particleground state a simplifed tight-binding Hamiltonian in a zero differential overlap (ZDO)basis has been used. The necessary integrals are calculated parameter-free, Le., ab initio, in a Slater-type basis.29 It has already been mentioned that the charge fluctuations and Ai elements are unambiguously defined parameters to measure the degree of atomic electron density localization. Nevertheless the present contribution should be accepted as a model theory which has the advantage of conceptual transparence and simplicity. We believe that the evaluation of general ordering principles in electronic structure theory is possible only in such a level of sophistication and not by pure collection of standard quantum chemical output data. To support the character of a model approach, an instability index (IS) for the considered nonalternant A systems on the many-body level is suggested by adding the standard deviation ai of the correlation strength parameters Ai to the corresponding mean value A,,, IS = 4" up The first term measures the net correlation strength in the ?r systems, and the second one the influence of individual localization centers. Small IS indexes characterize stable molecules. Large Ai elements are an indicator of T sites with sizeable
+
(21) Old, A. M.; Ptirsch, F.; Fulde, P.; BBhm, M. C. Z. Phys. B 1987,66, 359. (22) OleS, A. M.; Fulde, P.; BBhm, M. C. Chem. Phys. 1987, 117, 385. (23) Bbhm, M. C.; Bubeck, G.; OleS, A. M. Chem. Phys. 1989, 135,27. Bubeck, G.; Ole& A. M.; Bohm, M. C. Z.Phys. B 1989, 76, 143. (24) Bbhm, M. C.; Staib, A. Chem. Phys. 1991, 155, 27. (25) Hafner, K. Nachr. Chem. Techn. Lab. 1980, 28, 222. (26) Hafner, K.; Stowasser, B.; Grimmer, H.-P.; Fischer, S.; Bohm, M. C.; Lindner, H. J. Angew. Chem. 1986, 98, 646. (27) Stollhoff, G.; Fulde, P. Z . Phys. E 1977, 26, 257; 1978, 29, 231. (28) Stollhoff, G.; Fulde, P. J . Chem. Phys. 1980, 73, 4548. (29) Roothaan, C. C. J. J . Chem. Phys. 1951, 19, 1445.
3616 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992
5
Bohm and Schiitt
4
5
121 5 w * 6@2 5
El
8 m 3 4
6
5
Dl
161
6
3
7
2
2
7
9
5 '1 4 IO 3
6 14
tal
El
El
13
12 3
[II]
2
4 3
10 2 3
2
Figure 1. Studied nonalternant hydrocarbon T systems 1-17 together with the atomic numbering scheme. Only the molecular topology is displayed.
atomic electron density localization. High reactivity and low stability are here expected. The organization of the present work is as follows. The theoretical background of the many-particle approach is described in the next section; numerical results are then discussed in section 111. Pentalene has been selected as the model system to investigate the interplay between charge fluctuations/correlation strength and bond alternation. This problem is subject of section IV. Some conclusions are finally formulated in V. 11. Theoretical Methods
is the knowledge A prerequisite for the determination of of the independent-particle or SCF wave function I+SCF). In analogy to our previous investigation on alternant hydrocarbonskbm we make use of the simple tight-binding Hamiltonian (eq 11.1) to approximate )+scp).
fisc, = - C tij,aiu+aju
(11.1)
iJ,u
has been described in detail in previous work, it suffices to collect the essentials of the many-body procedure. In (11.2) the correlated ground state is derived from I+SCF) by application of the exponential operator exp(-Cijqi,Oij]lJlscF): I+O)
e x p k C qijOij)%CF)
(11.2)
ij
The qij in the above relation are variational parameters determined by energy minimization. The 0, are local projection operators which reduce the weight of unfavorable configurations that are part of I+xF) in the correlated ground state. In the present work prevailingly interatomic correlations are considered; but see also the next but one section. The latter refer to correlations accessible in a simple valence basis. They are described by virtual twoparticle excitations. The correlations are thus of the densitydensity type. For the discrimination between inter- and intraatomic correlations,see refs 34-36. In the subsequent calculations only the intra-atomic intraorbital projectors Oii Oi are taken into account. In (11.3) we define intraorbital projectors in terms of the abov_ea-electron number operators niu. The irreducible parts of the Oi are the Oi of eq 11.2:
oi
The tij in eq 11.1 are the conventional tight-binding hopping (kinetic energy) integrals between a AOs. In the present work a strict tight-binding scheme has been employed where nonvanishing tij are allowed only between bonded neighbors. In altemant = 2nipi7 (11.3) r molecules the tight-bindingwave function coincides exactly with the SCF state I+SCF) due to symmetry. This one-to-one relation The interrelation between both sets of local projectors is provided does not hold a priori in nonaltemant s systems. The one-particle by eq 11.4: density derived by (11.1) is nevertheless quite close to charge oi = oi - (Oi) (11.4) distributions calculated by ab initio SCF methods. The kinetic energy integrals tiJ and the subsequently described two-electron The brackets in the ( have the same meanhg as already defined repulsions are calculated ab initio in a Slater-type basis.29 The in section I in connection with the number operator n,. To atomic screening coefficient has been determined according to determine the variational parameters qp eq 11.2 is linearized. This Burns' rules.'O As the method of the LA in combination with ab initio Hami l t o n i a n and ~ ~ model ~ ~ ~ operators ~ ~ ~ ~ of the ZDO type6*8*2'*22*33-36 (33) 016, A. M.; Pfirsch, F.; Fulde, P. Bahm, M. C. J. Chem. Phys. 1986,
oi
oi)
~
(30) Burns, G . J . Chem. Phys. 1964, 41, 1521. (31) Stollhoff, G.;Vasilopoulos,P. J. Chem. Phys. 1986,84,2744. KBnig, G.; Stollhoff, G. Ibid. 1989, 91, 2993. (32) Rosczewski, K.;Chaumet, M.; Fulde, P. Chem. Phys. 1990, 143,47.
~~
85, 183. (34) Borrmann, W.; Ole& A. M.; Pfirsch, F.; Fulde, P.; Bahm, M. C. Chem. Phys. 1986, 106, 11. (35) Ole& A. M.; Pfirsch, F.; Borrmann, W.; Fulde, P.; Bohm, M. C. Chem. Phys. 1986, 106, 21. (36) OleS, A. M.; Pfirsch, F.; Bohm, M. C. Chem. Phys. 1988, 120, 6 5 .
Nonalternant Hydrocarbon
?r
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3677
Compounds
approximation is valid for not too strong correlations. Its validity has been analyzed recently.37 In (11.5) the 9ii are related to matrix C(0iHOj)c~ii= (H0j)c
(11.5)
i
Ear, =
XEcorr,i i
= -X(HOi)c~ii i
(11.6)
elements (HOi), and ( OiHOj)c. A fragmentation of the interatomic ?r correlation energy into a sum of individual intraorbital contributions is expressed in eq 11.6. The index c in the above equations symbolizes that only connected diagrams are taken into account in the determination of the expectation values. Adaptation of the linked cluster theorem guarantees proper normalization of the correlation energy and related many-particle indices;38see below. H abbreviates the T Hamiltonian employed to evaluate the matrix elements ( OiHOj), and (HOi),, respectively. The former ones define the residual u interaction; physically they measure the stiffness of the corresponding ground-state wave function. The ( O$IOj), elements can be interpreted as virtual “kinetic” excitations from the occupied to the unoccupied one-particle space. In the present adaptation of the LA we measure these elements in terms of the Hartree-Fock hopping; F,, is the m,n’th matrix element of the underlying single-particle operator: (HOj)c =
(HRB0j)c
= 2CVmtPjmDjmPjnDjn
(11.7)
m,n
Ai
Aimax= Pii/Dii Aimax= Dii/Pii
The V,, in (11.7) stand for on-site (m = n) and intersite (m # n) *-electron repulsion integrals. The single-particle matrix elements in the adopted framework are expressed in (11.9). The applicability of this approach has been quantified in recent in~estigation.~~,~~
F,, = T,, = t,, - 0.5Pm,V,,
(11.9)
(Ui,+Uj,)
Dij = (ai,aj,+)
((An;))min = 2PiiDi,(l - Dii/Pii)
In the employed ZDO framework both sets of matrix elements are interrelated by D.. IJ = 6.. ?I - p.. 1J
(11.12)
6, in (11.12) is the Kronecker delta. Next we summarize the expressions for the charge fluctuations ((An:)),, in I$o) and I$sCF)as well as the correlation strength parameter A,. Evaluation of eq I. 1 for the *-electron fluctuations in the independent-particle limits leads to ( (hi? )SCF =
2PiiDii
(11.13)
which amounts to 0.5 in altemant hydrocarbons. In the subsequent eq 11.14, which refers to the charge fluctuations in the correlated ((An?)
)cor,
= 2PiiDiA 1 - 4~iiPi@ii)
(11.14)
ground state, we have neglected the mutual coupling of the different correlation operators.’8~20Combination of the latter two relations allows for a straightforward definition of the correlation strength parameter Ai: (37) Bohm, M. C.; Bubeck, G.;Ole& A. M.Z.Narurforsch. 1989, 44a, 117. (38) Horsch, P.; Fulde, P. Z.Phys. B 1979, 36, 23.
(11.16)
for 0 I( n i ) I1.0 for 1.0 I( n i ) 5 2 . 0 (11.17)
In eqs 11.18-11.20 we have expressed the above occupation probabilities Pi(n) in the correlated ground state as a function of the variational parameters qii and the matrix elements Piiand Dip The associated occupation probabilities in the S C F picture are given in (11.21)-(11.23).
(11.10) (11.11)
for 1.0 5 ( n i ) 5 2 . 0
((An;))min = 2PiiDii(l - Pii/Dii)
Finally the calculation of the eqs 11.7-11.9 requires knowledge of elements Pij and Dip This is the only information in the many-body step that is required from the independent-particle precursor. As already mentioned in the introduction the Pijare the elements of the first-order density matrix per spin direction; they are expressed in eq 11.10. Matrix elements of type Dij are given in (11.1 1). Pij =
for 0 I( n i ) I1.0
The above pair reflects particle-hole symmetry in electronic structure problems. It is immediately seen that the correlation strength parameter Aimaxis largest for one ?r electron per carbon center. The simple physical picture associated with Aimax= 1.O has been rationalized in the preceding section. The reduction of Aimaxfor ( n,) # 1.O reflects the presence of residual fluctuation at the corresponding u centers. Aimaxfor ( n i ) < 1.0 is confined to a complete suppression of the probability of double occupancy Pi(2). Aimaxfor ( n i ) > 1.0 is coupled to the suppression P,(O). The minimum charge fluctuation associated to the Aimaxelements of (11.16) are given in (11.17).
m.n
(11.8)
(11.15)
Comparison of eqs 11-15and 11.6 indicates that the correlation strength depends on the ratio between two-electron interaction and kinetic hopping; see definitions 11.7 and 11.8. The interatomic correlation energy, on the other side, is defined by the ratio between interaction square and hopping. It is convenient and of theoretical transparence to sketch briefly the two marginal limits for the correlation strength parameter Ai, Le., AISCF = 0 and A, = Aimax.The lower limit is trivial and defines the independent particle/free electron boundary. The Aimax elements for completely correlated electrons are given in (11.16).
= 4PijDi,XFmn(PijD,,$pim - PimPj,,Di,)
( OiHOj)c = (OiH,,O,),
4vjiPjjDii
Pi(2) = Pi(2)scF - 4~ii(PiiDii)’
(11.18)
pi(1) = Pi(l)SCF + 8?ii(PiiDii)’
(11.19)
Pi(()) = Pi(O)scF - 4~ii(PiiDii)’ pj(2)SCF = 0.5Pjj
(11.20) (11.21)
Pi( 1)SCF = 2Pjj( 1 - Pij)
(11.22)
Pi(0)SCF = 1.0 - 2Pij
+ Pi;
(11.23)
For the subsequent discussion it is furthermore of some utility to define a second correlation strength parameter Z,. For different atomic electron densities, the Ai and Aimaxat the respective centers differ; see eq 11.16. This leads to the problem that we cannot compare the correlation strength parameters of 7~ centers with different electronic charge. To get rid of this, we renormalize the A, by Aimaxand obtain a second correlation strength parameter
xi:
Zi = Ai/Aimax
(11.24)
Zi has the boundaries 0 and 1 independent of the electron density at the respective carbon center i. Therefore the parameters Zi enable us to compare electron density correlations of A centers irrespective of their electron density. Large differences between the Ai and Zi parameters are realized at strongly correlated T centers where the electron density differs remarkably from 1.0. In alternant hydrocarbons Ai coincides of course with Zi. The difference between the Ai and Zi numbers is thus a measure for the charge influence on the many-body character of *-electron bonding. Finally we should mention that the many-particle elements E,,,, ((An?)),,,, Ai, and Zi are of different weight in judging the importance of many-particle effects on the a-electronic structure of hydrocarbons. The charge fluctuations can be adequately described only by adopting the correlated ground state. The “independent-particle” fluctuations are always “trivial”, de-
3678
The Journal of Physical Chemistry, Vol. 96,No. 9,1992
TABLE I: Correlation Strength Parameters Ai and Zi, Charge Fluctuations ( ( A n ? ) ) , and r-Electron Occupation Numbers ( 4 )in the Nonalternant r Systems 1-4
system
center
1
c, = c4 c2= c3 c5 cl = c4 = c5 = c8 c2 = c3 = c6 = c7
2
= CIO
c9
3
c, = c4 e2 = c, c5 = CIO c6 = c9 e7
= C8
c11
c12 4
c1 = c6 = c7 = c12 C2 = Cs Cg = Cl1 C3 = C4 = C9 = Clo c13
= c14
Ai 0.427 0.349 0.160 0.451 0.396 0.394 0.327 0.377 0.340 0.389 0.342 0.360 0.282 0.276 0.398 0.375 0.384 0.313
zi 0.509 0.406 0.177 1.000 0.481 0.414 0.439 0.529 0.471 0.487 0.434 0.453 0.362 0.377 0.468 0.410 0.398 0.390
((An?)), 0.284 0.324 0.419 0.235 0.299 0.303 0.329 0.303 0.321 0.302 0.324 0.316 0.354 0.353 0.299 0.312 0.308 0.339
(ni)
1.09 1.07 1.05 0.62 1.10 0.98 0.85 1.17 1.16 0.89 0.88 0.89 0.88 1.15 0.92 1.05 0.98 1.11
Bohm and Schutt TABLE Ilk Many-Body Parameters and *-Electron Occupation of the Nonalternant Model Systems 8-12
system 8
center CI = C, = C, = C,
9
10
11
TABLE 11: Many-Body Parameters and r-Electron Occupation of the Model Systems 5-7
system 5
center = c3 = c4 = c2 = c5 cl
c6
c7 = cg 6
C, = C3
c2
= c8 c5 = c7 c4
c6
c9 = G o 7
C l = C5 C6 = Clo C2 = C4 = C7 = C9
c3= c* ell
= c12
A; 0.430 0.310 0.264 0.380 0.345 0.382 0.352 0.364 0.290 0.407 0.335 0.387 0.281
Z 0.625 0.440 0.395 0.538 0.379 0.51 1 0.362 0.468 0.305 0.524 0.425 0.492 0.366
((An?))cn,, (ni) 0.275 0.334 0.353 0.301 0.327 0.303 0.324 0.314 0.355 0.292 0.328 0.302 0.353
0.81 1.17 1.20 1.17 1.05 0.85 0.98 0.88 1.02 1.12 0.88 1.12 0.87
termined by the respective *-electron density. Simply speaking, the localization properties of the T electrons are driven by many-particle interactions. The T correlation energy E,, on the other side, is a correction term to improve the results of the independent-particle description which is sometimes of quite reliable accuracy. This divergent behavior of independent-particle properties can be explained as follows: The electron density based on I+scp) and thus also the associated electronic energy are valid up to first order in perturbation in I+o), while the corresponding fluctuations are not. The many-particle influence on ((An:))” occurs here already in first-order terms. 111. Results and Discussion In the following model calculations a common CC bond length of 140 pm has been adopted throughout. The characterization of the present results is largely simplified by comparing them with the correlation strength parameter Ai = Zi and charge fluctuations ((An:))mrrin a localized T double bond. For ethylene at a hypothetical bond length of 140 pm, we find Ai = 0.438 and ((An:))corr= 0.281.’* These indexes can be accepted as characteristic values for “localized” r electrons. In the Tables I-IV the atomic many-particle indexes Ai, Zi, and ((An;)),, for the nonaltemant model compounds 1-17 are collected; the T densities ( n,) are also summarized for convenience. In the four tables the molecules have been arranged according to the scheme used in Figure 1. Large differences in the electronic charge fluctuations and Ai/2, numbers are predicted for pentafulvene (1). In the employed linearized version of the LA perfect interatomic correlations with Z,= 1.Oare found at the exocyclic carbon site Cs.In combination with (h)the calculated Zi number indicates “mixed valency” with P6(Z) = 0. On the basis of previous computational experienceZ3v3’ we believe that due to the linearization in the determination of the variational parameters the latter many-particle indexes are
12
AI 0.438 0.315 0.405 0.269 0.466 0.336 0.446 0.400 0.287 0.306 0.371 0.342 0.437 0.382 0.351 0.357 0.277 0.279 0.4 19 0.335 0.394 0.416 0.276 0.433 0.320 0.420 0.270 0.268
2, 0.482 0.391 0.640 0.332 0.486 0.386 0.486 0.586 0.359 0.308 0.600 0.389 0.442 0.554 0.373 0.509 0.306 0.286 0.459 0.389 0.432 0.576 0.320 0.457 0.382 0.598 0.316 0.293
((An?))arr 0.281 0.339 0.282 0.362 0.267 0.330 0.276 0.289 0.352 0.350 0.297 0.328 0.281 0.298 0.324 0.312 0.361 0.360 0.290 0.331 0.302 0.284 0.360 0.283 0.338 0.281 0.363 0.365
(ni)
0.95 1.11 0.77 1.11 0.98 1.07 1.04 0.8 1 1.11 0.99 1.24 1.06 1.oo 0.82 0.97 0.82 1.05 0.99 1.05 0.92 1.05 1.16 0.93 1.03 1.09 0.83 1.08 1.05
TABLE I V Many-Body Parameters and *-Electron Occupation Numbers of the Model Systems 13-17
system
14
15
16
17
center
Ai 0.367 0.396 0.389 0.349 0.291 0.275 0.322 0.390 0.359 0.346 0.404 0.376 0.285 0.277 0.337 0.402 0.349 0.421 0.278 0.3 19 0.385 0.396 0.299 0.311 0.391 0.337 0.381 0.285 0.326
xi
0.495 0.429 0.490 0.420 0.327 0.380 0.440 0.494 0.396 0.424 0.445 0.480 0.342 0.350 0.385 0.457 0.387 0.589 0.296 0.414 0.455 0.430 0.302 0.374 0.5 13 0.362 0.450 0.325 0.371
((An?))mr 0.309 0.302 0.301 0.323 0.303 0.353 0.331 0.301 0.320 0.324 0.297 0.308 0.355 0.356 0.330 0.298 0.325 0.28 1 0.361 0.335 0.305 0.302 0.350 0.341 0.299 0.330 0.307 0.356 0.336
(ni)
1.15 0.96 0.89 0.9 1 0.94 1.16 1.16 0.88 1.05 0.90 1.05 1.12 1.09 0.88 0.93 0.94 0.95 1.17 0.97 1.13 0.92 1.04 1.oo 1.09 0.87 1.03 1.08 1.07 0.94
overestimated by roughly 10%. The charge fluctuations at c6 are possible only as a result of the strong deviation in ( n 6 )from 1.0. Zi = 1.0 in combination with ( n , ) = 1.0 causes complete suppression of any fluctuations. The many-particle indexes in compound 1 as well as in all other T models show some interesting “alternancy symmetry”. If not frustrated by the molecular topology, r centers with strong correlations/sizeable atomic electron density localization have neighbors with the opposite many-body indexes. The described oscillations in ((An;)),rr and Ai are always pairwise attenuated or strengthened, i.e., a strong localization center is usually coupled to a neighbor atom with sizeable fluctuations. The many-particle
Nonalternant Hydrocarbon
A
Compounds
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3679
Figure 2, Possible s-electronic substructures in the linearly annelated polycycles 8-11 according to the calculated Ai, Xi, and ((An:)), numbers. The schematized effects are strengthened in structures with bond alternation. The hatched area symbolizes a “connected”s unit, thin lines stand for disconnections and the arrows connect parent fragments.
data for pentafulvene in Table I give a first hint to the high instability of this A system (Le., presence of a highly correlated A center). Fulvalene (2) is the first example which visualizes that correlation strength parameters and charge fluctuation depend strongly on the topology of the considered A center. The A electronic delocalization at sites with three bonded neighbors exceed generally the ((An:)),rr numbers at atoms with two neighbors. A further electron reduction in the ((An:)),rr is predicted at terminal functions with one bonded partner, Le., C6 in 1. In terms of a descriptive interpretation fulvene, 2 can be characterized as a collection of five isolated A bonds. The numerical results for system 3 indicate stronger correlations/larger atomic density localization in the five-membered ring in comparison to the seven-membered one. The heptafulvene results (4) show some analogies to the pentafulvene data. It is common to all model systems 1-4 that the charge fluctuation at a center with two neighbors is confined to a rather narrow interval between ca. 0.300 and 0.325. C,= C4 in fulvene is here an exception caused by the strong delocalization at C5. In Table I1 the computational results for the bicycles 5-7 are compared. Azulene (6) is a stable A system which is frequently characterized as “aromatic” compound.’ Pentalene systems are accessible only by ring substitution leading either to steric protection or electronic stabili~ation.~~ The A electrons at the carbon centers C,, n = 1, 3, 4, 6, in pentalene are sizeably correlated. ((An:)),, is here even smaller than in a localized A bond. Zi at the latter sites amounts to 0.625, a rather high value for A centers in a cyclic structural unit. The many-body indexes of the second 4n A system heptalene (7) reflect in somewhat attenuated form the pentalene data. In Table I11 the computational results for the linearly coupled polycycles 8-12 are summarized. In Figure 2 schematic representations of possible electronic (sub)structures in the many-body level are symbolized. For s-indacene, e.g., two possible “parent” structures are indicated, i.e., a single 12 A annulene unit (left diagram) or two weakly coupled pentafulvene fragments (right diagram). Comparison with the computational results for 1 and 5 indicates that the latter representation can be ruled out. The many-particle indexes AI, Z,,and ( (An:)),rr of s-indacene (8) show analogies to the pentalene data. Rather strong localization centers here are C, with n = 1,3,5,7 and C, with m = 4,8. Like pentalenes, stable s-indacenes require substituents at the corresponding A centers.25 For as-indacene (9) two possible A electron structures are suggested in Figure 2. The fulvalene-like fragmentation (right diagram) is incompatible with the calculated many-body elements. The tricyclic rings 10 and 11 reflect remarkable “memory effects” to their bicyclic parents azulene 6 and heptalene 7, Le., cycloheptv]indenecan be described as an azulene fragment interrupted by a diene unit. On the other hand, the A, and ((An:)),rr numbers in 10 differ principally from the sindacene (8) results. The charge fluctuationsin the fivemembered
I-\
Figure 3. Possible s-electronic structures in the nonalternant polycycles 13-17 according to the calculated A,, Zi, and ((An:)),, numbers; see legend to figure 2.
ring of 10 are larger than the fluctuations in the five-membered ring of 8. The correlation strength parameters A, and Zi and charge fluctuations ((An:)),rr of the polycycles 14-17 are collected in Table IV. Possible substructures within the total Aelectron ensemble are shown schematically in Figure 3. Cyclopent[dJazulene (13) can be decomposed formally into an azulene unit with a weakly coupled A double bond or into a pentalene fragment perturbed by diene network. The computational results indicate that the first substructure is more realistic. “Memory effects” to azulene units are more or less realized in the nonaltemant A compounds 14,15, and 17; see Figure 3. 14 corresponds to a diene-perturbed azulene and not to an ene-perturbed heptalene. In 16 the A electrons remember to their naphthalene parent. Finally we should mention that the described fragmentations are even more pronounced in structures with bond alternation; they are partially attenuated in the numerical results collected in the Tables I-IV, which refer throughout to conformations without bond-length alternation. The ((An:)),, and A, elements collected in the Tables I-IV indicate that the different A centers can be fragmented into characteristic subclasses according to their above many-particle parameters. The latter is in first place dependent on the topology/environment of the corresponding r center. In Figure 4 characteristic charge fluctuations and correlation strength parameters A, for the T centers indicated on the left-hand side of the diagram are displayed. The A functions in nonalternant hydrocarbons are divided into seven classes, A-G. For a comparable classification in alternant A systems, see refs 18 and 20. As already mentioned, maximum correlations in the “mixed valence” regime are predicted at the exocyclic A center of pentafulvene; comparable results are found for other nonalternant compounds with exocyclic units. The atomic *-electron density localization is here of maximum size. Next in the scale suggested in Figure 4 one finds A centers with two neighbors where each is bonded to three adjacent u functions. The enhanced charge fluctuations at the latter sites cause reduced ((An;)),, numbers
3680 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 A-""-
Bohm and Schutt TABLE V: Mean Values of the Correlation Strength Parameters A,v and E.,, as Well as the Electronic Charge Fluctuations for the Nonalternant Hydrocarbons 1-17' system Aav IS Z,, ((An2)),,,,, -E,,/eV status
A B C D E F G H
1 2
3 4
Q"t
00 2150
[
5 6 7 8 cmD
Prcj=0 i
9 10 11 12
13 14
I5 16
17 C,H6 C,&
H
SCF f l u c t u a tions
020
A B C D E F G H
J
Figure 4. Charge fluctuations and correlation strength parameters A, characteristic for the A centers x = A 4 in nonalternant hydrocarbons schematized on the left-hand side of the figure. H refers to the independent-particle indices for on-site r densities realized in the studied A systems (magnitude of the charge fluctuations).
at the reference atoms B. The a centers C are bonded to one neighbor with three partners and one with two T bonds. This topologic arrangement allows for a weak enhancement in the a-electron delocalization at the considered site. A remarkable enhancement of the fluctuations in predicted for class D atoms with direct neighbors bonded to only two carbon atoms. As can be seen in the display this arrangement leads to a rather narrow ((Ant))" width but to a remarkable A, distribution. The latter scattering is caused by larger differences in the on-site a density in the studied nonalternant materials. As already mentioned, a further enhancement in the charge fluctuations is possible at a centers bonded to three neighbors. Lowest in the ((Ant)), scale are bridging centers of type E. The fluctuations are then slightly enhanced at bridging atoms in linearly coupled polycycles. An exception are the type C a AOs bonded to an exocyclic carbon site. The corresponding correlation strength parameter A, is here much smaller than a t other A centers. It is of the same order of magnitude as A, elements in weakly correlated bonds formed by main-group elements.2'~22 The many-particles indexes ((An,2)), and A, in nonaltemant hydrocarbons differ in several ways from their pendants in altemant molecules: (i) The charge fluctuations at weakly correlated a centers with (n,) # 1.O are smaller than in alternant hydrocarbons; this Can be rationalized by adopting eq 11.14 as marginal, i.e., independent-particle, boundary. (ii) The minimum fluctuations ((Anx)2)mina t centers with (n,) # 1.0 in nonalternant materials are always finite; see eq 11.17. (iii) The combined influence of i and ii attenuate the accessible width between ( (An?))mIr and ( (An?))min in nonalternant a hydrocarbons in comparison to alternant ones. In Table V the mean values of the two correlation strength parameters A,/Z,, as well as ( for the studied model systems are collected together with the a-electron correlation energy (normalized with respect to two electrons). Additionally the "instability index" IS defined in the Introduction is given. The averaged charge fluctuation ( (An2) indicates roughly comparable net *-electron delocalization in all studied compounds. Furthermore the corresponding mean values are comparable with the ( (An?)mrr parameters derived for benzene and naphthalene (bottom of Table V), a systems which are frequently adopted as
0.360 0.382 0.348 0.375 0.359 0.352 0.359 0.356 0.373 0.350 0.359 0.354 0.348 0.352 0.350 0.353 0.347 0.354 0.352
0.467 0.412 0.385 0.404 0.437 0.387 0.407 0.433 0.445 0.405 0.418 0.431 0.392 0.396 0.404 0.398 0.391 0.354 0.352
0.500 0.446 0.457 0.420 0.518 0.428 0.459 0.443 0.435 0.430 0.418 0.427 0.428 0.421 0.410 0.393 0.413 0.354 0.352
0.312 0.307 0.320 0.311 0.309 0.320 0.316 0.317 0.311 0.320 0.318 0.320 0.322 0.321 0.323 0.322 0.324 0.323 0.324
0.621 0.671 0.555 0.563 0.553 0.559 0.563 0.552 0.613 0.547 0.564 0.548 0.544 0.562 0.551 0.581 0.545 0.677 0.583
U U S
U S U U
S
S S
S
"In the third column "instability index" (IS) has been given which is the sum of byand the mean-square deviation of the Ai. Additionally the corresponding values for benzene and naphthalene are given in the table (bottom). The ?r correlation energy E, has been normalized to two A electrons. In the last column we have given the status of the corresponding system, either the parent compound or derivatives. s stands for stable, u for unstable, and - for unknown. The following references have been adopted: refs 44-46, 25, 47, 48, 26, 49-51; sequence corresponds to the sequence of the compounds. Reference 49 refers to the A systems 13 and 14.
prototypes of strongly delocalized "aromatic" species. The largest total *-electron localization in the considered nonalternant a compounds is predicted for 1, 2,4,5, and 11. In summary, the *-electron delocalization in many nonbenzoid a compounds is almost comparable with that realized in, e.g., C6H6. This is a rather unexpected many-particle result. Also the mean values of the correlation strength parameter 4, coincide almost perfectly in the considered compounds, Le., the many-body character of the A bonding in 1-17 is comparable with electronic correlations in C6H6 Or C,,Hg. Comparison of the A,, and Z, columns in Table V, however, visualizes remarkable differences within the nonaltemant A systems as well as between the latter molecules and C6H6/CIOHB.Larger differences between the 4, and Z,, in the nonaltemant compounds indicate that the electron correlations cover a larger part of the maximum accessible many-body interactions in comparison to a hypothetical alternant A system ((n,) = 1.0). A,, in pentalene 1 amounts to 0.360. The associated 8, element, however, demonstrates that this corresponds to 50% of the maximum allowed interatomic correlations. In C6H6 the many-body parameter A, = Z, = 0.354 symbolizes directly that only 35.4% of the accessible correlations are exhausted. The normalized a correlation energy is roughly comparable in all considered K materials. Theoretical approaches to quantify a priori stabilities of hydrocarbon a systems have an outstanding history in chemistry. But most of the available techniques fail principally in certain classes of compounds or miss physical/chemical transparence. Early numerical attempts to relate the stability of a ?r system to the total a-electronic energy in the HMO picture lead frequently to dead ends. Normalized a energies of 2.489,2.560,2.614,2.673, and 2.667 in the HMO approximation are calculated for the nonalternant a systems 1, 2, 5, and 6 as well as for C6H6.39 The above numbers refer always to two R electrons and are given in units of Hiickel B or the hopping integrals ti) The R energies of 1, 2, and 5 are too large to explain the instability of the corresponding systems. Comparison with a energies of linear a compounds leads to even larger conceptual difficulties. Historically (39) Heilbronner, E.; Bock, H. Das HMO Modell und seine Anwendung Verlag Chemie: Weinheim, 1970.
Nonalternant Hydrocarbon
A
Compounds
The Journal of Physical Chemistry, Vol. 96, No. 9, 1992 3681
the missing interrelation between T energy and stability has been traced back to ”limitations” of the Hiickel model which, however, never have been quanMied. Also the concept of resonance energies (i.e., the energy difference between a cyclic structure and a hypothetically defined acyclic parent) to estimate stabilities of u systems is not free of some ambiguity caused by the definition of hypothetical reference structures. See refs 40-42 for detailed discussions. Graph-theoretical determinations of resonance energies are nevertheless quite s u c c e s s f ~ l . ~ ~ On the other hand, it is seen in Table V that the suggested instability indexes (IS) correlate well with experimental findings. In the last column we have given the present experimental status for the adopted nonaltemant hydrocarbons. Experimental work is available for 1 1 of the model systems or derivatives collected in Table V. To our best knowledge, studies on the compounds 4, 9-12, and 16 are yet not published. The label s in Table V > has been used to indicate that the correspondingsystem is at least of some stability. Most of the compounds belonging to this class are even remarkably stable; see below. The index u symbolizes unstable, highly reactive nonalternant systems. Of course it will become clear in the subsequent discussion that there is a continuous transition between the two extremes s and u. The suggested subclassificationis nevertheless of some utility. The six s systems in Table V have throughout smaller IS values than the five highly 0301 unstable representatives. For the former u molecules we calculate the following IS elements: 3 = 0.385,6 = 0.387,13 = 0.392,14 = 0.396,15= 0.404,and 17 = 0.398. With exception of system - 82 . 0 c, 0 c, 15, all IS numbers are smaller than 0.400. The corresponding elements are furthermore quite close to the SI indexes of naph0.26; ’ 3’ 4’ 5’ 6‘ 1’ 0 thalene and benzene. The IS numbers for the considered unstable 2 Arlpm systems are generally larger. We find 1 = 0.467,2 = 0.412,5 = 0.437,7= 0.407,and 8 = 0.433. Subsequently we summarize Figure 5. Variation of the charge fluctuations ((An:))m and normalized some experimental results to demonstrate the utility, but also the correlation strength parameter Zi in pentalene as a function of the limitations, of the suggested subclassification. bond-length alternation Ar. For the ‘equidistant” conformation a common CC bond length of 140 pm has been adopted. The central CC bond It is well-known that azulene (6) is stable until 250 OC; also has been kept fixed in course of the bond-length alternation. thermodynamic data of 6 indicate the high stability of this nonalternant hydrocarb~n.~’Azulene-like properties are also reof the a compounds 1 and 2, finally, has been known for many alized in the polycyles 14, 15, and 17. Diels-Alder reactions in On the basis of the present IS elements we expect that system 14, e.g., occur under conservation of the azulene ~tructure.4~ the yet unknown u molecules 4,10, and 16 should be synthetically This is consistent with the fragmentation displayed schematically feasible and fairly stable. The “stable” u substructures is the latter in Figure 3. The azulene u structure is also conserved in the two compounds have already been symbolized in Figures 2 and catalytic hydrogenation of system 15 with Pt/H2 in C2H50H.50 3. High reactivity and large instability is predicted for 9, 11, and The a compounds 3 and 13 are less stable than the other four 12. examples discussed above. For 3 this is obviously a result of the Finally we should emphasize that the suggested approach is repulsion between the hydrogen atoms at C1, C4, C5, and Cloin only a first step to describe/estimate stabilities of u systems in the planar conformation. Compound 13 polymerizes in the terms of many-body elements. The IS indexes combine several presence of acids and with increasing temperat~re.4~ Just the latter pieces of theoretical information. The underlying Aavnumbers polycycle is neither very stable nor sizeable unstable. characterize the net correlation strength in the u system and thus In the class of unstable nonaltemant u systems 5,7, and 8 have the global localization properties of the electrons. The standard focused considerable interest. As already mentioned derivatives deviation ui in IS gives weight to the intramolecular scattering of 5 and 8 are accessible only as sterically protected u networks or by chemical modifications in their electronic s t r u c t ~ r e . ~ ~ . ~of~the charge fluctuations and many-particle interactions. Molecular r systems with strong atomic electron density localization Synthesis of heptalene (7), on the other hand, has been possible.@ are intrinsically unstable. By means of the atomic ((An?)),, Nevertheless it is a highly reactive compound. The different or Ai parameters these centers can be identified and the design stabilities of 5 and 8, on one hand, and heptalene (7) on the other of properly substituted molecules should be simplified. The present hand, is also reproduced by the IS indexes in Table V; i.e., we data visualize also the limitations of one-electron models (i.e., have to compare 0.437and 0.433 with 0.404. The high instability Hiickel u electron energies) to quantify electronic structure
’ -
(40) (41) 1692. (42) (43) (44) (45) (46)
Aihara, J.-I. J . Am. Chem. SOC.1976, 98, 2750. Gutman, I.; Milun, M.; Trinajstic, N. J. Am. Chem. SOC.1977, 99, Zhou, Z.; Parr, R. G. J . Am. Chem. Soc. 1989, 1 1 1 , 7371. Zhou, Z.; Parr, R. G.; Garst, J. F. Tetrahedron Lett. 1988, 4843. Thiec, R.; Wiemann, J. Bull. Chem. SOC.Fr. 1956, 177. Badger, G. M.Aromaticity; University Press: Cambridge, 1969. Koerner von Gustorf, E.; Henry, M. C.; Kennedy, P. V . Angew.
Chem. 1967, 79,616. (47) Heilbronner, E. In Non-benzenoid aromatic compounds; Ginsburg, D., Ed.; Interscience: New York, 1959. (48) Dauben, Jr., H. J.; Bertelli, D. J. J. Am. Chem. Soc. 1961,83,4657. (49) Hafner, K.; Schneider, J. Jusrus Liebigs Ann. Chem. 1959,624, 37. (50) Reel,H.; Vogel, E. Angew. Chem. 1972, 84, 1064. (51) Anderson, Jr., A. G.; MacDonald, A. A,; Montana, A. F. J. Am. Chem. SOC.196&90,2993. Jutz, C.; Schweiger, E. Angew. Chem. 1971,83, 886.
properties of u systems. In this representation only the hopping is taken into account. The present many-body indexes, however, refer to ratios formed by hopping and two-electron interaction. IV. Bond-Length Alternation ia Pentalene In the Figures 5 and 6 we have shown schematically the above many-particle elements of pentalene as a function of the bondlength alternation Ar. In Figure 7 the Ar dependence of the u-electron population ( n i ) is given. The data shown several unexpected results. It has been realized experimentally that most of the synthesized pentalene descendants have structures with remarkable bond altemati0n.2~In chemistry there exists an almost universal agreement to identify structures with bond-length alternation as “localized structures”. The latter label is then a synonym for increasing localization of the electrons. In this context
Bohm and Schutt
3682 The Journal of Physical Chemistry, Vol. 96, No. 9, 1992
w
O
/
O&O&, 035
0 55
0
0
- -1 - - 2
3
,
. 3
4
5
6
6
1
2
3
4
5
0
1
2
3
4
-
1
L
10312
0 310 7 8 Arlpm
7 8 Arlpm Figure 6. Variation of the mean values A,,, E,,, and ((An2)),,,,,, in pentalene as a function of the bond-length alternation (top diagram). In the bottom diagram the variation of the A correlation energy is schematized; the numbers are normalized with respect to two A electrons. Full curve, only interatomic correlations; broken curves, correction due to intraatomic A correlations.
5
6
7 Arlpm
8
Figure 7. On-site A density ( n i ) in pentalene as a function of the bondlength alternation &.
a general comment seems to be appropriate. On the basis of general knowledge on the electronic origin of chemical bonding52 there is no reason to relate energetic stabilization to increasing electronic localization; covalent bonds assumed. Lowering of the electronic energy requires increasing interatomic sharing. To explain the latter theoretical elements in a form convenient to chemists, we make use of the well-known Heisenberg uncertainty principle 6,6, 1 tr. 6, is the momentum uncertainty and 6, the spatial uncertainty. Decreasing charge fluctuations are now equivalent to increasing electron localization, corresponding to decreasing 6, parameters. The uncertainty principle requires then an enhanced uncertainty in the momenta. As a result larger momenta become more probable and the kinetic energy is enh a n d . The 1:l correspondence between large charge fluctuations and high stability is thus evident. The numerical results in Figure 5 indicate that the charge fluctuations at C1 = C4of pentalene are sizeably enhanced with increasing bond alternation &, the atomic r-electron density localization at the corresponding centers is reduced. Increasing ((An;)),,, values with increasing Ar are also characteristic for the bridging atoms C7and C8.The net effect of the bond-length
alternation is displayed in Figure 6 in terms of the ((Anz))wmav, and E,, paFameters. On one hand, the r-electron fluctuations are an increasing function of Ar in the considered interval. As explained above, this leads to an increasing stabilization of the system. As,,, on the other hand, remains roughly constant while the normalized averaged correlation strength parameter Ea, is reduced. This is in the first place a result of the mutual assimilation in the electron density (n,)as a function of Ar. In the last section we have given the numerical explanation leading to such a relative variation of $,,and Ea,,. In the Dlh structure of pentalene the interatomic T correlations comprise a larger part of the maximum allowed ones. The accessible upper limit is enhanced with increasing bond alternation. As the numerical outcome, one finds decreasing 2,, numbers with increasing Ar. In contradiction to generally accepted interpretations, it has been demonstrated in the present work that bond-length alternation in T systems with sizeable electronic correlations does not lead to an increasing localization of the electrons measured in terms of the charge fluctuations in the correlated ground state I!&). We believe that the above results contribute to the microscopic explanation for the distortive nature of r The enhancement of the charge fluctuationswith increasing Ar elements is an increasing function of the correlation strength. Bond-length alternation is irrelevant as enhancement factor for ((An;)),m in weakly correlated bonds. The D6h structure of benzene, for instance, is forced by the superimposed u frame. In pentalene, however, the r-electronic correlations are so strong that they determine the structure; the u-electronic effect is here overcompensated for. The interrelation between the tendency of bondlength alternation in hydrocarbon T systems and the gap between the occupied and virtual one-electron functions ci, which depends of course on the correlation strength, has been known for a longer time.s3 The explanation given in the latter work, however, is microscopically incorrect. Analysis of the r correlation energy in Figure 6 (lower diagram) provides a further interesting result: the r electronic correlations are a driving force for the bond alternation in pentalene. Similar results have been evaluated in the physical community for polya ~ e t y l e n e ~ Jand ~ - ~by~ the present authors in some alternant polyenes and mono cycle^.'^^^^ The coincidence of increasing correlation energies IEw,l and increasing charge fluctuations/ decreasing correlation strength parameters is not expected a priori. Enlarged IEwmlnumbers are frequently equated with increasing localization of the electrons. The observed lEoorrland ((An?))” variation as a function of Ar is a consequence of the fact that the charge fluctuations depend on the ratio between the interaction and the HF hopping, while IE,,,l depends on the interaction square. The Ar dependence of the charge fluctuations is driven by the kinetic denominator formed by ( OiH04)cintegrals. The IE,I variation, however, is determined by the interaction square (OiH),Z* To guarantee that the 1EwJ curve in Figure 6 is not modified principally by intraatomic correlations, we have evaluated a second curve (dashed line) with inclusion of the latter many-particle interactions. In previous work of one of us8*33,35 an atom-inmolecules was used to determine intraatomic correlation energies in combination with a simple valence basis. The intraatomic correlations are thereby calculated separately. In this is expressed as level of sophistication
The probabilities Pi(n? to find in n’electrons at the center i have already been introduced in section 11. n‘differs from the above n numbers due to the presence of o-type orbitals. The X ( 5 3 ) Binsch, G.; Heilbronner, E.;Murrell, J. N. Mol. Phys. 1966,11,305. Heilbronner, E.; Yang, Z.-Z. Angew. Chem. 1987, 99, 369. (54) Lievin, J.; Breulet, J.; Verhaegen, G. Theor. Chim. Acta 1981, 60, 339.
(52) Ruedenberg, K. Rev. Mod. Phys. 1962, 34, 326. Kutzelnigg, W . Angew. Chem. 1973, 85, 551.
(55) Schmidt, M. W.; Lam, T. M. B.; Elbert, S.T.; Ruedenberg, K. Theor. Chim. Acta 1985,68,69. Lam, B.; Schmidt, M. W.; Ruedenberg, K. J . Phys. Chem. 1985,89, 221.
J. Phys. Chem. 1992,96, 3683-3688 summation is over the X possible electronic configurationsat center i. For n' = 4 the configurations s2p2and sp3 will have different weights at a given C atom. Determination of the W,(n',i) requires an analysis of I$,) with respect to the different atomic terms. The czn(i), at least, are tabulated atomic correlation energie@ for atom i (here only C sites) with n'electrons in configuration A. Previously it has been shown that eq IV.l can be reproduced almost quantitatively by simple interpolation f o r m ~ l a s ? ~@:F(i) J~ in eq IV.2depends only on the density (4') of the valence electrons and the hybridization parameter ri = ( n p ) / (n:) with ( n p ) symbolizing the net electron density in p AOs:
@gy(i)= -(0.119
+ l.020r[)(n,')1.23
(IV.2)
In the numerical determination of the intraatomic correlation energy in pentalene as a function of Ar', a Ar'hdependent u ribbon with sp2 configuration has been assumed. The corrected curve in Figure 6 indicates that the intraatomic correlations are without decisive influence on the shape of the correlation energy curve. Only the interatomic u correlations are one driving force for the bond-length alternation.
V. FmIRemarks of The charge fluctuations in the correlated ground state nonalternant hydrocarbons have been adopted as quantitative measure for the atomic density localization of (u) electrons. The theoretically transparent marginal boundaries are (i) the independent-particle limit with maximum mean-square deviations of the electronic charge around the respective mean value (n,) and (ii) the perfectly correlated regime with minimum fluctuations. The additional charge degrees of freedom in nonalternant u systems ( ( n , ) # 1.0)relative to alternant ones with a common ( n , ) of 1.0at any center i cause an attenuation of the accessible . densities ( n , ) # 1.0 in the weaker width of the ( ( A n t ) ) w r r A correlated regime reduce the charge fluctuations in comparison to the ( ( A n t ))wn numbers in alternant materials. On the other hand, nonvanishing residual fluctuations for perfect interatomic correlations are possible only in nonalternant u compounds. It has been demonstrated that the u electrons in the studied model systems are sizeably correlated and by no means free-electron-like. The calculated ( (Ant))wrrand Ai/& numbers provide characteristic topologic and nearest-neighbor information on the considered u center. This local property allows for the evaluation of general ordering principles concerning the importance of
3683
many-particle effects at the different u centers of a molecule. On the basis of the averaged correlation-strength parameters = (C;"P,A,)/mand the associated standard deviation ui, a new compound index has been suggested to estimate (in)stabilities of hydrocarbon 7r systems. In contrast to previous one-electron elements, i.e., total Huckel u energies, the many-particle character of u-electron bonding is explicitly taken into account. And in contrast to 'resonance energies", which are always based on energy differences between the considered compound and a hypothetically defined reference structure, our suggestion is independent of such references. Only information on the studied systems is necessary. Large standard deviations ui between the correlation strength parameters Ai are an indicator of strong u-electron localization centers in the studied hydrocarbons. The latter centers should cause the observed instabilities of the corresponding molecules. Therefore the above Ai or ( ( A n t ) ) , numbers are a transparent local probe to understand electronic structure effects in u systems. An analysis of the bond-length alternation in pentalene has shown that a-electronic correlations are one driving force for the transition from the equidistant Dzhstructure to the C, one. For this 4n annulene we have visualized that increasing bond-length alternation is accompanied not only by increasing correlation energies IE,I but also by enhanced charge fluctuation of the u electrons allowing for a reduction of the kinetic energy. Simply speaking, pentalene and also other more strongly correlated hydrocarbons with conjugated u units prefer bond alternation to maximize the delocalization of the u electrons. The latter behavior differs principally from the modulation of IEJ and ( (An;))mrr in 'two-center" systems with a single distance coordinate r'. Increasing interatomic separations are accompanied by increasing IE,I numbers but decreasing charge fluctuation. The origin lies in the reduction of the SCF, Le., independent particle, amplitude in I+,,) with increasing r'. Correlations tend to localize the electrons. In the pentalene problem the mean value of the CC bond length has been kept fixed. Only Ar' has been modified. Under these conditions it may be possible that IEwnl and ( (Ant))wnshow a mutual Ar'variation, which is surprising on the first sight.
Acknowledgment. This work has been supported by the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie. M.C.B. acknowledges support through a Heisenberg grant. The drawings have been kindly prepared by Mr. G. Wolf. Re&@ NO. 1,497-20-1; 2,21423-86-9; 3,51905-28-3; 4. 831-18-5; 5, 250-25-9;6, 275-51-4;7, 257-24-9;8, 267-21-0;9, 210-65-1;10,
(56)Verhaegen, G.;Moser, C. M. J . Phys. B 1970,3,478.Declaux, J. P.; Moser, C. M.; Verhaegen, G. Ibid. 1971,4, 296.
270-19-9;11,259-63-2; 12,257-56-7; 13,94386-70-6; 14,209-42-7; 15, 352644-3;16,187-78-0;17, 193-85-1; C6H6,71-43-2; CIOHB,91-20-3.
Alkali-Metal Dihallde Molecules Ju Guan-Zhit and Emest R. Davidson* Department of Chemistry, Indiana University, Bloomington, Indiana 47405 (Received: October 1 , 1991) This paper presents the optimized structure parameters, harmonic vibrational frequencies, and energies calculated using the UHF, MP and CI theoretical models at the 6-31G*,6-311G**,and 6-311+G**basis sets for MX2 molecules (M = Li, Na; X = F, CI). According to our calculated results, UHF calculations cannot determine that the C, configurationsare most stable for MF2 (M = Li, Na), but they can for MC12,and the post-UHF calculations can for both MF2 and MC12. All of these MX2species are predicted to be stable relative to the M + X2and MX + X asymptotes, even though the reaction of M and X2 to form MX is strongly exothermic. Introduction
The reactions of alkali metal with halogen molecules have studied experimentally since the 1930s.' Recently, using the EPR spectra: and infrared crossed molecular beam Present address: Institute for Theoretical Chemistry and Department of Chemistry, Shandong University, Jinan, People's Republic of China 250100.
0022-36S4/92/2096-3683$03.00/0
and Raman spectra,'+ chemists have obtained results which constitute an impressive case for the existence Of MX2 'molecules", (1) Polanyi, M.; Schay, G. Z.Phys. Chem. B 1928,1, 30. (2) Maya, J.; Davidovits, P.J . Chem. Phys. 1974,61, 1082. (3) Grice, R. In Aduonces in Chemical Physics; Lawley, K. P., Ed.; Wiley: New York, 1975;Vol. 30,pp 247-312.
0 1992 American Chemical Society
