J. Phys. Chem. 1993,97, 11427-1 1434
11427
Correlation Effects and Charge Fluctuations in T Systems. A Combined PPP and Localized Many-Particle Analysis Michael C. Bohm,’ Udo Schmitt, and Johannes Schiitt Institut fur Physikalische Chemie, Physikalische Chemie III, Technische Hochschule Darmstadt, 0-64287 Darmstadt, Germany Received: June 28, 1993”
A local correlation approach is employed to elucidate the many-particle character of the a electronic system in hydrocarbons, heterocompounds, and a collection of other a networks. The self-consistent-field (SCF)wave function I+scF), which is the prerequisite to derive the correlated ground state I+o), is calculated in the PariserParr-Pople (PPP) scheme. The matrix elements occurring in the SCF precursor as well as the many-particle part are calculated ab initio in a minimal basis of Slater-type atomic orbitals (AOs). In the determination of I+o), singly and doubly excited configurations are considered. The former ones optimize the one-particle density beyond the results from the SCF calculation. The atomic n-electron density localization is measured by the mean-square deviations of the electronic charge at center i ((An?))” around the corresponding mean-value ( n i ) . The many-particle character of a electron bonding is quantified by two correlation-strength parameters Ai and Zi, mapping the reduction in the charge fluctuations ((An?))mrrdue to electronic correlations. It is shown that the a electronic localization properties sensitively depend on the topology of the considered a center and the type of the atom. a systems are generally far from the independent-particle limit. Stronger electronic correlations lead to sizeable atomic localization of the a electrons. Heteroatoms on the right-hand side of C within the periodic table cause enhanced a electron correlations. Bond-length alternation is accompanied by an enhancement of the electronic charge fluctuations ((An?))”; the delocalization of the a electrons is thereby strengthened. This ( (An?))wrr enhancement as a function of the bond-length alternation is an increasing function of the correlation strength operative in the a network. I. Introduction The electronic structure of a systems has been studied by chemists and physicists for more than 60 years now. Especially hydrocarbon a networks have focused the interest of the scientific community. From the experimental branch this has been caused by a variety of synthetically accessible molecules which show a broad spectrum in their chemical properties. From the theoretical branch it has been a result of the uniform atomic composition facilitating comparative electronic structure investigations. If only the a subspace is taken into account, the electronicproperties of hydrocarbons are exclusively determined by a single a atomic orbital (AO) at each carbon site. For many years it had been assumed throughout that the A electronic structure of these compounds can be described adequately by free-electron or independent-particle models. Three of the most popular representatives are the Hiickel molecular orbital (HMO) approximation,’ the perimeter model of Clar and Platt,2 and the freeelectron (FE)approach of Kuhn.3 In the present context we will employ the common symbol ~+SCF) to label any wave function based on a single Slater-determinant. Since we are concerned with the pair properties of the a electrons only, details of the single-determinantal scheme are irrelevant. In this regard there is no discrimination between a simple one-electronapproximation of the HMO-type and an elaborate self-consistent-field (SCF) calculation of the Hartee-Fock (HF) type. In the past 15 years a number of theoretical contributions have been published, which unambiguously have demonstrated that the “exact” a-electronic ground state I+o) sizeablydiffers from any one-electron or singledeterminantal wave function I+SCF) .4-”J Electronic correlations are by no means negligible in these systems. Quite generally they are stronger in a bonds than in the u bond of a considered element combination. As a result of the considerable many-body interactions in the a subspace the corresponding electrons show remarkably high atomic localization. ~~~~~
~~
~
~~~~~~~~
*Abstract published in Advance ACS Absrracrs, October 1, 1993.
In previous theoretical contributions we have analyzed the localization properties of a electrons in many hydrocarbon compounds by many-particle quantities with transparent physical meaning and well-defined boundary~alues.11-~4The mean-square deviations of the a electronic charge at the ith atomic center ((An?)) around the respective mean-value ( n i ) have been used to measure the atomic delocalization of the a electrons. In eqs 1.1 and 1.2 the electronic charge fluctuations are defined in the
single-determinantal approximation (+SCF) and the correlated ground state In eq 1.1 the symbol (...) has been adopted to abbreviateexpectation values referring to the one-determinantal ground state J+scF); in short, (..) = ($SCFI ...I+SCF). This convention will be used throughout the manuscript. ni in the above definitions symbolizes the a electron number operator at center i; ni = ,Erniuwith ni, = ai,+ai,. u stands for electron spin t or 1. The ai,+, ai, are the conventional creation and annihilation operators for the a electrons at center i. Increasing electronic correlations lead to a suppression of the electronic charge fluctuations ((An?))wm,which reach their maximum in the independent-particlelimit ($scF).It is intelligiblethat the strength of the charge fluctuations determines the probabilities Pi(n) to find n = 2, 1, 0 electrons at the ith A center. Increasing twoelectron repulsions cause a reduction in the probability of double occupancy Pi(2). In alternant hydrocarbons15 without excess charge and a common a density ( n i ) of 1.0 at any a center i, it is possible at least in principle to suppress the charge fluctuations completely. The correlated ground state l+o) is then only defined by “covalent” configurations Pi( 1) = 1.0 (single occupancies). Then Pi(2) = Pi(0) = 0 must hold (prerequisite: alternant and neutral hydrocarbons). For a densities ( n i ) # 1.0, which are
0022-3654/93/2097-11427%04.00/0 0 1993 American Chemical Society
11428
Bahm et al.
The Journal of Physical Chemistry, Vol. 97, No. 44, 1993
accessible in nonalternant molecules and heteroderivatives, ((An?)) = 0 cannot be realized even for perfect n-electron correlations. In eq 1.3 we define a correlation-strength parameter Ai by relating the electronic charge fluctuations in the “exact” ground state I+o) to the ones realized in (+scF). Ai = 0 is obtained at the
independent-particle limit, Ai = Af““ for perfect interatomic correlations. The latter are feasible in a minimal A 0 basis. For more detailed information concerning this type of many-body interaction we refer to our previous work.l”1* Af““ is given in eqs 1.4 and 1.5 in terms of matrix elements Pii, Dii, which are expressedineqsI.6 and 1.7. ThePuin (1.6) aresimplytheelements
A?
= Pii/Dii = Dii/Pii
for 0 I( n , ) I1.0
(1.4)
for 1.0 I( n i ) 5 2 . 0
(1.5)
R Dij = ( airaid)
(1.7)
of the first-order density matrix per spin direction and the Dij refer to the virtual A 0 subspace. In ZDO (zero-differential overlap) models, see below, we have Di, = 6, - Pij. In addition to the correlation strength parameter Ai, which explicitly depends on the r density ( n i ) according to eqs 1.4 and 1.5, it is convenient to define a density invariant delocalization parameter Zi with allowed values between Zi = 0 (independent-particle limit) and Zi = Zf““ = 1.0 (perfect interatomic correlations):
w Figure 1. Studied 7r systems 1-40 together with the employed atomic numbering scheme. Note that the adopted enumeration sometimes differs from the conventionalstandard in order to simplifycomparisonsof related molecules. Only the molecular topology is indicated. The non-carbon centers are indicated by the corresponding symbols.
Our recent theoretical investigation~ll-l~ on hydrocarbon ?r a Pariser-Parr-Pople (PPP) H a m i l t ~ n i a nfor ~ ~the ~ ~derivation * systems have demonstrated that the many-body parameters Ai, of I+SCF) in the present investigation. Comparison with available Zi are highly specific atomic probes describing the strength of ab initio calculations has shown that this setup leads to *-electron electronic correlations as a function of the position of a specific densities of sufficient accuracy. Thereby we expect our present rcenteranditsneighborrelations. For investigationsof ((An?)), analysis of nonalternant hydrocarbons is to be more accurate Ai, Z ielements in other types of chemical bonds see refs 16, 17, than our previous study.13 To conserve the enhanced numerical and 20. Theoretical basis of our previous work on the correlated quality in the J+SCF) precursor also in the many-body part of our ground state of T hydrocarbons has been the method of the local theoretical work we have improved the description of the “exact” approach (LA), a many-particle model where the local nature of ground state I+o) in comparison to recent work. In addition to the correlation hole is explicitly taken into a c c o ~ n t . ~ . ~ ~ - ~ ~ density-density correlations described by two-particle excitations It is the purpose of the present work to extend our theoretical we have now considered single-particle excitations to optimize investigations on localization properties of ?r electrons and the the one-particle density in I+o). associated many-particle character of a-electron bonding to The organization of the present manuscript is as follows: In systems containing alternant and nonalternant hydrocarbons, section I1 we describe the PPP Hamiltonian for the derivation of heterosystems with B, N, 0 atoms and inorganic ?r networks (9, I+SCF) together with the adopted LA formalism. Computational 13,20). In Figure 1 the ?r systems studied below are portrayed. results are discussed in section 111, and final remarks are given Additionally we have considered model systems (10, 11, 12) in the last section. employedin recent work intended to analyze the question, whether or not “delocalization” is a driving force in chemistry.2’26 The 11. Theoretical Background models 10-12 formed by six-membered monocycles have been We start with the description of the PPP Hamiltonian to derive adopted by Shaik et al.25 to discuss this problem. Finally it can the SCF precursor (+scF), the knowledge of which is the be seen that we have selected a number of molecules with prerequisite for a subsequent many-body calculation. In eqs 11.1interesting *-electronic properties which have been studied in 11.4 the matrix elements of the mean-field operator in the ZDO large detail in the past years in the chemical community. The basis are s u m m a r i ~ e d . ~In~ .the ~ ~ first two expressions spin heteroatoms encountered in the model systems of Figure 1 summation has been performed. These equations are of the prevented the application of our recent strategy to investigate the electronic correlations in A hydrocarbons, where we have used “closed-shell” type (index cl) with denoting the ikth element of the “closed-shell” first-order density matrix. The spin-free P i k the simple HMO model to derive I+scF). In alternant ?r systems occurring in eqs 11.3 and 11.4 have been defined in the previous this is a sufficient formalism, as the one-particle density is completely determined by symmetry. To facilitate trustworthy section. The given spin-dependent expressions of the SCF Hamiltonian are necessary in the evaluation of the correlated calculations on the .rr systems collected in Figure 1, we have used
The Journal of Physical Chemistry, Vol. 97,No. 44, 1993 11429
Charge Fluctuations in u Systems
The underlying SchriMinger equation reads
The index c is the last element of eq 11.10 symbolizes that the linked-cluster theorem has been employed to derive theexpectation value E.32 This implies that only connected diagrams have been taken into account. Equation 11.10 is thus properly normalized. Evaluation of the expression is not possible without further approximations. In the present work we replace exp(S) by (1 S), which is valid in the case of not-too-strong electronic correlations. This degree of sophistication furthermore corresponds to the numerical conditions accepted in our previous work on hydrocarbon u systems.’1-14 With the above simplification the electronic energy E can be written as
+
ground state; see below. The single-site “core-integrals” hii occurring in the diagonal elements of the Fock operator have been approximated by valence orbital ionization potentials (VOIPs) for p electr0ns.2~ They are the only experimental parameters adopted in the present PPP model. All other integrals have been calculated ab initio in a minimal basis of Slater-type AOs. Bums rules have been adopted to define the atomic screening constants.30 The V, and v k symbolize on-site and intersite two-electron repulsion integrals and the tik stand for kinetic hopping elements. All matrix elements have been calculated according to formulas of Roothaan for a-type symmetry.” &abbrevia@ thecorechargeat center k. In thesystems we have studied z k = 1 does hold at any atomic site k. This leads to a net molecular charge q of -1.0 in the boron ring 14 and to q = +1.0 in the oxygen ring 15. In short, both examples are six u-electron systems. The method of the LA has been described in previous contributions in connection with ab initio formulations and in combination with semiempirical Hamiltonians of the ZDO type.53-23 Therefore it suffices to summarize the basic principles of this efficient and physically transparent many-body formalism. For a previous implementation under inclusion of single-particle excitations, see ref 16. In eq 11.5 the correlated ground-state wave function I$o) is written as where
s = - C t i n i - CvVoij i
(11.6)
E=
+
with EscFdenoting the mean-field energy and E, the interatomic correlation energy. The latter is defined in eq 11.12.
In eq 11.12 no matrix elements of the type (Hni),between the one-determinantal ground state I$SCF) and singly excited configurations occur as a result of the Brillouin theorem.33 In short, I$SCF) already is optimized with respect to the one-particle density. Variation in (11.12) leads to the set of linear equations expressed in (11.13):
A convenient representation of the solution of the above system of equations is given in the subsequent expression
ij
describes local interatomic correlations. The variational parameters & refer to the optimization of the one-particle density in I$o) via single-particle excitations. The ni are the associated number operators; see section I. The variational parameters vi, measure the reduction of unfavorable configurations in I$SCF) where the probabilityof double occupancyPi(2) is overestimated. The 0, are the associated projection operators. In analogy to our previous work on a m o l e c ~ l e s ~we ~ -have ~ ~ restricted the operator manifold to diagonal projectors Oii Oi. In eq 11.7 we
ai= 2niinit
(11.14) with the matrix ‘k
The different elements in (11.14) and (11.15) ((OiHOj)c etc.) symbolize blockmatrices of dimension i,jor i, k. The correlation energy E, can be written in the form of eq 11.16, which shows
(11.7)
define intraorbital projection operators Oi by the a-electron number operators ni, given in the foregoing section. The irreducible parts of the Oi are the Oi encountered in eq 11.6. The Oiand Oiareinterrelatedviaeq 11.8. Themeaningofthebrackets
oi= ai- (Oi)
(11.11)
(11.8)
in the second term on the right-hand side already has been explained in section I. The variational parameters ti and are determined by the variation of the electronicground-state energy E
that E, is expressed in terms of additive contributions caused by theintraorbital projectors Oi. Inversion of the matrk’f requires its nonsingularity. If this condition is not fulfilled the operators 0,and nk are not independent of each other and describe the same process. Under these circumstances the corresponding qii, t k are set equal to zero and the properly reduced T is inverted; see below. The different matrix elements occurring in the above manybody equations are determined as follows: (0iH)c = (0iHra)c = 2CVmpi$in$’ijPin
(11.17)
mSn
As can be seen the elements (OiH), are here exclusively
determined by the residual interaction of the u Hamiltonian,
B6hm et al.
11430 The Journal of Physical Chemistry, Vol. 97,No. 44, 1993 which contains the a-electron interaction beyond the mean-field approximation: ( 0iHoj)c = (0iFOj)C
= 4 ~ i p i j x ~ m n ( p , ~ i -s j n m,n
Pi,pjIpij) (11.18) In eq 11.18 we have approximated the a Hamiltonian H by the respective mean-field operator F,which corresponds to a secondorder perturbation expansion in the framework of the LA. This simplification is no longer possible in the derivation of the remaining matrix elements containing single-particlecorrections. For these terms we have (OlHnk)c = (OiHr-nk),
4x
Vmn[Pi$in(PikDi#mk
( OiHresnk)c
+ (OiFnk),
(11.19)
- DikPizmk) +
m,n
p,mDim(pi$ipnk - DikPi$nk)l (0iFnk)c =
xFmn(P#i$&ik)
(11.20) (11.21)
m.n
For the correction due to single-particle excitations we find (niHnk)c = (nfHr=nk)c + ("iFnk)c (niHreank)c = m,n
Vmn(PimDignkDnk + PipizmkDmk) (11.23)
repulsion in I+o) (=positive variational parameter qi) leads to an enhancement in the probabilityof singleoccupancy. Unfavorable doubly occupied configurationsPt(2) are suppressedaccordingly. a-electron number conservation then implies a reduction in the probability of zero occupancy. At theendofthissection wesummarizeallbondlengthsadopted in the studied model systems (see Figure 1). Bond-length alternation is measured relative to these numbers. The data correspondto standard elementsor to numbers adopted in previous model investigations.25 All values are expressed in picometers: CC = 140, CB = 144, CN = 134, CO = 132, N N = 129, BN = 144, LiLi (in Lib) = 277, HH (in Ha)= 99. For the studied two-center a bonds in CzH4, H2BNH2, CO experimental bond lengths have been adopted additi0nally.3~ 111. Results and Discussion
Subsequently we collect the formulas for the charge fluctuations ((An?) ) and a-electron localization parameters Ai already introduced in the foregoing section. In eqs 11.25 and 11.26 we (11.25)
summarize the electronic charge fluctuations observed in the independent-particle approximation I+~cF) and the correlated ground state I+o). In (11.26) we have neglected coupling terms between the different correlation operators; see also our previous work.11-14 TheprimedmatrixelementsP'ii,D'iiinthelast equation symbolize that the corresponding quantities are derived for the correlated wave function, where the density matrix is reoptimized via single-particle excitations. This reoptimization with respect to the single-particle excitations is possible as long as single and double excitationsare linear independent and commute with each other. In the case of noncommutativity of the different types of excitationsit would become necessary to calculate the correlation parameters by using explicit coupling terms connecting the respective excitations. In the present study we have considered corrections linear in (k. The P'ik confined to [,bo) read Pk:
= Pfk - ctm(pk$im m
+ pi$km)
(11*27)
Within this theoretical setupeqs 1.4 and 1.5 defining the maximum accessible density-dependent correlation-strength parameters AY read
,y = P;,/D:, Ay = D',,/P:,
for 0 I (n;.)I 1.0
(11.28)
for 1.0 S (n:) 5 2 . 0 (11.29) The (n;) are the corrected on-site a densities calculated in I+o). In eqs 11.30-11.35 we define the probabilities P,(n) for double, single, and zero occupancy (n = 2, 1,O) in the ith a A 0 in J+SCF) and I+o). Equation 11.34 clearly indicates that two-electron
To make the subsequent discussion more transparent, we have divided the studied model systems into characteristic subgroups as displayed in the different lines of Figure 1. The compounds 1-3 are examples for homopolar (1) and heteropolar (2,3) twocenter two-electron bonds. The simplest n-electron T systems considered with n > 2 are the linear and cyclic four and six electron molecules &7. Together with the two-center species 1-3 their theoretical results are summarized in Table I. Then we analyze the many-particle quantities of the six-ring systems 8-13; see Table 11. As already mentioned 9-12 are model compounds employed in previous ~ o r kto~study ~ ,'delocalization" ~ ~ as driving force in a-electronic networks. Closely related to the latter set of a molecules are the heterorings 14-19. Together with the benzene and borazole data they are considered in Table 11. In Table I11 the many-particle parameters for the bicyclic rings 20-22 are summarized. Then we discuss the nonalternant hydrocarbon compounds 23-32; seeTable IV. Thenext subgroups of the a systems is provided by the azapentalenes 33-37, the numerical results of which are collected in Table V. Finally we study the correlated ground-state properties of the azafulvenes 38-40 (Table VI). In Table I we have summarized the many-body elements ((An?)),,, Ai, Z,for the 'localized" (=two-center) a bonds in systems 1-3and in thea molecules 4 7 . Note that in thealternant hydrocarbons 1,4,5, and 7 A, = Z,always does hold. The charge fluctuations in ethylene at its equilibrium bond length amount to0.358, avaluethat exceedsourprevious ((A$)),parameter by ca. 15%.12 This can be traced back to the improved calculation of matrix elements in (II.18), which now containsan SCFoperator of the PPP type. In ref 12 the corresponding quantities have been calculated in the HMO scheme. Table I clearly indicates the reduction in the electronic charge fluctuations with increasing length of the"loca1ized" double bonds in 1-3. The atomic density localization of the electrons is thereby enhanced. We have considered the equilibrium bond length of the a systems 1-3 together with a geometry showinga 7-pm bond-length elongation. It can be seen that the electronic localization as a function of the internuclear separation critically depends on the typeof the atomic
The Journal of Physical Chemistry, Vol. 97, No. 44, I993
Charge Fluctuations in u Systems
TABLE k Charge Fluctuations ((An?)), and Correlation Strength Parameters AI in the Model Systems 1-7’ compound atom i ((An?)), Ai zi 0.358 0.283 0.283 1 c, = c2 0.315 0.315 1 CI = c2 0.342 2
B N
2
B N
3
C 0 C
3
0 4
c,
c1= c2 = c3
5
C1 Cz C3 C4
6
C N CI = c2
7
c3 = c 4 = c6
c5
0.448 0.261 0.448 0.227 0.427 0.248 0.423 0.206 0.253 0.426 0.343 0.337 0.281 0.243 0.439 0.239
0.101 0.476 0.101 0.544 0.144 0.503 0.152 0.587 0.492 0.147 0.313 0.324 0.435 0.312 0.120 0.520
0.108 0.508 0.106 0.572 0.153 0.533 0.157 0.606 0.492 0.147 0.313 0.351 0.472 0.312 0.120 0.520
For 2,3, and 6 also 2, numbers are given. For the diatomic systems 1-3 we have calculated the above many-particleindices for the equilibrium distance (first numbers) and for separationsthat are 7 pmelongated. The considered bond lengths are the following (in pm): 1, 133, 140; 2, 123, 130; 3,113,120. In cyclobutadiene, system 5, a bond-length alternation of 1 pm has been adopted.
TABLE Ik Charge Fluctuations ((An?)), Monocyclic Heterosystems 14-19. 8 13
14
c 2 = cs c3 = c 4
c5
15
0
16
N
c2 (26 c3 = cs c4
c 2 = c6 c3 = c 4
c5
17 18
N C N c 2=c 4 c3 c6
0
center. In ethylene the 8-pm variation is accompanied by a 4.5% reduction in the ((An?)),, values at each carbon site. In 2 and 3 ((An?)),,, is roughly conserved at the electropositiveatom in the two geometries studied. At the N atom of 2 a’l3%reduction of ((An?)), as response to the bond elongation is predicted. At the oxygen site of CO ((An?))co, is reduced even by 17%. The normalized correlation strength 21 here approaches almost 0.6 and indicates the strong electronic correlations at the 0 atom. The probability of double occupancy Pi(2) is thereby reduced to 0.1 10; the associated independent-particle value is 0.257. At N of the elongated system 2 electronic correlations lead to Pi(2) = 0.126. The I~SCF) parameter amounts to 0.262. Comparison of the ethylene and CO data indicates that the correlation strength at C is reduced in the neighborhood of a stronger correlated neighbor atom. This is a quite general phenomenon, which is known from our recent many-body in~estigations.l~-~~ Electronic localization at one center allows increasing delocalization at the neighboring site. The “polyatomic”hydrocarbons 4 and 7 with terminal C atoms visualize the stronger electroniclocalization at thesecenters. This effect closely resembles the well-known Anderson localization in the solid state.35 In cyclobutadiene, system 5, we predict charge fluctuations comparable to those of a localized u single bond at the same internuclear spacing. Compare with the ethylene data derived for 140 pm. The nitrogen atoms in 6 cause a still stronger atomic u localization in the four-membered ring. In comparison to C2H4 ((An?))w, is suppressed by ca. 10%. One major conclusion that can be drawn from the data summarized in Table I is the high A-electron localization in cyclobutadiene that is as strong as in a localized CC double bond. Atoms more electronegative than C lead to an enhanced atomic electronic density localization. Table I1 contains the atomic charge fluctuations ((An?)), together with the corresponding average ( (Anay2)),, for the sixmembered ring systems 8 and 13-19. The enhanced hopping contribution in C6H6 in comparison to a localized u bond (i.e., in units of the “resonance integral” we have 2.0 versus 2.67 for one A-electron pair) leads to enlarged electronic charge fluctuations in the six-membered rings. C6H6 with ((An?)) = 0.367 nevertheless deviates remarkably from the “free-electron” boundary with ((An?))SCF = 0.50. On the average the A electrons in borazole 13 are stronger localized than in C6H6. The data in Table I1 indicate that the A system of B3N3H6can be fragmented
B N B
19
N
0.367 0.383 0.257 0.402 0.362 0.368 01.366 0.227 0.366 0.368 0.363 0.319 0.370 0.367 0.367 0.322 0.369 0.3 18 0.369 0.365 0.372 0.324 0.376
11431
in the
0.367 0.320
0.371
0.343
0.360 0.354
0.352
C 0.350 a Additionally we have given thecorrcspondingmean-value ((An,$))averaged over all centers i. The C6H6 and B3N3H6 data have been added for convenience. into two subunits. We observe sizeable charge fluctuations at the B atoms, while the N sites act as strong “localization”centers. At these atoms we predict Pi(2) numbers of 0.164. In the onedeterminantal approximationthe probabilityof double occupancy is 0.284. We believe that the enhanced charge localization at N in comparison to C6H6 is responsible for the different stability of borazole and benzene. In a previous contribution we have explained the relation between charge delocalization (=interatomic sharing) and chemical stability in some detail.13 The H2BNH2 data in Table I visualize an interesting result. The charge fluctuations in the “localized” two-center systems remarkably exceed those encountered in the cyclic ring. But this is a “length” effect. The bond length in borazole exceeds the equilibrium separation in 2 by rougly 20 pm. In the six-rings 15-18 one can see again that heteroatoms more electronegative than C, and thus stronger correlated, tend to localize the A electrons. The ((An?)),,numbers calculated for these u models indicate some kind of damped additivity rule. With increasing number of N atoms the ( (Anpy2))uI,reduction becomes smaller. The theoretical results summarized in Table I1 thus coincide with the general trends introduced already in Table I. In Figure 2 we have portrayed the variation of the correlationstrength parameter Z,in 8-10 and 13 as a function of the bondlength alternation Ar. The sum of all length parameters is not modified in these simulations. In Figure 3 Zi for two electronic configurations of the Li6 ring is displayed. In the upper half, model 12 is considered, a six-electron system provided by Li 2pu AOs. In the lower half the six Li electrons occupy Li 2s AOs; see also ref 25. Both figuresvisualize that increasing bond-length alternation Ar is accompanied by a Zi reduction and thus by an enhancement in the electronic charge fluctuations. In any case it is evident that bond-length alternation allows increasing +electron delocalization, if delocalization is measured in terms of the dynamic property of charge fluctuations. We believe that the two figures show in transparent form the tendency of u electrons in cyclic networks to form alternant bonds. Increasing charge fluctuations (=uncertainty in position space in the terminology of the Heisenberg uncertainty principle) immediately imply decreasing uncertainty in momentum space. Small momenta become of larger probability. The associated reduction of the electronic kinetic energy leads to an energetic stabilization. A number of recent theoretical investigations have shown that
-
11432 The Journal of Physical Chemistry, Vol. 97, No. 44, 1993
Bbhm et al.
TABLE III: Charge Fluctuations ( (An?)) and Correlation Strength Parameters A h El in the Bicyclic Compounds 20-22' ((&"Z))mrr,
compound 20
0.52
7
0
.
3
t
4
u0.32
13 , 01 0.27 0.26
22
f3 i JB
0.25 0.24 0
8 Arlpm Figure 2. Modification in the correlation-strength parameter 21 as a functionof the bond-lengthalternation &in the model systems8-10 and 13. The topmost (broken) curve refers to the N atoms of borazole 13; the lower broken curve to the B centers of 13.
2
4
6
0.29
0.11 3 -
0.12
0
21
8
16
24 Ar/pm
Figure 3. Modification of the correlation-strenth parameter 21 = Ai as a function of the bond-length alternation & in the Lis ring. The upper dispersion corresponds to a 2pr configuration while the bottom one has been derived for Lis frozen in Li 2s AOs.
bond-length alternation is obstructed by the u frame.26.36 Note that the distortive nature of the a-electron system is already feasible in the I+SCF) picture as has been shown by LonguetHiggins and Salem in a fundamental c~ntribution.~' Figures 2 and 3 indicate that the reduction of the localization parameter Z i as a function of the alternation coordinate is more pronounced at strongly correlated atoms. This has been found already in the 'localized" double-bond systems 1-3; see Table I. Compare the &variation at N and B of borazole 13. The stronger a-electron localization a t the N centers of 13 in comparison to 9 leads to a larger X ireduction as a function of Ar compared to the Zi modulation in 9. In Figure 3 the difference between the weaker correlated Li 2s electrons and the Li 2 p set ~ becomes transparent. The gradients of the two Zicurves differ sizeably. Diagrams 2 and 3 contain an interesting result which should be emphasized again. Bond-length alternation in (cyclic) A systems is not accompanied by increasing localization of the a electrons. The opposite is true. The a system is distortive as a result of increasing A delocalization with increasing bond alternation. Thereby the a electronic energy is lowered. The driving force
atomi
((An?))0.386 0.366 0.404 0.237 0.258 0.292 0.309 0.371 0.368 0.356 0.356 0.368 0.368 0.357 0.389 0.387 0.357 0.322 0.370 0.355 0.356 0.369 0.369 0.356 0.388 0.388
Ai
0.223 0.262 0.181 0.552 0.486 0.408 0.381 0.256 0.263 0.287 0.286 0.262 0.262 0.286 0.221 0.224 0.287 0.353 0.257 0.288 0.287 0.261 0.261 0.287 0.223 0.223
21
Anw
2..
0.257 0.301 0.224 0.596 0.557 0.503 0.319,0.356,0.415 0.407 0.267 0.368 0.295 0.288 0.264 0.264 0.287 0.228 0.228 0.363,0.273,0.280 0.291 0.387 0.273 0.298 0.290 0.265 0.264 0.291 0.230 0.230 0.263,0.273,0.283
The atomicnumbering scheme is displayedin Figure 1. Additionally we have given the corresponding mean values ((&v2))m, &,., E,. of bond-length alternation in the a framework is an increasing function of the correlation strength. See also our previous discussion of this unexpected finding.12-14 In Table I11we have summarized the many-particle parameters Ai, Z,, ((An?))" of the bicyclic a systems 20-22 together with themeanvaluesaveragedover all ~ c e n t e r s i .Thelatter quantities are labeled by the index av. The data collection indicates that the atomic localization of the a electrons critically depends on the topology of the considered A atom. A 10%difference in the ((An?))" elements as a function of the atomic position is predicted for the B centers in 20 (minimum 0.366, maximum 0.404). The largest charge fluctations (=maximum electronic delocalization) are found a t B atoms bonded to three nitrogens. The width of the atomic ((An?))" elements is almost doubled (=18%) a t the more electronegative N atoms. Here the a electronic charge fluctuations are sizeably smaller than those observed at the B centers; compare with the borazole data. The minimum and maximum ((An?) )-, numbers amount to 0.237 and 0.292. In analogy to the boron centers the largest charge fluctuations at N are found at atoms forming three A bonds. A comparison of the 2,, numbers derived for 20 and the monocyclic BN system 13 (see Figure 2) indicates that the localization properties of the a electrons are roughly comparable in both molecules. The additional ring in 20 is not coupled to an increasing a delocalization. The enhanced charge fluctuations accessible at the bridging atoms 9 and 10 are attenuated largely by the ((An?)),,, suppression at the direct neighbors. The electronic charge fluctuations a t the nitrogen sites of 21 and 22 are larger than those encountered in 13 and 20. Again this demonstrates that enhanced fluctuations a t a neigbor atom (B in 13 and 20 versus C in 21 and 22) have a strong response on the considered more electronegative center (always N). C as neighbor of N prevents the strong a localization caused by B neighbors. The strong correlation between atomic topology and the ability to delocalize ?r electrons is also underlined by the model systems 2622. Next we move to Table IV, where we have collected the average many-particle elements ( (Antv2))-, Aav,Z,for the nonaltemant hydrocarbons 23-32. For the sake of convenience we have added
Charge Fluctuations in
A
Systems
The Journal of Physical Chemistry, Vol. 97,No. 44, 1993 11433
TABLE IV. Mean Values of the Charge Fluctuations ( (An,v*))mmand Correlation Strength Parameters A.,,
Z,in
the Nonalternant Hydrocarbon Systems 23-32’ compound 23 24 25 26 21 28 29 30 31 32
c6Hs
((An.,? 0.346 0.350 0.355 0.359 0.364 0.353 0.358 0.360 0.364 0.365 0.367
&V
0.305 0.297 0.285 0.278 0.267 0.270 0.269 0.267 0.265 0.264 0.264
TABLE V: Charge Fl~ctuation~ ((Ant) )- and Correlation Strength Parameters Ah ZIin Pentalene 28 as Well as the Diaza Derivatives 33-37.
2 ,
0.337 0.314 0.313 0.312 0.306 0.388 0.352 0.337 0.315 0.300 0.264
Averaging is over all atomic sites i. For convenience we have added the C6H6 data. a
the corresponding numbers for C6H6,a molecule that generally is employed as a prototype of a “delocalized” A system. We have already mentioned that many of the A systems considered in Table IV have focused considerable interest by experimental chemists. For experimental work see refs 38-47. It is wellknown that some of the A systems in Table IV are highly unstable. Characterization of pentalenes or indacenes is possible only by “steric protection” or electronic modification via electron-donating substituents. Inspection of the ( (Anavz)),, numbers indicates roughly comparable average charge fluctuations in the nonalternant A networks and in benzene. The 2,, elements in Table IV, however, indicate that the net electronic correlations are stronger in the nonalternant A molecules. This difference between C6H6 and the other A networks summarized in the table is a result of A electronic shifts in the nonalternant molecules violating the charge symmetry of alternant molecules with (n,) = 1.O at any center i. In our recent analysis of nonaltemant hydrocarbon^'^ we have explained in some detail the compatible A delocalization in 23-32 and C6H6 and the large differences in the stability of the corresponding molecules. In the highly reactive nonalternant A systems one has always A centers with strongly localized electrons and small interatomic sharing; see the pentalene data in Table V. Steric protection is necessary here to synthesize and characterize the corresponding A molecules. In ref 13 we have explained that high stability of a A system always implies small deviations within the atomic ( (Ant)),,, numbers. In Table V we have summarized atomic many-particle quantities as well as the associated mean values for pentalene 28 and different azapentalenes. The sizeable A electronic localization and strong correlation at certain centers is clearly seen. As already expected the more electronegative N atoms (reference: C) in 33-37 cause an enhancement in the interatomic A correlation strength. In the first place this modification is an on-site (N atoms) effect, as the A electronic localization properties at the carbon atoms are roughly conserved in the aza derivatives. The rather high localization parameters 2,indicate that azapentalenes should be highly reactive as a result of their low A delocalization. Also the pentafulvene 23 and azapentafulvene data 38-40 in Table VI show the strong A localization encountered in this class of compounds. We predict the strongest localization at the exocyclic A center where the normalized correlation strength exceeds 64% of the accessible maximum. At the end of this section we present in Table VI1 the normalized interatomic correlation energies of all 40 systems studied. Normalization always refers to two electrons. For the diatomic molecules 1-3 we have adopted the experimental equilibrium bond lengths, while the standard geometries described in section I1 have been accepted for all other A networks. The systems 1-3 indicate that the interatomic correlation energy E,,, is enhanced by heteroatoms more electronegative than C; compare the C2H4 and CO results. lEmrr1 of HzBNHz is roughly comparable to the
28
33
34
35
36
31
0.332
0.312 0.449
0.368 0.378 0.288 0.375 0.327 0.385 0.378 0.287 0.373 0.319 0.387 0.292 0.376 0.333 0.328 0.313 0.326 0.383 0.378 0.287 0.384 0.324 0.368 0.383 0.344
0.243 0.211 0.414 0.233 0.320 0.205 0.213 0.425 0.249 0.350 0.221 0.408 0.234 0.314 0.316 0.351 0.317 0.207 0.213 0.417 0.222 0.323 0.241 0.207 0.307
0.316 0.378
0.347 0.493 0.211 0.314
0.337 0.315 0.536 0.308 0.471 0.291 0.315 0.433 0.279 0.451 0.258 0.502 0.300 0.439 0.473 0.508 0.485 0.299 0.314 0.524 0.277 0.489 0.336 0.299 0.446
0.353,0.270,0.388
0.343,0.294,0.405
0.341,0.311,0.355
0.341,0.295,0.415
0.342,0.294,0.405
0.341,0.293,0.425
Additionally we have given the corresponding mean-values
((A~Z~,A . ,~ ) Z , ) ~ averaged ~, over all atomic sites i.
TABLE VI: Charge Fluctuations ((Ant)),
and Correlation Strength Parameters Ah 21in Pentafulvene 23 and in the Aza Derivatives 38-40.
compound atom i ((An?))” 23 CI =C4 0.341 C2 C3 0.365 CS 0.436 c6 0.231 38 N1 0.287 c 2 0.370 c3 0.365 c 4 0.340 CS 0.438 c 6 0.229 39 c1 0.357 Nz 0.305 CS 0.369 c4 0.337 CS 0.440 C6 0.217 40 C1 =Cq 0.342 Cz = C3 0.364 CS 0.440 N6 0.189
A, 0.315 0.269 0.127 0,533 0.422 0.259 0.268 0.317 0.122 0.536 0.283 0.387 0.260 0.323 0.117 0.561 0.313 0.371 0.115 0.618
Zi
0.349 0.274 0.134 0.641 0.485 0.261 0.277 0.343 0.131 0.639 0.309 0.408 0.261 0.364 0.125 0.682 0.346 0.273 0.128 0.693
((bV2))-, bV, 2,
0.346,0.305,0.337
0.338,0.321,0.356
0.338,0.322,0.358
0.340,0.317,0.343
Additionally we have given the corresponding mean values
((Anav2)),, Aav,Z., averaged over all atomic sites i. A correlation energy of ethylene. The lEcorrl enhancement due to N here is overcompensated by the lECorrlreduction due to B and the heteropolar charge distribution. Interatomic correlations are suppressed with increasing charge separation in two-electron bonds. They approach zero in a perfectly ionic bond.20 In the family of hydrocarbons cyclobutadiene,system 5, is an exception. The sizeable *-electron localization leads to a rather high Emu number. As can be seen in the table most of the C C increments are of the same order of magnitude. With some exceptions the width in the E,,, elements is rather small. The E , numbers in Table VI1 follow the trends discussed in connection with theother many-particle indices ( (Ani2)),,, A,, X i . N and 0 heteroatoms cause an enhancement of the interatomic correlation energy.
11434 The Journal of Physical Chemistry, Vol. 97, No. 44, 1993
TABLE MI: Interatomic Correlation Energies &,, in the Studied u Systems 1-40. -EcOrr/
-ECOtT/
system
eV
system
eV
system
1 2 3 4 5 6 7 8 9 10
0.687 0.686 0.984 0.690 0.769 1.029 0.647 0.440 0.724 0.458
11 12 13 14 15 16 17 18 19 20
0.225 0.079 0.607 0.409 0.594 0.491 0.540 0.543 0.571 0.601
21 22 23 24 25 26 27 28 29
0
30
-E"/
eV
system
0.469 0.466 0.579 0.554 0.525 0.469 0.424 0.406 0.413 0.408
31 32
33 34 35 36 37 38 39
40
-E-/
eV
0.420 0.419 0.493 0.591 0.486 0.499 0.473 0.638 0.632 0.661
Normalization always refers to two 7r electrons.
IV. Final Remarks The method of the local approach in combination with a PPP Hamiltonian has been used to investigate the many-particle character of *-electron bonding in a number of hydrocarbons, heteroderivatives, and other model systems. The correlated ground state I+o) has been derived by taking into account singly and doubly excited configurations. The T electronic charge fluctuations ((An?)),, in the correlated ground state have been adopted as quantitative measure for the atomic density localization of (r) electrons. ((An?)) has two unambiguously defined boundary values. ((An?)) ( (A~?))scF,Le., the maximum accessible charge fluctuations encountered in the independentparticle limit. As already emphasized this boundary is met in any single-determinantal framework. ((An?)) ( (An?))mx describes perfect interatomic correlations with the minimum possible electronic charge fluctuations. In altemant hydrocarbons with (ni)= 1.O a t any center i complete suppression of the charge fluctuations is possible at least in principle. The electrons are then atomically localized and no interatomic sharing takes place. It is intelligible that the ((An?)),,numbersdescribethebonding abilities of ( T ) electrons. Small ( (An?)),rr elements indicate low stability/high reactivity of a T system. In ?r systems with (nt) # 1.0 complete *-electron localization at the respective centers is no longer possible. Interatomic correlation effects are enhanced when moving from the left-hand side of the periodic system to the right-hand side. In the same direction bond-length alternation is accompanied by amplified charge fluctuations.
Acknowledgment. This work has been supported by the Deutsche Forschungsgemeinschaftand the Fonds der Chemischen Industrie. We are grateful to Dr. S.Philipp for critically reading the manuscript. The drawings have been kindly prepared by Mr. G. Wolf.
Bohm et al.
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