Hartree-Fock instabilities and electronic properties

Hartree–Fock Instabilities andElectronic Properties

DOMINIQUE DEHARENG, GEORGES DIVECentre d’Ingéniérie des Protéines, Institut de Chimie B6, Sart Tilman, B-4000 Liège 1, Belgium

Received 17 September 1999; accepted 8 December 1999

ABSTRACT: Hartree–Fock instabilities are investigated for about 80compounds, from acetylene to mivazerol (27 atoms) and a cluster of 18 watermolecules, within a double ζ basis set. For most conjugated systems, therestricted Hartree–Fock wave function of the singlet fundamental state presentsan external or so-called triplet instability. This behavior is studied in relation withthe electronic correlation, the vicinity of the triplet and singlet excited states, theelectronic delocalization linked with resonance, the nature of eventualheteroatoms, and the size of the systems. The case of antiaromatic systems isdifferent, because they may present a very large internal Hartree–Fock instability.Furthermore, the violation of Hund’s rule, observed for these compounds, is putin relation with the fact that the high symmetry structure in its singlet state hasno feature of a diradical-like species. It appears that the triplet Hartree–Fockinstability is directly related with the spin properties of nonnull orbital angularmomentum electronic systems. c© 2000 John Wiley & Sons, Inc. J ComputChem 21: 483–504, 2000

Keywords: Hartree–Fock instabilities; resonance; π-electronic systems;antiaromatic systems; violation of Hund’s rule

Correspondence to: D. Dehareng; e-mail: [email protected]/grant sponsor: Belgian program on Interuniversity

Poles of Attraction, initiated by the Belgian State, Prime Min-ister’s Office, Service Fédéraux des Affaires Scientifiques, Tech-niques et Culturelles; contract/grant number: P4/03

Contract/grant sponsor: Fonds de la Recherche ScientifiqueMédicale; contract/grant number: 3.4531.92

Contract/grant sponsor: Servier-Adir

Journal of Computational Chemistry, Vol. 21, No. 6, 483–504 (2000)c© 2000 John Wiley & Sons, Inc.

DEHARENG AND DIVE

Introduction

H artree–Fock (HF) instabilities have been wellknown since the sixties.1 – 22 In this introduc-

tion, we shall present only a brief summary.Let us consider even electronic systems, 2N being

the total number of electrons. The HF frameworkcan be considered at several restraint levels. Themost often used one is the restricted HF (RHF)scheme in real coordinate space (RRHF) where theelectrons are paired in molecular orbitals (MO); inthat case, the HF monodeterminantal wave function(w.f.) is constructed from N real one-electron func-tions each of which gives rise to two spin orbitals bymultiplication by the α or β spin function. This typeof w.f. is an eigenfunction of the total spin squaredoperator S2 and of the z-projection Sz, i.e.,

S2ψ = s(s+ 1)h̄2ψ , Szψ = msh̄ψ ,

with s = ms = 0, h̄ = h/(2π), h being the Planckconstant.

The variation calculus23 leads to a RRHF wavefunction that is a stationary solution of the Schrö-dinger equation within the constrained frameworkadopted.

Some of the restrictions of this RRHF scheme canbe lifted. One can use a distinct spatial function foreach α and β spin orbital (s.o.), thus working atthe unrestricted HF (UHF) level,24 one can considercomplex functions instead of real ones (CRHF orCUHF levels) and, at last, one can choose monoelec-tronic functions that are no longer pure α or β s.o.but are a mixture of both; this latter framework iscalled the generalized HF level (RGHF or CGHF).Except in CRHF, the w.f. obtained in these frame-works is no longer an eigenfunction of S2, but theUHF w.f. is still an eigenfunction of Sz.

The general strategy is to find a stationary solu-tion within a chosen HF level, to ascertain that it isa minimum, then to pass to a less restricted level, tocheck for stability of the wave function within thisnew framework, and to reoptimize it if it presentsan instability.

As pointed out by several authors (for instancerefs. 1, 12, 18, and 25), the instability matrices aredirectly related with those obtained in the randomphase approximation (RPA) or the time-dependentHF (TDHF) frameworks.1, 25

The HF instabilities were classified by Fuku-tome,11 according to the (spin rotation ⊗ time re-versal) subgroups, into eight classes in direct con-nection with those cited above, R(C)RHF, R(C)UHF,and so on.

Some of these instabilities are recognized as triv-ial ones. The first type are those related with the factthat the HF w.f. obtained corresponds to an excitedstate. These are denoted as “filling up” instabilitiesby Chambaud et al.15 and correspond to internalones, because the chosen HF framework, be it RHFor UHF, does not change. Other trivial instabilitiesare those found for nonsinglet states when passingfrom the restricted open shell (ROHF)26 to the UHFframework, as emphasized by Paldus and Cizek8

and by Hilbert and Coulson,17 because UHF alwaysproduces lower energies than ROHF; these instabil-ities are external ones, because the framework hasbeen changed (ROHF→ UHF). In this framework,the case of the allyl radical has been a matter ofdiscussion.27

For closed-shell systems, external (RHF→ UHF)instabilities were called triplet instabilities for thefirst time by Cizek and Paldus,5 and this nomencla-ture still remains.

For particular singlet entities named 1,3-dipoles,it is also well known that a UHF solution provides alower energy than the RHF one,19 – 22 due to the di-radicalar character of the singlet fundamental state.As a reminder, 1,3-dipoles are generally representedas resonant structures between zwitterions and sin-glet diradicals as in the following example:

The HF instabilities are also related to the sym-metry dilemma as emphasized by Löwdin3, 7 statingthat a w.f. with MOs belonging to a lower symme-try point group than that related with the geometrycould lead to a lower energy. This symmetry break-ing dilemma is also encountered28 in the frameworkof the density functional theory (DFT).29

The aim of this work is to search for some re-lationships between the nature of the molecularsystems and the existence or absence of HF insta-bilities. Such links cannot be found by consideringonly a few molecules. Thus, a large number ofsystems has to be investigated. The investigatedspecies are all even electronic molecules or ionsand most of our work deals with the so-calledtriplet, or external, instabilities. Moreover, RHF in-ternal instabilities were also investigated for thehigh-symmetry annulenes, i.e., cyclic polyenes, be-cause it was predicted, on a semiempirical basis,5, 16

that this type of instability should appear for largesystems. Also, UHF internal instabilities were in-

484 VOL. 21, NO. 6

HARTREE–FOCK INSTABILITIES AND ELECTRONIC PROPERTIES

CHART 1.

vestigated for some systems called antiaromaticsaccording to the Hückel classification,30 i.e., annu-lenes with 4n π electrons. Thus, the HF instabilitieswere investigated for some saturated compoundsand for unsaturated ones. In this latter series, thecompounds were considered within very distinctclasses, noncyclic compounds, aromatic, and an-tiaromatic entities. The HF instability was studied asa function of the number of π electrons, the numberof heavy nuclei underlying the π delocalization, thenature of the heteroatoms involved, as summarizedin Chart 1.

The second and third sections of this workpresent a brief theoretical background, the details ofwhich can be found in refs. 1 – 15, and the computa-tional tools used. The fourth section deals with somepeculiar points of the optimized geometries. Finally,the fifth section is split into subsections presentingthe results and their discussion, firstly as generalcomments, secondly as a function of the nature ofthe systems investigated.

Theoretical Background

Let 90 be the self-consistent HF monodetermi-nantal wave function and 9 the general multide-terminantal wave function for which 90 is expectedto be a good approximation. 9 can be expanded interms of all the determinants constructed with theoccupied and virtual molecular spin orbitals (m.s.o.)derived from the self-consistent field (SCF) schemein the LCAO–MO framework:

9 = 90 +occ∑

i

virt∑a

Di→a9i→a

+occ∑

i

occ∑j

virt∑a

virt∑b

Di→a,j→b9i→a,j→b + · · · . (1)

9i→a is the wave function, the determinant of whichcontains the virtual s.o. χa instead of the occupiedone χi, corresponding to a monoexcitation of the

reference state 90, 9i→a,j→b corresponds to a diex-citation χi → χa and χj → χb, and so on, theDi→a represent the expansion coefficients describ-ing single excitations from the occupied MO i to thevirtual MO a, the Di→a,j→b represent the expansioncoefficients describing double excitations from theoccupied MOs i, j to the virtual ones a, b.

In the framework of the single particle propa-gator used in the RPA,1, 25 the coefficients Di→a,j→b

factorizes into the product Di→aDj→b.The sign of the second-order energy term E2

obtained for 9 must be positive as a necessary con-dition for stability.1 – 15 In the RPA framework, oneobtains:

E2 = 12

(D, D∗)(

A BB∗ A∗

)(DD∗

)= 1

2(D, D∗)H

(DD∗

)(2)

D = {Ds}A = {Ast}, Ast = 〈s|V|t〉 = (εa − εi)δst + (aj||ib)B = {Bst}, Bst = 〈st|V|0〉 = (ab||ij),

the ∗ superscript referring to the complex conjugatematrix,

s ≡ i→ a, t ≡ j→ b

V = H − 〈0|H|0〉,H being the electronic hamiltonian, and |0〉 therepresentation of 90, εi are the eigenvalues of theone-electron Fock operator and (aj||ib) are the usualbielectronic integrals∫ ∫

χa(1)χb(2)(1/r12)[χi(1)χj(2)− χj(1)χi(2)

]dτ1 dτ2.

Thus, the stability of the HF w.f. is defined accord-ing to the inertia of theHmatrix, i.e., to the numberof its negative eigenvalues: no negative eigenvalueof the H matrix means that the investigated systemhas a stable HF w.f. The A matrix elements involvemonoexcited states couplings, while the B matrixelements couple diexcited states with the funda-mental one.

The eigenvector(s) corresponding to the negativeeigenvalue(s) of H are used to construct a gener-alized Fock matrix the diagonal elements of whichare molecular orbital energy differences, and the offdiagonal elements are functions of the instabilityeigenvector. By diagonalizing this new Fock matrix,a set of trial s.o. is obtained in the new framework.10

It can be shown8 – 10 that the A and B matricesfactorize into spin conserved (A′, B′ → H′) and spin-unconserved (A′′, B′′ → H′′) parts. The internal HFinstabilities result from the appearance of negative

JOURNAL OF COMPUTATIONAL CHEMISTRY 485

DEHARENG AND DIVE

eigenvalues for H′ and external ones from the oc-currence of negative eigenvalues for H′′. For evenelectronic systems, the internal instabilities are oftencalled singlet instabilities, while external ones arereferred to as triplet instabilities. Singlet instabilitiesappear when, remaining in the RHF framework, onecan find a lower energy RHF w.f. than that initiallyobtained in a self-consistent way. Triplet instabilitiesappear when one can find a lower energy UHF w.f.than that initially obtained in a RHF self-consistentway.

Computational Tools

All the calculations were performed with theUnix version of GAUSSIAN9431 on three comput-ers, Dec-α servers 4 and 8 processors, 4100 and8400, and a R4400 SGI. The excited states were de-termined either at the HF level, or at the singlyexcited configuration level (CIS),32 or even at a mul-ticonfigurational SCF level labeled CASSCF (com-plete active space SCF).33 A few calculations wereperformed at the post Hartree–Fock perturbationalMøller–Plesset34 level, either at the second order(MP2) or the fourth order (MP4).

Optimized Geometries: Energetical andGeometrical Considerations

The stability of the RHF-self-consistent (SCF)wave function was investigated in the UHF frame-work for seven fully saturated compounds as wellas for seventy nine unsaturated ones (see Appen-dix). The geometry of these molecules or ions, intheir lowest singlet state, was optimized, at theRHF–SCF level, within a chosen symmetry pointgroup (SG), but the nuclear arrangement obtained atconvergence was not necessarily a zero-order (min-imum) or even a higher order critical point (seethe following). The C2v-planar cis-butadiene//(RHF,6-31G∗∗) singlet structure is a transition state (TS)for which the imaginary frequency is equal to162.61 cm−1. The singlet state structures of the highsymmetry (Dxh) x-annulenes (x = 10, 14, 18) are notminima, but are higher order critical points, i.e., arecharacterized by a null gradient and a hessian withone or several negative eigenvalue(s). Finally, theantiaromatic systems RHF singlet state geometriesoptimized within the high SG restraints are not crit-ical points: it is not possible to obtain a null gradientwith these geometrical constraints.

The x-annulenes (x > 6) in their highest planarSG, corresponding to identical C—C bond lengths,are considerably much higher in energy than anylocal minimum. For instance, for the optimizedgeometries obtained at the RHF/6-31G level, An10(D10h) is 125.386 kJ/mol higher than An10 (C2),An14 (D14h) is 919.488 kJ/mol higher than An14(C1), An18 (D18h) is 1814.094 kJ/mol higher thanAn18 (C1). This is to be put in relation with thedistortive property of the π electrons that coun-terbalance the symmetrizing trend of the σ ones.Indeed, it has been shown by several authors35 thatthe σ electrons bring a nuclear system to a symmet-rical structure with all the C—C bonds equal, whilethe π electrons show a distortive behavior leadingto a single/double bond alternance. The equilib-rium geometry results from the equilibration of thetwo opposite trends. Thus, if benzene equilibriumstructure has a D6h SG, it is due to the stronger influ-ence of the σ electronic frame, and the π electronicdelocalization responsible for the resonance stabiliz-ing energy is actually a byproduct of the symmetricstructure.

At the RHF/6-31G level, An10 (D10h) and An14(D14h) are critical points of order 3 and 5, respec-tively. Among all the imaginary frequencies, theunique non degenerate one is the highest in absolutevalue. So, for An10 (D10h) (An14 (D14h)), the imagi-nary frequency equal to i294.0 cm−1 (i2863.8 cm−1)corresponds to the nondegenerate normal mode re-lated with the formation of alternant single anddouble bonds. The An18 (D18h) critical point of or-der 8 has only degenerate imaginary frequenciesrelated to out-of-plane deformations, and the nor-mal mode describing the single–double bonds alter-nation is a degenerate mode at the real frequencyof 1692.7 cm−1. From these energetical considera-tions, the first step in the distortion of An10 (D10h)and An14 (D14h) is the appearance of single/doublebond alternance, while for An18 (D18h), it is out-of-plane folding.

For several systems, the CASSCF level was alsoused for the geometry optimization as well as forthe determination of singlet and/or triplet excitedstates. Except in one case (the cyclopentadienyl an-ion Cpdm1), all the active molecular orbitals (MOs)were chosen to be π and π∗ ones. In the case ofunsaturated compounds, the lowest transitions aremost often of the type π → π∗ and the choice ofsuch an active space seems to be adequate. Never-theless, for some exceptions like Cpdm1, there are alot of π → σ ∗ transitions before or in between theπ → π∗ ones. Thus, several choices of active MOwere done with, at least the one containing σ MOs.36

486 VOL. 21, NO. 6


It remains true that the choice of a reliable activespace is a serious problem because the CASSCF con-vergence is very dependent on it.

Results and Discussion

The investigated systems are not all planar, andthe MOs cannot be systematically designated as σor π by reference to their symmetry according to aneventual symmetry plane, as usual in organic chem-istry. For instance, in the case of twisted-butadiene(C2), An10 (C2), An14 (C1), An18 (C1), or mivazerol,it is no longer sensible to speak of π MOs but onlyof MOs related with the unsaturated bonds becauseno symmetry plane exists. However, for the sake ofconciseness, they will also be refered to as π MOshereafter.

Before considering detailed observations as afunction of the size and the nature of the systems,some general comments are presented in the follow-ing subsection.

GENERAL COMMENTS

Among all the investigated compounds (see Ap-pendix), only those presenting an external (RHF→UHF) instability are listed in Table I.

The last ones are the mivazerol (27 atoms), andone of its protonated structure; they are benzylim-idazoles with an α2 agonist profile.37 The basissets used differ by the presence or the absence ofpolarization and/or diffuse functions and have incommon the 6-31G kernel. An exception is madefor the 10-annulene in its global minimum geome-try of symmetry C2 (ref. 38), An10 (C2), for whicha calculation was also done within the double ζ

Dunning D95 basis set.39 Table I provides the en-ergy gain 1E(reop) obtained after the optimizationof the RHF w.f. in the UHF framework. Because thatnewly reoptimized UHF(reop) wave function is nolonger an eigenfunction of the total spin squarredoperator S2, the mean value 〈S2〉(reop) of that op-erator for the UHF(reop) wave function is given.Moreover, the CIS triplet excitation energies areshown, as well as the energy difference 1E(L-H)between the lowest virtual MO (LUMO) and thehighest occupied one (HOMO), obtained at the RHFlevel. The negative eigenvalues λ0(tr) of the stabilitymatrix was determined for some compounds. Obvi-ously, the larger the λ0(tr) absolute value, the larger1E(reop).

Orders of Magnitude

The results of Table I clearly show that, for manyπ electron systems, the RHF w.f. is not stable inthe UHF framework. The energy gain 1E(reop) isof the order of 3–15 kJ/mol for most of the four orsix π electron compounds. This quantity becomessomewhat higher for the N6Hx+

x cations, the cy-clooctatetraene dication Cotp2 (D8h), and the five-and seven-membered rings containing boron. Forspecies with a larger number of π electrons, the en-ergy gain can be much higher.

In the case of the annulenes in their highest SG,finding an HF instability, either external (triplet) orinternal (singlet one in this work), becomes moreand more obvious as the cycle size grows up be-cause the system is then very high in energy, and ischaracterized as a critical point of increasing order.Nevertheless, if one compares these systems relaxeddown to a local minimum [An10 (C2), An14 (C1),An18 (C1)], the external HF instability remains large(Table I), and increases with the number of unsatu-rated bonds.

Relation with the LUMO–HOMO Gap

Fukutome11 argued that an HF instability shouldoccur when the energy difference1E(L-H) betweenthe HOMO and the LUMO becomes small, thus en-abling easy electronic transitions, usually π → π∗ones. As seen from Table I, the LUMO–HOMO gapranges in the order of 600–1300 kJ/mol when an HFinstability occurs. Such large numbers are of the or-der of magnitude of the A and B matrix elements[eq. (2)]. Thus, it remains that the relation betweenthe LUMO–HOMO gap and the occurrence of an HFinstability is qualitative, or semiquantitative whentaking the elements of A or B as references.

Appearance of Atomic Spin Density (ASD)

The reoptimized UHF w.f. presents, in nearly allcases, alternant spin densities on the atomic frame-work involved in the π delocalization. This cor-responds to the magnetically fully ordered loweststate, as emphasized by Yamaguchi:40 spins up andspins down are regularly alternating. Large atomicspin densities (ASDs) of the UHF reoptimized w.f.means a significant electronic localization aroundthe nuclear site. Schütt and Böhm35c showed thatthe localization of the π electrons around each nu-cleus was linked to the electronic correlation: thelarger the correlation, the stronger the localization,and thus, the larger the ASDs.


DEHARENG AND DIVE

TABLE I.Systems, among Those Listed in Appendix, Presenting a RHF→ UHF External Instability.

Molecular System (SG) λ0(tr) 1E(reop) 〈S2〉(reop) 1E(tr,CIS) 1E(L-H)

Ethylene (D2h) (a) −11.55 −0.10 0.03 349.4 1449.0trans-Butadiene (C2h) −70.82 −6.51 0.34 274.4 1192.9Twisted butadiene (C2) −45.04 −2.83 0.24 295.8 1230.9cis-Butadiene (C2v) (TS) −81.21 −8.70 0.40 252.3 1166.5Butadiyne −21.85 −1.05 0.19 1327.9Truncated cephalosporine (a) −3.94 −0.01 0.014 329.4 1168.2Cyclopropene (a) −17.69 −0.24 0.05 338.5 1395.5BC4H5 (C2v) −23.45 0.64 130.0 953.8trans-E,E hexatriene (C2h) −109.16 −20.38 0.67 228.8 1045.7cis-E,E hexatriene (C2v) −19.95 0.67 230.8 1055.9cis-E,Z hexatriene −81.14 −11.41 0.53 257.2 1108.6Fulvene (C2v) −68.50 −7.42 0.42 221.9 1031.0Furane (C2v) (a) −16.92 −0.38 0.09 340.6 1289.3Protonated furane (C2v) −2.40 0.21 319.3 1287.6Thiophene (C2v) −15.59 −0.35 0.09 325.5 1222.8Protonated thiophene (C2v) −2.05 0.20 319.7 1247.3Thiazole (Cs) −3.77 −0.02 0.02 344.5 1236.3Benzene (D6h) (a) −74.39 −9.12 0.47 332.3 1265.7

(b) −65.47 −7.16 0.42 334.6 1259.8(c) −5.02 0.36 336.0 998.4

Pyrimidine (Cs) −2.78 0.29 383.7 1281.0Pyridine (Cs) −6.34 0.40 351.6 1235.8Pyridazine (C2v) −11.22 0.50 349.6 1273.11,2,3 Triazine (C2v) −14.15 0.55 322.3 1324.0Hexazine N6 (D6h,TS) −66.70 1.04 161.6 1195.0N6H2+

2 (Cs) −51.84 0.97 328.4 1161.7N6H4+

4 (Cs) −74.96 1.22 313.8 1365.5C7H+7 (D7h) (a) −7.03 −0.10 0.06 326.4 1113.9

(b) −20.56 −0.84 0.16 315.2 1106.9C6H6N+ (C2v) −15.61 −0.69 0.17 331.1 1089.9C5H6NB (t1,Cs) (a) −23.17 0.74 246.5 1064.2

(b) −15.99 0.64 261.0 1075.6C5H6NB (t2,Cs) (TS) −18.17 0.65 238.4 1057.5C5H6NB (t2,C1) −19.07 0.69 237.5 1064.2BC6H7 (C2v) −19.53 0.69 233.8 1036.6Cotp2 (D8h) −28.22 0.87 212.4 958.5Naphtalene (D2h) −110.91 −30.52 1.01 267.1 1023.8Indole (Cs) −9.06 0.53 311.2 1094.0An10 (D10h) −111.33 1.63 173.8 897.6An10 (C2) (a) −29.95 1.14 242.0 1059.5

(d) −30.52 1.16 1034.1An11p1 (C1) −33.76 1.24 141.9 793.5An12p2 (C1) −53.32 1.44 162.9 798.8Anthracene (D2h) −84.02 1.76 189.2 850.8Phenanthrene (C2v) −57.06 1.58 259.2 998.9An14 (D14h) −247.74 2.54 84.9 690.2An14 (D7h) −160.53 2.36 138.8 829.2An14 (C1) −54.50 1.68 217.7 985.5Naphtacene (D2h) −141.56 2.41 134.0 736.9An18 (D18h) −359.30 −390.77 3.42 32.6 573.9An18 (D9h) −211.22 3.12 129.2 789.8An18 (C1) −105.74 2.53 190.6 905.1Pentacene (D2h) −205.76 3.05 92.2 655.1

488 VOL. 21, NO. 6


TABLE I.(Continued)

Molecular System (SG) λ0(tr) 1E(reop) 〈S2〉(reop) 1E(tr,CIS) 1E(L-H)

Hexacene (D2h) −274.03 3.69 60.1 594.3Hexaphene (Cs) −184.59 3.44 160.5 769.5Toluene −9.13 0.47 330.4 1233.8Aniline −5.30 0.38 338.1 1138.5Phenol (Cs) (a) −7.55 0.43 336.9 1200.2

(b) −4.88 0.36 341.8 1188.1Phenolium (Cs) −7.00 0.41 343.0 1223.9Phenone −10.84 0.56 295.2 1188.6Salicylamide −2.40 0.27 341.0 1085.83-Methyl-indole −8.95 0.53 310.7 1083.62-Methyl-indole −8.25 0.51 314.2 1085.4Mivazerol −6.36 0.42 328.4 1005.4Protonated mivazerol −63.96 −7.70 0.45 326.7 1102.7

When no symmetry point group (SG) is indicated, SG is C1. The first eigenvalue λ0(tr) of the spin-unconstrained instability matrixH′′ is given, in kJ/mol, for few systems.1E(reop) is the energy lowering, in kJ/mol, from the reoptimization of the RHF wave functionwithin the UHF framework. 1E(tr,CIS) is the energy difference, in kJ/mol, between the lowest singlet state and the triplet one relatedwith the RHF→ UHF instability, at the CIS level. 〈S2〉(reop) is the average value of the squarred total spin operator obtained for theUHF reoptimized wave function. Except for few cases (ethylene, truncated cephalosporine, cyclopropene, furane, benzene, C7H+7 ,C5H6NB (t1,Cs), An10 (C2), phenol), only the results obtained with the largest basis set are presented (see Appendix). Otherwise,see notes.1E(L-H) is the LUMO-HOMO energy difference, in kJ/mol. For the abbreviations, see Appendix.(a) 6-31G; (b) 6-31G∗∗; (c) 6-31++G∗∗; (d) D95.

Possible Relation with Excited States

It was found previously, in the case of miva-zerol,37 that the eigenvector of the H matrix relatedwith the instability was very similar to the mo-noexcitation CI expansion (CIS) vector of the lowesttriplet state, expressed in the basis of RHF–MO de-terminants. Thus, the energy of the triplet state,whose CIS expansion vector is the same as the insta-bility eigenvector, is calculated relative to the fun-damental singlet state, at the CIS level [1E(tr,CIS)].The instability seems to arise when the triplet state,obtained at the CIS level, lies under 350 kJ/mol[see 1E(tr,CIS) in Table I]. However, the CIS levelis known to provide very approximative excitationenergies, and the question remains of the existenceof a quantitative relationship between the occurenceof an instability and the value of 1E(triplet–singlet)evaluated at a higher degree of accuracy than CIS.The triplet energies, relative to the fundamental sin-glet state, are presented in Table II.

Moreover, the excitation energy of the first ex-cited singlet state was also determined for somemolecules. The results are presented in Table III.

Apart from the antiaromatic systems that will bediscussed later, and except for Cpdm1, the tripletexcited states are significantly lower than the sin-glet ones. For annulenes with a large number of πelectrons, like An18, the first excited singlet state,

becomes sufficiently near the fundamental one toperturb it and induce an internal RHF instability. Itis the only singlet instability found for the investi-gated systems. The H′ negative eigenvalue λ0(sg) isequal to −11.77 kJ/mol, and one obtains a1E(reop)of −0.467 kJ/mol at the D18h RHF(6-31G) optimizedgeometry. This was already found at the semiempir-ical level, within the π-electron model, by Cisek andPaldus,5 who predicted the appearance of a RHF in-stability for annulenes CnHn with n greater than 10.However, as emphasized by several authors,4, 5, 8 theappearance of the instability depends on the para-metrization of the semiempirical framework. At theab initio level, with a double ζ basis set like 6-31G, itonly occurs when n ≥ 18, at least for the highly sym-metrical planar structures. Let us also point out that,at the ab initio level also, the appearance of the HFinstability depends on the quality of the calculation,i.e., on the basis set expansion. For instance, ethyl-ene, furane, or the truncated cephalosporine RHFw.f.s are found unstable within the 6-31G basis setbut stable within the 6-31G∗∗ or 6-31+G∗ ones.

From Tables II–III, one can observe that the ex-cited singlet state obtained from the same π → π∗transitions than the triplet associated with the insta-bility is much higher in energy than the triplet, andeven can be only the second or third excited singletstate. Let us consider, for instance, trans-butadiene


DEHARENG AND DIVE

TABLE II.Energy Difference, in kJ/mol, between the Lowest Singlet State and the First Triplet One or That Related with theExternal Instability, at Several Calculation Levels.

Molecular System (SG) (bs) CaLe 1E SySt

Nonantiaromatic systemstrans-Butadiene (C2h) (6-31G∗∗) HF 165.2b 3Bu

CAS(4,4) 250.6b 3BuBC4H5 (C2v) (6-31G∗∗) HF 147.6a 3B2Cyclopropylium cation (D3h) (6-31G∗∗) HF 656.8a 3E′

CAS(2,3) 637.5b 3E′Cyclopentadienyl anion (D5h) (6-31+G∗∗) PP-CAS(6,8) 416.9b 3E′1(PP)(c)

PS-CAS(6,8) 304.7b 3E′′1(PS)Imidazole (Cs) (6-31G∗∗) HF 367.4a 3A′

CAS(6,6) 375.6b 3A′Pyrrole (C2v) (6-31G∗∗) HF 290.7b 3B2

CAS(6,6) 361.3b 3B2Benzene (D6h) (6-31G∗∗) CAS(6,6) 355.5b 3B1uBC5H−6 (C2v) (6-31+G∗∗) CAS(6,8) 263.6a 3B1(PS)(c)

284.1a 3A2(PS)303.3a 3A1(PP)

N6H2+2 (Cs) (6-31G) CAS(8,8) 314.8a 3A′′(SP)(c)

419.0a 3A′(PP)Cotp2 (D8h) (6-31G∗∗) CAS(6.8) 273.2b 3E1u

290.4b 3E3uAnthracene (D2h) (6-31G) HF 206.9a 3B1uPhenanthrene (C2v) (6-31G) HF 234.3a 3B2An10 (D10h) (6-31G) HF 147.0a 3B1u

CAS(10,10) 204.7b 3B1uNaphtacene (D2h) (6-31G) HF 130.3a 3B1u

Antiaromatic systemsC3H−3 (D3h) (6-31+G∗∗) CAS(4,7) −56.2a 3A′2Cyclobutadiene (D4h) (6-31G∗∗) UHF(alt)(sg) & UHF(tr) −128.7a 3A2g

UHF(reop)(sg) & UHF(tr) 102.9a 3A2gCAS(4,4) 44.6b 3A2g

C5H+5 (D5h) (6-31G∗∗) CAS(4,5) −67.7a 3A′2Benzene dication (D6h) (6-31G∗∗) UHF(alt)(sg) & UHF(tr) 34.4b 3A2g

UHF(reop)(sg) & UHF(tr) 35.1b 3A2gCAS(4,6) −58.1b 3A2gMP2/CAS(4.6) −48.6b 3A2g

(C2h) within the 6-31G∗∗ basis set. At the CIS level,the 3Bu state lies at 274.4 kJ/mol (Table I) from theground state, while the 1Bu lies at 674.7 kJ/mol. Atthe better calculation level CAS(4,4)/6-31G∗∗, this1Bu is not the first excited singlet state because itsenergy relative to the ground state is 832.2 kJ/molcompared with 644.5 for the 1Ag (Table III) corre-sponding to another π → π∗ excitation. This 1Ag

state is found at 905.4 kJ/mol at the CIS level. This isnot typical of trans-butadiene alone, because pyrrole(C2v), for instance, presents similar features. This

can be understood from the RPA expressions,25 be-cause the singlet transition matrix elements involvemore bielectronic integrals, and are thus alwayslarger than the triplet ones.

In a few cases the instability was not related withthe first triplet state but with one higher triplet state(1,2,3 triazine, N6H2+

2 , N6H4+4 , C6H6N+) or with sev-

eral [N6H4+4 , An11p1 (C1), An12p2 (C1), naphtacene

(D2h), An14, pentacene (D2h), An18, hexacene (D2h),hexaphene (Cs)]. In the former cases, the lower ex-cited states result from σ → π∗ transitions.

490 VOL. 21, NO. 6


TABLE II.(Continued)


C7H−7 (D7h) (6-31+G∗∗) CAS(8.8) −2.1a 3A′2Cot (D8h) (6-31G∗∗) UHF(alt)(sg) & UHF(tr) −17.1a 3A2g

UHF(reop)(sg) & UHF(tr) 190.7a 3A2gCAS(8,8) 66.0b 3A2g

An10p2 (D10h) (6-31G) UHF(reop)(sg) & UHF(tr) 136.5b 3A2gCAS(8,10) −24.2b 3A2g

An12 (D12h) (6-31G) CAS(12,9) 25.7b 3A2gCAS(10,10) 66.1b 3A2g

284.4b 3E1uUHF(alt)(sg) & UHF(tr) 13.2b 3A2gUHF(reop)(sg&tr) 227.5b 3A2g

The symmetry group (SG) is the same for the singlet (sg) and the triplet (tr), (bs) indicates the basis set used.CaLe = calculation level; HF means usual Hartree–Fock level for both entities, i.e., RHF(sg) and UHF(tr); UHF(alt) = UHF level forthe singlet where the last occupied β m.s.o. (initially the same as the last α one) was permuted with the first virtual one, leadingto a diradical singlet; UHF(reop) = UHF level for the singlet where the RHF wave function, found unstable, is reoptimized; SySt =symmetry of the triplet state:1E = E(triplet) − E(singlet).a Vertical triplet excitation energy.b Triplet excitation energy at the optimized geometry of the first triplet state, within the SG restraints.c The triplet state results from a σ → π∗ (SP) or a π → σ ∗ (PS) transition, instead of a π → π∗ one (PP) (default).36

TABLE III.Energy Differences between the Lowest Singlet States, in kJ/mol, at Several Calculation Levels.


Nonantiaromatic systemstrans-Butadiene (C2h) (6-31G∗∗) CAS(4,4) 644.5 1AgImidazole (Cs) (6-31G∗∗) CAS(6,6) 658.6 1A′Pyrrole (C2v) (6-31G∗∗) CAS(6,6) 643.6 1A1Benzene (D6h) (6-31G∗∗) CAS(6,6) 470.6 1B1uN6H2+

2 (Cs) (6-31G) CAS(8,8) 432.0 1A′′(SP)a

489.2 1A′′(SP)593.2 1A′(PP)

Cotp2 (D8h) (6-31G∗∗) CAS(6,8) 379.5 1E3uAn18 (D18h) (6-31G) CAS(4,4) 285.8 1B1u

CAS(8,8) 234.9 1B1u

Antiaromatic systemsC3H−3 (D3h) (6-31+G∗∗) CAS(4,7) 64.2 1A′1Cyclobutadiene (D4h) (6-31G∗∗) CAS(4,4) 216.0 1A1gC5H+5 (D5h) (6-31G∗∗) CAS(4,5) 74.8 1A′1Benzene dication (D6h) (6-31G∗∗) CAS(4,6) 45.4 1A1gC7H−7 (D7h) (6-31+G∗∗) CAS(8,8) 35.4 1A′1Cot (D8h) (6-31G∗∗) CAS(8,8) 131.6 1A1gAn10p2 (D10h) (6-31G) CAS(8,10) 18.0 1A1gAn12 (D12h) (6-31G) CAS(12,9) 65.6 1A1g

CAS(10,10) 140.6 1A1g

(SG) Denotes the symmetry point group of the molecular system, (bs) indicates the basis set used.CaLe = calculation level; SySt = symmetry of the excited singlet state;1E = E(excited singlet state) − E(fundamental singlet state).a The excited state results from a σ → π∗ (SP) or a π → σ ∗ (PS) transition, instead of a π → π∗ one (PP)/(default).


DEHARENG AND DIVE

In the latter cases, the mixing of several π∗ MOs,by the intermediate of several singly excited elec-tronic configurations in the w.f. expansion, may benecessary to provide the optimized UHF w.f. withthe best spin alternance related with the magneti-cally fully ordered state.

The importance of the perturbation leading tothe reoptimized UHF w.f. is also related (Table I)with the value of 〈S2〉(reop), which can even be-come higher than the normal value for a tripletstate (An14, naphtacene, An18, pentacene, hexa-cene, hexaphene). The greater the π electron num-ber, the greater the deviation to the singlet statedescription, which becomes highly contaminated byhigher spin states. Nevertheless, the UHF energy islower than the RHF one, although usually the con-tamination by the higher states provides a higherenergy.

RHF Internal Instability

The internal HF instability, which occurs whileremaining in the chosen HF (RHF or UHF) frame-work, is different from the external above men-tionned one. It can often be directly related withthe famous symmetry dilemma3, 7 when the ex-cited state of the same multiplicity has a differentsymmetry. As emphasized in a study on doubletstates in ET processes,41 this HF instability involvesnonadiabatic coupling, i.e., interaction between twoelectronic states that lie close to one another. Thisis what is found for An18 (D18h), where the firstexcited singlet state, about 230 kJ/mol above theground 1A1g state, is a1B1u. Thus, the reoptimizedRHF w.f. no longer belongs to the D18h symmetrypoint group because of the perturbation by the 1B1u.

Electronic Correlation

Fourth-order perturbation Møller–Plesset34 cal-culations including single, double, and quadrupleexcitations [MP4(SDQ)] were performed on bu-tadiyne (6-31G), trans-butadiene (6-31G), and ben-zene (6-31G∗∗), to estimate the electronic correlation

importance obtained with the zero-order Slater de-terminant being either the RHF w.f. or the reopti-mized UHF w.f. The results are shown in Table IV.

Though the (RHF → UHF) energy lowering1E(reop) lies around only −6.3 to −11.3 kJ/mol(results not all shown in Table I), the difference be-tween the correlation energies obtained for the RHFand UHF w.f.s lies around +40 to +80 kJ/mol. Thislargely lower correlation energy on the UHF w.f.emphasizes that this calculation level already takesinto account an appreciable amount of the electroniccorrelation. This has to be put in relation with theappearance of non zero ASDs, i.e., with a certaindegree of electronic localization that was preciselylinked35c to the electronic correlation.

HF INSTABILITY VS. THE NATURE OF THESYSTEMS AND THE π ELECTRONS NUMBER

As recalled in a previous section, the optimumgeometry results from an equilibrium between theπ “distortive” and σ “antidistortive” electronicforces;35 nevertheless, the π electrons conserve thepossibility of delocalizing over the backbone. Thisis directly related to the quantum mechanical res-onance energy (QMRE), a concept introduced byCoulson et al.,42 and widely used. For annulenes Xn,it was shown35a that the distortive energy of the πelectrons was proportional to the triplet excitationenergy of the X2 π bond Etr(X2): the higher this en-ergy, the higher the π electrons distortive energy tobring a single–double-bond alternance. Moreover,the localizing propensity of the π electrons is relatedto their correlation:35c the larger the π electronic cor-relation, the higher their tendency to localize eacharound one nuclear center.

Nonantiaromatic Systems

Geometry Optimization with the UHF Reoptimizedw.f. A further geometry optimization was per-formed with the UHF reoptimized w.f., for trans-butadiene (6-31G), butadiyne (6-31G, 6-31G∗∗), ful-

TABLE IV.Correlation Energies (kJ/mol) ERcorrx = E(RMPx) − E(RHF) and EUcorrx = E(UMPx) − E(UHF), x = 2 and 4,Obtained in the RHF or the Reoptimized UHF Framework, for Butadiyne, trans-Butadiene, and Benzene.

Molecule Basis set ERcorr2 EUcorr2 ERcorr4 EUcorr4

Butadiyne 6-31G −924.7 −833.0 −965.2 −908.5trans-Butadiene 6-31G −916.3 −846.6 −1021.1 −965.2Benzene 6-31G∗∗ −2076.0 −1993.6 −2172.5 −2127.5

492 VOL. 21, NO. 6


vene (6-31G∗∗), benzene (6-31G∗∗), naphtalene (6-31G), anthracene (6-31G), and phenanthrene (6-31G). The critical point obtained was always aminimum, as controlled by analytical frequency cal-culations, in the same symmetry point group asthe initial one, except for fulvene, though the MOsdid no longer belong to that symmetry. The ben-zene C—C bonds turned to be 0.00095 nm larger,the unsaturated bonds in butadiyne and butadienealso increased [δ(C≡C)= 0.00159 nm in 6-31G∗∗ and0.00334 nm in 6-31G, and δ(C=C) = 0.00356 nm],while the central C—C bond decreased to a smallerextent [δ(C—C) = −0.00140 nm, −0.00248 nm, in 6-31G∗∗ and 6-31G, respectively, for butadiyne, and−0.00257 nm for butadiene]. The C—H bonds weremuch less sensitive to the geometry reoptimiza-tion. The fact that the initially smaller bond lengthsincrease and that the larger ones decrease upongeometry optimization with the reoptimized UHFw.f. is also observed for naphtalene, anthracene,and phenanthrene, the systems thus tending to lesspronounced single/double bond alternance. If onerefers to previous works on σ–π electron trends,35

this should mean that the reoptimized UHF w.f. pro-vides a description of the π electrons possessing lessdistortive tendency. Moreover, the inclusion of elec-tronic correlation provides a smaller single/doublebond difference, as observed for trans-butadiene atthe MP2 level: C—C and C=C bond lengths passfrom 0.14672 and 0.13221 nm at the RHF level to0.14567 and 0.13431 nm at the MP2 one. The reop-timization of the HF w.f. takes into account a part ofthe electronic correlation.

The case of fulvene is somewhat different. Thefully optimized geometry obtained with the reop-timized UHF w.f. belongs to the Cs point group,the difference with C2v being significant and mostlyvisible on the saturated C—C bond lengths withthe ethylenic extremity; from the initial C2v valuesof 0.14770 nm, they become equal to 0.14494 and0.14961 nm in Cs. The bond lengths and ASDs onthe carbons are presented in Chart 2.

CHART 2.

The comparison of fulvene and benzene showsthat the HF instability is very similar in both sys-tems, but the geometry reoptimization with thereoptimized UHF w.f. produces quite different re-sults. As a matter of fact, for benzene, only a spinalternance results from the reoptimization of thew.f., and it does not perturb the global symmetrybecause this alternance is already a characteristicsof this system. In contradistinction, fulvene is anonalternant hydrocarbon, and the magnetically re-ordered spins cannot result in a regular alternantscheme. As seen from Chart 2, a triplet coupling ap-pears between the atomic centers linked through thelarger bond length at the UHF-optimized geometry.

Systems with Two π Electrons. In the case of sucha small number of π electrons (ethylene, cyclo-propene, truncated cephalosporine), the instabilityis very small within the 6-31G basis set and disap-pears with the 6-31G∗∗ or 6-31+G∗ ones. The cyclictension does not seem to perturb the instability.

Systems with Four π Electrons. The planarity ofthe system does influence the importance of the in-stability, by comparison of trans (or cis) and twistedbutadiene. The π electronic correlation is largerwhen the system is planar, due to the better over-lapping between the pz atomic orbitals on whichthe π MO are expanded. As already recalled, thelarger this electronic correlation, the more impor-tant the localization trend of the π electrons on eachnuclear site.35c The electronic localization manifestsitself, namely, via the ASDs derived from the re-optimized UHF w.f.: they are equal to −0.62 and+0.58 a.u. (e/bohr3) for the two distinct carbonsof trans-butadiene, and to −0.52 and +0.51 a.u. inthe case of twisted butadiene. Such an observationcould lead one to suppose the existence of a relationbetween the electronic π correlation and the HF in-stability: the higher the correlation, the higher theHF instability.

The presence of the boron in the five-memberedring BC4H5 highly increases the instability. Thecomparison of three of its properties (atomicnet charges, ASDs, and difference in C—C bondlengths) with those of cis-butadiene is presented inTable V.

The presence of the boron induces (i) a greaterbond length difference, exclusively due to thelengthening of the Cint—Cint bond; (ii) a largernegative net charge on its adjacent carbons, Cext;(iii) a larger ASD of the UHF reoptimized w.f. onthe carbons, meaning a stronger localization of theπ electrons, and thus, a larger electronic correlation.


DEHARENG AND DIVE

TABLE V.Net Charges (NC) from the Mulliken Population Analysis, on the Carbons Linked to the Heteroatom (Cext) andthe Carbons Linked to Two Other Carbons (Cint) in the Case in the Five-Membered Rings.

Molecular System NC(Cext) ASD(Cext) NC(Cint) ASD(Cint) δ(C—C)

cis-Butadiene (C2v) −0.280 −0.67 −0.110 +0.63 0.01361BC4H5 (C2v) −0.340 −0.74 −0.114 +0.80 0.01846Pyrrole (C2v) +0.078 −0.212 0.00688Furane (C2v) +0.169 −0.223 0.01016Thiophene (C2v) −0.357 −0.30 −0.119 +0.29 0.00915Protonated furane (C2v) +0.165 −0.47 −0.158 +0.46 0.01403Protonated thiophene (C2v) −0.323 −0.46 −0.092 +0.44 0.01168

A comparison is made with the cis-bytadiene (C2v,TS). The basis set used is 6-31G∗∗. The difference between the larger (Cint—Cint)and shorter (Cext—Cint) bond lengths is also presented: δ(C—C)(nm) = (Cint—Cint) − (Cext—Cint). The atomic spin densities (ASD)on the two carbons, obtained with the reoptimized UHF w.f., are given.

It seems that the presence of boron results in a re-strained space for the π electrons, probably linkedwith the geometry.

Systems with Six π Electrons. Influence of thegeometry: as already emphasized for four-π elec-tron systems, the planarity of the system influencesthe magnitude of the instability. The case of thehexatriene isomers clearly emphasizes this featurealso.

The instability of hexatriene, whatever its confor-mation, is greater than that of benzene. For trans-E,Ehexatriene, an additional geometry optimizationwas performed, at the HF/6-31G∗∗ level, under therestraint that all the C—C bond lengths are equal.This led to a structure that is not a critical point. Theoptimized bond length is rC—C(opt) = 0.13737 nm.This system has an even more unstable RHF w.f.: the1E(reop) is equal to −68.915 kJ/mol. Thus, it is notthe geometrical characteristics of the single/doublebond alternance that produces larger instabilitiescompared with symmetrical systems like benzene.It should rather be an electronic feature related withelectronic resonance. The two usual cyclic geomet-rical resonance forms for benzene leading to thewell-known idea of electronic delocalization canbe associated with two equivalent cyclic resonanceforms for spin singlet coupling pattern, as shown isChart 3.

Such coupling pattern resonance structures canbe drawn for noncyclic species like hexatriene, but

none of them should have a large weight becausethey are either diradicalar or zwitterionic; the dou-ble arrows must then show an equilibrium displace-ment towards one of the structures. Thus, it seemsthat the larger the resonance is, the lower is thetriplet HF instability.

Influence of heteroatoms: for six- or seven-membered rings, the replacement of one C—H unitby one nitrogen (benzene → pyridine, C7H+7 →C6H6N+) has nearly no influence on the instability,the “nitrogen π electron” participating equally wellto the appearance of the spin alternance. When morethan one nitrogen replace C—H units, their relativeposition in the ring is determinant in the way they

CHART 3.

494 VOL. 21, NO. 6


influence the instability. If they are connected alto-gether (pyridazine, 1,2,3 triazine, hexazine, N6Hx+

x ),the instability increases as a function of the nitro-gen number. In this series, only hexazine43 is not aminimum but a TS. Its elusiveness was explained byShaik et al.35a as coming from a π distortive strengthlarger than the σ symetrizing one. The reverse situ-ation is observed when the nitrogens are separatedby a carbon (pyrimidine, 1,3,5 triazine): the largerthe number of nitrogens, the lower the instability.In the case of 1,3,5 triazine, the RHF w.f. is stable,the instability root being positive though very small:λ0(tr) = 4.121 kJ/mol. The propensity to lead to amagnetically ordered lattice with alternating spins40

seems to be stronger when identical atoms are notseparated, at least when carbon and nitrogen areconcerned.

The influence of the number and the position ofthe nitrogens in six-membered rings was investi-gated in the spin-coupled (SC) w.f. framework44 byCooper et al. They calculated that the resonance en-ergy was the same for benzene and pyridine—thatit increased for pyrimidine and decreased for pyri-dazine. However, their definition of the resonanceenergy is not the usual one;35a they defined it asthe energy difference between the full SC descrip-tion and that corresponding to the most importantspin coupling scheme, which is the Kekule formfor all the studied structures. Given the geometry,the fact that the w.f. expands on important valencebond (VB) configurations other than the Kekuleones implies a large π delocalization: the greaterthe number of non-Kekule configurations with largeexpansion coefficients, the larger the electronic delo-calization. Thus, in this sense, pyrimidine is a moredelocalized system than benzene, while pyridine isa less delocalized one. As already deduced fromthe hexatriene/benzene comparison, it appears thatthe more delocalized the π system is, the loweris the HF instability. In contradistinction, entitieslike hexazine are π localized systems that shouldpresent a large electronic correlation,35c in relationwith the observed large HF instability.

The influence of nitrogen was also studied inseven membered rings. Chart 4 presents the waythe single/double bond alternance appears in thegeometry of the three systems C7H+7 , NC6H+6 , andBC6H7, as well as the ASDs on the heavy nuclei forthe reoptimized w.f.

The high symmetry of C7H+7 , coming from theequilibrium between the σ and π electron trends, al-lows, as a by-product, an electronic resonance, justlike benzene. This is no longer the case for nonsym-metrical species like BC6H7 and NC6H+6 . Nitrogen

CHART 4.

has a propensity to easily form four bonds. In thecase of C6H6N+, according to the calculated bondlengths, N forms two double bonds with its twoneighboring carbon atoms. This behavior produces,in return, a butadiene-like geometrical structure op-posite to N. The geometry of this system is not assymmetric as that of C7H+7 . However, the HF insta-bility is very weak also. When looking at the ASDsof the UHF reoptimized w.f., one can understandthat the degree of electronic localization is, on theaverage, of the same order as that in C7H+7 . Insteadof a null ASD on one carbon, there are two ASDsof −0.10 on two adjacent carbons, indicating thedelocalization of one π eletron around these twonuclei. For the other ASDs values, they are of thesame order, meaning a similar electronic localiza-tion degree, or conversely, a similar delocalizationor resonance tendency, for the two systems.

The boron has the characteristic of increasingthe instability in four of the cases studied. Its ASDremains null after the w.f. reoptimization. It onlymakes single C—B bonds (Chart 4) and, in BC6H7,for instance, it compels the C6 backbone to formalternant single/double bonds as in hexatriene. In-terestingly, the 1E(reop) of hexatriene is very sim-ilar to that of BC6H7. From the reoptimized UHFw.f. ASDs values, it appears that the π electrons arerather strongly localized. Thus, in seven-memberedrings, the boron seems to act as a π electron local-izer, as in five-membered rings by the fact that itforces the geometry to be of single/double bondalternant type. This boron behavior as a “nonpar-ticipant” to the π delocalization is also very clearin the fact that the six-membered ring BC5H−6 has astable RHF w.f. (λ0(tr) = +18.475 kJ/mol), and hasvery similar characterictics as the five-memberedring Cpdm1 (low π → σ ∗ transitions, for instance).In the case of borazine, Cooper et al.45 emphasized,in the spin-coupled (SC) w.f. framework, that borondid not bear a specific optimized atomic orbital. The


DEHARENG AND DIVE

situation turns to be, electronically, as if boron wasabsent, although it strongly influences the geometry.

Number of π sites: the number of nuclear sitesalso has an influence on the instability, the seven-membered ring C7H+7 being the least unstable andthe eight-membered one Cotp2 being the most un-stable, benzene lying in between. All the systemshaving six π electrons are characterized by five sin-glet coupling schemes.44 The relative weights of allthese configurations determine the magnitude ofthe resonance energy as defined by Cooper et al. Thelarger the resonance energy, the larger the π elec-tronic delocalization, and the less distinguishablethe atomic net spins become. On this basis, C7H+7should be the most aromatic system of the threementioned above.

The comparison of pyrrole and pyridine, respec-tively characterized by a stable and unstable RHFw.f., can lead one to think that the nature, or thenumber, of the nitrogen electrons involved in theelectronic delocalization is of primary importance.When two electrons are donated by the nitrogen,the HF w.f. is either stable or less unstable thanwhen the nitrogen donates only one electron to theπ network. This is confirmed by the high stabilityof borazine, despite the presence of three borons:each of the three nitrogens puts two electrons inthe π network. The peculiar behavior of nitrogenin five-membered rings was already emphasized byCooper et al.46 within the SC w.f. framework. TheSC description of borazine and boroxine by Cooperet al.45 brought to the fore the singlet coupling ofelectron pairs on the nitrogens and the oxygens, re-spectively. This is in complete opposition with thebehavior of benzene44 where the SC orbitals are verylocalized on each carbon atom, an observation quiteparallel with the spin alternance appearing in thereoptimized w.f.

Some of the five-membered rings are stable en-tities (pyrrole, imidazole, imidazolium, Cpdm1),furane being a limit case. Nevertheless, their firsttriplet state lies in the same energy range as thatof benzene, which is unstable. Thus, if there is aqualitative link between the relative CIS energy ofthe triplet state and the existence of a triplet in-stability, the extrapolation to a quantitative level isnot possible. For these systems, the number of πelectrons is higher than the number of nuclear cen-ters. For the highly symmetrical species Cpdm1, aspin alternance is not imaginable. Thus, the onlypossibility is a complete delocalization of the spindensity with a null atomic average. All the α andβ m.s.o. are the same, which corresponds to a per-fect singlet pairing between all the π electron cou-

ples. However, for systems containing one N—Hgroup or one oxygen giving two electrons to theπ system, N and O can break in some way thecomplete delocalization of the whole π cloud byimposing, in their environment, a singlet couplingof two electrons, as mentionned above.45 Thus, asfar as the remaining ω electrons are concerned, thesystems are very butadiene like, and should be un-stable. Table V presents a comparison of some oftheir properties with the butadiene-related ones.Sulfur perturbs the least the electric charge distri-bution compared with cis-butadiene, although thecentral bond is rather affected. This geometrical dif-ference become smaller when sulfur is protonated.The more the five-membered ring resembles cis-butadiene, the greater the HF instability. As a matterof fact, it is the only way to obtain a spin alter-nance in a five-membered ring with six π electrons.Nevertheless, the question of the pyrrole and furanstability remains. It appears that, even if the twoheteroatom π electrons are singlet coupled, theirpresence strongly influences (15–20% of the SC w.f.is not the Kekule form46) the π delocalization. Itseems that the furan 15% weight (or occupationnumber) of the non-Kekule VB configuration couldbe considered as a threshold for the appearance ofthe instability because thiophene, which has a RHFunstable w.f., is characterized by only 12% of thenon-Kekule form in its SC w.f.46 Let us point outthat furan and thiazole present the same occupationnumber for the most important VB configuration(R4), i.e., 85%, while this number becomes equalto 88% for thiophene. Thus, furan and thiazole aremore delocalized π systems than thiophene, whichhas a more unstable RHF w.f. than the two formers.We, again, observe a correlation between the delo-calization and the stability of the RHF w.f.

“Y” π systems: guanidine, guanidinium, and re-lated species, often referred to as Y aromatic sys-tems, all present a stable RHF w.f. Obviously, twoguanidine characteristics prevent the occurrence ofa HF triplet instability. First, the number of nu-clei that could bear a spin density is lower thanthe number of π electrons, like in five-memberedrings, in which a singlet pairing takes place whenheteroatoms are present. Second, by reference toseveral nitrogen-containing species studied here,like pyrrole, the singlet coupling preferentially takesplace on nitrogen rather than on carbon, thus lead-ing to two guanidine nitrogens with singlet cou-pling, the third one being able to bear a spin densityopposite to that on carbon. This possibility disap-pears for guanidinium where the three nitrogens areequivalent. A description in terms of a SC w.f.44 – 46

496 VOL. 21, NO. 6


CHART 5.

and of delocalization indices35c would be very in-teresting for these systems because it seems that thedelocalization is very peculiar.

Systems with More Than 6 π Electrons. All theprevious conclusions and remarks are confirmed bylarge systems.

(1) For a given number of electrons, and com-paring planar systems in a local minimum, theinstability is larger when the delocalization issmaller (cf. anthracene vs. phenanthrene, hexacenevs. hexaphene). Because the electronic π delocal-ization is a by-product of the symmetrical geome-try, this is shown schematically for anthracene andphenanthrene in Chart 5, by the representation ofthe single (r > 0.1425 nm), double (r < 0.1375 nm),and partially double (0.1375 nm < r < 0.1425 nm)bonds. As a reminder, at the RHF/6-31G∗∗ level, theC—C bond length in benzene is equal to 0.1386 nm,and in hexatriene, the mean values for the threestudied conformations are 0.1470 and 0.1327 nm.

Thus, geometrically, phenanthrene is a less sin-gle/double alternant system than anthracene be-cause the C—C are not as different. As a matterof fact, the ASDs of the reoptimized UHF w.f. aresmaller for phenanthrene than for anthracene. Thisis also true for geometries where all the C—C bondlengths are equal. This points out the importance ofthe nuclear arrangement.

(2) When a nitrogen electron pair is involved in acyclic polyene (indole), it breaks the delocalizationby a singlet spin pairing of its two electrons (nullspin density on N after the w.f. reoptimization). Asa matter of fact, indole, with virtually 10 π electrons,has a 1E(reop) similar to that of benzene.

Here above, it was pointed out that the pla-narity of the system was directly related with the1E(reop). This is confirmed by the comparison be-tween naphtalene and An10 (C2), the latter being

FIGURE 1. Optimized conformations of An14 (C1) andAn18 (C1), at the RHF/6-31G level.

not at all planar.38 By contrast, phenanthrene andAn14 (C1) have very similar 1E(reop) although onesystem is quite planar and the other is not at all(Fig. 1). Thus, several factors play a role in the im-portance of the HF instability, such as the planarityand delocalization or resonance scheme.

Isomeric comparisons: for the isomer pairs(anthracene, phenanthrene) and (hexacene, hexa-phene), the relative energies were considerablychanged when going from the RHF to the re-optimized UHF w.f. Within the RHF frame-work, the nonlinear molecules phenanthrene andhexaphene are much lower in energy than their re-lated linear isomers: 1E(phenanthrene–anthracene)(RHF) = −31.895 kJ/mol and 1E(hexaphene–hexacene) (RHF) = −98.136 kJ/mol. These en-ergy differences become much less importantwhen considering the reoptimized UHF w.f.:1E(phenanthrene–anthracene) (UHF) = −4.933kJ/mol and 1E(hexaphene–hexacene) (UHF) =−8.694 kJ/mol. This indicates that the correlation,leading to a certain degree of electronic localiza-tion, is much larger in anthracene and hexacenethan in phenanthrene and hexaphene. Calculationsat the MP levels should ascertain this hypothesis. Atthe MP2 and MP4 levels, considering a reduced setof MOs (from the 31st to the 66th) for generatingthe excited configurations and using the RHF w.f.(RMP2, RMP4) as zero-order reference, one obtains1E(phenanthrene–anthracene) (RMP2) = −26.522kJ/mol and 1E(phenanthrene–anthracene) (RMP4)= −21.995 kJ/mol. It appears that systems extend-ing preferentially in one direction (1D) and lookinglike wires (anthracene, naphtacene, hexacene) haveπ electrons that are much strongly correlated, andthen localized, than systems that span over 2D space(phenanthrene, hexaphene).

For the nonplanar systems An10 (C2), An11p1(C1), and An12p2 (C1), a small increase of the in-


DEHARENG AND DIVE

stability occurs when passing from An10 (C2) toAn11p1 (C1), but the difference between An10 (C2)and An12p2 (C1) is much larger, and of the sameorder as the 1E(reop) difference between benzene(D6h) and Cotp2 (D8h). The only common featuresof Cotp2 (D8h) and An12p2 (C1) are their charge andthe occurrence of two opposite null ASDs for thereoptimized w.f., thus dividing the cyclic backboneinto two parts containing an odd number of nuclei.

Antiaromatic Systems

The antiaromatic systems are some kinds of di-radicalar entities characterized by the fact that thetwo highest occupied MOs are degenerate andsingly occupied. Thus, their energies cannot be de-termined at the RHF level, but at least at the UHFone (denoted as UHF(alt) in Table II). Consequently,the HF instability investigated is these systems isan internal one, because one remains in the UHFframework. The internal UHF instability of their w.f.was thus investigated, because Yamagushi40 found,within the INDO framework, that C3H−3 , C4H4, andC5H+5 do present a “spin-flipping” internal UHFinstability. Internal UHF instabilities for some an-tiaromatic entities are presented in Table VI.

From these results, only the antiaromatic entitiesC4nH4n present a large internal UHF instability, theother systems being either stable or very slightlyunstable. The singlet lowest state of these C4nH4n

(D4nh) molecules is not a degenerate state, and is acritical point, i.e., its geometry is characterized by

TABLE VI.Energy Gain 1E, in kJ/mol, Obtained afterReoptimization of the UHF w.f. for AntiaromaticSystems, in the Symmetry Group (SG).

Molecular System (SG) 1E 〈S2〉b 〈S2〉a

C3H−3 (D3h) 0.0 1.06 1.06Cyclobutadiene (D4h) −231.7 1.00 1.25C5H+5 (D5h) −0.01 1.17 1.17Benzene dication (D6h) −0.69 1.25 1.25C7H−7 (D7h) 0.0 1.25 1.25Cot (D8h) −207.5 1.08 1.63C9H+9 (D9h) −1.99 1.54 1.54An10p2 (D10h) 0.0 1.64 1.64An12 (D12h) −279.6 1.15 2.27An14p2 (D14h) 0.0 2.22 2.22

The basis set used is the same as that for the CASSCF geom-etry optimization. 1E = E(UHF(reop)) − E(UHF); 〈S2〉b and〈S2〉a are the mean values of S2 for the UHF w.f. before andafter its reoptimization, in units of h̄.

a null gradient. In the case of cyclobutadiene andcyclooctatetraene (Cot), this point is a TS.47 For theionic antiaromatics, this singlet is a degenerate statewhose w.f. components, at the CASSCF level, can berepresented by

ψ (1) = c1[D1 −D2]+ · · ·ψ (2) = c2[D3 +D4]+ · · ·

where the Di are determinants for which the highestMO levels and their occupation numbers are repre-sented in Chart 6. The MO pair components (e(1)

x ,e(2)

x ) are, respectively, the HOMO and the LUMO ofthe HF level. They are degenerate when both occu-pied, but they are no longer degenerate when oneis doubly occupied and the other is virtual. Thus,four electronic configurations can be representedas D1, D2, D3, and D4 in Chart 6. At the CASSCFlevel, D1 and D2 are equivalent because the MO pair(e(1)

x , e(2)x ) is in the optimized active space, but this

is not the case at the RHF level, where only onecomponent is SCF, the other being virtual. Thesetwo components have different symmetry proper-ties, and the two RHF determinants D1 and D2 donot necessarily have the same energy. This is ob-served for the C4nH4n antiaromatics, for which theRHF energy difference between D1 and D2 lie in therange of 21–34 kJ/mol. Nevertheless, for antiaro-matic ions, D1 and D2 have the same RHF energy.This points out that, for these ions, the two mole-cular orbitals involved are actually equivalent, and

CHART 6.

498 VOL. 21, NO. 6


CHART 7.

any remixing between them could not change theirenergies, which are equal to the UHF energy of thediradicalar configurations D3 or D4. For the C4nH4n

species, the diradicalar singlet UHF w.f. obtainedbefore and after reoptimization is characterized bythe two highest occupied s.o. (HOSO) of the typeshown in Chart 7.

There is a remixing leading to new HOSOs thatdo not involve common atomic centers, as alreadydiscussed by several authors.40, 48, 49 This transfor-mation is accompanied by a large energy lowering,of about 210–290 kJ/mol, corresponding to a staticcorrelation effect. Moreover, the ASDs, equal to zeroon each atomic center before the w.f. reoptimization,become alternant, exactly as it is the case for ben-zene, each atomic center bearing alternatively a spin

up or a spin down as represented below:

For the antiaromatic ions, the spin densities onthe carbon atoms of the UHF w.f. before reoptimiza-tion (b.r.) and after (a.r.) are presented in Chart 8.

Let us point out that the UHF w.f., be it reopti-mized or not, has not the symmetry of the geometrypoint group, in contradistinction of the INDO re-sults by Yamagushi.40

The diradicalar character of the antiaromatic sin-glet state is apparent from the ASDs obtained eitherfor the stable UHF w.f. of C3H−3 , C7H−7 , An10p2,or for the nonreoptimized UHF w.f. of C5H+5 andbenzene dication. It does not appear in the nonreoptimized UHF w.f. of cyclobutadiene, Cot, orAn12, because the spin densities on the carbons arenull. However, this feature disappears in the reop-timized UHF w.f. spin densities of C4nH4n, C5H+5 ,and benzene dication, where the spin pattern be-comes completely or partially alternant. In the caseof the antiaromatic anions, because the number ofπ electrons is greater than the number of carbons,a completely alternant pattern is not possible. Thecase of C9H+9 presents a reversed situation becausethe best alternant spin scheme, without direct tripletcouplings, is found in the nonreoptimized UHF w.f.A stable UHF w.f. with noncompletely alternantspins is also found for An10p2 and An14p2. As seenabove, doubly charged nonantiaromatic annuleneshave RHF w.f. more unstable than the unchargedsystem with the same number of π electrons (com-paring benzene with Cotp2 and An10 with An12p2).The inverse is observed in the case of antiaromaticsystems. This is one of the differences between ex-ternal and internal HF instabilities. The RHF frame-work is not well suited to account for electroniclocalization leading to spin alternance, while UHFis.

The UHF w.f. of all the antiaromatic systems ishighly spin contaminated (see the 〈S2〉 values inTable VI), the value of 1 being equal to the exact av-erage of the S2 eigenvalues for a singlet and a triplet.This is to be related with the very proximity of thetriplet state (Table II), in the 42–63 kJ/mol range inmost cases.

The singlet ground state of the C4nH4n species,though characterized by the fact that the two last


DEHARENG AND DIVE

CHART 8.

valence electrons occupy two distinct degenerateMOs, are not truly diradicalar entities, by contrast tothe ionic antiaromatics. The internal HF instabilityoccurring in ionic antiaromatics results in a simplerearrangement of the ASD alternance around the cy-cle, and this constitutes a little energetic effect. Onthe contrary, passing from null to alternating ASDshas a dramatic energetic effect. This type of spindistribution, quite similar to that of benzene, for in-stance, was already pointed out by Karadakov etal.50 within the SC w.f. theory.

In the case of the ionic antiaromatic systems, fromthe ASDs before and after the w.f. reoptimization(see above), the diradicalar character seems to de-crease upon reoptimization. For the C4nH4n systems,the w.f. reoptimization leads to a spin alternancelike for the aromatics or the conjugate species, thusresembling more an external instability. However,the magnitude of 1E(reop) is unusual, either forexternal instabilities (Table I) or the internal ones ob-served for doublet states,41 and mainly due to nonadiabatic coupling. Thus, one can conclude that thestatic correlation effect mentionned above, leadingto reoptimized MOs that are combinations of AOon noncommon nuclei, is the main energetic com-ponent of 1E(reop).

The case of antiaromatic anions is particularlyinteresting. These systems are characterized by anumber of π electrons greater than the heavy nucleinumber, just as the aromatic system Cpdm1. Con-sequently, by analogy, one should find null ASDsbecause there is no way to accommodate a spin al-

ternance corresponding to all π electrons. However,the observation is quite different, as seen in Chart 8.The UHF w.f. of these species present a spin den-sity that seems to imply one singlet pairing, thesum of the ASDs being about two units lower thanthe number of π electrons. The analysis of the re-optimized α and β m.s.o. LCAO expansion clearlyshows that, in both cases of C3H−3 and C7H−7 , oneα π m.s.o. is very similar to one π β, leading toa quasi-perfect singlet pairing between them andmaking them disappear from the spin density dis-tribution, just like the case of Cpdm1.

Table II shows that, at the CASSCF level, someof the antiaromatic systems violates the multiplicityHund’s rule.51 This rule stipulates that, because ofthe degeneracy of the two highest occupied MOs,the triplet state should lie lower in energy thanthe singlet (meaning a negative 1E). This viola-tion occurs for the annulenes C4nH4n, for whichthe triplet state lies higher than the singlet. Theionic antiaromatics obey the rule. This behavior wasfound by Yamagushi,40 Kollmar and Staemmler,48

and Gallup49 for fourfold symmetric systems, andrelated with the possibility of obtaining the last twodegenerate orbitals built on nonadjacent centers,which provides a static correlation effect stabiliz-ing the singlet. Moreover, they emphasized that themagnitude of the bielectronic integrals also plays arole in the rule violation. All these studies and con-clusions were based on properties of the local D4nh

point group. Nevertheless, a more nonlocal reason-ing could be performed. For these C4nH4n systems,

500 VOL. 21, NO. 6


there exist as many π electrons as heavy nuclei;thus, one can easily imagine that a closed-shell rep-resentation would be preferred to a diradicalar onewhen a nondegenerate MO pair (e(1)

x , e(2)x ) is con-

cerned, i.e., when leaving the D4nh SG. This is notthe case for the ionic antiaromatics that can equallyaccept both closed-shell and diradicalar representa-tions, as seen from the two equivalent representa-tions of the degenerate states (see above). Then, ona large variation domain of the internal coordinatesof the C4nH4n systems, these are “usual” closed-shellsystems for which the triplet state is energeticallyhigher than the singlet one. When arriving at theD4nh symmetry, there is no reason why they shouldabruptly have a triplet state lower than the singletone; this would be an unphysical discontinuity.

For the antiaromatic ions, the correct rela-tive triplet/singlet energy is obtained only at theCASSCF level (Table II). As a matter of fact, thesinglet state is degenerate for all these species intheir highest SG and HF calculations are not ad-equate. For the neutral antiaromatics C4nH4n, thereoptimization of the UHF singlet w.f. provides thecorrect ordering singlet/triplet, but the magnitudeof the 1E is too large compared with CASSCF re-sults. The static electronic correlation correction inthe reoptimized UHF w.f. is thus largely overesti-mated, due to a very poor zero-order UHF startingguess for such systems.

RELATION WITH A SPIN-ORBIT COUPLING

Most insaturated compounds present an externaltriplet instability. The MOs corresponding to the in-saturation are expanded essentially on atomic basisfunctions of type p. These corresponds, in the atomiccase, to atomic orbitals with nonnull orbital angu-lar momentum L, and are characterized by namelyone nodal plane passing through the nucleus. Inthe absence of external field, the three p atomicorbitals are degenerate. The threefold degeneracyreflects the three possible orientation of one L com-ponent along a chosen axis. In the molecular case,the threefold degeneracy related with the orbital an-gular momentum orientation is replaced by a x-folddegeneracy related with the symmetry properties ofthe monoelectronic MO functions. The x value canbecome larger than two only in very high symme-try point groups (cubic, octahedral, icosahedral, . . .)for which the threefold degeneracy of the usual 3D(x, y, z) axes is restored. For linear or planar systems,one axis (the highest symmetry axis z) is usuallyfavored, breaking down the latter threefold degen-eracy. Then, the highest x value is 2, related with

the so-called π MOs, which all possess one nodalplane containing all the nuclei because they areexpanded on p-like atomic basis functions. As anextension of the atomic case, and because the an-gular momentum is a constant of the motion if thespin is neglected, one would think that the twofolddegeneracy of a MO is related with the angular mo-mentum the system would have if this MO wassingly occupied. In the more general case, where nosymmetry exist but where insaturations are present,x can only be equal to one, i.e., no degeneracy canbe found except those purely accidental. Neverthe-less, there exist MOs that are characterized, as πones in the planar case, by one nodal surface thatpasses through all the nuclei involved in the insat-urations. Thus, they behave locally as π MOs, andit is very temptating to associate an orbital angu-lar momentum to the electrons that occupy theseMOs. In turn, these electrons could be subject toa spin-orbit coupling,52 from the following reason-ing. The UHF framework provides a w.f. that is nolonger an eigenfunction of the S2 operator. For manysystems, the mean value of S2 is equal to or verynear its eigenvalue S(S+ 1)h̄2, S being the total spinquantum number. However, for other systems likeinsaturated ones, relaxing the w.f. constraint to be aneigenfunction of S2 by optimizing separately α andβ spin orbitals provides mean values of S2 ratherdifferent from S(S + 1) a.u. In these cases, the sys-tem behaves as if the spin was no longer a “good”quantum number. This automatically implies thatthe spin angular momentum is coupled to anothervectorial property such as the orbital angular mo-mentum. Due to the relativistic origin of the spin,such interactions appear naturally in the relativis-tic expression of the Hamiltonian, in which all theenergetic terms related with spin-orbit coupling areproportional to (−geµBh̄2/(2m2

e c2r3ij))(Si ·Lj), where

ge is the Lande factor (equal to 2.002319), µB is theBohr magneton, me is the electron mass, c is thelight speed, rij is the distance between electrons iand j, Si is the spin angular momentum vector ofelectron i (in a.u.), and Lj is the orbital angular mo-mentum vector of electron j (in a.u.). The factor(−geµBh̄2/(2m2

e c2r3ij)) is equal to 2.666 × 10−5 a.u.,

which corresponds to about 5 cm−1 if one takes(Si ·Lj) = 1. In closed-shell systems, all the termsshould cancel out by electron pairing, giving rise tototal angular momenta L, S, and J = L+ S equal tozero. However, it could be imagined that, becauseof spin-orbit coupling, some individual componentsSi ·Lj are such as they remain after summation on allthe electrons, resulting in nonzero L and S total vec-


DEHARENG AND DIVE

tors. This effect should be small, of the order of fewtenths of a cm−1, or a few cm−1 in the case of smallmolecules like benzene. But maybe is it sufficientto be in part responsible for the larger correlationfound between the π electrons in insaturated sys-tems. This assumption could be seen as supportedby the low-lying triplet state found in all the studiedsystems.

Conclusions

The existence of an external HF instability is obvi-ously a characteristic of most unsaturated systems.Although qualitatively linked with the proximity ofthe excited π∗ levels, it is not quantitatively relatedwith the excitation energy of the triplet state de-spite its denomination. It does concern, instead, thespin-coupling framework. The fact that it is a spin-linked property is already clear from the fact that theFukutome classification11 results from properties ofpeculiar groups, the symmetry group consisting ofspin rotation and time reversal. Moreover, it wasshown that the π electronic ensemble obey unusualbosonic intersite statistics under certain symme-try conditions.53 Consequently, it appears that thetriplet HF instability is related with spin proper-ties of systems that possess electrons with nonnullorbital angular momentum, typically π electronicsystems. The spin coupling pattern is maybe influ-enced by this orbital angular momentum, in whichcase one can speak of a spin-orbit coupling. If this isso, one can expect the occurrence of triplet HF insta-bilities in systems that do not present unsaturatedsites as such, but that acquire, in a certain regionof nuclear conformations, a nonnull orbital angularmomentum.

A relation between the triplet HF instability andboth the electronic correlation and the π electronsresonance, or delocalization, was pointed out: thelarger the electronic correlation, the larger the HFinstability; the larger the resonance, the lower theHF instability.

The study of triplet HF instabilities emphasizedthe existence of an electronic correlation, and therelated electronic localization, which is differentfor “1D” condensed aromatic compounds like an-thracene and “2D” ones like phenanthrene.

Heteroatoms influence triplet instabilities eitherthrough the geometry that they impose (boron, sul-fur), or through their chemical nature or, more prob-ably, through both features (oxygen, nitrogen).

Antiaromatic systems are subject to internal UHFinstabilities. These are rather small for ionic sys-tems, while they are very large for neutral C4nH4n

molecules. This feature is related to an importantstatic correlation effect.

Appendix

Here follows the list of the investigated mole-cular systems, within particular symmetry pointgroups (SG); when no SG is indicated, SG is C1. Thecalculation level for both the geometry optimiza-tion and the wave function stability study is alsoindicated for the antiaromatics, being otherwise theRHF one. The lowest level basis set used was 6-31Gbut, for several systems, one or two other basis setswere also used, including polarization or/and dif-fuse functions. When another basis set than 6-31Gor more than one basis set was used, it is indicatedin parentheses by (bs n), with n as follows: n = 1: [6-31G∗∗]; n = 2: [6-31+G∗∗]; n = 3: [6-31G, 6-31G∗∗];n = 4: [6-31G, 6-31+G∗]; n = 5: [6-31G, 6-31G∗∗, 6-31++G∗∗].

The only system not optimized at the ab initiolevel was a complex of 18 water molecules, opti-mized at the molecular mechanics (Amber) level.54

For antiaromatic systems, UHF(alt) means UHFlevel where the last β spin-orbital from a RHF guessis permuted with the first β virtual one, thus leadingto a diradical singlet state; UHF(reop) means UHFlevel from a reoptimized wave function.

ABBREVIATIONS

TS = transition state; Cpdm1 = cyclopentadi-enyl anion; C5H6NB (t1, Cs) = C5H6NB (Cs) (Nand B separated by two carbon atoms); C5H6NB(t2, Cs) = C5H6NB (Cs) [N and B adjacent (TS)];C5H6NB (t2, C1)= C5H6NB (C1) (N and B adjacent);Cot = cyclooctatetraene; Cotp2 = cyclooctatetraenedication; An10 = 10-annulene (or cyclodecaten-taene); An10p2 = 10-annulene dication; An11p1 =11-annulene monocation; An12 = 12-annulene (cy-clododecahexaene); An12p2 = An12 dication; An14= 14-annulene; An14p2 = 14-annulene dication;An18 = 18-annulene.

FULLY SATURATED SYSTEMS

Complex of 18 water molecules, cyclopropane(D3h), tertbutyl-methane, NH2CH3, NH2BH2,BH2CH3, cyclohexane.

FULLY OR PARTIALLY UNSATURATEDNONAROMATIC-LIKE SYSTEMS

Dimethyl formamide, acetylene (D∞h) (bs 3),ethylene (D2h) (bs 3), trans-butadiene (C2h) (bs 3),

502 VOL. 21, NO. 6


twisted-butadiene (C2) (bs 3), cis-butadiene (plane-TS) (C2v) (bs 3), butadiyne (bs 3), neutral arginine,side-chain protonated arginine, guanidine (bs 5),guanidinium (bs 5), methyl-guanidine (bs 5), trun-cated cephalosporine (bs 4), cyclopropene (bs 3),BC4H5 (5-membered ring) (C2v) (bs 1), fulvene(bs 3), trans-e,e hexatriene (C2h) (bs 1), cis-e,e hexa-triene (C2v) (bs 1), cis-e,z hexatriene (bs 1).

UNSUBSTITUTED AROMATIC-LIKE SYSTEMS

Cyclopropylium cation (D3h) (bs 1), cyclopenta-dienyl anion (D5h) (bs 2), pyrrole (C2v) (bs 3), furane(C2v) (bs 3), protonated furane (C2v) (bs 3), imida-zole (Cs) (bs 3), imidazolium cation (Cs), thiophene(C2v) (bs 1), protonated thiophene (C2v) (bs 1), thi-azole (bs 3), benzene (D6h) (bs 5), pyridine (C2v)(bs 3), pyrimidine (Cs) (bs 3), pyridazine (C2v) (bs 1),1,2,3 triazine (C2v) (bs 1), 1,3,5 triazine (D3h) (bs 1),BC5H−6 (C2v) (bs 2), borazine (D3h), hexazine (D6h),N6H2+

2 (Cs), N6H4+4 (Cs), C7H+7 (D7h) (bs 3), C6H6N+

(C2v) (bs 3), C5H6NB (t1, Cs) (bs 3), C5H6NB(t2, Cs),C5H6NB (t2, C1), BC6H7 (C2v) (bs 1), Cotp2 (D8h)(bs 1), naphtalene (D2h) (bs 3), indole (Cs), An10(D10h, C2), anthracene (D2h), phenanthrene (C2v),An11p1 (C1), An12p2 (C1), An14 (D14h, D7h, C1),naphtacene (D2h), An18 (D18h, D9h, C1), pentacene(D2h), hexacene (D2h), hexaphene (Cs).

SUBSTITUTED AROMATIC-LIKE SYSTEMS

Toluene, aniline, phenol (Cs) (bs 3), phenoliumcation (Cs), phenone, 2-methyl-indole, 3-methyl-indole, salicylamide (bs 3), mivazerol, protonatedmivazerol.

ANTIAROMATIC SYSTEMS

Molecular systems (SG) Calculation levels

Cyclopropyl anion or C3H−3 (D3h) CAS(4,7)/6-31+G∗∗,UHF(alt)/6-31+G∗∗,UHF(reop)/6-31+G∗∗

Cyclobutadiene (D4h) CAS(4,4)/6-31G∗∗,UHF(alt)/6-31G∗∗,UHF(reop)/6-31G∗∗

Cyclopentadienyl cation CAS(4,5)/6-31G∗∗,or C5H+5 (D5h) UHF(alt)/6-31G∗∗,

UHF(reop)/6-31G∗∗

Benzene dication (D6h) RHF/6-31G∗∗,UHF(alt)/6-31G∗∗,UHF(reop)/6-31G∗∗,CAS(4,6)/6-31G∗∗

7-annulene anion or C7H−7 (D7h) CAS(8,8)/6-31+G∗∗ ,UHF(alt)/6-31+G∗∗,UHF(reop)/6-31+G∗∗

Cot (D8h) RHF/6-31G∗∗,UHF(alt)/6-31G∗∗,UHF(reop)/6-31G∗∗,CAS(8,8)/6-31G∗∗

An10p2 (D10h) RHF/6-31G,UHF(alt)/6-31G,UHF(reop)/6-31G,CAS(8,10)/6-31G

An12 (D12h) UHF(alt)/6-31G,UHF(reop)/6-31G,CAS(10,10)/6-31G

An14p2 (D14h) UHF(alt)/6-31G

Acknowledgments

GD is chercheur qualifié of the FNRS, Brussels.

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504 VOL. 21, NO. 6

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Hartree-Fock instabilities and electronic properties