modern valence bond theory

8/13/2019 Modern Valence Bond Theory

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Mod ern valence bond theory

J. Gerratt,a D. L. Cooper,bP. B. Karadakovc and M. RaimondidSchool of Chem istry, University of Bristol, Can tock s C lose, Bristol, U K BS8 I T S

c Chemistry Department, University of Surrey , Guild ford, U K GU2 5 X Hd Dipartimento di Chimica Fisica ed Elettrochimica and C entro del CN R per l o studio delle relazioni tra struttura ela reattivitci chimica , Universita di Milan o, Vi a Golgi 19 , 20133 M ilano, Italy

Department of Ch emistry, University of Liverpool, PO Box 147 , Liverpool, U K L69 3BX

The spin-coupled (SC) theory of molecular electronicstructure is introduced as the modern development ofclassical valence bond (VB) theory. Various importantaspects of the SC wave function are described. Attention isparticularly focused on the construction and properties ofdifferent sets of N-electron spin functions in different spinbases, such as the Kotani, Rumer and Serber. Applications ofthe SC description to a range of different kinds of chemicalproblems are presented, beginning with simple examples: theH2 and CH4 molecules. This is followed by the descriptionoffered by the SC wave function of more complex situationssuch as the insertion reaction of H2 into CH2(lA1), thephenomenon of hypervalence as displayed by moleculessuch as diazomethane, CH2N2, SF6 and XeF2. The SC

description of the ground and excited states of benzene isbriefly surveyed. This is followed by the SC description ofantiaromatic systems such as C4H4and related molecules.

1 IntroductionThe description of the behaviour of electrons in moleculesinvolves the application of quantum mechanics to very complexsystems. Our ultimate objective is not simply to confirmtheoretically what we already know from experiment. Thismerely assu res us that quantum m echanics is correct. What weseek is much more: we seek insight into the behaviour of theelectrons in a molecule, an explanation of the formation of

Joe w as born in 1938 and obtained his first degree in chemistryfro m Hertford Colle ge, Oxford in 1961 and his PhD i n 1966 withProfessor I . M. Mills in Reading. He was a post-doctoral fellowwith Professor W . N . Lipscomb at Harvard, where he laid thefounda tions of spin-coupled theory , culminating in a major paperin Advances in Atomic and Molecular Physics in 1971. He hasbeen at Bristol since 1 968 , where is is Reader in TheoreticalChemistry.Mario was born in Gravedona (Co mo ),near Milan (Italy) n 1939.He obtained his first degree fro m the University of Milan. He wasa post-doctoral fellow with Professor M . Karplus at Harvard. Hecollaborated with Professor M. Simonetta. In 1976, he met Joe atCE CAM , at O rsay, where they first began their collaboration. Hewas appointed to a Ch air in Physical Chemistry in 199 4.David was born in Leeds in 1957 . He obtained hi sfirst degree fro mBrasenose Colle ge, Oxford in 1979 and h is DPhil in 1981 with

Graha m Richards, also in O xford. He was a Sm ithsonian Fellowat Harvard College Observatory fr om 198 1, where he met Joeand joined force s with him and Mario. He returned to Oxfordin 1984 as a Royal S ociety lecturer, before taking up an appoint-ment in Liverpool where he is now Reader in PhysicalChemistry.Peter was born in Sofia , Bulgaria in 19 59. He graduated inChemistry fr om the University of Sofia in 1981 and obtainedhis PhD there in 1983. From 1981 to 1986 he worked as aresearch assistant and as an assistant professor at the Facultyof Chemistry of Sofia U niversity. In 198 6, he moved to theBulgarian Academy of Sciences, where he w as a research jellowuntil 199 4. In 1990-1 995 he wa s a visiting research associate withJoe Gerratt at the University of Bristol. In 1995, Peter moved to theUniversity of Surrey where he is now a lecturer in PhysicalChemistry.

Chem ical Society Reviews, 1997 87

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chemical bonds, the characteristics of such bonds, theirstrength, type, how they form and how they break.However, one should not lose sight of the fact that adescription, if it is to carry any conviction, must also providereliable numerical results which, in addition, must be capable ofrefinement if one so wishes.The solutions to these problems are not simple, nor evenunique. Quite certainly, no answers of much value areobtainable by straightforward application of the considerablecompu ting power now available to find some kind of numericalsolution to the Schrodinger equation. Instead the problemremains with ourselves and as a result-unsurprisingly-several different app roaches have b een tried.Over the last forty or mo re years, the most fruitful approachhas seemed to be the mo lecular orbital (MO ) or self-consistentfield (SCF) approach, in spite of the fact that the MO wavefunction does not describe correctly even the most basic ofchemical processes: the breaking of a chemical bond. Never-theless, many developments have flowed from the MOapproach, som e of them of great conceptual importance, such asthe rules governing the conservation of orbital symmetry inpericy clic reactions, others of a technical natur e which allow usto progress to more complex w ave functions in which electroniccorrelation effects missing from the SCF approach (includingthose which are responsible for providing a correct descriptionof bond breaking) are taken into acco unt. This last is the methodof configuration interaction (CI method), which has nowreached a stage where ca. 108 or 109 configurations can behandled by various comp uter codes. Sophisticated extensions ofthe SC F method, such a s the multiconfigurational SC F(MCSCF) approach and the complete active space SCF(CASSCF) approach, have been developed and these areembodied in highly efficient computer codes, such asGAUSSIAN96, GAMESS, MOLPRO, MOLCAS and otherpackages which are widely available.While these techniques have benefited from several genera-tions of developmen t work by man y talented research workersto produce code s that must surely be close to optimal for scalar,vector- and even parallel-processing machines, the effect of thelarge numbers of configurations, which are inevitably involved,seriously affects our vital chemical and physical insight into theproblem.

More recently, density functional theory (DFT) a techniquethat has been in use for many years by the solid state physicscommunity (see e . g . ref. 7), has caught the attention of manyquantum chemists.8 A great deal of developm ent work has beencarried out in recent times, as is obvious to anyone w ho attendsquantum chemistry conferences, both nationally and interna-tionally. DFT clearly has a num ber of advantages as comparedto the ab initio techniques based upon MO theory mentionedabove, but questions concerning the foundations of DFT,particularly the origin of the all-important exchang e correlationpotential, remain and have indeed become more urgent.Concurrent with the introduction of MO theory and itsvariants, is the theory of Heitler and London, or valence bond(HL o r VB) theory. In fact, it was Heitler and London w ho firstshowed convincingly that the explanation of the strength ofcovalent bonding lay with qu antum theory.2 Just as important,was the clarity of the description offered by this approach. Inparticular, the HL theory identifies the exchange effect as thefundamental phenomenon responsible for those propertieswhich we associate with a covalent chemical bond: itscapability of holding together two electrically neutral atoms,valency itself, the saturation of valency and the idea of thedirectonality of chem ical bonds; concep ts which lie at the veryheart of chem istry. On the basis of these ideas, Heitler and hisstudents were able to produce a co mpelling explanation, at leastat a qualitative and even at a sem i-quantitative level, of many,i f not most , a spec ts of chemica l b ~ n d i n g . ~ ? ~Heisenberg further showed that this very same approach iscrucial to the understanding of the many different forms of

magnetism. T o this day, the Heisenberg theory remains the onlyexplanation of this central phenomenon of the physics ofcondensed matter.It would therefore seem natural for VB theory to havereceived most attention and development effort. For a shorttime, it did so. Howev er, the origin of the ex change effect liesin the ov erlap between the wave functions of the participatingatoms. This o verlap, or non-orthogonality, between the relevantatomic wave functions has been the source of serious technicaldifficulties in the wide application of the Heitler-Londonapproach. Such problems remained until new algorithmsimplemented on modern workstations with large memory,extensive disk storage and high speed I/O, effectively overcamethem.An important extension to the HL theory was the introductionof ionic structures into the wave function, i .e . the introductionof chem ical structures in which the distribution of the electronsis such that two or more of the participating atoms b ear formalpositive and negative charges. Nevertheless, the introduction ofionic structures gives rise to sever e problem s, not least from theinterpretational point of view. Even in the simplest case of H2,in order to obtain reasonable quan titative accuracy, it is crucialto add to the original Heitler-London (covalent) wave functioneqn. (1.1):yc .- {@lsA(rl)@lSB(r2)@lSB(rl)@lSA(r2)}C(a1C32 1a2> , (1.1)ionic structures of the form eqn (1.2):yl - { @lSA(rl)@lSA(r2) @lSB(rl)@lSB(r2)C ( a 1 8 2 1a21, (1.2)giving as the total wave function for the H2 molecule a linearcombination of w ave functions (1.1) and (1.2): (1.3).

Y, = CY, +C*Y,. (1.3)Here @ls A ( r l ) and @IsB (r2 ) denote 1s-like orbitals for electrons1 or 2, centred on hydrogen atoms A or B. Coefficient C 2 issmall but by no means negligible, C2/C1 = 0.25 for H2 near itsequilibrium geometry.W e are thus invited to view the H2 molecule, which as far asevery chemist is concerned, is quintessentially covalent, as aresonance mixture between a covalent contribution, representedby wave function (l.l), and an ionic part, represented by wavefunc tion (1.2), a physical picture w hich flies in the face of onesevery chemical instinct.For larger molecules, many more ionic structures can beformed. From a chemical perspective, most of them undoub-tedly appear rather unlikely if not extraordinary. Nevertheless,this mode of description is still widely used in a number ofcontemporary texts in inorganic and organic chemistry (see e . g .ref. 5) . But as the number of valence electrons increases, thepossible nu mber and type of ionic structures grows to such anextent as to obscure the original clarity of the VB description.How ever, in organic chemistry, there are som e situations inwhich ionic structures play an altogether more positive role. Forexample, resonance between covalent and ionic structuresprovides a direct explanation of the ortho-/para- or meta-directing properties of different substituents of a benzene ringunder electrophilic attack by various substituents. There is nodou bt that still today, org anic chem ists, at least in the privacy oftheir laboratories, find that this explanation is the simplest andmost satisfying.How ever, in a remarkably under-valued paper, Coulson andFischer,6 using the H2 molecule as a simple example, show edthat the ionic structures express nothing more than thedeformation of the atomic orbitals that occurs when theyparticipate in chemical bonds. They showed that wave fun ction(1.3) can be rewritten as eqn. (1.4),

q t o t = @A(rl)@B(r2)k @B(r l)@A(r2) f l (a l ( 32 la2>,(1.4)88 Chemical Society Reviews, 1997

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which is precisely of the form of the original HL function, butwhere the orbitals @A and @*,nstead of simply being atomic,are now (in unnormalized form): eqns. ( I .5) and (1.6)@A = @isA + W i s g ,@B = i sg + h+is.A

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(1.6)andthe mixing parameter h being the same as coefficient C2 infunction (1.3).1- Note that orbitals @A and @B overlap: eqn.(1.7).

(@A I @B) = A A B (1.7)Thus we see that in cases such as this, the occurrence of ionicstructures does in fact do no more than to allow the originalatomic orbitals to delocalize somewhat into the neighbouringatoms as the molecule forms, which is of course perfectlyreasonable. A s the internuclear distance R increases, h + 0 andthe orbitals revert to pure atomic form.Wave function (1.3) or (1.4) contains left-right correlationwhich is necessary for a correct description of the dissociationof the Hz m olecule. It yields 85% of the observed value of D,,the binding energy of H2, compared to 77% for the Hartree-Fock or molecular orbital wave function. This is not the onlyform of electron correlation in H 2. In particular, angularcorrelation about the internuclear axis is missing. But the typeof non-dynamic correlation present in wave function (1.4)ensures that molecular dissociation, however complex, isalways correctly described.Generally speaking, the deformations of the atomic orbitalsin a molecule are large or sm all, depending upon such factors asthe type of chem ical linkage (single, double, triple, aromatic oranti-aromatic), the disparity in the electronegativity of theatoms concerned and the bond length: in all cases, as thedistance between the atoms becomes large, i.e. as the bondbreaks, the deformation of the atomic orbitals decreases to zero,usually occurring quite suddenly at a critical internucleardistance, and the isolated atom form is regained.The passage from w ave function (1.3) to (1.4) gives us a newperspective on the role of the ionic structures and suggests anentirely novel direction for constructing electronic wavefunctions for molecules, to which we now turn.2 Spin-coupled wave functionsGeneralization of the foregoing leads us to propose thefollowing wave function for a molecular system: eqn (2.1)Y S C = V S M = d{ -. v:, 1@2 . * .@N@$@:M~ (2-1)which is known as a spin-coupled (SC) wave function. Itincorporates a number of features which do not arise in MO-based wave functions and these are described below.In the following Section, the construction of spin functions@:,M will be briefly discussed and after this we shall be readyfor a description of the physical interpretation of the spin-coupled wave function.Function (2.1) describes a system with a total number ofelectrons N, : eqn. (2.1 i).

N , = 2nc + N . (2.1 i)Of these, 2nc electrons are inactive or core electrons,described by n, doubly occupied orbitals vl, 2, . , vn,.Theyare not considered to take part in the chemical process understudy. In addition we have N active or valence electrons,which ar e the objects of our investigation. They are described byN distinct, singly occu pied orbitals @ I , $2, . @p, . @N. Theseorbitals are non-orthogonal, i.e. they overlap: eqn. (2.l.ii).

(+p @v) = A p v (2.1 ii)

They are determined in the familiar way as linear combinationsof basis functions (approximate atomic orbitals) X,, chosenbeforehand and sited on all the atomic nuclei in the molecule.Thus: eqn. (2.2)m

p=lwhere rn is the total number of basis functions. The coefficientscpp re determined by m inimising the total energy of the systemE , as we shall see. Note that ACLp 1 , i.e. the orbitals arenormalised.The core or inactive orbitals vr re similarly determined aslinear combinations of basis fu nctions, eqn. (2.3),

m

p=lbut with the add ed proviso that they are not only norm alised, butare also orthogonal to one another: eqn. (2.4).

(2.4)This property of the core orbitals simplifies many of thesubsequent formulas considerably and may always be imposedwithout changing the form o f the total wave function (2.1). Notethat in addition there is a further simplification: The coreorbitals vlmay always be taken to be orthogonal to the activeorbitals @p, again without changing the form of our assumedtotal wave function (2.1): eqn. (2.5).

(2.5)These properties of the orbitals enable us to write the totalenergy in a comp act form with a clear physical m eaning, as weshall see.We now turn to the functions @ and @ M which alsoappear in the total wave function (2.1) and play an importantrole in the theory. These are many-electron spin functions. Thefunction 0i;lc describes the coupling of the spins of the 2ncelectrons in the core. It has the simple form eqn . (2.5.i)

0:; = G ( a 1 f i 2- i l a 2 ) G(a3f i4 B3a4) ... 2.5.i)showing that the electron spins form n, pairs, each pair havinga net spin of zero. W henever there are o rbitals that are doublyoccupied, this spin function, known as the perfectly paired spinfunction, is the only o ne permitted by the Pauli principle.Function KM is different. It is an N-electron spin functionfor the N active electrons. The subscripts indicate that the netspin of these electron s is S with z-component M . A characteristicfeature of the spin-coupled approach now appears. Since the Nvalence orbitals are singly occupied, there are several distinctways of coupling the individual spins of the electrons to eachother in order to form the required overall resultant spin S. Thisnumb er is denoted byf; and is given by the simple formula (seeref. 9): eqn. (2.6).

(@p v,)= 0 (p, = 1,2, ...,N ; i = 1,2, ...,n,)

* * x m a 2 n c C32nc 2nc 1 a2 n c )

( 2 s + 1)Nf = ($/+S+l) (;N-S)

More will be said about the imp ortant topic of spin functions inthe next Section.Thus spin function OFMoccurring in eqn. (2.1) has the formof a linear combination of all the linearly independent spinfunctions, @ g M k = 1,2, .....,f? qn. (2.7).(2.7)

7 CoefficientC , is equal to (1 + h2). k=lChemical Society Reviews, 1997 89

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The coefficients CSk are known as spin-coupling coefficientsand are also determined by minimising the total energy. Theirphysical significance will be described shortly.The use of sets of spin functions as in (2.7) is one of the mainfeatures of the spin-coupled approach.Finally the operator SQ stands for the antisymmetrisingoperator. It ensures that the entire function following in brackets{ ... in eqn. (2.1) is antisymmetric, i . e . obeys the Pauliprinciple; that is when any pair of space and spin coordinates inthe total wave function Y s M re transposed, Y s M hangessign.

Wav e function (2.1) incorporates a number of parameters thatare to be optimised by m inimising the total energy. It is usefulat this stage to sum marise these:There-are the coefficients cpp in eqn. (2.2) for the spin-coupled orb itals. Since the orbital index p ranges over thevalues 1 to N and the index for the basis functions, p , from1 to rn there are Nrn such coefficients. They are not allindepend ent, since it should be recalled that each orbital pis normalised, a condition which, in effect, fixes onecoefficient per orbital.Similar considerations apply to the coefficients c lp 2.3) forthe core orbitals, though in this case, the constraints ofnormalisation and orthogonality, eqn. (2.4), reduce thenumber of independent coefficients c


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electrons i and coupling the associated spins oi,oj to singletseqn. (3.2.i).i - j ) = * aipj - piaj . (3.2.i)

But simply pairing up the spins of all electrons i, j in all possibleways will produce an over-complete set of spin functions. Thiscan be seen most easily in the case of four electrons. For a netspin S = 0, eqn. (2.6) tells us there are two linearly independentspin functions, but following the elementary Rumer prescrip-tion, we would obtain three, the spins paired up as (1-2)(3-4),or (14)(2-3), or (1-3)(24). However, there is a simplegraphical technique, devised by Rumer, to eliminate theredundant spin functions, leaving just the required number, eqn.(3.2.ii).

NN -- gV ($N+l) * (3.2.ii)We place N numbered dots in sequence around a circle, andjoin them up in pairs such that the joining lines do not cross.Thus for four electrons we have:

and this leads to two spin functions only (3.2.iii)= fi(alP2 Pla2)fl(a3P4 - P3a4), (3.2.iii)

and (3.2.i ~)@:,0;2 = G ( a l P 4 4 a l ) f l < a 2 P 3 P3a2) (3.2.iV)

as required.A general Rumer function is constructed by pairing the spinof electrons p and q , Y and s , t and u, etc. to singlets. We labelthe resulting spin function by: eqn. (3.2.v)k - ( p - q , r - s , t - u ,...). (3.2.v)

The Rumer basis of spin functions has found extensive use inorganic chemistry in the description of the mechanisms of agreat variety of organic reactions such as aromatic electrophilicand nucleophilic attack and a host of addition reactions. Evenafter forty or more years of development of molecular orbitalmethods, this mode of description obstinately remains a majorpart of theoretical organic chemistry.

The Rumer basis has proved to be particularly convenient indescribing aromatic systems, where the well-known Kekul6structures play a fundamental role. The two KekulC structuresfor benzene are illustrated as(1) and (2),together with the threecorresponding structures for naphthalene (3)-(5]. An im-

In the case of benzene, three more spin functions-those corresponding tothe so-called 'Dewar' or para-bond structures-are needed to complete theset of five spin functions for six JC electrons with net spin S = 0. In the caseof naphthalene with ten JC electrons,fAO = 42, so that in addition to the threeKekulC structures shown, there are no less than 39 further possible spinpairings-a fact which most organic chemists do not wish to know. As willbe seen, an important result of spin-coupled theory is that the role of theunwanted extra 39 structures may be considered negligible when theorbitals are optimized.

portant result which arises out of our extensive use of differentbases of spin functions, is the great utility of the little-knownSerber basis.12 This set of spin functions is constructed byconsidering pairs of electrons (1,2), (3,4), ....., ( N - 1,N in asimilar manner to that of Rumer. The pairs of spins are thencoupled to form either a singlet (S = 0) or triplet (S = 1 ) spin,which are subsequently coupled successively together to formthe final spin. A particular function in this basis is identified bythe quantum numbers, eqn. (3.2.vi), . )S (3.2.vi)in which s12, s34, ..., etc. is equal to 0 or 1, depending onwhether electrons 1 and 2, or 3 and 4 form a singlet or triplet. S4,S g .... is the net spin for four, six, etc., electrons.

k = ((. . ((s12,s34)S4; S6)S6;

The Serber function eqn. (3.2.vii),((.. (0,0)0;0)0.... O (3.2.vi )

in which the electron pairs 1 and 2, 3 and 4, . N 1 and Nform singlets is identical to the Rumer spin function eqn.(3.2.viii),( 1 - 2 , 3 - 4 ,....., ( N- l ) - N) (3.2.viii)

and to the last Kotani spin function eqn. (3.2.i~).k = OiOi.. . ) (3.2. x)

The Serber spin functions are particularly useful in displayingthe spatial symmetry properties of spin-coupled wave functions,when it is obvious that those spin functions which do not lead tothe required overall symmetry of the total wave function havezero spin-coupling coefficients. This has turned out to be ofgreat utility in cases where the introduction of electroncorrelation leads to unexpected additional symmetries of, forexample, theo lectrons in a planar system. Examples of thiswill be presented in due course.From a more general point of view, it is often physically andchemically meaningful to divide the electrons into groups and itis very convenient if this division is reflected in the mode ofconstruction of the spin functions. Consider a system in whichwe wish to focus attention upon a group consisting of N1electrons and another containing N2 electrons, where N1 + N2= N , the total number of active electrons. For example, thisdivision might refer to N1 electrons in o orbitals and N2electrons in a n system, or reflect the fact that in one particularmode of dissociation, the molecule forms two fragmentsconsisting of N 1 and N2 electrons. For such purposes it is usefulto form a set of spin functions of the type eqn. (3.2.x),

M l , 4in which the index k of the total spin function is characterised byeqn. (3.2.xi),

k = (SJ2klk2) (3.2.xi)and k l , k2 describe the 'internal couplings' of the componentspin functions and @ &,;,. The symbol( SM S1S2M1M2) stands for the vector coupling coefficientwhich couples together the two angular momenta S1 and S2.As an example of this, a spin-coupled calculation was carriedout of the dissociation of the HCN molecule in its ground state,involving the dissociation of the C-N triple bond.'? eqn.(3.2.xii).

HCN(X12+) CH(42-) +N(4S). (3.2.xii)The 14-electron molecule of HCN thus decomposes into twofragments, each consisting of seven electrons. A very conve-nient basis of spin functions is one in which the seven electronsof the CH radical are coupled to a definite spinS1 (equal to t inthe combined product, CH + N, of lowest total energy) and theN atom is in its ground state, also with a spin S2 of +.Since the

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total spin S = 0 in this example, the two sub-spins S 1 ,S2 mustbe equal. Th e other possible value of S1, S2, which may be ofinterest, 4, i),would for large C-N distances correspond to thedissociation of the molecule into the ground 2n) tate of CHand the excited 2P state of N, which corresponds overall to anexcited state of the products. U sing this 7 0 7 basis at C-Ndistances close to equilibrium, tells us how much this excitedstate contributes to the total ground state of H CN.Of course for equilibrium internuclear distances, HCN isbetter described in terms of electron-pair bonds, i.e. in theSerber or Rumer basis of spin functions. This brings us to theproblem of transforming from one set of spin functions toanother. In practical terms the problem is as follows: Acalculation has been carried out using, e . g . the Kotani basis(which may be the most conven ient) and we have obtained a setof sp in-coupling coefficients CSk ( k = I , . fi orrespondingto this basis. What are the values of the spin-couplingcoefficients csk in som e other basis?This is not merely a question of theoretical interest, since it isintimately connected to important physical and chemicalinformation which a spin-coupled calculation can yield about agiven system. Certainly such a transformation is alwayspossible and it remains to develop an efficient algorithm forcarrying it out. This has been accomplished by means of thecode SPINS (see ref. 15), which also runs on a personalcomputed and which transforms a set of spin-couplingcoefficients between the Kotani, Rurner and Serber bases ofspin functions. It is also possible to combine any of thesetransformations with a reordering of the active orbitals in a spin-coupled wave function in any manner.It is, hopefully, clear from the foregoing that there are manypossible w ays of coupling the individual electron spins to forma given resultant S and the choice is mainly dictated by theactual system under consideration and the process ( e . g . eactionor dissociation, etc.) under study. An appropriate choice of basisof spin functions sheds much light upon the behaviour of thewave function, in a very compact and physically meaningfulmanner. Unsuspected symmetries are often exposed and,combined with the shapes of the orbitals, a great deal of physicsand chemistry of the system is revealed. Such information oftensuggests, for example, energetically the most favourablereaction path, or, in the case of a molecule in an electronicallydegenerate state, frequently suggests the most likely Jahn-Teller distortion of the molecule.4 The physical interpretation of the spin-coupled wavefunctionHaving discussed the various distinctive features of the spin-coupled wave function, we are in a position to describe itsphysical and chemical significance and to assess its generalquality and reliability in relation to other available ab initioquantum chemical approaches.The spin-coupled approach, as we have seen, describes asystem with N active electrons, by N distinct, singly occupied,non-orthogonal orbitals, the spins of which are coupled togetherin all allowable ways to form the required overall resultant S.The single occupancy allow s the electrons to avoid one another,thus incorporating a significant am ount of correlation betweenthem. The overlap between the orbitals, on the other hand,allows for quantum interference effects which are crucial for agood description of bonding.Perhaps the mo st characteristic feature of the wave functionis the linear combination of many spin functions. The pairing ofthe electron s in all possible wa ys, together with the optimization8 States with ( S , ,S,) equal to 3, and ( ) are too high in energy to beof interest and are almost certainly repulsive as far as the interaction of CHand N are concerned.Copies of which may be obtained by contacting [email protected] or from http://rs2.ch.liv.ac.uk/dlc/SPINS.html

of the shapes of the orbitals, introduces further correlationeffects. These two attributes always allow for a correctdescription of molecular dissociation, however complicated. Asthe interatomic distances increase, the orbitals regain their pureatomic shape, and the m ode of cou pling of the spins also reflectsthe separation of the parts of the molecule. This aspect of thebehaviour of electronic wave functions is perhaps the mostimportant in chemistry, since the making and breaking ofchemical bonds, or their rearrangement, constitutes the veryessence of chemistry at the molecular level.The spin-coupled wave function thus incorporates a con-siderable amount of chemically significant electron correlationin a compact and highly visual form.One phenom enon which made its appearance early on in thiswork is the fact that chemical bonds do not appear to breakgradually, but on the contrary, are little affected by increasingbond distances until a critical value is reached, when largechanges are observed to occur in the wave function over a veryshort internuclear distance: ca. 0 . 5 ~ 0 , hen the orbitals rapidlylose the deformation characteristic of bond formation and at thesame time, the spin coupling coefficients also vary rapidly. Thisoccurs at an interatomic distance of c a. 4 . 5 ~ 0 , hich remainssurprisingly constant for many different species and processes.Indeed, one m ight assert that in a chem ical reaction,

A + B C +D,to a good approximation, nothing occurs between the reagents Aand B until they approach to within 4 . 5 ~ 0 f one another, whenthe reaction occurs with surprising suddenness. Hence a fairlyuniversal total reactive cross-section might be estimated to be4n (4.5a0)2 - 70 X 10-20 - 7 X 10-19 m*.As we have seen, the spin-coupled wave function is veryflexible, but it is important to understand the limits of thisflexibility: we have the freedom to order the orbitals in any w aywe please, we may choose the set of spin functions in a widevariety of w ays. Som e of this freedom can be w ell utilized inorder to reduce the am ount of com putational effort necessary indetermining the wave function (see below). However, aconcomitant danger in this apparent freedom is the opportunityfor deriving formalisms which may appear different but do notintroduce anything new into the theory.

It is also important to appreciate clearly the distinctionbetween SC theory and the older or classical valence bondtheory. In classical VB theory, the orbitals are taken to bepredetermined, either as simple atomic orbitals or hybrids ofatomic orbitals. These hybrids, m oreover, are fixed, for exam pleeither as sp, sp2 or sp3, etc. type orbitals. In SC theory, incontrast, no such preconceptions are im posed. The o rbitals areoptimized as linear combinations of basis functions (usuallyapproxim ate AOs) much as in MO-based approaches. However,in comm on with classical VB theory , the spin coupled orbitalsin general overlap with one another (except, of course, in thecase of orbitals of different sym metry), or, since the S C orbitalsare often localized , by virtue of the physical sep aration betweenthem. Generally speaking, no constraints, apart from normal-ization, are applied to the SC orb itals and as a result they maybe as localized or as delocalized as the situation demands.Bearing in mind that the S C orbitals are always singly occupied,this last means that their shapes are determined by whateverproduces the optimum balance between the greatest extent ofavoidance of the electrons in different orbitals and quantuminterference effects, which arise from the overlap betweenorbitals. In practice, we h ave found that this invariably mean sthat the SC orbitals turn out to be localized and indeed oftenresemble atomic or hybrid atomic orbitals, or semi-localized,meaning that the SC orbitals spread over two or, at most threecentres. We have found a greater degree of delocalization thanthis to be rare.One should also bear in mind that, in contrast to classical VBdescriptions, the SC orbitals remain singly occupied, the92 Chemical Society Reviews, 1997

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optimization of the orbitals, as explained above, obviating theneed to introduce ionic structures for covalent molecules.The determination of the SC wave function, the orbitals andvalues of the spin-coupling coefficients, is attained by the usu alprocedures in quantum m echanics. According to the variationalprinciple, we mu st calculate the total energy E and minimize itwith respect to these same param eters, cpp,c , ~nd CSk. This isdone by calculating the first and second derivatives of E withrespect to cpp, Ip nd CSk. Much of the algebraic detail is set outin the original paper20 and here we do no more than quote oneor two equations. The total energy corresponding to the spin-coupled wave function is given by eqn. (4.1).

= p p l 4 o p I h l o u ) + ;p u=l Ir.~ ~ p ~ , i . ) o . B . . I s , o i o l1 =l )(4.1)

Here, X stands for the usual non-relativistic electronic Ha-miltonian, containing the operators for the kinetic energy of theelectrons and all the Coulomb interactions between theelectrons and the nuclei making up the molecule.The norm alisation integral (Ys, Y s M ) is written on the righthand side of (4.1) as A and the usual one- and two-electron

The D ( p Y) and D ( p v IAt are elements of the one- and two-electron density matrices. This, together with the normalizationintegral A is where all the effects of the non-orthogonalitybetween the orbitals occurs.If inactive orbitals are present, it is sufficient to modify theone-electron operator h which occurs in (& h I )./IThe sp in-coupled wave function as represented by eqn. (2.1)is, of course, based on a single spatial configuration. There areseveral situations which are not covered by this form of wavefunction. In particular, degenerate electronic states, frequentlyaccomp anied by a Jahn-Teller distortion of the nuclearframework, pose interesting problems (as indeed such statesalso do in the MO framework). The further systematicrefinement of the spin-cou pled wave function an d the treatmentof excited states also present areas not covered by wave function(2.1). Excited states in particular have proven to be a difficultyfor the traditional valence bond approach, but are fullyaccounted for in the spin-coupled approach. This matter ismentioned briefly in Section 8.The spin-coupled wave function does not-cannot-incor-porate all the different types of electron correlation effects. Anobvious extension is the development of a multiconfigurationspin-coupled wave function and steps in this direction werealready attempted w ith the earliest applications of spin coupledtheory. 17.18 Since the spin-coupled wave function incorporatesas much radial correlation as is possible within the framew orkof a single-configuration form, the main improvements to thewave function must stem from the inclusion of doubly excitedconfigurations in which two occupied orbitals are replaced byorbitals which differ from them , in some sense, by symmetry. Inthe case of the very simple diatomic m olecules H2, LiH and L iz,(see ref. 17 and 18 above ), whose occup ied orbitals are all of 0symm etry, it is fairly clear that the main im provem ent is due to

integrals as (@p h I @v) and (@p@v I g @h@t).

I This is achieved by replacing h by the operator F,, which is the Fockoperator for the core.F , = c 2 h l c(J , -K l ) ,

/ = I / =Iwhere J , and K , are the usual Coulomb and exchange operators of Hartree-Fock theory, but here are constructed from the inactive orbitals only.

replacement of the two 0 orbitals involved in the bond by twoorbitals of n symmetry, n+n-, if complex o rbitals are used, orby n X d x J T ~ ~ ,f real orbitals are employed.In the remaining parts of this Review, w e present a selectedseries of applic ations of spin-co upled theory to different parts ofchemistry.5 Simple examplesThe m ost elementary exam ple of course is just the H2molecule.As described by eqns. (1.4) or (2.9), there are two orbitals, @Aand @B which overlap. These are displayed in Fig. 2 below. O nthe right of Fig. 2, contour plots of the two o rbitals are shown ,while on the left, orbital @A is shown as a three-dimensional

shape with the intemuclear axis superimposed. As the inter-nuclear distance R increases, the deformation of each orbital @*and @B decreases to 0, leaving just a pure hydrogen 1 orbital oneach atom . The potential curve fo r H2 given by the spin-coupledwave function is compared to the results from a number of otherwave functions in Fig. 3.We see that the spin-coupled result remains close to that fora full configuration-interaction (full-CI) wave function** for allvalues of R and then it becomes identical with it as R OAlso shown on this diagram is the potential curve given bythe original Heitler-London wave function, eqn. (1. l) , and thatgiven by MO theory (the self-consistent field (SCF) function).It can be seen that the SCF wave function does not describedissociation correctly.The potential curv e of the low est triplet state of H2, which isrepulsive, is also shown. According to eqn. (2.1), the wavefunction for this state ( N = 2, S = 1) is given by eqn . (5.1).

** A full-CI wave function is the most general variational wave functionthat can be constructed from a given basis set, either in the MO frameworkor that of VB ( in which case it is called a full-VB wave function). The twowave functions (full-CI and full-VB) are entirely equivalent.Chemical Society Reviews, 1997 93

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-0.2rlbohrFig. 3

which for A4 = 0, can be obtained simply from that for thesinglet state by reversing the signs in eqns. (1.1) or (1.4).A somewhat more complicated example is provided by themethane molecule CH4. In this case we have eight valenceelectrons and two core electrons, so that N = 8, n, = 1. Theground state is a singlet, S = 0, so that now there is a total offourteen spin functions. The SC wave function is determined byminimizing the total energy as described in Section 2. Using avery large basis set,?? the energy of the spin-coupled wavefunction is ca. 0.065 Hartrees lower than that of a SC F functioncalculated with the same basis, which is a fairly substantialdifference. This indicates that a significant am ount of the effectsof electron correlation is incorporated in the SC wavefunction.On examining the wave function, we find that the eight SCvalence orbitals form four sets of two orbitals each. The orbitalpairs are symmetrically equivalent and are permuted amongstthemselves by operations of the tetrahedral group, T d . Withineach pair, one of the orbitals is very largely localized on one ofthe H atoms and clearly resembles a H(1s) orbital with smalldeformations, almost exclusively onto the C atom. The secondorbital of the pair is localized mainly on the C atom, but withsome small degree of delocalization, almost exclusively ontothe H atom of the pair. This is shown in Fig. 4.The deformation of the C-based orbital can be seen quiteclearly. Thus, without the imposition of any preconceptions onour part, the SC wave function for methane produces what areclearly fou r, somewh at deformed, sp 3 hybrid orbitals and alsofour deformed H(1s) orbitals. It goes without saying that thiscorresponds closely to our usual concept of four-valent carbonwith bonds to four H atoms.In addition, however, we m ust consider the spin functions ofwhich there are fourteen. This exam ple illustrates very clearlysome of the features of different spin functions. Table 1 showsthem, using the Serber basis, with the orbitals ordered as ( 1, 2)(c$~,$~), 5, 6), ( 7,&J. The notation of column 2 has beenexplained in Section 3.Of these, spin function 14 corresponds to the perfectly pairedcase and this has a weight of 89.4%. This of course is what we

~~ ~ ~

t t 'Triple-zeta valence plus polarization' (TZVP).94 Chemical Society Reviews, 1997

imagine to be the case in a simple molecule such as methane. A llthe other spin functions represent different possible ways ofcoupling the eight valence electrons. It can be seen that anumber of them are zero by symmetry and that six others areexactly equal (as long as the molecule has tetrahedral sym-metry). These represent the case when there are two tripletbonds, the two triplets giving a zero net spin. Th e weight of suchan unlikely spin arrangement is naturally rather small: 1.7%.With four symmetrically equivalent bonds, there are six suchunique pairs of triplet-coupled bonds. Together with theperfectly paired spin function, this constitutes the entire spinfunction, for 89.4 + 6 X 1.7 = 100%. However, once the Hatoms mov e away from their equilibrium position in CH4, spin-coupling patterns other than the perfectly p aired case assume amuch more important role.Table 1Spin coupling coefficients for CH4 (Serber basis)

Spin-Coupling pattern Coefficient Weight12345678910

1 1121314

0.0487-0.12980.0-0.12980.00.00.00.0542- .1298- .12980.0-0.1298-0.1298

0.9453

0.00230.01680.00.01680.00.00.00.00290.01680.01680.00.01680.01680.8936

If we transform these spin-co upling coefficients to the R umerbasis, we find that the perfectly paired spin function nowcontributes ca. 80% to the total spin function. The rema inder ismade up almost entirely from structures containing a singleH-H pairing and only two C-H bonds. There are four suchstructures which, at the equilibrium geometry of CH4, are allsymmetrically equivalent. The occurrence of this type of spinpairing with significant contributions (ca. 20%), provides uswith some insight into the topic of the following section: theinsertion of H2 into CHZ(lA1).6 The CHZ(lA1) + H2 insertion reactionFollowing on from this discussion of methane, it is interesting tostudy a related process: that of the insertion of the H2 moleculeinto the carben e radical CH2 in its low-lying 'A1 excited state toform C H4, eqn. (6.1).

H2 +CHz('A1) CH4. (6.1)This reaction is obviously more co mplex than the simple bond-breaking process, eqn . (6.2)

CH4 CH3(2A'I) +H (6.2)

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and involves a fairly complex rearrangemen t of bonds. Clearlythe spin-coupling of the electrons on the right hand side of eqn.(6.1) s completely different to that of the left hand side. In orderto understand the pathway for this reaction, it is necessary tolook briefly at the SC description of C H2 in this excited state.24This is shown in Fig. 5.

;

H

H H

I I I IFig. 5 Orbitals (a) 1; (b) 2 ; (c) 3; (4 4; ( e ) 5 ; (f) 6

are plotted in the plane of the molecule. Weclearly see two C-H bon ds ( 1, $2) and ( 3 , +4). Orbitals +5and 6 are plotted in a plane perpendicular to the molecularplane and bisecting the H-C-H angle. If the two C-H bond s areregarded as pointing into the corners of a tetrahedron, then 5and @6 point approximately in the direction of the tworemaining corners of the same tetrahedron. Thus, according tothe SC description, the 'A1 state of CH2 closely resemblesmethane itself from which two H atoms have been plucked.This is quite different from the simple MO description of thisstate, according to which the two no n-bonding electrons of CH2are considered to occupy a single lone pair orbital of a1 0)symmetry, stemming from the C atom and pointing along the C2axis away from the H atoms. The following discussion focusesupon the utility of the S C description.It would appear that the most obvious approach of an H2molecule to CH2( A l) is a simple symmetric path where the axisof the incoming H2 is perpendicular to the C2 axis of themolecule an d in a plane perpen dicular to that of CH2, bisectingthe H-C-H angle . How ever, this implie s the simu ltaneou sbreaking of the H2 bond and the formation of two new C-Hbonds. According to SC calculations, this path encounters avery high-energy barrier and thus is very unfavourable. But onoptimising the various parameters which define the H2 + CH2system (see ref. 23), we find the minimum energy path is asshown schematically in Fig. 6.Th e H2 molecule HI-H2 approaches along the line of the lobeof one of the singly occupied orbitals of CH2, 5, say. Then as

Orbitals

Fig. 6 Schematic drawing of the pathway for the insertion of H2 intoCHZ(I'41)the H1-C distance decreases to ca. 3ao, the H1-H2 bond b eginsto break and a new HI-C bond begins to form-as shownclearly by the sharp variation in the coefficients of the two mo stimportant spin-couplings. Simultaneously, the second inco mingH atom , H2, now swings right around so as to interact with theother singly occupied orbital of CH2, 6, so forming the secondnew C-H bond. There is no barrier along this path and the valueof the exothermicity of the reaction, calculated from theenergies of the S C wave fun ctions for reactants and products is474.5 kJ mol-1, compared to an experimental value of 490kJ mol-1.With the benefit of hindsight, this reaction path is entirelyreasonable: one new C-H bond starts to form, followed swiftlyby a second. Energetically, this is obviously more favourablethan the simultaneous formation of two new C-H bonds. Theoverall process is nonetheless synch ronous. Although one bondstarts to form first, this process is not completed before thesecond bond begins to form. However, it would be hard topredict this highly asymmetric reaction path without knowingthe form of the SC orbitals for CH2( lA1).If one were to reverse the m otions along this path and studythe disintegration of CH4 by, say, the pumping of much energyinto appropriate vibrational modes, then this picture predictsthat as the first H atom starts to depart from CH4, it takes asecond H atom with it and, moreover, that the nascent H2molecule w ill possess a great deal of angular momen tum, i.e. theH2 will be in a highly excited rotational state. There is indeedexperimental eviden ce for this.7 HypervalenceThe im possibility of drawing satisfactory Lewis structures for awhole range of molecules, such as NO2, HCNO, N20, 03 swell-known. These are often referred to as 1,3-dipolar mole-cules. In order to bring their electronic structure and apparentvalency in line with other, less unusual systems, many chemistsfall back upon the concept of 'resonance'. The resonancestructures that are comm only draw n for NO2 are well-known toall chemists. For diazomethane (CHZNZ), which is planar, it iscommon to indicate resonance between the structures shownbelow, which indicate that diazomethane has some diradicalcharacter. Although it is unstable and tends to dissociateexplosively into N2 and an excited state of CH2, there isotherwise nothing in its vibrational or rotational spectrum toindicate that this molecule possesses any kind of unpairedelectron character.


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Spin-coupled calculations on diazomethane and a number ofother 1,3-dipolar molecules listed above have been carried outat various levels, the simplest example to consider being just thefour n electrons of CH2N2. According to the SC model, they areaccommodated in four distinct orbitals, shown in Fig. 7. The

I I I

contours are plotted in a plane parallel to the molecular planeand la0 above it. It can be seen that orbital $1 is largely aorbital centred on the C atom of CH2N2,but distorted somewhattowards the neighbouring N atom, N,. Similarly, 2 is centredon N, and is deformed towards l . However, there is a secondorbital $3 centred on N, but distorted towards the terminal Natom, N, and finally there is $4, which is a JI orbital on N, butdistorted towards $3. The spin-coupling coefficients show that,in spite of the very large overlap between the two orbitalscentred on N,, 2 and $3 (A23 = 0.785, using a very large basisset of Gaussian-type orbitals$$), the electrons occupying theseorbitals are not paired with each other, but instead there is analmost perfect pairing of the electron spins in the two orbitalpairs @ I , 2) and ($3, $4). This corresponds to the chemicalstructure:

H \ C = N E NH in which the central N atom takes part in five electron-pairbonds.The large overlap A23 between 2 and 3 should not beforgotten. We tend to think that if a pair of electrons forms onebond, then it cannot contribute to another bond, i.e. that pairs oforbitals forming a bond are orthogonal to all other pairs. Here,we see that this is not so: in spite of the clear formation of fivefully fledged electron-pair bonds, the overlaps between themare such that the average number of electrons around N, (theMulliken population) is very close to 7: i.e. the central N atomis almost exactly neutral. The same is true of N,. However, theC atom is slightly negative and the two H atoms arecorrespondngly slightly positive. This provides the H atomswith a distinct acidic character, all of which is fully in accordwith the known chemistry of CH2N2.

A triple zeta plus valence-shell polarization (TZVP) basis.96 Chem ical Society Reviews, 1997

Similar considerations hold for other 1,3-dipoles. The N 20molecule is best presented asN = N = O

in which the central nitrogen atom again participates in fiveelectron-pair bonds. The obseryed bond lengths of NzO( R N - N = 1.128,R N = 1.184 A) are certainly reminiscent ofan N-N triple bond and a normal N=O double bond.Thus, from the perspective of SC theory, the problempresented by these molecules resolves itself in a remarkablysimple fashion. Contrary to what we were all taught asundergraduates, the nitrogen atom does indeed form fivecovalent linkages and the availability or otherwise of d orbitalshas nothing to do with this state of affairs. This usually showsitself in terms of multiple bonds, such as in CH2N2,N 2 0 , F3N0,etc., rather than as five single bonds, simply because the smallsize of the N atom normally precludes the presence of fivenearest neighbours. This is not so for the phosphorus atom. Theapparent difference in valency between the first and secondrows of the periodic table is therefore a consequence of the sizeof the atoms and is not primarily due to the availability orotherwise of 3d orbitals for bonding.This conclusion is reinforced by spin-coupled and otherstudies on SF6 and PF5. The nature of the bonding in thesemolecules presents a direct challenge to conventional views onvalency. According to the SC calculations, the results for SF6,for example, show that the 12 valence electrons are accommo-dated in twelve well-localized orbitals which form six highlypolar S-F bonds. Furthermore, the nature of the orbital pairswhich form these bonds is very little affected by whether or notd orbitals on the S atom are included in the basis set. Two of theresulting orbitals, forming one of the S-F bonds, are shown inFig. 8. Once more it should be emphasized that these orbitals arethe straightforward outcome of an ah initio calculation, with noconstraints imposed as to the final form of the orbitals, nor thetype of pairing of the electron spins.It remains to rationalize this result in simpler terms.It is worth recalling that explanations of the bonding in SF6which are based upon the supposed d2 sp3 hybridization of theorbitals on the S-atom cannot explain properly why SH6, forexample, which is unknown, should not be just as stable as SF6itself. The polarity of the S-F bonds is clearly an essential partof the answer.This can be simply visualized as follows. Consider one of thebond pairs 2 ) of SF6, illustrated above. Orbital $1 consistsof a combination of a 2p orbital on one of the F atoms and an sphybrid Xs(sp) on the central S atom, where Xs(sp) is of the formin eqn. (7.1).

XS(SP) - %3s) S(3P) (7.1)1 = W P ) hXs(sp> (7.2)

so that eqn. (7.2) holds.

(see the upper orbital in Fig. 9). The other member of the pair,2 , is an almost pure F(2 p) orbital [eqn. (7.3)].

$2 = W P ) (7.3)(see the lower orbital in Fig. 9). Hence the bond pair ( 1, $2) isof the form eqn. (7.3.i).

( , $2) = { (F(2p) +hXs(sp)), W P )= ( m p ) , F(2p)) h(Xs(sp), W P ) ) (7.3.i)In other words the S-F bond has significant ionic character, butwith sufficient covalency to provide directionality.

3 The calculations on the twelve valence electrons included all 132 spinfunctions, as specified by eqn. (2.6). However, at least at the equilibriumgeometry, only the perfectly paired spin function plays any significantrole.

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F

F F

FF

F

Fig. 8

It is important to note that we may form six sp hybrids of theform (7.l), according to whether we choose the x- -, or z-axis.These hybrids however are linearly dependent, since we startedfrom just four orbitals, S(3s) , S(3px),S(3pY),S(3pJ and arrivedat the six functions (7.1). This is resolved by the incorporationof F(2p) character.The polarity of the S-F bonds in SF6 is therefore a necessarypart of the SC description.Similar considerations apply to PFs and the original paper(ref. 25) should be consulted for further details.It remains to add that almost precisely the same descriptiongoes through for XeF2, Fig. 9. There are two bonding pairs oforbitals, each one of which is very similar to the pair (Ql,2described above for SF6, leading to very polar Xe-F bonds.It is clear from the results presented in this section, that thetime has come from the much-loved octet rule to be superseded.Presented with sufficient energetic incentives, almost allvalence electrons can take part in bonding. We need retain onlyan 8-electron rule, similar to the 18-electron rule of transitionmetal chemistry. Polar bonds which shift density away from thecentral atom appear to be favoured, particularly if the formalnumber of bonds is very high. Hence differences in electro-negativity and the size of the central atom can be useful firstguides to the possible existence of a particular hypervalentspecies.

Xe

Fig. 98 Aromaticity and antiaromaticityThe concepts of aromaticity and antiaromaticity lie at the veryheart of organic chemistry. The first useful description ofbenzene was due to Kekul6 who drew the structures (1)-(2) inSection 3 (see also the footnote) and whose ideas of resonancebetween the different C-C bonds were later justified andclarified on the basis of quantum theory by Pauling in terms ofdifferent spin-pairings of the electrons in C(2p,) orbitals, i . e . nterms of resonance between the so-called Kekul6 and Dewar (orpara-bond) structures.Molecular orbital theory, however, gives an entirely differenttype of description: that of n orbitals delocalized around thebenzene ring. The associated MO energy level diagram isshown below, with the appropriate labels for the point group ofthe molecule, D6h(Fig. 10).

a2c

Fig. 10Accordingly, the electron configuration of the ground state is(a:" etg ). By generalizing this diagram, a simple but very useful


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rule was obtained by Hiickel which predicts that if n is thenumb er of carbon atom s in a ring system, those molecules with4n + 2 C atom s will be aromatic, while those with 4n C atomsare predicted to be a ntiaromatic. Over the last forty or so yearsthis has become the accepted description of benzene, while thatin terms of Kekulk and para-bond structures has becomesomewhat less common.Howev er, in 1 986 a spin-coupled calculation was carried outon the n electrons of benzene.26 As emphasised previously insection 4, no constraints or preconceptions were imposed uponthe form of the orb itals, nor upo n the type of cou pling betweenthe spins. The result showed six n orbitals, each of which islocalized around one of the C atoms constituting the ring. O neof them is depicted in Fig. 1 1 in which the co ntours of the orbital

H

Fig. 11are drawn in a plane parallel to the molecular plane and laoabove it. It can be seen that although this orbital is clearlylocalized, there are obvious deformations towards the neigh-bouring C atoms on each side. The remaining spin-coupledorbitals are obtained from the one shown by successiverotations of 2x/6 about the principal symmetry axis of themolecule. The spin-coupling coefficients in the Rumer basishave the values shown in Table 2. These numbers change onlyvery slightly with different basis sets for the orbitals. Spinfunctions 1 and 4 correspond to the two Kekulk structures andwe se e that they each make a contribution of ca . 40.5% to thetotal wave function. The remaining three spin functions, 2 , 3 and5 are the Dewar or para-bond functions and each of themcontributes ca. 6.4% to the total.Table 2

Spin-coupling pattern Coefficient Weight1 (1-2,3-4,5-6) 0.51638 0.40462 (1-4, 2-3, 6-5) - .09461 0.06363 (2-5, 3-4, 6-1) - .09461 0.06365 (1-2, 6-3,54) - .09461 0.06364 (2-3,4-5, 6-1) 0.51638 0.4046

The m ost significant feature of this result is that the energyimprovement (energy lowering) obtained by the spin-coupledwave function over that of the MO wave function (for a givenbasis set) is no less than ca. 92% of the maximum attainableimprovement using a wave function of whatever type, MO orVB, constructed from six electrons and six orbitals. In otherwords, a fully correlated wave function for the electrons ofbenzene approximates closely to the spin-coupled wave func-tion. There is thus very much more to the K ekulk description ofbenzene than was hitherto realized. This calculation has sincebeen repeated many times with basis sets of varying size andwith comp lete optimization of valence and all inactive orbitals.The results vary very little.Spin-coupled calculations have subsequently also beencarried out on many aromatic systems, such as heterocyclicfive- and six-membered rings, on naphthalene and on azulene.For naphthalene an d azulene, with ten n electrons, the orbitalsobtained are very similar to those of benzene, with the exception

of the two orbitals localized a t each of the C atoms which bridgethe two rings. These orbitals display a three-way deformation,towards each of the three adjacent carbon atoms. In additionformula (2.6) sho ws that for a ten-electron system there are 42possible spin functions which should be taken into account. Butsince the spin-coupled orbitals are fully optimized, it turns outthat the only spin functions which play any significant role inthese molecules are those corresponding to the Kekulk struc-tures [in the case of naphthalene , structures (3), 4) and (5) n thediagram in Section 31 and that the contribution of the other 39structures may be neglected.The M O description also predicts a number of excited statesof benzene. Thus, a single excitation of an electron from anoccupied MO (a2uor el,) to one of the unoccupied MOs shownon the diagram above ( e . g . o th e e2" or bZg orbitals) gives riseto a number of valence excited states. In addition, there is alsoa large numb er of Rydb erg states with energies below that of thefirst ionization potential.A constant aim of theoretical studies is to determine theseexcited states, preferably without going beyond the o/napproximation. Certainly for the ground state, to abandon theo/n separation would be to ignore a vast body of chemicalexperience, but the situation may be different in excitedstates.It turns out that the excited states of benzene (with energiesless than the first ionization potential) fall into three classes:covalent, ionic and Rydberg. An example of a covalent state isthe first singlet excited state, lBzu, ying at an energy of 4.90 eVabove the ground state. It may be represented to an excellentapproximation (see ref. 27) simply as eqn. (7.3.ii)

Y( BZu) = K 1 - K2 (7.3 ii)i . e . as the negative combination of the two Kekulk structures ofthe ground state. Covalent states are in general fairly easilydescribed within the o/x approximation.On the o ther hand, ionic states require linear combinations ofstructures of the type:

in which two orbitals occupy one C-atom site, while on theneighbouring C atom, there are none. Ionic states of benzene aremuch harder to describe within the o x ramewo rk. Physically,it is obvious that this is due, in part, to the existence of positiveand negative charges in the n-electron distribution, whichcauses a static polarization of the 0 ore, so that the core differsfrom that of the covalent states. In addition, there are furtherdynamic o-x interactions which fall outside the o/xapproxima-tion. In any case, a reliable description of these effects requiresextensive basis sets which include diffuse atomic orbitals.Lastly, there are the Ry dberg states. These are characterizedby one orbital which is very diffuse and extends a significantdistance from the molecule. Given a basis set which includessuch diffuse atomic orbitals (even to the extent of centring themat the midpoint of the m olecule), su ch states are not too difficultto describe w ell.Th e spin-coupled description of the excited states of benzenethus leads to an important and useful classification: the valencestates are covalent or ionic, the latter being significantly harderto describe than the covalent states, and Rydberg states, whichdiffer physically from the valence states, but otherwise are notdifficult to determine accurately.In the MO description, all the valence states arise from one ortwo singly excited reference co nfigurations, such as (gu:g e2Jor (a$ue:g b2& and it is not at all clear from this why the v ariousvalence excited states should be so physically distinct.The simplest antiaromatic system is cyclobutadiene, C4H4. Asimilar diagram as for benzene for the en ergies of the molecularorbitals of C4H4, assuming a square-planar geometry, gives:

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C4from which it follows that the electron configuration of the nelectrons in the ground state is (a;, ei) . However, the eg MO isdoubly degenerate and according to Hunds rules, the twoelectrons with this energy will distribute themselves one in eachMO and with spins aligned parallel. MO theory thus unambigu-ously predicts that the ground state of square-planar cyclobuta-diene should be a triplet (3Al.J.This is not so. It is now well-established that the groundelectronic state of cyclobutadiene is a singlet. In the square-planar geometry (D4h symmetry), the state has the symmetrylBlg,with the 3A2gstate lying at an energy of ca. 0.43 eV aboveit. Furthermore, the square-plane of the ground state is unstable,the molecule preferring to distort to a rectangular geometry withtwo short C-C double bondsnn and two longer C-C singlebonds. However in the 3A2, state, the square plane geometry isstable.From the spin-coupled wave function for the ground state,one obtains four orbitals of n symmetry without imposing anypredetermined form on these orbitals, nor on the type of spincoupling. Two of the resulting orbitals are shown in Fig. 12.

I HFig. 12Orbitals (a) ,; (h ) 2

They are plotted in a plane parallel to the molecular plane butla0 above it. It can be seen that and $2 are centred about thetwo horizontal C atoms, Cl and Cz. Orbitals and $4 are thesame as , and $2, but are rotated by x/2 about the mainsymmetry axis and instead are centred about C3 and C4 in thediagram, where C3 lies vertically above C4. We note that orbital$2 possesses an extra nodal plane passing through C3-C4,compared to $1 (similarly $4 and $3).Even more remarkable are the spin couplings, for these showthat orbitals 42) re almost exactly coupled to a triplet andthe same holds for ($3, $4), the two triplets coupled to give anoverall value of the spin S for the four electrons of zero.l(((Analysis of how orbitals $,-$4 behave under the operations ofD4h, shows that the total wave function for the ground state ofC4H4 has the correct lB1, symmetry. Furthermore, the squareplanar geometry is not stable and the molecule distorts to arectangle. In the course of this distortion, orbitals rapidlybecome localized over the four atoms C1-C4 and clearly showthe formation of two C-C double bonds which are shorter thanthe remaining two C-C single bonds.17 A second order Jahn-Teller instability.I[[ The phrase almost exactly is important here (spin coupling coefficients0.999865 and 0.0166468), for If GI , $ 2 ) and ( 3, 4) were each exactlycoupled to triplets (with spin coupling coefficients 1.0 and O.O), then theoverall wave function would remain unchanged by the replacement of

The four orbitals of the triplet state are remarkably similar tothose of the singlet ground state. The spin pairing is also verysimilar, orbital pairs ( , $2) and ($3, Q4) each forming a triplet.These two triplets, however, are now coupled to form an overalltriplet, as required for this state. This is found to have an energy0.410 eV higher than that of the ground state, which compareswell with the experimental value of - 0.43 eV (see above).It thus appears that antiaromatic character is connected to theformation of a triplet spin from a pair of electrons in two distinctorbitals, such as and $2 above. We refer to such acombination of orbitals as an anti-pair.In order to place the concept of anti-pairs found forcyclobutadiene within a wider context, several related systemswere studied, one of which is 2 4-dimethylenecyclobutane-1,3-diyl (DMCBD), shown below:

This molecule has six electrons in six orbitals ofn symmetry (itis an isomer of benzene) and can be regarded as being derivedfrom C4H4by removing two H atoms from cyclobutadiene andsubstituting them with methylene groups. From this, one wouldpredict that only one of the anti-pairs found in cyclobutadienewould remain. This is indeed the case. The ground state ofDMCBD is a triplet. Orbitals $2) and ($3, 44) are, asindicated in the diagram above, four highly localized C(2n)orbitals and (together with the appropriate 0 orbitals) formnormal C-C double bonds. The two remaining orbitals of the C4ring, denoted by a and b, form an anti-pair very similar to oneof those in cyclobutadiene itself.Even more remarkable is the bismethylenebiscyclobutyl-idene molecule (BBB), shown below. This is similar toDMCBD, but with an extra C4 unit, plus methylene group,added.

b dOrbitals ( 5, +J, c ~ ,8) and (&,, form the n componentsof fairly conventional C-C double bonds. However, while theterminal orbitals Q7 and $9 are deformed towards theirrespective partners, 8 and 10, the other orbitals centred on Cg,Cg, C8 and Cl o are deformed in three directions, due to thepresence of three C-atom neighbours.BBB turns out to have anti-pairs in both C4 rings, i . e .orbitals(a,b) and (c,d). That is, each ring has associated with it a netelectron spin of S = 1. The spins of the two C4 units, however,are aligned antiparallel with each other, giving the ground stateof BBB a net spin of zero. This is therefore very much akin toan antiferromagnetic system, except that the spins stemmingfrom each C4 unit each have the value of unity. It is not too hardto imagine an organic polymer consisting of an infinite numberof such C4 units and displaying this kind of antiferromagneticbehaviour.According to the Hiickel4n rule, cyclooctatetraene (CSHS)isthe next member of the antiaromatic series after cyclobuta-diene. Consequently, one would expect that the SC picture ofbonding in this molecule would, in some way, remind one ofthat observed for C4H4. However, SC calculations recentlycarried out at the lowest-energy tub-shaped 0 2 d geometry ofCsHs, as well as at two idealized geometries: a D 8 h regularoctagon and a D4h octagon with alternating carbon-carbon bondlengths show something different;29 see Fig. 13.The eight active orbitals at the D 2 d and D4h eometries formfour identical, largely independent olefinic carbon-carbon JC (or


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Dt3h D4h

D2d(view fiomabove)Fig. 13

D2d(view from side)

at the tub-shaped geometry, almost n;) bonds. Resonance isinsignificant (perfect pairing within the bonds represents by farthe most important spin function) and the conclusion is that, atthese two geometries which include the one experimentallyobserved, cyclooctatetraene is definitely non-aromatic. Anti-aromaticity is restricted to the idealized regular octagonalstructure. However, the nature of the SC wavefunction at thisgeometry is different from that for cyclobutadiene. The eightequivalent SC orbitals are localized and very similar to those inbenzene; there are no antipairs. The key to the low stability andhigher reactivity of the molecule-the two main characteristicfeatures of antiaromatic systems-is in the spin-couplingpattern: in the Serber spin basis, spin functions involving tripletpairs are responsible for 81% of the spin function, with 75contributed by a spin function made up of triplet pairs only,(((1,112;1)1;1).The comparison between the SC descriptions of cyclobuta-diene, benzene and cyclooctatetraene clearly indicates that thereason for the lower stability and higher reactivity of anti-aromatic systems is due to a simultaneous unfavourablecoupling of the spins of all valence orbitals to triplet pairs,which discourag es bonding interactions and sug gests diradicalcharacter.9 Summary and conclusionsIn this short survey we have attempted to describe a range ofdifferent chemical systems to which spin-coupled theory hasbeen applied and, hopefully, demonstrated the clarity andfreshness of the chemical insights that the theory offers. Thewhole style of description used in this article differs radicallyfrom that traditionally em ployed by the more orthodox methodsof quantum chemistry.Inevitably, a different choice of topics could have been m ade,so that those covered fall far short of the many applications sofar of spin-coupled theory. Among the topics that have beenquite arbitrarily excluded is the description of degenerate statesand the study of Jahn-Teller distortions, the application of spin-coupled theory to electron-deficient compounds such as theboranes, a mo re detailed account of virtual orbitals and their use

in refining the ground state wave function and the determinationof excited states of molecules, intermolecular forces, electro-cyclic addition reactions, N-S heterocyclic ring systems andcharge-transfer collisions in plasmas.On the other hand, there still remain many technicaldevelopments and improvements that could be incorporatedinto the spin-coupled codes. Since these mostly do not involveany fundamentally new theory, but straightforward extensionsof known methods (e.g. gradients, see ref. 28), such develop-ments have taken second priority to the wide application ofspin-coupled theory to many different types of chemicalsystems.It has long been considered that the use of non-orthogonalorbitals would lead to a formalism of immense complexity,which in turn would require computing resources that wouldmake such an approach hopelessly inefficient. In fact, we seethat the opposite is true: the formalism leads to a description ofmolecules and chem ical systems that is extremely c ompact andhighly visual, and hence long expansions of the wave functionin terms of different configurations, which obscures all our vitalinsight, are avoided.This, finally, is the major success of spin-coupled theory.10 References

I R. B. Woodward and R. Hoffmann,Angew. Chem., nt. Ed. Engl., 1969,8, 781.2 W. Heitler and F. London, 2. Physik, 1927, 44, 455.3 W.Heitler, Marx f fand h .Radiologie, 1934,11,485.4 G. Nordheim-Poschl, Ann. Physik., 1936, 26, 258.5 N. N. Greenwood and A . Eamshaw, Chemistry of the Elements,6 C. A. Coulson and 1. H.-Fischer, Phil. Mag., 1949, 40, 386.7 J. C. Slater, Quantum Theory of Molecules and Solids. Vol. IV,8 R. G. Parr and W . Yang, Density Functional Theory fo r Atoms and9 M. Kotani, A . Amemiya, E. Ishiguro and T. Kimura, Tables of

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McGraw-Hill Book Co., New York, 1974.Molecules, 1989.Molecular Integrals, 2nd edn., Maruzen, Tokyo 1963.10 R. Pauncz, Spin Eigenfunctions, Plenum Press, New York, 1979.11 G. Rumer, Gottingen. Nachr., 1932, 337.12 R. Serber, Phys. Rev., 1934,45,461;J . Chem. Phys., 1934,2, 697.13 M. Sironi, D. L. Cooper, J. Gerratt and M. Raimondi, J . Mol. Struct.

14 J. C. Manley and J. Gerratt, Computer Phys. Commun., 1984,31, 75.15 P. B. Karadakov, J. Gerratt, D. L. Cooper and M. Raimondi, Theoretica16 G. A. Gallup, R. L. Vance, J. R. Collins and J. M. Norbeck, Adv.17 S. Wilson and J. Gerratt, M o l . Phys., 1975,30, 777 .18 N. C. Pyper and J. Gerratt, Proc. Roy. Soc., 1977, A355, 407.19 J. Gerratt and M. Raimondi, Proc. Roy. SOC., 980, A371, 525.20 J. Gerratt, Adv. Atomic Mol . Phys., 1971,7,141.21 P. A. Hyams, J. Gerratt, D. L. Cooper and M. Raimondi,J . Chem. Phys.,1994,100,4408; 1994,100,4417.22 P. B. Karadakov, J. Gerratt, D. L. Cooper and M. Raimondi, J . Chem.Phys., 1992, 97, 7637.23 M. Sironi,D. L. Cooper,J. Gerratt and M. Raimondi,J . Am. Chem. SOC.,1990,112,5054.24 M. Sironi, D. L. Cooper, J. Gerratt and M. Raimondi, J . Chem. Soc.,Faraday Trans 2 , 1987,83,1651;S. C. Wright, D. L. Cooper, J. Gerrattand M. Raimondi, J . Chem. Soc. Perkin Trans. 2 , 1990, 369.25 D. L. Cooper, T. P. Cunningham, J. Gerratt, P. B. Karadakov andM. Raimondi, J . Am. Chem. SOC., 1994, 116,4414.26 D. L. Cooper, J. Gerratt and M. Raimondi, Nature, 1986, 323, 699.27 E. C. da Silva, J . Gerratt, D. L. Cooper and M. Raimondi, J . Chem.28 J. Gerratt and I. M. Mills, J . Chem. Phys., 1968, 49, 1719.29 P. B. Karadakov, J. Gerratt, D. L. Cooper and M. Raimondi, J . Phys.

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Chem., 1995, 99, 10186.Received, 2nd October I996Accepted, I7th December 1996

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