A Quantum Theory of Molecular Structure and Its Appllcatlons

8/8/2019 A Quantum Theory of Molecular Structure and Its Appllcatlons

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Cywmr. Rev. 1991. 91. 93-928 893

A Quantum Theory of Molecular Structure and Its AppllcatlonsRICHARD F. W. BADER

oepamnnt of amn&y, m s w uwersmy, t i a m ,"io ~ S I.am&Recdved oclober 24. 1990 Revlsed Manuscr&It Recdved MBrch 28, 1991)

cmtents1. Inlroduction 893893

8 9 38 9 48 9 48 95

A. An Atom in a Molecule as an OpenB. Outline of the Theory and It s Appllcations11. The Quantum Mechanics of an Open SystemA. The N e e d for a Quantum Description of anB. The Actlon Principle InOuantum Mechanics

Quantum System

Open Systemand Schwinger's Principle of StationaryActionC. Atomic Principle of Stationary ActionD. Open Systems and Fluxes in VectwE. Consequences of the Zero Flux BwndaryCurrentsConditionA. Variatbnal Derivallon of Atomic Force andVirlai Theorems6. Energy of an Atom in a MoleculeIV. Transferability of Atcfnlc PropertiesA. Energy Additivity in Normal HydrocarbonsB. Origin of Strain Energy in Cyclic

111. Definition of Atomic ropert ties

HydrocarbonsV. Molecular Structure and Structural stabilityA. The Notion of Struclure in ChemistryB. Molecular GraphsC. A Theory of Molecular SbuctweVI. Applicatlons of the Quantum Theory ofMolecular StructureA. Bond Order, Bond Path Angle. BondEllipticity. and Structural Stability6. Atomic Populations and MomentsVII . Ropertles of th e Laplaclan of the ElectronicCharge DensityA. Role of the Laplaclan in the Theory of6. Laplaclan of the Charge Density and theC. Classification of Atomic Interactions

Molecular StructureLewis Electron Palr Model

VII I . ConclusionsI. Introduction

8 9 88998999 0 19 0 19 0 39 0 49 0 49 069 0 79 079 089 109 139 1 39 1 79 1 99 1 99209 2 29 2 8

A. An Atom In a Molecule as an Open QuantumSystemTh e role of physics in chem istry is to predict whatcan be observed and to provide an und erstanding ofthese observations. T he dom inant operational conceptof chemistry is that of an atom in a molecule with adeterminable and characteristic set of properties. Thusth e physics of chemistry is necessarily the phy sics ofan a tom in a molecule, th at is of an open system , one

I._

1i

Richard F. W. Bader was banh 1931. He recaivad MS B.SC. andMSc. from McMaster University and hlsW.D. in 1957 from theMassachusetts Institute of Technolosy wwking wHh Professa C.G. Swah. He dd postdoaorai wak at MIT fmm 1957 to 1958 andfrom 1958 to 1959 at Cambridge University in the laboratow ofProfessor H. C. Longuet-Higglns. He began his sclentlflccareeras a physical organic chemist. He was at the University of Ottawafrom 1960 to 1963 and then moved to McMader University.whereh e remains today.which is free to exchange charge and momentum withneighboring atoms. To extend the predictions ofphysics to the domain of chemistry it is therefore,necessary to generalize quantu m mechanics to a sub-system of a total system. Such a generalization is in-deed possible, but only if th e open system satisfies aparticular boundary condition. When this condition ismet, one obtains a definition of an atom in a moleculeand a prediction of its properties. Inseparable from th equan tum definition of a n atom in a m olecule is thedefinition of th e bonds which link th e atoms to yielda molecular structure. In essence, the quantum de-scription of an open system recovers the molecularstru ctu re hypothesis-that a molecule is a collection ofatoms each w ith a chara cteristic set of properties, thatare linked by a network of bonds. Th e emergence ofthis hypothesis from 1 9th century experimental chem-istry is thus seen as having been an inevitable conse-quen ce of physics.B. Outllne of the Theory and It s Appllcatlons

It is the purpose of this article to review the deriva-tion of th e quantu m mechanics of an open system an dto illustrate its use in the development an d applicationof a theory of atoms in molecules.' The article beginswith a review of the generalization of quantum me-chanics th at leads to the definition of an open systemand to a prediction of it s properties.'" Th is generali-zation is accomplished through an extension ofSchwinger's principle of sta tionary action, an extension0009-2665/91/0791-0893$09.50/0 0 99 1 A n " chenacalSooiay


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894 Chemlcal Revlews, 1991, Vol. 91, No. 5which is possible only if a c ertain bou ndary conditionis satisfied. Th e boundary condition demands that th eflux in the grad ient vector field of th e charge densityp(r) vanish a t every poin t of th e surface S(Q,r) hichbounds an open systern Q. Th at is, the surface is oneof zero flux in O P . ~ s a consequence of th e boundary

Vp(r).n(r)= 0 V r E S(Q,r) (1)being stated in terms of a property of the electroniccharge density, quantum subsystems are defined in realspace. Because of the dom ina nt topological prop ertyof a molecular charge distribution-that it exhibitsmaxima at the positions of the nuclei-the bou ndarycondition leads to the partitioning of a m olecular systeminto a set of disjoint sp atia l regions, each region con-taining in general, a single nucleus. The se regions areidentified with th e chemical atoms. Th e properties ofthe gradient vector field also contain th e informationneeded for a definition of molecular str uct ure an d itsstability,6 by using the ma them atics of qu alitative dy-namics. Th e result is a theor y of atom s, bonds, struc-ture, and structural stability.It is a primary purpose of this paper to dem onstra tethat quantum mechanics predicts the properties ofatoms in molecules just as it predicts the properties ofthe to tal system. Following the review of th e general-ization of qu an tum mechanics to an open system whichyields a definition of an atom a nd its properties, exam-ples of the application of the resulting theory of atom sin molecules to chemical problems are presented . Theseexamples are chosen to illustrate the principal featuresof the theory:(a) Th e demonstration that each atom makes an ad-ditive contribution to th e average value of every prop-erty of a molecular system. Th is is the p rinciple un-derlying the cornerstone of chemistry-that atom s andfunctional groupings of atoms make recognizeablecontributions to the total properties of a system. Onepredicts the properties of some total system in termsof the properties of the functional groups it containsan d conversely, one confirms the presence of a givengroup in a molecule through the observation of itscharacteristic properties. In those limiting situationswherein a group is essentially the same in two differen tsystems, one obtains a so-called additivity scheme forthe total properties, for in this case the atomic con-tributions as well as being additive are transferablebetween molecules. It will be shown th at th e methyland methylene groups as defined by the theory of atomsin molecules predict th e additivity of the energy whichis experimentally observed in normal hydrocarbons.Th e deviations in this add itivity which are found forsmall cyclic molecules and which serve as the experi-mental definition of strain ene rgy are also predicted bytheory. Th e recovery of these experimentally mea-sureable properties of atom s in molecules by the a tom sof theory confirm s th at they are the atom s of chem istry.(b) The definition of bonds, molecular structu re, andstruc tural stability as determined by the gradient vectorfield of th e charge density is exemplified in a num berof system s, including those whose charge distribution sare accessible to experimental measurement. Th e im-portance of distingu ishing between molecular geom etryand the generic concept of molecular structure is il-lustrated an d discussed, and it is shown tha t a theoryof m olecular structu re is obtain ed without recourse to

Baderthe Born-Oppenheimer approxim ation.Second only to the molecular structu re hypothesis inthe ordering, understanding, and predicting of chemicaleven ts is th e Lewis model of the electron pair.' Th ismodel an d its assoc iated models of molecular geometryand chemical reactivity find physical expression in thetopological properties of the Laplacian of the electroniccharge d e n ~ i t y . ~ ~ ~his scalar field, defined by thesecond derivatives of the electronic charge density,determ ines where elec tronic charge is locally concen-trated and depleted, and it plays a dominant rolethrou gho ut the theory of atom s in molecules. Th us thereview also illustrates the following:(c) T he recovery of t he Lewis model of the electronpair in term s of the topological properties of the La -placian of the charge density and the use of the La-placian to predict molecular geom etries and chemicalreactivity.(d) The ab ility of the Laplac ian of th e charge density,when used in conjunction with the definition of achemical bond and t he local mech anics governing thecharge density as afforded by theo ry, to yield a classi-fication of atomic interaction s. Th is classificationscheme is directly applicable to experimentally mea-sured charge distributions.I I . The Quantum Mechanics of an Open SystemA. The Need for a Quantum Description of anOpen System

It is a postulate of qua ntum mechanics th at every-thing th at can be known abo ut a system is containedin the sta te function \k. Th e value of a p hysical quan -tity is obtained through t he action of a correspondingoperator on \k. Th us qu antum mechanics is concernedwith observables, the linear Hermitian operators asso-ciated with the physical properties of a system, andtheir equations of motion. T he theorems of quan tummechanics that yield relationships between variousobservables, such as the virial and Ehrenfest theorems,are derived from the Heisenberg equation of motion.Questions we have abou t a qua ntu m systenl are there-fore, answered in terms of th e values and equa tions ofmotion for the relevant physical observables. The sevalues and relationships refer to the to tal system. Th euse of the a tomic concept in our attem pts to understandand predict the properties of molecules and solids,however, requ ires answers of a more regional nature andit would appear that to find chemistry within theframework of qu antum mechanics one mus t find a wayof determining the observables and the ir prope rties forpieces of a system . How is one to choose the pieces?Is there one or are there many ways of partitioning asystem into pieces in such a way that quantum me-chanics predicts the ir properties? If there is an answerto this problem then the necessary information mustbe contained in the state function \k, for \k tells useverything we can know about a system.Therefore, the question, "Are there atoms in mole-cules?" requires the asking of two equally importantquestions: (a) Does the sta te function predict a uniquepartitioning into subsystems? (b) Does quan tum me-chanics provide a complete description of th e subsys-tems so defined? To answer questions a and b one mustturn to a development of physics that introduces the


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Quantum Theory of Molecular Structurequan tum observables and th eir equations of motion ina no narbitrary way, as opposed to one based on w hatwould be an arbitrary extension of the correspondenceprinciple. Such is Schwinger's principle of sta tionaryaction." It replaces the conventional array of assum p-tions based on classical Hamiltonian dynamics an d th ecorrespondence principle with a single quantum dy-namical principle. Th e approach is also a very generalone, one tha t enables the asking of questions a an d b.B. The Actlon Prlnclple in Quantum Mechanicsand Schwinger's Principle of Stationary Actlon

In 1933 Dirac published a paper entitled "The Lan-grangian in qua ntu m Mechanics". After presenting adiscussion as to why the Langrangian formulation ofclassical mechanics could be considered to be morefundamental than the approach based on the Ham il-tonian theory, Dirac went on to say, "For these reasonsit would seem desirable to take u p the question of whatcorresponds in the quan tum theory to th e Lagrangianmethod of classical mechanics."A knowledge of th e transform ation fun ction or tran-sition amp litude (qn, tz lqrl , t l ) uffices to determine thedynam ical behavior of a system with time , since it re-lates that st ate function \k(qr2,tz) a t t ime t z o t h a t a tt ime t l according tolo( q r z , t z l q ) = q ( q r 2 , t Z ) = J q r z , t z t q r l , t l ) dqrl*(qrl,tl)

(2 )The symbol Qrl is used to den ote the complete set ofcomm uting position operators for the particles at timetl and qr l , heir eigenvalues. Dirac was the inventor oftransformation theory and throu gh repeated use of themultiplicative law of transfo rma tion functions he wasable to express the function connecting states a t timest l an d t zby a sequence of transform ation functions fortimes interm ediate between the in itial and final times.Take n to the limit of the successive intermediate timesdiffering only infinitesimally one from the next, themu ltiplicative law yields a pro duct of all the transfor-mation functions associated with the successive infi-tesimal increments of time. Dirac then sta ted tha t thetransformation function associated wth th e time dis-placement from t to t + dt corresponds to exp[(i/ h ) Ldt] , where the Lagrangian L was to be considered as afunction of th e coordinates at tim e t and the coordinatesa t t ime t + dt , ra ther than of the coordinates and ve-locities. Th e transformation function then becomesexp[(i/h)W ] where W, he action integral equal to JLdt between the limits tland t2 , s interpreted as the sumover all the individual coordinate-depende nt terms inthe succession of values of t. W ith this construct Diracwas able to answer the question of what in qu antummechanics corresponds to the classical principle ofstationary action.Feynman built on this work and in 19481 it culmi-nated in his path integral formulation of quantummechanics. In th e classical limit considered by Dirac,only one trajectory connects the system at tim e t l tothat a t t ime t2an d he lim ited his discussion to this case.W hat Fey nma n did was to consider all the trajectoriesor paths t ha t connect the state s at the initial and finaltimes, since he wished to obtain the correspondingquan tum lim it. Each pat h has its own value for the

Chemical Reviews, lQQl,ol. 91, No. 5 895action W and a ll the values of ex p [( i / h) w must beadded together to obtain the total ransition amplitude.Th us the expression for the transition am plitude be-tween the states l q r l ) and lqr2) is the sum of the ele-mentary contributions , one from each trajectory passingbetween tjrl at t ime tl and tj n at t ime tB.Each of thesecontributions has the same modulus, but its phase isthe classical action integral (l / h ) JL dt for the path.This is expressed as

Th e differential 6qr(t) ndicates tha t one must integrateover all paths connecting qrl a t t l an d qn a t t2an d 1/Nis a normalizing factor.'lSchwinger's quantu m action principle pu t forth in19514 s also concerned w ith the de term ination of th etransformation function. A statem ent of this principleisa ( q r 2 , t 2 1 q r l , t l ) = (i/h)(qrz,t216Wlzlqrl,tl) =

where W12 s the action integral operator and 2 is theLag-range function operator. Equ ation 3 is a differentialstatem ent of Feynman's pat h integral formulation, andwhile Schwinger developed i t independ ently, it can beobtained as a consequence of Feynman's principle (seefor example, Yourgrau and M andelstam"). Th e actionprinciples afford conceptual advantages in formulatingthe laws of quantum mechanics and represent morethan alternative formulations of the laws of quantummechanics. In fact, they may provide the real founda-tion of qu antu m mechanics and thu s of physical theo-ry.12The quantum action principle (eq 3) embodiesSchwinger's postulate th at if variations are effected ina quantum mechanical system, the correspondingchange in the transformation function between theeigenstates lqrl,tl) and lqrz, tz) s ( i / h ) imes the m atrixelement of the variation of the action integral W lzcy ne cti ng the two states. Th e action integral operatorWlz is defined as

(4)where 2 [ t ] is an invariant Hermitian function of thefield \k and its first derivatives. Th e principle of sta-tionary action is obtained from eq 3 by noting tha t aninfinitesimal unitary transformation can also be usedto obtain a differential charac terization of a transfo r-mation function. Th e operator U

( 5 )and its inverse 0-1= i + ( i e / h ) G (6)where t denotes an infinitesimal real quantity and dis a linear Hermitian operator, induce infiqitesimalunitary transformations. In what follows, tG will berepresepted by the infinitesimal unitary operator F,where F is referred to as the gen erator of th e trans -formation. Th e infinitesimal transformation inducedon an observable & is defined to be4

(7)

0 i - ( i c / h ) &

a& = & - &' = ( ic /h)[&,&]= ( i / h ) [ P , a ]


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896 Chemlcal Reviews, 1991, Vol. 91, No. 5and the sam e transformation w hen applied to the as-sociated state vector yields

la) = la') - la ) = - ( i / h ) R a ) (8)T he effect of altering the tw o comm uting sets of posi-tion operators at times t l an d t 2 n the transformationfunction (qn,tzlqrl,tl) n to 4,' - a&, and Qn -64, by theaction of t he two infinitesimal generating operators F( t Jan d F ( t 2 ) s given by6 ( q r 2 , t 2 1 ~ r l , t l ) = (6qr2,t21qrl,tl) (qrd2lQr1,tl) =(i/h)(qrZ,t#(t2) - F ( t l ) ( q r l , t l ) 9)If the parameters of a system are not altered, then thevariations of th e action integ ral in eq 3 arises only frominfinitesimal chang es of th e sets of com mu ting observ-ables a t the two times tl an d t 2 However, by eq 9, sucha transformation is characterized in terms of the gen-erators of infinitesimal unitary transformations F(t,)and F ( t 2 )acting on the two eigenvectors. Th us bycomparing eqs 3 an d 9 one obtains for such variationsthe result

(10)which is the operator principle of stationary action. Itstates th at th e action integral operator is unaltered byinfinitesimal variations in state functions between thetimes tl and t2, being affected only by the action ofgenerators a t the two time endpoints.In the principle of stationary action, the variation ofthe action integral does not vanish a s it does in H am-ilton's p rinciple, but instead e quals the difference in th eeffects of infinitesim al generators acting a t the tw o timeendpoints. This esult requires tha t the variation of theaction integral appearing in eqs 3 an d 10 be generalizedto include the variations of the sta te functions and ofthe time a t the time endpoints. Th e principle of sta-tionary action then implies the equation of motion ofthe systems as obtained in Hamilton's principle, andthe endpoint variations define the generators of theinfinitesimal canonical transformations which inducechanges in the dynamical properties of the system. Inthis way a single dynamical principle recovers not onlythe equation of motion, but also defines the observables,their equations of motion, and the Heisenberg com-mutation relations.Th is generalized variation of the ac tion integral maybe illustrated and its analogy with the correspondingclassical principle made clear by expressing the La-grangian operator in terms of the commuting set ofposition operators Qr t and their time derivatives. Asumm ary of the more com plete discussion given in ref1 is presented here. Th e action integral operator is

6%0'2 = P(t2 )- P ( t J

Bader

To first order in the infinitesimals, the required gen-eralization of th e action integral reduces to the changein L long the varied path between the unvaried tim eendp oints and the unvaried integrand times th e varia-t ion in the t ime a t the two t ime e nd ~ 0i n ts . l ~hat is

After using aq integration by pa rta to rid th e resultingvariation in f of the variations one obtains theresult

6Q12 = c(Tltz{(d2/dQrt)- d(t32//&,)/dt)6& dt +R t 2 ) - P(tJ (13)

where the genera tors are defined in term s of the vari-ations in 4, and the time at the ti%e endpoints, andfollowing a Legendre transform of L hey may be ex-pressed as

Th e symbol A& denotes the complete change in thecoordinate operatorABrt = sa,, + d,6t (15)

A comparison of eq 13 with th e principle of stationaryaction, eq 10, yields the equation of motion:s l / a a r , = d ( a 2 / a i j r t ) / d t (16)

since satisfaction of this principle requires that thequan tity under the tim e integral multiplied by the ar-bitrary variations 6&, must vanish, as it does in therestricted variation of the action integral where one seta6W12 = 0.Th e generator defined in eq 1,4 is composed of twoparts: the temporal generator -Hat yields the Heisen-berg equation of motion for q bservable & when usedin eq 7, while the generator (dL/dQ,)aQ, is the generatorof purely spatial changes.'J3 Th e use of this lattergenerator in eq 7 to indue% infinitesimal changes in theoperators sart an d Prt= t3L/dQrt yields th e H eisenbergcom mutation relations.'*'3 Th us the principle of sta-tionary action provides a complete description of aquantum dynamical system. Th e demonstration th atthis p rinciple applies to a properly bounded subsystemof some total system is sufficient to establish thequantum mechanics of a subsystem.A number of alternative expressions of the principleof statio nary action will prove useful in its applicationto a subsystem. Th e first is a restatem ent of eq 10 togive

P ( t ) = ((d2/dBrJAQr, - R (& , , d 2 / d t r J 6 t ) (14)

By dividing both sides of this e quation by t2- l andsubjecting th e result to the limit At - , one obtainsan expression for the principle of stationary action interm s of a variation of the Lagrange functiona l [ t ]= d P / d t (18)

This result can be equivaleptly expressed, by usingHeisenberg's equation for dF/dt, aswhere it is understood that t he variation in2 s effectedby the a ction of th e infinitesima l generator F. Equation19 is the operational statement of the principle of sta-tionary action.' It determines the equations of motionfor the observables and the related theorems, such asthe Ehrenfest and virial theorems, which determine themechanics of a given system.C. Atomic Prlnclple of Statlonary Actlon

Th e generalization of the principle of stationary ac-tion to a subsystem is necessarily stated in the coor-dinate basis, as he bo undary condition is defined in real

a l p ] = ( i / h ) [ R , P l (19)


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Quantum Theory of Molecular Structurespace and th e Schrodinger representation of th e sta tevector is employed in what follows.Th e action integral W I2[\k] for the total system is

Chemical Revlews, 1991, Vd. 91, No. 5 687single-particle natur e of th e op erator in eq 26, one candefine a related single-particle density f "(r,t). Thisis accomplished by a summ ation over the spin s of allthe electrons, followed by a n integ ration over all coor-dinates bu t those of one electron, a process denoted bythe symbol J d f . When this result is multiplied by N,the nu mber of electrons in the system , this is the sameprocedure used to obtain the charge density p(r, t) from\k*9. The density obtained in this manner is

L o ( r , t ) = l d + L o = -(h2/4mN)V2p(r,t) (27)Because of the natural boundary condition that \k*V-\k.n and \kV\k*-nvanish on the b oundaries of the sys-tem a t infinity, the vanishing of the Lagrangian integralf" [\ k ,t ] can be taken to be a consequence of the van-ishing of the flux in the gradient vector field of thecharge density a t the infinite boundary of the system,sincef o [ \ k , t ] = - ( h 2 / 4 m N ) l d r V 2 p ( r , t )=

- ( h 2 / 4 m N ) ~ d S ( r , t ) V p ( r , t ) . n ( r , t ) 0 (28)In anticipation of the identification of a quantum

subsystem with an atom, the subsystem Lagrangian andaction integrals are referred to as atomic integrals. T heatomic Langrangian integral is obtained from the La-grangian density in eq 21 by the summ ing of all spinsand integ ration over the coordinates of all electrons bu tone, followed by the integration of the final electroniccoordinate, denoted by r , over the basin of the a tom Q,as indicated in eq 29. Correspon dingly, th e atomic

WI2[\k]= x I t * f [ Y , t ] t =J ,tzdt I d ' L[\k,V\k,+, t] (20)

where the Lagrangian integral f [ \k ,t ] s obtained by theintegration of the Lagran gian density over the coordi-nates of all the p articles in the system. In the absenceof extern al fields, the Lagrangian density for the systemof many particles interacting via a many-particle po-tential energy operator V isL[\k,VS,\k,t] = ( ih/2)( \k*+ - +*\k) -(h2/2m) zivi\k*-vi\k - v\k*3 (21)Th e variation to first-order of this action integral withrespect to the independ ent variables \k an d \k* and with69 and 69* 0 a t the time endpoints, yields for theextrenum condition that 6W12 = 0, Schriidinger'sequationswhere the H amiltonian fi s given byEi = - (n2 /2m)Civ f - CiCJae2(lri - xa1)-1

ih& = H\k and -ih+* = H\k* (22)

CCe2(lr i - rj1l-l + C C 2ZJ&lx, - X,& (23)Te rms of the form V**&\k appear in an integra l overthe surface of the system when an integration by partsis used to rid the variation in W12of term s of the form6 V q . Th us to obtain eqs 22 as he E uler equations inthe variation of the action integral requires that oneeither demand that 69 vanish on the boundaries of thesystem at infinity or, tha t th e stat e function satisfiesthe so-called natural boundary conditions, that VWn= 0 and V\k*-n= 0 on the same infinite boundaries.Th e Lagrangian density and th e integrals it definesexhibit an imp ortan t property at the point of variationwhere Schrodinger's equ ations hold, Le., where 6WI2=0. Denoting by Lo he Lagrangian density obtained a tthe poin t of variation, one has, using eqs 22,L O = - ( h /4m)Ci{\k*Vf\k*+ \kvf\k* + 2vi\k*-vi\k}(24)

This can be further simplified by using the followingidentity which relates the kinetic energy as it appea rsin Schrodinger's equation with that appearing in theLagrangian

i


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898 Chemical Revlews, 1991, Vol. 91, No. 5boundaries of the system and a t the time endpoints, theconstra int imposed on the va riation of th e action inte-gral in Hamiltons principle t o obtain th e equations ofmotion. Thu s the variation of \k in the atomic actionintegral with the necessary retention of 6\k on theboundaries and hence a t th e time endpoints neces-sarily leads to the generalization of the variation of theaction integral that yields Schwingers principle ofstationary action.Such a generalized variation of th e atom ic action in-tegral given in eq 30 is de tailed elsewhere1i2 and theresult is

&derIn the case that P s a vector, J F isa second-rank tensor.T o reexpress the result given for AW in eq 35 in theform analogous to eq 17, we need the Heisenbergequation of motion for F(Q,t) . This is obtained by usingeq 22 for Schrodingers equation of m otion to givedF(Q, t ) /d t = (1 /2 ){( i /h ) ([A&), + cc) +

#ds(Q,r)((ss/st)pF(r) - [(1/2)(J~(r)cc)l.n(r))(39)

Comparison of this exp ression with that for the changein action in eq 35 shows that the term s subtracted fromthe endpoint averages of t he generator are just thosewhich account for the surface contributions to thisdifference, integrated over the time-like surface con-necting the two time endpoints. Th us what remains isthe difference in the values of th e generator a t the twotim e endpoints averaged over the interior of the sub-system, th e essence of the principle of statio nary action.Th e change in th e atomic action integral can be ex-pressed entirely in terms of th e interio r averages of th egenerator asThe quantity j is the vector current an d it is definedas

j(r) = (h/2mi)Sd7( \k*V\k - V\k*\k) (33)Its variation 6&r) as it ap pears in eq 32 is6*j(r) = ( h / 2 m i ) S d r ( q * 6 V I - V\k*6\k) (34)This term is obtained by combining th e surface termarising from the variations with respect to V\k, with thesurface term arising from the imposition of the varia-tional constraint (eq 31). Thus the variation of th esurface of the subsystem tog ether with the restrictionth at t he subsystem be bounded by a zero flux surfacecauses the quan tum mechanical current density j toapp ear in the variation of the action integral, a termwhose presence is a necessary requirem ent for the de-scription of th e prop erties of an open system.By proceeding a s before, the v ariations in the st atefunction are replaced by operators which act as gen-erators of infiQitesimal ni ta ry transformations. T ha tis, G\k=(-il_h)- where F s an infinitesimal Hermitianoperator (F= tG). Introducing the n otion of generatorsinto the result for th e variation of the atom ic actionintegral yieldsAW12[\k,Q]= F(Q,t2)- F(Q,t l )- S d t # d S ( P , r )

{(6S/at)pF- [(1/2)(J&j + cc)]*n(r)) 35)T he resu lt is expressed in terms of property averagesfor N electrons, so AW = N6W. Th e atomic averagesof the generator at the tim e endpoints F(Q,t) and thecorresponding property density pF are defined asF(Q, t )= S d r P F ( r , t )=R( N / 2 ) J d r l d ~ ( \ k * h+ (&)*\k] (I)* (36)and

&,t) = ( N / 2 ) 1 d r { \ k * h + (f3\t)*\k] (37)The contzibution to the current density for the ob-servable F is

JF(r , t ) = (Nh/2m i) Id . { \k *V ( lb ) (V\k*)P \k ](38)

a result equivalent to th e statem ents of stationa ry ac-tion obtained for the total system, eq 17. T he principleof station ary action for a subsystem can be expressedfor an infinitesimal time interval in terms of a variationof the Lagrangian integral,as given in eq 19 for the totalsystem. For the atomic Lagrangian this statemen t is6L[*,Q,tl = (1 /2 )Ki /h ) ( [p , f i )n + cc) (41)

For stationary state, th e Lagrangian integral, apartfrom t he presence of a Lagrang e multiplier to ins urenormalization of $, reduces to the energy functionalused by Schrodinger14 n th e derivation of th e station -ary-sta te wave equa tion. For an atom in a molecule ina stationary state this energy functional isB[$,Ql =& d r S d ~ { ( h ~ / 2 m ) ~ ~ V ~ $ * . V ~ ~(p+X)$*$)42)where denotes the full many-electron potential energyoperator, and A, the variational constraint on th e nor-malization of $, is identified with -E , th e negative ofthe total energy. Th e atomic statement of the sta-tionary action in terms of this functional is1p3

WWI = - (1 /2 ){( i /h ) ( [ f iA)n + cc) (43)Th e derivation of th e principle of station ary actionfor an atom in a molecule in the time-d epend ent caseor in a stationary state, or in the presence of a n elec-tromagnetic field,15 yields the correspondingSchrodinger equation of motion for the total system,identifies the observables with the variations of thesta te functio n, defines their av erage values, and givestheir equations of motion. Th e state m ents of the at-omic principle of statio nary action as expressed in termsof variations in L[\k,Q] nd G[$,Q] are variationalstate me nts of H eisenbergs equa tion of motion and ofthe hy pervirial theorem for a generator F, espectively.They yield the theorems and relations governing themechanics of an atom in a molecule. Because of thevariational derivation of the se atomic statem ents of th eprinciple of statio nary actio n, they a re satisfied by th e


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Quantum Theory of Molecular Structuresame class of approx imate w ave functions, RH F andUHF, as satisfy the derived theorems, such as thegeneralized Hellmann-Feynman and hypervirial theo-rems, for a total system. Th e reader is referred toEpstein's book on the variational m ethod in quan tumchemistry for a discussion of the validity of thesetheorems for a to tal system.16*Equations 40-42 represent a generalization of quan-tum mechanics. They enable one to obtain a quantummech anical description of the p ropertie s of any regionof space bounded by a surface of zero flux in the gra-dien t vector field of the charge density. In this sense,the mechanics of a total system is obtained as a specialcase of these more general equations.D. Open Systems and Fluxes in Vector Currents

Corresponding to eq 35 for the variation in W12[!J!,fl],the s ubsystem pro jection of the v ariation of the en ergyfunctional for a stationary st ate is equal to the infini-tesimal flux in the cu rrent density through the surfaceof the subsystem, eq 44 .M[+,fl] = -(1/2)($dS(n,r)jF(r).n(r) + CC I (44)Th e same surface integral appears in the subsystemstatem ent of the hypervirial theoremlsb (the stationarysta te analogue of eq 39 )W / h ) ( + , [fiA$)n cc) =

I$dS(Q,r)jp(r)-n(r) + cc] (45)and because of th e dependence of the both q uantitieson the c urrent flux one obtains the atomic statem entof the principle of stationary action for a stationarystate as given in eq 43 . Thi s principle forms the basisfor the discussion of the mechanics of an atom in amolecule.Th e nonvanishing of the flux of a q uantum mechan-ical current is what distinguishes the mechanics of asubsystem from th at of the total system in a stationarystate . Th e flux in the curre nt density will vanishthrough any surface on which $ satisfies the naturalboundary condition, V + - n = 0, a condition which issatisfied by a system with boundaries at infinity. Thu sfor a total system the energy is stationary in the usualsense, as[+] = 0, and the usual form of th e hypervirialtheorem is obtained w ith th e vanishing of the commu-tator average ($ , [ f iA$)0 (46)E q y t i o n 46 is a consequence of t he Herm itian propertyof H, a property no t enjoyed by a subsystem. Th edifference between the average of the Ham iltonian andits Herm itian conjugate equals the flux in the cu rrentdensity through the suiface bounding th e systems2When an observable G does not possess a shar p valuein a statio nary state , i.e., its commutator with H doesnot vanish, there is a nonvanishing curre nt whose netoutflow from any infinitesimal region is determined bythe corresponding commutator

V*jG= ( i / h)$*[fi,bl$ (47)The energy is not stationary over a volume s2 in sucha situation, i ts ghange being determined by the flux ofthe cu rrent of G hrough th e surface, eq 44 , or equiva-

Chemlcal Reviews, 1991, Vol. 91, No. 5 899lently, by the average of the co mm utator, eq 43. Fromthis discussion it is clear that H etains th e propertyof a H ermiticity over a subsystem in the case

only when fi an d 6 commute.In sum mary, a subsystem is an open system, free toexchange charge and mom entum with its environpent.Thus the current density jG or any observable G is ofparticular im portance in the mechanics of a subsystem,since a nonvanishing flux in this current implies afluctuation in the subsystem average value of theproperty G. Because of the presence of the surface termin eq 45 , the hypervirial theorem for a subsystem leadsto important physical results which have no c ounterpartfor the total system.E. Consequences of the Zero Flux BoundaryCondition

Th e fundamental result of the theory, as containedin eq 40 for a time-dependen t system and in eq 43 fora stationa ry state, is tha t the properties of a region ofspace bounded by a surface of zero flux in th e grad ientvector field of the charge density are predicted byqua ntum m echanics. These are th e only physicallyrealizable quantum subsystems defined by the actionprinciple.lP2 Th e question stil l to be answe red iswhether such regions exist and w hether they correspondto the a toms of chemistry. Affirmative answers to thesequestions are obtained as a consequence of the principaltopological property exhibited by t he e lectronic chargedistribution-that in general, it exhib its local maxim aonly at the positions of the nuclei. Thi s is illustratedin Figure 1 by the charge density for the m olecule SF6which exhibits behavior tha t is typical of th e vast m a-jority of systems. Show n is a plot of the trajectorie straced ou t by the gradient vectors of the charge density,each vector originating a t infinity. Every trajectory orgradient path terminates a t a nucleus and this behavioris found in all three dimensions. Th e nuclei are theattractors of the gradient vector field of the chargedensity and the result is a partitioning of th e total spaceof a system into a set of disjointed mononuclear regionsor basins, a basin being the open region of space trav -ersed by all of the trajectories of Vp terminating at agiven attracto r. This is a partitioning into atoms wherean atom is defined as the union of an attractor and itsbasin.'V6 It is clear from the figure tha t each such regionis bounded by a zero flux surface in Vp and that i t sproperties are therefore, predicted by quantum me-chanics.Every trajectory of Vp originates and term inates a ta critical point in this field, a point w here Vp = 0. Acritical point, with coordinate rc, s characterized by thenumber of zero eigenvalues of the associated Hessianmatrix, the matrix of second d erivatives of p(rJ whichdetermines its rank Q, and t he algebraic sum of theirsigns which determine its signature A. Th e local max-ima, as found at the positions of the nuclei, behavetopologically as do (a,X)= (3,-3) crit ica l points! The reis a (3,-1) critical point between sulfur and each fluorinenucleus, but no t between the fluorines themselves. Th eeigenvectors associated with th e two negative eigenva-lues of such a critical point generate a set of gradient


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900 Chemical Revlews, 1991, Vol. 91, No. 5

Figure 1. Displays of the charge density in th e form of a relief (uppe r) an d contour map s (left -hand side) and of the gra dient vectorfield of th e charge dens ity (right- hand side) for a symmetry pla ne contain ing the sulfur and four of the fluorine nuclei in SFB.T he(3,-1) critical points (d ots) are indicat ed on the lower diagrams. Th e upper gra dient vector field ma p shows only those trajectorieswhich originate a t infinity and termi nate a t the nuclei whose positions are denoted by crosses. The property of a nucleus to act a san attra ctor in this field leads to a disjoint part itioning of space into a se t of atom ic basins each of which is bounded by a zero fluxsurface (eq 1). Th e lower diagram also shows (in bold) the pair of gradient pat hs which originate at each (3,-1) critical point (whereV p = 0) and define the atomic interaction lines, and the pair of gradient paths of the set of paths which terminate a t each criticalpoint and define the intersection of th e interatomic surface with th is plane. The molecular graph consists of S octahedrally linkedby bond paths to six F nuclei. Th e outer contour of the charge density equals 0.001 au. Th e succeeding contours increase in valuein the order 2 x IO", 4 X IO", 8 x 10" with n beginning at -3 and increasing by unity. Th e same set of contours is used throughoutthe paper .paths all of which termina te a t the critical point an ddefine a two-dimensional manifold in three-d imension alspace-an interato mic surface, Figure 1. Each atom isbounded by one or more su ch surfaces, which are clearlyzero flux su rfaces, since Vp is tangent to a trajectorya t any point on the surface.T he positive eigenv alue of a (3,-1) critical point de-fines a unique pair of eigenvectors each of which ori-ginates at the critical point and terminates a t a neigh-boring nucleus. The y define a line linking the nucleiwhose basins share a interatomic surface and alongwhich the charge density is a maximum with respectto any neighboring line. Such a line is called an atomicinteraction line,8Figure 1. T he presence of such a linelinking two nuclei in a molecule which exists in a min-imum energy geometry implies that the two atom s arebonded to one ano ther and in this instance the line iscalled a bond ~ a t h . ~ J ~his topic is expanded upon insection V, which presents the developm ent of the theoryof molecular structure.Th e discussion of t he general topological propertiesof the charge distribution has served to dem onstrateth at th e application of the bo undary condition for the

definition of a q uan tum subsystem yields a partitioningof a mo lecule or solid int o a set of basins each with asingle nuclear attracto r, a partitioning in to atoms. Inthe great majority of cases, the nuclei are the sole at-tractors of a charge density. Qu antum mechanics statestha t the properties of a to tal system are determ ined bythe properties of these individual forms and the successof t he a tomic concept in th e classification an d predic-tion of chemical knowledge is accounted for by thiscongruence in a dominant physical form and its pre-dicted qua ntu m mechanical consequences. I t is possiblein some system s however, to observe local maxim a inthe charge density without the presence of a nuclearattractor.l8Jg In such cases the zero flux boundarycondition is still satisfied and quantum mechanicsidentifies such nonnuclear attractors or pseudoatomsashaveing a definable set of propertie s which contribu teto the properties of the total system. Examples of suchpseudoatoms are found in clusters of group I atoms.The quantu m theory of stru cture describes these sys-tems as consisting of positively charged atomic coreswith very localized charge distributio ns bound by anintermeshed network of negatively charged pseudoa-


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Quantum Theory of Molecular Structure Chemical Revlews, 1991, Vol. 91 , No. 5 901mode of integration indicated by NJdr$*$ as used inthis definition of an atom ic average is the same as h a temployed in the definition of the electronic chargedensity, p(r). It implies a sum mation over all spins andan integ ration of th e spat ial coordinations of all electronbut one. From this point on the subscript 1will bedropped from the coordinates of the electron whosecoordinates are integrated only over 52 and all single-particle, unlabeled coordinates and o perators will referto this electron.

Th e correspond ing variation of $[$,fl], subject to th econstra int which gives rise to th e zero flux boundarycondition (eq 1) is given by the surface integral in eq50

toms. Th e pseudoatom s are regions of very diffuse andloosely bound electronic charge density. Th e atomiccores are not linked to one another directly, but onlythrough intervening pseudoatoms which form a con-nected network throughout the cluster. Th e absolutevalue of p at a maximum in a pseudoatom a nd the ex-te nt by which it exceeds the values of p at neighboringcritical points is extraordinarily sma ll and , in accord-ance with th e uncertainty principle, the electron densityof the pseudoatoms is loosely bound and unconfinedwith a very low kinetic energy per electron. It is thepseudoatoms which ar e responsible for the binding andfor the cond ucting properties of these systems.lgThese exam ples illustrate the ability of the quan tumtheory of structu re to always identify those compo nentsof a system that are responsible for determining itsproperties a t the atomic level, and we now pursue thedevelopm ent of th e mechanics of an atom in a molecule.I I I . Deflnltlon of Atomlc PropertlesA. Varlatlonal Derlvatlon of Atomic Force andVlrlal Theorems

T he present discussion will be limited to systems instationary states. Th e derivations of the same theoremsfor the general time depen dent case can be found in ref1 and 2. The atomic statement of the principle ofstation ary ac tion, eq 43, yields a variational deriva tionof the hypervirial theorem for any observable F , a de-rivation which applies only to a region of space flbounded by a surface satisfying th e condition of zeroflux in the gra dient vector field of th e charge density(eq 1). This principle will be used to obtain a varia-tional definition of the force acting on an atom in amolecule and of the atom ic virial theorem. Th e deriv-ations will illustrate the important point th at the def-inition of an atomic p roper ty follows directly from theatomic stateme nt of stationary action. A full discussionappears in ref 1.Th e Hamiltonian is taken to be the many-electron,fiped-nucleus Ham iltonian given in eq 23. Th e symbolV will be used to denote th e complete poten tid energyope rato r, th e sym of the electro n-n ucl ep V,,, elec-tron-electron V, and nuclear-nuclear V, potentialenergy operatorsQ = Q,, + Q,, + Qm (48)Th e commutator of this Hamiltonian and the mom eq-tum operator of a single electron is equal to ihVV.The method of obtaining the subsystem average ofthe comm utator and hence of the force acting on theatom fl is determin ed by the definition of the func tional9[$,52]via eq 43. It has been demonstrated that themode of integration used in the definition of the sub -system functional $[$,Q] eq 42) is the only one whichleads to a physically realizable boundary condition.Because of eq 43, this same mode of integration (see eq36) defines the atomic average of th e com mutator a ndthus of the atomic force, F ( n )( N / 2 ) ~ ( ~ / h ) ( $ , [ ~ , ~ l I $ ) *CCI =NJdrlJdr{$*(-Vlfi+) = F(Q) 49)The result is multiplied by N , the total number ofelectrons, in the definition of an atomic property. T he

-$ S(Q,r)Z( )m (r) (50)where F(r)is the q uantu m m echanical stress tensor. Itis defined asz(r)= ( Nh 2 / 4 m) l d r {V( V$ * ) $ +a result which may be expressed in terms of the one-electron density matrix r(l) sF(r)=

$*VV$ - V$*V$ - V$V$*) (51)

( h 2 / 4 m ) { ( ~ ~vv) - (VV + vv)]r(l)(r,r)lr+(52)Th e stress tensor is a symmetric dyadic. It has thedimen sions of pressure, force/unit area, or equivalentlyof an energy density. Th e quantu m stress tensor playsa dominant role in the description of the mechanicalproperties of an atom in a molecule and in the localmechanics of the charge density.

Combining eqs 49 and 50 yields eq 53, he atomicforce law for a statio nary(53)

The force may be equivalently expressed by usingGausss theorem as an integration of the force density-V-F(r)over the basin of the atom

F( 2) = -$S(52,r)Z( ) a ( )

Equation 53 has a classical analogue which states t ha tthe force exerted on th e m atter contained in a region52 is equal to the negative of th e pressure acting on eachelement of the surface bounding the region. A localform of th e force law is readily obtain ed from th e timederivative of th e current de nsity, and for a stationarystate the result is

F(r)= NJdr+*(-V@$ = -V-Z(r) (55)which is clearly th e differential form of th e integratedforce law in eq 54. Th e integrated and differential forcelaws have a num ber of imp ortant consequences whichare now explored. whose gradient isaveraged in eq 49 and 55 is the many-particle operatordefined in eq 48. Th e operator -VV, eq 56, is the force

Th e potential energy operator


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002 Chemical Reviews, 1991, Vol. 91, No. 5exerted at the position r of electron 1by all of the otherelectrons and the nuclei in th e system, each of th e otherparticles being held fixed in some a rbitrary configura-tion (V 5 V1 nd r rl)

BaderMu ltiplication of th_e com mu tator average app earingin eq 45 by N /2 for F = i - 6 yields the result

( ~ / 2 ) { ( i / f i ) ( r ~ . , [ A , i . b l ~ / ) nCCI =2NSdrSd7 ( -h2 /4m){$*V2$ + (V2$*)$) +

The integration implied by dr in eq 55 averages thisforce on the electron at r over the motions (i.e., posi-tions) of all of th e rem aining particles in the system ,and the result is the force density F ( r ) , he force ex-erted on the electron at r by th e average distributionof the remaining particles in the tota l system. Inte-gratio n of th is force dens ity over the basin of th e atomQ hen yields the average electronic or Ehrenfest forceexerted on the ato-m in the system . Even though t heforce operator -V V involves the coordinates of all theparticles in the system, and includes their m utual in-teraction , the m ode of integr ation employed in eq 55yields a corresponding density in real space whose in-tegration over an ato m with a boundary defined in realspace yields the force acting on the atom (eq 54).The direct evaluation of the average value of thisoperator requires the information contained in thetwo-electron density ma trix, yet acco rding to eq 53 an d55 , this force, in both its differential and integratedforms, is determined by th e stress tensor which requiresonly th e one-electron density matrix for its evaluation.One can view eq 55 as a statem ent tha t the forces actingon a particle arising from the electrostatic interactionsbetween the particles and describable in terms of thegradie nt of a po tential energy operator are balanced bya force - V 3 , which is purely quantum in origin. Thevirial of the Ehren fest force, which determines th e po-tent ial energy of the electrons, is also describable interms of the stress tensor Z, and thus the m echanicsof a quan tum system is determined by the informationcontain ed in the one-electron density m atrix.An atomic surface for an atom Q s the union of somenumber of interatomic surfaces denoted by S(Q(Q,r),the re being one such surface for each bonded neighbora. Th us the force acting on an atom is given in eq 54can be expressed as a sum of surface terms

F ( Q )= -C.,,$dS(QlQ,r)~(r).n(r) (57)T he s u m n this equation runs over the surfaces share dwith atoms bonded to Q, he atom s linked to R by at-omic interac tion lines. Th is expression for the forceacting on an atom provides the physical basis for themodel in which a molecule is viewed as a se t of i nter -acting atoms. It isolates, throug h the definition ofstruc ture , the s et of atomic interac tions which deter-mines the force acting on each a tom in a molecule forany con figuration of th e nuclei.We now co nsider the use of th e virial oper ator r.p inthe atom ic state m ent of th e principle of stationary ac-tion eq 43 , to obtain the atom ic statem ent of the virialtheorem. Th e virial theorem may be obtained by ascaling of t he electronic coordinates,21 n d the use ofthe virial operator as the generator of a n infinitesimalunitary transform ation is indeed equivalen t to a scalingof th e electronic coord inate r.lv2

The first term is twice the average electronic kineticenergy of the a tom T(Q)xpressed in term s of th e usualLaplacian operator. Th e second term, arising from thecommutator a nd labeled Y,(Q), is the integrated av-erage of th e virial of th e E hrenfe st force acting on anelectron in th e basin of th e atomYb(Q) = N L d r l d r $ * ( - r - V @ $ (59)

Starting from the identity given in eq 25 one maydefine two kinetic energy densities both of which in-tegrate to th e average kinetic energy when integr atedover all space. Thu s-( h2/4m)NJd7{$*V2$ + $V2$*) =

(fi2 / 2m)N s ~0+*.V$- ( i2 / 4 m ) N sd+V2($*$)(60)Equation 60 may be expressed in symbols as

K ( r ) = G(r ) + L(r) (61)Integration of the final coordinate r in eq 61 over aregion of space Q yields

&K( r ) d r = J G ( r ) d r - ( f i 2 / 4 m) L V* Vp ( r ) r(62)

orK ( Q )= G(Q) - (fi2/4m)$dS(Q)Vp(r).n(r) =

G(Q) + UQ)63)Because of the zero flux surface condition (eq 1) theterm L ( Q )vanishes when the integration in eq 63 iscarried out over an atomic basin. Thu s for a quantu msubsystem

K(Q) G(Q) = T(Q) (64)as found for the to tal system and T(Q)s a well-definedquantity.Th e variation in !?[$,a] aused by the g enerator i.6is given by th e surface integral in eq 65

-$dS(Q,r)rG(r)-n(r) -(fi2/4m)$dS(Q,r)vp(r).n(r) (65)

where the final line is obtained through t he use of theidentity V(rgV$) = V$ + r.VV$. T he negative of th efirst term on the right-hand side of eq 65 is labeledYs(n)and is the virial of th e Eh renfest forces exertedon the surface of the atom. Th e quantity 8.n is theoutwardly directed force per unit area of surface andr.Z.n is the virial of this force


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Quantum Theory of Molecular Structure Chemical Reviews, 1991, Vd. 91, No. 5 SO5Y,(Q) = #dS(Q, r)r .Z(r ) -n(r ) (66)

Th e second term in eq 65 is L ( Q ) s defined in eq 63.Equating the com mutator a nd surface results followedby some rearranging of terms yields-2T(Q) = Yb(Q)+ Y,(Q) + L ( Q ) (67)

Since the atom Q is bounded by a surface of zero flux,L ( Q )= 0 and one obtains the atomic statem ent of thevirial theorem -2T(Q)= Y,(Q) + YE@) (68)

-2T(Q) = Y(Q) (69)where Y(Q) , he sum of the surface and basin terms,is the total virial for the atom . While the partitioningof the virial into basin and surface contributions isdependent upon the choice of region (an origin canalways be found which causes the surface virial tovanish), the value of the total virial Y(Q) s, as evidentfrom its e quality with twice the kinetic energy, inde-pendent of this choice.

Equation 69 is identical in form with the virialtheorem for a tota l system-the negative of twice theaverage kinetic energy of t he elec trons, equals the virialof the forces exerted on them. It is worthwhile here tosummarize the ways in which this result is dependentupon th e zero flux boundary cond ition (eq 1): (a) Theuse of the principle of stationary action to obtain avariational derivation of this theorem is restricted toa region satisfying eq 1. (b) Satisfaction of eq 1 insuresthe vanishing of the term L ( Q )which arises from thesurface flux of the current density jpp. (c) Th e van-ishing of L ( Q ) s also necessary for th e kinetic energyT(Q)o be well defined. There is no statem ent corre-sponding t o eq 69, variational or otherwise, for a sub-system with arbitrary boundaries.For a stationary stat e, a local statem ent of th e virialtheorem can be ob tained by using the identity

V * ( & ) TFZ+ p V * T (70)T he trace of the s tress tensor is given in term s of thekinetic energy densities defined in eq 63 by

TrZ(r) = - K ( F )- G ( F ) (71)

or

or equivalently asTF;(F) -2G(r) - L ( F ) (72)

and su bstituting this final result into eq 70 and rear-ranging yields-2G(r) = -r.VG + V-(r .3)- ( h 2 / 4 m ) V 2 p ( r ) (73)For a stationary state the local virial -r.V-P equals thevirial of the Ehrenfest force density F ( r )as can be seenby taking the virial of eq 55:

r.F(r) = NSdr$*(-r-V@$ = -r .VG (74)Th us the local statem ent of the virial theorem is termfor term, the d ifferential form of th e integrated theoremin eq 67. Because of this correspondence, one can definethe density correspondingto th e total virial Y(Q)as and

V(r )= -r.VG + V - ( r G ) (75)the local form of the virial theorem c an be written a s

( h 2 / 4 m ) V 2 p ( r ) 2 G ( r ) + V(r ) (76)Th e kinetic energy density G ( r ) s necessarily positiveand eq 76 demonstrates that in those regions whereelectronic charge is locally concentrated, i.e., where th eLaplacian of the charge density is negative, th e elec-tronic potential energy density V(r) is in local excessover the ra tio of 2:l for the average value of T to Y inth e virial theorem . Equ ation 76 is unique in relatinga prop erty of the electronic charge density to the localcomponents of the total energy. It will be used exten-sively in the characterization of bonding and in theprediction of the mechanisms of generalized Lewisacid-base reactions.From eq 75 it is clear th at the virial of th e electronicforces, which is the electronic potential energy, is totallydetermined by the stress tensor P and hence by theone-electron density matrix. T he atomic statemen t ofthe v irial theorem provides the basis for the definitionof th e energy of a n atom in a molecule.B. Energy of an Atom in a Molecule

Th e method of averaging an operator over the statefunction to obtain t he corresponding atomic average isdictated by the mode of integration defined by theatomic statem ent of the principle of stationary action:summation over all spins and integration over thespatia l coordinates of all electrons but th e one whosecoordinates appear in generator and which are inte-grated over the atomic basin. This imparts a basicone-electron natu re to an atomic property, a s each isdetermined by the integration of a correspondingproperty den sity over the basin of th e atom , as previ-ously indicated in eqs 36 and 37 for the_determinationof the atom ic averages of t he generator F. This remainstrue even for many-electron operators, as llustrated ineq s 54 and 55 for the atomic force F ( Q ) nd in eqs 59and 74 for the electronic potential energy density, thevirial of th e Eh renfe st force.Th e most importan t consequence of the definition ofan atomic property is th at the y e ra g e value of an ob-servable for the total system ( A ) s given by the sumof its atomic contributions A ( Q ) . Equation 77 is true

( A )= CnA(Q) (77)for both one-particle and two-particle operators. Itstates th at each atom makes an additive contributionto the value of every property for a total system. Thisis the principle underlying the cornerstone ofchemistry-that atoms and functional groupings ofatoms make recognizable contributions to the totalproperties of a system. In practice, we recognize agroup and predict ita effect upon th e static and reactiveproperties of a system in terms of a se t of propertiesassigned to the group. In the limiting case of a groupbeing essentially the sam e in two different systems, oneobtains a so-called additivity scheme for the totalproperties, for in this case the atomic con tributions aswell as being additive in the sense of eq 77 are trans-ferable between molecules.Even a property not represented by a linear Hermi-tian operator can be expressed as a sum of atomic


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BO 4 Chemical Revlews, 1991, Vol. 91, No. 5contrib utions , as in eq 77. Th e polarizability of amolecule for example, which is determined by thefirst-order response of th e charge density to an electricfield, is not directly expressible as an average over acorresponding operator. This is no t to say however, tha tthe polarizability c anno t be expressed as an additiveatomic property, a s is indeed done empirically. Th eatomic contributions to the molecular polarizability andmagnetic susceptibility are defined and discussed in refs15 and 22.

Th e theory of atom s in molecules is founded upontwo important observations that resulted from thestudy of molecular charge distribution^.^ First, theatom s of theory are th e most transferable pieces of asystem that can be defined in real space and whichexhaust the space of the system. They therefore,maximize the transfer of atom ic information betweenmolecules at the level of th e charge density. Second andmost important, the constancy in th e average values ofan atom 's obse rvables, including its contribution to th etotal energy of a system, is found to be directly deter-mined by t he constancy in its distribution of charge. Asa consequ ence of this o bservation an d eq 77, when th edistribution of charge over an atom is the same in twodifferent molecules, Le., when the atom or some func-tional grouping of atom s is the same in the real spaceof two different systems, then it makes the same con-tribution to the total energy in both systems. It isbecause of the direct relationship betw een the spa tialform of an atom and its prope rties th at we are able toidentify them in different systems. Thu s whether theform of an atom changes by a little or by a lot, its energyand other prope rties change by corresponding amo unts.This observation has obvious consequences for densityfunctional theory and these have been discussed in ref23.Along with the discovery th at th e m ost transferableatomic unit of the charge density is a region of spacebounded by a zero flux surface5 was the observationthat when the charge density was nearly unchangedover such a fragment in two different systems, the ki-netic energy density exhibited a corresponding degreeof transferability. Th us transferability of the distri-bution of charge over an atom lead s to a correspondingconstancy in its kinetic energy. If one postulated theexistence of th e virial theorem for such an atom , thenthis observation implies that when the charge distri-bution of an atom is identical in two different systems,the atom will contribute identical amounts to the to talenergies of both systems. Th is postulate has beenproven true by the atomic state me nt of stationary ac-tion, and t he atom ic virial theorem yields the definitionof an energy in an atom.Th e electronic energy of a n atom in a m olecule, thequantity E,(Q) , s defined as

E , ( Q ) = T(Q) Y(Q) (78)Because of t he atomic virial theorem (eq 69), the atomicenergy E,(Q)satisfies the following relationships whichare the direct analogues of th e all space results:

(79)Because of eq 69 an d th e vanishing of th e Laplacian ofthe charge density over an atomic basin, th e followingiden ti ies hold:

E , ( Q ) = - T ( Q )= (1/2)Y(Q)

BaderE,(Q2)= - K ( Q ) = -G(Q) = ( 1 / 2 ) $ T t ~ ( r ) d 7 ( 8 0 )It is to be emphasized, th a t all of th e above relation-ships, together with the atomic statements of th e virialtheorem (eq 79) remain true w hen Q refers to the totalsystem. It is in this sense th at an atom is a quantumsubsystem.From its definition it is clear tha t like other atomicproperties, the sum of the energies of th e atoms in asystem equals the total electronic energy E,:

E , = &Ee(Q) = T + Y (81)and when there are no forces acting on any of th e nucleiin the system, this sum equals the to tal m olecular en-ergy E as obtained by averaging the Hamiltonian in eq23.As emphasized above, it is the energy E,(Q) definedin terms of the virial theorem th at possesses the prop-erty of paralleling the constancy exhibited by th e chargedistribution of an atom w hen it is transferred betweensystems. This theorem equ ates the electronic potentialenergy to the virial of th e Ehren fest forces acting on theelectrons in the basin of the atom , and it is this st epwhich makes possible the partitioning of the totalenergy into a sum of atomic contributions. It is thepoten tial energy of interaction between particles andeventually between subsystems that is the stumblingblock to obtaining an nonarbitrary and physical par-titioning of a to tal energy. How, for example, does onepartition the energies of repulsion between pairs ofelectrons and between pairs of nuclei appearing in t heHamiltonian in eq 23? As discussed in d etail in refs 1an d 2, the use of th e virial to define a potential energyovercomes this difficulty. A force is local, and as il-lustrated in eq 55, it is possible to define the force ex-erte d on an electron by all of the o ther particles in thesystem, a result th at is given deeper physical signifi-cance by th e fact t ha t the same force is expressible interms of th e quantu m stress tensor. By taking the virialof this force density, one obtains a local potential energydens ity (eq 75) one which is also expressible locally interm s of the stress tensor. Th us the definition of th eenergy of a n atom proceeds not through a spatial par-titioning of th e Ham iltonian, which would violate theindistinguishability of th e electrons, or of th e elem entsof the ab stract H ilbert space on which the H amiltonianacts, but rather through a partitioning of the Hamil-tonian into a s u m of effective one-electron contrib utionsby using the v irial operator.'V2 In su mm ary, forces un-like energies, are local and by defining the potentialenergy in terms of the virial of a force, one obtains alocal expression for the p otentia l energy. All of theexpressions are obtained directly from quantum me-chanics and it is not necessary to rationalize or justifythe resulting equations or their consequences. T heabove discussion is simply to make clear how physicsdoes provide an answer to a long standing problem.I V. Transferability of Atomic PropertiesA. Energy AddRivRy in Normal Hydrocarbons

Th e above discussion emphasized th at t he use of thezero flux surface for the definition of an atom orfunctional grouping of atom s maximizes the exten t ofthe transferability of its properties between systems,


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Quantum Theory of Molecular Structwea characteristic essential to the role of th e atomic con-cept in chemistry. By defining a group and its prop-erties and thereb y enabling one to determine the effectof its presence on th e properties of another grou p, thetheory of atoms in molecules parallels the most im-portant of all chemical codifiers, th e subs tituen t effect.Studies have illustrated the ability of the theory toquantify and make understandable the effects of arange of substituents on the properties of the ethyl,formyl, and phen yl group^,^*^^ for example. It is thepurpose of the present discussion to explore the limitingcase where a group is transferable with little or nochange in i ts properties and correspondingly, its per-turbation of the remainder of th e system is minimized.It is possible to experimentally mea sure the energyof an atom in a molecule as an additive contribution tothe molecule's heat of formation in those instanceswhere a class of molecules exhib its an additiv ity schemefor the energy. Essential to the theoretical predictionand un derstanding of this experimental observation isthe property of the ato ms expressed in eq 77 that theirproperties, including their energies, be additive to yieldthe total prope rty value for a molecule. It is demon-strated here that the energies of the methyl andmethylene groups as defined by theory, predict theadditivity and transferability of the group energies asis observed experimentally in normal hydrocarbonmolecules. Th eir properties also predict an d accountfor the deviations in this additivity scheme that areobserved for small cyclic molecules, deviations whichserve to define the strain energy. Th e ultimate test ofany theory is its ability to predict what can be exper-imentally measured. By appealing to the limiting caseof near transferability of atomic properties, one candemo nstrate tha t the atoms of theory are the atom s of

Th e stud y of the molar volumes of th e norma l hy-drocarbons by Kopp in 1855 provided the earliest ex-amp le of th e additivity of group properties. Th e ex-perimentally determined heats of formation for thesam e homologous series of molecules, CH3(CH2),CH3,also obey a gro up additiv ity scheme.*a2 It is possibleto fit the experimen tal heats of formation for this series,beginning with m = 0, with th e expression

AH~f"(298) 2A + mB (82)where A is the co ntribution from the methyl group andB th at from the methylen e group. Th e generally ac-cepted values for A and B a t 25 "C are -10.12 and -4.93kcal/mo l, respectively. Th e group enthalpy correctionsfrom 298 to 0 K a re additive for the n-alkanes,%as arethe group zero point energy corrections.M Th us thecalculated en ergies of th e vibrationaless molecules intheir equilibrium geom etries should exhibit the sam eadd itivity of th e energy as represented by eq 82 and theadditivity is indeed mirrored by the single determi-nantal SC F energies a t both th e 6-31G*/6-31G* and6-31G**/6-31G* levels of approxim ation. Th e calcu-lated m olecular energies E for the n-alkanes satisfy therelationship

E = 2E(CH3)+ mE(CH2) (83)The quantity E(CHJ is one-half the energy of ethane,equal to -39.61912 au, and E(C H2 ) is the energy in-crement per methylene group equal to -39.037 79 au ,when the 6-31G**/6-31G* calculated results were used.

Chemical Revkws, 1991, Vol. 91, No. 5 901

F m . Contours of the charge density for the minimumenergygeometries of t he pentane (top) and hexane (bottom) moleculeain the plane containing the carbon nuclei and the two terminalmethyl protons. The projected positions of the out-of-planesymm etrically equivalent pairs of protons are indicated by opencrosses. The central maps are for a (perpendicular) symm etryplane containing the C and H nuclei of th e central methylenegroup in pentane (left -han d ide) and for one of two equivalentsuch groups in hexane (right-hand side). The bond paths areshown, as are the positions of the interatomic surfaces. Theposition of out-of-planenuclei are indicated by open ~ 0 8 8 8 8 .Th econtour displays of th e charge distributions of the methyl andcorresponding methylene groups are superimposable on one an-other.These group values fit the calculated energies to withinf 0.00014 au, an average deviation smaller tha n t heexperimental one. Th e calculated results indicate th atthe corrections to th e energy arising from th e correla-tion of th e electronic motions, a con tributio n neglectedin a single determinantal calculation, should also obeya group additivity scheme. Th is indeed appears to bethe case, as is demo nstrated an d discussed later.The distributions of charge for the five- and six-carbon membe rs are illustrated in Figure 2 in the formof con tour maps of p. The m aps show the bond pathslinking th e nuclei and ind icate th e intersection of th einteratomic surfaces with the plane of the diagram. Th ediagrams show qualitatively what the atomic propertieswill demon strate quantitatively: th at the methyl andmethylene groups in this series of molecules aretransferable with little change in their form and, hence,with little change in their properties.Th e physical properties of the n-alkanes indicate tha tthe molecules are nonpolar and this is reflected in th esmall magnitu des of th e ne t charges of th e carbon an dhydrogen atoms and of the molecular mom ents. Hy-drogen is slightly more electronegative than carbon in


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006 Chemical Revkw8, 19Q1,Vol. 91, No. 5satu rate d hydrocarbons, and th e order of group elec-tron-withdraw ing ability in hydrocarbons without geo-metric strain is H > CH 3 > CH2> CH > C. In ethane ,methyl is bonded to methyl while in the other moleculesof the n-alkanes it is bonded to methylene from whichit withdraw s charge. T o within the accuracy of thenumerical integrations of the atomic properties (whichin general are i0.001 e an d f l kcal/mol), one finds (seerefs 1,27 or 28) the energy and population of the methylgroup to be cons tant when it is bonded t o a methylenegroup. Th us the m ethyl group is essentially the samein a l l the members of th e homologous seriespast ethane.Thi s transferable meth yl group is more stable relativeto methyl in ethane by an am ount AE = -10.5 f 0.5kcal/mol and its electron population is greater by anamount AN = 0.0175e.Th e charge and energy gained by the m ethyl groupis taken from the methy lene group. W hat is remarka-ble, and w hat accounts for the additiv ity observed inthis series of m olecules, is th at the energy gained byme thyl is equal to the energy lost by methylene. Inpropane, where the methylene group transfers chargeto two m ethyl groups, its energy relative to the incre-ment in eq 83 is E(CH2 )- 2AE, and its net charge is(necessarily) +2AN where AE and AN the quan tit iesdefined above for the m ethyl group. Thu s the energyas well as the charge is conserved relative to the groupenergies defin ed in eq 83. In butan e, a methylene groupis bonded to a single methyl group and correspond inglyits energy is E(CH 2)- AE and its net charge is +AN.Th e corresponding methylene groups in pentane an dhexane, those bonded to a single methyl group, have thesame energies and net charges as a methylene g roup inbutane. Thu s the charge transfer to m ethyl is dampedby a single methylene group, and the central methylenegroup in pentane and th e two such groups in hexane(see Figure 2) should have a zero net charge and a nenergy equal to the increm ent AE(CH2). Th is is whatis found to within the unce rtainties in the integratedvalues, their calculated net charges being 0.0005 f0.0002e an d th e maximum deviation on the energy fromthe standard value being within th e integration errorof -1 kcal/mol. (I t should be kept in mind th at thetota l energy of a methylene group is -25 X lo3kcal/mol). There fore, methylene groups bonded only toother methylenes, as found in pe ntane, hexane, and allsucceeding m embe rs of the series possess a zero netcharge and contribute the standard increment E(CH 2)to the total energy of the molecule. Th e underlyingreason for the observation of addi tivity in this se ries ofmolecules is the fact that the change in energy for achange in population, the qu antity AEIAN, is the samefor both the methyl and m ethylene groups. The smallamount of charge shifted from methylene to m ethylmakes the same contribution to the total energy.

It is to be emphasized that the energies assigned tothe methyl and methylene groups are independentlydetermined by the theory of atom s in molecules. Th efact that this assignment leads to an energy for thetransferable methylene group equal to the value E(CH2)in eq 83 , an equation which mirrors the experimentaladditivity of the energy eq 82 , confirms that th e theo-retically defined atom s are responsible for the experi-mentally measured increm ents to the he at of formation,and that quantum mechanics predicts the properties

Badofof ato ms in molecules jus t as it does th e properties ofth e total molecule. It is a straightforward ma tter to usequantum mechanics to relate a spectroscopically de-termined energy to the theoretically defined differencein energy between two sta tes of a system. In a lessdirec t, bu t no leas rigorous manner, quantum mechanicsalso relates the difference in t he experimentally de-termined h eats of formation of bu tane an d pentane toth e corresponding theoretically defined energy of th emethylene group.

Th e additivity of the energy in the n-alkanes is ob-tained in spite of sm all differences in group prop erties,differences which necessarily result from a change inthe na ture of the bonded neighbor. Thus here are twokinds of meth yl groups: th e one uniq ue to ethane andthe transferable methyl group which is bonded to amethylene group. Th ere are thre e kinds of methylenegroups: the one unique to propane and two transferableforms, one bonded to a m ethyl and the other bondedonly to other m ethylene groups. Oth er properties ofthese groups exhibit the same pattern of transferablevalues as do their energies and populations. This hasbeen for the atomic first mom ents, theatomic volumes, and the atomic contributions to theelectronic correlation energy as determined by densityfunctional theory. This latte r result indicates tha t eachof th e transferable methyl and m ethylene groups shouldmake a characteristic and essentially consta nt contri-bution t o the tota l correlation energy of a normal hy-drocarbon molecule, a result anticipated on the basisof the ability of the SCF calculations to recover theexpe rime ntal additiv ity of th e energy. It has also beendemon strated that the m ethyl and methylene groupscontribute characteristic contributions to the meanmolecular polarizabilities of normal hydrocarbon^.^^It must be considered remarkable that a methylgroup with a to tal en ergy in excess of 25 000 kcal/mol,can be tran sfer red between molecules-in reality an din theory-with changes in its energy of approx imately1 kcal/mol. It is still more remarkable when i t is re-alized tha t th e individual contributions to th e energyof a carbon ato m change by 2-5000 thousand kcal/molbetw een m em bers of th e ~ e r i e s . ' - ~ ~ - ~ ~B. Origln of Strain Energy in CyclicHydrocarbons

Th e hybridization model predicts that th e smallerbond angles found in a molecule with angular strainshould result in an increase in the p c harac ter of th estrained C-C bonds and hence in an increase in the scharac ter of the associated C-H bonds.36 O rbitalmodels relate a n increase in electronegativity of a car-bo n atom relative to th at of a bonded hydrogen to a nincrease in th e s character of its bonding hybrid orbital.Thus it follows th at the presence of ge ome tric stra in ina hydrocarbon molecule should resu lt in an increase inthe electronegativity of carbon relative to hydrogen. Intheir classic study of strain in the cyclopropane mole-cule, Coulson an d M ~ f f i t t ~ ~mphasized this point byshowing tha t the bond lengths and bond angles of th emethylene group in cyclopropane resemble those forethylene. Th e argument for an increase in electroneg-ativity w ith increasing s character is based on energy,an s electron being more tightly bound than a p elec-tron. Th e theory of atom s in molecules shows th at th e


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Quantum Theory of Molecular Structureelectronegativity of a carbon atom does indeed increaseand ita energy decreases as the extent of geometricstrain increases.Relative to ita population in the stand ard m ethylenegroup, each hydrogen in cyclopropane transfers 0.045eto carbon, reducing the net charge on the carbon atomfrom +0.196e to +0.106e. While this charge transfe rleads to a n increase of 15.6 kcal/mol in the stability ofthe carbon atom, it results in a decrease of 12.5 kcal/mol in the stability of each hydrogen atom. Th us themethy lene group in cyclopropane is calculated to be 9.4kcal/mol less stable than the standard transferablemethy lene group. Th is yields a total strain energy forthe molecule three tim es this or 28.2 kcal/mol, in goodagreement w ith the generally accepted value based onth e experim ental hea ts of formation of 27.5 kcal/mol.T he m ethylene group in cyclopropane is more stableth an t he same group in ethylene by only 2.1 kcal/mol.In terms of th e charge transfer within the group andita energy, it resembles more closely the ethylene frag-ment (where q(C) = +O.O80e) than it does the standardmethylene group.

In the less strained cyclobutane, the transfer of chargefrom hydrogen to carbon relative to th e populations inthe stand ard methylene group is reduced to 0.014e foraxial H and 0.012e for the other and the charge oncarbon is +0.170e. T he hydrogens are destabilized by9.0 kcal/mol, and the carbon stabilized by only 2.5kcal/mo l, t o give an energy increa se of 6.5 kcal/mol foreach methylene group. Th is yields a predicted strainenergy of 26.1 kcal/mol, a value which again is inagreement with the experimental value of 26.5 kcal/mol. Experim entally, the heat of formation of cyclo-hexane is found to be six times t he h eat of formationof the standard methylene group and to possess nostra in energy. An axial hydrogen in this molecule iscalculated to possess 0.007 more electrons, (N(H)1.099e) and be m ore stable by 1.6 kcal/mol th an anequatorial hydrogen. Th e atomic populations and en-ergies of a methy lene group in cyclohexane differ littlefrom their values in the stand ard group and the energyof th e group differs by only O.OOO1 au or 0.06 kcal/m olfrom the standa rd value. Th us in agreement with ex-periment, cyclohexane is predicted to be strain freewhen its energy is compared w ith six times the energyof the stand ard methylene group. Th e reader is referredto ref 27 for furth er examples of the relation betweenincreasing strain energy and an increasing degree ofcharge transfer from H to C in bicyclic systems and th epropellane molecules. Th e few examples discussed hereare introduced t o emphasize tha t the stra in energiescalculated for cyclopropane and cyclobutane, and thepredicted absence of str ain in cyclohexane, all of whichare in agreement with experiment, are predicted by th etheory of atoms in molecules. T he energy of thestandard transferable methylene group, as defined bythe zero flux boundary condition and as found in thepenta ne an d hexane m olecules, serves as the basis forthe dete rmination of these results. Not only does theorypredict the transferability of atoms and groupings ofatoms without change, it also correctly predicts themeasured changes in their energies when these groupsare perturbed.While the excellence of th e agreem ent of the relativeenergies of th e m ethylene group in the cyclic molecules

Chemlcal Revlews, 1991, Vol. 91, No. 5 907with the m easured str ain energies may be to som e ex-ten t due t o the fortuitous cancellation of e rrors in thecontributions not specifically considered, namely thecorrelation energy, the zero point energy, and A (AHF )between 0 and 298 K, the nature of the results leavesno doubt as to the correctness of the interpretation thathas been given: th e atoms of theory recover the ex-perimentally measured properties of atoms in mole-cules.V. Molecular Structure and Structural StabllltyA. The Notion of Structure in Chemistry

Th e essential understanding and original intent as-sociated with the notion of struc ture in chem istry is thatit be a generic property of a system. Struc ture impliesthe existence of a particular network of bonds whichwas presumed to persist over a range of nuclear dis-placements until some geometrical param eter atta ineda critical value at which point bonds w ere assumed t obe broken and/or formed to yield a new structure.However, the word "structure" has over the years BC-quired a duality of meanings. Thi s has occurred for tworeasons. Th e first was a result of our inability to un-ambiguously assign a network of bonds to a given sys-tem. T he second was a result of our ever-increasingability to experimentally measure the geometrical pa-rameters which characterize the minimum energy nu-clear configuration of a system within the Born-Op-penheimer model.It is most important to distinguish clearly betweellmolecular geometry and th e original inte nt an d use ofthe notion of structu re in the molecular structure hy-pothesis. Geometry is a nongeneric property since anyinfinitesimal change in a set of nuclear coordinates,denoted collectively by X, esults in a different geom-etry. Molecular structure, on the other hand, was as-sumed to be a generic property of a system. Any con-figuration of the nuclei X' in the neighborhood of agiven configuration X, hile it has a different geometry,should possess the same structure, that is, the samenuclei should be linked by th e sam e network of bondsin both X and X'. Difficulties ascribed t o th e notionof m olecular suutr ucture are the inabilities to assign asingle geometrical structu re, average or o therwise, torotation or inversion-related isomers, to a molecule inan excited vibrational state, or to a molecule in a"floppy" st at e wherein the nuclear excursions cover awide range of geome trical parameters. In reality theseare shortcomings of attempts to impose the classicalidea of a geometry on a qua ntu m system. T he nuclei,like the electrons, cannot be localized in space and in-stead are described by a corresponding distributionfunction. Th e definition of structu re proposed hererecognizes this essential point and associates a m olec-ular structure with an open neighborhood of nuclearc o n fi ia ti o n space, with a corresponding average beingtaken over the nuclear d istribution function.It h as been showne th at the topological properties ofa system's charge distribution enable one to assign amolecular graph to each point X n the nuclear con-figuration space of a system. Th is assignmen t corre-sponds to defining a unique network of atomic inter-action lines to each molecular geometry. Th e chargedistribution can be obtained from a wave function be-


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808 Chemical Revlews, 1991, Vol. 91, No. 5yond th e Born-Oppenheimer approximation by thegenerator coordinate method for example, to yield awave function which, while no longer associated witha specific geometry, is associated w ith a specific struc-ture.6b A molecular structure is then defined as anequivalence class of molecu lar graphs. Th is definitionassociates a given structure with a n open neighborhoodof the m oat probable nuclear geometry, an d removes theneed of invoking the Bo rn-Opp enheimer approximationfor the justification or rationalization of structure in amolecular system ? By defining all possible structuresfor a given system, the theory shows th at a change instructure must be an abr upt a nd discontinuous process,one which is described in terms of the mathematicaltheory of dynam ical systems and their stabilities. Th ereader is referred to th e original papers6 or to ref 1 forfull discussions of this aspect of the theory.B. Molecular Graphs

T he princ ipal topological prope rties of an electroniccharge distribution-ma xima, which ar e topologicallyhomeomorphic to (3,-3) critical points a t the positionsof the nuclei and (3,-1) critical points between atomswhose basins share a common s u r f a c n e r e introducedin section II.E. The first of these critical points definesan atom and its basin while the second defines theinteratomic surface an d atomic interaction line. Th enuclei of two atom s which share a common interato mi csurface are linked by a line along which the chargedensity is a maximum with respect to any neighboringline.Th e existence of a (3,-1) critical point and its asso-ciated atomic interaction line indicates tha t electroniccharge density is accumulated between th e nuclei th atar e so linked. Th is is ma de clear by reference to th edisplays of the charge den sity for such a critical point,as given in Figure 1 for example, and particularly inFigure 3, which em phasizes the fact t ha t the chargedensity is a maximum in an nteratomic surface at th eposition of th e critical point. Thi s is th e point wherethe atomic interaction line intersects the interatomicsurface and charge is so accumulated between the nucleialong the length of this line. Both theory a nd obser-vation concur that the accumulation of electronic chargebetween a pair of nuclei is a necessary condition if twoatoms are to be bonded to one another.' This accu-mulation of charge is also a sufficient cond ition whenthe forces on the nuclei are balanced and the systempossess a minimum energy equilibrium internuclearseparation. Th us the presence of an atomic interactionline in such an eq uilibrium geometry satisfies both t henecessary and sufficient conditions tha t th e atom s bebond ed to one another. In this case th e line of maxi-mum charge density linking the nuclei is called a "bondpath" and he (3,-1) critical point referred to as a "bondcritical point".*J7For a given configuration X of the nuclei, a"molecular graph" is defined as he union of the closuresof the bond paths or atomic interaction lines. Picto-rially the m olecular graph is the network of bond p athslinking pairs of neighboring nuclear attractors. Themolecular graph isolates the pair-wise interactionspresent in an assembly of atom s which dom inate andcharacterize the properties of the system be it atequilibrium or in a state of change.

Figure 3. Relief and contour maps of the charge density anda display of t he asso ciated gradient vector field for th e planecontaining the C-C nteratomic surface in ethene i.e ., the planebisecting and perpendicularto the C-C bond path. In this planethe (3,-1) critical point appears as a two-dimensional attractor.Note the elliptical nature of the contours with the major axisperpendicular to the plane containing the nuclei. (The projectedp it io n s of the nuclei on this plane are ndicated by o n croeees.)This property of a charge distr ibution is measured6 he bondellipticity (section V1.A). The less rapid rate of falloff in p in the"r"plane is reflected in a corresponding paucity of trajectoriesin the display of the gradient vector field of p.

A molecular graph is the direc t result of th e principaltopological properties of a system's charge distribution:th at local maxima, (3,-3) critical points, occur at thepositions of the nuclei thereby defining th e atoms, andtha t pairs of trajectories which originate a t (3,-1)criticalpoints are found to link certain, but not all pairs ofnuclei in a molecule. Th e network of bond paths th usobtained is found to coincide with the network gener-ated by linking together those pa irs of atom s which areassumed to be bonded to one another on the basis of


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Quantum Theory of Molecular Structure Cbmlcal Reviews, 1991, Vol. 91, No. 5 SO9through a synthe sis of o bservations on eleme ntal com-bination an d models of how atom s combine, particularlymodels of chemical valency. A great deal of chemica

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A Quantum Theory of Molecular Structure and Its Appllcatlons