Theoretical Definition of a Functional Group and ... - Wiley Online Library

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Theoretical Definition of a Functional Group and the Molecular Orbital Paradigm Richard Frederick William Bader," Paul Lode Albert Popelier, and Todd Alan Keith ______

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I t is the purpose of this review to demon-

strate that the empirical classification of the observations of chemistry in terms of the properties assigned to functional groups is a consequence of and is predicted by physics. This is accomplished by showing that the atoms and functional groups of chemistry can be identified with bounded space-filling objects

whose properties are defined by quantum mechanics. The quantum mechanical definition of a group is combined with a new pictorial representation of its form to obtain a unified picture which should make it eminently recognizable to chemists. This picture. when combined with the demonstrated ability of these groups to recover the measured

properties of atoms in molecules, is offered as one which meets the expectations a chemist associates with the concept of a functional group. The manner in which this physical definition of a group differs fundamentally from models of functional groups based upon molecular orbital theory is discussed.

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1. Introduction 1.1. Does Chemistry Have a Basis in Physics? It is reasonable to ask why it has taken considerable time to arrive at a possible quantum mechanical definition of a functional group. and why there has evolved instead a prevailing opinion that this fundamental operational concept of chemistry, while unquestionably useful. would find no firm basis in physics. First. it has taken a reformulation of physics, as afforded by the work of Feynman"] and Schwinger,['] to provide us with the necessary quantum mechanical framework for asking the question "are there atoms in molecules?" Second, as so eloquently expressed and illustrated by KuhnL3Iwith examples from all branches of science. is the phenomenon of the resistance accorded a new theory because of the conceptual bias exerted by existing paradigms. The paradigm of molecular orbital theory effectively filters out the hope of establishing a quantum mechanical basis for functional groups. This rekiew summarizes the evidence in favor of the identification of quantum subsystems with the functional groups of chemistry. Quantum subsystems are defined as regions of real space. A new pictorial representation of a quantum subsystem is presented to complement the demonstration of their equality with a chemist's mental and physical model of a group. This summary is prefaced with a discussion of why molecular orbital theory, whose components are rooted in the mathematical Hilbert space. appears to be at variance with this identification. [*] Prof. R. F. W Bader. Dr. P L. A . Popelier. T. A . Keith

Department of Chemistry. McMaYter Unikcrsi ty Hamilton, Ontario. LXS 424 I (Canada) Telefzcx: Int. code + (Y05)522-2509

It is shown that this conflict is a result of insisting that all chemical concepts be cast in the orbital mold, thereby requiring that the orbital model be applied beyond its intended scope,

1.2. Classification as a Definition of Science A scientific discipline begins with the empirical classification of observations. It becomes exact in a predictive sense, when these observations are classified in such a way as to reflect the physics governing its behavior. In this limiting situation, prediction and classification are synonymous and the classification is based upon what physics predicts can be observed. This situation is exemplified in quantum mechanics by the classification of a given state in terms of the eigenvalues of the maximal commuting set of observables which constitute the measurable properties specific to that state.'"' In this sense, science is classification.

1.3. Classification in Chemistry The observations of chemistry are classified empirically in terms of the properties exhibited by functional groups. In chemistry we recognize the presence of a group in a given system and predict its effect upon the static and dynamic properties of the system in terms of the characteristic set of properties assigned to that group. The functional group is the central concept in the working hypothesis of chemistry, the molecular structure hypothesis. According to this hypothesis, which rests upon the accumulated observations of experimental chemistry. the properties of matter are a consequence of the atoms it contains and of the network of bonds that link them. It is the purpose of this

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Theor! of Functional Groups article t o demonstrate that the classification based upon the properties of functional groups is a consequence of and is predicted by physics. and that the classification is therefore, fundamental.[’]

2. The Molecular Orbital Paradigm The molecular orbital point of view and its associated paradigm that is used to argue against the possibility of establishing ;I physical basis for functional groups. is illustrated by a quotation from a paper by Libit and Hoffmann.[‘] It follows a paragraph in uhich they stress that it is the logic of substituent effects that has made possible the great strides in synthetic and mechanistic organic and inorganic chemistry: “Nothing like this logic comes out of molecular orbital calculations. Every molecule is treated iis a whole and no set of transferable properties associatcd with a funclional group emerges.” This statement appears a s true or false depending upon the role one assigns to the molecular orbitals (MOs) used in the calculation. If the orbitals are used not only to calculate the state function, but are in addition used in the search for properties to be assigned to an individual atom or group which forms a part of the real space of the molecule. the statement is true, since thc orbitals, like the state function they are used to approximate, are necessarily delocalized over the whole system. If. however. the role of the molecular orbitals is confined to the determination of the state function and the electronic structure of the system. the statement is false. A state function contains the necessary information for the exhaustive partitioning of the real space of a molecule into separate regions whose properties reflect the very transferability associated with functional groups.[5, It is the demonstration of this fact that will relieve chemists of the dilemma of having to contend with the coexistence in their minds of two apparently mutually exclusive models: one experimentally based. which provides the framework for the ordering of their observations and another, mathematically based, which when improperly applied. suggests that this framework has no basis in theory.

2.1. Molecular Orbital Theory as a Model of Electronic Structure Mullikenr8’and Hund[‘] developed molecular orbital theory to obtain a classification and understanding of band spectra.

Molecular orbitals provide the basis for a model of the electronic structure of molecules. The molecular orbital model provides an excellent example of a classification scheme which parallels the predictions of theory. This model. when applied to the oxygen diatomic molecule for example, enables one to predict without the necessity of calculation that this sixteen-electron system will possess a ground state of ‘ZC, symmetry followed by ‘ A , and ‘Z: excited states. This classification is precisely what quantum mechanics predicts can be observed experimentally, the spin and orbital angular momentum operators. along with the symmetry operations, forming a maximal commuting set with the Hamiltonian operator.[”] The use of orbital theory for the calculation of approximate state functions was initiated by Hartree” I ] who introduced the concept of the self-consistent field (SCF) in the numerical determination of atomic orbitals. The incorporation of exchange was accomplished by Fock[l2]and by Slater,[”I the latter through the specific variation of the spin orbitals in a determinant,‘I I wave function. The possibility of applying the Hartree- Fock method to the calculation of molecular orbitals was accomplished by R ~ o t h a a n [ ’ and ~ l Hall,[”] by showing how the orbitals could be expanded in a set of basis functions. Their work and its extension by others, has made it practicable to calculate state functions for molecules to any desired degree of accuracy within the extended framework of orbital theory. Molecular orbitals are used for purposes other than as a basis for the classification and calculation of electronic states. The orbitals themselves can be classified as bonding or antibonding, a classification which correlates with observed changes in bond lengths and dissociation energies of new states generated by changes in occupation of a particular orbital.””’ Because of their delocalized nature. there was an original reluctance on the part of chemists to embrace the molecular orbital model and use i t in place ofthe valence bond approach which provided a quantum mechanical basis for the Lewis model of the electron pair and its directed proper tie^.^' 1 8 1 This reluctance was overcome in part by the introduction of equivalent orbitals by LennardJones[’91and their subsequent evolution into localized orbitals. Lennard-Jones emphasized that any unitary trxnsforination amongst a set of orbitals leaves all properties, including the total charge density, unchanged and that such transformations must be interpreted with caution.[201 Wigner and Witmer introduced the use of group theory into chemistry in 1928 by deriving a set of rules for determining what types of states for diatomic molecules result from given states

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horn bi Kitcliener. Otiturio in 1931. H e receiwd liis B . 2 . r i n d M.Sc. his Plz. D. in 1957,from the Massacliust~ttsInsiifute of’ i7~c~litiolog.v i M I T ] bvorking with Professor C. G. Suzin. He didpost doctornl biwk ut M I T f i o t n 19.57 to 19.58 mid ut Cunihridge Diiiversitj*,fi.on?i1958 to 1959 in the Laborutory qf ProftJssor H . C. Longiiet- Hic?gins. He began his scientific cureer us u phj’sicul orgunic cliet?iist. H e IWS at the C‘tiivt~r.sit~~ qf O t i r r ~ i ~ i fioni 1960 f o 1963 and then moved t o M c Muster Uniwrsitj~,where. lie reniains todcrj.. His re.sratdi intcresis are ccwtered on the continued development of the tlieorj, of‘ atoms in nio1ec.ulc.s and its upplication to cl~rniisrrj~. Dr. Richurd F W. Bader

\ t ~ ~ s

fioni McMuster Utiiversitj. and

REVIEWS of the separate atoms.[211The first symmetry rule for molecular reactions appeared in 1962.[221It predicts the reaction coordinate in terms of the symmetry of the transition density obtained by mixing in the first excited electronic state in a secondorder perturbation expansion of the energy, expressed in terms of the nuclear displacements. This model likens the instability of a system for motion along a reaction coordinate to a pseudo Jahn-Teller effect. It is the nature of the first excited state of proper symmetry which determines the course of a chemical Molecular orbital theory in combination with group theory is indispensable in predicting the ordering and symmetries of electronic states, as first demonstrated by Mulliken.['"' and the transition density itself ciin be usefully approximated in terms of the product of relevant orbitals which in inany cases corresponds to the HOMO-LUMO product. In this manner the symmetry rule justifies the assumptions of frontier molecular orbital theory.['"' The same phase inforination of molecular orbitals provides the basis for the success of the concept of the conservation of orbital or state symmetry in the prediction of the course of a chemical reaction.["] In these and other related approaches to reactivity and discussions of bonding. the orbitals are properly used in their capacity to provide inforination regarding the electronic structure of a system.

2.2. Looking for Atoms in Molecular Orbitals

A significant breach of the model occurs when physical significance is ascribed to particular choices of nuclear-centered basis functions used in the expansion of the molecular orbitals. An orbital describes the motion o f a single electron in the field of all the nuclei and in the self-consistently determined average field of the remaining electrons. It is obtained by solving a set of coupled integral -differential equations whose solutions for an atom are called atomic orbitals and for a molecule, molecular orbitals.["1 The solutions to these equations are independent of the method used to obtain them. whether it be numerical or expressed in terms of an expansion. The acronym LCAO-MO (linear combination of atomic orbitals for molecular orbitals) grew naturally out of early attempts to gain some understanding of the form of an M O in terms of its cori-elated one-electron states at infinite separation,r261an idea first introduced by Lennard-Jones in 1929.[2'1 The pioneer workers. however, very much appreciated that this manner of representing a molecular orbital was meant to provide no more than a crude model of its form and that it did not imply the presence of an atomic component in a molecular orbital. Mulliken described how atomic orbitals (AOs) to be used as basis functions in the expansion of a molecular orbital. should be suitably modified for the molecule and not identified with the atomic case. He then continued with the statement,'"] "However. it should be kept in mind that the concept of an SCF M O is an independent one, and thus the use of a linear combination of AO's or STO's (Slater type orbitals) a s building blocks to construct an M O is merely a convenience." In this same paper Mulliken presents an explanation as to why no physical significance should be ascribed to an atomic population based on an LCAO expansion. This acronym, for the sake of clarity of thought, should be changed to LCBF (linear cornbination of 622

R . F. W. Bader et al. basis functions). In a later paper['"' Mulliken et al. completely disowned the concept of a population analysis upon the demonstration that it bore no relation to the forin of the total charge distribution, a s evidenced by the statement "It should be possible to avoid such inconsistent and unsatisfactory results by computing atomic charges directly from the molecular charge distribution." I t is more than a question of semantics to understand that basis functions are not atomic orbitals. There are no atomic orbitals in a molecule, nor is therc any overlap of atomic components. It is perhaps asking too much of some that they give up the notion that a molecular orbital is a linear combination of atomic orbitals and indeed it is not necessary that they do so. It is only necessary that they not insist that the only chemical concepts capable of definition are those which survive the filtering through the successive levels of approximation within the molecular orbital model, a s outlined above. The delocalized nature of molecular orbitals and of the basis functions employed in their expansion. which are associated with truncated portions of the mathematical Hilbert space. should not be used to exclude the possibility of identifying atoms or functional groups with bounded regions of real space. There is no question concerning the possibility of obtaining a quantum mechanical definition of uniquely defined regions of real space.[. 3"1 The answer to the question of whethcr or not these regions correspond to the atoms of chemistry should not to be prejudiced by any existing orbital-based paradigm. but rather be determined solely by comparison of their predicted properties urith the observed properties of matter.

3. Observational Constraints on the Definition of a Functional Group Before proceeding with the quantum mechanical development. it is imperative to state what are the essential requirements imposed upon the dcfinition of an atom in a molecule by the observations of chemistry. The first stems from the necessity that two identical pieces of matter exhibit identical properties. Since the form of matter is determined by the distribution of charge throughout real space, two objects are identical only if they possess identical charge distributions. This parallelism of properties with form exists not only at the macroscopic level, but as originally postulated by Dalton, is assumed to exist at thc atomic level as well. It demands that an atom be a function of its forin in real space. requiring that it be defined in terms of the charge distribution. It follows that if an atom exhibits the same form in the real space of two systems, or at two different sitcs within a crystal, it contributes identical amounts to the total value of every property in each case. This requirement demands in turn that the atomic contributions sum to yield the total value for the system, that is, the atomic contributions to a property must be additive. Only atoms meeting these two requirements can account for the observation of so-called additivity schemes wherein the atomic properties appear to be transferable as well dditive. Finally. the boundary defining an atom or functional group must be such 21s to maximally preserve its form and thereby inaxiniize the transfer of chemical information from one system to another.

Theory of Functional Groups These are the there requirements that must be satisfied if one is to predict that atoms and functional groupings of atoms contribute characteristic and measurable sets of properties to every system in which they occur It is because of the direct relationship between the spatial form of a group and its properties that we are able to identify it in different systems. Since the purpose of quantum mechanics is to predict the measurable properties of a system, it is reasonable to suppose that these observational constraints will be met by the quantum definition of an atom.

4. From Dirac’s Theory to the Theory of Atoms in Molecules Our goal here is not to review the mathematical foundation of the theory of a quantum subsystem. This has been done previously in a number of r e f e r e n ~ e s . [ ~ *Here ~ ’ I we wish to concentrate on its underlying ideas. We preface this discussion with a brief synopsis of this discussion for readers who wish to proceed directly to Section 4.3. Central to the new approach to physics developed by Feynman“] and Schwinger[’] is the principle of least action, the idea that there exists a quantity called the action whose minimization yields the equation describing the motion of a system. t h e action itself is a quantity which is determined by the evolution in time of a system in real space and it thus defines a space-time volume. Schwinger showed how one can obtain all the laws of physics by extremizing this space-time volume in a general way. If one desires a definition of an atom based on physics, the atom must generate its own space-time volume and its boundaries must therefore, be defined in real space. This is the reasoning underlying the basic tenet of the theory of atoms in molecules, which is that atoms be defined as pieces of real space.r51

4.1. From Dirac’s to Schwinger’s Theory The historical development of the ideas that led to Schwinger’s formulation of physics began with Dirac. Dirac demonstrated the equivalence of Schrodinger’s wave mechanics and Heisenberg’s matrix mechanics, both of which are firmly rooted in classical Hamiltonian dynamics.[321Dirac’s work went beyond this demonstration of equivalence, and in particular he introduced the theory of unitary transformations to parallel the canonical transformations of classical theory. Unitary or canonical transformations enable one to obtain equivalent descriptions of a system by passing from one representation to another. In 1933, in what was to be a paper of singular importance,[33] Dirac posed and answered the question of what would correspond in the quantum theory to the Lagrangian method of classical mechanics and its associated action integral, a formulation of mechanics he considered to be more fundamental than that based on Hamiltonian theory. The answer to this question is provided by the so-called multiplicative law. This law provides the possibility of expressing a probability amplitude connecting two states as a product of contributions connecting intermediate states. As a consequence of this theorem, the

REVIEWS associated action can be expressed as a sum of corresponding contributions. It is upon these ideas that both Feynman“] and Schwinger[*]based their new formulations of quantum mechani c ~ . These [ ~ ~approaches ~ ~ are equivalent in that Schwinger‘s “fundamental dynamical principle” is a differential statement of Feynman’s “path integral”, but Schwinger’s approach is specifically formulated to enable one to ask new questions of physics. It is this aspect of Schwinger’s work that is not fully appreciated. One cannot find atoms in pre-Schwinger quantum mechanics. If one tries. one is simply led to his formulation.

4.2. From Schwinger’s Theory to the Theory of Atoms in Molecules Central to the Lagrangian approach is the principle of least action described above. Within quantum mechanics, the extremization of the action can be accomplished by varying the state function in the action integral. In extending the principle to a subsystem of a total system, one must necessarily retain these variations on the surface of the subsystem. But, retaining the variations o n the boundary of the system is the very step which Schwinger purposefully makes in his generalization of the action principle. He does this to introduce Dirac’s transformation theory, a step he accomplishes by identifying the variations with the generators of unitary transformations. Such generators describe all possible transformations, and hence all physical changes a system can possibly undergo. With their introduction, Schwinger generalizes the action principle to obtain quantum mechanics from a single dynamical principle.[34b1In particular, the extended action principle defines the observables of quantum mechanics, their equations of motion and their expectation values, in addition to the commutation relations. These are precisely the properties which must be established and defined for a subsystem if one wishes to obtain a quantum prescription for an atom in a molecule. The possibility of defining the action for a subsystem is a consequence of its fundamental additive nature. The variation of the action for a subsystem leads one down the same road traversed by Schwinger, but one does not arrive at the same destination and recover his principle unless the subsystem satisfies a particular boundary condition. This condition is imposed as a constraint on the variation of the action and it requires that the subsystem be bounded by a surface through which there is a zero flux in the gradient vector field of the electron density ~ ( u ) [Eq. (a)], where n is the unit vector normal to the surface.

V@(r). n(v) = 0 at every point on the surface

(a)

Because the boundary condition is stated in terms of the electron density, a quantum subsystem is defined in real space, as it must be if its properties are to be defined as quantum mechanical expectation values. Indeed there is no such thing as an isolated system, and all systems are subsystems with varying degrees of interaction through their shared zero-flux surfaces. 623

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REVIEWS 4.3. Defining a Group by Applying the Quantum Boundary Condition The application of the boundary condition defining a quantum subsystem, the zero-flux requirement in Equation (a). leads to a disjoint partitioning of the space of a molecule into a set of atomic basins.". 351 This is a consequence of the principal topological property exhibited by the electron density; that is. it exhibits local maxima at the positions of the nuclei. A nucleus thus serves iis an attractor in the V Q field. and the region of space traversed by the trajectories of V Q that terminate at a nuclear maximum defines the basin of the atom. The basin is necessarily bounded by ii zero-flux surface.". 3 s 1 A representation of a group as a space-filling object with zero-flux surfaces separating it from the remainder of the molecule, is shown in Figures 1 ii and 1 b. The group is the central

graph which in turn defines the molecular structure.". 351 That portion of the molecular graph which defines the methylene group up to its interatomic surfaces is shown in Figure 1 c. In agreement with the chemical definition. a functional group is a linked set of atoms. bounded by a zero-flux surface. Aside from the interatomic surfaces being curved rather than flat, this representation of the methylene group is similar in all respects to the space-filling models of atoms that chemists link together to represent a group or molecule. The boundaries between the a t o m in the van der Waals cnvelope dcfined by thc electron density are in general, less pronounced than whcn they are approximated by using van der Waals atomic A group identical to that in Figure 1, hound only to other rnetliylenes, is found in all 12-hydrocarbons following and including pentane. In terms to i t 5 spatial representation and its other properties. it fulfills the chemist's expectations of the transferable inethylene group of the hydrocarbons; it possesses a zero net charge. as it must if it is to be the repeating building block of the rz-hydrocarbons. It makes a constant contribution to the molar volume a s anticipated by the experiments of Kopp in 1855. Its energy equals the energy increment between successive pairs of hydrocarbons in agreement with the heats of formation of the hydrocarbons measured by Rossini et al.r371Its polarizability and magnetic susceptibility reproduce the measured values of these field-induced properties for the methylene g r o u p . i ~ ~ 391 .

Fig. I . A Lie\&of thc trnnsferable methylene group of the hydrocai-bonswith t i ) the van der Wxil, surface of the t \ r o hydrogen xoiiis. and b) ihe two C / C surfaces di\pla!ed in the Ibregi-ound. The bond critical point is denoted by a dot for one C 1 C surliice. c ) Thc portion of the molccultii- graph that defines the iiiethylcne gi-oiip up t o the t \ c o bond critic:il points (denoted by dots) in the CIC ~nteratomicsiirf,ices. d ) A di\pl;rq o f thc indiiidual trajectories of PL,nhich terminate at 3 bond critickil point and deline ii CIC interatomic \ut-lhce. Also illusti-kited is the unique pair of triijeciorres which origin;itcor thecritical point tirid define the bond path. The .;htipe 01 thi\ \urlirce I, charxcterisiic of \urlhce\ between wtiiriited carbon atoms. including t h a t found i n the diamoiid lattice.

This group serves as an example ofthe groups that are defined by theory and demonstrates their ability to reproduce the quantitative as well a s the qualitative properties of the groups of chemistry. The classification and identification of a group in terms of its properties is the central idea in the application of quantum mechanics to the study of functional groups. Thus. the quantum mechanical concepts underlying the definition of the measurable properties of a system deserve careful consideration.

5. Definition of Group Properties 5.1. Observables and Expectation Values

methylene in pentane. It is obtained by separating the basins of the two neighboring carbon atoms in the hydrocarbon chain at their common interatomic surfaces. The group is represented symbolically by /C(H2)J.the vertical bars denoting the C JC interatomic surfaces, and graphically (see Fig. 1 a. b) by showing thc two interatomic surfaces and their intersection with a surface on which the electron density attains a prescribed value, in this case 0.001 au. This envelope of electron density determines the van der Waals shape of the group["] and is used to replace those portions of the atomic surfaces which occur infinitely far from the nuclei. Illustrated in Figure 1 d is the manner in which a C / C interatomic surface is defined by the set of trajectories of VQ which terminate at a bond or (3,-1) critical point of the electron density (a point where Pq = 0), as well as the unique pair of trajectories which originate at this point and terminate at each of the neighboring nuclei. They define the bond path, the line of maximum charge density linking the carbon nucleus to its bonded C neighbors. The network of bond paths determines thc molecular 624 624

C

Associated with every measurable property of a system is a linear Hermitian operator. an observable. An oft repeated statement concerns the impossibility of defining an atom or its properties, a n atomic population for example, because they are not "quantum obser\~ables".[sO1 Part of the problem with such statements arises from their misuse of the language of quantum mechanics, and one again turns to Dirac for a clear statement of the principles involved. Dirac defines an ~ h ~ ~ r wtoi hbek a~ real dynainical variable, that is, a linear Hermitian operator, whose eigenstates form a complete set.[321The nicuszirahle value of a property is one of the eigenvalues of the corresponding observable, an expectation value being the average value obtained by repeated measurement.'"] Properly stated, the electron density is the measurable c..\pcc.intior? value of the density operator. a quantum mechanical oh. s ~ ~ r ~ ~Thus ~ h a/ n~atom ~ . ~is~defined ' ~ in terms of the expectation value of a quantum observable. To state. a s some do. that "an atom or its population is not a quantum observable" is to say

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Theorq of Functional Groups that neither is a linear Hermitian operator, which while true, is not a ineaningful statement. The quantum boundary condition defining an atom [Eq. (a)] can be stated in terms of the density operator and thus an atom can be defined directly in terms of the action of a quantum Similarly, an electron population is defined as the expectation value of the number operator, the integral form of the density operator. and again a quantum observable.[41' It will yield an atomic population if one can obtain ;I quantum definition of its expectation value for atoms in molccules.

5.2. Atomic Expectation Values As discussed above, Schwinger's formulation of quantum mechanics"] does indeed enable one to define expectation values for a subsystem which satisfies the boundary conditions [Eq. ( a ) ] .The required statement obtained from the generalized variation of the action integral is given in Equation (b).r5,3 0 . 3'1 In this equation the symbol W[Y,R.t] denotes the Lagrange integral for ii system Q bounded by a zero-flux surface, the quantity whose integral over time yields the action. The Hamiltonian operator is denoted by H, and G denotes any quantum observable. The important point to note about Equation (b) is that the commutator when multiplied by i:h. gives the time rate-of-change of G, the Heisenberg equation of motion. Thus, Equation (b) is a variational derivation of the equation of motion for the average value of an o bsei-vable.

6.2. Atomic Force Theorem The time rate-of-change of the momentum is force. and setting the generator G equal t o j , the momentum operator for an electron, yields the atomic force theorem. This i s the Ehrenfest force,[431the average force exerted on the electron density. In a stationary state this theorem establishes that the force acting on an atomic basin is determined by the flux in the force through its atomic surface. and is equal and opposite to that exerted by its bonded neighbors. Thus an atom and its properties respond only to changes in the total Ehrenfest force acting on each element of its surfxe. and hence only t o changes in its surface. By preserving the interatomic surface characteristic of a given interaction. the properties of the group iire prcserved whether or not the bonded neighbor is considered to be present. Using this property it is not necessary to "cap" ii group when only a fragment of a larger system is to be studied. as exemplified by the isolation of the single peptide group from a polypeptide shown in Figure 2 b. The alanyl group is obtained by scpa-

[H.c],

The equation applies to any region of space bounded by a surface of zero-flux in the gradient vector field of e [Eq. (a)]. It applies equally to the total system and to every atom within the system. Thus an atomic expectation value is defined on an equal footing with the expectation value of the same property for the total system. An important consequence of the averaging implied in Equation (b) i s that the expectation value of any observable M for the total system is given by the sum of the corresponding atomic values M(R [Eq. (c)].

6. Atomic Theorems 6.1. Atomic Populations Setting the observable G i n Equation (b) equal to the number operator yields not only a definition of an atomic population, as the integral of the electron density over the atomic basin. but also an equation for its time rate-of-change. the integrated equation of It is a general result that a single variation principle [Eq. (b)] defines an observable's expectation value and yields a theorem governing its behavior, for both the total system and for its constituent a t o m ~ . [ ~ , ~ ~ ]

Fig 2. a) A represenlation of the rransferable methyl group oFsiiIiirated hydrocarbons with its single interatomic surfiice. This group a n d its properlies i\ found in :ill n-hydrocarbons following and including piopane. h) A repre\eiitiition (Ji'lhc a l a n y l group. lNHCHRC(=O)I. R = CH,. The -C(=O)l surf~iceisonthe right, the -HN1 surface on the left. and the CHR group is in the foregound with [lie nir.th\l group pointing down. Figure 2 h I F drawn to half scalc rclative t o Figure 221

rating the basins of neighboring N and C atoms at their common interatomic surfaces in the two amide bonds which link the group to a polypeptide chain. The alanyl group, like the other peptide groups investigated so far. possesses a zero net ~ h a r g e , [ and ~~> a peptide ~ ~ ' unit can serve as a building block of a polypeptide chain. The CIN interatomic surface of the amide bond undergoes only minor change, primarily in its outer reaches far from the bond critical point, as the substituent R is changed and different peptide units can be linked together to form a polypeptide with only minor modifications in their form and properties.[44'4 5 1 The sensitivity of an interatomic surface to changes in neighboring atomic basins acts as probe for spatially isolating changes within a system. Thus a rotation about the C - N bond of forinainide causes only minor changes to the CIO and CI H surfaces, but results in a substantial shift in the CIN surface towards the N nucleus. As anticipated on the basis of these findings, the properties of the carbonyl oxygen and hydrogen atoms, their charges, energies, and associated bond lengths. are largely unaffected by the rotation. the major changes occurring within the basins of the C and N atoms and primarily in the region of their interatomic surface.[46' 625

R. F. W. Bader et al.

REVIEWS 6.3. Atomic Virial Theorem Setting the generator equal to the operator i . j , where i is an electronic position coordinate. yields the atomic statement of the virial theorem. Since this generator has the dimensions of action, its time derivative is energy. Thus this theorem defines the atomic averages of the electronic kinetic and potential energies. The potential energy, which is determined by the interdctions between all the particles in a system, is defined to be the virial of the Ehrenfest force. a local property. Thus the total energy, like all properties, is given by a sum of atomic contributions. An atomic energy exhibits the necessary property of being as transferable as is the electron density itself. As discussed above, when the physical form of any object, including an atom, is the same in two situations. observation requires that all of its properties, including its energy, must remain unchanged.[471This requirement is fulfilled by the energies of the transferable methyl and methylene groups depicted in Figures 1 and 2a.f48-511 While there is no known functional relation between the electron density and the total energy, the properties of the quantum atom speak for its existence and ultimate discovery. The total energies of the methyl and methylene groups are in the tens of thousands of kJmol- yet they are transferable amongst the n-hydrocarbon molecules, both theoretically and experimentally, with changes in energy of less than four kJmo1-l. This is observed in spite of the individual contributions to the potential energy of a group changing by thousands of kJ mol- ' between succeeding members of the homologous series. The glycyl group, with an energy of approximately 524000 kJmol-', can be transferred from formylglycylamine to triglycyl with a change of only 24 kJ mol By reason of their definition in terms of the zero-flux surfacec5]and by demonstration,". 3 6 . 4 4 . 4 5 . 4 8 - 5 1 1 the atoms and groups of theory are the most transferable pieces of a system definable in an exhaustive partitioning of real space. If perturbations are relatively small, groups respond so as to minimize the total change resulting from their interaction. Thus, the methyl group of ethane necessarily withdraws a small amount of electronic charge from the methylene group when transferred to propane, but not only is charge conserved. so is the energy. The energy lost by the methylene group equals the energy gained by each methyl group, and a similar conservation of charge and energy occurs for methylene groups bonded to a single methyl group in succeeding members of the homologous series. Thus additivity of energy is preserved in spite of the methyl group in ethane necessarily being 511

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atom.c421The contribution of this flux to the mean atomic magnetic susceptibility j ( Q ) , becomes vanishingly small when the current flow, like the charge density. is localized within the atomic basins, as illustrated for the ionic system LiH in Figure 3 a. It becomes appreciable and Compdrabk in magnitude to

a

Fig. 3. a) The current density induced by an externally applied magnetic field for a symmetry plane containing the nuclei in LiH; the Li nucleus is on the left. The direction of flow within a current loop, for a field applied perpendicular to and into the plane. is indicated by a n arrow. The positively charged Li atom has both diamagnetic and paramagnetic currents induced in its basin, the paramagnetic loops being isolated from the diamagnetic flow in the basin of the negatively charged H atom by the interatomic surface. This localized atomic behavior of the current flows and the presence of a paramagnetic component in the basin of the electropositive atom is typical ofmolecules with closed-shell (ionic) or intermediate (polar) interactions [ 5 ] and rellects the correhpondmg localization of the electron density. b) The current flow in an alternating plane of symmetry containing the carbon nucleus in the methane molecule, showing the intersection of the C / H interatomic surfaces. In this molecule with shared (covalent) atomic interactions [5] there is a substantial flow of current across the interatomic surtaces.

the basin contribution to z(Q) in systems with shared atomic interactions, as found in methane (Fig. 3b). The flux in the current flow across the interatomic surfaces for a linked ring of atoms dominates j in molecules such as benzene.[52.5 3 1 Equation (b) also demonstrates that in a steady magnetic field there is local conservation of the current over each atomic basin. The electronic current density and its derived properties induced by a magnetic field for an atom in a molecule exhibit the same degree of transferability as does the charge density itself. Thus the methyl and methylene groups of Figures 1 a, b and 2 a contribute constant amounts to the magnetic susceptibilities of the members of the homologous series of hydrocarbons, confirming Pascal's values[391for the group s u ~ c e p t i b i l i t i e sThe .~~~~ same constancy in group contributions is found for the mean polarizability when the molecules are in the presence of an applied electric field.f501Thus even in the presence of external fields, the atoms of theory behave as required by observation; their field-induced properties reflecting the form of the atom in the field-free case.

6.5. Atoms in Crystals 6.4. Atomic Current Theorem The time rate-of-change of position is velocity and setting the generator in Equation (b) equal to E. yields the theorem governing the average velocity of the electrons in the basin of an atom. In the presence of a constant uniform applied magnetic field, this velocity corresponds to the average of the electronic current induced over the basin of the atom. Equation@) shows this atomic average of the electronic velocity to be equal to the flux in the position-weighted current through the surface of the 626

The idea of individual atoms being the source of the scattering of X-rays in a crystal dates back to Laue and Bragg.'541A carbon atom in the face-centered cubic structure of diamond is shown in Figure 4.[551It is completely bounded by its four interatomic surfaces, each of which is curved in the form of a chaiselongue. This atom, of Td symmetry, linked to a mirror-image neighbor at one of its interatomic surfaces, defines a WignerSeitz cell of D,,symmetry for the crystal. This cell is the smallest connected set of atomic basins which preserves the translational Angm C%WI Int Ed Engl 1994. 33, 620 631

Theor! of Functional Groups

REVIEWS

Fiy. 4. A representation 01‘ ii cat-bon a t o m of T, syinmetr! in the diamond lilllice I t i5 linked by bond pxtha 10 four other ;Itotms. each bond p i l h originatineat thecriticiil point delinine the a s m a t e d iiitcratotnic w-face.

invariance of the lattice. The single atom depicted in Figure 4 determines all of the bulk properties of the lattice in terms of its quantum mechanical expectation values, including the scattering of X-rays and the appearance of the ”forbidden” 222 reflection. Any and all cxpectation values and their associated theorems are defined by Equation (b) for an atom. a linked grouping of atoms, o r for the complete molecule or crystal. The above examplcs illustrate how the theory of atoms i n molecules enables one to apply the theorems and principles governing the behavior of quantum mechanical observables to the functional groups of chemistry. and do so in precisely the same manner as is done for the total system. In what follows a survey of the form and properties of some atoms and common functional groups is presented to demonstrate how they recover and expand upon our understanding of this basic chemical concept.

7. Form and Properties of Functional Groups The periodic variations in the form of the A and H atoms in the second-row hydrides AH,, are displayed in Figure 5 and selected properties are given in Table 1. The properties of the atoms in LiH. BeH,. and BH, reflect the hydridic nature of these compounds. There is an abrupt change in the form of the H atom a t methane, reflecting a change to a n o n p o l x interaction that is characterized by a nearly equal sharing ofthe valence charge density. The decreasing size of the H atom in NH,, H 2 0 , and H F reflects the increasing polarity of the atomic interactions and the resultant acidity in these latter members. The atomic tirst moment M(Q) measures the dipolar polarization of an atomic density relative to its nucleus. The directions of M(H) reflect the interatomic charge transfers. the H atoms being polarized towards the positively charged A atoms preceeding carbon and away from those bearing a negative charge following it. The atomic susceptibilities also parallel the characteristics of the atomic charge distributions. The currents localized in the basins of the A atoms in the hydridic molecules correspond to opposing paraI11;lgnetic and diamagnetic flows, as illustrated for LiH in the Figure .?a. The resulting cancdhtion leads to a sinall negati\ value of X(A)for ~i and positive or paramagnetic contributions for Be and B. The boron and hydrogen atoms in B,H,, and the apical B , ~ r o u pi l l B,H; are displayed i n Figure 6, The stability of thesc electron-deficient compounds is a consequence of the delocalization of electronic charge over the surfaces of the threeand four-mcmbrred rings of atoms.[”’ A boron atom in B,H, is a mcmber ol‘a single four-membered ring. The apical boron ~

p13.j. Representiitioiis of the atoms in the second-I-ow hydride, AH,, I n the 11)dridic tneiiibers. LiH, UeH?. and BH,, the A a t o m consists pritiiartl). ol:i core o f decreasing radius. dressed with w i n e residual Lalcnce density. The Ihrm and prcipcrtiec of the atoms undergo :I iiiarkcd change at methane. a nonpolar iiiolcculc: i i o Cole is visible on the C iitoin. and the H atoms. considerabl! rcduced iii w e a n d populiition. nou lie on the convc‘r side of the interatomic aurlhce T h e incre;iaing polarity o f t h e remaining members is rellected in the dccrcehinp v i e ol’the H iilotn a n d the increasing convexit! of its interakmic \urfacc.

atom in BjHu shares in four four-membered rings, possesses 0.90 more electrons on average. and is more stable by

R . F. W. Bader et al.

REVIEWS Table 1 . Atomic properties of hydrides AH, computed at 6-31 1

LiH

i1.91 2

- 7.3662

BeH

1.725

- 14.2353

BH,

2.112

-23.7822

CH4

0.175

-37.6601

I.04x

- 54.7472

NH , OH,

F 1-I

-

-

1.254

-0.779

-75.3719

- 99.8149

30 79 (99.30) 19 54 (100.00) 23.17 (99.92) 76 85 (99.70) 137.64 (99.33) 149.83 (99.39) 130.44 (99.51)

+ +G(ld.?p)

6-31 1 + +G(?d.ZpI level [a]

+ 0.005

-0 81

- 0.9 I2

- 0.6 I98

0 000

1 .66

- 1) 862

-

0.000

I.94

-0.704

0 000 -0 172

-

15.X

-21.9

+0.351

-20.X

fl1.456

-16.7

- 0 044

-

0 7682

11.8723

-O.h381

195.82 ( 9 I .97j 139 16 I 4 5 96) XX 14 ( 97.44 I

0 4905

5 1 .45 (97 621 21 17

11.627

-0 3-128

fY7.52) 18.97

0.779

-0.2416

0.349

-

l97.3?) I I 49 197.20)

-

0.407

-0.566

-11 8 -

6.39

-0.395

-3.79

+(I126

- 3 02

i. 0. I X6

~

2 04

fO.153

-1.11

+I)102

-0.80

[a] Thc cliarge y, energ) E. roluine i', and first iiioiiient M a r e i i i atomic units (au), the magnetic susceptibility Xis iii 10."' J ( T e s l a - '. [h] Volume is deter-mined b) the l1.001 du density envelope. The iiuniher in parentheses is the fraction oS the iitoinic population conrained within this enrclope [c] A ncgati\s s i g i denotes .I polariution o f t l i c electron density of A along the principal symmetry B Y I C i n a direction awe). from the protons. ii poiitive \ign thc ac\t'i-se. id] A i i c g x i n e %ignc1cnott.s ii poinl-ir;rtioii of thc electron density of H towards A along the A H bond direction. ii positive sign the reverse.

1070 kJmol-', while a boron atom in B,Hf participates in four three-membered rings, possesses 1.66 more electrons on average. and is more stable by 1730 kJ mol-

'.

Fig. 6 a).Tlle boron atom in dihorane. The two terminal hydrogen atoms lie i n the plane of the diagram a n d one such BI H interatoiiiic surface is shown oii the left An interatomic s u r h c e for a bridging hydrogen. which will project out of the plane of the diaprain. is also s h o a n . b) and c) A terminal rind a bridging hydrogen atom. respectively. in diborane. d ) A €+-Bl group ofB,H;- shohing two of its four B I B iiiteriltomic surfaces. A cage critical point is located at the apex where the four surfaces intersect.

The carbon and hydrogen atoms of ethane, benzene, ethene, and ethyne are displayed in Figure 7 and properties are given in Table 2. The value of g at the Cl C bond critical point. the quantity Q,, increases throughout this series and yields bond orders of 1.0, 1.6. 2.0, and 3.0, respectively. The decreasing extent of charge transfer from c to H evidenced in the atomic charges (,(H), reflects the increase in the electronegativity of the carbon atom with change in hybridization from 'p3 to 'p' The 'p2 carbon atom in benzene is more stable than that in ethene.

F ] ~7.. T h e c and H atoms in ethane (;I). bcivene ( b ) , ethene ( c ) . and cthyne ( d j . respectively. T h e CIC surl'ace i i i ethane is siinilar in shape t o that fouiid i n methql o r methylene groups and diamond (Figs. 2 a . l a . b. 41 The CIC surfaces i n the remaining inolecules 'tit' p l m a r a s :I conyequencc or \yiiiineti-y. Note That the H atom is largest in ethane and cmallsst i n ethyne

REVIEWS

Theory of Functional Groups Table 2 . Atomic nroDerties of C and H atoms in hydrocarbons computed at 6-311

CIH,

0.1x4

-37.6720

C ‘ A

0.020

-37.8246

C2HI

0.035

-

C,HI

-0.136

37.7651

- 37.8642

CH;

0 I79

- 37.7439

ct1:\

-0 035

- 37 7399

CH.;

-0.422

-37.5359

64.74 (99.85) 83.08 (99.68)

96.X

(99.06) 126.04 (99.12) 93.41 (99.52) 116.81 (99.22)

182.24

+ fC(2d.2~):6-311+

+G(?d.Zp) lebel [a]

-0.029

-13.5

-0.061

-0.6522

1 0 051

- 13 6

- 0.020

-0.6371

+0.006

-11.2

-0.017

-0.6328

+n.i26

- 15.8

0.000

-

-0.001 - 1.513

-

(97.95)

0.136

-0.5593

0.274

-0.5000

0.012

-0.h113

-0.193

-0.6609

51.45 (97.76) 49.76 (97.76) 49.89 (97.64) 41.92 (95.87) 35.42 (97.73) 49 22 (97.53) 68.45 (96.37)

+0.127

-3.39

f0.119

- 7 94

+0 I20

-2.96

1 0 111

-2.61

+ 0. I (16 +0.131 f0.25X

In]. [b] See footnotes to Table 1 . [c] A negative sign denotes a polarization of the electron density of C along the principal symmetry axis in a direction iiwiiy from the proIon\. ii positive \ign the reberse. [d] A neplive sign denotes a polarization o l t h e electron density of H towards C along the C-H bond direction. a positive ~ g hn e reverse.

Bridgehead carbon atoms in [I .l.l]bicyclopentane and [1.1 .l]propellane are shown in Figure 8 a and c. In addition to the C / H surface, the carbon atom in the bicyclic compound has

Fig. 8.a ) .A bridging C atom and b) its H atom in [I.l.l]bicyclopentane. c) a bridging C atom i n [l.l.l]propellane.and d ) a methylene group in cyclopropane. For details sce text. The carbon atom in (a) IS capped by electron density bound to a proton in the basin of a hydrogen atom (b). The carbon atom in (c) exhibits a bulge of nonbonded density. which the Laplacian of the charge density shows to be a nonbonded charge concentration. a site of electrophilic attack [5].

three CI C interatomic surfaces with the methylene carbon atoms. Their common point is a cage critical point where the electronic charge density attains a local minimum value of0.098 ~ u . [ ~ ’ I The carbon atom in the propellane molecule, in addition to three similar C I C surfaces with methylene carbon atoms, shares a fourth surface with the second brighehead carbon atom. This surface, the flat inner triangle bounded by three ring critical points, while of rather limited spatial extent, represents a significant accumulation of electronic charge. The value of Q at the ring critical points is only 0.004 au less than its value of 0.203 au at the central bond critical point, of bond order 0.7 (0.6 with a GVB wave function[571).where Q attains its maximum value in the interatomic surface. The result is a triangular accumulation of electronic charge linking the bridgehead carbon atoms’ in place Of the usual line Of Of charge density‘ Note the significant accumulation of electronic charge in the Aii,Cvii

C/iw?i.

In! Ed. Ei7pl. 1994, 33. 620-631

nonbonded region of the propellane carbon atom[”] whose presence accounts for its susceptibility to electrophilic attack.[”] In Figure 8 d a methylene group from cyclopropane is shown. It differs from the standard transferable methylene groups displayed in Figure 1 by having planar C 1 C interatomic surfaces. Subjecting the standard group to the geometrical constraints of a three-membered ring results in a strain-induced transfer of charge from H to C within the CH, group. This leads t o an increase of 38.7 kJmol-’ in its total energy relative to the standard group and to a total strain energy three times this, or 116.1 kJmol- 1,[48, 501 in agreement with the experimental value of 115.1 k J ~ n o l - ’ . This ~ ~ ~is] another example of an experimentally measured property being recovered in terms of a group defined as a quantum subsystem. Figure 9 gives representations of the carbon atom in the methyl cation. the methyl radical, and the methyl anion. Atomic properties are given in Txble 2. The atomic forms provide a vivid display of the growth of the nonbonded charge on the carbon atom.

Fig. 9. The carbon atoms in the methyl cation (a), radical ( h ) . and iinioii (c). respectively. A comparison of (a) with (c) adds to the understanding of the differing properties of cationic nd anionic carbon atoms.

629

R . F. W Bader et al.

REVIEWS 8. Summary and Outlook This article has presented the quantum mechanical definition of a subsystem and its properties made possible by Schwinger's reformulation of physics. It has shown that the properties of these subsystems meet the constraints imposed on the definition of atoms in molecules by the observations of chemistry. Their properties and physical forms are compatible with a chemist's expectations regarding the behavior of a functional group. 111 particular, the subsystem expectation values for the historically important additive and transferable properties of volume. energy. polarizability, and magnetic susceptibility. agree with the experimental values obtained for the homologous series of 17-hydrocarbons. Agreement with experiment is the ultimate test of any theory. It has been demonstrated that the classification and prediction of chemical behavior afforded by the concept of a functional group is contained within the quantum mechanical framework of observables and their equations of motion. This extension of the predictions of quantum mechanics to include the atoms that make up a total system increases the input of physics into chemistry and complements the information obtained from theories and models of electronic structure. The literature now contains many examples of how the properties of atoms i n molecules can be used to increase our understanding ol' chemistry. Perhaps because of its use of a new and unfamiliar principle of physics, there is a tendency to ignore the quantum basis of the theory as exemplified in reference [40], or to overlook this basis and instead refer to it as an "electron density analysis".'"01 The theory is a generalization of quantum mechanics to a subsystem. enabling one to apply physics to the total molecule or to its constituent atoms on an equal footing. The electron density enters through the statement of the boundary condition."'" ii condition necessarily expressed in terms of the density if the theory is to apply to real physical systems. The boundary condition is identical to that used by Wigner and Seitz'"" in the definition of an atomic cell in their historic quantum investigation of the properties of metallic sodium within the one-electron model. The boundary condition is of profound physical importance. It not only defines atoms in molecules, but further study of the gradient vector field of the electron density le,I' d s to ;I complete theory of structure and structural stability."] The resulting unified theory of molecular structure is obtained because the definition of an atom given in Equation (a) is a consequence of the fundamental form exhibited by matter at the atomic level. In outlook, one can anticipate an increase in the effectiveness of the functional group concept, through the replacement of-the present qualitative and largely empirically derived knowledge of group properties by the predictions of quantum mechanics. Since theory predicts what can be observed. one should eventually be able to do everything with the functional groups of theory that one does with them in the laboratory, including their use[44.4s3a s synthons.[63]It is perhaps our ability to design the synthesis o f a desired molecule in terms of the successive linking of chemical transformations. each step depending upon the particular characteristics of ii given functional group, that represents the crowning achievement of the functional group concept. The synthesis of a given molecule from ;I bank of 630

theoretically defined groups will require further advances in subsystem quantum mechanics in the form of a variation or perturbation theory to predict the changes undergone by ;I group when it is placed in a new as required in the synthesis. I t is to be hoped that the recognition of the theory of atoms in molecules as an extension of quantum mechanics to the atomic doniain'651 will spur workers entering the field to further explore and develop subsystem quantum mechanics, enabling physics to be put to its fullest use i n chemistry, in the design and synthesis of new molecules.

Reccived December 17. 1992 [A 934 IE] A t i ~ y r i t .C ' h i . 1994, IO6. 647

Geriiiaii v s r m n :

[I] R . I'

Feynmm. Kr>i,..Mod. P i i i ~1948. 20. 367 387. [2] J. Schv.inper. Phi 5 . /?cis. 1951. 87. 914-927 131 T S. Kuhn. Thr S i r m i i i r ~ .ril Si.roirrl/(.l < m i / l i t m 1 t . s . 7 / 1 d d Uni\crvty d' Chicaeo Pi-esr. Chic;igo. 1970. I strite o f II system is completely dctermincd by specifying the the complete sct of commuting obsei-b,ihles. This is the maxiion which can be gained ahout a quantized system. A. Mesyiah, Qi(irirfiiiii MI,(/ I O I I / < 5 . L i d 1. Wiley. New York. 1958, p. 204 aitd 301 ;P. Roman, . A ~ / w I I ~ c Y / c)iiiiir/iiiri T/iiwri . Addison-Wesley. Reading, MS. USA, 1965. p. 13. [S] R. F. N Bader. .A/oI?I.! I I I Ilo/wiili~.s- A Q i i u i i ~ i i i ~Thcvrj. i Oxford University Preas. O s f o i i l . 1990. 161 L . Libit. R. Hoffninnn. .I ,4171. C/WIH. S w . 1974, 96, 1370 -1383 Beddnll. 1 C%CWI. P/JJ,S. 1972. 56. 3330 3329. Rcr. 1928. 32. 186; R ~ Y:kJm/. . PIIJY.1930, 2. 60 1 15. 1928. i/. 759;

Spi-iiiSci-. Berlin. 1933. p. 561

Hriiidhiiih

[I01 In 1115 ireiniiii~ceiice\.Mullikeii notes that

rler Pitj~srh,24. 3 r d editioti.

.outh he spent much tiiiie examining the loc,il I1ol.a o i i Plum island. near to his home m d soes o n to state that "I h i i d n e i i become i i i i aiiiiiteiii. taxonomist". defined a s one who i s ii -'scient i f i c classifier" R . S. Mullikcn i i i Sclwfcrl Pr/piw of Rohcrt S.,Zlii/lrXoi. P w / / (Ed,.: D. A. Ramsny. J. H i i i x ) , Thc Univcrsity of Chicago Press. Chicago. 1975. p 4. [ I I] D. R . Hal-tree. Pro1 .