Atomic Charges Within Molecules

ADVANCES IN ATOMIC AND MOLECULAR

.

PHYSICS VOL. 20

ATOMIC CHARGES WITHIN MOLECULES G . G. HALL* Department of Mathematics University of Nottingham Nottingham. England I . Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Electron Density . . . . . . . . . . . . . . . . . . . . . . . C . Electrostatic Potential . . . . . . . . . . . . . . . . . . . . . I11 . Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Atomic and Overlap Populations . . . . . . . . . . . . . . . B. Modified Weighting . . . . . . . . . . . . . . . . . . . . . . C . Conserving the Bond Moment . . . . . . . . . . . . . . . . D . DividingSpace . . . . . . . . . . . . . . . . . . . . . . . . E . Localized Hybrids . . . . . . . . . . . . . . . . . . . . . . . IV . Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Bisection . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Loges . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Topological Atoms . . . . . . . . . . . . . . . . . . . . . . V . Pointcharges . . . . . . . . . . . . . . . . . . . . . . . . . . A. Empirical . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Localization . . . . . . . . . . . . . . . . . . . . . . . . . C . Shrinking Gaussians . . . . . . . . . . . . . . . . . . . . . . VI . Purport and Prospect . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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41 42 42 43 44 45 45 48 48 49 49 53

53 54 54

57 57 57 58 60 62

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I Prologue Understanding the structure and properties of a molecule is not accomplished merely by calculating its wave function. The contemporary wave function is generally too complicated to be allowed to emerge from the

* Present address: Division of Molecular Engineering. Kyoto University. Sakyo.ku. Kyoto 606. Japan . 41 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003820-X

42

G. G. Hall

computer. What is required is a set of derived quantities which are relatively few in number, unambiguous in definition, and rich in meaning. One candidate for this important role is the charge of an atom within the molecule. The atomic charge has an immediate intuitive appeal for the chemist who has used electrical models of molecules for many years. Unfortunately it is not easy to find a definition of the atomic charge which is free of arbitrary features and delivers all the desired meanings. This article records the progress made toward this goal. The concept of an atomic charge is not of relevance for all molecular properties. Section I1 will argue that two properties, the electron density and the electrostatic potential, have a special relation to the concept and can participate in its definition. The first attempts at defining the atomic charge used simple wave functions in which behavior near an atom was described by means of a fixed set of atomic orbitals. The contribution of an atomic orbital to the density, its population, is the definition examined in Section 111. A contour map of the electron density over a molecule is dominated by the high peaks at the heavy nuclei. This observation motivates the next class of definitions, in Section IV, which partitions space into atomic regions and defines the charge by integration. Since the atomic charge simplifies a continuous distribution into a charge at a point, another starting point for a definition is to approximate to the density by a sum over point charges. This option is discussed in Section V. Many attempts to define the atomic charge have been made. Some have become modified in later work as ambiguities and difficulties emerged. Some persist and continue to perform a useful role in understanding and conveying information about molecules. Section VI summarizes the advantages and disadvantages of the alternatives.

11. Properties A.

ENERGY

The geometrical structure of a molecule is determined by its energy E(R). This is the energy, in the Born-Oppenheimer sense, when the nuclei are given a fixed conformation Rand the electrons take up the minimum energy distribution around them. The equilibrium configuration is the one for which this energy is minimized with respect to changes in the nuclear coordinates: aE/aR

=0

(1)

ATOMIC CHARGES WITHIN MOLECULES

43

The energy E can be defined as the mean value of the Born - Oppenheimer Hamiltonian H if the wave function is known. It is written more simply if the one-electron density matrix y(x, 1 x',) and the two-electron density matrix r ( x , x 2 I xixi), where x, is the position and spin coordinates ofthe rth electron, are given or can be deduced. For a full discussion of these matrices see Davidson (1 976). The Hamiltonian H depends on the nuclear conformation as a set of parameters. It may also contain other parameters. I f a is one such parameter the generalized Hellmann - Feynman theorem shows that aEpa = (aH/aa)

(2) where the angle brackets indicate the mean value with respect to the wave function used to calculate E (Hellmann, 1937; Feynman, 1939; Epstein, 1974). The theorem is true not only for exact wave functions but also for approximate wave functions which are stable (Hall, 1961). When a is a nuclear coordinate, (-dE/dcu) is the corresponding force on the nucleus. H depends on a through the nuclear-electronic and nuclear-nuclear terms so the theorem states that the force on a nucleus is given by the classical repulsive force from the other nuclei and the attractive force between that nucleus and the electrons. The electrons behave like a classical continuous distribution of charge with density where r is the spatial part of x and the sum is over the spins. Thus, for the force on the nucleus, only the density p is needed. A much stronger result than this has been proved by Hohenberg and Kohn (1964). They show that, under suitable conditions, the energy is a unique functional of p. This has been a tantalizing result since it is an existence theorem and does not construct the functional. Much research effort is directed toward the discovery of effective functional forms. Recently, for example, an expansion formula has been given by Freed and Levy (1 982). The significance of this result in the context of this article is that properties derived from the energy, such as the equilibrium configuration and the binding energy, must be explicable in terms of p.

B. ELECTRONDENSITY The electron density is an observable quantity, though indirectly. It can be deduced from X-ray diffraction and electron scattering. For recent reviews see Coppens and Stevens (1977) and Bonham et al. (1978). Corresponding

G. G. Hall

44

plots of theoretically calculated electron densities have been given by Wahl ( 1964) and Bader el al. ( 1967). Some molecular properties are immediately related to the electron density. These include the moments

M(r,s,t)=

I

xrysz‘pd.5

(4)

Among these are the spherical moments which include the dipole moment with components

M( 1 ,O,O) MQ,1 ,O) M(O,O,1) and the quadruple moment with components M( 1,1 ,O)

M( 1 ,o, 1) M(O,l,1) M(2,0,0) - M0,2,0) M(2,0,0) M(0,2,0) - 2M(0,0,2)

+

These are the moments which are required to give the asymptotic form for the electrostatic potential.

C. ELECTROSTATIC POTENTIAL The electrostatic potential +(r) at a point r can be defined in terms of the electron density as r

where ra is the position and 2, the charge of nucleus a.The gradient of this is the electric field intensity E. An obvious use for this potential is in the evaluation of the electrostatic energy of interaction between two molecules. If molecule A has the potential 4Aand molecule B has a charge density, including the point nuclei, pBthe energy of interaction is

where the second form shows the underlying symmetry of the relationship. This equation also indicates that charge and electrostatic potential are closely related concepts. I The symbol 4 is used for the electrostatic potential in preference to the more usual symbol Vsince the latter is easily confused with the potential energy.

ATOMIC CHARGES WITHIN MOLECULES

45

The chemical significance of the electrostatic potential is that it indicates, for any molecule, the regions most likely to be attacked in a reaction or to be involved in strong intermolecular forces. Thus lone pairs, which typically show negative values of c$ with a minimum inside the lone pair region, are strongly attractive to protons. A full calculation of this attractive energy would have to allow for the effect of the proton field in perturbing the electron distribution. A review of calculations of C#I including numerous figures has been given by Scrocco and Tomasi (1973, 1978). A concept closely related to this is the electrostatic potential at a nucleus. At nucleus p, this is defined as c$/9(rp)=

2 Z a / I r a - rpl -

atB

I

P(r')/Irp-

r/l

(7)

where the term with (Y = is omitted from the sum. This remains finite whereas c$ itself diverges at the nucleus. This is the quantity that is needed in the relation, which is another example of the generalized HellmanFeynman theorem,

aElazp= ( a H / a z p )= c$p

(8) where H is the Born- Oppenheimer Hamiltonian. Politzer (1 98 1) has shown that this relation can be used to develop useful relations between molecular energies. The use of electrostatic potentials in a variety of chemical situations is described in the book edited by Politzer and Truhlar (198 1). The interesting possibility that electrostatic forces, rather than steric forces, may be the dominant ones in the recognition of proteins by one another, has been raised by Warshel ( 1981).

111. Population A. ATOMICAND OVERLAP POPULATIONS

The best known definition of an atomic population is due to Mulliken (1935). If the diatomic molecule AB has a molecular orbital

+

aA, bB, (9) where A, is a normalized atomic orbital on A and B, on B while a, b are numerical coefficients, then the atomic population of A is uz. Since A, and B, are on different centers their overlap integral, S, is not zero. The total y/=

46

G. G. Hall

charge is found by integrating l(yI2 and so is

a2 + 2abS + b2

and, if this is normalized to 1, it can be seen that there is also an overlap population (2abS)which must be included to achieve conservation of total charge. Mulliken resolved this by dividing the overlap term equally between the atoms. Thus the gross atomic population of A was defined as qA

= a2

+ abS

(1 1)

The atomic charge is the nuclear charge of A minus q A . Obviously, the complete discussion allows for the contributions of several atomic orbitals on each atom and of all the molecular orbitals. The popularity of this definition is due in part to its straightforwardness and in part to its conformity to basic quantum ideas about charge distributions, or their probability distributions, when a linear superposition occurs. It is easy to calculate and it is tempting to take the atom as an observable state of an electron within the molecule with qA as its probability. Nevertheless the definition has a number of disadvantages. Dividing the overlap population equally is a simple but arbitrary device. It can result, when a, b have opposite signs, in q becoming negative or exceeding unity. The prior definition of the atomic orbitals is assumed. In principle, a complete set of orbitals can be selected which are all centered on one atom and which can be used to express all of the molecular orbitals. This choice would put all the electrons on that center. It is implicit, therefore, in the definition that the orbitals used are selected from those on the different atoms in some “balanced” way. The use merely of those occupied in each free atom is one attempt to balance the basis set. There is also a mathematical difficulty which becomes apparent at a later stage. This concerns the choice ofaxes within the system. Except for S-type orbitals atomic orbitals have an angular dependence relative to given axes. If the axes are rotated the orbitals transform among themselves. It is highly desirable that the definition of a quantity should be invariant to this kind of axis rotation. More precisely a good definition of atomic charge should be independent of any choice of axes at that atom (also at its neighbor) and indeed should not change if the occupied molecular orbitals undergo a unitary transformation. As a simple example of the problem let the orbital B, be replaced by (- B,) (i.e., a phase change), then b becomes (- b) and q becomes a2 abS only because the overlap also changes its sign. For diatomic molecules these problems can be avoided by conventions on the choice of axes and of phase (i.e., S > 0) but this is not possible for polyatomics. In the Mulliken theory the overlap population is itself significant. It mea-

+

ATOMIC CHARGES WITHIN MOLECULES

47

sures the electron density which is attracted to both nuclei and so gives rise to bonding. The total overlap population is then a quantity which should correlate with bond strength. The first attempt to generalize these ideas to polyatomic molecules was made by Chirgwin and Coulson ( 1950)in the context of a K electron theory of conjugated molecules. For the gross atomic population their definition is the natural generalization of the diatomic definition and involves overlap populations from all other atoms. For their bond-order definition they generalized the Huckel theory definition which assumed orthogonality. If the density matrix in terms of the atomic orbitals A, is rs

then the atomic population on A, is

and their bond order between A, and A, is

Mulliken ( 1955) himself made a generalization of this to arbitrary molecules. His formulas allow for sums over all the atomic orbitals associated with each atom in such a way that the result is invariant under an axis rotation. A second attempt to provide a generalization was made by Lowdin (1 950) in connection with his method of orthogonalizing a set of functions in a symmetrical way. The orthogonalized atomic orbitals can be defined as A:

=

2 As(S-1/2)sr S

where S-' is the matrix inverse of the overlap matrix S.The density matrix becomes P=

rs

P:sA:*A:

(16)

where

The diagonal elements are used by McWeeny (195 1) to define the atomic populations and the off-diagonal elements the bond order. Although these definitions are not the same as the Chirgwin-Coulson and Mulliken ones

G. G. Hall

48

[see, e.g., the numerical comparisons for diborane given by Politzer and Cusacks ( 1968)],Davidson ( 1976)has shown that when the overlap integrals between atoms are small they are the same to first order.

B. MODIFIED WEIGHTING A number of authors have attacked the Mulliken definition of the gross atomic population on the ground that an equal division ofthe overlap term is arbitrary. One alternative possibility is to divide it in proportion to the net atomic populations of that orbital. Formulas ofthis type have been suggested by Stout and Politzer ( 1968) and by Christoffersen and Baker (197 1). These achieve the desirable object of ensuring that the net contribution to an atomic population from any molecular orbital is always positive and less than 2. In terms of a diatomic and in the simple notation given above, the Christoffersen -Baker definition of the gross atomic charge becomes

+ abS[2a2/(a2+ b2)]

(18) As might be expected, these charges give a more realistic representation of the charge distribution in the sense that, if localized as point charges on the nuclei, the resulting molecular dipole moment is closer to the accurately calculated one. As has been pointed out by Grabenstetter and Whitehead (1972), these formulas are not invariant to a unitary transformation of the occupied molecular orbitals. An alternative form suggested by Ross and Schmit ( 1966) does retain this invariance but has not attracted such support because of its complexity. qA = u2

C. CONSERVING T H E BOND MOMENT The use of the atomic populations themselves results, as B shows, in expressions that become complicated to use and to justify. Lowdin ( 1953) suggested that another criterion for the division might be the conservation of the bond dipole moment. An immediate disadvantage of this idea is that it introduces new integrals, the moments, into the calculation. Attempts to use this idea have been made by Doggett ( 1969)and by Hillier and Wyatt (1969). The results are not greatly changed from the original Mulliken populations though, as Hillier and Wyatt point out, the change is sometimes enough to reverse the charge on an atom. Although the proposal does improve the significance of the calculated atomic charges by ensuring both the conservation of total charge and of total dipole moment, it pays the price of becoming rather more difficult to imple-

ATOMIC CHARGES WITHIN MOLECULES

49

ment. The problem of invariance under rotations is not eased and as a result the charges become basis dependent.

D. DIVIDING SPACE The artificial element in these definitions of atomic charge led Pollak and Rein ( 1967)to suggest a major change in approach. They proposed to define the atomic charge directly by an integration of the electron density over a volume appropriate to the atom. For a diatomic they suggested dividing space by a plane through the midpoint of the bond. At a stroke this gives charges which are invariant to basis and axis transformations. It also gains considerably in intuitive appeal. The use of the midpoint in this definition was seized on by several authors as unnecessarily arbitrary. Bader et al. (197 1) moved the plane to the point along the internuclear axis where the density has a minimum (i.e., to the col point). They pointed out that as a bonus, a radius could then be assigned to the atom so that the internuclear distance is strictly the sum of the radii. Politzer and Hams ( 1970) preferred to define their atomic region by reference to the free atom. Maclagan (1 97 1) offered three definitions. The first was the point where the bare nuclear potentials were equal, i.e., z A / r A = ZB/rBand the second where the forces were equal, i.e., z A / r i = ZBIri. The first gave larger and unrealistic results but the second, which has some justification in terms of the Hellmann -Feynman theorem (see Berlin, 195I), is much more reasonable. A third point obtained by using(2 - 2) for Z for first-row atoms gives qualitatively more acceptable results. While these definitions do solve some of the problems of the Mullikentype approach, they each have their own limitations. Finding a convincing argument for the choice of dividing point will not be too difficult as soon as sufficient evidence for the reasonablenessof the final results is produced! It is much more difficult to see how the definition can be extended to polyatomic molecules since the use of planes to divide space is then no longer sufficient. A recent paper by Grier and Streitwieser (1982) points out that differences in electron distribution following substitution are often localized and so much less sensitive to the definition of the boundary. By an integration differences in the electron populations can be calculated even though the totals remain in some doubt. E. LOCALIZED HYBRIDS

Because part of the problem arises from the need to make the result invariant under rotation of axes at an atom, McWeeny (1 960) suggested that

G. G. Hall

50

there should be a preliminary phase to the calculation in which a “best” hybrid atomic orbital for use in describing the bond is determined. This was developed by Davidson ( 1967) who started by calculating the best least squares fit of the molecular charge density by atomic orbitals. Localized atomic orbitals have also been defined by Ruedenberg et al. ( 1982) not only to describe bonding but to describe its modification along a reaction path. These ideas have been expanded into a full discussion by Roby ( 1974). His work is based on several new ideas. In the first place he rejects the idea of atomic populations implied in all previous work. He associates with an atom all the charge which affects it. When charge is shared by two atoms it is counted in both. This, he claims, gives a more realistic impression of what each atom is contributing to the bonding. Obviously, to conserve total charge, the sum of his atomic occupation numbers has to have the shared parts subtracted so that they are included only once in the molecular total. In terms of the notation in Section III,A, the Roby occupation number is

n A = az

+ 2abS + b2S2

(19)

and the shared occupation number is ,s

= 2abS

+ (a2+ bZ)S2

(20)

so that (21) n~ + n~ - SAB is conserved by the normalization condition. As might be expected these atomic occupation numbers exceed the nuclear charges in all the quoted examples (see Table I). Another aspect of Roby’s work is his systematic use of projection operators. This has several major advantages which are important to his argument. The projection operator is defined in a way that makes it easier to preserve invariant properties of the various kinds that are required. It also makes it possible to generalize from a molecular orbital wave function to any one TABLE I

Rosy OCCUPATION NUMBERS ~~

Liz N, F, LiH

FH

3.870 8.355 9.225 3.562 9.510

3.870 8.355 9.225 1.845 1.396

1.741 2.710 0.447 1.407 0.906

ATOMIC CHARGES WITHIN MOLECULES

51

involving the same set of atomic orbitals. Thus if the natural spin orbitals for the molecules are li) with the occupation number ni then the one-electron density is written

Similarly, the projection operator for the atom A can be defined as

where IAr) are the orthonormal orbitals required to define completely the state of the free atom A. The occupation number of A is then defined as

where Tr is the trace operation applied to the matrix. Since the numbers ni are positive, it is apparent that the atomic occupation numbers must all be positive. IflABt ) is the tth member ofthe set ofatomic orbitals on A followed by those on B and if S,, is the matrix of their overlap integrals then the projection operator into the joint space of A and B is n,

The corresponding occupation number is and from this the shared occupation number is defined as sA,

= nA

+ n B - nAB

(27)

It is readily seen that, if A and B are so remote that all their overlap integrals vanish, S,,becomes the unit matrix and nAB= nA nB so that sM = 0. Roby is also concerned with finding the most appropriate description of the atoms. The matrix S, is in block form with unit matrices in the diagonal positions. The remaining blocks can also be made diagonal. This is done by applying a unitary transformation to the orbitals of each atom separately exactly as in the theory of corresponding orbitals (Amos and Hall, 1961). The significanceof these transformations is that they construct the hybrids in pairs on each center such that their overlap is a maximum. This obviously solves the phase problem at the same time. The Roby theory has been modified by Cruickshank and Avramides (1982). They point out that if the set of atomic orbitals on each atom is extended toward completeness, then each atomic occupation number will tend to the total number of electrons in the system. Their solution is to restrict the atomic orbitals to those which span the Hartree-Fock model of

+

G. G. Hall

52

TABLE I1

ROBYAND MULLIKEN ATOMICCHARGES Mulliken

Roby 4.4

CH, NH3 OH, FH

-0.27 -0.43 -0.46 -0.32

qB

SAE

q.4

4B

Overlap

0.07 0.14 0.23 0.32

1.461 1.341 1.201 1.048

-0.76 -0.92 -0.78 -0.47

0.19 0.31 0.39 0.47

0.760 0.685 0.562 0.461

the atom. This achieves the object of producing values of the occupation numbers that rapidly become independent of basis size though it leaves unresolved the question of what is the appropriate state of the atom. Cruickshank and Avramides also proceed to calculate an atomic charge in the more conventional sense. They reduce the atomic occupation numbers by halfthe shared charge to produce an atomic charge which is conserved. In simple terms, their gross atomic charge is qA = nA - isAB= a2

+ abS - +S2(a2- b2)

(28) which differs from Mulliken bJ terms in the square of the overlap. They also claim that the shared occupation number is a measure of bond strength. For single bonds, sM is approximately (2s)so the operation offinding hybrids on each atom to maximize the overlap is a return to the older valence concept of bonding being determined by the principle of maximum overlap of hybrids. A comparison between the net atomic charges and those of the original Mulliken analysis is shown in Table 11. As is evident, the Roby charges are more moderate than those of Mulliken. Very similar considerations to these arise in the discussion of atomic charge initiated by Rys et al. (1976) and continued by Yanez et al. (1978). They are concerned with the electron density rather than with individual orbitals and so can use a linear combination of contracted spherical Gaussians to fit the density by means of a least squares criterion. The number of these functions is fixed and equal to the number of electron shells but the contractions are determined by an optimized fit to a spherically averaged atomic density. Some provision for the effect of charge transfer and bonding is made by allowing for an adjustable scale factor on the valence shell function. The constraints in this fitting are clearly designed to avoid the difficulties mentioned above of allowing too much freedom to the atomic basis functions while trying to obtain a close fit to the electron density. The

ATOMIC CHARGES WITHIN MOLECULES

53

TABLE 111 COMPARISON OF CHARGES OBTAINED BY FITTING PROCEDURES Density fitting"

CH4 NH,

OH, FH

Field fittingb

q,4

qH

qA

qH

-0.033 -0.658

0.008 0.219 0.307 0.392

0.728 -0.252 -0.650 -0.507

-0.182 0.084 0.325 0.507

-0.614 -0.392

Yanez et al. (1978). Hall and Smith ( 1 983).

resulting net atomic charges appear to be consistent from molecule to molecule and from one wave function to another. Some examples are shown in Table 111.

IV. Partitioning A. BISECTION

Even a cursory inspection of the experimental charge densities of molecules obtained from X-ray crystallography or by electron scattering experiments shows the strong peaks of charge associated with the heavy atoms and the thin threads of charge that hold them together. The natural consequence is a definition of the atom by bisecting the bonds, using planes, at their col points. For simple enough molecules these planes divide up space into atomic regions and the atomic charge can be defined as the integral over this region. Various authors have suggested the extension of this process of bisecting bonds to calculated electron densities so that direct comparisons could be made between theory and experiment. The idea works well for diatomic and linear molecules since the planes remain parallel and the definition is unambiguous. It can be extended in a plausible way to planar molecules but it becomes increasingly difficult, for more complicated molecules, to allocate all the volumes cut offby the planes

54

G. G. Hall

in a nonarbitrary way. An alternative analysis of experimental densities is given by Hirshfeld ( 1977). B. LOGES

The Lewis ( 1916) theory of atomic structures saw the electrons within an atom as localized in inner shells or in specific bonding directions. The prequantum electronic theory of valence (see, e.g., Remick, 1943) built this up into an interpretive scheme which correlated a mass of empirical observations and had considerable predictive power. Daudel ( 1953)and his collaborators (for a recent review and references see Aslangul et al., 1972) have set themselves the task of finding the rigorous quantum analog ofthis theory. They propose the partitioning ofthe space of the molecule into loges such that the probability of a fixed number of electrons being separated is greatest. Thus for an atom such as Li there is a definite radius of a sphere such that the probability of two electrons being inside it and one outside is a maximum. The loge boundaries are generally either spheres (for inner shell loges) or planes. The loge partitioning has considerable chemical appeal with its visual interpretation of the two-electron bond and the lone pair. It suffers from the disadvantage that it requires the N particle probability functions and that it uses an information theory criterion which is difficult to manipulate. A simpler approach using pair distributions has been discussed by Bader and Stephens (1975). ATOMS C. TOPOLOGICAL A more fundamental attack on the problem of partitioning the electron density in a theoretically sound way has been initiated by Bader and his collaborators (Bader and Beddall, 1972;Bader et al., 1973).They considered the orthogonal trajectories ofthe electron density, i.e., the collection of paths each ofwhich represents the motion of a point starting off, usually at infinity, and climbing up the density by the steepest route possible, which will cut the level surface ofp orthogonally, i.e., in the direction of Vp. Almost all of these trajectories terminate at one or other of the nuclei. The trajectories which terminate at a nucleus fill up the space assigned as belonging to that atom. In the terminology of Thom (1975), the nucleus is an attractor and the atom its basin. The behavior ofthe electron density at a nucleus is dictated by the fact that the potential becomes infinite there. To satisfy the Schrodinger equation

ATOMIC CHARGES WITHIN MOLECULES

55

exactly this divergence has to be eliminated by a canceling term. The source of this is the kinetic energy operator. If the wave function has a discontinuity in slope at the nucleus then it has a divergent kinetic energy contribution of exactly the right kind. For an isolated atom the wave function will be symmetrical so the electron density has a conical cusp at the nucleus (Kato, 1957).For an atom inside a molecule the electric field at the nucleus due to the rest of the molecule has to be included since this has the effect of pulling the cone to one side. In the Hartree-Fock theory, where the effect of the other electrons is averaged, the electric field at the nucleus is the force acting on the nucleus. By the Hellmann-Feynman theorem this will vanish when the nucleus is in equilibrium and consequently the electron at the nucleus will experience no other force there and so will have a symmetrical conical singularity. In nonequilibrium configurations and particularly for H atoms the cone may be so pulled by the field that the nucleus ceases to be a local maximum. In general, orthogonal trajectories start and finish at critical points, i.e., points where V p = 0, or at infinity. Although the nucleus is, strictly speaking, a singularity rather than a critical point (Lea,V p is not defined at the nucleus) it behaves, topologically, exactly like a maximum. In the neighborhood of a critical point the leading term in the local approximation to p is quadratic, i.e.,

p-po=r+Hr where H i s the Hessian matrix with elements

(29)

H~ = gplax, axj By classifying the forms of H the different species of critical point are identified (Collard and Hall, 1977),H has three eigenvalues, and all are real since it is symmetrical. Each can be positive or negative. The number of nonzero eigenvalues is the rank and the difference in the number of positive over negative eigenvalues is the signature of H. The maximum has rank 3 and signature - 3 which is abbreviated to (3,- 3). A minimum would have (3,3) since all its eigenvalues are positive. There are two kinds of col: (3,- 1) and (3,l). The (3,- 1) col is the one which lies between nuclei. There are two trajectories which originate there and one goes to each of the bound nuclei. It is this which Runtz et al. (1977)define as the bond. The negative eigenvalues at the col define a surface which is the boundary surface between the two atoms. The (3,l)col is found inside a molecular ring. Its positive eigenvalues define a finite surface, or cap, whose edges are the bonds while its negative eigenvalue is an axis. A minimum occurs inside a cage where these ring axes originate. This analysis of the electron density has the advantage of being completely

56

G. G. Hall

independent of axes and of basis sets provided these are sufficient to describe the critical points. Not only does it define a partition of space into atomic volume it also defines bond paths and ring caps whose shapes contain important information about the bonding. For a planar molecule containing a ring of identical atoms the situation raises new complications. At the center of a three-membered ring, p must have three-fold symmetry and consequently H will be degenerate. If the ring is large enough p may become rather flat there with zero eigenvalues. This would turn the point into a rank 1 maximum ( 1 ,- 1). In the cap the significant term in the approximation to p will be the cubic one. The form of the trajectories at the center is that ofFig. 3 in Collard and Hall ( 1977).Similarly, for a four-membered ring the major term is quartic with four hills and valleys meeting at the center. There are some disadvantages in this solution to the partitioning problem. In the first place there are some features that do not exactly match chemical ideas. Thus, for example, two approaching He atoms produce a col and a bond between them just as two H atoms would in forming H, and despite their inability to form a chemical bond. To distinguish these situations topology alone is insufficient and quantitative criteria have to be added. Another difficulty has been that since the atomic boundaries are often complicated surfaces it is necessary to develop new numerical techniques to manipulate them. Even the atomic charges, defined by an integration over the atomic volumes, are far from trivial to calculate. A suggestion to overcome this has been made by Hall and Smith ( 1984) who revert to an earlier device of taking planes through the col points and pointing out that there is very little density in the outer disputed regions. Their calculations involve a fitting ofthe molecular density by Gaussians so the outer parts have even less charge than in the true density. Some examples are shown in Table 111. A major argument in favor of this partitioning is that it fits naturally into molecular quantum mechanics. Bader has shown (for a recent review see Bader and Nguyen-Dang, 1981) that a virial theorem can be proved for each atomic region and that various quantum mechanical results true for the molecule can be extended to apply to the regions because of the special properties of the dividing surfaces. He also shows that many molecular properties, in particular the total energy, can be expressed as additive functionals of the atomic regions. Stutchbuq and Cooper ( 1 983) have demonstrated the practical utility of this Bader definition by discussing the atomic charges and dipoles of a series of alcohols and amines. In particular, they point out that the role of methyl groups in neutral saturated molecules is minimal whereas in ions they play an important role as electron sources or sinks.

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V. Point Charges A. EMPIRICAL

The idea of using point charges and point dipoles to represent the electron distribution over a molecule is a classical idea (cf. Polya and Szego, 1951) and has been applied by many people without using quantum theory. Julg ( 1971) has reviewed these attempts, which rely principally on dipole moments, and shown their utility in relation to NMR and ESCA observations. He also demonstrated the existence of a minimum in the electrostatic potential around H20inside the lone pairs. Reproducingthis minimum is a severe test for a simple model because Earnshaw’s theorem ensures that the potential cannot have a minimum at a point in space free of charge. A systematic attempt to build point models of simple molecules has been made by Kollman (1978). He uses the electronegativity of the atoms to partition charges between atomic centers. Lone pairs are represented by point charges located at the van der Waals radius, or perhaps half of it, from the nucleus. The results are used in calculations of intermolecular forces. Noel1 and Morokuma ( 1976) are concerned by the hydration of ions and complexes. They develop point charge models with the aim of reproducing intermolecular forces derived from calculations. These calculations show that the electrostatic terms are the controlling ones in the angular dependence of the forces. B. LOCALIZATION The idea of using the Mulliken populations to give a point charge model has appealed to many authors. Replacing the electronic density of an atom by a delta function implies that the atomic orbitals are well localized and close to spherically symmetric. In some molecules this may be true but it is not always so. A molecule containing 0 or N atoms frequently has significant lone pairs. For these atoms, at least, the point charges must be supplemented by point dipoles to achieve a plausible representation of the electron density. The possibility of using moments at a single convenient central point to represent the molecular charge density has been investigated by Rein ( 1973). He is interested in calculating molecular interaction energies and demonstrates that the expression for these in terms of molecular moments becomes divergent even when they are well separated. By comparison, using an ex-

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pansion on every atom produces much more satisfactory and compact results. This work suggests that one criterion for a suitable set of point charges and dipoles must be the accuracy of the resulting electric potential at typical intermolecular distances. A very similar approach to the problem has been made by Scrocco and his colleagues(for a recent review see Scrocco and Tomasi, 1978).They prefer to expand around the charge centers of the localized molecular orbitals. This avoids the need for any arbitrary division of an overlap density. It also abbreviates the expansion by locating the atomic charge at the center so that the dipolar term vanishes. In a recent paper (Etchebest et al., 1982)a modification ofthis procedure is suggested. The K shell electrons are absorbed into the nucleus. The other localized orbital distributions are expanded about their centers of charge up to the octuple. These additional moments have the effects of reducing the error in the electrostatic potential to about one-quarter of its value obtained using the Mulliken point charges and of allowing the representation to be used up to 2.5 (A) of any nucleus. On the other hand, Huzinaga and Narita (1980) have preferred to return to the Mulliken populations. Since p is a quadratic form in the atomic orbitals they take the charge and dipole of each orbital product and allow charges to be transferred by adding dipoles to preserve the moment. This gives a model consistingof point charges and point dipoles which are determined to ensure a good local representation. C. SHRINKING GAUSSIANS The process of deriving point charge models received a considerable boost when Hall ( 1973;also Tait and Hall, 1973)showed that spherical Gaussians could be shrunk into delta functions without changing any of the spherical moments. Since this preserves an infinite number of moments it is a major advance over a fitting which retains only a small number of them The derivation ofthe model begins with the electron density expressed as a quadratic form in Gaussians multiplied by powers of the coordinates. Now the product of two Gaussians is itself a Gaussian but its center is moved to a point on the line joining the original centers. The remaining factors can easily be transformed to this new center. By this technique the double summation is turned into a linear summation. For an FSGO wave function, i.e., one in which only spherically symmetric Gaussians are used but their locations are optimized, all the products will themselves be spherical Gaussians and the summation takes the form

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The point charge model of this is then

A factor (x - x,) can be introduced into an orbital expressed in terms of Gaussians by applying the operator (d/ax). When higher Gaussians are used (see Martin and Hall, 1981) the single summation can be written as

where Ds(V)is a function of the operators (dldx), (d/ay),and (a/&) so that the correct power factors are produced. The point model then replaces this by

Thus, for example, a p function will introduce the point dipole VS. By using the identity

it is readily shown that

"

"

for any function F which satisfies V 2 F= 0 (37) This includes all the spherical moments. The first moment which is not of this kind has F = r2. This point charge model provides approximate electrostatic potentials which agree closely with the true ones outside a radius of about 6 bohr from the nuclei. Its disadvantage is that the number of points can become large. N Gaussians in the original basis can lead to as many as N ( N + 1)/2 points when the products are rewritten as single Gaussians. This has led a number of authors to suggest ways of combining many of the smaller terms into larger units to achieve a more compact result without too much loss of accuracy. Shipman (1975) retains the charges corresponding to the squared terms and adds to them the overlap charges weighted by the exponents of the Gaussians. This has the effect of retaining both the total charge and the total dipole moment. Huzinaga and Narita (1980) have rediscovered the same allocation and extended it to higher Gaussians. For the special case ofa Frost FSGO wave function, where the number of Gaussians is exactly half the

G. G. Hall

number of electrons, Amos and Yoffe (1975) show that an even simpler model is possible in which two electrons are placed at the center of each Gaussian. Stone ( 198 1) has given formulas for relating moments at one point to moments at a different point and comments on the advantages of a choice which remains compact but ensures that the neglected higher order moments do not diverge. He advocates that squared terms, which are naturally centered on the atoms, should be retained in full but that the overlap terms should be redeveloped around a center, probably the centroid, and terminated. An alternative procedure for abbreviating the point charge expressions has been developed by Hall and Smith (1984). They use a variation principle, developed by Hall and Martin (1980) and Hall (1983), to optimize the fitting of the electron density by a linear combination of Gaussians on the atoms. The shrinking ofthese into delta functions is then possible without introducing new centers. In effect this redistributes the charges in an optimal way. When the exponents of the Gaussians are also optimized, some become small corresponding to diffuse functions spreading over several nuclei. They suggest that these functions should be partitioned spatially at the col points using planes and that charges and dipoles within each atomic region should be centered on the nuclei. This theory seeks to combine the utility of the point charge expression with the conceptual clarity of the topological partitioning.

VI. Purport and Prospect In strict terms, atoms do not exist in molecules as peas do in a pod and the atomic charge is an artificial concept which can be validly defined in different ways. Earlier sections have outlined three alternative ways of looking at the problem. The key issue on which they differ is their understanding of localization. Clearly the atomic charge must presuppose some definition of a localized electron density around a particular nucleus. In the population approach this is defined using atomic orbitals since these do concentrate in the appropriate neighborhood and are natural to the form of the wave function. The partitioning approach provides a definite volume around each nucleus to constitute the atomic region so the localization is strictly nonoverlapping and precise. The point charge approach cames localization to its ultimate form by using distributions such as delta functions and their derivatives. Distributions themselves are not very meaningful but they produce simple expressions for molecular moments and for the electrostatic potentials. The three approaches can also be contrasted in what they regard as the

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purpose of the atomic charge. In the population approach the atomic charge occurs as a major term in the total energy. It quantifies the extent of ionic bonding in the molecule and leaves the overlap population to relate to covalent bonding. On the other hand, in partitioning it is the electron density which is dominant and its differential topology which motivates the definitions. Virial theorems show that a partitioning of the energy into atomic contributions does result but these contributions depend on the shape of the boundary surfaces and on various functions defined over them. The direct comparison possible, when exactly the same atomic orbitals are used to define the atom in any molecule, has been lost. The point charge approach is not intended to explain molecular binding but rather to simplify the calculation of the intermolecular forces. Each of the approaches has its own technical problems which are slowly being solved. The problem of providing definitions which incorporate the desired invariance properties has been a long-standing one in population studies. The systematic use of density matrices and projection operators has led to a solution though there are problems remaining in the specification of the atomic states and their orbitals. Calculations of atomic chargesand other properties in the partitioning approach are relatively few because of the difficulties of specifyingand using the bounding surfaces. The use of strongly localized functions to fit the electron density and the recognition that some contributions are too small to merit accurate calculation have reduced the magnitude of this problem. Any use of point charges is subject to obvious mathematical problems since the self-energy of a point charge is divergent and some other physical quantities cannot be expressed since distributions cannot be multiplied. The device of carrying out the approximations using Gaussians and shrinking these into delta functions as the final step has decoupled the two operations and permits conventional techniques to be used most of the time. Although it might seem that at present there are three essentially different definitions of the atomic charge, each with its own advantages and disadvantages, some convergence in the approaches can be discerned. There is the recognition within the population approach that localization in the Hilbert space of atomic orbitals is not enough and that spatial localization is also necessary. In the partitioning approach the converse applies since spatial localization alone brings computational problems which are considerably eased by the use of localization in the Hilbert space. Even the point-charge approach benefits by an intermediate stage in which localized functions play an important role. Finally, the use of a variation principle to determine fitting seems to open up the possibility of an eventual unified definition. The goal of defining an atomic charge, within a molecule, which can be calculated readily and used in discussions of molecular bonding and reactivities has not yet been reached. As the different definitions are refined in

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response to objections and become numerically closer, hope rises that it will eventually be achieved. This process of deriving a concept, the atomic charge, from the wave functions and mechanics of the electrons is an important example of scientific synthesis, the deduction of a higher level concept from lower level ones. The difficulties of this process have been discussed by Hall (1 959) and, more recently, by Primas ( 1981). A chemical concept, which does not enter into the Schrodinger equation, has been shown to have a real existence, though with limitations because the definition is not yet precise, within the solutions describing the electronic motion inside the molecule.

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