Chemical Bonding: From Lewis to Atoms in Molecules
R. F. W. BADER,1J. HERNA´ NDEZ-TRUJILLO,2F. CORTE´ S-GUZMA´N21
Department of Chemistry, McMaster University, Hamilton, Ontario L8S 4M1, Canada
2Facultad de Quimica, Universidad Nacional Auto´noma de Me´xico, Me´xico D. F., 04510
Received 30 March 2006; Revised 20 June 2006; Accepted 22 June 2006 DOI 10.1002/jcc.20528
Published online 23 October 2006 in Wiley InterScience (www.interscience.wiley.com).
Abstract: The Lewis electron pair concept and its role in bonding are recovered in the properties of the electron pair density and in the topology of the Laplacian of the electron density. These properties provide a bridge with the quantum mechanical description of bonding determined by the Feynman, Ehrenfest, and virial theorems, bonding being a consequence of the electrostatic forces acting within a molecular system.
q 2006 Wiley Periodicals, Inc. J Comput Chem 28: 4–14, 2007
Key words: bonding; electron density; forces; virial theorem; Feynman theorem
From Lewis to Quantum Mechanics
‘‘The 1916 paper of Lewis,1introducing the concept of the elec-tron pair, is one of the most important papers to be published in chemistry in this century. It is a revelation to read this paper and realize the extent to which his ideas continue to dominate chemical thinking. One must marvel at the intuitive leap that was involved with advancing the notion of the importance of the electron pair, coming as it did before the development of quan-tum mechanics or the still later introduction of the concept of electron spin.’’ This quotation is taken from ‘Atoms in mole-cules: a quantum theory’2 from the chapter that introduces the properties of the Laplacian of the electron density, the quantity !2(r), and the remarkable pictorial mapping of the Lewis
elec-tron pair concept onto real space that it provides. The chapter progresses quite naturally from a discussion of the topological properties of the Laplacian and its description of the spatial pair-ing of electrons to the characterization and understandpair-ing of chemical bonding that are obtained when the properties of the Laplacian are coupled with the virial and electrostatic theorems of quantum mechanics. The present paper progresses in a similar manner: from a review of the electron pair concept as obtained from the pair density and its physical revelation through the Laplacian to a discussion of chemical bonding in terms of the theorems of quantum mechanics, a discussion complemented through the use of the atomic statements of these theorems.
Lewis was so intent on having electrons pair in spite of the repulsion that exists between them, he felt compelled to state1: ‘‘. . . a study of the mathematical theory of the electrons leads, I believe, irresistibly to the conclusion that Coulomb’s law of inverse squares must fail at small distances.’’ However, with the
advent of wave mechanics in 1926 and the latter introduction of the concept of electron spin that, in hands of Slater and Pauling, led to valence bond theory and to a model of directed valence, Lewis stepped back from the extreme stance he held in support of his model and in the first issue of the Journal of Chemical Physics, which appeared in 1933, he states:3 ‘‘There can be no question that in the Schro¨dinger equation we very nearly have the mathematical foundation for the solution of the whole prob-lem of atomic and molecular structure. . . .’’
Spatial Localization of Electron Pair and Number Densities
Localization and the Pair Density
In 1975, it was shown that the quantum mechanical requirement for the spatial localization of an electron in a many-electron sys-tem, when expressed in terms of the pair density, is that the Fermi correlation hole for the electron be totally contained within the same spatial region. Correspondingly, the extent to which this requirement is not met provides a quantitative mea-sure of its delocalization over the remaining space of the sys-tem.4 The concepts of localization/delocalization transcend or-bital models, being instead consequences of the exclusion princi-ple introduced by Pauli in 1925.5 The exclusion principle is embodied in the pair density and made evident in the density of the Fermi hole. The role of the pair density is best made evident in the properties of the conditional same-spin density obtained
by dividing the-spin pair density (r1, r2) by the spin
den-sity(r1), a number density, eq. (1),
ðr
1; r2Þ ¼ ðr1; r2Þ=ðr1Þ ¼ ½ðr2Þ þ hðr1; r2Þ (1)
with a similar expression holding for the electrons. The quan-tityh(r1, r2) describes the density of the Fermi hole, eq. (2), a
term identical to Slater’s exchange charge density and the term he modeled to obtain his1/3expression for the exchange energy.6,7
hðr1; r2Þ ¼ ð1=2Þ X i X j f iðr1Þiðr2Þjðr2Þjðr1Þg=ðr1Þ: (2) It equals the exchange density for -spin electrons divided by (r
1). It is a negative quantity that describes the extent to
which the density of the second electron is excluded from the neighborhood of r1and thus(r1, r2) determines the extent to
which the exclusion principle decreases the density of an elec-tron at some position r2 subject to another electron being
assigned the position r1. [It must be understood that neither
elec-tron is fixed at either of these points, the charge of each being spread out in space in the manner determined by the spin den-sity(r)].
The Fermi hole is like a shadow of ever changing shape in three-dimensional space, cast by and following the position of a reference electron, whose form is determined in the many-dimensional orbital space. The integral of h(r1, r2) over r2
equals1, corresponding to the removal of one -spin electron. The exclusion is locally complete for r1¼ r2, since h(r1, r1)
¼ (r
1). If h(r1, r2) ~(r2) for positions removed from
r1, that is, the density of the Fermi hole is localized about this
point, then all other -spin electrons will be excluded from the space corresponding to the exclusion of one electronic charge. In a closed-shell system, the Fermi hole density of an electron of-spin will be similarly localized, resulting in a pair of elec-trons being localized about the point r1. Thus Fermi correlation
does not act directly to ‘pair up’ electrons. Rather, since there is no Fermi correlation between electrons of opposite spin, an, pair is obtained by default, a result of all other electrons of both spins being excluded from a given region of space.4 It is worth
noting that in 1933 Lewis echoed this point of view in the state-ment3: ‘‘There is no force that draws two electrons into a pair.’’ With the understanding afforded by the properties of the Fermi hole, one may employ it to give physical substance to the ‘Pauli attractions or repulsions’8 used to account for the operation of the Pauli principle in chemistry.
Localization and Delocalization Indices
The exclusion of density of given spin from a point in space is obtained by weighing the density of the Fermi hole with the spin density, the product (r1)h(r1, r2), an operation that by eq.
(2), yields the exchange density. The integration of the absolute value of this quantity over the basin of a given atom A for both spins (as assumed in the remainder of the discussion), a quantity denoted by |F(A, A)|¼ (A) gives the total number of electrons
excluded by the electrons present in atomA and this a measure of the number of electrons localized toA, a quantity whose lim-iting value is clearlyN(A), the average electron population of A. At this limit, theN(A) electrons in A, necessarily a whole num-ber, would be totally localized within the basin of atomA, as all other electrons would then be excluded. The ratio of the local-ization index (A) to the population of A, the quantity l(A) ¼ (A)/N(A), gives the fraction of the electron population of A that is localized to A.4
It is important to note that while complete localization is an unattainable limit for an atom in a molecule, it is approached quite closely in ionic interactions, with the 10 core electrons on Na in NaF and NaCl exhibiting percentage localizations of 99.5 and 98.9% respectively, for atomic basins defined by their zero-flux interatomic surfaces.
A value of (A) < N(A) implies that the remainder of the electrons of A are delocalized, and the exclusion of same spin electrons occurs over an extended region of space. Their deloc-alization over two atoms,A and B, is obtained by a correspond-ing integration of the exchange density over both atomic basins to yield |F(A, B)|þ |F(B, A)| ¼ (A, B), the delocalization index.4,9 If the exchange of electrons is largely confined to within a given atomic basin, then the electrons are correspondingly localized on that atom, while if the electrons exchange between atomic basins, then the electrons are delocalized over both atoms or, equivalently, are shared by both atoms. Electrons that exchange are indistinguishable, and consequently, the physical picture underlying electron delocalization is exceedingly simple—it is determined by the extent to which the electrons exchange atomic basins. The definition of(A, B) applies even in the case of H2
where there is but a single electron of each spin that exchanges atomic basins to yield a delocalization index of unity.
Since the Fermi correlation necessarily integrates to –N, one has the useful book-keeping device given in eq. (3) that determines how the average population of atomA is distributed over the entire molecule; how many electrons are localized withinA, and how many are delocalized over the basin of every other atomB,4
1ðAÞ þ ð1=2ÞX
B6¼A
ðA; BÞ ¼ NðAÞ: (3) The exchange indices are readily calculated. At the Hartree-Fock level, they are given by the simple product of the overlap integrals of each pair of spin-orbitalsSij(A), over the basin of the atom in
question. ThusF(A,B) is given by FðA; BÞ ¼ X
i;j
Si;jðAÞSi;jðBÞ: (4)
Beyond Hartree-Fock, the summation is extended to include the weighted contributions from the excited configurations.9 Wang and Werstiuk have shown that an acceptable approximation to the exchange indices is obtained when theSij(A) are evaluated using
natural orbitals and their occupation numbers generated in a corre-lated calculation.10The delocalization index(A, B) is of particular importance in bridging classical notions of bonding and quantum mechanics.11 The role of exchange is to reduce the electron– electron Coulomb repulsion between a pair of bonded atoms, and
(A, B) counts the number of pairs contributing to this reduction, the ‘, spin-exchange resonance’ or ‘covalency’ of valence bond theory. The delocalization index (A, B) and the interatomic exchange energyVx(A, B) it determines provide a clear indication
of the progression of bonding from ‘covalent’ to ‘polar’ to ‘ionic,’ the increasing localization of the electrons within the atomic basins paralleling the increasing interatomic charge transfer, causing a reduction in both quantities.
The delocalization indices for bonded atoms (atoms linked by a bond path) at the Hartree-Fock level may be interpreted as bond orders in relation to the Lewis electron pair model.9One, two, or three pairs of electrons equally shared between two atoms yield delocalization indices of 1, 2, or 3, respectively. Significant charge transfer between the bonded atoms increases the degree of localization within each atomic basin, and hence charge transfer reduces the value of (A, B). In any case, the value of (A, B) always gives the number of electrons that are delocalized or exchanged between the basins ofA and B, inde-pendent of any model.
The Laplacian and the Spatial Representation of Fermi Correlation
The exchange indices translate the consequences of the Fermi correlation into atomic contributions, but they do not provide a spatial representation of electron localization that one can asso-ciate with models of the bonded and nonbonded electron pairs of the Lewis model. Spatial displays of the density of the Fermi hole, a distribution function in six-dimensional space, are of lim-ited use.2What is needed is a clear representation of the pairing of electrons in real space, a requirement fulfilled by the Lapla-cian of the electron density.12
As demonstrated by Morse and Feshbach, the Laplacian of a scalar function such as(r) has the intrinsic property of locating where the function is locally concentrated, where !2(r) < 0, and where it is locally depleted, where !2(r) > 0.13 Since density is concentrated where!2(r) < 0, one defines the func-tion L(r) ¼ !2(r), such that a maximum in L(r) denotes a maximum in the concentration of the density. One distinguishes between charge concentrations (CCs) and charge depletions (CDs) determined by the topology ofL(r), and charge accumula-tions and charge reducaccumula-tions determined by the differing topology of(r). L(r) exhibits multiple valence maxima that are found to coincide with the number and relative positions of the localized electron pair domains that have been invoked in chemical mod-els.12 The much simpler topology of (r), while defining the atom and molecular structure, in general exhibits maxima only at the positions of the nuclei.14
The topology exhibited by the Laplacian of the electron den-sity in real space is a consequence of the electron pairing deter-mined by the conditional pair density in six-dimensional space.15 It is clear from eq. (1) for the conditional pair density, that in a case wherein the Fermi hole of an electron of given spin is strongly localized about r1, the conditional same-spin density
will approach the single-particle spin density in regions r2
removed from the region of localization. In such regions, the conditional pair density approaches the total density for a closed-shell system and consequently, its Laplacian distribution
approaches the Laplacian of the electron density. Thus the topol-ogies of the two Laplacians exhibit a homeomorphism, one that has been shown to approach an isomorphic mapping of one field onto the other.15 The local concentrations of the Laplacian of the conditional pair density indicate the positions where the remaining electron pairs will most likely be found relative to a reference pair, and correspondingly, the valence shell CCs dis-played inL(r) signify the presence of regions of partial pair con-densation, that is, of regions with greater than average probabil-ities of occupation by a single pair of electrons.
Comparative displays of the Laplacian of the density with that of the conditional pair density have been previously illustrated15 for a number of molecules that exhibit both bonded and non-bonded CCs show that the mapping of one field onto the other approaches an isomorphism.15It should be realized that while the Laplacian of the conditional pair density changes with every change in the position of the reference electron, the mapping obtained for the case of maximum localization of the Fermi den-sity is irretrievably implanted onto the Laplacian of the denden-sity. ThusL(r) condenses the essential pairing information determined by the conditional pair density in six-dimensional space, into a to-pology that is experimentally measurable in real space.
The reduction from the average pair population denoted by a CC imbues the region with the properties associated with the bonded and nonbonded ‘electron pairs’ of chemistry, as made understandable in terms of the local expression for the virial the-orem,2,16eq. (5)
ðh2=4mÞr2ðrÞ ¼ 2GðrÞ þ VðrÞ: (5)
G(r) is the kinetic energy density andV(r) the virial field, the electronic potential energy density, whose sum equals H(r), the energy density. SinceG(r)> 0 and V(r) < 0, a negative value for!2(r), as found at a CC in L(r), indicates a region where V(r) overrides the kinetic energy making the electronic potential energy density maximally stabilizing, thus accounting for the role of bonded and nonbonded CCs as sites of nucleophilic ac-tivity or Lewis bases. Contrawise, minima in L(r), denote sites that perform as Lewis acids and the approach of reactants is determined by the alignment of the CCs and CDs on one reac-tant with the complement of CDs and CCs on the other, the Laplacian complement of Fischer’s ‘lock and key’ analogy of binding.12 A recent example of this property of L(r) is the accounting of the structure and formation of donor–acceptor complexes of transition metal carbonyls. The dominant interac-tion is the alignment of the pronounced nonbonded CC on C of each CO with a region of CD on the metal atom.11,17–19 How-ever, a secondary interaction is also defined byL(r), in terms of the alignment of the CCs on the metal with the regions of CD on each C whose form mimics the* LUMO of CO,11in agree-ment with the Chatt-Duncanson model of d-p* back bond-ing.20 These features of L(r) are displayed in Figure 1 for the Mn(CO)6þcomplex. The topology of L(r) is summarized by its
atomic graph giving the number and relative locations of the CCs and CDs in the valence shell of charge concentration. The crystal field model of complex formation is recovered in its en-tirety by the atomic graph of the metal atom.11 Thus the eight
vertices (the CCs) of the atomic graph for a metal atom in an octahedral complex correspond to the occupied t2g set and the
six faces (the CDs) to the vacant egset.
The Laplacian was early on shown to give physical expres-sion to atomic shell structure in terms of pairs of alternating
shells of CC and CD, the innermost ‘shell’ of charge concentra-tion being a spike-like concentraconcentra-tion at the nucleus.21The shell structures exhibited by the Laplacians of the density and of the conditional pair density predict not only the same number of shells, they are essentially isomorphic in their recovery of the radii and values of the maxima and minima in the two Lapla-cians that characterize each shell.15Thus, the atomic shell struc-ture exhibited by the Laplacian of the density is a faithful repre-sentation of the real-space structure imposed by the six-dimen-sional electron pair density.
The correspondence of the topology of L(r) with the spatial properties determined by the pair density was originally empiri-cally surmised because of the remarkably faithful mapping of the CCs ofL(r) in the recovery of all the nuances of the VSEPR model in terms of the assumed number, relative size, and angular orienta-tion of the bonded and nonbonded electron pair domains,12,22a model23originally rationalized on the investigations of the same-spin pair density by Lennard-Jones.24,25 This mapping is now widely utilized, particularly in the analysis of densities obtained from accurate X-ray diffraction experiments.26,27
The density and its Laplacian at a bond critical point, b(r)
and !2b(r), together with the densities for the kinetic Gb(r),
potentialVb(r), and total energy Hb(r), are widely used to
char-acterize atomic interactions.28 Gatti has provided a critical review and summary of the use of these indices in bonding clas-sification.27The Laplacian, the kinetic and potential energy den-sities are inter-related by the local statement of the virial theo-rem, eq. (5), and the Laplacian of the density thus bridges the discussions of bonding that focus on the properties of the density with the theorems of quantum mechanics.
Quantum Mechanical Basis of Chemical Bonding
Forces in Chemistry
The bonding between atoms is accounted for in terms of the electron–electron (e-e), nuclear–nuclear (n-n), and electron–nu-clear (e-n) electrostatic forces, the only forces operative in a field-free molecule. The e-n force is the sole force of attraction between atoms, a view forcefully stated in a quotation from Lennard-Jones and Pople,29(‘‘There is only one source of attrac-tion between two atoms, and that is the force between electrons and nuclei. But there are three counteracting influences: the nuclei repel each other, the electrons repel each other, and the kinetic energy of the electrons increases when a chemical bond is formed.’’) and it is thus the force responsible for the binding predicted by both the Feynman and Ehrenfest theorems. These forces determine the potential energy operators in the Hamilto-nian. The resulting wave function enables one to determine the average of the electrostatic force that acts on the electrons and on the nuclei, using theorems of quantum mechanics. The force acting on the electrons, the Ehrenfest force, is obtained from the equation of motion for the electronic momentum operator p, while the force acting on a nucleus, the Feynman force, is obtained from the corresponding equation for a nuclear gradient operator (The commutator average in the equation of motion for an operator D that contains derivatives of a parameter appearing in the Hamiltonian, such as the derivative of a nuclear coordi-Figure 1. Envelopes of the Laplacian distribution for the Mn atom
in the complex Mn(CO)6þ and for the entire complex. A value of
L(r) ¼ 17 au is used for the Mn atom to clearly delineate the 12 vertices of CC and the six faces of CD. The L(r) ¼ 0 surface is shown for the ligands, the surface that separates the regions of CC from those of CD. Note the pronounced nonbonded CC on each car-bonyl carbon and the manner in which it is directed at a centre of CD on Mn. The torus of charge removal bounding the carbon CC is the Laplacian complement of the* orbital.
nate, must include the commutator [D, E] in addition to [H, D]). Both forces contribute to the virial of the forces acting on the electrons as described by the virial theorem, the equation of motion for the virial operator r p. The molecular virial theorem relates the virial of the Ehrenfest forces acting on the electrons to their kinetic energy. Thus through the Ehrenfest and Feynman theorems, one has the tools that are needed to describe the forces acting in a molecule and through the virial theorem, to relate these forces to the molecule’s energy and its kinetic and potential contributions in the manner initiated by Slater.30
Feynman’s Electrostatic Theorem
A molecule consists of some number of point-like nuclei embed-ded in a static distribution of negative charge described by the electron density (r). Every element of electronic charge con-tained in some infinitesimal volume element dr exerts an attrac-tive force on each of the nuclei. The Feynman electrostatic theo-rem states that the force acting on a nucleus is the resultant of the electrostatic forces of attraction exerted by the density and of repulsion exerted by the other nuclei.31 Feynman’s electro-static theorem is a special case of a more general theorem expressing the dependence of the derivative of the energy with respect to an external parameter appearing in the Hamiltonian, as first stated in 1933 by Pauli and discussed in his later book.32 The general statement was also derived by Hellmann in 1933.33 The electrostatic consequences of this theorem on bonding are obtained by setting the external parameter equal to a nuclear coordinate, as done by Feynman.
The total molecular energy E within the Born-Oppenheimer procedure is a function of only the nuclear coordinates and is therefore, the potential energy function governing the motion of the nuclei. In a diatomic molecule, E is a function of the inter-nuclear separation R and the Feynman force, which is directed along the internuclear axis may be expressed asF(R)¼ dE(R)/ dR. The force vanishes at Re, the equilibrium separation where
dE(R)/dR¼ 0. It is attractive for R > Re, F(R)< 0, and
repul-sive forR < Re. The behavior ofF(R) and E(R) as functions of
R for a bound state are shown in Figure 2. The force attains its maximum attractive value at Ri, the inflexion point in E(R)
where d2E/dR2¼ dF/dR ¼ 0.
The bonding energy of a diatomic molecule, the depth of the potential well at Re, is given by the integral of F(R) over its
attractive region, the area lying between theF(R) curve and the R axis from R¼ ? to R ¼ Re, eq. (6)
bonding energy¼ Z Re
1 FðRÞ dR: (6)
Thus a ‘bond energy’ is a consequence of the existence of Feyn-man attractive forces exerted on the nuclei acting over a range of internuclear separations.
Berlin pointed out that one can divide the space of a diatomic molecule into a binding region, the region between the nuclei where density exerts forces drawing the nuclei together, terming the remaining space the antibinding region.34Density in the anti-binding regions exerts a larger force on one nucleus than the other, drawing it away and acting in concert with the nuclear force of repulsion. Bonding is thus achieved when the force resulting from the accumulation of electron density in the binding region between
the nuclei is sufficient to exceed the antibinding force over a range of internuclear separations to yield an integrated bond energy, eq. (6) and a molecule in electrostatic equilibrium. Silberbach33has argued that Berlin’s expression for defining the binding–antibind-ing regions, while correct and most natural, is not unique. All defi-nitions, however, lead to the single conclusion that density in the internuclear region leads to binding. One can do no better than quote Feynman: ‘‘It now becomes clear why the strongest and most important attractive forces arise when there is a concentration of charge between two nuclei.’’31
All chemical bonding is the result of the net electrostatic force of attraction exerted on the nuclei by the electron density accumulated between the nuclei, including the long-range ‘dis-persion forces’. These forces, as pointed out by Feynman, are a result of the density of each atom being polarized in the direc-tion of the other and the van der Waals force is the result of ‘‘the attraction of each nucleus for the distorted charge distribu-tion of its own electrons that give the attractive 1/R7force.’’31 This prediction regarding the long-range distortion of the density on the approach of neutral atoms was confirmed in detail by Hirschfelder and Eliason35 through the use of the ‘‘electrostatic Hellmann-Feynman theorem’’ applied in an exact calculation of the leading term in the 1/R expansion of the force of attraction between two well-separated hydrogen atoms, the term6C6/R7.
The inwardly directed polarization of each H atom at a separa-tion of 8 au (atomic units) is clearly displayed in Figure 7.22 of ref. 2 obtained from a highly correlated wave function.36 Schro¨dinger perturbation theory for two interacting atoms leads to changes in the wave function weighted by matrix elements that may be interpreted as instantaneous dipoles on each atom. Ascribing the attraction to these correlatively induced and opposed dipoles is an attempt to reduce the two-electron interac-tion in pair space that appears in the perturbed wave funcinterac-tion, to a physical interaction between the atoms in real space, whereas Figure 2. Variation of the total molecular energy E(R) and the Feynman force F(R) for a diatomic molecule. The force attains its maximum attractive value atR¼ Ri, the inflection point in theE(R)
curve. The larger the attractive force on the nuclei, the greater is the bonding energy, the area between theF(R) curve and the R axis up to Re, the equilibrium separation. These curves apply to all bound
diatomic states, interactions differing in the magnitude of the maxi-mum attractive force and thus in the depth of the potential well.
as Feynman points out ‘‘it is not the interaction of these dipoles which leads to van der Waals’ force,’’ but rather the attraction of each nucleus for its own distorted charge cloud. Approximate wave functions constructed from large basis sets satisfy Hell-mann-Feynman theorem to a degree sufficient for discussions of chemical binding, as exemplified by the study of the forces in diatomic molecules, using wave functions from the Mulliken-Roothaan group at the university of Chicago,37–39 the errors in
the total electronic contribution to the force on a nucleus being of the order of 0.1%.40,41
It is the accumulation of density between the nuclei of bonded atoms that results in the formation of ‘a bond path, the line of maximum density linking bonded atoms.’42,43Every bond path in a stable geometry removed from a structural instability in the density is mirrored by a virial path, a line of maximally negative potential energy density linking the same nuclei,44their simultaneous presence emphasizing that bonding is a result of the lowering of the potential energy in the bonding region caused by the accumulation of density that attracts the nuclei. The criticism is sometimes made that a bond path should not be found in He2, which is deemed a repulsive interaction. However,
the He2 molecule has been detected experimentally in electron
impact ionization of a supersonic beam, its detection being described in a paper entitled ‘‘The weakest bond: Experimental observation of helium dimer.’’45 Thus in its equilibrium separa-tion, He2is bound (liquid helium does exist) and exhibits a bond
path, but like all molecules, it becomes repulsive when com-pressed and the bond path is transformed into an atomic interac-tion line. One may quote Slater in this regard: ‘‘The writer believes that there is no very fundamental distinction between van der Waals binding and covalent binding. If we refer them back to the electronic charge distribution, which as we have seen is solely responsible for the interatomic forces, we have the same type of behavior in both cases.’’46
The changes in density that lead to bonding may be illus-trated in the form of density difference maps between the molec-ular density and the overlapped densities of the separated atoms, later termed ‘bond density maps.’ Such maps for A2,40 AH,41,47
and AB48–50were constructed from near H-F wave functions for 300 neutral, charged, and excited states of molecules obtained from the Mulliken-Roothaan group, as referenced above. The maps demonstrate the universality of the electrostatic under-standing of chemical bonding, illustrating that all bonding is the result of charge accumulation in the binding region. The maps exhibit a progressive change from the density increase in the binding region being equally shared in A2to becoming
increas-ingly more localized within the basin of the most electronegative atom in AB, to being completely so localized in the ionic limit. Thus covalent, polar and ionic interactions differ only in the manner in which the density increase in the binding region is disposed, giving substance to the 1916 statement of Lewis1: ‘‘. . . according to the theory which I am now presenting it is not necessary to consider two extreme types of chemical combi-nation, corresponding to the very polar and the very nonpolar compounds as different in kind, but only as different in degree.’’ The LMSS wave functions use an STO basis of s, p, d, and f functions capable of satisfying the nuclear cusp conditions on the density. The orbital exponents were separately optimized for the
free atoms and for the molecular states from the first three rows of the periodic table. These studies pointed out the very different charge reorganizations found for H as opposed to atoms in the sec-ond- and third-rows of the periodic table. In particular, the density difference maps clearly demonstrate the reduction in the electron density at and in the vicinity of nuclei of atoms beyond He, a result of the decrease in the optimized exponents of the basis functions on molecular formation, a behavior just the opposite to that found for the hydrogen molecule and used as a model of ‘contractive promo-tion.’51For example, the value ofD(0), the change in the density at a nucleus, equalsþ0.112 au for H2, but0.188 au for C2, and
0.384 au for N2.
A consistent physical choice of atomic reference state can be made in diatomic molecules by appeal to physics through the use of the valence state of the atom determined by perturbation theory in the limit of a vanishing axial electric field. At this limit, the quantum numberl begins being a good quantum number, break-ing the ml degeneracy into, . . .. sets. The resulting D maps
yield a single pattern for the redistribution of density for atoms past He: a quadrupolar polarization with a density accumulation along the axis in both the bonded and nonbonded regions of the atom, and its removal from perpendicular torus-like region encir-cling the axis at the position of the nucleus. This pattern of charge reorganization is the universal response of an atom to an axial electric field, be it an applied static field or a dynamic one arising from nuclear displacements, vibrational, or bond formation. The density of a F atom 13 au from an approaching Li atom, for example, exhibits a quadrupolar polarization along the axis of approach consistent with the valence state configuration 2p12p,4
the same configuration that yields a build-up of density in the binding region of the bond density map for F2. The maps point
out an important feature of the charge reorganization accompany-ing bondaccompany-ing that density is accumulated in the both the bindaccompany-ing and antibinding regions of a nucleus, the increase in density in the vicinity of the nucleus leading to the reduction in the e-n potential energy. In a bound state, the force exerted by the density accumu-lation in the binding region exceeds that from the density in the antibinding region.
Ehrenfest Force Theorem
The Ehrenfest force density,F (r), the force exerted on the elec-tron density at some point in space is obtained by the appropri-ate averaging of negative gradient of the total potential energy operator ^V with respect to the coordinates of ‘1’ over the wave function, that is, by averaging over the motions of the electrons other than ‘1’, as done to define the electron density,
F ðrÞ ¼ N Z
d 0 ðr1V^Þ : (7)
F (r) is N times the force exerted on an electron at position r by the average distribution of the remaining electrons and by the rigid nuclear framework, the force exerted on the electron den-sity. The force density can be converted into a potential energy density by forming the ‘virial of the force operator—the force acting through a distance,’ by taking its scalar product with the position vector r. Integration of the resulting viral density over
all space it yields the molecular virial , the virial of all the forces acting on the density, the quantity defined in eq. (8)
V ¼ Venþ Vee X F e (8) V equals the electron–nuclear Ven, and electron–electronVee
poten-tial energies together with the sumSXFe, the virial of the elec-tronic contribution to the Feynman forces exerted on the nuclei, X being the coordinate of nucleus . This latter term can be equated to the sum of the virials of the Feynman forces exerted on the nuclei, XF, and the nuclear–nuclear repulsion energyVnn¼
SXFn.2It is in this manner thatVnnenters into the molecular
virial to yield the result that V, the virial of the Ehrenfest force acting on the electron density, equals the total potential energyV and the virial of the Feynman forces on the nuclei,2eq. (9)
V ¼ Venþ Veeþ Vnn X F¼ V X F: (9) The Virial Theorem
The molecular virial theorem relatesT, the kinetic energy of the electrons, to the virialV of the forces acting on them, eq. (10)
2T¼ V: (10)
For a diatomic molecule the theorem takes the form52
2T¼ V RðdE=dRÞ ¼ V þ RFðRÞ (11) with the virial of the Feynman force given by RF(R). Equation (11) holds for allR. The virial theorem inter-relates the kinetic and potential energies of the electrons with the virial of the Feynman forces acting on the nuclei, all as a function ofR. For R ¼ ? and for the equilibrium separation, F(R) vanishes and one obtains the result 2T ¼ V or equivalently, since E ¼ T þ V, T¼ E(Re), the equilibrium energy. The same relationships
hold for the separated atoms and defining the difference between the energy for the molecular minus the separated atom values by D, one obtains the well-known constraints on the energy changes accompanying the formation of a bound state for whichDE < 0: DE ¼ ½V and DE ¼ DT at R ¼ Re. Thus molecular stability
is a result of a decrease in the potential energy that necessarily requires an increase in the kinetic energy. This interpretation is consistent with the requirement of the electrostatic theorem of bonding that electronic charge be accumulated in the bonding region, since next to the nuclear positions, the internuclear region exerts the lowest potential. Thus the virial and electro-static theorems give mutually consistent interpretations of chem-ical bonding, an anticipated result since, as pointed out by Sla-ter30 ‘‘. . . both the virial theorem and Feynman’s theorem are exact consequences of wave mechanics. Both interpretations agree in pointing to the existence of the overlap charge density as the essential feature in the attraction between the atoms.’’ Sla-ter regarded these as ‘two of the most powerful theorems appli-cable to molecules and solids.’46
Equation (11) enables one to relate the variations in E, T, andV to the Feynman forces acting on the nuclei over the entire
range ofR, variations illustrated in Figure 3 for H2, N2, CO, and
Ar2 obtained from highly correlated wave functions with large
basis sets.53 Such diagrams are given in Slater’s book for H2þ
and H2.30 Equation (11) yields differential statements for dT(R)
and dV(R). These expressions yield constraints on the signs of the slopes dT(R)/dR and dV(R)/dR, which are seen from Figure 3 to change sign with decreasing R. The derivation of these equations and their full discussion applied to H2 have been
given previously,2,54their consequences being summarized here for a range of interactions from shared to polar to closed-shell. For large values ofR lying in the range? > R > Ri(the
inflec-tion point shown in Fig. 2), the signs of the derivatives are not uniquely determined, being the result of the two competing con-tributions, F(R) and RdF(R)/dR, being of opposite sign. How-ever, for the approach of neutral atoms at large values ofR lying in the range ? > R > Ri, RdF(R)/dR will dominate with the
result that T must decrease and V increase upon the initial approach of two neutral atoms (or the approach of one neutral and one charged atom).55 These changes are understandable in terms of the charge reorganization previously shown for H2at a
separation of 8 au.2 Density is removed from the immediate vicinities of both nuclei, whereV is maximally negative and T is maximally positive, as well as from the antibinding regions, and accumulated in a diffuse distribution in the binding region resulting in relaxation of the gradients in. Thus V is increased and T is decreased and an attractive Feynman force acts on the nuclei. This is the origin of the initial 1/R6long-range attraction between neutral atoms.
AtR¼ Ri, (Ri 2 au for H2and 2.5 au for N2), the signs
of the derivatives are uniquely determined, dT/dR < 0 and dV/ dR > 0, and T must increase and V decrease upon further approach of the atoms, changes that actually take effect forR> Ri in both molecules. Recall that F(R) attains its maximum
attractive value atR ¼ Riand thus the charge reorganization in
this range ofR is dominated by the accumulation of density in the binding region, a process that continues forR < Ri, leading
to DT > 0 and DV < 0, as required by the virial theorem at R¼ Re. For R< Re, DT > |DE|, the Feynman forces are
repul-sive and the contribution of the nuclear virial to the potential energy, eq. (11) is positive and destabilizing. Thus forR > Re,
DT < DE, the forces are attractive and the nuclear virial is sta-bilizing while forR < Re, DT > DE, the Feynman forces are
repulsive and the nuclear contribution to the viral is positive and destabilizing. Feinberg and Ruedenberg criticize the Hellmann-Feynman theorem by stating56 ‘‘An obvious limitation of the theorem is that it deals only with properties of dE/dR: it does not even make statements regarding the properties ofdV/dR and dT/dR,. . .’’ A reader may judge whether or not this statement is at variance with the unified picture of bonding afforded by the viral theorem, which demonstrates that the Feynman force and the kinetic and potential energies are all inter-determined, the behavior of each being an inescapable consequence of the others, the Feynman force in particular playing a leading role in determiningdV/dR and dT/dR.
Slater points out that the behavior of theE, T and V vs. R for the approach of oppositely charged ions will differ from case for the neutral atoms for largeR.30In the ionic case, the long-range behavior is dominated by the force of attraction between the
ions and one finds just the opposite behavior from the approach of neutral atoms; F(R) now overwhelms the term RdF(R)/dR with the result that V must decrease and T increase even for large values of R, a process that continues for further decrease in R. However, for distances in excess of the charge transfer, 10 au in LiF for example, T decreases and V increases as found for the approach of neutral atoms. Figure 3 illustrates the universality of the mechanics of bonding from shared to polar to closed-shell when viewed in terms of the theorems of quantum mechanics. One may employ different methods for the calcula-tion of the energies of interaccalcula-tion in these separate cases, but they all are consequences of a single force: the electrostatic force of attraction of the nuclei for the electrons.57 One is
reminded of Slater’s belief quoted above ‘‘that there is no very fundamental distinction between van der Waals binding and covalent binding,’’ which followed his description of the inter-play ofE, T, V and F(R) in bonding.46
Role of Ehrenfest Force in Bonding
An atom in a molecule is an open system bounded by its zero-flux surface and the force acting on the electrons in the basin of the atom, the Ehrenfest force (EF) F (A), is equal and opposite to the force exerted on its surface, eq. (12).2,16
F ðAÞ ¼ Z A drF ðrÞ ¼ I dSðr; AÞ $ðrÞ (12) The surface force is expressed in terms of the quantum stress ten-sor, $ðrÞ, which has the dimensions of pressure, or force/unit area. Thus the stress tensor determines the force exerted on each element of the atomic surface and its product with an element of the surface dS(r), as in eq. (12), gives the force acting on that element.F (A) can be attractive, drawing two atoms together to-ward their interatomic surface, or repulsive, tending to draw each atom away from the surface. The EF forces on the atoms are necessarily equal and opposite and no net force acts on an atom or on any element of its density in a stationary state, but the direction of EF acting on an atom is important in determin-ing the mechanics of atomic interactions. The general behavior of EF has been shown to be typical of all interactions, from homopolar to polar to closed-shell, including ionic and van der Waals interactions58and is illustrated for CO in Figure 3.F (A) is repulsive for large separations, attaining a maximum value in the region of the inflexion point in the molecule’s potential energy curve for the approach of neutral atoms. For ionic inter-actions, it is repulsive only for the values of R preceding the transfer of electronic charge, reflecting the corresponding decrease in the potential energy for largeR. It becomes increas-ingly attractive for all interactions before the equilibrium separa-tion is attained, a result of the accumulasepara-tion of density in the binding region, whether shared or localized within a single atomic basin. The attraction of the nucleus external toA for the density ofA, is the principal stabilizing force in the formation of a molecule A|B and its virial is the principal source of energy lowering.11,58In general, the single largest contribution toF (A), the force on atom A in a diatomic AB in the vicinity ofRe, is
the attraction of the B nucleus for the density on A and the larg-est single contribution to the lowering of the energy is the potential energy of interaction of the nucleus of one atom inter-acting with the density of the other.11The virialV(A) appearing in the atomic statement of the virial theorem, 2T(A) ¼ V(A), consists of virial of the EF acting within the basin of the atom and from the virial of the forces acting on its surface. An attrac-tive EF yields a stabilizing surface virial, and thus the viral of the EF force acting on the interatomic surface in an equilibrium separation leads to the lowering of the molecule’s potential energy. The definition of bonding between atoms denoted by a bond path may therefore, be extended to include the properties of the EF: the presence of a line of maximum density linking a Figure 3. The variation of the changes in E, T, and V on bonding
as a function ofR in atomic units obtained from highly correlated wave functions. The calculations for H2, N2, and CO have been
pre-viously described:51H
2is QCISD with SCVS (self consistent virial
scaling), N2and CO are MRCI using a large active space, and Ar2
is from a QCISD SCVS/cc-pVZT calculation, which includes f func-tions. Energy and distance scales are in atomic units. The diagrams illustrate that the bonding in homopolar, polar, and van der Waals interactions all exhibit a common underlying quantum mechanism. Even an ionic interaction such as LiF will exhibit an initial decrease inT and increase in V for large R before the onset of charge transfer.
pair of nuclei in an equilibrium geometry of a bound state or one lying within the attractive region of a potential well, implies not only the absence of repulsive Feynman forces on the nuclei but also the presence of an attractive Ehrenfest force acting across the interatomic surface drawing the two atoms together.58 There are no other forces acting in a field free system.
The curves for(R) vs. R bear a remarkable similarity to the curves shown for corresponding behavior of the atomic Ehrenfest force F (A) in terms of the values of R where they attain their maximum values and change into an attractive interaction. One may question as to whetherF (A) will continue to become increas-ingly more attractive asR decreases, toward the united atom (UA) limit, or whether it will eventually reflect the repulsion generally associated with the merging of ‘inner shell electrons’ defined by the Laplacian. This question is readily answered by pushing any two atoms together—with or without charge transfer—and moni-toring the change not only inF (A), but also in the total energy E and in theelectronic energy He¼ E Vnn¼ Vne þ Veeþ T.59
AsR is decreased from Re, E increases but the electronic energy
Hedecreases monotonically to the united atom (UA) limit. The
ki-netic energyT, as required by the virial theorem, attains its limit-ing value equal to the negative of the energy of the UA. One might have expected F (A) to show some ‘bumps’ as closed inner-shells come into contact—‘closed-shell or Pauli repul-sions’—but this does not happen. Instead, it parallels He and
exhibits a monotonic decrease. Thus, the attractive potential energyVenoverwhelms the increasingT and Veerepulsions in the
approach of He to the UA limit. SinceE and He differ only by
Vnn, the repulsion upon close approach of two atoms is a result of
the overriding increase in the nuclear–nuclear repulsive energy. Thus, according to physics, when repulsions do exist upon the close approach of atoms, they are a consequence of the nuclear– nuclear repulsions dominating the total energy. The sole repulsive contribution to the Ehrenfest force is from the e-e interaction mak-ing understandable the parallelmak-ing nature of EF with the behavior ofHeand its becoming increasingly attractive as the UA limit is
approached. Some texts5ascribe the increase in E for separations in the neighborhood of R Reto a dominant increase inT rather
than inVnn. This is, however, not true. While the ‘compression’ of
the density asR is decreased causes T to increase, it also results in an increase in the accumulation of density between and in the vicinities of the nuclei causingVento decrease by an amount that
counters both the increase in T and in the e-e repulsion. It is for this reason that the electronic energyHeundergoes its continuous
decrease to the UA energy. T is a component of He and its
increase is inseparable from the accompanying decrease in Vne.
Since T increases to a finite limit—the negative of the UA energy—while Vnn increases exponentially to infinity, one
antici-pates and finds the rate of increase inVnnto numerically outweigh
the rate of increase inT for R< Re. Within the B-O
approxima-tion, there is a clear distinction between the electronic Hamiltonian that defines the Eigenvalue that we have called He, and Vnnthat
appears as an additive constant.
Local Behavior of T
A more detailed accounting of the role of the kinetic energy on bonding may be obtained by comparing the topology of the
ki-netic energy density with that of the electron density. Does the accumulation of density in the internuclear region result in a corresponding increase in T? Such questions were posed and answered by Bader and Preston (BP) in 196960through a study of the kinetic energy density expressed in its positive definite form, as given in eq. (13),
GðrÞ ¼ ð1=8Þiir iðrÞ r iðrÞ=iðrÞ (13)
wherei¼ ’i*’iis the density of theith natural orbital’iwith
occupationli. For the Hartree-Fock ground state of H2, only the
1g orbital is occupied and the topology ofG(r) in this case is
determined by the gradients of the electron density, ¼ (1g).2
The density is a minimum at the bond mid-point rm,!(rm) ¼
0 and henceG(rm)¼ 0. The diminished gradients in resulting
from the accumulation of density over the bonding region cause G(r) to exhibit markedly low values over the entire internuclear region, a result that is only minimally affected by electron corre-lation, the 1g2configuration accounting for 98% of the electron
density. The significant reduction in the kinetic energy density G(r) in the bonding region of H2calculated from a highly
corre-lated function36is displayed in its profile in Figure 4. The reader is asked to compare the contrasting behavior of the two distribu-tions(r) and G(r) in the internuclear region. This study demon-strated how the accumulation of electron density in the bonding region, an accumulation critical to the lowering of the potential energy V necessary for bonding, is accomplished with a simulta-neous associated decrease in the kinetic energy.60 The kinetic energy is increased in the antibonding regions and in this man-ner the virial theorem, that requiresT to increase by an amount equal to one-half the decrease in V, is satisfied even with the lowering of G(r) in the bonding region. BP presents the same Figure 4. Profiles maps along the internuclear axis of the electron density(r) and the kinetic energy density G(r) in atomic units (au) for H2 at a separation of 1.4 au. The centre point in (r) is a (3,
1) critical point while the same point in G(r) is a (3, þ3) critical point, a local minimum and G(r) increases in all direction away from the critical point. The discontinuous drop inG(r) at the posi-tions of the nuclei, which is most pronounced for a hydrogen atom, is a consequence of the nuclear-electron coalescence cusp condition on(0).21,60
comparison for the unbound state of He2. Unsurprisingly, since
is decreased in the bonding region of He2incurring steep
gra-dients in the density, the kinetic energy density in this region is dramatically increased for He2.
The reduction in the gradients of in a direction parallel to the axis leads to a considerable reduction in the parallel compo-nent T|| ¼ (1/8) h(@=@zÞ2=i, and hence to the significant
decrease inT in the internuclear region. The perpendicular com-ponents T\, on the other hand, are greatly increased over the original atomic value, a result of the contractions of the density in directions perpendicular to the bond axis. This behavior is reflected in the value of !2(r) at the bond critical point, a quantity appearing in the local expression for the virial theorem, eq. (5). For a shared interaction,!2(r) < 0, the magnitudes of the two perpendicular curvatures of overwhelming the softened curvature along the axis.
Though the topology ofG(r) for molecules beyond H2
exhib-its a more complicated behavior,61 the contrasting behaviors G(r) and of !2(r) for H2 and He2 at the bond critical point
carry over to other molecules and bond critical point data are used in the characterization of bonding. Thus shared interactions are found in general to haveGb/b< 1 and !2b(r)< 0 while
closed-shell interactions in general, haveGb/b 1 and !2b(r)
> 0.2,28
The continuing comparative study of the topologies of G(r) and(r), coupled with the use of the virial theorem, pro-vided the observational basis for the theory of atoms in mole-cules in 1972.21
Discussion and Conclusions
There is tendency to describe the use of the electrostatic force in the interpretation of chemical bonding as an ‘electron density analysis,’ which is unfortunate. The recognition of the electro-static force as the only force operative in a field-free molecule focuses one’s attention on the physics of bonding, using the the-orems of quantum mechanics, including their atomic counter-parts. The consequences of the accumulation of density in the binding region on the total energy and its kinetic and potential contributions are fully accounted for by the virial and the Feyn-man and Ehrenfest force theorems. Let there be no doubt that bonding is a result of the lowering in the potential energy as demanded by the virial theorem. The role of the kinetic energy is fully accounted for, its role being quantified not only through the virial theorem expressed as a function of internuclear separa-tions, but also through the study of its local properties in terms of the physical field G(r). Out of all the interactions within a molecule, only the exchange-correlation contribution to the elec-tron–electron interaction requires information beyond the nuclear charges and the electron density, the Coulomb portion being ex-pressible as ½(r1)(r2)/|r1 r2|. The role of exchange in
bond-ing, the factor responsible for incorporating the Lewis electron pair model into quantum mechanical discussion of bonding, is brought to the fore through the study of the Fermi correlation and its physical consequences. These are summarized by the atomic localization/delocalization indices and the intra- and inter-atomic contributions to the exchange energy and they are given physical expression in terms of the topology of the
Lapla-cian of the density. The Ehrenfest force condenses all of this in-formation into an expression for the force acting on the electron density that is realized in real space.
Lewis’ concept of an electron pair bond1prepared chemists for what was to come from quantum mechanics, from VB and MO theory and from our present abilities to determine the physical consequences of the spatial pairing of electrons from studies on the electron density and the pair density. The study of the topol-ogy14 and the mechanics of the electron density62 have extended our view of bonding beyond the Lewis model by offering a unified view of atomic interactions in terms of the properties of the bond path.63 The result is a definition of structure that encompasses all interactions in all systems, including those that lie beyond the scope of the electron pair model of bonding, as illustrated in its many applications across the spectrum of chemistry.
References
1. Lewis, G. N. J Am Chem Soc 1916, 38, 762.
2. Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Oxford University Press: Oxford, 1990.
3. Lewis, G. N. J Phys Chem 1933, 1, 17.
4. Bader, R. F. W.; Stephens M. E. J Am Chem Soc 1975, 97, 7391. 5. Pauli, W. Z Phys 1925, 31, 765.
6. Slater, J. C. Phys Rev 1951, 81, 385.
7. Slater, J. C. Quantum Theory of Atomic Structure, Vol. 2; McGraw-Hill: New York, 1960.
8. Bader, R. F. W. Chem Eur J 2006, 12, 2896.
9. Fradera, X.; Austen, M. A.; Bader, R. F. W. J Phys Chem A 1999, 103, 304.
10. Wang, Y. G.; Werstiuk, N. H. J Comp Chem 2003, 24, 379. 11. Corte´s-Guzma´n, F.; Bader, R. F. W. Coord Chem Rev 2005, 249,
633.
12. Bader, R. F. W.; MacDougall, P. J.; Lau, C. D. H. J Am Chem Soc 1984, 106, 1594.
13. Morse, P. M.; Feshbach, H. Methods of Theoretical Physics I; McGraw-Hill: New York, 1953.
14. Bader, R. F. W.; Nguyen-Dang, T. T.; Tal, Y. Rep Prog Phys 1981, 44, 893.
15. Bader, R. F. W.; Heard, G. L. J Chem Phys 1999, 111, 8789. 16. Bader, R. F. W. J Chem Phys 1980, 73, 2871.
17. Macchi, P.; Sironi, A. Coord Chem Rev 2003, 238/239, 383. 18. Bo, C.; Poblet, J. M.; Be´nard, M. Chem Phys Lett 1990, 169, 89. 19. Bo, C.; Sarasa, J. P.; Poblet, J. M. J Phys Chem 1993, 97, 6362. 20. Chatt, J.; Duncanson, L. A. J Chem Soc 1953, 2329.
21. Bader, R. F. W.; Beddall, P. M. J Chem Phys 1972, 56, 3320. 22. Bader, R. F. W.; Gillespie, R. J.; MacDougall, P. J. J Am Chem Soc
1988, 110, 7329.
23. Gillespie, R. J.; Hargittai, I. The VSEPR Model of Molecular Geom-etry; Allyn and Bacon: Boston, 1991.
24. Lennard-Jones, J. E. Proc R Soc Lond Ser A 1949, 198, 14. 25. Lennard-Jones, J. J. Chem Phys 1952, 20, 1024.
26. Koritsanszky, T. S.; Coppens, P. Chem Rev 2001, 101, 1583. 27. Gatti, C. Z Kristallogr 2005, 220, 399.
28. Bader, R. F. W.; Esse´n, H. J Chem Phys 1984, 80, 1943.
29. Lennard-Jones, J.; Pople, J. A. Proc R Soc Lond Ser A 1951, 210, 190.
30. Slater, J. C. Quantum Theory of Molecules and Solids I; McGraw-Hill: New York, 1963.
31. Feynman, R. P. Phys Rev 1939, 56, 340.
33. Hellmann, H. Z Phys 1933, 35, 180. 34. Berlin, T. J. Chem Phys 1951, 19, 208.
35. Hirschfelder, J. O.; Eliason, M. A. J. Chem Physics 1967, 47, 1164. 36. Das, G.; Wahl, A. C. J Chem Phys 1966, 44, 87.
37. Cade, P. E.; Huo, W. M. At Data Nucl Data Tables 1973, 12, 415. 38. Cade, P. E.; Wahl, A. C. At Data Nucl Data Tables 1974, 13, 339. 39. Cade, P. E.; Huo, W. M. At Data Nucl Data Tables 1975, 15, 1. 40. Bader, R. F. W.; Henneker, W. H.; Cade, P. E. J Chem Phys 1967,
46, 3341.
41. Bader, R. F. W.; Keaveny, I.; Cade, P. E. J Chem Phys 1967, 47, 3381.
42. Bader, R. F. W. In M. T. P. International Series of Science, Vol. II: Theoretical Chemistry; Coulson, C. A.; Buckingham, D. A., Eds.; Butterworths: Guilford, Surrey, 1975; pp. 43–79.
43. Runtz, G. R.; Bader, R. F. W.; Messer, R. R. Can J Chem 1977, 55, 3040.
44. Keith, T. A.; Bader, R. F. W.; Aray, Y. Int J Quantum Chem 1996, 57, 183.
45. Luo, F.; McBane, G. C.; Kim, G.; Giese, C. F.; Gentry, R. J Chem Phys 1993, 98, 3564.
46. Slater, J. C. J Chem Phys 1972, 57, 2389.
47. Cade, P. E.; Bader, R. F. W.; Henneker, W. H.; Keaveny, I. J Chem Phys 1972, 50, 5313.
48. Bader, R. F. W.; Henneker, W. H. J Am Chem Soc 1966, 88, 280. 49. Cade, P. E.; Bader, R. F. W.; Pelletier, J. J Chem Phys 1971, 54,
3517.
50. Bader, R. F. W.; Bandrauk, A. D. J Chem Phys 1968, 49, 1653. 51. Ruedenberg, K. Rev Mod Phys 1962, 34, 326.
52. Slater, J. C. J Chem Phys 1933, 1, 687.
53. Herna´ndez-Trujillo, J.; Bader, R. F. W. J Phys Chem A 2000, 104, 1779.
54. Bader, R. F. W. In The Force Concept in Chemistry; Deb, B. M., Ed.; Van Nostrand Reinhold: New York, 1981; pp. 39–136. 55. Hurley, A. C. Int J Quantum Chem 1982, 22, 241.
56. Feinberg, M. J.; Ruedenberg, K. J Chem Phys 1971, 54, 1495. 57. Bader, R. F. W.; Matta, C. F.; Corte´s-Guzma´n, F. Organometallics
2004, 23, 6253.
58. Bader, R. F. W.; De-Cai, F. J Chem Theor Comp 2005, 1, 403. 59. Herna´ndez-Trujillo, J.; Corte´s-Guzma´n, F.; Fang, D. C.; Bader,
R. F. W.; Faraday Discuss (in press).
60. Bader, R. F. W.; Preston, H. J. T. Int J Quantum Chem 1969, 3, 327.
61. Bader, R. F. W. Lec Ser Comp Comput Sci 2006, 4, 1230. 62. Bader, R. F. W.; Nguyen-Dang, T. T. Ad Quantum Chem 1981, 14,
63.
