![[CO-Borane]](./f12-2msi.gif)
|
(r). The electron density describes the manner in which the electronic charge is distributed throughout real space. The electron density is a measurable property and it determines the appearance and form of matter. This is illustrated in the following figures. Figure 1 displays the spatial distribution of the electron density in the plane containing the two carbon and four hydrogen nuclei of the ethene molecule. The electron density is a maximum at the position of each nucleus and decays rapidly away from these positions. When this diagram is translated into three dimensions, the cloud of negative charge is seen to be most dense at nuclear positions and to become more diffuse as one moves away from these centres of attraction, as illustrated in Figure 2. The presence of local maxima at the positions of the nuclei is the general and also the dominant topological property of (r). Figure 3 illustrates the same feature for the 110 plane of carbon nuclei in the diamond lattice.
Figure 1. (a) The electron density in the plane containing the two carbon and four hydrogen nuclei of the ethene molecule, portrayed as a projection in the third dimension and in the form of a contour map. The absolute maxima in
Figure 1. (b) Same as in Figure 1a, but for a plane obtained by a rotation of 90° about the C-C axis, a plane containing only the carbon nuclei.
Figure 1. (c) Again, the same portrayal as in Figure 1a, but this time for a plane perpendicular to the C-C axis at its mid-point. What appears as a C-C saddle in (a) is seen to be a maximum in the plane perpendicular to the C-C axis. The point exhibits two negative curvatures perpendicular to this axis and one positive curvature along the axis.
Figure 2. Envelopes of the electron density for the ethene molecule for values (in atomic units) of 0.36 in (a), 0.20 in (b) and 0.002 in (c). Matter consists of point-like nuclei embedded in a spatial distribution of negative charge that becomes increasingly diffuse for points progressively removed from the nuclei.
Figure 3. The electron density for diamond as projection above a 110 plane. The second diagram displays the tetrahedral structure of the bond paths linking the carbon nuclei in diamond, lines that are a consequence of the topology exhibited by the electron density.
To determine what physics has to say about this property of the electron density one must consider not the density itself but the field one obtains by following the trajectories traced out by the gradient vectors of the density. Starting at any point, one determines the gradient of
(r). This is a vector that points in the direction of maximum increase in the density. One makes an infinitesimal step in this direction and then recalculates the gradient to obtain the new direction. By continued repetition of this process, one traces out a trajectory of 
(r). A gradient vector map generated in this manner is illustrated in the upper diagram of Figure 4 for the same plane of the ethene molecule shown in Figure 1. Since the density exhibits a maximum at the position of each nucleus, sets of trajectories terminate at each nucleus. The nuclei are the attractors of the gradient vector field of the electron density. Because of this fundamental property, the space of the molecule is disjointly and exhaustively partitioned into basins, a basin being the region of space traversed by the trajectories terminating at a given nucleus or attractor. Since a single attractor is associated with each basin, an atom is defined as the union of an attractor and its basin.
Figure 4. Maps of the gradient vector field of the electron density for the same plane containing the nuclei shown in Figure 1. Each line represents a trajectory traced out by the vector
| Figure 4. (a) A display of the trajectories that terminate at the nuclei. Each trajectory is arbitrarily terminated at the surface of a small circle centered on the nucleus. The set of trajectories that terminate at a given nucleus (attractor) cover the basin of the attractor.
|
|
| Figure 4. (b) The same as (a) but including the sets of trajectories which terminate and originate at the bond critical points (denoted by dots). Only one pair of an (infinite) set of trajectories that terminate at the critical point lie in this plane. |
|
| Figure 4. (c) A contour map of the electron density overlaid with the bond paths that define the molecular graph and with the trajectories that mark the intersection of the interatomic surfaces with this plane and define the boundaries of the atomic basins. |
|
The second gradient vector field map in Figure 4 includes the trajectories, shown in bold, that both originate and terminate at the critical points found between nuclei that appear linked by a saddle in (r) in Figure 1. A critical point denotes an extrenum in (r), a point where (r) = 0. Associated with each such critical point is a set of trajectories that start at infinity and terminate at the critical point, only two of which appear in the symmetry plane shown in the figure. They define an interatomic surface, a surface that separates the basins of neighbouring atoms. There is a unique pair of trajectories that originate at each such critical point and terminate, one each, at the neighbouring nuclei. They define a line through space along which the electron density is a maximum. The two sets of trajectories associated with such a critical point, a bond critical point, the set that terminates at the critical point and defines the interatomic surface and the pair that originates there and defines the line of maximum density, are shown in Figure 5.
| Figure 5. A three-dimensional display of the set of trajectories of (r) that terminate at a bond critical point and define an interatomic surface and of the unique pair of trajectories that originate at the same point and define the bond path. Only one pair of each set that terminates at the critical point appears in the plane illustrated in Figure 4 (b) and (c).
|
|
In an equilibrium geometry the line of maximum density is called a bond path because the set of bond paths for a given molecule, the molecular graph, faithfully recovers the network of chemical bonds that are assigned on the basis of chemical considerations. Thus a pair of bonded atoms are linked by a line along which the electron density, the glue of chemistry, is maximally concentrated. Molecular structures predicted by the molecular graphs determined by the electron density are shown in Figure 6.
Figure 6. Molecular graphs - lines of maximum electron density linking bonded nuclei - in hydrocarbon molecules in the upper two diagrams, and boranes and carboranes in the lower diagram. Bond critical points, where the trajectories defining the bond path originate, are denoted by dots. Note that the bond paths can be curved away from the internuclear axis in strained or in electron deficient molecules. A molecular graph and the characteristics of the density at the bond critical points provide a concise summary of the bonding within a molecule or crystal.
The molecular graph undergoes discontinuous and abrupt changes if the nuclei are displaced into critical configurations. When this occurs, one makes or breaks certain of the bonds and changes one structure into another. These changes are described and predicted using the mathematics of qualitative dynamics and the resulting theory of structural stability is illustrated in Figure 7 for the very strained molecule called [1,1,1]propellane.
Figure 7. Diagrams illustrating changes in structure induced by the dynamics of the nuclei. The molecular graph in a is for the highly strained [1,1,1]propellane molecule, C6H6 (the two hydrogens attached to each apical carbon atom are not indicated). The gradient vector field maps are for the symmetry plane containing the C-C bridgehead bond critical point and the three apical carbon atoms. When the separation between the two bridgehead nuclei is increased to a critical value, the bond critical point coalesces with the three neighbouring ring critical points to form a singularity in (r), as depicted by the gradient vector field map in b. The singularity is unstable and its formation signifies the breaking of the C-C bridgehead bond. Further separation of the nuclei causes it to bifurcate into a cage critical point yielding a new structure in which the bridgehead carbon atoms are not bonded to one another, the cage structure depicted in c.
The definition of an atom in terms of the topology of the electron density is neither valid nor useful if its properties are not predicted by quantum mechanics. The fundamental nature of the atom, as the building block of matter, follows from the demonstration that the topological and quantum definitions of an atom coincide. An atom can be alternatively defined as a region of real space bounded by surfaces through which there is a zero flux in the gradient vector field of the electron density. This is clear from Figure 5, which shows that an interatomic surface is defined by the set of trajectories that terminate at a point where (r) = 0. Thus an interatomic surface satisfies the "zero-flux" boundary condition stated in equation (1)
(rs)·n(rs) = 0,
for every point rs on the surface S(rs) (1)
where n(rs) is the unit vector normal to the surface at rs. In words, the surface is not crossed by any trajectories of (r). An atom, as a constituent of some larger system, is itself an open system subject to fluxes in charge and momentum through its bounding surface.
Around 1950, Feynman and Schwinger, following along a path first traversed by Dirac, developed a new formulation of quantum mechanics based upon the classical principle of least action. Their work enables one to ask and answer questions that could not be answered using the Hamiltonian-based approach to quantum mechanics. Schwinger's generalization of the action principle, as contained in his principle of stationary action, in addition to determining the field equation, yields a variational derivation of Heisenberg's equation of motion for any observable 
. This principle equates the change in action to the infinitesimal (
) transformations caused by the generators -(i/
) acting in the space-like and time-like surfaces that bound the space-time volume swept out by a system, as well as to displacements in these surfaces. A time-like surface describes the temporal evolution of the spatial boundary enclosing a portion of some total system. Thus Schwinger's principle enables one to derive a quantum description of an open system. In doing so one obtains the remarkable result that only an open system bounded by surfaces satisfying the `zero-flux' boundary condition stated in equation (1) yields an expression for the change in action that is the same in form and content to that for an isolated system and in addition, yields equations of motion for the observables that are identical to those predicted by the field equation. Thus the definition of an open system at the atomic level is not open to choice but is determined by physics. Consequently, an open system satisfying equation (1) is termed a proper open system.
A total isolated system also satisfies equation (1). Thus a single principle of physics determines the properties of the total system and its constituent atoms. The properties of the topological atoms coincide with those ascribed to the atoms of chemistry; (i) they are additive to yield the corresponding property for the total system and (ii) they are transferable from one system to another to the extent that the transfer leaves their distribution of charge unchanged. It is known from experiment that atoms and functional groupings of atoms can be transferable to a remarkable degree and that when this occurs, one can determine the atomic or group contributions to the total properties of a system. The theory of atoms in molecules recovers the experimentally determined characteristic and additive group contributions to all properties, as has been demonstrated for the volume, energy, electric polarizability and magnetic susceptibility. These are the atoms of chemistry.
The envelope of the electron density, of value 0.001 atomic units, contains almost all of the electronic charge of a system and provides a measure of its van der Waals size and shape. The intersection of the interatomic surfaces of an atom or a group with this envelope thus yields a display of the group as a space-filling object, examples of which are shown in Figures 8 and 9. All proper open systems are transferable to some extent, this property underlying the usefulness of the atomic model in chemistry.
Figure 8. Depictions of atoms and functional groupings of atoms as space-filling objects - regions bounded by the intersection of the interatomic surfaces and the van der Waals envelope of the electron density. The second-row hydrides AHn where A = Li, Be, B, C, N, O and F. Note the change in the size and form of the hydrogen atom, from one characteristic of the hydride ion in LiH to the positively charged one in HF, wherein the atom has been stripped of more than half of its electron density.
Figure 9. (a) The transferable methyl group of the normal hydrocarbons. This group remains essentially unchanged throughout this series of molecules and contributes a fixed amount to each property, including the volume, energy, electric polarizability and magnetic susceptibility. (b) The alanyl peptide unit bounded by its two amidic interatomic surfaces. This group, like all groups that serve as transferable building blocks, has a zero net charge and a fixed set of properties.
There is no such thing as an isolated system, and all systems are open systems that experience varying degrees of interaction through their shared zero-flux surfaces. Thus the statement of the principle of stationary action for a proper open system is simply a generalization of quantum mechanics that applies to all physical systems. The operational statement of this theory is most elegantly and simply stated using the language of field theory. The principle, when stated in terms of a variation of the Lagrange-function operator 
[
,t,
] for the observable , and in a form that is applicable to any region of space bounded by a zero-flux surface, is
[,t,] =
(/2){(i/
<|[
,]|
> + cc }
(2)
The observable multiplied by -(i/), where denotes a real infinitesimal, is the generator of a corresponding unitary transformation in the Lagrange-function and one sees that Schwinger's principle combines the action principle with Dirac's transformation theory. The variation in [,t,] may be alternatively expressed as
(3)
that is, by the time rate-of-change of the property density for G together with a term accounting for the flux in JG(r), the vector current of this density, through the surface S(rs) bounding the region . This latter term vanishes for an isolated system.
The Table lists the atomic theorems obtained using equation (2) for a number of important generators. The Schrödinger representation is used in these equations and the expression 
d
' denotes a summation over all spins and an integration over the spatial coordinates of all electrons but the one whose coordinates appear in the observable . For example, the time rate-of-change of momentum is force and when = 
, one obtains the Ehrenfest atomic force theorem. In this case [,] = i
, where is the total potential energy operator. The vector current J(r) is the momentum density divided by the mass of the electron m and the integral of its time rate-of-change multiplied by m yields the force acting on the atom. This force is equated to the sum of a basin and a surface contribution: the basin contribution is the commutator average and the surface term is the momentum flux through the surface, a term described by the quantum stress tensor 
(r). In a stationary state, the basin term for a given is balanced by its surface flux contribution.
The d' averaging of the commutator yields the force exerted on an electron at a point r in , as determined by averaging over the motions of all the remaining particles in the system. Thus it describes a force density and its integral over yields the average force exerted on the basin of the atom. The basin contributions to all properties are determined by a corresponding property density. Thus when = 
· the commutator yields -· whose d' averaging yields the electronic potential energy density. This density expresses in a rigorous manner, the local potential energy experienced by a single electron determined in the average field of all the remaining particles in a many-particle system.
The kinetic energy T() and the potential energy defined by the atomic virial theorem have the remarkable and necessary property that they are as transferable as is the electron density. Proper open systems are the most transferable pieces of matter one can define in an exhaustive partitioning of the real space of any system, leading to their most important property; if the distribution of electronic charge is essentially unchanged for a given atom in two different systems, then its properties, including its energy, are transferable as well as additive, the atom contributing the same amounts to all properties in both systems. It is the energy defined by the atomic virial theorem that exhibits the essential physical requirement that two identical pieces of matter possess identical energies, be they of macroscopic or microscopic dimensions, in the form of equivalent atoms at different sites within a crystal or two identical peptide units in a polypeptide.
When a molecule is placed in a uniform external magnetic field, a flow of electric current is induced within the molecule. This current is described by the quantum current density J(r) introduced above and appearing in the atomic current theorem. This theorem states that the atomic average of J(r), the average velocity of the electrons within the atom as induced by the applied field, is equal to the flux through its atomic surface of the tensor corresponding to the position weighted current. Because of this theorem, the magnetic susceptibility of a molecule can be equated to a sum of atomic or group contributions and the values so defined recover Pascal's experimentally determined values for the corresponding group increments for the hydrocarbons. The same current determines the shielding or deshielding of a nucleus from the applied magnetic field in an NMR experiment, and this shielding is also expressible as a sum of atomic contributions; see references 6 and 7.
Figure 10. Display of the trajectories of the current induced in the carbon dioxide molecule by a magnetic field directed out of the plane of the diagram. The interatomic surfaces separating the basins of the carbon and oxygen atoms are also indicated. Note how these surfaces isolate the paramanegtic current flow in the basin of the carbon atom.
Figure 10 illustrates the current induced in the carbon dioxide molecule for a magnetic field directed out of the plane of the diagram. This display of the calculated current is made possible through the use of the theory of atoms in molecules in overcoming the "gauge origin" problem. One sees the presence of both diamagnetic (clockwise) and paramagnetic (counter-clockwise) currents with the former dominating. The current induced by an externally applied magnetic field determines all of the magnetic response properties exhibited by a molecule and understanding these properties requires an understanding of the induced current and its atomic contributions. Thus the presence of the paramagnetic currents within the basin of the carbon atom in carbon dioxide provides a clear physical explanation of the magnetic susceptibility and the magnetic shielding of the nuclei in this molecule.
While the topology of the electron density provides a faithful mapping of the concepts of atoms, bonds and structure, it provides no indication of maxima in r corresponding to the electron pairs of the Lewis model. This model is secondary only to the atomic hypothesis itself in understanding chemical bonding and reactivity and the geometry of molecules, the latter as predicted in terms of the localized electron pairs assumed in the VSEPR model. The physical basis of this most important model is one level of abstraction above the visible topology of the electron density, appearing in the topology of the Laplacian of the density. This function is the scalar derivative of the gradient vector field of the electron density, the quantity 2, and it determines where electronic charge is locally concentrated, 2 < 0, and depleted, 2 > 0, the local charge concentrations providing a mapping of the electron pairs of the Lewis and VSEPR models.
The Laplacian of the electron density recovers the shell structure of an atom by displaying a corresponding number of alternating shells of charge concentration and charge depletion. The uniform sphere of charge concentration present in the valence shell of a free atom is distorted upon chemical combination to form local maxima and minima. The number of local maxima in -2 in the valence shell, the local valence charge concentrations, together with their relative positions and magnitudes, coincide with the number and corresponding properties of the localized electron pairs assumed to exist in the VSEPR model of molecular geometry.
Figure 11. A relief map of the Laplacian of the electron density for the ClF3 molecule in the equatorial plane (a) and in the plane containing all four nuclei (b).
(a)(b)
The realization of the VSEPR model in terms of the Laplacian of
is illustrated in Figure 11 showing a relief map of the Laplacian for the ClF3 molecule. The chlorine atom exhibits three shells of charge concentration, the fluorine atom two such shells, corresponding to the presence of three and two quantum shells respectively. The VSEPR model predicts a T-shaped geometry for ClF . This geometry enables the two non-bonded electron pairs, which are assumed to be larger than the bonded pairs, to occupy the least crowded equatorial positions. The equatorial plane (top diagram) shows the presence of two non-bonded and one smaller bonded charge concentrations. In the axial plane there are three bonded charge concentrations and a fourth apparent maximum which is actually another view of the (3,-1) critical point between the two non-bonded maxima, that is, a critical point with one positive curvature. Thus the valence shell charge concentration of the chlorine atom possesses two non-bonded and three bonded charge concentrations in agreement with the five electron pairs assumed in the Lewis model and in this geometry, they maximally avoid one another, in agreement with the VSEPR model. The sizes of the charge concentrations decrease in the order, nonbonded > equatorial bonded > axial bonded, as assumed in the VSEPR model, the bonded sizes reflecting a greater net charge on the axial fluorines.
Figure 12. The zero value surfaces of 2 for carbon monoxide (blue) and borane, BH3 (red). These surfaces enclose regions where the electronic charge is maximally concentrated and they define the reactive surface. The same surfaces also show the inner shell charge concentrations on the carbon and boron nuclei, evident as small inner spheres. The molecules are orientated so that the `lump' in the valence shell charge concentration (VSCC) of carbon is aligned with the `hole' in the VSCC of boron. The VSCC of boron is reduced to a belt-like distribution lying in the plane of the hydrogen nuclei, giving the base atom direct access to the core of the boron, the feature that makes BH3 a strong Lewis acid. Note the torus of charge depletion encircling the carbon nucleus. This feature corresponds to the localization of the 
* antibonding orbital on carbon.
The Lewis model is also used to rationalize chemical reactivity. In addition to a local charge concentration in the valence shell that behaves as Lewis base or nucleophile, there are also local charge depletions, and such charge depletions behave as Lewis acids or electrophiles. A chemical reaction corresponds to the combination of a charge concentration in the valence shell charge concentration of the base with a charge depletion in the valence shell charge concentration of the acid, the reactivity paralleling the magnitude of the charge concentration and the depth of the charge depletion. The geometry of approach of the acid and base molecules is predicted through the alignment of the corresponding "lumps" and "holes" in their Laplacian distributions, as illustrated for the approach of the non-bonded charge concentration on carbon of the CO molecule to the hole on the boron atom in BH3 , Figure 12. This predictive property of the Laplacian has been illustrated in many different reactions including those as diverse as the approach of methane to the oxygen atom of a metal oxide surface and the geometries of hydrogen bonded complexes.
The final Figure illustrates the charge concentrations present in the outer core of the barium atom in BaH2, a bent molecule. The use of the Laplacian of the electron density to account for the bent geometries of the hydride, halide and methylide molecules of calcium, strontium and barium, in terms of a distortion of the outer core of the electron density of the metal atom is discussed in reference 12.
Go to: Richard Bader's Home Page | Chemistry Faculty | Chemistry Welcome Page.
Last update: 08jan96, mgm
