The two faces of chemistry: can they be reconciled?

Found Chem DOI 10.1007/s10698-012-9172-y The two faces of chemistry: can they be reconciled? Mark E. Eberhart • Travis E. Jones Ó Springer Science+Business Media Dordrecht 2012 Abstract Shortly before his death, Richard Bader commented in this Journal on the dichotomy that exists within chemistry and between chemists. We believe that the dichotomy results from different goals and objectives inherent in the chemical disciplines. At one extreme are designers who synthesize new molecules with interesting properties. For these chemists, the rationale underpinning molecular synthesis is far less important than the end product—the molecules themselves. At the other extreme are the chemists who seek a fundamental understanding of molecular properties. We suggest that the Quantum Theory of Atoms in Molecules, by virtue of the rich hierarchical structure inherent in the theory, offers a bridge through which to unite these two groups. However, if there is to be reconciliation, it falls to the theorists to develop ‘‘quantum mechanically’’ correct tools and concepts useful to the synthetic and applied chemist. Keywords Quantum theory of atoms in molecules Introduction Chemistry is a discipline of two faces, one applied and the other theoretical. The applied face focuses on the design and synthesis of molecules and solids, while the theoretical face looks for explanations of a molecule or solid’s properties. At first blush, this observation may not seem to warrant note, yet it does set chemistry apart from its sister sciences. The applied and theoretical components of physics, for example, have been subsumed into separate areas of study and specialization. Classical physics is largely concerned with the theories of statics, dynamics, electricity, magnetism, fluid dynamics, and so forth, while the application of these theories forms the basis of civil, mechanical, electrical, and aeronautical engineering. The formal recognition of the applied and theoretical components of science as distinct is to some degree driven by the different values of designers and theoreticians. From these M. E. Eberhart (&)
T. E. Jones Molecular Theory Group, Colorado School of Mines, Golden, CO 80401, USA e-mail: [email protected] 123 M. E. Eberhart, T. E. Jones values, vocabulary arises that is nuanced to the specific disciplines; the physicist and the civil engineer may both use the word mass, but the image evoked by the word is likely to be quite different. For chemists, however, where there has been no clear separation of the science into its design and theoretical parts, our value differences can result in confusion, misunderstanding, and controversy. But from this controversy there also comes an opportunity to align theory to the needs of the applied chemist. This is exactly the situation that presents itself to the community advancing the Quantum Theory of Atoms in Molecules (QTAIM) (Bader 1990). That there is controversy and misunderstanding is clearly evidenced by the recent article in which Richard Bader (2011) ‘‘presents thoughts on the divide that exists in chemistry between those who seek their understanding within a universe wherein the laws of physics apply and those who prefer alternative universes wherein the laws are suspended or bent to suit preconceived ideas.’’ Before leaping headlong into this controversy, however, it is worth taking a deeper look at the value difference that place the laws of physics in opposition to the ideas of some chemists. Dirac (1929) was probably the first to articulate the values of the modern quantum chemist when he wrote, The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation. One can only imagine what Dirac would consider to be ‘‘too much computation,’’ nonetheless, for nearly 80 years the emphasis has been to exploit growing compute power to implement increasingly complex but computationally feasible approximate methods that allow the properties of molecules and solids to be calculated. For many theoreticians, the ability to calculate a property from first principles serves as a complete explanation of its origins. On the other hand, the designer is interested in identifying molecules with a given set of properties. As has been argued before (Eberhart 2002), this ability is conferred by what are known as structure-property relationships, which express the control that structure plays in mediating properties. The search for structure-property relationships is an essential component of scientific inquiry and usually begins with a precise hierarchical description of structure, where each level is characterized by a different length scale (Smith 1981). However, Cohen (1976) has argued that sometimes representations of structure are mere constructs that allow one to rationalize properties. Cohen called structure-property relationships of this type, ‘‘reciprocity relationships,’’ which abound in the contemporary picture of molecular structure. Known to every first year chemistry student is the common classification of molecules and solids as ionic, covalent, Van der Waals, or metallic. These classifications grew exclusively from the desire to see structure as the origin of a particular property. It was Arrhenius’ (ca. 1885) need to explain the property of electrical conductivity of some solutions that gave rise to the ion and the description of a crystal as ionic. Lewis (1916) originated the covalent bond as a way to explain the existence of binding forces in nonionic molecules. The need to explain the formation of condensed phases by molecules whose atoms possessed a full octet of electrons (an impossibility according to the Lewis picture) led to the Van der Waals bond (ca. 1922). The metallic bond (ca. 1925) grew from the need 123 The two faces of chemistry: can they be reconciled? to explain the differences in conductivity between nonionic solids. All of these representations of the bond were devised before the discovery of quantum mechanics. However, they were so effective in explaining the chemical phenomena of interest, and so useful in the synthesis of new molecules, that during the first half of the twentieth century they became the principal descriptors of molecular-structure and the basis for many, if not most, of the structure property relationships used by the applied molecular scientist. Hence, many of these structure property relationships cannot be reconciled with quantum mechanics, i.e. ‘‘wherein the laws of physics apply.’’ Simply stated, if the goal is to bring chemistry in its entirety under the umbrella of physics, it is unlikely that providing a quantum mechanically correct description of molecular structure will be sufficient. To achieve this goal, one must build an entirely new set of quantum mechanically correct structure-property relationships that are as useful as the reciprocity relationships that they will replace. The Hohenberg–Kohn theorem (Hohenberg and Kohn 1964) provides insight as to the form that some of these relationships may take. It posits that all ground-state molecular properties are a consequence of the charge density. And, of course, QTAIM provides an extremely rich—quantum mechanically correct—description of charge density. It seems only reasonable, therefore, to seek relationships between charge density, as descried by QTAIM and, at a minimum, ground state properties. And as Cyril Smith (1981) has argued, this search should begin with a precise hierarchical description of structure. The QTAIM structural hierarchy Though little noted, intrinsic to QTAIM is a structural hierarchy derived from volumes bounded by surfaces on which the gradient of the charge density vanishes at every point. Such surfaces are referred to as zero flux surfaces or ZFSs. By virtue of being bounded by ZFSs, these volumes are quantum observables, that is, they are characterized by properties that are in principle measurable (Bader 1990). Though there are infinitely many such volumes, the initial formulation of QTAIM was concerned only with those where the bounding ZFSs did not intersect an atomic nucleus. These distinct volumes, called Bader atoms or sometimes atomic basins, partition the charge density of a molecule or solid into space filling regions each of which encloses a single nucleon—hence the name, ‘‘atom.’’ QTAIM’s structural hierarchy is a consequence of the topological connections between Bader atoms, which are determined by the charge density’s rank 3 critical points, CPs. These are the places where the charge density, a three-dimensional scalar field, achieves extreme values in all directions. As with all 3D scalar fields, the charge density possesses at most four kinds of CP: local minima, local maxima, and two types of saddle points. These CPs are denoted by an index, which is the number of principal positive curvatures minus the number of principal negative curvatures. For example, at a minimum, the curvature in all there principal directions is positive; therefore it is called a (3, ?3) CP. The first number is simply the number of dimensions of the space and the second is the net number of positive curvatures. A maximum is denoted by (3, -3), because all three curvatures are negative. A saddle point with two of the three curvatures negative is denoted (3, -1), while the other saddle point is a (3, ?1) CP. The charge density at the atomic nucleus is always a maximum a (3, ?3) CP (within the approximate Coulomb Hamiltonian the charge density at a nucleus is cusp with the curvatures undefined), hence it is also called a nuclear CP. The other CPs, which must be present in a molecular system, sit on the ZFSs bounding the Bader atoms and mediate their 123 M. E. Eberhart, T. E. Jones connectivity (Bader 1990; Zou and Bader 1994). The simplest topological connection results from a shared (3, -1) CP between two Bader atoms, which is indicative of a charge density ridge originating at the (3, -1) CP and terminating at nuclear CPs. In essence, this charge density ridge possesses the topological properties imagined for the chemical bond, which motivated studies showing the presence of such a ridge between atoms that conventional wisdom assumed to be bound. Accordingly, this ridge is descriptively referred to as a bond path and the accompanying (3, -1) CP as a bond CP. Thus, through QTAIM, a structure was associated with the strictly heuristic concept of the bond as a simple link. Other types of CPs have been correlated with other features of molecular connectivity. A (3, ?1) CP is required at the center of ring structures (rings of bond paths). Accordingly, it is designated a ring CP. Cage structures must enclose a single (3, ?3) CP and are given the name cage CPs. In these cases, the Bader atoms of a ring or cage share ring or cage CPs with the other atoms of the ring or cage, which is indicative of the more complex nature of the topological connections between these atoms. Though only recently noted (Eberhart 2001; Jones and Eberhart 2009, 2010), CPs, bond paths, and the ZFSs of Bader atoms are elements drawn from the larger set of extremal points, lines, and surfaces. These extremals are generically referred to as ridges and valleys. Incorporating all members of this set into QTAIM provides a more robust, elegant, and unified topological theory of molecular structure. In 2D, a ridge is a familiar topographic feature, the path (gradient path) connecting mountain passes to neighboring peaks, for example. There is only one such gradient path, and it is a path of locally least steep ascent terminating at the local maximum. Consequently, it is an extremum with respect to all neighboring paths. Similarly, a valley is an extreme gradient path connecting a saddle point to a local minimum, and because valleys and ridges differ only by the sign of the curvatures along the path, both are often referred to as ‘‘ridges.’’ In 3D fields (the electron charge density), ridges are the points, gradient paths, and zero flux surfaces that are extreme with respect to all neighboring points, gradient paths, and zero flux surfaces respectively. They are denoted by an index, n - d, where n is the dimensionality of the space and d is the number of principal directions in which the charge density is extremal (Eberly et al. 1994). Thus, a 0-ridge is nothing more than one of the four types of critical points. A 1-ridge is an extremal gradient path, of which the bond path is an example. And a 2-ridge is an extremal gradient surface, of which the ZFSs bounding Bader atoms are examples. For an extended system1 there will always be four kinds of charge density CPs (0-ridges), six kinds of 1-ridges, and four kinds of 2-ridges. The 1-ridges pairwise connect the four kinds of critical points and the 2-ridges are surfaces containing three distinct CPs. The ridge structure forms a set of space filling volumes homeomorphic to a tetrahedron. Coincident with the four vertices of each tetrahedron is a nuclear, bond, ring and cage CP, respectively. The six edges of the tetrahedron are 1-ridges, and the four faces are 2-ridges, Fig. 1. The resulting tetrahedra are simplices, which means that they are the most basic unit of charge density retaining local topology. Simplices may be glued together to form a simplical complex that is homeomorphic to the charge density topology of any molecular system. Accordingly, these simplices have been designated irreducible bundles, IBs, where bundle is used to evoke an image of a bundle of gradient paths. 1 All examples are drawn from extended systems where the ridge structure is somewhat easier to visualize than it is in finite molecules. In solids all possible ridges are present and all gradient paths are of finite length. Though only examples drawn from extended systems will be given here, the arguments are generalizable to open systems, i.e. molecules and surfaces Jones and Eberhart (2010). 123 The two faces of chemistry: can they be reconciled? Fig. 1 An irreducible bundle of fcc Cu. The CPs are shown as spheres: nuclear CP top, ring CP middle left, bond CP middle right, and cage CP bottom. The dark rods are gradient paths (differential gradient bundles) originating at the cage CP and terminating at the atom CP. The gradient paths in the IB are confined to the tetrahedral volume. The faces of this tetrahedron are 2-ridges each containing three CPs By way of illustration, consider an fcc copper crystal. The topology of this structure is distinguished by five symmetry-unique CPs: a ring CP, a bond CP, a nuclear CP, and two cage CPs–one each at the center of the tetrahedral and octahedral holes. These CPs give rise to two distinct irreducible bundles, shown in Fig. 2. The first has as its vertices the nuclear, ring, bond, and cage CP from the tetrahedral hole and the second the nuclear, ring, bond, and cage CP from the octahedral hole. Since the two IBs have three vertices in common, they must share a tetrahedral face, i.e. a 2-ridge. Together these two IBs form the symmetry unique wedge of the fcc crystal structure, which, under the operations of the fcc space group, will generate an extended simplical complex whose physical realization is the full charge density of fcc copper. From a molecule or solid’s simplical complexes one can construct subcomplexes, called d-skeletons, which recover the charge density at various topological levels. In particular, the 0-skelton of the simplical complex is the set of all of its CPs. Its 1-skeleton consists of all 1-ridges and is called the underlying graph of the complex. The 2-skeleton is the set of all 2- ridges, and the 3-skeleton is the full simplical complex. A molecule or solids underlying graph will contain as a subset the molecular graph common to modern depictions of molecules. However, the molecular graph depicts only bond paths, i.e. the connections between atoms, while the underlying graph (1-skeleton) depicts the full set of 1-ridges and captures the topology of the atomic connections, providing a more complete representation of bonding than the traditional picture of a bond as a simple connection. In addition to the extended simplical complexes, local structures can be generated by gluing together a finite number of IBs. The most interesting of these are given through the union of IBs sharing a single CP. The union of all IBs sharing the same nuclear CP will generate Bader atoms. The union of all IBs sharing the same cage CP will yield the repulsive basin first noted by Pendás, Costales, and Luaña (1997). Additionally, one can construct the union of all IBs sharing the same ring point. Finally, there is the union of all IBs sharing the same bond CP. This volume will contain a single bond critical point and its associated bond path and is referred to as a bond bundle, again to stress the fact that as a bundle of gradient paths the volume is bounded by ZFSs and hence has well defined properties, for instance an energy. 123 M. E. Eberhart, T. E. Jones Fig. 2 The two symmetry unique simplices, IBs, of the fcc crystal structure. The IBs are shown to the right as shaded polyhedra relative to the fcc tetrahedral hole. Top right is the IB containing the cage CP located in the octahedral hole while bottom right is the IB containing the cage CP from the tetrahedral hole. These two IBs may be glued together to form the symmetry unique wedge of the fcc structure (left), which, under the operations of the fcc space group will generate the charge density of the full fcc crystal Fig. 3 The irreducible and bond bundles of the B2 structure. The Al atoms are located at the corners of the cube and the Ni atom at its body center. Shown (left) is half of the Ni–Ni second neighbor bond bundle resulting from the union of the 16 IBs (darker wedge). The first neighbor bond bundle and the IBs from which it is formed are shown on the right To illustrate, consider the B2 ordered intermetallic of NiAl, Fig. 3. There are two symmetry unique bond CPs in this structure. One is located on the interatomic axis joining aluminum atoms to their eight nearest-neighbor nickel atoms. The other bond CP is on the interatomic axis joining nickel atoms to their six second-neighbor nickel atoms. Consequently, there are fourteen bond paths terminating at each nickel nuclear CP and eight terminating at each aluminum nuclear CP. These two types of bond CP necessitate two different bond bundles, which are shown in Fig. 3. The Ni–Ni second neighbor bond bundle is constructed from the union of sixteen IBs of one type and the Ni–Al bond bundle from the union of six IBs of two types. 123 The two faces of chemistry: can they be reconciled? Because a bond-bundle is a simplical complex, it also may be represented in terms of a skeletal structure. Its underlying graph (1-skeleton) provides a compact representation of the connection between atoms. While the IB is the basic structural element reflecting the connections between atoms, it too has a structure built from volumes bounded by ZFSs. To help with visualization, begin by noting that sufficiently close to a nuclear CP all charge density gradient vectors are radial and of the same magnitude. As a result, it is always possible to find a spherical isosurface, S, of radius dR centered on each nuclear CP. In the conventional spherical coordinate system, every point on such a sphere may be specified in terms of a polar and an azimuthal angle, S(h, /). Through each point of this surface there corresponds a gradient path, G, that originates from any number of cage CPs and terminates at the nuclear CP at the center of S. Hence, each gradient path may be specified by a pair of coordinates, G(h, / ). An IB will intersect S to produce a triangle. The set of points on this triangle are denoted as SIB(h, /). The gradient paths terminating at the nuclear CP and passing through the points interior to SIB(h, /) will originate at the same cage CP and together form a compact open set that is the interior of the IB. Obviously, 2-ridges form the boundary of this open set and its closure is the IB. Imagine covering SIB(h, /) with a set of nonintersecting differential elements of area dA. The gradient paths passing through the points comprising each of these area elements gives rise to a family of differential volume elements bounded by ZFSs. These differential gradient bundles, dG(h, /), (see Fig. 1) are the smallest structures admitted by QTAIM that, in principle, possess measurable properties, and for each nucleon, are parameterized by h and / only. The properties of larger structures, such as IBs or Bader atoms, are found by integrating over the properties of the differential gradient bundles. But more significantly, the underlying value of a property derives from a 2D property distribution that is itself well-defined within the QTAIM formalism. Discussion The existence of QTAIM specific 2D property distributions sheds light on what has been one of the controversial aspects of QTAIM, the assumption that a bond path and CP is indicative of a bonding (energy lowering) interaction. Bader and Preston (1969) have argued that the nature of the bonding interaction derives from the relative curvatures of the charge density parallel and perpendicular to a bond path and not simply from the existence of a bond CP. In support of this argument, Bader and Preston investigated the charge density of dimers including He2, which served as an example of a ‘‘unbounded’’ interaction, along with several ‘‘covalent’’ dimers such as H2. Similarly, Fig. 4 shows the charge density and gradient paths for ‘‘unbounded’’ Ar2 and ‘‘covalent’’ N2 molecules. Note that the family of gradient paths for these molecules are quite distinct. For Ar2 the gradient paths are atomic like—radial—except in the immediate neighborhood of the 2-ridge that is the ZFS of the Ar Bader atoms. In contrast, for N2, the gradient path curvature is distributed along a greater length of the path, and particularly, much closer to the nucleus. But significantly, all gradient paths, and hence all differential gradient bundles, sample both the region parallel and perpendicular to the bond path. Values of charge density properties at a point, or along a path other than a differential gradient bundle, have no physical significance within the QTAIM formalism. The implication that a bond CP or a bond path carries information about the ‘‘bond’’ is misguided and contrary to QTAIM formalism. 123 M. E. Eberhart, T. E. Jones Fig. 4 The gradient paths of Ar2 (left) and N2 (right) superimposed on a charge density contour plot. As discussed in the text, the number of electrons contained in the volumes produced by revolving the white ‘‘wedges’’ about the internuclear axis will give a QTAIM consistent representation of the electron distribution Fig. 5 Representative gradient paths of two IBs from the octahedral hole of fcc elements. For each IB the nuclear CP is at the top , the cage CP to the right, the ring CP is center foreground, and the bond CP on the left. The structure of the family of gradient paths is fully determined by specifying their curvature and torsion at each point in the IB. As in the case of the dimers, we find examples where the curvature and torsion are confined to the regions close to the Bader atom boundaries, e.g. Ar (left) and those where the curvature and torsion are distributed along the gradient path length e.g. Rh (right) As an illustration of this point, consider the 2D charge density distribution of N2 and Ar2. This distribution is found by integrating the charge-density within the differential gradient bundles. In the case of dimers, these are volumes produced by rotating about the internuclear axis a wedge produced by two gradient paths contained in the same internuclear plane and separated by an apex angle of dh. Figure 4 shows two such wedges for both N2 and Ar2, one wedge contains the bond path and the other is perpendicular to the bond path. Consistent with our calculations, and as is apparent, in the case of N2 it is the wedge containing the bond path with the greatest charge density. On the other hand, for Ar2, the charge density in the wedge perpendicular to the bond path contains the greatest charge density—though the difference is small and falls within the range of computational error. Still, the difference in the distribution is telling and provides a quantitative distinction between a ‘‘Van der Waals’’ and a ‘‘covalent’’ bond. Obviously, the 2D property distributions provides a more detailed and complete descriptions of atom-atom interactions than does the current vocabulary, e.g. ionic, 123 The two faces of chemistry: can they be reconciled? covalent, etc. However, it seems reasonable that these common classes of interactions will manifest distribution similarities. What remains then, is to divine the metrics that will be most useful in characterizing the structure of the family of differential gradient bundles and demonstrating that there are relationships between these structural metrics and the property distributions. As a first step in developing these relationships, we will hazard a guess as to the structural metrics that might characterize the family of differential gradient bundles by noting that the members of this family are basically space curve, each of which is fully specified by its curvature and torsion at each point along the curve. Inspections of Figs. 4 and 5 drive the point home that these structural features are sensitive to the classes of atomatom interactions. However, building the convincing case that there are indeed relationships between these structural metrics is clearly beyond the scope of this, or any single investigation, and we leave it as a clarion call to the QTAIM community to confirm the existence of these relationships. Acknowledgments We gratefully acknowledge support of this work under ONR Grant No. N00014-10-10838 and under ARO Grant No. 421-20-18. References Bader, R.F.W.: Atoms in Molecules. A Quantum Theory. Clarendon Press, Oxford (1990) Bader, R.F.W.: On the non-existence of parallel universes in chemistry. Found. Chem. 13, 11–37 (2011) Bader, R.F.W., Preston, H.J.T.: The kinetic energy of molecular charge distributions and molecular stability. Int. J. Quant. Chem. 3, 327–347 (1969) Cohen, M.: Unknowables in the essence of materials science and engineering. Mater. Sci. 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