Quantum topology phase diagrams for molecules, clusters, and solids


  • Samantha Jenkins

    Corresponding author
    1. Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research, College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha, Hunan 410081, China
    2. Key Laboratory of Resource Fine-Processing and Advanced Materials of Hunan Province of MOE, College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha Hunan 410081, China
    • Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research, College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha, Hunan 410081, China
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The need to make more quantitative use of the total electronic charge density distribution is demonstrated in this short perspective. This is framed in the perspective of the ground breaking early work of Bader and coworkers, along with mathematicians who captured the essential nature of a molecule in a suitably compact form in real space. We see that this simple form is the Poincaré–Hopf relation for molecules and clusters and the Euler–Hopf relation in solids. Thom's theory of elementary catastrophes combined with the Poincaré–Hopf relation provides the inspiration for the new quantum topology. An alternative use of the Poincaré–Hopf relation, molecular recognition, is discussed. Quantum topology is then used to create a topology phase diagram for both molecules and solids. The author adds their perspectives of the huge potential of the quantum topology approach by demonstrating the ease with which new theoretical ideas can be generated. © 2013 Wiley Periodicals, Inc.


All of quantum chemistry is based on the solution of the many-body Schrödinger equation. This solution consists of the eigenvalue (the energies typically obtained from ab initio calculations) and eigenfunction (the wave function, where Ψ(r, t)·Ψ*(r) = ρ(r, t) and is a quantum mechanical observable).

However, the majority of existing quantum chemistry calculation methodologies that map the important features of the potential energy surface (PES) place their main focus on the eigenvalues, for example, for time dependent (TD)/density functional theory (DFT), in the form of Kohn–Sham energies, and disregard a quantitative treatment of the eigenfunctions in the form of the total charge density distribution ρ(r). As hugely successful as an energetics-driven approach to computational chemistry has been, there remain issues that need to be addressed, for example, a problem for DFT is the approximate exchange-correlation functional.1 The realization that the full Schrödinger equation solution consists of eigenfunctions as well as the energy eigenvalues leads to the understanding that to better investigate, for instance, the PES of structural isomers, a fully quantitative treatment of the eigenfunctions should also be included. Conventionally, the detail of the topology of conformers does not use quantum mechanics to guide computational searches to find isomeric conformers. In this short perspective, the author demonstrates that Bader's quantum theory of atoms in molecules (QTAIM)2 provides the basis for a search for the spanning set of isomeric topologies. QTAIM only requires the charge density ρ(r) distributions that can be obtained from either theory, for example, TD-DFT, configuration interaction (Hartree Fock (HF)/cis, complete active space self-consistent field (CASSCF), and multi-reference configuration interaction (MRCI)) or experiment, for example, X-ray diffraction3 and 1H NMR spectroscopy data.4 QTAIM works by proving that the topological condition of zero-flux (ρ(rn(r) = 0) of the charge density ρ(r) serves as the boundary condition for the application of Schwinger's principle of stationary action in the definition of an open system.

In this perspective, the author shows how the theory of elementary catastrophes has inspired a new non-Euclidean quantum mechanics-based geometry to describe the topology of molecules and clusters. We discuss some recent articles representative of current research that explore the relationship between the topology of the molecules and the phase space for both the molecular and the solid-state topologies as well as an alternative application of the Poincaré–Hopf relation; “molecular recognition.” The author's perspective on future directions is included in each subsection. The conclusions and outlook are then presented.

The Poincaré–Hopf Relation

Before a topological phase space of sets of isomers can be constructed then a satisfactory way of capturing the essential nature of a molecule into a suitably compact form in real space is needed. Early work of Bader and coworkers5 drew on the mathematical work of Collard and Hall,6 Johnson,7 and Smith8 to form a topological theory of molecular structure. This work states that for a distribution of electronic charge associated with a nuclear configuration X the topological properties can be condensed into the number of different types of critical points of ρ(r, X), that is, where the associated gradient vector ρ(r, X) ≡ r ρ(r, X) vanishes. We can identify critical points in the total charge density distribution where ρ(r) = 0 and further classify these points according to the properties of the Hessian matrix of ρ(r). Diagonalizing this matrix gives the coordinate invariant (ordered) eigenvalues λ1 < λ2 < λ3, and the critical points are conventionally labeled using the notation (ω, σ) where ω is the rank (the number of distinct eigenvalues) and σ is the signature (the algebraic sum of the signs of the eigenvalues). In three dimensions, there are four types of stable critical points; these are denoted as (3, −3) [local maxima, usually corresponding to nuclear positions, the nuclear critical points (NCPs)], (3, −1) and (3, +1) [saddle points, called bond critical points (BCPs) and ring critical points (RCPs), respectively], and (3, +3) [cage critical points (CCPs)]. The set of critical points and associated bond-paths is called the “molecular graph.”

The Poincaré–Hopf relation, a fundamental theorem of topology,6 is used universally for clusters and molecules within the QTAIM to account for all the four types of three-dimensional critical points that may be present. For molecules and clusters, the relation is expressed as:

equation image(1)

where n, b, r, and c are the numbers of NCPs, BCPs, RCPs, and CCPs, respectively. The topological complexity ∑BRC can be defined for isomers as the sum of the numbers of BCPs, RCPs, and CCPs.

A similar relation exists for the determination of the correct numbers of critical points in infinite solids modeled with periodic boundary conditions. This is sometimes referred to as the Morse relation, although the term Euler–Poincaré relation9 is also used:

equation image(2)

The Poincaré–Hopf Relation and Molecular Recognition

An interesting alternative application of the topography mapping and Poincaré–Hopf relation is offered by the scalar field of molecular electrostatic potential (MESP), denoted as V, a useful entity for describing molecular recognition as well as reactivity in chemistry.10, 11 Like QTAIM, there are four types of nondegenerate critical points, for a three-dimensional function such as MESP, at which ∇V = 0. However, due to existence of asymptotic minima as well as maxima, the Poincaré–Hopf relation for the electron density, see Eq. (1) needs to be modified. The MESPs used by Gadre and coworkers were evaluated and their respective topography mapped using their in-house developed codes.12

A suggestion for future work could be to combine the approach of using quantum topology phase diagram and molecular recognition, this could be either using as the Euler–Poincaré relation with the electron density as done in QTAIM or to use the alternate scalar field, namely MESP for this purpose. This means that we could use MESPs and ∇V to examine neighboring points on the topology phase diagram, see Figure 2.

A Non-Euclidean Geometry for Molecules Inspired by Thom's Catastrophe Theory

The molecular graph summarizes the topology of a molecule created by the critical points of ρ(r) and the concepts of geometry and dimensionality used should reflect this. Instead of relying on vague intuition from experience with Euclidean geometry, a quantum mechanically consistent description based on the critical points of ρ(r) should be used.

The notation 3-DQT, 2-DQT, or 1-DQT is used to refer to all possible molecular geometries consistent with QTAIM, following on from recent work.13, 14 The presence of a CCP is a necessary condition for a structure to be considered to be quantum topologically 3-DQT. A structure containing a CCP possesses a region that is completely bounded by bond-paths with an RCP in the plane of each enclosing face. Structures with 3-D Euclidean geometries can be either 2-DQT or 1-DQT, but without a CCP they do not contain an enclosed region in the topology of the gradient of the charge density. The six water molecule cluster labeled “PRISM_BOOK” in Figure 3g of recent work13 has an “open box” form with no “lid” or CCPs and is therefore a 2-DQT quantum topological object; structures without CCPs but with RCPs are determined to be 2-DQT and structures with neither CCPs nor RCPs are 1-DQT. A hypothetical spiral-shaped molecule could be 1-DQT in the absence of any RCPs or CCPs, strongly defying expectations from Euclidean geometry that such a molecule be a 3-D object.

The dimensionality of a molecule within the framework of quantum topology is also related to the coalescence and annihilation of critical points. If a 2-DQT molecular graph with a single RCP coalesces with a single nearby BCP through some electronic structure perturbation, then both critical points are annihilated and a 1-DQT molecular graph results. Similarly, if a 3-DQT molecular graph with a single CCP coalesces with a nearby RCP, again through some perturbation in the electronic structure, both critical points are annihilated and a 2-DQT structure results. If a 1-DQT molecular graph loses its BCPs, for example, for a dissociated dimer, then a 0-DQT molecular graph results, simply isolated unbound nuclei. It is also possible to increase the dimensionality and topological complexity, ∑BRC of a molecular graph; a string-like 1-DQT molecular graph can be rolled up to create an RCP and hence a 2-DQT molecular graph. A 2-DQT molecular graph can also be rolled up to create an CCP and hence a 3-DQT molecular graph.13

The inspiration for the new non-Euclidean quantum topology is to be found in Thom's theory of elementary catastrophes,15 which provides a mathematical model for structural changes in the neighborhood of a bifurcation point, where the critical points (ω, σ) where ω < 3 are degenerate critical points. It was Collard and Hall6 that suggested several decades ago that this theory be applied to changes in molecular graphs such as the coalescence, creation, and annihilation of critical points. Catastrophe theory is limited to the description of bifurcation catastrophes involving singularities of corank two or lower, where the corank is defined as τ = 3 − ω. An article by Bader16 invites the reader to extend the applicability of Thom's classification to include a bifurcation catastrophe involving a singularity of corank larger than two. However, unlike the catastrophe theory of molecular structures, quantum topology is not limited to the particulars of geometry of the molecular graph under consideration.

The connection between quantum topology and elementary catastrophes can be explained in terms of ellipticity. When a BCP and an RCP are on the point of coalescing and annihilating both critical points, the smallest negative eigenvalue of the BCP λ2, the most sensitive to changes in the electronic charge density, approaches zero closely. This occurs, while the positive RCP λ2 eigenvalue also approaches zero. In addition, the ellipticity ε = |λ1|/λ2 − 1, will dramatically increase, approaching infinity at the point of coalescence. When an RCP and a CCP are on the point of coalescing and annihilating, an equivalent effect occurs; the negative RCP λ1 eigenvalue and the positive CCP λ1 eigenvalue again approach zero closely. Equivalent relations to the BCP ellipticity can be introduced for the RCP eigenvalues that will be called “ring-asymmetry,” εrcp = (λ32) − 1 and for the CCP eigenvalues, the “cage-asymmetry,” εccp = (λ31) − 1.

Examples of molecular graphs that are 2-DQT and 3-DQT, are shown in Figures 1a and 1b and Figures 1c and 1d respectively for clusters of water molecules.13 Examples of 1-DQT molecular graphs include isolated water molecules and string-like hydrogen carbons, for example, alkanes, alkenes, and alkynes. There are examples of weak bonding interactions with a BCP and a nearby RCP close to coalescing, see Figures 1a 1c. On casual inspection, if one imagines that molecules are Euclidean objects then Figure 1b appears to be 3-D. However, from QTAIM, it can be seen that there are no CCPs present for this molecular graph, so it is 2-DQT. In Figure 1c, a 3-DQT molecular graph is close to becoming a 2-DQT molecular graph, because the sole CCP is very close to an RCP, see Figure 1c as an example.

Figure 1.

A selection of molecular graphs for 2-DQT and 3-DQT topologies are shown in subfigures (a and b) and (c and d), respectively. The large red spheres represent the oxygen nuclei, the white spheres the hydrogen nuclei. The small red and green spheres show the locations of the RCPs and BCPs, respectively. In subfigures (c and d) the large blue spheres correspond to the locations of the CCPs. The solid lines connecting the water molecules represent hydrogen bonding with a degree of covalent character.17, 18 The dashed lines correspond to hydrogen bonds without a degree of covalent character. (From S. Jenkins, et al., Phys. Chem. Chem. Phys. 2011, 13, 11644. © Royal society of chemistry publishing (RSC), reproduced by permission.) [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

A Spanning Phase Space of Isomer Topologies

The solutions of the Poincaré–Hopf relation of 2-DQT and 3-DQT molecular graphs for any isomer set can be considered with this new approach.13, 14 In Table 1, predictions for the form of undiscovered Poincaré–Hopf solutions for conformers are made by providing the exhaustive sets of sum rules to construct topology phase diagrams. The “forbidden” solutions are trivially recognized as molecules with a different number of nuclei from the isomer set under consideration, see Figure 2. The construction of the topology phase diagram starts by considering the least complex structure, that is, with the lowest value possible of the topological complexity ∑BRC, where there are one or more RCPs and the numbers of NCPs and BCPs are equal. Next, the least topologically complex compact regular polyhedron is searched for as the minimum condition for an upper bound of the topology phase diagram, see Figure 2. In Table 1, it can be seen that although only odd values of ∑BRC are permitted for even numbers of NCPs, only the (2n, 2n, 2n) and (2n + 1, 2n + 1, 2n) combinations are forbidden. The converse is true for odd numbers of NCPs. Parallel diagonal bands of allowable topologies that intersect at NCP = BCP correspond to the solutions of the Poincaré–Hopf relation with zero CCPs. The successive parallel bands correspond to CCP = 1, 2, 3… Note that although in previous work the “Missing” label is applied to some of the less complex topologies, for example, for (BCP = 18, RCP = 2), (BCP = 19, RCP = 3), and (BCP = 20, RCP = 4),13 it is more likely that instead these topologies should be labeled as “Unstable.” As it is unlikely that molecular graphs with the lowest topological complexity ∑BRC would be stable if they included a CCP, only molecular graphs with ∑BRC greater than that of the stable structure at (BCP = 21, RCP = 5) are feasible.

Figure 2.

Phase diagrams of the solutions of the Poincaré–Hopf relation and the sum rules of the topological complexity, ∑BRC for the six molecule water cluster W6. Permitted topologically and energetically stable solutions are denoted by black circles or red squares in the electronic version. The upper left plot area marked with asterisks denotes the region of topological instability based on there being increasingly more CCPs. The region marked as forbidden is excluded, as no valid solutions exist for the Poincaré–Hopf relation. In addition, we exclude regions with no RCPs, as these solutions to the Poincaré–Hopf relation correspond to string-like 1-DQT structures as opposed to clusters. (From S. Jenkins, et al., Phys. Chem. Chem. Phys. 2011, 13, 11644. © Royal society of chemistry publishing (RSC), reproduced by permission.) [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Table 1. Permitted combinations of even (2n) and odd (2n + 1), (2n − 1) numbers of critical points and their sum the topological complexity, ∑BRC, for conformers, for NCP = 2n or NCP = 2n + 1.
  1. It is assumed that BCP ≥ NCP as we consider only 2-DQT and 3-DQT molecular graphs.

2n2n2m − 12n2n + 1
2n2n2m2n + 12n + 1
2n2n + 12m2n2n + 1
2n2n + 12m + 12n + 12n + 1
2n + 12n2m +12n + 12n
2n + 12n2m2n2n
2n + 12n + 12m2n + 12n
2n + 12n + 12m − 12n2n

One use of the topology phase diagram is to clearly indicate when a preference for 2-DQT and 3-DQT molecular graphs, respectively. It is known that clusters of four (W4) and five (W5) water molecules possess an energetic preference for planar structures whilst six (W6) water molecule clusters are most energetically stable in a compact arrangement. From the topology phase diagrams and the calculated energies of the W4 and W5 clusters, it can be seen that the most stable structures have the sparsest possible topologies; NCP = BCP and both possess 2-DQT topologies. However, the most energetically stable W6 cluster possesses a 3-DQT molecular graph and the corresponding point on the topology phase diagram (BCP = 21, RCP = 5) resides mid-way along the BCP axis between the most sparse and most compact topologies.

The next practical step forward for the topology phase diagram will be a computational implementation to generate molecular graphs from the associated topology phase diagram. A suggested start is to begin with the sparsest possible molecular graph and use Monte Carlo methods to generate molecular graphs corresponding to neighboring points in the topology phase diagram. An alternative procedural method would be to use a “greedy”' algorithm, that is, start with a ring and then find a way to increase the number of BCP's by one and then repeat.

Enclosing Phase Boundaries for the Solid State

Further nonmolecular/cluster applications of quantum topology could include phase transitions in solids19 or as a tool for the discovery of new allotropes.20–22 A simple example of such an application, a variety of phases of water ice23–25 subjected to varying pressure, is presented in this perspective, see Figure 3 and the accompanying figure caption. The phase diagram is constructed by using Eq. (2) to determine consistency in the numbers and types of critical points. Next, one calculates the ratio RCP/BCP = 1, for example, for ice Ic this slope is drawn and the specific data (BCP = 16, RCP = 16) point added, where other points would simply correspond to integer multiples (periodic repetitions) of the primitive unit cell.

Figure 3.

Phase diagram for solid ice phases, ice Ih, ice Ic, ice IX, ice VI, ice XI, and ice VIII shown by dotted and solid lines. Additionally, the changing topology of a pressure-induced phase transition is shown by the dashed lines for the evolution of ice XIh at 0 kbar to an ice VIII-like phase at 100 kbar. The fine grid is added as a guide to the eye. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

As different phases of ice (or any other solid state substance under study) will have differently sized primitive unit cells, we regard only the slope as having meaning and not the position of the data point. The solid and dotted lines plotted in Figure 3 can be considered to separate out the phase boundaries, whereas the dashed lines separate out the pressure boundaries. It can be seen for the solid state that instead of uniquely defining a topology at each point, a slope RCP/BCP is instead defined. This is due to differences in the primitive unit cells that will naturally occur for different phases of a solid. Larger super-cells enable the details of phase transitions to be followed in more detail as can be seen from the increasing separation between adjacent phase and pressure boundaries.

Another application could be to define a measure of “crystal recognition,” as opposed to molecular recognition, using QTAIM. This could work, for example, by examining the angle of separation of the RCP/BCP slopes during a phase change. There may be something we can learn from the rate at which the slope RCP/BCP changes. In the case of water ice, there is a known phase transition that occurs in the applied pressure 80–85 kbar range24 for ice XI where the slope RCP/BCP = 1 for 80 kbar and RCP/BCP > 1 for a pressure of 85 kbar, see Figure 3. Therefore, a suggested threshold that is characteristic of a transition is RCP/BCP > 1. Pressure increases will cause more RCPs to form relative to the number of BCPs. If we use the same size super-cells either side of, or through, a transition of interest, the position of the data points can be continually compared to see how much the relative separation of the data points changes as the slope RCP/BCP varies.

The final suggestion for future work is to understand Linus Pauling's26* rules for determining the crystal structures of complex ionic crystals specifically in terms of the Euler–Poincaré relation for solids, see Eq. (2), within the quantum topology framework. In particular of the five rules,26* first, third, fourth, and fifth rules could be considered, the second rule being the exception as it requires the calculation of both the charge of the anion integrated over the atomic basin and the strength of electrostatic bonds reaching the anion, which is outside of the scope of this approach. However, we can apply the Euler–Poincaré relation within the quantum topology framework to the remaining four rules by developing quantum topology analogs. For instance, for the fifth rule we can quantify the local environment using the Euler–Poincaré relation by considering the solid as being built from rings and cages. We can illustrate this for molecules, but the principle is equally applicable to solids; within QTAIM the numbers of nuclei associated with each ring or cage is precisely determined. Once we can quantify the local environments of each cage, and ring then, we can consider the different types of rings and cages of the periodically repeating unit of the solid contains.

Conclusions and Outlook

In this short perspective, the Poincaré–Hopf relation is used to construct a topology phase diagram to locate a spanning set of unique topologies and has been discussed in the context of the new non-Euclidean geometry and the historical background that inspired it. There is a huge scope for developing the theory of the topology of QTAIM (which was started in the mid-1970s, abandoned in the early 1980s and only very recently revisited) for both molecules and solids. In this short perspective, the author has attempted to show the huge potential in developing the general area of quantum topology by:

  • Introduction of ring-asymmetry, εrcp = (λ32) – 1 and for the CCP eigenvalues, the cage-asymmetry, εccp = (λ31) – 1. This would also be applicable to the solid state as well as to molecules and clusters.

  • Introduction of the solid-state topology phase diagram, and the idea of using the RCP/BCP slope to characterize physical phase transitions.

In addition, the following proposals are made:

  • Combining the approach of using the quantum topology phase diagram and molecular recognition with MESPs.

  • Explore the hierarchy of critical points in electron momentum density by the use of Poincare–Hopf relation and the topology phase diagram.

  • Rules to generate allowable topologies could inform automated searches.

  • The use of solid-state topology phase diagrams to understand physical phase transition phenomena.

  • Probe folding behavior; computer generation of spanning topology phase diagrams of a particular amino acid residue to discover “missing” residue topologies.

  • Revisit Linus Pauling's first, third, fourth, and fifth rules for determining the crystal structures of complex ionic crystals within the quantum topology framework.


S.J. gratefully acknowledges the support of the One Hundred Talents of Hunan program and the aid program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province. Thanks also to the useful comment from S. Gadre on the nature of the MESPs and their use in molecular recognition, also an anonymous referee for helpful comments.

  1. 1

    First rule: a coordinated polyhedron of anions is formed about each cation, the cation–anion distance determined by the sum of ionic radii and the coordination number (C.N.) by the radius ratio. Second rule: the electrostatic valence rule. Third rule: the sharing of edges and particularly faces by two anion polyhedra decreases the stability of an ionic structure. Fourth rule: in a crystal containing different cations, those of high valency and small coordination number tend not to share polyhedron elements with one another. Fifth rule: the rule of parsimony: the number of essentially different kinds of constituents in a crystal tends to be small.

Biographical Information

original image

Samantha Jenkins obtained her PhD in 2000 for a thesis on computational ice physics. She then undertook postdoctoral fellowships at Sussex University and then McMaster University. She then moved to Sweden, being awarded a Docent in chemical physics in 2006 while obtaining a track record of industrially sponsored research funding. In 2010, she moved to Hunan China where she is now a professor at the College of Chemistry and Chemical Engineering, Hunan Normal University, where she has gained funding from the National Natural Science Foundation of China. She is a recipient of the 100 Talent Award of Hunan. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]