### Abstract

- Top of page
- Abstract
- Introduction
- Pseudosymmetry, Quasisymmetry, and CSMs
- Geometric CSMs
- Generalized CSMs
- Application of CSMs in Quantum Chemistry
- Conclusions
- Acknowledgments
- Biography

The introduction of symmetry-related arguments in scientific theories since its early development has led to an apparent paradox: highly symmetric models are routinely being applied to situations where this symmetry is not present (or only present in an approximate way). In this article, the formalism of continuous symmetry measures is presented as a solution to this situation, making a special emphasis on its application to problems related to quantum chemistry. © 2012 Wiley Periodicals, Inc.

### Introduction

- Top of page
- Abstract
- Introduction
- Pseudosymmetry, Quasisymmetry, and CSMs
- Geometric CSMs
- Generalized CSMs
- Application of CSMs in Quantum Chemistry
- Conclusions
- Acknowledgments
- Biography

Everyone has some intuitive idea of what symmetry is: we recognize bilateral symmetry in our bodies, we enjoy the rotational symmetry of many flowers, admire the fragile regularity of snowflakes, and are fascinated by the complex but at the same time simple polyhedral shape of crystals. As we encounter plenty of symmetrical objects in our everyday life, it should be no surprise that symmetry arguments have entered, either consciously or not, as a basic ingredient in the conceptual models developed to explain the physical world. Symmetry acquired its basic place in science through the deep influence of Greek philosophy in western culture. Prominent examples of this important role of symmetry along the history of science are, for instance, Plato's attempt to relate the properties of matter with those of the highly symmetrical regular solids,[1] the polyhedral model of the universe described by Kepler in his *Mysterium Cosmographicum*,[2] or the actual models of particle physics, all of them relying heavily on symmetry-based arguments.[3]

The influence of symmetry in science has been profound at all levels, not only in theories trying to explain the overall behavior of nature. The definitive boost to this tendency came at the end of the XIX century with the formalization of symmetry through the development of group theory.[4] In this respect, crystallography was probably the first scientific field where group theory was applied systematically to rationalize and explain the origin of the geometrical regularity in crystals which turned out be strongly related to the behavior of matter at the atomic level. These ideas permeated into related fields such as chemistry where the postulate of van't Hoff[5] and Le Bel,[6] that the four valencies of carbon where arranged in a tetrahedral disposition opened definitively the door for applying symmetry in chemistry. The prominent role of symmetry in modern chemistry came with the development of quantum mechanics and its application to explain chemical bonding. In modern quantum chemistry, symmetry is not only applied to molecular geometry but also to molecular vibrations, operators, wavefunctions, or to molecular orbitals. For instance, simple arguments based on the symmetry of molecular orbitals along a reaction path allow the classification of certain reactions as being either symmetry-allowed or forbidden. The proposal of the Woodward-Hoffmann rules and the concept of “Conservation of Orbital Symmetry,”[7] some 40 years ago established in this sense an indissoluble link between theoretical chemistry and symmetry that led to the introduction of group theory and its applications in any modern standard chemistry curriculum.

From the point of view of a modern chemist with access to the wealth of structural information contained in data bases, it should be remarkable that these theories derived for idealized symmetric molecular geometries can be applied without major revisions to obtain rational explanations for many experimental observations, even if we know that the vast majority of molecules occurring in nature or artificially created in a laboratory have no symmetry at all. It is precisely when trying to understand this apparent contradiction that one of the major shortcomings of the traditional approach to symmetry becomes evident: symmetry has been usually defined as a “black or white” property, that is, a molecule either has a certain symmetry or not. In this respect, small atomic displacements with an imperceptible influence in the physical properties are, however, sufficient to destroy its symmetry. A possible solution to this problem is based on the generalization of the concept of symmetry to describe it as a continuous property allowing a progressive series of “gray shades” between the “black” or “white” situations.

### Pseudosymmetry, Quasisymmetry, and CSMs

- Top of page
- Abstract
- Introduction
- Pseudosymmetry, Quasisymmetry, and CSMs
- Geometric CSMs
- Generalized CSMs
- Application of CSMs in Quantum Chemistry
- Conclusions
- Acknowledgments
- Biography

A central question that needs an answer before trying to incorporate the lack of symmetry of the real world to science is indeed the real essence of symmetry. What is in reality symmetry, a property that every one is able to recognize in their environment although the existence of perfectly symmetric objects is practically impossible? In a recent book on the intimate relation between science and symmetry,[8] J. Rosen defines symmetry as the immunity of a situation to a possible change. According to this definition, when we have a situation for which it is possible to make a change under which some aspects of the situation remain unchanged, then this situation is said to be symmetric under the change with respect to those aspects. This broad definition of symmetry does not only apply to the most common idea of symmetry, that of geometrical symmetry of a physical object, where the situation would be the shape of that object, and the possible changes any shape-conserving transformation (translations, rotations, reflections, or isotropic scaling), but it does also apply to other more abstract kinds of symmetries relevant in modern physics. In this sense, the situation could be the realization of an experiment and the possible changes could be the performance of the same experiment at different locations, at different times, or in different states of motion. In these cases, the immunity of the situation to a possible change would be obtaining the same result in the experiment regardless of the specific conditions of the experimental setup.

Let us, however, focus our interest in purely geometrical symmetry and for this it will be useful to discuss a simple example: we perceive the human body to have bilateral symmetry as we consider that it conserves its appearance when it is reflected through the (imaginary) plane separating the body into two similar halves. This definition introduces some interesting concepts that are essential to reconcile the apparent contradiction between the use of symmetry in our explanation of the world and its practical absence in the “real” world. On one hand, to have a symmetric situation it must be possible to perform a change, although this change does not actually have to be performed, and on the other hand, we must judge if some aspects of the situation would remain unchanged if the change was performed. Note that the situation needs only to be immune to change in some aspects. In this sense, we say that the human body has bilateral symmetry even if we definitively know that if we focus our attention on the distribution of its internal organs this is certainly not the case. The other important question related to our perception of symmetry is that we must judge the similarity in the situation before and after the change. In this sense, our recognition of symmetry depends strongly both on the precision of our observations and on our capacity to separate essential features of the situation from symmetry breaking details that we consider to be merely accidental. In this respect, although we state that the human body has bilateral symmetry, we are in reality meaning that it has approximate bilateral symmetry or that it is quasisymmetric with respect to mirror symmetry.

To precisely define quasisymmetry, we must introduce the concept of an approximate symmetry transformation, by which we mean a transformation that changes every state of a system to a state that is nearly equivalent to the original state. In the classical treatment of symmetry, for a given object the answer to its degree of a determined *G*-type symmetry is dichotomic, that is, the object is either *G*-symmetric or it is not. The way of allowing approximate symmetry in our theories is to find a metric for a set of states of the systems that gives a quantitative answer to the question of what does “nearly equivalent” mean. This metric is nothing else than a distance between every pair of states such that a null distance means equivalence and positive distances represent increasing degrees of nonequivalence. Although there is not a unique way of defining this metric, we will center our attention in one of such approaches, the so-called continuous symmetry measures (CSMs) proposed by Avnir and coworkers[9] in 1992.

Once we have introduced a metric that measures the degree of symmetry content in an object, we can extend the concept of quasisymmetry to introduce that of false symmetry or pseudosymmetry, which makes reference to the use of a given symmetry to study aspects of an object which obviously does not have this symmetry. In this sense, quasisymmetry is just a particular case of pseudosymmetry in which the studied object does not have the desired symmetry, but it is close to have it.

CSMs were originally formulated to deal with the problem of quasisymmetry in the geometry of molecular configurations defined as a set of points corresponding to the position of atomic nuclei, but it evolved soon to a more general formulation[10, 11] that allowed the consideration of more complex mathematical objects of common use in quantum chemistry such as operators, matrices, or wave functions.[12] Although the focus of this perspective is in this later development, we find it instructive to start our presentation of CSMs in their original geometric version and then generalize them for other purposes.

### Geometric CSMs

- Top of page
- Abstract
- Introduction
- Pseudosymmetry, Quasisymmetry, and CSMs
- Geometric CSMs
- Generalized CSMs
- Application of CSMs in Quantum Chemistry
- Conclusions
- Acknowledgments
- Biography

As mentioned above, we will talk about pseudosymmetry when the configuration (given by the frozen positions of the set of atomic nuclei) of a molecule can be treated as a distorted configuration of higher symmetry. Mathematically speaking, a molecule with a configuration **Q** belonging to a symmetry point group *G* is pseudosymmetric if the positions of the nuclei in **Q** can be described by with being a set of displacements and a set of virtual atomic positions corresponding to a configuration **P** that has a higher symmetry described by point group *G*^{°}. In this case, we will say that molecule **Q**, with real symmetry *G*, is also pseudo *G*^{°}-symmetric or, in other words, that *G*^{°} is a pseudosymmetry group for molecule **Q**. Following this line of reasoning, we will say that a pseudosymmetry turns into quasisymmetry if the displacement vectors are sufficiently small. In the definition of pseudosymmetry given above, although the actual symmetry group *G* of the molecule is unique, the pseudosymmetry group *G*^{°} can be arbitrarily chosen, although it is obvious that the use of pseudosymmetry to analyze a problem will only be judicious if there is some geometrical relation between the problem structure **Q** and the reference structure **P**. By this, we mean that while it is sensible to use the D_{3h} group to study some properties of a molecule with a triangular configuration even if this triangle is not a perfect equilateral triangle, it does not make to much sense to use the D_{3h} pseudosymmetry for a linear trinuclear molecule, even if there is no mathematical contradiction in doing it.

The basic idea behind the CSMs[9-11] is to use the distance between and to gauge the *G*^{°} symmetry content of configuration **Q**. Following this idea, the CSM for an object **Q** relative to a reference symmetry point group *G*^{°}, abbreviated S(**Q**, *G*^{°}), is defined as:

- (1)

where correspond to the coordinates of the geometrical center of **Q** and *N* is the number of vertices. For this expression to correspond to a measure of the *G*^{°} symmetry contents of **Q**, we can, however, not take any arbitrary *G*^{°}-symmetric configuration. The minimization process indicated in Eq. (1) refers to the fact that this expression corresponds only to a symmetry measure if are the coordinates corresponding to **P**, the *G*^{°}-symmetric configuration closest to **Q**. The mathematical requirements involved in this minimization process, discussed extensively in the literature,[9-11] can be summarized stating that we must minimize S(**Q**, *G*^{°}) with respect to the relative position, orientation, and size of **Q** and **P**, and, as the two configurations have been defined as a set of *N* vertices, we will need also to minimize S(**Q**, *G*^{°}) with respect to all relative vertex labelings. The factor 100 in Eq. (1) is simply introduced for convenience.

It can be shown that S(**Q**, *G*^{°}) can only take values between 0 and 100. A value of S(**Q**, *G*^{°}) = 0 means that **Q** is invariant under the symmetry operations of *G*^{°}, while increasingly larger values of S(**Q**, *G*^{°}) indicate that **Q** progressively departs from having *G*^{°} symmetry. In the language of CSMs, a given molecular configuration **Q** is nearly *G*^{°}-symmetric when the corresponding S(**Q**, *G*^{°}) value is sufficiently small. As a rule of thumb, for molecules, we can consider that it is sensible to consider a configuration **Q** to be G^{°}-quasisymmetric for values of S(**Q**,*G*^{°}) below 1.

As an illustration of the potential usefulness of Geometric CSMs and the closely related continuous Shape Measures (CShMs)[10, 11] in stereochemistry, let us consider a straightforward application, that is the problem of deciding the best geometrical description for the coordination environment of an atom. Although it is relatively easy to assign a tetrahedral geometry for the disposition of the four atoms bonded to an sp^{3} carbon atom in organic compounds, the situation is not straightforward for coordination compounds, especially when higher coordination numbers are considered.

Let us take a simple example. From visual inspection, the coordination environment of cobalt in [Co{PC_{6}H_{5}(EtS)_{2}}_{2}] seems to correspond to a distorted octahedron (Fig. 1). One simple question is to quantify the degree of distortion, that is, the departure of the actual coordination polyhedron from a perfect octahedron. The answer to this question is straightforward in the language of CSMs: S(**Q**, Oh) = 5.61. Of course to get a feeling of the meaning of this result, it is necessary to compare it with other compounds. The use of CSMs and CShMs allows us to directly compare the symmetry (shape) contents of a same structure with respect to different references. In this case, it is obvious that a plane hexagonal coordination environment of Co is not adequate, and the corresponding large symmetry measure confirms this. It is, however, interesting to note that if we use CShMs to test if a description of the coordination environment using a trigonal prism as the reference could be adequate we get an unexpected answer: the trigonal prism is not only adequate but also even better than the octahedron for this case were our intuition is mislead by the visual representation of the molecular configuration.

Geometric CSMs and CShMs have been successfully applied to a variety of stereochemical problems such as the description of the coordination environement,[13-20] chirality,[21-23] and reactivity of chemical compounds.[24-26] The usefulness of these measures is especially evident when they are applied in conjunction with a statistical analysis to a large set of structural data obtained from a search in databases such as the Cambridge Structural Database.[27] In all these studies, CSMs are applied to molecular configurations defined by a set of points corresponding to the equilibrium positions of the nuclei in the molecule. To extend the power of the CSMs formalism to chemistry-related problems, it is, however, necessary to go beyond this simple description of a molecule. Geometric CSMs do not give satisfactory results when we try to apply them to molecules that differ in the substitution of some atoms. If we try to compare, for example, the degree of hexagonality of benzene and fluorobenzene, the D_{6h} CSM applied to the geometric structure of fluorobenzene indicates just a minor distortion, basically due to the a larger C—F distance. A comparison of the electronic density for both molecules shows, however, that the substitution of one H atom by F is inducing changes in the rotational symmetry of the molecule that are by far more important than those predicted by the geometric CSM. A seminal contribution in the direction of extending the usefulness of CSMs in chemistry was been given by Dryzun and Avnir in a recent paper[12] where the concept of a CSM is generalized to deal with mathematical objects beyond a simple set of points in space, showing how to calculate CSMs for matrices, vectors, or functions such as the electron density or molecular orbitals.

### Generalized CSMs

- Top of page
- Abstract
- Introduction
- Pseudosymmetry, Quasisymmetry, and CSMs
- Geometric CSMs
- Generalized CSMs
- Application of CSMs in Quantum Chemistry
- Conclusions
- Acknowledgments
- Biography

As mentioned above, CSMs are defined on the basis of the distance of a given object from having certain symmetry. For a given symmetry group *G*, we define an operator , the symmetry-group operator, that applies all *h* symmetry operations of *G* to an object and then averages the results:

- (2)

where are unitary operators representing the symmetry operations in *G*. If we consider a general mathematical object |Ψ〉 such as a matrix, a vector, or a function that represents a point or a set of points in some metric space, and a symmetry point group *G*, then we can define the CSM for |Ψ〉 with respect to group *G* as:

- (3)

where a minimization over the possible orientations of the symmetry operations of *G* must be performed. In this minimization, it is necessary to impose a fixed relative orientation of the different symmetry elements of the group to retain its group structure. Equation (3) always returns values between 0 (if the object is *G*-symmetric) and 100. The application of the general definition of the CSM in Eq. (3) to obtain particular expressions for the CSM for different mathematical entities |Ψ〉 such as vectors, matrices, or functions has been extensively treated in the literature[12, 28-31] and will not be discussed here in detail. When Eq. (3) is applied to an object described as a set of points in a Euclidean space, it is easy to show that it simplifies to give Eq. (1), showing that geometrical CSMs are nothing else than a particular case of the generalized CSMs.

Before showing some of the possible applications of CSMs in quantum chemistry, let us discuss some interesting features of the CSMs that appear when trying to use them to measure the symmetry contents of molecular orbitals. The relevant equation in this case is

- (4)

where are the operators that transform the MO under the symmetry operations of *G*, and if the orbital is normalized, as it is usual, 〈φ|φ〉 = 1. Following this definition, a given orbital is *G*-symmetric when *S*(φ,*G*) = 0. If the orbital is not fully *G*-symmetric, then *S*(φ,*G*) > 0 with the maximum value *S*(φ,*G*) = 100 indicating that the orbital is totally devoid of the symmetry of *G*. Note that as the symmetry operator implies an average of all symmetry operations, *S*(φ,*G*) is a measure of the loss of *G* symmetry as a whole. By this, we mean that we may have different orbitals with the same *S*(φ,*G*) > 0 value but the departure from the ideal *G* symmetry may have different origins, with the symmetry descent due to the loss of different sets of symmetry operations.

The expectation values for each individual symmetry operation, , play a crucial role in the CSMs formalism and they have been termed symmetry operation expectation values (SOEVs).[28] If the molecule is exactly *G*-symmetric, the SOEV for a nondegenerate orbital may be either 1 (if the orbital is symmetric with respect to the symmetry operation) or −1 (if it is antisymmetric). If *G* is only a pseudosymmetry, the values of the SOEVs will lie in the range between −1 and 1. For small symmetry-breaking distortions, approximately “symmetric orbitals” will have SOEVs close to 1, whereas approximately “antisymmetric” ones will have SOEVs close to −1. The case of degenerate orbitals is somewhat more involved. As a given symmetry operation will, in general, transform each orbital into a linear combination of the degenerate set, for an orthonormalized set of degenerate orbitals, the SOEV will be ±1 only for those operations leaving the orbital unchanged or just changing its sign, but in the general case the SOEVs for degenerate MOs will adopt values between −1 and 1 even in the case that the molecule has the full symmetry of *G*.

It has been shown that a convenient way to proceed when applying the CSMs formalism to molecular orbitals is to define a measure for each irreducible representation of the pseudosymmetry group *G*.[31] The group symmetry operator in Eq. (2) is nothing else than the projection operator corresponding to the totally symmetric irreducible representation of *G*. Generalizing this relation, it is possible to define a CSM associated with each irreducible representation Γ of the group by substituting in Eq. (3) the expectation value of by that of the projection operators corresponding to each irreducible representation:

- (5)

where χ_{Γ}^{*} (*R*) is the character corresponding to operation R in the irreducible representation Γ of G. Equation (5) is strictly valid only for unidimensional representations. The treatment of degenerate representations is somewhat more involved[31] and will be omitted here.

### Application of CSMs in Quantum Chemistry

- Top of page
- Abstract
- Introduction
- Pseudosymmetry, Quasisymmetry, and CSMs
- Geometric CSMs
- Generalized CSMs
- Application of CSMs in Quantum Chemistry
- Conclusions
- Acknowledgments
- Biography

An example of the application of CSMs in quantum chemistry is given by the analysis of the influence of the electronegativity difference in a simple chemical bond.[32] The well-studied transition from a covalent to a ionic bond as the electronegativity difference between the two atoms is increased can be easily translated into a problem where the initial centrosymmetric charge distribution is progressively distorted as a consequence of the electronegativity difference. Using a simple Hückel model for the chemical bond (Fig. 2), it is easy to obtain inversion symmetry measures for the Hamiltonian (represented by a matrix), the molecular orbitals, and the charge density as a function of the energy difference between the energies of the two atomic orbitals involved in the formation of the bond.

If we consider the inversion center located on the middle of the A-B bond, the effect of the inversion operator is to switch the position of the two orbitals, locating φ_{A} on atom B and φ_{B} on atom A. As in the Hückel model, all orbitals in the basis set are considered to be equal except for their location in space, in this case the inversion operation just swaps the two orbitals. The inversion CSM for the 2 × 2 Hamiltonian matrix turns out to depend on both the electronegativity difference, 2δ, and the interaction energy, β:

- (6)

If the two atoms are identical then δ = 0 and *S*(**H**, *C*_{i}) = 0, regardless of the value of β. As the difference between the energy of the two atomic orbitals (2δ) increases, for a given value of the interaction energy β, the inversion measure becomes larger (see Fig. 3a) which means that the system progressively looses inversion symmetry as δ increases. Note that in the limiting case of two atoms with a high energy difference (δ >> β) or in the case of noninteracting atoms (β = 0), the inversion measure for the Hamiltonian is always *S*(**H**, *C*_{i}) = 100, indicating a total loss of inversion symmetry, a result that is fully consistent with the idea that two different, unrelated atoms are not equivalent—they cannot be interchanged by a symmetry operation.

The calculation of the inversion symmetry measure for the two MOs gives:

- (7)

where the negative sign corresponds to the bonding combination φ_{1} and the positive one to its antibonding counterpart φ_{2}. For a homonuclear diatomic molecule with δ = 0, Eq. (7) indicates that the inversion content of the bonding orbital is *S*(φ_{1}, *C*_{i}) = 0 (it is fully centrosymmetric) while that for the antibonding orbital is *S*(φ_{1}, *C*_{i}) = 100 (it is totally devoid of inversion symmetry). As the value of δ increases, the inversion symmetry of the Hamiltonian is gradually lost and, as a consequence, the inversion symmetry measure of the bonding orbital increases and that of the antibonding orbital decreases (Fig. 3b). For any value of the interaction energy β, for sufficiently large values of δ, the inversion of both orbitals reaches a value of 50. This limit corresponds to a purely ionic bond (a situation that is also achieved for a vanishing interaction energy β) where the “bonding” orbital φ_{1} = φ_{B} is identical with the atomic orbital on the more electronegative atom and the “antibonding” orbital φ_{2} = φ_{A} coincides with the atomic orbital on the more electropositive atom.

A second example I would like to present is an analysis of how the rotational symmetry of the π-electron system in benzene is progressively lost when a varying number of CH groups is replaced by N (Fig. 4a).[33] The analysis is based on the *C*_{6} symmetry measure for the ground state electron density. As it has been shown that, in general, symmetry measures for the electron density are only slightly affected by the computational method[30] we used in this case the HF/STO-3G level of theory to lower the computational cost.

The analysis of the calculated symmetry measures as a function of the number *m* of N atoms (Fig. 4b) reveals some interesting results. As it could intuitively be expected, the symmetry loss increases with the number of substitutions, and a maximum symmetry breaking for the cases with 3 CH groups and 3 N atoms is more or less apparent, although this symmetry breaking is of the same magnitude as that calculated for rings with two N atoms. A totally unexpected result that is evident in Figure 4b is that the differences in the symmetry measure for the different substitution patterns for a given C to N ratio are small, so that the symmetry content for the electron density depends practically only on the number of nitrogen atoms in the system and not on their spatial distribution. A second point that is also not intuitive and only revealed by the use of a quantitative tool such as the CSMs is that the symmetry loss induced by the replacement of a CH group by N in benzene is somewhat larger than when a N atom is being replaced by CH in hexazine, an effect that is also evident comparing the two sets with 2 and 4 N atoms. This behavior is easily rationalized using a Hückel model. As we have seen in the example above, the symmetry breaking in a single bond does not only depend on the electronegativity difference between the two atoms (in the present case the electronegativity difference between N and C) but also on the interaction energy. In other words, the effect of a symmetry breaking perturbation depends not only on the strength of the source of asymmetry but also on the resistance of the symmetric system to symmetry breaking. Analyzing the off diagonal elements of the Hückel Hamiltonian, we find that the average interaction energy in pyridine, β_{av} = −7.17 eV, is lower (in absolute values) than that in pentazine, β_{av} = −7.93 eV, and, hence, it should be no surprise that the effect of a CH/N substitution on the rotational symmetry of the system is larger in the former case.