Continuous Symmetry Measures for ComplexSymmetry Group

Chaim Dryzun*

Symmetry is a fundamental property of nature, used extensively

in physics, chemistry, and biology. The Continuous symmetry

measures (CSM) is a method for estimating the deviation of a

given system from having a certain perfect symmetry, which

enables us to formulate quantitative relation between symmetry

and other physical properties. Analytical procedures for calculat-

ing the CSM of all simple cyclic point groups are available for

several years. Here, we present a methodology for calculating

the CSM of any complex point group, including the dihedral,

tetrahedral, octahedral, and icosahedral symmetry groups. We

present the method and analyze its performances and errors.

We also introduce an analytical method for calculating the CSM

of the linear symmetry groups. As an example, we apply these

methods for examining the symmetry of water, the symmetry

maps of AB4 complexes, and the symmetry of several Lennard-

Jones clusters. VC 2014 Wiley Periodicals, Inc.

DOI: 10.1002/jcc.23548

Introduction

“Since the beginning of physics, symmetry considerations have

provided us with an extremely powerful and useful tool in our

effort to understand nature. Gradually, they have become the

backbone of our theoretical formulation of physical laws.”[1]

These words of the Nobel winning physicist Tsung-Dao Lee

summarize the common conception regarding symmetry

among scientists. Symmetry is a fundamental property of nature

and it is widely used to get a simple and elegant description of

various systems.[2–6] Symmetry considerations can be found in

the heart of all major physical theories.[3] Symmetry is a limiting

factor, and the existence of symmetry implies the existence of

conservation laws.[3–5] Mathematically, symmetry is represented

using the strict laws of group theory, in which objects are cate-

gorized by the appropriate symmetry groups.[6] An object can

belong to a certain symmetry group or not, which means that

any symmetry-based analysis is only qualitative: if the system

possesses certain symmetry, then a certain physical quantity is

conserved, but if the system is not perfectly symmetric, the

only observation we can make is that the related physical prop-

erty can vary.[3–5] To get quantitative symmetry–property rela-

tion, perturbation theory must be used to assess the size of the

distortion from perfect symmetry. Several methodologies were

suggested during the years for measuring the symmetry con-

tent of a given system.[7]

One common method for estimating the symmetry content

of a given system is the continuous symmetry measures (CSM)

introduced by Zabrodsky et al.[8] This symmetry measure is

defined as the normalized square of the distance between the

original object and the closest symmetrical object. The CSM

methodology was used to investigate the symmetry depend-

ence of different systems in various areas of science for more

than two decades.[9] A few years ago, the method was gener-

alized, enabling the treatment of various mathematical descrip-

tions.[10] This generalization included the explicit analytical

expression for the closest symmetrical object in terms on the

symmetry elements of the symmetry group. For structures

which are described using sets of vectors, analytical solutions

were given for all the simple cyclic symmetry groups.[11] For

more complex groups, the exact expression is too complex to

be solved analytically. A specific numeric solution for the Cnv

and Dn groups was described recently,[12] but this solution

cannot be applied to other complex symmetry groups.

In this article, we introduce a different (although related)

approach. Instead of trying to minimize the measure with

respect to all the symmetry elements of the symmetry group

at the same time, we treat each symmetry element sepa-

rately, with constraints on the relative position of the ele-

ments. This general approach allows us to treat all the

complex symmetry point groups under the same formalism

(except the linear groups, which will be treated separately).

We will present the basic equations of the CSM and intro-

duce our approach in details. As the method is based on

some approximations, we will discuss the error causes and

compare our results with more precise methods whenever it

is possible. To complete the picture, we also introduce meth-

ods for calculating analytically the CSM for the linear symme-

try group, C1v and D1h. We will apply these methods for

calculating the symmetry of water, AB4 type complexes, and

Lennard-Jones clusters (LJ clusters).

Theory

Background

The original paper presenting the CSM methodology,[8]

referred to an object which is described by a set of vectors: �Q i

C. Dryzun

Department of Natural Sciences, The Open University of Israel, Raanana,

43107, Israel

E-mail: chdnew@yahoo.com

VC 2014 Wiley Periodicals, Inc.

748 Journal of Computational Chemistry 2014, 35, 748–755 WWW.CHEMISTRYVIEWS.COM

FULL PAPER WWW.C-CHEM.ORG

(i 5 1, 2,. . ., N). The symmetry measure for this object, with

respect to a given G symmetry group, is defined as:

S Gð Þ5100 � j�Q i2 �W ij2

j�Q i2�Q0j2(1)

where �W i is a set of vectors representing the coordinates of

closest object which is G-symmetric (aka, the geometric dis-

tance between the structures is minimal) and �Q0 is the center

of mass of the original structure. The outcome is a number

between 0 and 100, where S Gð Þ50 indicates that the original

object is perfectly G-symmetrical, and the value of the mea-

sure increases as the original object departs from perfect

G-symmetry.

The heart of the methodology, and the main computational

challenge, is the algorithm for finding the coordinates of the

closest symmetrical object, �W i , which is unknown a priori[8]

(The exception is the continuous shape measures (CShM)[13] in

which the original structure is compared to a predefined struc-

ture—comparison of the different approaches can be found in

Supporting Information). A few years ago, the CSM was gener-

alized and a general algorithm for finding the closest symmet-

rical object was presented.[10] In this formalism, the system,

jWi, is represented as a point (or as a set of points) in a metric

space. The symmetry group is represented by an orthogonal

projection operator, G, which operates all the symmetry opera-

tion of the relevant group and average the results:

G5 1h �Ph

j51 g j , where h is the order of the group and g j is an

operator representing the jth symmetry operation (this is the

formalism for all the finite symmetry groups, which are rele-

vant to this article. A similar, more general, operator for infinite

compact groups was also defined[10]). Applying this operator

on the original system return the closest symmetrical system

and the symmetry measure is defined as:

S Gð Þ5100 � 12hGi� �

5100 � 12hWjGjWihWjWji

" #

5100 � 12

Xh

j51hWjgjWi

h � hWjWji

24

35

(2)

This equation represents the symmetry measure of the

whole group. When we apply this methodology to structures

which are composed by a set of Euclidian vectors, the symme-

try measure becomes[10]:

S Gð Þ5100 � 12

Xh

j51

XN

i51�Q i � g �nð Þj �QPi;j

h �XN

i51�Q i � �Q i

24

35 (3)

where �Qi is a set of vectors representing the original structure,

h is the order of the group, g �nð Þj is an operator representing

the jth symmetry operation, �n is a unit vector representing the

direction of the relevant symmetry element, and Pi;j is the per-

mutation for the ith atom and jth symmetry operation. For a

given permutation, the only unknown is the direction of the

symmetry element, �n. For all the simple cyclic symmetry

groups, there are analytical solutions for finding the unit vec-

tors which minimizes eq. (3)[11] (a brief discussion regarding

the subject of permutations can be found in Supporting

Information).

For complex symmetry groups, which contain more than

one symmetry element, analytical solution cannot be formu-

lated as the dependencies between the different symmetry

elements make the expressions too complicated. Recently, an

article was published where the explicit expressions for the

Cnv and Dn groups were rearranged and solved numerically.[12]

Although this method was proven to be successful, it has

some limitations: First of all, this scheme was designed espe-

cially for the Cnv and Dn groups and it cannot be applied or

generalized to other complex symmetry groups. Second, the

numerical procedure will give the precise answer only if the

right permutations are supplied. To get these permutations,

the program solves the analytical procedure for the simple

cyclic subgroups (the main Cn subgroup and the orthogonal

Cs/C2 subgroups), and uses the resulting permutations as input

for the numerical solution. The implicit approximation is that

the directions of the symmetry elements of separate cyclic

subgroups and the related permutations are identical to the

corresponding directions and permutations in the complex

group. This implicit approximation is usually fine and the

results are usually exact, as was demonstrated by the

authors,[12] but in cases where the deviations from perfect

symmetries are not small, the approximation can break and

the results will not be accurate.

A general methodology for calculating the CSM

for complex groups

Before introducing the new approach in details, we need to

remind ourselves two facts about symmetry groups.[6] First,

each of the simple cyclic symmetry groups can be generated

using one basic symmetry operation, g: all the symmetry oper-

ations of the group are powers of this basic operation: g j

where j 5 1, 2,. . ., h. This is also true for the permutations

which minimize the distance between the structure before

and after the symmetry operations had been operated. If the

permutation P is related to the basic symmetry operation, g,

then the jth power of this permutation, Pj, is related to the jth

symmetry operation, g j . Second, every finite complex symme-

try group can be expressed as a multiplication of two or three

cyclic groups: G5G1 � G2 or G5G1 � G2 � G3. The symmetry

operations can be expressed as a multiplication of powers of

the basic operations of the cyclic subgroups: g j1 g l

2 or g j1 g l

2gm3 ,

respectively. The permutation related to a specific symmetry

operation can also be expressed as a multiplication of powers

of the basic permutations of the cyclic subgroups: Pj1Pl

2 or

Pj1Pl

2Pm3 , where the powers are identical to the power of the

relevant symmetry operations of the cyclic subgroups.

The heart of our approach is the assumption that at least

one of the symmetry elements of the complex group is

located in the same direction (or at least very close to the

direction) of that symmetry element in the relevant cyclic sub-

group alone. For example, for Dn5Cn � C2, we assume that

FULL PAPERWWW.C-CHEM.ORG

Journal of Computational Chemistry 2014, 35, 748–755 749

the direction of Cn axis in the Dn group is the same as its posi-

tion if we evaluated only the Cn subgroup or that the direc-

tion of the C2 axis in the Dn group is the same as its position,

if we evaluated only the C2 subgroup. If the object is symmet-

ric, this is of course true—the positions of all the symmetry

elements of the complex group are identical to their position

for the relevant subgroup. We assume that this is at least par-

tially true for nonsymmetric objects. For objects with small

deviations from perfect complex symmetries, we expect this

approximation to be valid, but as we will demonstrate later in

this article, the assumption holds also for objects with larger

deviations from perfect complex symmetries.

We will now describe in detail the algorithm for calculating

the CSM for complex symmetries. We will start by describing the

procedure for the G5G1 � G2 complex groups and then we will

generalize it for the G5G1 � G2 � G3 complex groups. We start

by choosing one of the cyclic subgroups, for example, G1, and

calculate the best permutation and axis for it using the analytical

procedures[11] (or the fast approximation method for large struc-

tures[14]). Now, we need to find the axis for the second cyclic

subgroup, G2, with the constraint that the angle between it and

the axis of the first cyclic subgroup equals to a known angle.

We used searching techniques for finding the direction of the

axis and the permutation which minimizes the CSM (using

Brent’s rule[15] —an one-dimensional minimization techniques

which uses inverse parabolic interpolation[16] if the underlying

change is sufficiently regular and the golden ratio bisection[17] in

other cases). By having the symmetry axes and the permutations

for both cyclic subgroups, we can construct all the symmetry

operations of the complex symmetry group and their related

permutations, as described earlier. By applying all the symmetry

operations on the original structure and averaging the results

[eq. (3)], we can calculate the CSM value. We can now repeat

this procedure, by first finding analytically the axis and the per-

mutation of the G2 cyclic subgroup and use this information to

find the axis and the permutation of the G1 cyclic subgroup and

calculate the CSM for the complex group. The lower of these

two CSM values will be the CSM value for the complex group.

This procedure can be easily generalized for G5G1 � G2 � G3

complex groups. After finding the directions for the axes, for the

G1 and G2 cyclic subgroups, using the above procedure, the

direction for the symmetry axis of G3 is uniquely determined by

its angles with the symmetry axes of the G1 and G2 subgroups.

All we are left to do is to find the permutation for this sub-

group, which can be evaluated using the procedure described in

Ref. [14]. Now, we can build all the symmetry operations of the

complex group and their relevant permutations and calculate

the CSM. In this case, we have six different combinations to

check in order to find the minimal CSM value.

Error estimation

We ran several tests to estimate the error of the methodology.

First, we compared the results of our program with the results

of the Cnv/Dn program, which is based on the numerical solu-

tion.[12] We calculated the Cnv and Dn symmetry of 500,000 dif-

ferent structures using both methodologies. In �95% of the

cases, the results were the same (up to floating point arithme-

tic errors). In the other cases, the difference between the

methods was always lower than 1%. In some cases, our

method returned the lower value and in other cases, the Cnv/

Dn program performed better. This implies that in practice,

both programs present similar performance, up to implemen-

tation issues. This is no surprise, as both programs are based

on the same approximation, as explained earlier. We also

observed that for continuous changes—the symmetry meas-

ures changes continuously and smoothly, as expected. A

detailed analysis can be found in Supporting Information.

A second test was to compare the results of our method

with the results of the SHAPE program.[13,18] This program is

based on the CShM methodology,[13] which calculates the nor-

malized distance of the given structure from a predetermined

structure (we used the following reference structures: an icosa-

hedron, a dodecahedron, an octahedron, a cube, a tetrahe-

dron, a C60 fullerene, a planar square, a trigonal bipyramid, a

C3v symmetric ammonia molecule, and a C2v symmetric water

molecule). The structural changes are kept small compared to

the original symmetrical structures, so the CSM and CShM

should be the same. We checked this assumption during the

calculations: cases where our results were lower than the

results of the SHAPE program were excluded. Also, excluded

were all the cases where the closest symmetrical structures

were different from original symmetrical structures. We calcu-

lated the complex symmetries of 250,000 different structures

using both methodologies. In �92% of the cases, the results

were the same (up to floating point arithmetic errors). In �6%

of the cases, the difference between the methods was lower

than 1% and in the other cases, the difference between the

methods was lower than 12% (in all of these cases, the results

of the SHAPE program were lower than our results).

As the symmetry group is larger and more complex (higher

order and more symmetry elements) the probability of having

an error increases: For the simple Cnv and Dn groups, �98% of

all cases had errors of 0.00% and for all the other cases, the

errors were less than 1%. For more complex groups (Dnd, Dnh,

and Td), �96% of all cases had an errors of 0.00% and for all

the other cases, the errors were less than 2% (actually, most of

these structures had errors smaller than 1% and only five

cases, which are 0.15% of all cases, had errors in the range 1–

2%). In the case of the high symmetry groups (Oh and Ih),

there were much more errors: only 84% of all cases had errors

of 0.00%, other 10% had errors smaller than 1%, and in all

other cases, the errors could get up to 12%. This indicates that

the approximation is good for cases where there are less sym-

metry elements and symmetry measures. This is logical, as we

expect more errors as the complexity of the group raises—the

probability that the axes of the complex group will coincide

with the position of the axes of the subgroups are smaller.

This is because in these cases there are more “degrees of free-

dom”—more ways to build the complex symmetry group.

In all the cases where the relative error was higher than 1%,

the true value of the CSM was higher than 5, indicating that

the structure are highly asymmetrical with respect to the cho-

sen symmetry group. As the CSM can be referred to as a

FULL PAPER WWW.C-CHEM.ORG

750 Journal of Computational Chemistry 2014, 35, 748–755 WWW.CHEMISTRYVIEWS.COM

perturbation theory, measuring the symmetry of nonsymmet-

ric objects is not very productive, getting the CSM value up to

12% in these cases is good enough. In any case, these large

errors occurred only for <2% of all cases. Interestingly, all of

these cases were of structures which deviate from high sym-

metry group—like Ih and Oh. We can also bind the general

error of structures with symmetry measure lower than 10 to a

maximum relative error value of 1%. More details regarding

the error analysis can be found in Supporting Information.

The symmetry groups of linear structure: C‘v and D‘h

In order for us to be able to calculate the CSM for all the sym-

metry point groups, we have to introduce a procedure for cal-

culating the CSM for the C1v and D1h symmetry groups. It is

impossible to use the folding-unfolding algorithm[8] or the

analytical solutions,[11] as they are unable of dealing with infi-

nite point groups. The generalized formalism of the CSM[10]

offers us a way to cope with compact infinite group, but it is

impractical as the expression is too complex to be solved ana-

lytically or numerically. Here, we present simple analytical pro-

cedures for computing the CSM for the C1v and D1h

symmetry groups. We construct the inertia tensor of the given

object,[19] which is a 3 3 3 matrix (if we want only geometrical

measure, we will use a mass of 1 for all points and if we want

a measure with physical and chemical meaning, we can use

the actual masses of the atoms). Eigen-decomposition techni-

ques[20,21] are used to find the eigenvalues and eigenvectors

of the inertia matrix (as this is a symmetric, real, positive defi-

nite, 3 3 3 matrix, it can be decomposed easily using various

techniques. We used the explicit analytical solution for the

characteristic polynomial[21]). The lowest eigenvalue multiplied

by a factor of 100 is the S(C1v) value (as it represent the aver-

age distance of the points from the main axis of inertia) and

the corresponding eigenvector is the direction of the symme-

try axis. In this case, the permutation is always the identity

permutation—in which each atom is interchanged with itself.

The closest symmetrical object can be obtained by projecting

the points on the symmetry axis. In order of calculating the

S(D1h) value, we use the procedure for S(C1v), and then use

the analytical S(Ci) procedure[11] on the resulting C1v-symmet-

rical object. The outcome is the closest D1h-symmetrical

object and the S(D1h) can be calculated using eq. (1).

Testing and Demonstrating the Methodology

The symmetry of AB2 structures

We start with a simple example—the C2v symmetry of bent

AB2 type structures. If both AAB bonds are identical, the struc-

ture has perfect C2v symmetry. If the bond length are different

(because of external force or as a part of asymmetric vibra-

tion), the C2v symmetry is broken. We started with a configura-

tion of water (both bond lengths are 0.97 A and the angle is

104.75�[22]), changed the ratio of the bond lengths and meas-

ured the S(C2v) and S(C1) along this asymmetric vibration

mode (Fig. 1). As one bond becomes larger than the other, the

S(C2v) measure increases until we reach a ratio where the clos-

est symmetrical structure is linear (C1v symmetry) instead of

the original C2v symmetric structure. From this point, as the

ratio changes, the molecule becomes more linear and the

S(C2v) measure equals the S(C1) measure.

The next step was to run an ab initio molecular dynamics

simulation of water, to check the real S(C2v) values for water.

We used the CP2K program.[23] We took a cubic cell

(a 5 b 5 c 5 14.2102 A) with 96 water molecules, optimized it,

ran a short simulation at 300 K (using Nose–Hover chain[24]

with length 3 and time constant of 50.0 fs) to bring the sys-

tem to equilibrium, and then we ran a simulation of 15 ps,

using a time step of 0.5 fs. We used periodic boundary condi-

tions, BLYP functional,[25] norm-conserving pseudopotential,[26]

and 300 Ry as energy cutoff for the wave-plane expansion.

Energy and forces were converged to 1026 Hartree and 1025

Hartree/Bohr, respectively. The structure was saved every 10 fs

(every 20 steps), so at the end we had the structure of 3000

cells—a total of 288,000 water molecules. We calculated the

S(C2v) for all of these molecules, and the histogram is pre-

sented in Figure 2. As can be seen, the water molecule is

highly symmetric. The average S(C2v) is 0.017, and the Root

Mean Square (RMS) value is 0.029, which means the water

molecules only slightly deviate from perfect C2v symmetry dur-

ing the asymmetric vibration. We run similar simulation with

different temperatures in the range 275–370 K. As can be seen

in Figure 3, there is a linear correlation between the tempera-

ture, the average, and RMS S(C2v) values. This is consistent

with earlier work using the CSM as a criterion for the melting

of clusters.[27] This is only logical, as the CSM, the measure of

deviation from perfect symmetry, is a measure of geometrical

disorder, and should behave as structural entropy.

We estimated the error by comparing the values of our pro-

gram with the outcome of the Cnv/Dn program, which is based

on the numerical solution[12] and SHAPE program[13,18] (the ref-

erence structure in this case was a perfectly symmetrical water

Figure 1. The S(C2v) and S(C1v) measures of an AB2 type structure as a

function of the ratio of the bond lengths. Also shown are the closest sym-

metrical structures for the different cases.

FULL PAPERWWW.C-CHEM.ORG

Journal of Computational Chemistry 2014, 35, 748–755 751

molecule). All the values calculated by our program matched

exactly the corresponding values of the numerical Cnv/Dn pro-

gram and the SHAPE program.

D4h-Td symmetry maps of AB4 structures

A second example is the symmetry of AB4 type structures.

Symmetry maps[13,28] are effective for presenting and under-

standing relations between different symmetries. They are

used to understand the geometrical constraints, which gov-

ern the shape of different compounds. We will look on the

D4h-Td symmetry maps of AB4 structures. The spread pathway

is the shortest distortion pathway between a perfect tetrahe-

dron structure (Td symmetry) and a perfect planar square

structure (D4h symmetry).[29] This is also the minimal energy

pathway between these structures.[30] It was shown that this

pathway is important, as many metal complexes follow this

distortion pathway.[29,30] This pathway represents the minimal

distortion limit—meaning one cannot construct a structure

whom will have a set S(Td) and S(D4h), which will be lower

than the Spread pathway.[29,30]

We constructed 100,000 random AB4 structures, calculated

their S(Td) and S(D4h) and compared

with the values of Spread pathway (Fig.

4). As can be seen, all the points are

above the line representing the Spread

pathway, as seen before[29] (more

detailed discussion about the behavior

of these systems can be found in Refs.

[28–30]). Previous studies showed that

although it seems that the symmetry

values for random deviations are found

over most of the symmetry map area, in

practice, many metal complexes tend to

distort using the Spread pathway.[29] In

Figure 5, we bring the S(Td) and S(D4h)

for 20 [Cu(II)Cl4]22 and 10 [Ni(II)(CN)4]22

complexes (their structure was taken

from the Cambridge structural data-

base.[31] More details can be found in

Supporting Information). As can be seen,

they are all distorted, but all the distor-

tions are along the Spread pathway. We

compared the results of our program

with the results of the SHAPE pro-

gram[13,18] and in all cases the values

were the same.

Figure 2. The S(C2v) distribution of water at 300 K. [Color figure can be

viewed in the online issue, which is available at wileyonlinelibrary.com.] Figure 3. The average and RMS S(C2v) values of water as a function of the

temperature. [Color figure can be viewed in the online issue, which is avail-

able at wileyonlinelibrary.com.]

Figure 4. A S(D4h)-S(Td) map for AB4 structures. The solid line represents the spread distortion path-

way. The dots represent the symmetry 100,000 random AB4 structures. [Color figure can be viewed

in the online issue, which is available at wileyonlinelibrary.com.]

FULL PAPER WWW.C-CHEM.ORG

752 Journal of Computational Chemistry 2014, 35, 748–755 WWW.CHEMISTRYVIEWS.COM

Symmetry analysis of LJ

clusters

To demonstrate the ability of

the method to identify and

quantify the symmetry content

of the high symmetry groups,

we analyzed the symmetry of

LJ clusters.[32] These clusters

have been explored intensively

and it is known that their mini-

mum energy conformations

have icosahedral symmetry or

one of its subsymmetries,

except for some exceptions

which have tetrahedral and

octahedral symmetries.[32] We

chose several representatives

LJ clusters (see Figure 6) and

calculated their symmetry con-

tenta the results are collected

in Table 1. As can be seen, the

program identifies correctly the

right symmetry and its sub-

symmetries. An interesting

observation is that the S(Td)

and S(Oh) generally become

smaller as we add more atoms

Figure 5. A S(D4h)-S(Td) map for AB4 structures. The solid line represents the spread distortion pathway. The

green triangles represent the symmetry of 20 [CuCl4]22 complexes. The brown squares represent the symmetry

of 10 [Ni(CN)4]22 complexes. [Color figure can be viewed in the online issue, which is available at wileyonlineli-

brary.com.]

Figure 6. The minimum energy structure for some LJ clusters: (a) LJ-5 (D3h symmetry), (b) LJ-9 (C2v symmetry), (c) LJ-13 (Ih symmetry), (d) LJ-38 (Oh symme-

try), and (e) LJ-55 (Ih symmetry).

FULL PAPERWWW.C-CHEM.ORG

Journal of Computational Chemistry 2014, 35, 748–755 753

to the cluster. This relates to the question of the transition

between the icosahedral symmetry of small LJ clusters to the

cubic (octahedral) symmetry of the Face Centered Cubic (FCC)

Lennard-Jones crystal.[32] A previous review showed that in

many cases latent octahedral symmetry can be found within

icosahedral structures.[33] We believe that our results show this

is true for the LJ clusters, and as the cluster grows, this octa-

hedral symmetry becomes gradually more dominant.

We run Metadynamics[34] simulations using the SPRINT col-

lective variables[35] for LJ clusters with 13 atoms and 38 atoms

to explore different possible conformations, using the proce-

dure described in Ref. [35]. We calculated the S(Oh) and S(Td)

for the trajectory and compared them with the energy (Fig. 7).

As can be seen, more symmetrical conformations have lower

energies. For LJ-38 cluster, where the minimal energy confor-

mation has Oh symmetry, the values drop to zero for the mini-

mal energy conformation. For LJ-13 cluster, where the minimal

energy conformation has Ih symmetry, the values drop down

for the minimal energy conformation, although they cannot

reach zero. This means that symmetry measures, and especially

the S(Oh) and S(Td) measures, are good for finding the minimal

energy of a cluster, and they have a potential use as collective

variables for enhanced dynamics simulations.[34,36]

Conclusions

We introduced in this article a general approach for estimating

the CSM of complex symmetries. Unlike earlier works, this

approach can be applied to all the complex symmetry point

groups and it does not assume that the closest symmetrical

object is known a priori. Beside the possible practical applica-

tions, which will be discussed in the next paragraph, this work

represents a closer to the voyage that began more than two

decades years ago.[8] For the first time, the CSM for any point

group symmetry can be calculated in practice. There are still

more challenges—calculating the CSM for function, for crys-

tals, for dynamical systems, and more. Some of these issues

are currently discussed and developed, and we are sure that in

time these goals will be achieved.

Now that the CSM for all the point groups can be calcu-

lated, we can use them to investigate the symmetry of various

systems and formulate quantitative relations between symme-

try and other physical and chemical properties. As mentioned

earlier, symmetry is vastly used in science, and the existence of

symmetry implies the existence of a conservation rule or selec-

tion rule.[2–6] The CSM methodology can be implemented in

these cases to estimate the degree of deviation from the per-

fect rule as a function of the deviation from perfect symmetry.

The CSM methodology was successfully applied for various

cases in the past[9] and now we can expand the scope of

investigation. Some simple examples for possible applications

are presented in this article and there are many more—in the

fields of spectroscopy, inorganic chemistry, biochemistry, nano-

chemistry, theoretical physics, and chemistry, and much more.

Acknowledgments

The author thanks Prof. Michele Parrinello and his group for their

hospitality and the fruitful discussions. The author also thanks Mr.

Amir Zait for the helpful comments and technical help.

Keywords: symmetry � symmetry measures � high symme-

try � symmetry maps � water dynamics � Lennard-Jones clus-

ters � metal-complexes symmetry

Table 1. Symmetry analysis of several Lennard-Jones clusters.

S(C2v) S(C3v) S(C5v) S(D5h) S(Td) S(Oh) S(Ih)

LJ 5 D3h 0.0 0.0 37.3 42.1 33.9 68.1 47.3

LJ 6 Oh 0.0 0.0 28.9 33.1 0.0 0.0 67.1

LJ 7 D5h 0.0 11.9 0.0 0.0 28.0 28.4 68.0

LJ 8 Cs 5.5 8.7 10.5 17.1 14.6 15.4 55.0

LJ 9 C2v 0.0 6.0 16.4 21.8 29.7 30.3 49.5

LJ 10 C3v 4.4 0.0 21.5 30.1 24.5 30.0 43.4

LJ 11 C2v 0.0 3.9 6.9 21.7 23.8 40.5 53.2

LJ 12 C5v 2.2 5.0 0.0 18.2 25.4 34.5 42.8

LJ 13 Ih 0.0 0.0 0.0 8.4 5.3 5.3 0.0

LJ 26 Td 0.0 0.0 7.1 18.3 0.0 10.3 32.0

LJ 38 Oh 0.0 0.0 2.9 5.9 0.0 0.0 9.2

LJ 45 C1 6.8 7.1 8.3 20.9 19.0 24.8 36.4

LJ 55 Ih 0.0 0.0 0.0 2.2 3.0 3.0 0.0

Figure 7. The S(Td) and S(Oh) as a function of the energy for LJ-13 (the first

graph) and LJ-38 clusters (the second graph).

FULL PAPER WWW.C-CHEM.ORG

754 Journal of Computational Chemistry 2014, 35, 748–755 WWW.CHEMISTRYVIEWS.COM

How to cite this article: C. Dryzun. J. Comput. Chem. 2014, 35,

748–755. DOI: 10.1002/jcc.23548

] Additional Supporting Information may be found in the

online version of this article.

[1] T. D. Lee, Particle Physics and an Introduction to Field Theory; Har-

wood Academic Publisher: London, UK, 1981.

[2] A. V. Shubnikov, V. A. Koptsik, Symmetry in Science and Art; Plenum

Press: New York, 1997.

[3] J. F. Cornwell, Group Theory in Physics; Academic press: London, 1988.

[4] M. Hargittai, I. Hargittai, Symmetry through the Eyes of a Chemist, 3rd

ed.; Springer: New York, 2009.

[5] E. Noether, Nachr. D. K€onig. Gesellsch. D. Wiss. Zu G€ottingen, Math-

phys. Klasse 1918, 235. (English translation by M. A. Tavel, Transp.

Theory Stat. Phys. 1971, 1, 183).

[6] (a) J. Rotman, An Introduction to the Theory of Groups, 4th ed.;

Springer-Verlag: Free Software Foundation New York, 1994, p. 517; (b)

T. W. Judson, Abstract Algebra: Theory and Applications, Math Depart-

ment, Virginia Commonwealth University, 2009.

[7] (a) P. G. Mezey, J. Am. Chem. Soc. 1990, 112, 3791; (b) G. Gilat, J. Math.

Chem. 1994, 15, 197; (c) P.G. Mezey, Mol. Phys. 1999, 96, 169; (d) I. L.

Garz�on, J. A. Reyes-Nava, J. I. Rodr�ıguez-Hern�andez, I. Sigal, M. R.

Beltr�an, K. Michaelian, Phys. Rev. B 2002, 66, 073403; (e) M. Petitjean,

Entropy 2003, 5, 271; (f ) A. Potts, D. M. Bangnall, N. I. Zheludev, J. Opt.

A: Pure Appl. Opt. 2004, 6, 193; (g) I. S. Mili, D. C. Sorensen, SIAM J.

Matrix Anal. Appl. C 2006, 28, 749; (h) D. Milner, S. Raz, H. Hel-Or, D.

Keren, E. Nevo, Pattern Recognit. 2007, 40, 2237.

[8] (a) H. Zabrodsky, S. Peleg, D. Avnir, J. Am. Chem. Soc. 1992, 114, 7843; (b)

H. Zabrodsky, S. Peleg, D. Avnir, J. Am. Chem. Soc. 1993, 115, 8278; (c) H.

Zabrodsky, D. Avnir, J. Am. Chem. Soc. 1995, 117, 462; (d) H. Zabrodsky,

S. Peleg, D. Avnir, IEEE Trans. Pattern Anal. Mach. Intell. 1995, 17, 1154.

[9] (a) D. R. Kanis, J. S. Wong, T. J. Marks, M.A. Ratner, H. Zabrodsky, S.

Keinan, D. Avnir, J. Phys. Chem. 1995, 99, 11061; (b) I. Saragusti, I.

Sharon, O. Katzenelson, D. Avnir, J. Archeol. Sci. 1998, 25, 817; (c) S.

Keinan, M. Pinsky, M. Plato, J. Edelstein, D. Avnir, Chem. Phys. Lett.

1998, 298, 43; (d) O. Katzenelson, J. Edelstein, D. Avnir, Tetrahedron:

Asymmetry 2000, 11, 2695; (e) S. Alvarez, M. Pinsky, M. Llunell, D.

Avnir, Crystal Eng. 2001, 4, 179; (f ) K. B. Lipkowitz, S. Schefzick, D.

Avnir, J. Am. Chem. Soc. 2001, 123, 6710; (g) E. Estrada, D. Avnir, J. Am.

Chem. Soc. 2003, 125, 4368; (h) S. Alvarez, S. Schefzick, K. Lipkowitz, D.

Avnir, Chem. Eur. J. 2003, 9, 5832; (i) J. Breuer, D. Avnir, J. Chem. Phys.

2005, 1292, 074110; (j) A. Steinberg, M. Karni, D. Avnir, Chem. Eur. J.

2006, 12, 8534; (k) D. Yogev-Einot, D. Avnir, Tetrahedron: Asymmetry

2006, 17, 2723; (l) C. Dryzun, Y. Mastai, A. Shvalb, D. Avnir, J. Mater.

Chem. 2009, 19, 2062; (m) C. Dryzun, P. Alemany, D. Casanova, D.

Avnir, Chem. Eur. J. 2011, 17, 6129; (n) I. Tuvi-Arad, D. Avnir, J. Org.

Chem. 2011, 76, 4973; (o) E. F. Sheka, B. S. Razbirin, D. K. Nelson, J.

Phys. Chem. A 2011, 115, 3480; (p) H. Elgavi, C. Krekeler, R. Berger, D.

Avnir, J. Phys. Chem. C 2012, 116, 330.

[10] (a) C. Dryzun, D. Avnir, Phys. Chem. Chem. Phys. 2009, 42, 9653; (b) C.

Dryzun, D. Avnir, ChemPhysChem 2011, 12, 197.

[11] (a) M. Pinsky, D. Casanova, P. Alemany, S. Alvarez, D. Avnir, C. Dryzun, Z.

Kizner, A. Sterkin, J. Comput. Chem. 2008, 29, 190; (b) M. Pinsky, C. Dryzun, D.

Casanova, P. Alemany, D. Avnir, J. Comput. Chem. 2008, 29, 2712.

[12] M. Pinsky, A. Zait, M. Bonjack, D. Avnir, J. Comput. Chem. 2013, 34, 2.

[13] (a) M. Pinsky, D. Avnir, Inorg. Chem. 1998, 37, 5575; (b) S. Alvarez, D.

Avnir, M. Llunell, M. Pinsky, New J. Chem. 2002, 26, 996; (c) J. Cirera, E.

Ruiz, S. Alvarez, Organometallics 2005, 24, 1556; (d) D. Casanova, M.

Llunell, P. Alemany, S. Alvarez, Chem. Eur. J. 2005, 11, 1479.

[14] C. Dryzun, A. Zayit, D. Avnir, J. Comput. Chem. 2011, 32, 2526.

[15] (a) R. P. Brent, Algorithms for Minimization without Derivatives, Chapter

4; Prentice-Hall: Englewood Cliffs, NJ, 1973; (b) H. M. Antia, Numerical

Methods for Scientists and Engineers, Birkh€auser, Birkhauser Verlag,

Boston, Basel, Berlin, 2002, pp. 362–365.

[16] G. E. Forsythe, M. A., Malcolm, C. B. Moler, Computer Methods for

Mathematical Computations; Prentice-Hall: Englewood Cliffs, NJ, 1977.

[17] E. K. P. Chong, S. H. Zak, An Introduction to Optimization. Series in Dis-

crete Mathematics and Optimization, 3rd ed.; John Wiley & Sons, Inc.,

New York, Wiley-Interscience, 2008.

[18] The SHAPE program can be downloaded from the web-site of the

Electronic Structure Group at the University De Barcelona: http://

www.ee.ub.es/.

[19] L. W. Tsai, Robot Analysis: The Mechanics of Serial and Parallel Manipu-

lators; Wiley: New York, 1999.

[20] (a) Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, H. van der Vorst, Eds. Tem-

plates for the Solution of Algebraic Eigenvalue Problems: A Practical

Guide, SIAM: Philadelphia, 2000; (b) J. W. Demmel, Applied Numerical

Linear Algebra; SIAM: Philadelphia, 1997; (c) G. W. Stewart, Matrix Algo-

rithms, Vol. II; Eigensystems, SIAM: Philadelphia, 2001.

[21] (a) T. S. Blyt, E. F. Robertson, Basic Linear Algebra; Springer-Verlag Lon-

don, 1998; (b) G. Strang, Linear Algebra and its Applications; Thomson,

Brooks/Cole: Belmont, CA, 2006; (c) T. S. Shores, Applied Linear Algebra

and Matrix Analysis, Springer Science1Business Media, LLC, Springer,

New York, 2007.

[22] J. B. Hasted, In Water A Comprehensive Treatise, Vol. 1; F. Franks, Ed.;

Plenum Press: New York, 1972, pp. 255–309.

[23] (a) R. Car, M. Parrinello, Phys. Rev. Lett. 1985, 55, 2471; (b) J.

VandeVondele, M. Krack, F. Mohamed, M. Parrinello, T. Chassaing, J.

Hutter, Comput. Phys. Commun. 2005, 167, 1032128.

[24] (a) S. Nose, J. Chem. Phys. 1984, 81, 511; (b) S. Nose, Mol. Phys. 1984,

52, 255; (c) W. G. Hoover, Phys. Rev. A 1985, 31, 1695.

[25] (a) A. D. Becke, Phys. Rev. A: At. Mol. Opt. Phys. 1988, 38, 3098; (b) C.

Lee, W. Yang, R. G. Parr, Phys. Rev. B: Condens. Matter Mater. Phys.

1988, 37, 785.

[26] (a) N. Troullier, J. L. Martins, Phys. Rev. B 1991, 43, 1993; (b) S.

Goedecker, M. Teter, J. Hutter, Phys. Rev. B: Condens. Matter Mater.

Phys. 1996, 54, 1703.

[27] V. Buch, E. Greshgoren, H. Z. Hel-Or, D. Avnir, Chem. Phys. Lett. 1995,

247, 149.

[28] S. Alvarez, P. Alemany, D. Casanova, J. Cirera, M. Llunell, D. Avnir,

Coord. Chem. Rev. 2005, 249, 1693.

[29] (a) S. Keinan, D. Avnir, Inorg. Chem. 2001, 40, 318; (b) S. Keinan, D.

Avnir, J. Chem. Soc. Dalton Trans. 2001, 941.

[30] D. Casanova, J. Cirera, M. Llunell, P. Alemany, D. Avnir, S. Alvarez, J.

Am. Chem. Soc. 2004, 126, 1755.

[31] F. H. Allen, Acta Cryst. B 2002, B58, 380.

[32] (a) M. R. Hoare, Adv. Chem. Phys. 1979, 40, 49; (b) J. A. Northby, J.

Chem. Phys. 1987, 87, 6166; (c) R. E. Kuntz, R. S. Berry, J. Chem. Phys.

1995, 103, 1904; (d) R. S. Berry, R. E. Kuntz, Phys. Rev. Lett. 1995, 74,

3951; (e) J. Pillardy, K. A. Olszewski, L. Piela, J. Phys. Chem. 1992, 96,

4337; (f ) N. J. A. Sloane, R. H. Hardin, T. D. S. Duff, J. H. Conway, Dis-

crete Comput. Geom. 1995, 14, 237; (g) T. Coleman, D. Shalloway, Z.

Wu, J. Global Optim. 1994, 4, 171; (h) C. D. Maranas, C. A. Floudas, J.

Chem. Phys. 1992, 97, 7667.

[33] J. Echeverr�ıa, D. Casanova, M. Llunell, P. Alemany, S. Alvarez, Chem.

Commun. 2008, 2717.

[34] A. Laio, M. Parrinello, Proc. Natl. Acad. Sci. USA 2002, 99, 12562.

[35] F. Pietrucci, W. Andreoni, Phys. Rev. Lett. 2011, 107, 085504.

[36] (a) L. Rosso, P. Minary, Z. Zhu, T. E. Tuckerman, J. Chem. Phys. 2002, 116,

4389; (b) L. Maragliano, E. Vanden-Eijnden, Chem. Phys. Lett. 2006, 426,

168; (c) J. B. Abrams, M. E. Tuckerman, J. Phys. Chem. B. 2008, 112, 15742;

(d) G. N. Patey, J. P. Vallea, J. Chem. Phys. 1975, 63, 2334; (e) P. Bash, U.

Singh, F. Brown, R. Langridge, P. Kollman, Science 1987, 235, 574; (f ) A. M.

Ferrenberg, R. H. Swendsen, Phys. Rev. Lett. 1988, 61, 2635; (g) S. K.

Kumar, J. M. Rosenberg, D. Bouzida R.H. Swendsen, P.A. Kollman, J. Com-

put. Chem. 1995, 16, 1339; (h) T. Huber, A. E. Torda, W. F. van Gunsteren, J.

Comput. Aided Mol. Des. 1994, 8, 695; (i) B. Roux, Comput. Phys. Commun.

1995, 91, 275; (j) A. F. Voter, Phys. Rev. Lett. 1997, 78, 3908; (k) E. Darve, A.

Pohorille, J. Chem. Phys. 2001, 115, 9169; (l) S. Marsili, A. Barducci, R.

Chelli, P. Procacci, V. Schettino, J. Phys. Chem. B 2006, 110, 14011.

Received: 14 November 2013Revised: 29 December 2013Accepted: 2 January 2014Published online on 6 February 2014

FULL PAPERWWW.C-CHEM.ORG

Journal of Computational Chemistry 2014, 35, 748–755 755