FULL PAPER
WWW.C-CHEM.ORG
Continuous Symmetry Measures for Complex Symmetry 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 calculating 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 LennardC 2014 Wiley Periodicals, Inc. Jones clusters. V
Introduction
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 separately, with constraints on the relative position of the elements. 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 introduce 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 methods for calculating analytically the CSM for the linear symmetry group, C1v and D1h. We will apply these methods for calculating the symmetry of water, AB4 type complexes, and Lennard-Jones clusters (LJ clusters).
“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 categorized 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 property can vary.[3–5] To get quantitative symmetry–property relation, 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 content 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 dependence of different systems in various areas of science for more than two decades.[9] A few years ago, the method was generalized, enabling the treatment of various mathematical descriptions.[10] This generalization included the explicit analytical expression for the closest symmetrical object in terms on the 748
Journal of Computational Chemistry 2014, 35, 748–755
DOI: 10.1002/jcc.23548
Theory Background The original paper presenting the CSM methodology,[8] i referred to an object which is described by a set of vectors: Q
C. Dryzun Department of Natural Sciences, The Open University of Israel, Raanana, 43107, Israel E-mail: [email protected] C 2014 Wiley Periodicals, Inc. V
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: SðGÞ5100
i 2W i j2 jQ i 2Q 0 j2 jQ
(1)
i is a set of vectors representing the coordinates of where W closest object which is G-symmetric (aka, the geometric dis 0 is the center tance between the structures is minimal) and Q of mass of the original structure. The outcome is a number between 0 and 100, where SðGÞ50 indicates that the original object is perfectly G-symmetrical, and the value of the measure 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 i , which is unknown a priori[8] closest symmetrical object, W (The exception is the continuous shape measures (CShM)[13] in which the original structure is compared to a predefined structure—comparison of the different approaches can be found in Supporting Information). A few years ago, the CSM was generalized and a general algorithm for finding the closest symmetrical 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 ^ which operates all the symmetry operaprojection operator, G, tion of the relevant group and average the results: P ^ 1 h g ^ ^ G5 j51 j , where h is the order of the group and g j is an h operator representing the jth symmetry operation (this is the formalism for all the finite symmetry groups, which are relevant 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: " # ^ ^ 5100 12 hWjGjWi SðGÞ5100 12hGi hWjWji 2 3 Xh ^ jWi hWjg j51 5 5100 412 h hWjWji
(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 symmetry measure becomes[10]: 2 SðGÞ5100 412
Xh XN j51
h
i51 XN
i g^ ðnÞ Q Q j Pi;j
i Q Q i51 i
3 5
(3)
i is a set of vectors representing the original structure, where Q h is the order of the group, g^ ðnÞj 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 permutation 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 vectors 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 formulated 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 especially 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 ^ : all the symmetry operusing one basic symmetry operation, g ^j ations of the group are powers of this basic operation: g 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 ^ j . Second, every finite complex symmesymmetry operation, g 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 ^ j1 g ^ l2 g^ m the basic operations of the cyclic subgroups: g^ j1 g^ l2 or g 3, 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: Pj1 Pl2 or Pj1 Pl2 Pm 3 , 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 subgroup alone. For example, for Dn 5C n C 2, we assume that Journal of Computational Chemistry 2014, 35, 748–755
749
FULL PAPER
WWW.C-CHEM.ORG
the direction of Cn axis in the Dn group is the same as its position if we evaluated only the Cn subgroup or that the direction 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 symmetric, 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 partially 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 structures[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 permutation 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 subgroup, 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 solution.[12] We calculated the Cnv and Dn symmetry of 500,000 different structures using both methodologies. In 95% of the 750
Journal of Computational Chemistry 2014, 35, 748–755
cases, the results were the same (up to floating point arithmetic 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 implementation 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 measures 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 normalized distance of the given structure from a predetermined structure (we used the following reference structures: an icosahedron, a dodecahedron, an octahedron, a cube, a tetrahedron, 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 calculated 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 symmetry 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 freedom”—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 chosen symmetry group. As the CSM can be referred to as a WWW.CHEMISTRYVIEWS.COM
WWW.C-CHEM.ORG
FULL PAPER
perturbation theory, measuring the symmetry of nonsymmetric 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
WWW.C-CHEM.ORG
Continuous Symmetry Measures for Complex Symmetry 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 calculating 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 LennardC 2014 Wiley Periodicals, Inc. Jones clusters. V
Introduction
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 separately, with constraints on the relative position of the elements. 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 introduce 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 methods for calculating analytically the CSM for the linear symmetry group, C1v and D1h. We will apply these methods for calculating the symmetry of water, AB4 type complexes, and Lennard-Jones clusters (LJ clusters).
“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 categorized 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 property can vary.[3–5] To get quantitative symmetry–property relation, 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 content 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 dependence of different systems in various areas of science for more than two decades.[9] A few years ago, the method was generalized, enabling the treatment of various mathematical descriptions.[10] This generalization included the explicit analytical expression for the closest symmetrical object in terms on the 748
Journal of Computational Chemistry 2014, 35, 748–755
DOI: 10.1002/jcc.23548
Theory Background The original paper presenting the CSM methodology,[8] i referred to an object which is described by a set of vectors: Q
C. Dryzun Department of Natural Sciences, The Open University of Israel, Raanana, 43107, Israel E-mail: [email protected] C 2014 Wiley Periodicals, Inc. V
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: SðGÞ5100
i 2W i j2 jQ i 2Q 0 j2 jQ
(1)
i is a set of vectors representing the coordinates of where W closest object which is G-symmetric (aka, the geometric dis 0 is the center tance between the structures is minimal) and Q of mass of the original structure. The outcome is a number between 0 and 100, where SðGÞ50 indicates that the original object is perfectly G-symmetrical, and the value of the measure 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 i , which is unknown a priori[8] closest symmetrical object, W (The exception is the continuous shape measures (CShM)[13] in which the original structure is compared to a predefined structure—comparison of the different approaches can be found in Supporting Information). A few years ago, the CSM was generalized and a general algorithm for finding the closest symmetrical 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 ^ which operates all the symmetry operaprojection operator, G, tion of the relevant group and average the results: P ^ 1 h g ^ ^ G5 j51 j , where h is the order of the group and g j is an h operator representing the jth symmetry operation (this is the formalism for all the finite symmetry groups, which are relevant 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: " # ^ ^ 5100 12 hWjGjWi SðGÞ5100 12hGi hWjWji 2 3 Xh ^ jWi hWjg j51 5 5100 412 h hWjWji
(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 symmetry measure becomes[10]: 2 SðGÞ5100 412
Xh XN j51
h
i51 XN
i g^ ðnÞ Q Q j Pi;j
i Q Q i51 i
3 5
(3)
i is a set of vectors representing the original structure, where Q h is the order of the group, g^ ðnÞj 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 permutation 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 vectors 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 formulated 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 especially 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 ^ : all the symmetry operusing one basic symmetry operation, g ^j ations of the group are powers of this basic operation: g 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 ^ j . Second, every finite complex symmesymmetry operation, g 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 ^ j1 g ^ l2 g^ m the basic operations of the cyclic subgroups: g^ j1 g^ l2 or g 3, 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: Pj1 Pl2 or Pj1 Pl2 Pm 3 , 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 subgroup alone. For example, for Dn 5C n C 2, we assume that Journal of Computational Chemistry 2014, 35, 748–755
749
FULL PAPER
WWW.C-CHEM.ORG
the direction of Cn axis in the Dn group is the same as its position if we evaluated only the Cn subgroup or that the direction 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 symmetric, 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 partially 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 structures[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 permutation 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 subgroup, 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 solution.[12] We calculated the Cnv and Dn symmetry of 500,000 different structures using both methodologies. In 95% of the 750
Journal of Computational Chemistry 2014, 35, 748–755
cases, the results were the same (up to floating point arithmetic 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 implementation 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 measures 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 normalized distance of the given structure from a predetermined structure (we used the following reference structures: an icosahedron, a dodecahedron, an octahedron, a cube, a tetrahedron, 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 calculated 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 symmetry 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 freedom”—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 chosen symmetry group. As the CSM can be referred to as a WWW.CHEMISTRYVIEWS.COM
WWW.C-CHEM.ORG
FULL PAPER
perturbation theory, measuring the symmetry of nonsymmetric 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
