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Pere Alemany - One of the best experts on this subject based on the ideXlab platform.
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Revisiting the foundations of Symmetry Operation measures for electronic wavefunctions
Chemical Physics Letters, 2011Co-Authors: David Casanova, Pere AlemanyAbstract:Abstract An extension of the continuous Symmetry measures for N-electron wavefunctions is proposed in order to solve some of the problems encountered with earlier definitions for this property that appear when dealing with molecules that present quasisymmetries close to a point group with degenerate irreducible representations. The mathematical origin of these problems is analyzed and it is shown, via some examples, how the newly defined Symmetry Operation measures are able to overcome them and give a correct answer for any type of Symmetry Operation.
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quantifying the Symmetry content of the electronic structure of molecules molecular orbitals and the wave function
Physical Chemistry Chemical Physics, 2010Co-Authors: David Casanova, Pere AlemanyAbstract:A scheme to quantify the Symmetry content of the electronic wave function and molecular orbitals for arbitrary molecules is developed within the formalism of Continuous Symmetry Measures (CSMs). After defining the Symmetry Operation expectation values (SOEVs) as the key quantity to gauge the Symmetry content of molecular wavefunctions, we present the working equations to be implemented in order to carry out real calculations using standard quantum chemistry software. The potentialities of a Symmetry analysis using this new method are shown by means of some illustrative examples such as the changes induced in the molecular orbitals of a diatomic molecule by an electronegativity perturbation, the breaking of orbital Symmetry along the dissociation path of the H2 molecule, the changes in the molecular orbitals upon a geometrical distortion of the benzene molecule, and the inversion Symmetry content in the different spin states of the [Fe(CH3)4]2− complex.
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Symmetry Operation measures.
Journal of Computational Chemistry, 2007Co-Authors: Mark Pinsky, David Casanova, Pere Alemany, Santiago Alvarez, David Avnir, Chaim Dryzun, Ziv Kizner, Alexander SterkinAbstract:We introduce a new mathematical tool for quantifying the Symmetry contents of molecular structures: the Symmetry Operation Measures. In this approach, we measure the minimal distance between a given structure and the structure which is obtained after applying a selected Symmetry Operation on it. If the given Operation is a true Symmetry Operation for the structure, this distance is zero; otherwise it gives an indication of how different the transformed structure is from the original one. Specifically, we provide analytical solutions for measures of all the improper rotations, S, including mirror Symmetry and inversion, as well as for all pure rotations, C. These measures provide information complementary to the Continuous Symmetry Measures (CSM) that evaluate the distance between a given structure and the nearest structure which belongs to a selected Symmetry point-group. © 2007 Wiley Periodicals, Inc. J Comput Chem, 2008
David Casanova - One of the best experts on this subject based on the ideXlab platform.
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Revisiting the foundations of Symmetry Operation measures for electronic wavefunctions
Chemical Physics Letters, 2011Co-Authors: David Casanova, Pere AlemanyAbstract:Abstract An extension of the continuous Symmetry measures for N-electron wavefunctions is proposed in order to solve some of the problems encountered with earlier definitions for this property that appear when dealing with molecules that present quasisymmetries close to a point group with degenerate irreducible representations. The mathematical origin of these problems is analyzed and it is shown, via some examples, how the newly defined Symmetry Operation measures are able to overcome them and give a correct answer for any type of Symmetry Operation.
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quantifying the Symmetry content of the electronic structure of molecules molecular orbitals and the wave function
Physical Chemistry Chemical Physics, 2010Co-Authors: David Casanova, Pere AlemanyAbstract:A scheme to quantify the Symmetry content of the electronic wave function and molecular orbitals for arbitrary molecules is developed within the formalism of Continuous Symmetry Measures (CSMs). After defining the Symmetry Operation expectation values (SOEVs) as the key quantity to gauge the Symmetry content of molecular wavefunctions, we present the working equations to be implemented in order to carry out real calculations using standard quantum chemistry software. The potentialities of a Symmetry analysis using this new method are shown by means of some illustrative examples such as the changes induced in the molecular orbitals of a diatomic molecule by an electronegativity perturbation, the breaking of orbital Symmetry along the dissociation path of the H2 molecule, the changes in the molecular orbitals upon a geometrical distortion of the benzene molecule, and the inversion Symmetry content in the different spin states of the [Fe(CH3)4]2− complex.
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Symmetry Operation measures.
Journal of Computational Chemistry, 2007Co-Authors: Mark Pinsky, David Casanova, Pere Alemany, Santiago Alvarez, David Avnir, Chaim Dryzun, Ziv Kizner, Alexander SterkinAbstract:We introduce a new mathematical tool for quantifying the Symmetry contents of molecular structures: the Symmetry Operation Measures. In this approach, we measure the minimal distance between a given structure and the structure which is obtained after applying a selected Symmetry Operation on it. If the given Operation is a true Symmetry Operation for the structure, this distance is zero; otherwise it gives an indication of how different the transformed structure is from the original one. Specifically, we provide analytical solutions for measures of all the improper rotations, S, including mirror Symmetry and inversion, as well as for all pure rotations, C. These measures provide information complementary to the Continuous Symmetry Measures (CSM) that evaluate the distance between a given structure and the nearest structure which belongs to a selected Symmetry point-group. © 2007 Wiley Periodicals, Inc. J Comput Chem, 2008
Vojtěch Kopský - One of the best experts on this subject based on the ideXlab platform.
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Seitz symbols for Symmetry Operations of subperiodic groups
Acta Crystallographica Section A Foundations and Advances, 2014Co-Authors: Daniel B Litvin, Vojtěch KopskýAbstract:Recently adopted International Union of Crystallography conventions for the notation of Seitz symbols of Symmetry Operations require a revision of the Seitz notation used in International Tables for Crystallography Vol. E, Subperiodic Groups. This paper gives the subperiodic group Symmetry Operations blocks of Vol. E with the Seitz symbol for each included Symmetry Operation in the recently adopted conventions for Seitz notation.
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Seitz notation for Symmetry Operations of space groups.
Acta crystallographica. Section A Foundations of crystallography, 2011Co-Authors: Daniel B Litvin, Vojtěch KopskýAbstract:Space-group Symmetry Operations are given a geometric description and a short-hand matrix notation in International Tables for Crystallography, Volume A, Space-Group Symmetry. We give here the space-group Symmetry Operations subtables with the corresponding Seitz (R∣t) notation for each included Symmetry Operation.
Massimo Nespolo - One of the best experts on this subject based on the ideXlab platform.
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The chromatic Symmetry of twins and allotwins
Acta Crystallographica Section A Foundations and Advances, 2019Co-Authors: Massimo NespoloAbstract:The Symmetry of twins is described by chromatic point groups obtained from the intersection group H Ã of the oriented point groups of the individuals H i extended by the Operations mapping different individuals. This article presents a revised list of twin point groups through the analysis of their groupoid structure, followed by the generalization to the case of allotwins. Allotwins of polytypes with the same type of point group can be described by a chromatic point group like twins. If the individuals are all differently oriented, the chromatic point group is obtained in the same way as in the case of twins; if they are mapped by Symmetry Operation of the individuals, the chromatic point group is neutral. If the same holds true for some but not all individuals, then the allotwin can be seen as composed of twinned regions described by a twin point group, that are then allotwinned and described by a colour identification group; the allotwin is then described by a chromatic group obtained as an extension of the former by the latter, and requires the use of extended symbols reminiscent of the extended Hermann-Mauguin symbols of space groups. In the case of allotwins of polytypes with different types of point groups, as well as incomplete (allo)twins, a chromatic point group does not reveal the full Symmetry: the groupoid has to be specified instead.
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About the concept and definition of "noncrystallographic Symmetry"
Zeitschrift für Kristallographie, 2008Co-Authors: Bernd Souvignier, Massimo Nespolo, Daniel B LitvinAbstract:The definition of "noncrystallographic symme- try" given in Volume B of the International Tables for Crys- tallography actually corresponds to the concept of "local Symmetry". A new definition of "noncrystallographic sym- metry" is proposed, which fully complies with that of "crys- tallographic Symmetry" in Volume A of the International Ta- bles for Crystallography. The concept of "noncrystallographic Symmetry" is, quite obviously, directly related to that of "crystallographic Symmetry" so that once a definition of the latter is gi- ven, that of the former is obtained spontaneously. This is probably the reason why no explicit definition of "non- crystallographic Symmetry" is given in Volume A of the International Tables for Crystallography. Even the con- cept of noncrystallographic point groups, presented in section 10.1.4, is introduced without an explicit defini- tion but only as the groups differing from the crystallo- graphic point groups presented in Sections 10.1.2 and 10.1.3. As a matter of fact, one should not even need an expli- cit definition of "noncrystallographic Symmetry" once that of "crystallographic Symmetry" is given. Unfortunately, in the literature, and especially in the structural biology lit- erature, the term "noncrystallographic Symmetry" is used in a manner in striking contrast with what is directly im- plied by the definition of "crystallographic Symmetry". This contradiction is so fundamental that it causes serious misunderstandings and misinterpretations. We will show that the use of "noncrystallographic Symmetry" in structur- al biology is inconsistent with the accepted definition of "crystallographic Symmetry" and suggest that an alterna- tive terminology should be used. To understand the problem let us start with the defini- tion of "crystallographic Symmetry Operations" given in (1) Section 8.1.5: A motion is called a crystallographic Symmetry opera- tion if a crystal pattern exists for which it is a Symmetry Operation. A motion is an isometry, a transformation keeping an- gles and distances unchanged, i.e. a transformation with- out deformation. A crystal pattern is the extension of a crystal structure to a periodic arrangement of whatever ob- ject, concrete or abstract, constitutes the structure. The atoms forming a crystal structure represent a special case of a crystal pattern. This definition applies to the n-dimen- sional Euclidean space E n and can be expressed in a quan- titative way with the aid of group theory. With respect to a basis of E n , the Symmetry Operations of the space group of the crystal pattern are represented by (n þ 1) � (n þ 1) augmented matrices where the top-left nn block repre- sents the linear part of the Operation (the part that leaves the origin fixed) and the additional column represents the vector part of the Operation (the part which gives the trans- lation component of the Symmetry Operation). The above definition of a crystallographic Symmetry Operation im- plies that with respect to a suitable basis of E n (namely a primitive basis of the periodic pattern) the linear part of the matrix representing the Operation is an integral matrix. In E 2 and E 3 this results in the well-known crystallo- graphic restriction according to which only rotations (di- rect or inverse) of order 1, 2, 3, 4 and 6 are compatible with the existence of a crystal pattern. This restriction is extended to include also Operations of order 5, 8, 10 and 12 for the four-dimensional space (2). By an obvious contraposition, one is led to the defini- tion of a noncrystallographic Symmetry Operation as a mo- tion for which no crystal pattern exists allowing this mo- tion as a Symmetry Operation. In particular, in E 2 and E 3 , a noncrystallographic Symmetry Operation is a motion whose linear part is different from rotations (direct or in- verse) of order 1, 2, 3, 4 and 6, and in E 4 different also
Daniel B Litvin - One of the best experts on this subject based on the ideXlab platform.
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Seitz symbols for Symmetry Operations of subperiodic groups
Acta Crystallographica Section A Foundations and Advances, 2014Co-Authors: Daniel B Litvin, Vojtěch KopskýAbstract:Recently adopted International Union of Crystallography conventions for the notation of Seitz symbols of Symmetry Operations require a revision of the Seitz notation used in International Tables for Crystallography Vol. E, Subperiodic Groups. This paper gives the subperiodic group Symmetry Operations blocks of Vol. E with the Seitz symbol for each included Symmetry Operation in the recently adopted conventions for Seitz notation.
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Seitz notation for Symmetry Operations of space groups.
Acta crystallographica. Section A Foundations of crystallography, 2011Co-Authors: Daniel B Litvin, Vojtěch KopskýAbstract:Space-group Symmetry Operations are given a geometric description and a short-hand matrix notation in International Tables for Crystallography, Volume A, Space-Group Symmetry. We give here the space-group Symmetry Operations subtables with the corresponding Seitz (R∣t) notation for each included Symmetry Operation.
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About the concept and definition of "noncrystallographic Symmetry"
Zeitschrift für Kristallographie, 2008Co-Authors: Bernd Souvignier, Massimo Nespolo, Daniel B LitvinAbstract:The definition of "noncrystallographic symme- try" given in Volume B of the International Tables for Crys- tallography actually corresponds to the concept of "local Symmetry". A new definition of "noncrystallographic sym- metry" is proposed, which fully complies with that of "crys- tallographic Symmetry" in Volume A of the International Ta- bles for Crystallography. The concept of "noncrystallographic Symmetry" is, quite obviously, directly related to that of "crystallographic Symmetry" so that once a definition of the latter is gi- ven, that of the former is obtained spontaneously. This is probably the reason why no explicit definition of "non- crystallographic Symmetry" is given in Volume A of the International Tables for Crystallography. Even the con- cept of noncrystallographic point groups, presented in section 10.1.4, is introduced without an explicit defini- tion but only as the groups differing from the crystallo- graphic point groups presented in Sections 10.1.2 and 10.1.3. As a matter of fact, one should not even need an expli- cit definition of "noncrystallographic Symmetry" once that of "crystallographic Symmetry" is given. Unfortunately, in the literature, and especially in the structural biology lit- erature, the term "noncrystallographic Symmetry" is used in a manner in striking contrast with what is directly im- plied by the definition of "crystallographic Symmetry". This contradiction is so fundamental that it causes serious misunderstandings and misinterpretations. We will show that the use of "noncrystallographic Symmetry" in structur- al biology is inconsistent with the accepted definition of "crystallographic Symmetry" and suggest that an alterna- tive terminology should be used. To understand the problem let us start with the defini- tion of "crystallographic Symmetry Operations" given in (1) Section 8.1.5: A motion is called a crystallographic Symmetry opera- tion if a crystal pattern exists for which it is a Symmetry Operation. A motion is an isometry, a transformation keeping an- gles and distances unchanged, i.e. a transformation with- out deformation. A crystal pattern is the extension of a crystal structure to a periodic arrangement of whatever ob- ject, concrete or abstract, constitutes the structure. The atoms forming a crystal structure represent a special case of a crystal pattern. This definition applies to the n-dimen- sional Euclidean space E n and can be expressed in a quan- titative way with the aid of group theory. With respect to a basis of E n , the Symmetry Operations of the space group of the crystal pattern are represented by (n þ 1) � (n þ 1) augmented matrices where the top-left nn block repre- sents the linear part of the Operation (the part that leaves the origin fixed) and the additional column represents the vector part of the Operation (the part which gives the trans- lation component of the Symmetry Operation). The above definition of a crystallographic Symmetry Operation im- plies that with respect to a suitable basis of E n (namely a primitive basis of the periodic pattern) the linear part of the matrix representing the Operation is an integral matrix. In E 2 and E 3 this results in the well-known crystallo- graphic restriction according to which only rotations (di- rect or inverse) of order 1, 2, 3, 4 and 6 are compatible with the existence of a crystal pattern. This restriction is extended to include also Operations of order 5, 8, 10 and 12 for the four-dimensional space (2). By an obvious contraposition, one is led to the defini- tion of a noncrystallographic Symmetry Operation as a mo- tion for which no crystal pattern exists allowing this mo- tion as a Symmetry Operation. In particular, in E 2 and E 3 , a noncrystallographic Symmetry Operation is a motion whose linear part is different from rotations (direct or in- verse) of order 1, 2, 3, 4 and 6, and in E 4 different also
