In this note, we indicate the coincidence as abstract groups of some point groups which belong to different molecular orbitals. This elucidates somewhat vague presentation in many existing textbooks on molecular orbitals, thus abridging between group theory and quantum chemistry

Kanemitsu Shigeru

Kitamura H.

Sun Yong

Pure and Applied Mathematics Journal

In this note, we indicate the coincidence as abstract groups of some point groups which belong to different molecular orbitals. This elucidates somewhat vague presentation in many existing textbooks on molecular orbitals, thus abridging between group theory and quantum chemistry.journal articl

孫, 勇

16755

Sun, Yong

60274560

ソン, ユー

35231625400

Kanemitsu, Shigeru

16770

Kitamura, H.

16771

Kyutacar : Kyushu Institute of Technology Academic Repository

On some point groups著者 Sun Yong, Kanemitsu Shigeru, Kitamura H.journal orpublication titlePure and Applied Mathematic Journalvolume 4number 2-1page range 42-46year 2015-02-12URL http://hdl.handle.net/10228/5853doi: info:doi/10.11648/j.pamj.s.2015040201.18 Pure and Applied Mathematics Journal 2015; 4(2-1): 42-46 Published online February 10, 2015 (http://www.sciencepublishinggroup.com/j/pamj) doi: 10.11648/j.pamj.s.2015040201.18 ISSN: 2326-9790 (Print); ISSN: 2326-9812 (Online)  On Some Point Groups Y. Sun1, S. Kanemitsu2, H. Kitajima3 1Dept. of Electr. Engrg, Kyushu Inst. Techn., 1-1 Senshuimachi, Tobata, Kitakyushu, Fukuoka 804-8550, Japan 2Grad. School of Adv. Tech., Kinki Univ.,11-6 Kayanomori, Iizuka, Fukuoka 820-2555, Japan 3Grad. School of Sci. & Engrg, Saga Univ.,1  Honjocho, Saga840-8502, Japan Email address: sun@ele.kyutech.ac.jp (Y. Sun) To cite this article: Y. Sun, S. Kanemitsu, H. Kitajima. On Some Point Groups. American Journal of Software Engineering and Applications. Special Issue: Abridging over Troubled Water---Scientific Foundation of Engineering Subjects. Vol. 4, No. 2-1, 2015, pp. 42-46.  doi: 10.11648/j.pamj.s.2015040201.18  Abstract: In this note, we indicate the coincidence as abstract groups of some point groups which belong to different molecular orbitals. This elucidates somewhat vague presentation in many existing textbooks on molecular orbitals, thus abridging between group theory and quantum chemistry. Keywords: Point Group Molecular Orbitals, Symmetry Transformations: Characters  1. Introduction Point groups are indispensable tools for analysis of molecular orbitals. However, there is one point missing in most of the textbooks (given in references) to the effect that one group is associated to several orbitals, cf. e.g. Example 4, which has been a tradition in the theory because there is always associated a figure of the molecule. Since one group is associated to many molecules, it would be preferable to have a clear description of the correspondence between the molecules and the point group. In this note we shall show that some of the point groups which have different labels are isomorphic as abstract groups. Classification being the strong point of mathematics, this manifestation of the classification of point groups hopefully gives rise to better and clear understanding of the theory of point groups. We have treated the easiest case and apparently, we may go on classifying all the point groups, which task will be conducted subsequently. In the remainder of this section, we assemble fundamentals on group theory for readers' convenience. We are concerned with finite groups only. We take basic definitions of a group for granted. The number of elements of a group G is called the order of G denoted |G|. A subgroup H of a group H is a subset of G, which itself forms a group under the operation (called product) of G. A group with commutative product is called an Abelian group (or a commutative group). For finite groups, the following is fundamental. Lemma 1 (Lagrange). Let G be a finite group and let H be its subgroup. Then the order of H divides that of G: |H| | ||. Lemma 1 is used e.g. in the following context. Recall that the cyclic subgroup <a> generated by a in a group G is the set of all powers of a: <  >= {| ∈ ℤ} Since there are only finitely many different powers of a, we must have  =  , whence we may conclude that  = { ∈ ℕ| = 1} ≠ ∅ . Choosing  = min  , we may conclude that l divides all members of X and that all elements 1, , … ,  are distinct, so that | <  > | = , | <  > | = { ∈ ℕ| = 1}. The order of the subgroup generated by <a> is called the order of a and denoted by o(a). By Lemma 1, l | |G| and so | | = (|"#$|)| |/|"#$| = 1. Thus, we have the generalized Fermat's little theorem: | | = 1.                   (1.1) Corollary 1. A group G of prime order is cyclic. For, any element a of G generates a cyclic subgroup whose order o(a) divides the prime ' = || , and so either (() = 1 or (() = ' . The former case leads to  = + (identity) and the latter case leads to  =<  >. Lemma 2 (Sylow) For the highest prime power ', that divides the order of a group G, there exists a subgroup S of order ',, called a Sylow p subgroup of G. All the subgroups  Pure and Applied Mathematics Journal 2015; 4(2-1): 42-46  43  of G of prime power order '- are contained in the Sylow p subgroup S. All Sylow p subgroups are conjugate. Notation. To avoid confusion with the rotary reflection ./, we denote the nth symmetric group by nS . Care must be taken also in treating the dihedral group 0/  which is denoted by 2nD  in mathematics. 2nD is the semi-direct product of a cyclic group of order n and one of order 2: 2nD  =< 1 >⋊<  >, where o(b)=n, o(a)=2 and there is the relation 1 = 1/.  2. Point Groups By [7, p.8] point groups are rotational parts of the space group which leave the molecule (lattice) invariant and in the case of crystals there are 32 of them. Definition 1. We choose the principal axis of symmetry as the one which goes through the maximum number of molecules (lattice points). If there are some of them, then we choose the one. If there is no such an axis we choose it arbitrarily. The plane of symmetry is chosen so that it contains the maximum number of molecules. The intersection of the principal axis and the plane of symmetry is chosen to be the origin.  3/ is the rotation w.r.t. the principal axis through 25/.  6  is a reflection. 67  (h=horizontal) is a reflection through a plane perpendicular to the axis of symmetry; 68  (v=vertical) is a reflection through a plane containing a principal axis of symmetry; 69 (d=dihedral) is a reflection through a plane containing a principal axis of symmetry and bisects the angle between two 2-fold axes perpendicular to the principal axis. They are all of order 2.  i is the inversion w.r.t. the origin.  ./  is the rotation through 25/  followed by a reflection through a plane perpendicular to the axis of rotation, and is called rotary reflection (improper rotation). In what follows we often write the group itself by its typical element. Remark 1.  (1) In the case of crystals, rotations are only those which rotate through integral multiples of 5/3, 5/2 or the products of such rotations and the inversion. 6 = 3;, where 3; is the rotation through 25/2. (2) It seems that the introduction of the rotary reflection ./ leads to confusion because it is the product of two operations: ./ = 673/                  (2.1) and ./ = 673/ for any integer m. Hence ./ = 3/     m  even,             (2.2) ./ = 673/   m  odd and . = 673 = 67. Example 1. We have 3;8 ≅ 3=. Example 2. The group 3=  consists of the following 4 elements whose multiplication table is given below. (i) 3= is the rotation through 25/4 and 3== = + is the identity.  (ii) 3=; = 3; is the rotation through 5.  (iii) 3=? is the rotation through 35/2.  Table 1. Character table of 3= @ e AB ABC ABD EF 1 1 1 1 E 1 -1 1 -1 E; 1  ; ? E? 1 ? ;  Example 3. The group 3G consists of 6 elements whose multiplication table is given below. (i) 3= is the rotation through 5/6 and 3GG is the identity.  (ii) 3? is the rotation through 25/3 and 3? = 3G;.  (iii) 3; is the rotation through 25/2 and 3; = 3G?.  (iv) 3?; is the rotation through 45/3 and 3?; = 3G=.  (v) 3?I is the rotation through 55/3. Table 2. Multiplication table of 3G (r indicates 3G). o e K K; K? K= KI e e K K; K? K= KI K K K; K? K= KI e K; K; K? K= KI e K K? K? K= KI e K K; K= K= KI e K K; K? KI KI e K K; K? K= Table 3. Multiplication table of 3S  o e K K; s K;L KL e e K K; s K;L KL K K K; e KL s K;L K; K; e K K;L KL s s s K;L KL E K K; K;L K;L KL S K; E K KL KL s K;L K K; e Table 4. Regular multiplication table of 3S  o e M? M; M K; K e e M? M; M K; K M? M? e K; K M; M M; M; K e K; M M? M M K; K e M? M; K K M; M M? e K; K; K; M M? M; K e 44 Y. Sun et al.:  On Some Point Groups   Theorem 1. We have 3 3vC D= , and they are 3≅ S . Example 4. To 3?8  belong ammonia NH? , chloroform CHCl?, cyclopropenylcation radical, phosphorus sesqui -sulfide P=.? , etc. To 0?  belong Boron trifluoride BF? , triphenylmethyl radical, trans-perhydrotripolyphenylene, staggered conformation of ethane, etc. We note that the following table appears on [9, p.29] and the elucidation appears on [9, p.6]. Table 5. Multiplication table of 3?7 o e TDU TDV ADU = TDB ADC = TDC WX = TDD e e .? .?I .?= .?; = 3?; .?? .? .? 3?; e .?I .?; 3? .?I .?I e .?= = 3? .?? = 67 .? .?; = 3?; 3? = .?= 3? .?I .?? = 67 .?; = 3?; e .? 3?; = .?; 3?; .?? = 67 .? e .?= = 3? .?I 67 = .?? 67 .?= = 3? .?; = 3?; .? .?I e Theorem 2. 3?7 ≅ 3G.                 (2.3) Theorem3. We have 3 6 12hD D≅ ≅D  and 3 3 6vD D≅ ≅D  is its subgroup. Consequently, we may treat planar molecules belonging to 0?7 as those belonging to 0?. 3. Characters Definition 2. In a (matrix) representation Γ  of a finite group G, the trace of the matrix .Z  corresponding to 6 ∈  is called a character of Γ and denoted by χ(6). χ(6) = Tr(.Z) There are |G| characters of G. Those which corresponding to irreducible representations are called simple characters. χF(6), … , χ7(6). Theorem 4. Any irreducible representation of a finite Abelian group must be of degree 1, and so the representation ^ itself is equal to the trace of the representation matrix, i.e. it is a character. Theorem 5. Any finite Abelian group is expressed as a direct product of cyclic groups of prime power order. Lemma 3. Any cyclic group 3/ of order n is isomorphic to the additive group of residue classes modulo n: 3/ ≅ _/_. Theorem 6. Suppose 3/ =<  > be a cyclic group of order n. Since / = +  it follows that the values of a character χ ∶  3/ →  ℂ× must be the nth roots of 1. Hence it suffices to restrict the range to the torus group d = {z ∈ ℂ ||z| = 1} . Then the character χ is determined by its value at a, say χ() = +;fg// , the piervotny primitive root of 1. The correspondence χ → h + ℤ gives rise to the isomorphism X(3/) ≅ ℤ/ℤ ≅ 3/. This theorem follows from Lemma 3 since the correspondence a ↔ +;fg//  give rise to an isomorphism between the groups <a> and l/ of the group of nth roots of 1. Now combining Theorem 5 and Theorem 6, we deduce Theorem 7. Any finite Abelian group is isomorphic to its character group X(G) = {χ: G → d|χ is a homomorphism}:  ≅ X(). Let ^ = +;fg/? be the piervotny primitive 3rd root of unity and let o = +;fg/G the piervotny primitive 6th root of unity. Table XXI [7, p.16] after changing the columns of 3? and .? reads. Table 6. Character table of 3? @ e pD TDC = ADC TDD = WX TDB = AD TDB= WXADC χF 1 1 1 1 1 1 χ; 1 ^ ^; 1 ^ ^; χ? 1 ^; ^ 1 ^; ^ χ 1 -1 1 -1 1 -1 χI 1 −^ ^; -1 ^ −^ χ= 1 −^; ^ -1 ^; −^; Table 7. Character table of 3G  @ e Ar ArC = AD ArD = AC ArB = AD − C ArV χF 1 1 1 1 1 1 χ 1 -1 1 -1 1 -1 χ; 1 o; = ^ o= = ^; 1 o; = ^ o= = ^; χ? 1 o= = ^; o; = ^ 1 o= = ^; o; = ^ χ= 1 ω = −^; o; = ^ -1 −ω = ^; −o;= −^; χI 1 −o; = −^ −ω = ^; -1 o; = ^ ω = −^ By changing the 5th and 6th rows in Table 5, it coincides with Table 6, so that these character groups are isomorphic. By Theorem 7, we reprove Theorem 2. 4. Rotations as Matrices In this section we elucidate the rotations and reflections as matrices. Theorem 7. Let cos sinsin cosAθθ θθ θ− =               (4.1) and cos sin.sin cosBφφ φφ φ =  −            (4.2)  Pure and Applied Mathematics Journal 2015; 4(2-1): 42-46  45  Then the linear transformation Aθ=y x by Aθ  means the rotation (in the positive direction, i.e. counterclockwise) w.r.t. the origin byθ and that the linear transformationBφ=y x  by Bφ means the reflection w.r.t. the line ℓ  which goes through the origin and is subtended to the positive direction of the x -axis by the angle 2φ. Proof. We give a proof that makes full use of Euler's identity. Expressing the Cartesian coordinates as the complex number in polar coordinates: coscos sin ,sinr xz x iy r irr yΘ   = = ⇔ = + = Θ + Θ   Θ   z  we see that the rotation i ie z zeθ θ=  of z byθ is ( ) ( ) ( )( )( )cos sincos cos sin s insin cos cos sincos sin sin cos ,iie z re r irr ri r rx x i x yθθ θ θθ θθ θθ θ θ θΘ+= = Θ + + Θ += Θ − Θ −+ Θ − Θ= − + + which is the linear transformation  cos ysin.sin ycos yx xAxθθ θθ θ−   =   +    Of course,  arg .A zθ θ= Θ +             (4.3) For the treatment of Bφ we note that 0 ,B A Bφ φ=               (4.4) which we now check. Since 0B means the reflection w.r.t. the x -axis, it corresponds to taking the complex conjugate. Therefore, the effect of Bφ on z corresponds toie zθ . Hence with arg z = Θ , we have 0arg B z θ= − Θ .             (4.5) Now since B zθ is the vector obtained from z by the rotation by 12θ − Θ −  , it follows that 1 1arg2 2B zθ θ θ = − Θ −  , which is equal to (4.5). Hence (4.4) follows. Corollary 1. We have the following identities which have the geometric meaning. A A A A Aθ φ φ θ θ φ+= = ,            (4.6) A B Bθ φ θ φ+= ,                (4.7) B A Bθ φ θ φ−= ,                (4.8) and B B Aθ φ θ φ−= .               (4.9) It is easy to deduce (4.7) etc. from (4.3) and (4.5). Indeed, we find that arg A Bθ φ θ φ= + − Θz ,          (4.10) which is Bθ φ+ , i.e. (4.7).  Similarly, since arg Aφ φ= Θ +z , we have ( ) ( )arg B Aθ φ θ φ θ φ= − Θ + = − − Θz , whence (4.9).  Finally,  ( ) ( )arg B Bθ φ θ φ θ φ= − − Θ = − + Θz , whence (4.10) follows.. Lemma 4. We have the following identities 0B A Bθ θ−= ,               (4.11) 0B B Aφ φ= .               (4.12) Proof. We make use of (4.4) to express Bφ  as 0A Bφ . E.g. (4.12) follows by writing 0 0 0B B A B Bφ φ= and noting that 20B E= . With aid of Lemma 4, we may give a lucid proof of Corollary 1. 0 0A B A A B A B Bθ φ θ φ θ φ θ φ+ += = = ,    (4.13) i.e. (4.7). To prove (4.8), we use (4,4), (4.11), (4.7) to deduce that 0B A A B A A B Bθ φ θ φ θ φ θ φ− −= = = ,    (4.14) whence (4.8). Finally, 0 0 0B B A B A B A B B A A Aθ φ θ φ θ φ θ φ θ φ− − −= = = =   (4.15) by (4.11), (4.12) and (4.6) successively. The following examples give rudiments of matrix representations of the group 3vC . Example 5. Given a regular triangle ABC, let O be the center of gravity and let the mid points of AB, BC, CA be D, E, F, respectively and choose DO to be the x -axis. Let 2E E= , 23 2 3 43 3,C A C Api pi= = ,            (4.16) and 46 Y. Sun et al.:  On Some Point Groups   ( )( )( )432301 32 21 ,3 12 21 32 223 12 21 03 .0 1vvvBBBpipiσσσ − −  = = −   −  = =    = =  − ,     (4.17) Then ( ) ( ) ( ){ }23 3, , , 1 , 2 , 3v v vG E C C σ σ σ=     (4.18) forms a group w.r.t. the multiplication of matrices and is isomorphic to the group 3vC or the symmetric group of order 6 (cf. Theorem 1). Example 6. Let 3E E= , 23 30 1 0 0 0 10 0 1 , 1 0 0 ,1 0 0 0 1 0C C      = =              (4.19) and ( ) ( )( )1 0 0 0 0 11 0 0 1 , 2 0 1 0 ,0 1 0 1 0 00 1 03 1 0 0 .0 0 1v vvσ σσ      = =           =        (4.20) Then ( ) ( ) ( ){ }23 3' , , , 1 , 2 , 3v v vG E C C σ σ σ=     (4.21) forms a group w.r.t. the multiplication of matrices and is isomorphic to the group 3vC  (cf. Example 5). Example 7. We find a regular matrix 111 1 1a bX b a  =    such that 13cos sin 0sin cos 0 ,10 0 1AX C Xθθ θθ θ−−    = =       oo   (4.22) i.e. similar to 3C .  To this end, it suffices to consider 31AC X Xθ =   ooor 1 a cos sin sin cos 11 1 1 a sin cos a cos sin 11 cos sin sin cos 1b a b a bb ba bθ θ θ θθ θ θ θθ θ θ θ+ − +      = + −      + − +     whence a cos sinsin cosbθ θθ θ= += − +.  References [1] M. A. Armstrong, Groups and symmetry, Springer, New York-Berlin etc. 1988. [2] D. M. Bishop, Group theory and chemistry, OUP, Oxford 1973; 2nd ed. Dover, New York 1993. [3] A. Carbone and M. Gromov, A mathematical slices of molecular biology, Supplement to volume 88 of Gazette des Mathématiciens, French Math. Soc. (SMF), Paris 2001. [4] J. H. Conway, Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Cambridge UP, Cambridge 1985. [5] Theo Hahn, International tables for crystallography, Reidel Publ. Co. 1983. [6] H. Kitajima and S. Kanemitsu, Math-Phys-Chem approaches to life, Intern. J. Math. Math. Sci., Volume 2012, Article ID 371825, 29 pages (doi:10.1155/2012/371825), published May 13, 2012. [7] G. F. Koster, Space groups and their representations, Academic Press, New York1957. [8] K. Morikawa, A new view point of solid structure of proteins, Kodan-sha, Tokyo 1991 (in Japanese) [9] M. Nakasaki, Symmetry of molecules and group theory Tokyo Kagaku-dohjin, Tokyo1973 (in Japanese). [10] M. Ohiwa, Group theory and molecules, Kagaku-dohjin, Tokyo 1969 (in Japanese). [11] M. Ohya and S. Matsunaga, Coding and genes, J. Electr. Inf. Comm, Soc. J74-A (1991), 1075-1084 (in Japanese). [12] A. D. Thomas and G.V. Wood, Group tables, Shiva Publ. Ltd, Orpington 1980.  

On some point groups

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On some point groups

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