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Group Theory and Symmetry of Complex Compound and Its Properties Journal of Interdisciplinary Cycle Research ISSN NO: 0022-1945 Group theory and symmetry of complex compound and its properties 1 2 3 Dr. Alok Kumar Singh , Dr. Shwet Nisha and Dr. R.K.Singh 1,2 Department of Applied Mathematics, C.I.T. Ranchi, Jharkhand (India). 3 Department of Chemistry BIT ,Sindri, Jharkhand (India). [email protected] 2 [email protected] 3 [email protected] ABSTRACT Group theory provides a powerful tool for exploiting symmetry in the analysis of physical system. The symmetry [1] of a molecule is related to its physical properties and provides a quick simple method to determine the relevant physical information of the molecule. In this work, the character table of complex compound is studied and the polarity of compound is examined. Penta aquo phallate chromium (I) complex is prepared by the interaction of phthallic acid and di- tertiary amyl chromate in suitable solvent. Also the subgroup and conjugate classes of same complex compound is found. Keywords: Point group, conjugate classes, Character table, irreducible representation, wave function. 1. INTRODUCTION The different arrangement of atoms and their respective symmetry to each other greatly influences the chemical properties of the whole molecule [2]. More interestingly, symmetry can take place within the molecule itself or between two or more molecules. The symmetry in chemistry is mostly recognizable in molecular structures and isomers. As a rule, the symmetry in molecular structures determines if a molecule is polar or non-polar, which places a great effect on many properties of the molecule. In order for a molecule to be polar it has to be non-symmetrical, which is simply obtained when the atoms bonded to the central atoms are different. A dipole moment [3] is instantly formed as a consequence of different atoms possessing different electronegativities. Furthermore, the more highly electronegative atoms attract electrons more strongly, which places the molecule in an Volume XII, Issue VIII, August/2020 Page No:1785 Journal of Interdisciplinary Cycle Research ISSN NO: 0022-1945 unbalanced form. On the other hand, non-polar molecules are the ones that are symmetrical. Several researchers have done work on this type of simple and complex compound. In this paper group we have studied symmetries of Penta aquo phallate chromium (I) complex. And the character table of the group [4] is computed by using following properties of characters (the great Orthogonality theorem): The number of irreducible characters is equal to the number of conjugacy classes (it is the dimension of the center of the group algebra). There are orthogonality relations between characters; specifically, between the rows and the columns of the table. The sum of the squares of all characters in any irreducible representation is equal to the order of the group. The characters of all operations in the same class are equal in each given irreducible (or reducible) representation. The sum of the squares of the dimensions of the irreducible representations is equal to the order of the group. The dimensionality of the irreducible representation [5] of the group can be determined by a simple rule which state that the sum of the squares of the dimension of the representations equals the order of the group (order of the group is the number of its elements) [6].When matrices G(g) are given, it is useful to be recognize whether the representation is irreducible if following conditions is satisfied : Σg χ(G(g)) * χ(G(g)) = n Where χ(G(g)) is the character of the group element g [ trace of the matrix G(g) ].Irreducible representations also satisfy two orthogonality relations: Σg χ(G(gi)) * χ(G(gj)) = n δij Σg χ(G(gj)) * χ(G(gk)) = n δjk. Volume XII, Issue VIII, August/2020 Page No:1786 Journal of Interdisciplinary Cycle Research ISSN NO: 0022-1945 2. MAIN RESULT AND DISSCUSSION We use different experimental facts to obtain the structure of complex [Cr (C8H5O4) (H2O) 5] which is depicted below 2 2 3 It is a square planer, so it is of C4v group. Its elements are E, C4, C 4 = C , C 4, σv, σ′v, σd,σ′d where E = The identity C4 = A clockwise rotation through 90. 2 C 4 = A clockwise rotation through 180. σv = Reflection in line x-axis. σd = Reflection in line x=y axis. And so on. We find among the element of the group C4v there exist a relationships -1 C4 σvC4 = σ′v. Volume XII, Issue VIII, August/2020 Page No:1787 Journal of Interdisciplinary Cycle Research ISSN NO: 0022-1945 Show that σv and σ′v are conjugate to each other. We know if A is conjugate B and B is conjugate C then A and C are conjugate to each other. It immediately follows that we can split a group into sets such that all the elements of a set are conjugate to each other. In fact such set are called conjugate classes or simply classes of a group. 2 3 The conjugate classes for the group C4v is [E], [C 4], [C4,C 4], [σv,σ′v], and [σd,σ′d], The multiplication table for group C4v is: Volume XII, Issue VIII, August/2020 Page No:1788 Journal of Interdisciplinary Cycle Research ISSN NO: 0022-1945 Now the character table of C4v is: 3. CONCLUSION Only the molecule which belongs to the Cn, Cnv, and Cs group can have a permanent dipole moment. In molecule belonging to Cn or Cnv the dipole moment must lie along the axis of rotation and we know polar molecule is one having permanent dipole moment .Therefore the group C4v is also polar .The method described in this paper appears to be more efficient in dealing with the construction of the character table of symmetry group of the molecule. 4. REFERENCES: 1. F. Albert Cotton, Wiley -Interscience, 1990, “Chemical Applications of Group Theory” New York, QD461.C65 1990. 2. Robert L. Carter, J. Wiley, 1998, “Molecular Symmetry and Group Theory” New York,QD461.C32 1998. 3. .Daniel Harris & Michael Bertolucci Symmetry and Spectroscopy New York, Dover Publications 1989 . 4. Rowland, Todd; Weisstein, Eric W, "Character table" from Mathworld 5. Lowell H. Hall, Mcgraw-Hill, 1969, “Group Theory and Symmetry in Chemistry”New York, QD 461.H17. 6. David S. Schonland, van Nostrand, 1965,“Molecular Symmetry; An Introduction to Group Theory and Its Uses in Chemistry”,London,QD 461.S35. Volume XII, Issue VIII, August/2020 Page No:1789.
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