MATRIX REPRESENTATIONS OF THE COORDINATES OF THE ATOMS OF THE MOLECULES OF H 2 O AND NH 3 IN OPERATIONS OF SYMMETRY

Transcription

1 Trakia Journal of Sciences No 4 pp opyright 7 Trakia Uniersity Aailable online at: ISSN -769 (print) ISSN -55 (online) doi:.5547/tjs.7.4. Original ontribution MATRIX REPRESENTATIONS OF THE OORDINATES OF THE ATOMS OF THE MOLEULES OF H O AND NH IN OPERATIONS OF SYMMETRY M. Ianoa * L. Dospatlie Department of Informatics and Mathematics Trakia Uniersity Stara Zagora Bulgaria Department of Pharmacology Animal Physiology and Physiological hemistry Trakia Uniersity Stara Zagora Bulgaria. ABSTRAT The molecule is a spatial arrangement of atoms that make it up. This arrangement defines its geometric structure. Knowing the geometry and ways to describe to great adantage in studying the electronic structure of molecules as between geometric and electronic structure there is a close connection. Key words: operations of symmetry matri H O NH INTRODUTION The geometric structure of a molecule is described by the theory of point group symmetry. A fundamental concept in the theory of point groups is of geometric configuration. Under operation of symmetry along action which transforms a geometric configuration of a gien molecule in another configuration which is identical or equialent to the original. From this definition it follows that in carrying out operation of symmetry equialent atoms can be swapped or at best remain in place (- 5). The operations of the symmetry are carried out with respect to the elements of symmetry. The elements of symmetry are geometric concepts - aes planes points or combinations thereof. Obiously not eery point ais and plane that we can spend in space it occupies a molecule is an element of symmetry (-7). In Table are presented the elements operations of symmetry and symbols which are used for their determination. Table. Elements and operations of symmetry Element Symbol Operations Symbol Ais Rotation Plane Reflection From the geometry it is known that each point in -dimensional space is represented by three coordinates using the orthogonal coordinate system are ( y ) in the polar coordinate system are ( r ) etc. For a description in the space of one molecule of N atom needed N coordinates ( y ; y ; ; y ). N N N Let us eamine what happens to the *orrespondence to: M. Ianoa Department of Informatics and Mathematics Trakia Uniersity Stara Zagora 6 Bulgaria; mirkai@yahoo.com coordinates of the atoms in the molecule in carrying out any operation of symmetry of the molecule as a whole through the following two eamples. The water molecule Let us consider the molecule of water as the local coordinates of the atoms in the molecule and the global coordinates of the entire molecule coincide with those in the oygen atom as represented in Figure. 9 Trakia Journal of Sciences Vol

2 IVANOVA M. et al. Figure. Local coordinates of atoms in a molecule of water and the global coordinates of the entire molecule (coincide with those on the oygen atom). ais is chosen to coincide with the ais of atoms of oygen i.e. H : ( y ); O : ( y ); H : ( y ). () In operation atoms H and H swapped places. Then the new coordinates of the atoms resulting from are:... y. y y. y y. y.... () This is a little unusual and etraagant way of recording since we put the + and - instead of just characters in respectie equations. Obiously equations () would look differently if we were oriented aes at the respectie atoms (local coordinate systems) pointing in different directions. Then the coordinate of an atom after surgery symmetry would become a miture of y in the most general case. The choice of local coordinate systems was designed so as to obtain the simplest possible draws. The orientation of the coordinate aes in space is irreleant and therefore it is best to choose the most conenient. Equations () can be written in the following matri form: y y y y y y () In () the old and new coordinates of the atoms are represented as a matri column (this is the matri representation of a ector) right and left of the equality respectiely. A square matri of numbers - and shows how the coordinates are conerted into new ones. Therefore it is called the transformation matri in this case - transformation matri of the coordinates of the atoms of water molecules under the influence of the operation of symmetry. From () to get as coordinate appropriate corresponding row of square matri can be multiplied by the column of the coordinates i.e... y... y... y. Trakia Journal of Sciences Vol

3 The equations () and () are the same which could easily be coninced if you do the multiplication of the transformation matri and matri column in the old coordinates. onersely the transformation matri in () was written using the equations (); we put a + or - only where need to be replicated equations (). The square matri of type (9 9) in () is written so that it is clear which group of matri elements which refers atom. Let us remark that this matri contains ero elements in that block which corresponds to an atom which becomes atom displacements. For eample H (the first atom) under the effect of becomes H and the third block of the transformation matri are non-ero elements. Atom remains in place and in the second block on the second row are obtained non-ero matri elements. In other words the diagonal of the matri are obtained non-ero elements only to those atoms which during operation of symmetry remain in place. For each of the operations of the symmetry of the molecule of water (group operations E ) we can write transformation matri similar to that of in (). A total of four operations molecule of water will get four transformation matrices of type (9 9). Een for such a small molecule such as that of water four matrices are large. For each molecule of E. Since oygen atom stays in place and the four operations of its symmetry aes remain the same as or maintaining orientation (come matri element ) or routed (matri element ) but not translated into coordinates other atoms. The adantages offered by the recording of the operations of symmetry by means of transformation matrices are large. Rather than track what happens to indiidual atoms when performing an operation of symmetry we can do the same with matrices of these operations. For eample the product we could find by multiplying matrices and namely: 94 Trakia Journal of Sciences Vol IVANOVA M. et al. N atoms transformation matrices hae dimensions (N N). Let us now from the water molecule to consider only the three aes (coordinate functions) y of the atom of oygen which is the fied point in all operations of symmetry for the molecule consideration. Then () passes in y y which we can write as y y y which epresses the fact that the new functions are obtained by the action of on old functions like the action is represented by the matri (4) We obtained the transformation matri of the three coordinates of the atom of oygen under the action of the operation. Similarly for the other three operations of symmetry we get (5). The result is a transformation matri of operation. Table shows the result of multiplication operations of symmetry of a molecule of water. Table. Table of multiplication operations of symmetry in the molecule of the water

4 In general if the molecule rotates about an ais of symmetry of the angle then the transformation matri of the three aes of an atom which lies on the ais and which ais coincides with the ais of symmetry is: cos sin y ( ) y sin cos y. This matri shows that the alues of which is sin different from ero the new and y represent a miture of old and y namely: cos. sin. y. y sin. cos. y. When (rotation around ) we get y y y. Analogously to the eample of the water we can be presented and an eample of an IVANOVA M. et al. ammonia molecule (NH ) whose molecule is made up of four atoms. The ammonia molecule The ammonia molecule belongs in its groundstate equilibrium geometry to the point group. Its symmetry operations consist of two rotations (rotations by and 4 respectiely about an ais passing through the nitrogen atom and lying perpendicular to the plane formed by the three hydrogen atoms) three ertical reflections and the identity operation. orresponding to these si operations are symmetry elements: the three-fold rotation ais and the three symmetry planes and that contain the three NH bonds and the -ais (Figure ). Figure. The Table shows the result of multiplication operations of the symmetry of the ammonia molecule. symmetry group of ammonia Table. Table multiplication operations of symmetry in the ammonia molecule Trakia Journal of Sciences Vol

5 Note the reflection plane labels do not moe. That is although we start with H in the plane H in and H in if H moes due to the first symmetry operation IVANOVA M. et al. remains fied and a different H atom lies in the plane. Take ( y ) as the basic the matri of all the operations in group E are shown in Table 4. Table 4. Reducible representations of Operations Reducible Representati ons ONLUSION A matri representation of the operations of symmetry helps us understand molecular structure some chemical properties and characteristics of physical properties (spectroscopy) used with group theory to predict ibrational spectra for the identification of molecular shape and as a tool for understanding electronic structure and bonding. REFERENES. Wiberg E. Wiberg N. Holleman A. F. Inorganic hemistry. Academic Press pp han G. K.-L. and Sharma S. The Density Matri Renormaliation Group in Quantum hemistry. Annual reiew of physical chemistry 6: Marti K. H. and Reiher M. New electron correlation theories for transition metal chemistry. Physical hemistry hemical Physics : Legea O. Rohwedder T. and Schneider R. Numerical Approaches for High- Dimensional PDE s for Quantum hemistry in Encyclopedia of Applied and omputational Mathematics B. Engquist. 5. han G. K.-L. Low entanglement waefunctions. omputational Molecular Science. : otton F. A. hemical Applications of Group Theory. Third ed. John Wiley & Sons: anada : Tagliacoo L. Eenbly G. and Vidal G. Simulation of two-dimensional quantum systems using a tree tensor network that eploits the entropic area law. Physical Reiew B. 8: Trakia Journal of Sciences Vol