Computer Animations of Molecular Vibration Michael McGuan and Robert M. Hanson Summer Research 2004 Department of Chemistry St. Olaf College Northfield, Minnesota Vibrating molecules can be displayed in a web browser with the Java applet Jmol. In order for Jmol to display a vibrating molecule, it must be given a vector for each atom in the direction that the atom moves when the molecule is vibrating. To facilitate this process, we have developed a web site that uses a JavaScript application to allow the user to construct a molecule with simple geometry. The user communicates with the JavaScript application by entering data into an html page. The JavaScript application then calculates the vectors on each atom for the selected mode of vibration and displays the vibrating molecule using Jmol. The web site is available at Abstract Goals of the Project JavaScript application Jmol output html page user The goal of our project was to create a web page that will display vibration modes for user-created molecules. These modes are calculated from user input for molecular geometry, bond lengths, and bond angles. Introduction Matter is composed of particles called atoms, which combine by forming chemical bonds to make entities known as molecules. Real bonds are not rigid like those depicted in ball and stick models. Rather, bonded atoms behave as though they were connected by a spring, and are therefore free to oscillate in space. This type of motion is called vibration, and it results in stretching of bonds and deformation of the molecule’s shape without changing the molecule’s center of mass or its angular momentum. Each mode of vibration occurs at a specific frequency that depends on the masses of the vibrating atoms and the strength of the bond between them. Determining the Number of Vibrations Each atom in a molecule can move independently in the x, y, and z directions, so there are 3n possible motions for a molecule containing n atoms. Of these, 3 correspond to translation of the entire molecule in the direction of one of the axes. For nonlinear molecules, an additional 3 motions correspond to rotation about an axis. In linear molecules, only 2 rotations are possible. The remaining motions are vibrations, and they are called the normal modes of vibration. 3n-6 normal modes are available to a nonlinear molecule, while 3n-5 are available to a linear molecule. x x x y y y zz z z zz translation rotation x y z x y z translation rotation nonlinear linear Types of Vibration There are two types of molecular vibration. In a stretching vibration, atoms move parallel to their bond axes and the bond lengths change while the angles generally do not change significantly. In a bending vibration, atoms move perpendicular to their bond axes and the bond angles change while bond lengths remain the same. stretchingbending Molecular Symmetry Normal modes of vibration involve highly symmetrical movement of atoms. These modes can be determined by considering the symmetry of the vibrating molecule. A molecule’s symmetry consists of a set of operations which, when performed, leave the molecule in an orientation that is indistinguishable from its original orientation. Symmetry Operations C n : rotate molecule 360/n degrees about axis  :  reflect molecule through plane i: center of inversion; sends each point with coordinates (x,y,z) to (-x,-y,-z) S n : rotate molecule 360/n degrees about axis, then reflect through plane perpendicular to axis C3C3 E: do nothing to the molecule S4S4 Symmetry of Vibrations The set of symmetry operations that can be performed on a molecule is called its point group. The symmetry of a vibration mode is related to the vibrating molecule’s point group. Every point group has a character table that describes ways in which vibration modes can behave when subjected to the symmetry operations. The rows in the character table are called representations, and the entries in each cell are called characters. The dimension of a representation is defined to be the character under the identity operation E. Vibrations always have the same symmetry as one of the rows of the character table. Character table for point group D 3h Representations with A or B symmetry are 1-dimensional and always have characters of 1 or –1. A character of 1 indicates that the representation is symmetric with respect to the operation. A character of –1 means that the representation is antisymmetric with respect to the operation. This is a 1-dimensional bending vibration mode for a molecule with D 3h symmetry. To determine the symmetry type of this mode, operate on it with each of the symmetry operations in the D 3h point group and see whether the direction of the vectors is reversed or not. E2C32C3 3C2’3C2’ hh vv operation result character 11 The characters for this vibration match the A 2 ” column of the character table, so this mode is called A 2 ” bend. 2S32S3 1 Degenerate Modes Sometimes sets of vibrations occur at exactly the same frequency. These vibrations are said to be degenerate. A degenerate set of n vibrations transforms as one of the n-dimensional representations in the character table, but individual members of the degenerate set do not transform like any 1-dimensional representation. When a symmetry operation is performed on one member of a degenerate set, it need not be converted into itself or minus itself as would be the case for a 1-dimensional mode. Symmetry operations may turn one mode in a degenerate set into one of the other modes or a linear combination of the other modes. In order to determine the symmetry of a degenerate set of vibrations, it is necessary to first represent members of the set by vectors. Here are two degenerate vibrations for a square pyramidal molecule with C 4v symmetry. A vector has been assigned to each each vibration vector vibration The effects of each symmetry operation on the degenerate set can be expressed as a matrix. The first column of the matrix is the effect of the symmetry operation on the first mode, and the second column is the effect of the symmetry operation on the second mode. The character of the symmetry operation is the sum of the diagonal elements in the matrix. EC4C4 C2C2 vv dd operation 1010 effect on 0101 matrix character effect on This set of degenerate modes has the same characters as the E representation in the C 4v character table. In general, for an n-dimensional mode, each vibration is assigned an n x 1 vector, and each symmetry operation is represented by an n x n matrix. This is the character table for the C 4v point group. Displaying Vibrating Molecules with Jmol After a molecule’s vibration modes have been determined by a symmetry analysis, these modes can be displayed in Jmol. First, it is necessary to specify each atom’s location in space with x, y, and z coordinates. These can be calculated from experimentally determined bond lengths and angles. Once coordinates have been assigned, a vibration can be added with vectors. These vectors determine the direction in which atoms move during vibration Å 2.01 Å (0,0,0) (1.70,-0.46,0) (-0.85,-0.46,-1.47) (-0.85,-0.46,1.47) (0,2.01,0) experimental datacoordinates (0,1.6,0) (-0.38,-1.4,0) (0.19,-1.40,0.33) (0.19,-1.40,-0.33) (0,1.6,0) 105° vectors Data about molecular structure and vibration vectors is included in the html code which invokes the Jmol applet. Jmol reads this code and uses it to display a vibrating molecule. coordinates vectors atomic numbers