Ideas of Quantum Chemistry P82 shows how quantum mechanics is applied to chemistry to give it a theoretical foundation. The structure of the book (a TREE-form) emphasizes the logical relationships between various topics, facts and methods. It shows the reader which parts of the text are needed for understanding specific aspects of the subject matter. Interspersed throughout the text are short biographies of key scientists and their contributions to the development of the field. | 776 14. Intermolecular Motion of Electrons and Nuclei Chemical Reactions Fig. . a The three equivalent Jacobi coordinate systems. b The Euler angles show the mutual orientation of the two Cartesian coordinate systems. First we project the y axis on the X y plane the result is the dashed line . The first angle a is the angle between axes X and z the two other ft and y use the projection line described above. The relations among the coordinates are given by H. Eyring J. Walter . Kimball Quantum Chemistry John Wiley New York 1967. The three Jacobi coordinate systems are related by the following formulae cf. Fig. ri Ri cos ßij - sin ßij sin j r l. cos ßij Rj Mk tan ßij - V ßij -ßji- The Jacobi coordinates will now be used to define what is called the more convenient hyperspherical democratic coordinates. Democratic hyperspherical coordinates When a chemical reaction proceeds the role of the atoms changes and using the same Jacobi coordinate system all the time leads to technical problems. In order Accurate solutions for the reaction hypersurface three atoms 777 not to favour any of the three atoms despite possible differences in their masses we introduce democratic hyperspherical coordinates. democratic First let us define the axis z of a Cartesian coordinate system which is perpen- hypersphericai dicular to the molecular plane at the centre of mass . parallel to A Xr x R coordinates where r and R are any just democracy the result is the same of the vectors rk Rk. Note that by definition A represents the area of the triangle built of the atoms. Now let us construct the axes x and y of the rotating with molecule coordinate system RMCS cf. p. 245 in the plane of the molecule taking care that the Cartesian coordinate system is right-handed the axes are oriented along the main axes of the moments of inertia 18 with Iyy ptf R2 Ixx p X RX . Finally we introduce democratic hyperspherical coordinates equivalent to RMCS the first coordinate .
