Ideas of Quantum Chemistry P22 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. | 176 4. Exact Solutions - Our Beacons RIGID ROTATOR A rigid rotator is a system of two point-like masses m1 and m2 with a constant distance R between them. The Schrodinger equation may be easily separated giving two equations one for the centre-of-mass motion and the other for the relative motion of the two masses see Appendix I on p. 971 . We are interested only in the second equation which describes the motion of a particle of mass equal to the reduced mass of the two particles and position in space given by the spherical coordinates R e where 0 R rc 0 e n 0 0 2n. The kinetic energy operator is equal to - A where the Laplacian A represented in the spherical coordinates is given in Appendix H on p. 969. Since R is a constant the part of the Laplacian which depends on the differentiation with respect to R is In this way we obtain the equation equivalent to the Schrodinger equation for the motion of a particle on a sphere sin e 1y EY sin ede de sin e 2 d 2 where Y e is the wave function to be found and E represents the energy. This equation may be rewritten as Y is also an eigenfunction of J2 J2Y 2 R2EY where J2 is the square of the angular momentum operator. Eq. may be also written as 1 1 1 d . dY 1 d2Y1 y sinede sin de sine 2d 2j where A - R2E. The solution of the equation is known in mathematics as a spherical spherical harmonic 47 it exists if A J J 1 J 0 1 2 . harmonics y 0 Njm PJM 1 cos ff A- exp M 2n 46This reasoning has a heuristic character but the conclusions are correct. Removing an operator is a subtle matter. In the correct solution to this problem we have to consider the two masses with a variable distance R with the full kinetic energy operator and potential energy in the form of the Dirac delta function see Appendix E on p. 951 8 R R0 . 47There are a few definitions of the spherical harmonics in the literature see . Steinborn K. Rue-denberg Advan. Quantum Chem. 7 1973 1 . The Condon-Shortley convention often
