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Ideas of Quantum Chemistry P29

Ideas of Quantum Chemistry P29 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. | 246 6. Separation of Electronic and Nuclear Motions ity. Therefore the middle part of the above formula for kinetic energy represents an analogue of mr and the last part is an analogue of 2pm It is not straightforward to write down the corresponding kinetic energy operator. The reason is that in the above expression we have curvilinear coordinates because of the rotation from BFCS to RMCS40 whereas the quantum mechanical operators were introduced Chapter 1 only for the Cartesian coordinates p. 19 . How do we write an operator expressed in some curvilinear coordinates qt and the corresponding momenta pt Boris Podolsky solved this problem41 and the result is T 2 g-2 p Tg 2 G-1p metric tensor where pt -th-pq G represents a symmetric matrix metric tensor of the elements grs defined by the square of the length element ds2 r s grs dqr dqs with g det G and grs being in general some functions of qr. 6.8.5 SEPARATION OF TRANSLATIONAL ROTATIONAL AND VIBRATIONAL MOTIONS Eq. 6.35 represents approximate kinetic energy. To obtain the corresponding Hamiltonian we have to add the potential energy for the motion of the nuclei Uk to this energy where k labels the electronic state. The last energy depends uniquely on the variables ga that describe atomic vibrations and corresponds to the electronic energy Uk R of eq. 6.28 except that instead of the variable R which pertains to the oscillation we have the components of the vectors ga. Then in full analogy with 6.28 we may write UkGii Zn Uk 0 0 . 0 Vk osc 1 ÏN where the number Uk 0 0 . 0 Eel may be called the electronic energy in state k and Vk osc 0 0 --- 0 0. Since after the approximations have been made the translational rotational and vibrational internal motion operators depend on their own variables after separation the total wave function represents a product of three eigenfunctions translational rotational and vibrational and the total energy is the sum of the translational rotational and vibrational energies fully analogous .