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Ideas of Quantum Chemistry P30

Ideas of Quantum Chemistry P30 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. | 256 6. Separation of Electronic and Nuclear Motions __ _ where Vj V R fij and coupling of diabatic states JEz R r Ro Ho R i r Ro Ei R Vu R . 6.44 The crossing of the energy curves at a given R means that E E_ and from this it follows that the expression under the square root symbol has to equal zero. Since however the expression is the sum of two squares the crossing needs two conditions to be satisfied simultaneously E1 - E2 0 6.45 IV121 0. 6.46 Two conditions and a single changeable parameter R. If you adjust the parameter to fulfil the first condition the second one is violated and vice versa. The crossing E E_ may occur only when for some reason e.g. because of the symmetry the coupling constant is automatically equal to zero V12 0 for all R. Then we have only a single condition to be fulfilled and it can be satisfied by changing the parameter R i.e. crossing can occur. The condition V12 0 is equivalent to IH12I 1 Ho R 2 0 because H0 R H0 R0 V and Ho Ro 12 0 due to the orthogonality of both eigenfunctions of H0 R0 . Now we will refer to group theory see Appendix C p. 903 . The Hamiltonian represents a fully symmetric object whereas the wave functions 1 and 2 are not necessarily fully symmetrical because they may belong to other irreducible representations of the symmetry group. Therefore in order to make the integral H12 0 it is sufficient that 1 and 2 transform according to different irreducible representations have different symmetries .48 Thus the adiabatic curves cannot cross if the corresponding wave functions have the same symmetry. What will happen if such curves are heading for something that looks like an inevitable crossing Such cases are quite characteristic and look like an avoided crossing. The two curves look as if they repel each other and avoid the crossing. If two states of a diatomic molecule have the same symmetry then the corresponding potential energy curves cannot cross. 48H12 transforms according to the representation being the direct .