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Ideas of Quantum Chemistry P75

Ideas of Quantum Chemistry P75 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. | 706 13. Intermolecular Interactions Table 13.1. Situation i Fig. 13.6 Interaction energy Ejnt i 1 a -2 2 2 b R3 -2 - 2 R3 3 c 2 - 2 R3 4 d 2 2 2 R3 Here p. 0 0 a for electrons i 1 2 according to definition 13.17 and . a in a.u. Note that if we assume the same probability for each situation the net energy would be zero i.e. i Eint i 0. These situations have however different probabilities pi because the electrons repel each other and the total potential energy depends on where they actually are. Note that the probabilities should be different only because of the electron correlation. In this total energy there is a common contribution identical in all the four situations the interaction within the individual atoms the remainder is the interaction energy Eint i . If we could somehow guess these probabilities pi i 1 2 3 4 then we could calculate the mean interaction energy of our model one-dimensional atoms as Eint ipiEint i . In this way we could see whether it corresponds to net attraction Eint 0 or repulsion Eint 0 which is most interesting for us. Well but how to calculate them 15 We may suspect that for the ground state we are interested in the ground state of our system the lower the potential energy V x the higher the probability density p x . This is what happens for the harmonic oscillator for the Morse oscillator for the hydrogen-like atom etc. Is there any tip that could help us work out what such a dependence might be If you do not know where to begin then think of the harmonic oscillator model as a starting point This is what people usually do as a first guess. As seen from eq. 4.16 the ground-state wave function for the harmonic oscillator may be written as f0 A exp -BV x where B 0 and V x stands for the potential energy for the harmonic oscillator. Therefore the probability density changes as A2 exp -2BV x . Interesting. Let us assume that a similar thing happens16 for the probabilities pi of finding the electrons 1 and 2 in small cubes of volumes dV1 .