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Ideas of Quantum Chemistry P62 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. | 576 11. Electronic Motion Density Functional Theory DFT Fig. 11.3. The principle of the Bader analysis of electronic density distribution. d2f The quantity f j 2 works as a magnifying glass for almost invisible humps of the function f x . Fig. a shows a decreasing function f x that is similar to the dependence of the electron density of an atom as a function of the distance from the nucleus with a trace of a hump near X2. Figs. b c show the derivatives f and f respectively. The almost invisible hump on f turns out to be very visible when the function f has been plotted. This is why in the Bader approach A plays the role of a magnifying glass . R.F.W Bader Atoms in Molecules. A Quantum Theory Clarendon Press Oxford 1994 by permission from the Oxford University Press. maximum close to x2. If the cusp at x 0 were absent 10 f would also exhibit a maximum at x 0. We may say that dXf can detect some subtle features of the f x plot and gives maxima where the original function f x has only almost invisible humps . 10Non-zero size of the nucleus or and Gaussian type orbitals. 11.2 Bader analysis 577 There is a similar story with the function -Ap x y z - 2 r ex-dx2 dy2 dz2 cept that here we have three Cartesian coordinates. The way we choose the directions of the Cartesian axes is irrelevant because at any point of space -Ap x y z does not depend on such a choice. Indeed the coordinate systems which we may choose differ by an orthogonal transformation which is peculiar for it does leave the trace of the Hessian i.e. Ap invariant. Imagine now p of an atom decaying with the distance to the nucleus as f x similar to the decay of a smoke cloud Fig. 11.4.a dense in the centre and vanishing outward. Let us calculate the Hessian at every point along the radius. It is easy to calculate Ap x y z simply by summing up the diagonal terms of the Hessian. If we diagonalized the Hessian i.e. rotated the axes in a particular way its eigenvalues would correspond to the curvatures of the .
