tailieunhanh - Ideas of Quantum Chemistry P103

Ideas of Quantum Chemistry P103 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. | M. SLATER-CONDON RULES The Slater determinants represent something like the daily bread of quantum chemists. Our goal is to learn how to use the Slater determinants when they are involved in the calculation of the mean values or the matrix elements of some important operators. We will need this in the Hartree-Fock method as well as in other important methods of quantum chemistry. Only the final results of the derivations presented in this Appendix are the most important. Antisymmetrization operator The antisymmetrization operator is defined as A N D-1 where P represents the permutation operator of N objects in our case - electrons while -1 p stands for the parity of the permutation P even odd - if a given permutation P can be created from an even odd number p of transpositions . exchanges of two elements. The operator J has some nice features. The most important is that when applied to any function it produces either a function that is antisymmetric with respect to the permutations of N elements or This means that A represents a sort of magic wand whatever it touches it makes antisymmetric or causes it disappear The antisymmetrizer is also idempotent . does not change any function that is already antisymmetric which means A2 1. Let us check that A is indeed idempotent. First we obtain A2 N -1 -1 pP N -1 -1 pP N -2 -1 p p PP . P P PP Of course PP represents a permutation opera tor 2 which is then multiplied by its own parity -1 p p and there is a sum over such permutations at a given fixed P . 1 In the near future these elements will be identified with the electronic coordinates one element will be represented by the space and spin coordinates of a single electron x y z a . 2The permutations form the permutation group. From Solid State and Molecular Theory Wiley London 1975 by John Slater on the permutation group . It was at this point that Wigner Hund Heitler and Weyl entered the picture with their Grup- 986 M. SLATER-CONDON RULES 987 .