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Ideas of Quantum Chemistry P104

Ideas of Quantum Chemistry P104 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. | 996 M. SLATER-CONDON RULES Fig. M.1. Four Slater-Condon rules I II III IV for easy remembering. On the left side we see pictorial representations of matrix elements of the total Hamiltonian H. The squares inside the brackets represent the Slater determinants. Vertical lines in bra stand for those spinorbitals which are different in bra and in ket functions. On the right we have two square matrices collecting the h s and ij ij - ij ji for i j 1 . N. The dots in the matrices symbolize non-zero elements. and G12 0. This happens because operators F and G represent the sum of at most two-electron operators which will involve at most four spinorbitals and there will always be an extra overlap integral over the orthogonal spinorbitals.9 The Slater-Condon rules are schematically depicted in Fig. M.1. 9If the operators were more than two-particle the result would be different. N. LAGRANGE MULTIPLIERS METHOD Imagine a Cartesian coordinate system of n m dimensions with the axes labelled xi x2 . xn m and a function1 E x where x x1 x2 . xn m . Suppose that we are interested in finding the lowest value of E but only among such x that satisfy m conditions conditional extremum conditional extremum Wi x 0 N.1 for i 1 2 . m. The constraints cause the number of independent variables to be n. If we calculated the differential dE at point x0 which corresponds to an extremum of E then we obtain 0 dxy N.2 where the derivatives are calculated at the point of the extremum. The quantities dxj stand for infinitesimally small increments. From N.2 we cannot draw the conclusion that the jxE 0 are equal to 0. This would be true if the increments dxj were independent but they are not. Indeed we find the relations between them by making differentials of conditions Wi n m x dW Af Vxj j 1 j dxj 0 0 N.3 for i 1 2 . m the derivatives are calculated for the extremum . This means that the number of truly independent increments is only n. Let us try to exploit this. To this end let us multiply each .