tailieunhanh - Ideas of Quantum Chemistry P38

Ideas of Quantum Chemistry P38 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. | 336 8. Electronic Motion in the Mean Field Atoms and Molecules Variation is an analogue of the differential the differential is just the linear part of the function s change . Thus we calculate the linear part of a change variation s e - ELj j 0 using the yet undetermined Lagrange multipliers Lij and we set the variation equal to The stationarity condition for the energy functional It is sufficient to vary only the functions complex conjugate to the spinorbitals or only the spinorbitals cf. p. 197 yet the result is always the same. We decide the first. Substituting 8 in and retaining only linear terms in 8 l to be inserted into the variation takes the form the symbols 8i and 8j mean 8 and 8 N 1 E I 8i h i - 2 8i j ij i 8j ij - 8i j ji - i 8j ji - 2Lj 8i j j 0. i 1 ij Now we will try to express this in the form N 8i . 0. i 1 Since the 8i may be arbitrary the equation . 0 called the Euler equation in variational calculus results. This will be our next goal. Noticing that the sum indices and the numbering of electrons in the integrals are arbitrary we have the following equalities 22 i 8j ij U 8i ji Ty8i jj ij ij ij i 8j jt j 8i ij 8i j ji ij ij ij and after substitution in the expression for the variation we get 8i h i - 8i j if 8i j if - 8i j ji - 8i j ji - -Lij 8i j H 0. ij 15Note that 8 8tf 0. The Fock equation for optimal spinorbitals 337 Let us rewrite this equation in the following manner 8i h 1 dT2 2 j 2 i 1 -j dT2 2 i 2 j 1 - L jd 0 where 8i . 1 means integration over coordinates of electron 1 and dT2 refers to the spatial coordinate integration and spin coordinate summing for electron 2. The above must be true for any 8i 8 which means that each individual term in parentheses needs to be equal to zero h i 1 dT2 2 j2 i 1 - I d T2 2 i 2 j1 j 12 12 L j 1 . j The Coulombic and exchange operators Let us introduce the following linear operators a two Coulombic operators the total operator J 1 and the orbital operator Jj 1