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Ideas of Quantum Chemistry P17

Ideas of Quantum Chemistry P17 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. | 126 3. Beyond the Schrödinger Equation N 2 1s pz jpz 1s 1s p. N 2 1s pz pz 1s 1s px - ipy 1 px ipy 1s j 11 ipy r px ipy 1s x- 1s 1pzpz 1s r x - ipy px ipy 1s . 3.63 In the second row the scalar product of spinors is used in the third row the Hermitian character of the operator p. Further p2 py p2 1s 1 1s px - ipy px ipy 1s 1 1s py 0py 1s - i 1s p px 1s 1 px k H 1s py 1s 1 1s -A 1s 1s px- px 1s r r 1 3.64 r r We used the atomic units and therefore p2 -A and the momentum operator is equal to -iV. The two integrals at the end cancel each other because each of the integrals does not change when the variables are interchanged x y. Finally we obtain the following formula 1 -A 1s 1s r -Z-2 -3Z3 2Z3 Z where the equality follows from a direct calculation of the two integrals.33 The next matrix element to calculate is equal to c n . We proceed as follows please recall kinetic balancing and we also use Appendix H p. 969 c a n Ncl a wh o 1s 0 l a n 33In the first integral we have the same situation as awhile before. In the second integral we write the nabla operator in Cartesian coordinates obtain a scalar product of two gradients then we get three integrals equal to one another they contain x y z and it is sufficient to calculate one of them by spherical coordinates by formula H.2 in Appendix H p. 969. 3.4 The hydrogen-like atom in Dirac theory 127 N ci Pz 1s Px ipy 1s J Pz 1s Px iPy 1s J Nc p z 1s I Pz 1s Px ipy 1s I px ipy 1s Nc 1s p 2 1s 1 cZ2 cZ- The last matrix element reads as ÿ c a n 0 Nd a n 2 1s 0 JI N c ip20W1s I 2l 1 2 0 p2 0 l Nc1s IP 1 czZ cZ Dirac s secular determinant We have all the integrals needed and may now write the secular determinant corresponding to the matrix form of the Dirac equation WV - 8 0 c ff n W c a n p W V - 2c2 - 8 and after inserting the calculated integrals ZC 8 cZ cz Z 2c2 8 0 Expanding the determinant gives the equation for the energy 8 82 8 2ZZ 2c2 ZZ ZZ 2c2 c2Z2 0. Hence we get two solutions 8 c2 ZZ c4 Z2c2. Note that the square root .