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Ideas of Quantum Chemistry P25

Ideas of Quantum Chemistry P25 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. | 206 5. Two Fundamental Approximate Methods We insert the two perturbational series for Ek A and i k A into the Schrodinger equation H 0 AH 1 40 A2 k2 E iE E . 0 A 1 a2 2 - Ek AEk A Ek n An An and since the equation has to be satisfied for any A belonging to 0 A 1 this may happen only if the coefficients at the same powers of A on the left- and right-hand sides are equal. This gives a sequence of an infinite number of perturbational equations to be satisfied by the unknown E and t . These equations may be solved consecutively perturbational allowing us to calculate Ek and r k with larger and larger n. We have for example equations 0. rr 0 0 __ - 0i 0 for A . H k - Ek k 1- fr 0 1 J 1 0 __ 0i 1 i I7 l 0 for a . h k w -Ek k Ek n 5.20 2. 77 0 2 J 1 1 __ 0i 2 i I7 1 1 i C 2 0 t r A . W . W . Ek iAk Ek Ak Ek iAk XV XV XV XV XV XV XV XV etc.20 Doing the same with the intermediate normalization eq. 5.17 we obtain W 50 - 5.21 The first of eqs. 5.20 is evident the unperturbed Schrodinger equation does not contain any unknown . The second equation involves two unknowns and Ek1 . To eliminate we will use the Hermitian character of the operators. Indeed by making the scalar product of the equation with i k0 we obtain 0 H 0 -E 0 i 1 H 1 -E 1 i 0 Wk l H Ek k H Ek k I i 0 I H 0 -e 0 W1 1 i 0 I H 1 -E 1 i 0 Wk I H Ek k 1 Wk I H Ek k I 0 W0 I H 1 - EM 0 i.e. 20We see the construction principle of these equations we write down all the terms which give a given value of the sum of the upper indices. 5.2 Perturbational method 207 the formula for the first-order correction to the energy Ek1 Hkk . 5.22 where we defined H 1 W0 H 1 W0 Hkn Wk lH Wn 5.23 Conclusion the first order correction to the energy E 1 represents the mean first-order value of the perturbation with the unperturbed wave function of the state in which correction we are interested usually the ground state . 21 Now from the perturbation equation 5.20 corresponding to n 2 we have22 fi 0 H 0 -E 0 fi 2- l. 0 H 1 -E 1 fi 1- -E 2