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Ideas of Quantum Chemistry P107

Ideas of Quantum Chemistry P107 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. | 1026 U. SECOND QUANTIZATION One-electron operators The operator F i h i is the sum of the one-electron operators7 h i acting on functions of the coordinates of electron i. The I Slater-Condon rule says see Appendix M that for the Slater determinant ty built of the spinorbitals the matrix element F hu where hij h j . In the second quantization f E jj. ij Interestingly the summation extends to infinity and therefore the operator is independent of the number of electrons in the system. Let us check whether the formula is correct. Let us insert F j hji into F . We have F Ehijt j l Ehj ÿ F F Ehijôij Eha. i i i This is correct. What about the II Slater-Condon rule the Slater determinants 1 and 2 differ by a single spinorbital the spinorbital i in 1 is replaced by the spinorbital i in 2 We have 1 F 2 E hij 1 F 2 - ij The Slater determinants that differ by one spinorbital produce an overlap integral equal to zero 8 therefore 1 F 2 ha . Thus the operator in the form F j hiji j ensures equivalence with all the Slater-Condon rules. Two-electron operators Similarly we may use the creation and annihilation operators to represent the two-electron operators G 1 Yj g i j . In most cases g i j -1 and G has the form g 2 l 1 2 tn . ij ij ijkl 7Most often this will be the kinetic energy operator the nuclear attraction operator the interaction with the external field or the multipole moment. 8It is evident that if in this situation the Slater determinants 1 and 2 differed by more than a single spinorbital we would get zero III and IV Slater-Condon rule . U. SECOND QUANTIZATION 1027 Here also the summation extends to infinity and the operator is independent of the number of electrons in the system. The proof of the I Slater-Condon rule relies on the following chain of equalities A G A 2 ij kl fif kfa 2 ij kl lij kl 1 E kl Sj - stlsit 2 E ww - wn . ijkl ij because the overlap integral tj kl of the two Slater determinants ijty and kl is non-zero in the two cases only either if i k j l or