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Ideas of Quantum Chemistry P58

Ideas of Quantum Chemistry P58 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. | 536 10. Correlation of the Electronic Motions In the classical MC SCF method we 1. take a finite CI expansion the Slater determinants and the orbitals for their construction are fixed 2. calculate the coefficients for the determinants by the Ritz method the orbitals do not change 3. vary the LCAO coefficients in the orbitals at the fixed CI coefficients to obtain the best MOs 4. return to point 1 until self-consistency is achieved 10.13.2 UNITARY MC SCF METHOD Another version of the MC SCF problem a unitary method suggested by Levy and Berthier61 and later developed by Dalgaard and Jorgensen62 is gaining increasing importance. The eigenproblem does not appear in this method. We need two mathematical facts to present the unitary MC SCF method. The first is a theorem If A is a Hermitian operator i.e. A A then U exp iA is a unitary operator satisfying U U 1. Let us see how U looks U exp iA 1 iA 2 iA 2 3 iA 3 1 i A 2 3 - i A 3 . f1 i A 2 iA 3 iA ----- exp iA . Hence ÛÛ 1 i.e. Û is a unitary operator.63 61 B. Lévy G. Berthier Intern. J. Quantum Chem. 2 1968 307. 62E. Dalgaard P. Jorgensen J. Chem. Phys. 69 1978 3833. 63 Is an operator C of multiplication by a constant c Hermitian dCW C l.h.s. p c c r.h.s. cpW c p . Both sides are equal if c c . An operator conjugate to c is c . Further B iA what is a form of B B B p iA iA B iA . 10.13 Multiconfigurational Self-Consistent Field method MC SCF 537 Now the second mathematical fact. This is a commutator expansion - . - . i i . e-AHeA H H A 2 H A A 3 H A A A 10.29 This theorem can be proved by induction expanding the exponential functions. Now we are all set to describe the unitary method. We introduce two new oper- ators  A j j 10.30 where i and j are the creation and annihilation operators respectively associated to spinorbitals i j see Appendix U. Further S E SIJ I I J I- 10.31 We assume that A ij and Sjj are elements of the Hermitian matrices A and S their determination is the goal of the method are determinants from the