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Ideas of Quantum Chemistry P95

Ideas of Quantum Chemistry P95 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. | 906 C. GROUP THEORY IN SPECTROSCOPY This means that the displacement in space of function f r is simply equivalent to leaving the function intact but instead inversing the displacement of the coordi-7 nate system. Operators TR rotate functions without their deformation therefore they preserve the scalar products in the Hilbert space and are unitary. They form a group isomorphic with the group of operators R because they have the same multiplication table as operators R if R R1R2 then TR TRi7t2 where TR1 f r f R-ir and TR2f r f R-ir . Indeed 7 8 TRf TR1TR2 f r f R-1R-1r fiR r . 2 2 1 UNITARY VS SYMMETRY OPERATION Aunitary operation is a symmetry operation of function f r when RRf r f r . Example 6. Rotation of a point. Operator RR a z of the rotation of a point with coordinates x y z by angle a about axis z gives a point with coordinates x y z Fig. C.i.a x r cos f a r cos f cos a - r sin f sin a x cos a - y sin a y r sin f a r sin f cos a r cos f sin a x sin a y cos a Z z the corresponding transformation matrix of the old to the new coordinates there- fore is cos a sin a 0 U sin a 0 cos a 0 0 1 We obtain the same new coordinates if the point remains still while the coordinate system rotates in the opposite direction i.e. by angle -a . Example 7. Rotation of an atomic orbital. Let us construct a single spherically symmetric Gaussian orbital f r exp - r - r0 2 in Hilbert space for one electron. Let the atomic orbital be centred on the point indicated by vector r0. Operator RR a z has to perform the rotation of a function9 by angle a about axis z 7Motion is relative. Let us concentrate on a rotation by angle a. The result is the same if the coordinate system stays still but the point rotates by angle a or the point does not move while the coordinate system rotates by angle -a. What would happen if function f ri r . rtf is rotated Then we will do the following RRf ri r2 . rN f R-1r1 R-1r2 . R-1rN . 8This result is correct but the routine notation works in a quite .