By Carl J. Ballhausen
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In addition, we shall use the subscripts g and u. They stand for the behavior of the representation when the inversion operator t is applied. If the representation does not change sign, we designate it by a subscript g; if it does change sign, the subscript is u. 42 INTRODUCTION TO LIGAND Fl > THEORY . molecule has the following symmetry operations: E, the identity operation. 8 Ca axes. (One of them is shown in the figure. ) 3 C2 axes (the "coordinate axis" in the figure). 6 C, axes (the "coordinate axis" with a rotation plus or minus 90°).
The splitting factor for a many-electron system, and Enz, the one-electron parameter. We have for the diagonal element: (L,M L,S,M s) IL Hr;)l; . ML . Ms (2-80) i On the other hand, since the state (L,M L,S,M s) is a determinantal wave function built up of single orbitals, we get (L,M L,S,M s) I L Hr,)l; . (L,S) in a stepwise fashion with the same limitation as previously: that if more than one term with a given (L,S) is present, we can only get the sum of the X's. H - 1 . 2. · 1·3 3 · 30 INTRODUCTION TO LIGAND FIEI THEORY For the ground state only we have, according to Hund's rule for a shell less than half full, 2:m1 = ML = L and l:m, = M 8 = S.
However, applying two of them in succession is always equivalent to the use of a single symmetry operator. This feature is due to the general property of groups that the product of two elements is contained in the group. By applying the various symmetry operators of the point group, we see, 38 INTRODUCTION TO LIGAND FIE! rHEORY for example, that 0~(1) . 0~'(2) = 0,(1) (3-8) AB before, such equations should be understood in the following sense: first apply 0~' (2) to the molecule and then 0~(1).