By W. G. Harter, C. W. Patterson
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Extra resources for A unitary calculus for electronic orbitals
MIXED ORBITAL CONFIGURATIONS (21)nl (~2)n2 ... The following discussion of the unitary basis for several sub-shells is meant to explain additional calculational procedures, and to show further structure of the bases and operators. If a single electron can occupy several orbital subshells n~ I, n'~ 2 .... there will be m = (2~i+I) + (242+1) + i @ e orbital states for it. 36. 1 and gives the matrices in Tables II-IIl. 37 below. = E63+ ~ = E36+J~E47+J~E58 E74÷~E85 The numbering for Eij reflects the choice of numbers 1-5 for d states and 6-8 for the p states.
39 then give all other orderings. help explain the properties To of the permutation projectors (These properties were first discovered by Goddard 14. ) we shall make an analogy with the quantum theory of rigid rotators. 40 P ~ has been applied to a state 1000> of definite mn orientation (000). By (000) we mean that the rotator body axes 1, 2, and 3 are lined up with laboratory axes x, y, and z. 40 then give all other orientations. Now the quantum numbers m and n are eigenvalues of angular momentum operators Jz andJ3 for the lab z-axis and body 3-axis components respectively, In fact, two commuting groups of rotation operators and ~(~w) ~(~v) can be defined.
31(b-c) are derived Ia w i t h Ib, and that operators from products of U3xU 2 are v k0 q0 and v 00 ~X " Now the exclusion principle only totally anti-symmetric permits us to have multi-electron states with tableau of one column only. e. elements states or an i a l r e a d y pre- -1 if the new i must be m o v e d an odd of boxes to get "lexical" a c Ecf d = 0 , e f U6states Slater D e t e r m i n a n t are 0 if there is no j to be destroyed sent in initial state, of tables order, +1 if the new i is of order.
A unitary calculus for electronic orbitals by W. G. Harter, C. W. Patterson