The biaxial theorem

It is a mathematical property of spherical harmonic functions Yem(6, f ) that they obey an addition rule which is known as the biaxial theorem if ri and r2 are two vectors, with directions described by the polar angles (6i, 4 ) and (62,4 2), and if a is the included angle between the two vectors, then 

This result is proved in textbooks on mathematics - see for example Jeffreys and Jeffreys [3]. 

The significance of this last equation is that, although it contains a full summation over spherical harmonics on the left hand side, each one of which individually contains a complicated angular dependence, the right hand side is completely independent of either 0 or f . 

The Pauli principle allows each orbital Ygm(0, f ) to be occupied at most by two electrons, one with spin up and the other with spin down. If we fill all the individual angular wavefunctions which are solutions of the independent electron central field equations for a given value of , by putting all 2(2 + 1) (the factor of 2 arises because there are two spin states) electrons into a given subshell, then the resulting charge shell, given by 

It is also possible to produce a spherically symmetric half-filled subshell, in which each orbital contains only one electron, and the spins are all aligned parallel to each other. This situation arises, for example, in the element Mn, which has five electrons in the d subshell, all five of which have their spins pointing in the same direction. The charge density for the half-filled subshell is spherically symmetric, and it therefore has zero total orbital angular momentum L = 0 and a total spin S = 5/2 (its multiplicity is 25 + 1 = 6). Its ground state is therefore 6S5/2- 

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