Unsolds Theorem

White quotes only J2m=-i [ fa(9)] = C because the exponents cancel when d m( )) = — is 

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Unsold s theorem. 19 Unstable complexes, 547-548 Unsymmetric hydrogen bonding. 302... 

It is an interesting fact that just as the single s orbital is spherically symmetric, the summation of electron density of a set of three p orbitals, five d orbitals, or seven / orbitals is also spherical (Unsold s theorem). Thus, although it might appear as though an atom such as neon with a filled set of s and p orbitals would have a lumpy" electron cloud, the total probability distribution is perfectly spherical. 

Clearly this discrepancy is of some consequence, especially since Ambergeret al. (15) used Fischer s (7) Gml expressions in their recent treatment of the magnetic properties of U(Cot)2. Thus Amberger et al. (15), commenting on the differences between their numerical GML values and those of Hayes and Edelstein (27), point out that only the former meet the requirements of Unsold s theorem (47), but as this test involves the sums of the squares of the coefficients of the X components of Gml it would naturally give no check on the relative signs of S0 and Sw. Moreover,... 

Mowery (48) has confirmed the numerical values obtained by Warren (26), and Hayes and Edelstein also agree (27) in the relative signs of S0 and. In fact Hayes (49) has also informed the author that the Gml values of Hayes and Edelstein (27) were based on 6 = 45° (not 43.40°) for U(Cot)2 and that the S contributions were too large by a factor of /2. When allowance is made for this the results of Warren (26) and Hayes and Edelstein (27) are in full agreement and both naturally satisfy Unsold s theorem. 

A useful theorem, due to Unsold,1 states that the sum of the probability distribution functions for a given value of l and all values of m is a constant that is,... 

For any given value of /, the summation of the electron density probabilities for the complete set of orbitals will be a sphere. This is known as Unsold s theorem. Thus, for example, the sum of the electron density for the Ip, 2py, and 2px orbitals is a sphere, as is the case for the lone 2s orbital. 

The term symbols for a variety of possible electron configurations are shown in Table 4.3. For each entry, the ground-state term symbol is listed first. Note that the completely filled subshells s, p, and are all spherically symmetric, as required by Unsold s theorem and have the Sterm. Half-filled subshells (such as ord are also spherically symmetric. The term symbols for the pairs p and p, p and p", and d , and so on, are identical to one another because the p electron configuration, for instance, can be viewed equivalently as a p (hole) configuration. 

Unsold s theorem asserts that the charge distribution in a filled hydrogen-like subshell is spherically symmetric (independent of 0 and ). 

Where above the spherical symmetry of the average density (9.27) is demonstrated for a single CSF thanks to Unsold s theorem, it can be demonstrated in the general case by combining (9.27), (9.18) and the 3-j sum rule [13]... 

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