Quartet contaminant

Exercise 3.41 Assume the unrestricted Hartree-Fock (UHF) calculations of Table 3.26 contain only the leading quartet contaminant. That is,... 

As already discussed in the section on UHF, the doubly excited configuration, Irtf rtf rtf), provides correlation between the pair of electrons in rt. The presence of the term 2 X3 in Eq. [10] provides correlation between the p electron in rtj and the a electron in rt2 by confining them to different terminal carbons of allyl radical. The presence of this term makes Pg -F X Pj in Eq. [10] a pure doublet wavefunction, and its absence from pfjhf j-gsuits jn quartet contaminant in the latter wavefunction. This is easily demonstrated, as follows. 

The two sets of coefficients result in two sets of Fock matrices (and their associated density matrices), and ultimately to a solution producing two sets of orbitals. These separate orbitals produce proper dissociation to separate atoms, correct delocalized orbitals for resonant systems, and other attributes characteristic of open shell systems. However, the eigenfunctions are not pure spin states, but contain some amount of spin contamination from higher states (for example, doublets are contaminated to some degree by functions corresponding to quartets and higher states). 

The amount of spin contamination is given by the expectation value of die operator, (S ). The theoretical value for a pure spin state is S S + 1), i.e. 0 for a singlet (Sz = 0), 0.75 for a doublet (S = 1/2), 2.00 for a triplet (S = 1) etc. A UHF singlet wave function will contain some amounts of triplet, quintet etc. states, increasing the (S ) value from its theoretical value of zero for a pure spin state. Similarly, a UHF doublet wave function will contain some amounts of quartet, sextet etc. states. Usually the contribution from the next higher spin state from the desired is... 

BM (room temp.)] were considered consistent with a Mn(II) complex with the uncommon quartet ground state (400). These results were later disputed by Hill et al. (325), who attributed the low value of the magnetic moment to contamination and reported a sextet ground state for this molecule. The same authors confirmed the x-ray isomorphism of Mn(Et2Dtc)2 to Cu(Et2Dtc)2 as was reported previously (236). 

Fig. 27.—Resolution-enhanced, Overall, 360-MHz, H-N.m.r. Spectrum of a Mixture Containing Compounds 35, 35, and 21 in the Ratios of 75 7 12. [The bold numbers in the spectrum refer to the corresponding residues in the structures, and the italic numbers to the compounds in the mixture, each of which occurs as an anomeric mixture in the ratio of a /3 = 2 1. Signals of corresponding protons in the various components of this mixture coincide, unless otherwise indicated. The signals marked by (doublet at 8 1.32 quartet at 8 4.11) originate from a frequently occurring nonprotein, noncarbohydrate contaminant of unknown structure. In addition, the sample contained small proportions of the asialo analogs of 35 and 35, that is, 17 and 18 (see also, footnote 80a).]...

That doublets and triplets are not contaminated by triplets and quartets, respectively, is simply due to the fact that the contaminating multiplicities must be associated with determinants containing the same number of electrons. Hence, doublet states (which must have an odd number of electrons) can only be contaminated by states with multiplicities four, six, and so on. Similarly, triplet states (which must have an even number of electrons) can only be contaminated by states with higher and odd multiplicities. 

The table also contains expectation values of One of the deficiencies of an unrestricted calculation is that it does not produce a pure spin state. The ground state ofthe methyl radical is a doublet with = S(S-h 1)=. The unrestricted calculations produce a doublet wave function, which is contaminated with small amounts of a quartet, sextet, etc, as discussed in Section 2.5. The expectation values of are close to the correct value of J, showing that these contaminants are not large. 

Equation [12] demonstrates that the terms in Eq. [7] that introduce spin polarization into the UHF wavefunction for the k electrons in the allyl radical simultaneously contaminate it with the quartet wavefunction. In fact, the greater the amount of spin polarization in q UHF coeffi-... 

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