Closed-shell molecules spin contamination

By including electron correlation in the wave function the UHF method introduces more biradical character into the wave function than RHF. The spin contamination part is also purely biradical in nature, i.e. a UHF treatment in general will overestimate the biradical character. Most singlet states are well described by a closed-shell wave function near the equilibrium geometry, and in those cases it is not possible to generate a UHF solution which has a lower energy than the RHF. There are systems, however, for which this does not hold. An example is the ozone molecule, where two types of resonance structure can be drawn. Figure 4.8. 

In this paper we will refer specifically to the application of DFT methods to problems involving radicals. These open-shell molecules are more difficult to treat than closed shell ones at the conventional level due to the often inaccurate HF reference configuration. Conventional ab initio calculations of radicals is often complicated by spin contamination, a problem showed to be less severe in DFT calculations (24,37). Therefore, it is useful to apply this methodology to the determination of structure and reactivity of radicals. 

Let us say a little more about these methods. The HF methods are divided into spin-restricted HF (RHF) and spin-unrestricted HF (UHF) methods. Closed-shell systems are almost always calculated using RHF. In this procedure, one set of molecular orbitals is calculated and pairs of electrons are entered to the lowest-energy orbitals. If the molecule has an odd number of electrons, one orbital will be singly occupied and the species is a radical (spin = 0.5, expectation value of the spin-squared operator =0.75). Inmost cases, however, radicals are calculated using the UHF formalism. UHF calculations determine two sets of molecular orbitals, one for each type of spin named alpha and beta. These MO sets are similar but not identical. A radical, for instance, has one more a than P electron. The UHF procedure is more flexible than RHF because the paired a and P orbitals, which correspond to doubly occupied MOs in the RHF formalism, need not be identical. So UHF allows for spin polarization but, on the other hand, spin-contamination occurs (i.e., states of higher spin are mixed into the wave function). 

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