Symmetry breaking spin contamination

There are a number of other technical details associated with HF and other ah initio methods that are discussed in other chapters. Basis sets and basis set superposition error are discussed in more detail in Chapters 10 and 28. For open-shell systems, additional issues exist spin polarization, symmetry breaking, and spin contamination. These are discussed in Chapter 27. Size-consistency and size-extensivity are discussed in Chapter 26. 

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

Cilice 0 < c < 1 the first term shows tliat UHF orbitals reduce the ionic coutribution relative to the covalent structures, compared to the RHF case, eq. (4.18). This is the same effect as for the Cl procedure (eq. (4.19)), i.e. the first term shows that the UHF wave function partly includes electron correlation The fii t term can he written as a linear combination of the 0 and 1 determinants, and describes a pure singlet. state. The last part of the UHF determinant, however, has terms identical to two of those in the triplet combination (4.22). If we had chosen the alternative set of UHF orbitals A ith the alpha spin being primarily on centre B in eq. (4.20), we would have obtained the other two terms in i.e. the last term in (4.24) breaks the symmetry. The UHF determinant is therefore not a pure spin state, it contains both singlet and triplet spin ates. This feature is known as spin contamination. Foi c — 1 the UHF wave function i... 

The fundamental difference between CN and Nj is simply that one molecule is centrosymmetric while the other is not. The lowest-energy UHF wave function in both cases suffers from unphysical spin localization, and it is illusory to believe that, 2 is the easier of the two molecules to calculate. The low spin-contamination solution to the UHF equations exists simply because of the molecular symmetry, while in CN the lower symmetry of the molecule allows the equations to converge to the lowest energy solution. This is a somewhat unappreciated difficulty in calculations on open-shell molecules. If one has appropriate elements of symmetry, then unphysical solutions can be avoided by enforcing the constraints on the wave function. Even if the constraints are not enforced, problems with the reference function are easily identifiable nonzero dipole moments along directions where the exact value must vanish by symmetry, unsymmetric spin densities, and so on. However, the issue is more diabolical in lower-symmetry species where localization does not break the framework molecular symmetry. In these cases, UHF and ROHF... 

EXX methods can be extended to treat the correct many-particles symmetries in open-shell systems and transition metals in order to avoid spin-contamination and symmetry-breaking." " ... 

In summary, CASSCF represents a viable method for calculations on radicals in which ROHF symmetry breaking and/or pronounced UHF spin contamination are encountered. As we will see, CASSCF is indispensable whenever a multiconfigurational wavefunction is required (e.g., for singlet diradicals and for some excited states). Because of the sometimes rather arbitrary choice of... 

The strength of CISD is that, by virtue of its variational nature, it can rectify rather severe shortcomings of the starting wavefunction, such as spin contamination with UHF and symmetry breaking with ROHF. Even when symmetry breaking is encountered, judicious use of CISD allows geometry optimizations. CISD is free of the problems encountered in CASSCF calculations that are associated with the need to choose a subset of MOs for inclusion in the active space. 

CC calculations do provide a safe means for navigating between the Scylla of UHF spin contamination and the Charybdis of ROHF symmetry breaking for radicals. Cost is the main disadvantage of CC calculations. If performed with large polarized basis sets of the type required to obtain results of chemical accuracy, CCSD(T) and even CCSD calculations rapidly become prohibitively expensive as the size of the molecules on which they are performed increases. When they cannot be applied directly to the molecules under study, CC calculations on smaller model systems can sometimes be employed to validate the use of lower level procedures on the actual molecules of interest. This protocol has, in fact, proven useful in several recent studies. ... 

If one attempts to avoid the Scylla of spin contamination, which plagues calculations based on UHF wavefunctions, one encounters the Charybdis of symmetry breaking in ROHF wavefunctions. Symmetry breaking often makes ROHF geometry optimizations and energies even less reliable than their UHF counterparts, and RMP2 calculations will not solve the problems caused by symmetry-broken ROHF wavefunctions. 

A stability analysis can be carried out for clo.sed shell state SCF and DFT descriptions this shows whether the corre-.sponding ansatz for the wavefunction is stable with respect to variations which break spin (triplet contamination) or spatial symmetry. 

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