Symmetry dilemma

This behaviour is a nice example of the symmetry dilemma in the conventional Hartree-Fock scheme and is intimately connected with the question of Hartree-Fock instability. 

All the basis functions were given independently variable orbital exponents (CH4, 9 NH3,8 H20,7) and all exponents were optimised by the quadratically convergent direct search method of Fletcher (19). For comparison, the calculations were repeated with the GHOs constrained to have the symmetry of the molecule three independent variables for CH4 (1 sc, sp3, 1 sH) four for NH3 (1 sN, sp3, sp3, 1 sH) and four for H20 (1 sQ, sp3, sp3,1 sH). The most striking qualitative result is the confirmation of the results quoted earlier for H2 when the orbital exponents are all optimised, the GHO basis has the symmetry of the molecule there is no spatial symmetry dilemma.9)... 

The fact that the GHOs, when optimised in a molecular environment, generate an atomic orbital basis which has just the symmetry of the molecule (i.e. not higher or lower symmetry) is extremely encouraging — it tends to reassure that the GHO basis does refelct the physics of the interactions adequately. Thus, within the approximation scheme of a minimal GHO basis, the symmetry dilemma is actually a choice of basis dilemma and can be satisfactorily solved by using near-optimum scale factors for the GHOs. 

Figure 5-3. The symmetry dilemma in present-day DFT starting from the cylindrically symmetric molecular K-density (a), the dissociation into atomic fragments can either be computed with correct atomic densities but a wrong energy (b) or a correct energy, but wrong (because symmetry broken) atomic densities (c) (isodensity surfaces at 0.01 a. u. constructed from the p-orbital space adapted from Savin in Recent Developments of Modem Density Functional Theory, Seminario, J. M. (ed.), 1996, with permission from Elsevier Science).

It is a peculiarity of the Hartree-Fock scheme that the properties of the approximate eigenfunction Ca does not always reflect the properties of the exact eigenfunction C. A well-known example is given by the symmetry dilemma 8, which says that if an eigenfunction C has a special symmetry property characterized by the projector O. so that... 

Perhaps the greatest need for Brueckner-orbital-based methods arises in systems suffering from artifactual symmetry-breaking orbital instabili-ties, " ° where the approximate wavefunction fails to maintain the selected spin and/or spatial symmetry characteristics of the exact wavefunction. Such instabilities arise in SCF-like wavefunctions as a result of a competition between valence-bond-like solutions to the Hartree-Fock equations these solutions typically allow for localization of an unpaired electron onto one of two or more symmetry-equivalent atoms in the molecule. In the ground Ilg state of O2, for example, a pair of symmetry-broken Hartree-Fock wavefunctions may be constructed with the unpaired electron localized onto one oxygen atom or the other. Though symmetry-broken wavefunctions have sometimes been exploited to produce providentially correct results in a few systems, they are often not beneficial or even acceptable, and the question of whether to relax constraints in the presence of an instability was originally described by Lowdin as the symmetry dilemma. ... 

In fact, symmetry requirements on molecular orbitals introduce in a variational calculation certain constraints, which raise the total energy 18>. This problem, called the symmetry dilemma , has been studied for some n electron systems 19). It is not important for the present discussion because for a system of closed-shell type the NO s associated with a total wave function of correct symmetry are automatically symmetry-adapted. 

Fukutome11 in a series of papers has made a thorough analysis of the symmetry properties of the GHF spin orbitals. As pointed out by Low-din,12 there is no reason why a solution D of (2.23) should be symmetry adapted to the symmetry of the Hamiltonian. We can impose symmetry constraints on the spin orbitals, but that will in general raise the total energy. To characterize this situation Lowdin has coined the term symmetry dilemma. ... 

Symmetrised density-functionals, which have been proposed recently [88] as the correct solution of the symmetry dilemma in Kohn-Sham theory, also naturally lead to fractional occupations. The symmetry dilemma occurs because the density or spin-density of KS theory may exhibit lower symmetry them the external potential due to the nuclear conformation. This in turn leads to a KS Hamiltonian with broken symmetry, leading to electronic orbitals that cannot be assigned to an irreducible representation... 

Since (for fixed -I- nj P " i( f > n m = 0) is an even function ofthis alternative interpretation encounters no LSD or GGA spin-symmetry dilemma, In the separated-atom limit for H2, it correctly makes Px i r,r) = 0 for in the vicinity of either atom, since (by the Pauli exclusion principle) P"= (Hf, n u = 0) vanishes when either or vanishes. 

FUKUTOME CLASSES Symmetry dilemmas and the Fock operator... 

It should be emphasized that both the RHF solution and the UHF solution in Figure 10.6 are approximations to the same wave function. The 2 RHF wave function transforms in the same manner as the true wave function but is higher in energy than the symmetry-broken UHF solution. The difference in energy is very small, however, compared with the tme excitation energies of the system. Neither wave function may be classified as best and, for a satisfactory resolution of this symmetry dilemma of Hartree-Fock theory, a more advanced wave function must be used, as described in Section 12.8. 

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