Hartree-Fock theory instability

Recently, there have been several successful attempts at the calculation of spin-spin coupling constants at the density functional theory (DFT) level. " In view of the success of the DFT methodology, it would in particular be interesting to see how well DFT performs with respect to the calculation of spin-spin coupling constants. In particular, recent results demonstrated that DFT does not suffer from the triplet-instability problems that have plagued the application of Hartree-Fock theory to the calculation of spin-spin... 

In OCC and BCC theories, the wave function has the form of a standard coupled-cluster wave function with vanishing singles amplitudes. However, unlike standard coupled-cluster theory, where the orbitals are determined in a separate optimization of the reference state HF), the OCC and BCC orbitals are determined simultaneously with the optimization of the cluster amplitudes, making them more suitable for the description of correlation, in a manner reminiscent of MCSCF theory. In practice, the differences between the standard coupled-cluster wave functions and the BCC and OCC wave functions are small, except in systems characterized by Hartree-Fock singlet instabilities such as the allyl radical in Section 10.10.6. In such cases, the Harlree—Fock instability makes the standard approach unsuitable - the BCC and OCC models, by contrast, suffer from no such instabilities. 

As a final note, be aware that Hartree-Fock calculations performed with small basis sets are many times more prone to finding unstable SCF solutions than are larger calculations. Sometimes this is a result of spin contamination in other cases, the neglect of electron correlation is at the root. The same molecular system may or may not lead to an instability when it is modeled with a larger basis set or a more accurate method such as Density Functional Theory. Nevertheless, wavefunctions should still be checked for stability with the SCF=Stable option. ... 

For quantum chemistry, first-row transition metal complexes are perhaps the most difficult systems to treat. First, complex open-shell states and spin couplings are much more difficult to deal with than closed-shell main group compounds. Second, the Hartree—Fock method, which underlies all accurate treatments in wavefunction-based theories, is a very poor starting point and is plagued by multiple instabilities that all represent different chemical resonance structures. On the other hand, density functional theory (DFT) often provides reasonably good structures and energies at an affordable computational cost. Properties, in particular magnetic properties, derived from DFT are often of somewhat more limited accuracy but are still useful for the interpretation of experimental data. 

Thus, the stabilization of the benzene molecule is due to the formation of the six-electron bond. From the MO point of view, the stability would be attributed to the 71 system delocalization. However, as previously mentioned, the Hartree-Fock wave function for benzene (and for all aromatic systems) is unstable [32], and the delocalization effect is exactly a manifestation of this instability [9], Therefore, MO theory does not really explain the stability of these molecules. 

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

In all cases where the question concerning the relative stabilities of equidistant versus bond alternating structures arises (polyyne [20,21, polyacetylene 22-27, polymethineimine 28,29 ) the latter are more stable within the framework of the restricted Hartree Fock approximation. For polyyne and polyacetylene this issue is in accord with the well known concept of a Peierls distortion jsoj. The occurence of Hartree Fock instabilities (see e.g. refs. 31,32 ) in the case of the equidistant, metallic structures of polyyne (cumulene) and all-trans polyacetylene points, however, to the need for improved methods going beyond the independent particle model. First efforts in this direction 27 show that at the level of second order Moller-Plesset perturbation theory the alternant configuration of polyacetylene is still preferred energetically although as expected the energy difference to the equidistant structures becomes smaller. 

To illustrate the problems that may arise in connection with singlet instabilities, we consider the allyl radical C3H5, a planar molecular species of C2v symmetry. When the geometry of this system is optimized at the Hartree-Fock level with no spatial symmetry restrictions applied, a minimum of Cs symmetry is obtained - in disagreement with experiment and also with more elaborate calculations. In the following, we shall study the allyl radical at the RHF and UHF levels of theory in order to understand this behaviour [26]. The allyl radical is shown in Figure 10.4, which also illustrates the coordinates used in this study. 

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