Variational collapse prevention

However, the problem of variational collapse typically prevents an equivalent SCF description for excited states. That is, any attempt to optimize the occupied MOs with respect to the energy will necessarily return the wave function to that of the ground state. Variational collapse can sometimes be avoided, however, when the nature of the ground and excited states prevents their mixing within the SCF formalism. This simation occurs most commonly in symmetric molecules, where electronic states belonging to different irreducible representations do not mix in the SCF, and also in any situation where the ground and excited stales have different spin. 

The requirement that the wave function should be stationary with respect to a variation in the orbitals, results in an equation that is formally the same as in non-relativistic theory, FC = SCe (eq. (3.51)). Flowever, the presence of solutions for the positronic states means that the desired solution is no longer the global minimum (Figure 8.1), and care must be taken that the procedure does not lead to variational collapse. The choice of basis set is an essential component in preventing this. Since practical calculations necessarily use basis sets that are far from complete, the large and small component basis sets must be properly balanced. The large component corresponds to the normal non-relativistic wave function, and has similar basis set requirements. The small component basis set is chosen to obey the kinetic balance condition, which follows from (8.15). 

Separate initial state and final core-hole state calculations provided ASCF values of the CEBEs. The maximum overlap method [67, 68] was used to prevent variational collapse of the final hole state. This simply replaces the usual aufbau criterion for occupying orbitals in each iteration with a criterion that the occupied orbitals be selected to overlap as much as possible with those of the previous iteration. The Ahhichs VTZ basis set [69] was used, based on the very good results it provided in a recent MCSCF-... 

It has been argued that in order to prevent variational collapse, it is necessary to introduce a projection onto some set of positive-energy states (Sucher 1980, 1984). This, as we have just seen, is not the case. Even if it were the case, there is no prescription for choosing the set of positive-energy states onto which a projection must be made, and in the absence of any such prescription, one might as well choose the self-consistent field states (Mittleman 1981). 

Moreover, the negative sign of the mass-velocity term inevitably leads to a variational collapse in the sense that there is no repulsive term n H = f/nomei + hat could prevent the energy from going to —oo upon free variation on a domain that contains functions of arbitrarily high momentum. These undesirable features of the straightforward expansion get worse in higher orders. ... 

Extremely wide variation in solvent strength with temperature and pressure. Collapse of structure due to capillary forces is prevented during solvent removal. 

Shortly after we had run this comparison study, our own aged freeze-drier collapsed into obsolescence. In order to make this method work, the freeze-drier must be specially constructed, without resin in the vacuum chamber and with traps placed in the vacuum line to prevent the back-diffusion of oil vapors from the pump to the vacuum chamber. While we have been awaiting the rejuvenation of our own instrument, rebuilt to these specifications, Michael McKinnon, of our laboratory, has developed a variation of the Russian evaporation method. In this method, as in the freeze-drying method, the great problem is avoiding contamination. Fortimately, when contamination does occur, it seems to affect an entire batch of samples. It is therefore possible to detect the contamination by the judicious use of standards. This method gives values for DOG of the same order as the lowest freeze-drying values or the Sharp (27) direct injection values. 

With the Coulomb and exchange parts of the MP discussed so far, the core-like solutions of the valence Fock equation would still fall below the energy of the desired valence-like solutions. In order to prevent the valence-orbitals collapsing into the core during a variational treatment and to retain an Aufbau principle for the valence electron system, the core-orbitals are moved to higher energies by means of a shift operator... 

Variations on the simple pleated form include corrugating the paper before pleating, which has the effect of increasing both the surface area and to some extent the stiffness or dimpling the surface or the attachment of separator strips to maintain constant spacing and prevent the collapse of the pleats. Collapse of a pleated element will reduce the effective surface area and, if excessive, may lead to tearing. 

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