Negative energy continuum/solutions

Thus, the negative-energy electron solutions have now become positive-energy positron solutions, and because of this there is no more danger of radiative decay of bound electrons into the negative continuum. 

V is negative definite for most atomic and molecular potentials. Assuming that we are looking for solutions above the negative-energy continuum, where E > —2mc, it is permissible to invert the term in brackets in the seeond of these equations to get... 

A natural way to generalize the non-relativistic many-body Schrodinger equation is to combine the one-electron Dirac operators and Coulomb and Breit two-electron operators. However such an equation would have serious defects. One of them is the continuum dissolution first discussed by Brown and RavenhaU [36]. This means that the Schrodinger-type equation has no stable solutions due to the presence of the negative energy Dirac continuum. A constrained variational approach to the positive energy states becomes therefore necessary. 

Several conclusions can be drawn from Table 3. First, in accordance with the two-state model, /So and jSj all increase with decreasing HOMO-LUMO gap. Second, the intrinsic second-order polarizability of p-nitroaniline is increased by two-thirds when the solvent is changed from p-dioxane to methanol or A-methylpyrrolidone, even when the values are corrected for the differences in (A ). As we have adopted the value for p-nitroaniline in dioxane as a standard, it should therefore be noted that molecules that truly surpass the best performance of p-nitroaniline should have a second-order polarizability of l. p-nitroaniline (dioxane). As a third conclusion, there is a poor correlation between and the static reaction field as predicted by (91). This is in part due to the fact that the bulk static dielectric constant, E° in (89), differs from the microscopic dielectric constant. For example, p-dioxane has long been known for its anomalous solvent shift properties (Ledger and Suppan, 1967). Empirical microscopic dielectric constants can be derived from solvatochromism experiments, e.g. e = 6.0 for p-dioxane, and have been suggested to improve the estimation of the reaction field (Baumann, 1987). However, continuum models can only provide a crude estimate of the solute-solvent interactions. As an illustration we try to correlate in Fig. 7 the transition energies of p-nitroaniline with those of a popular solvent polarity indicator with negative solvatochromism. 

In the numerical solution of the SCF orbital equations kinetic balance restrictions are not required, as this condition will be satisfied exactly. However, in the numerical solution of MCSCF equations for purely correlating orbitals, difficulties may arise if the orbital energy , becomes too negative (Bierori etal. 1994 Indelicate 1995,1996 Kim et al. 1998). Here it is suggested that we use projection operators to eliminate the functions that correspond to the negative continuum. 

In principle problems of relativistic electronic structure calculations arise from the fact that the Dirac-Hamiltonian is not bounded from below and an energy-variation without additional precautions could lead to a variational collapse of the desired electronic solution into the positronic states. In addition, at the many-electron level an infinite number of unbound states with one electron in the positive and one in the negative continuuum are degenerate with the desired bound solution. A mixing-in of these unphysical states is possible without changing the energy and might lead to the so-called continuum dissolution or Brown-Ravenhall disease. Both problems are avoided if the Hamiltonian is, at least formally, projected onto the electronic states by means of suitable operators (no-pair Hamiltonian) ... 

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