Variational collapse principle

If the kinetic balance condition (5) is fulfilled then the spectrum of the L6vy-Leblond (and Schrodinger) equation is bounded from below. Then, in each case there exists the lowest value of E referred to as the ground state. In effect, this equation may be solved using the variational principle without any restrictions. On the contrary, the spectrum of the Dirac equation is unbounded from below. It contains the negative ( positronic ) continuum. Therefore the variational principle applied unconditionally would lead to the so called variational collapse [2,3,7]. The variational collapse maybe avoided by properly selecting the trial functions so that they fulfil the boundary conditions specific for the bound-state solutions [1]. 

The ADFT/ASCF-DFT scheme has been met with considerable reservation. Thus, ADFT/ASCF-DFT assumes implicitly that a transition can be represented by an excitation involving only two orbitals, an assumption that seems not generally to be satisfied. Also, the variational optimization in ASCF-DFT of the orbitals makes it difficult to ensure orthogonality between different excited state determinants when many transitions are considered, resulting ultimately in a variational collapse. Finally, it has been questioned [110] whether there exists a variational principle for excited states in DFT. In spite of this, some of the first pioneering chemical applications of DFT involved ASCF-DFT calculations on excitation energies [36, 113-116] for transition metal complexes and ASCF-DFT is still widely used [117-121]. 

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) ... 

A formal solution is the so-called minimax principle [354], which states that the problem of variational collapse is avoided by determining the minimum of the electronic energy with respect to the large component of the spinor, while guaranteeing a maximum of the energy with respect to the small component. How such a saddle point may look has been shown by Schwarz and Wechsel-Trakowski [217]. The minimax principle has also been discussed in great detail for the complicated two-electron problem [355] (see also Refs. [356,357]). 

From what has been said already with respect to the variational collapse and the minimax principle, it is clear from the beginning that the standard derivation of the Hohenberg-Kohn theorems [386], which are the fundamental theorems of nonrelativistic DFT and establish a variational principle, must be modified compared to nonrelativistic theory [383-385]. Also, we already know that the electron density is only the zeroth component of the 4-current, and we anticipate that the relativistic, i.e., the fundamental, version of DFT should rest on the 4-current and that different variants may be derived afterwards. The main issue of nonrelativistic DFT for practical applications is the choice of the exchange-correlation energy functional [387], an issue of equal importance in relativistic DFT [388,389] but beyond the scope of this book. 

Models of hot isentropic neutron stars have been calculated by Bisnovatyi-Kogan (1968), where equilibrium between iron, protons and neutrons was calculated, and the ratio of protons and neutrons was taken in the approximation of zero chemical potential of neutrino. The stability was checked using a variational principle in full GR (Chandrasekhar, 1964) with a linear trial function. The results of calculations, showing the stability region of hot neutron stars are given in Fig. 7. Such stars may be called neutron only by convention, because they consist mainly of nucleons with almost equal number of neutrons and protons. The maximum of the mass is about 70M , but from comparison of the total energies of hot neutron stars with presupemova cores we may conclude, that only collapsing cores with masses less that 15 M have... 

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... 

In particular, in the early days of relativistic quantum chemistry, attempts to solve the DHF equations in a basis set expansion sometimes led to unexpected results. One of the problems was that some calculations did not tend to the correct nonrelativistic limit. Subsequent investigations revealed that this was caused by inconsistencies in the choice of basis set for the small-component space, and some basic principles of basis-set selection for relativistic calculations were established. The variational stability of the DHF equations in a finite basis has also been a subject of debate. As we show in this chapter, it is possible to establish lower variational bounds, thus ensuring that the iterative solution of the DHF equations does not collapse. 

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