Variational collapse

The radial functions Pmi r) and Qn ir) may be obtained by numerical integration [16,17] or by expansion in a basis (for recent reviews see [18,19]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [20,21], where admixture of negative-energy solutions may yield energies much below experimental. To avoid this failure, the basis sets used for expanding the large and small components must maintain kinetic balance [22,23]. 

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]. 

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. 

Except for the efforts mentioned above, relativistic calculations of shielding evaluate the main relativistic effects using one or two component limits of the four-component formalism, quasi-relativistic approaches. These avoid the variational collapse in the calculation of the scalar relativistic terms by employing frozen cores, or effective core potentials. Some include the one-electron spin-orbit terms, and sometimes the higher order spin-orbit terms too. Others include both scalar and spin-orbit terms. Ziegler... 

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]. 

Conceptually, one of the simplest ways to study excited states at the correlated level of theory is by means of A methods, in which is obtained with some post-HF method. While the problems of variational collapse discussed above severely complicate efforts to calculate excitation energies by ASCF methods, it is actually straightforward to obtain ACI excitation energies, provided the same reference function is used for both states. In this case, the... 

There are other reasons why methods based on Dirac Hamiltonians have been unpopular with quantum chemists. Dirac theory is relatively unfamiliar, and the field is not well served with textbooks that treat the topic with the needs of quantum chemists in mind. Matrix self-consistent-field equations are usually derived from variational arguments, and as a result of the debates on variational collapse and continuum dissolution , many people believe that such derivations are invalid for relativistic problems. Most implementations of the Dirac formalism have made no attempt to exploit the rich internal structure of Dirac... 

The matrix form of the atomic Dirac-Hartree-Fock (DHF) equations was presented by Kim [37,95], who used a basis set of modified radial Slater-type functions, without the benefit of a balancing presciption for the small component set. A further presentation of the atomic equations was made by Kagawa [96], who generalized Kim s work to open shells and discussed matrix element evaluation. An extension to include the low-ffequency form of the Breit interaction self-consistently in an S-spinor basis was presented by Quiney [97], who demonstrated that this did not produce variational collapse. Our presentation of the DHFB method is based on [97-99]. 

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