Variational collapse method

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]. 

We could not show here the results of solving the relativistic Dirac-Coulomb equation. The FC method can be extended to the case of the Dirac-Coulomb equation with only a small modification [36]. It is important to use the inverse Dirac-Coulomb equation to circumvent the variational collapse problem which often appears in the relativistic calculations [37]. 

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-... 

Systematic location errors could occur due to high deformation of the rock specimen. To minimize the travel-time residuals, systematic location errors associated with picking errors and the velocity variations due to microcracking were removed by the application of the joint-hypocenter determination (JHD) method (Frohlich 1979). Using the JHD method, "station corrections" can be determined that account for consistent inaccuracies of the wave velocity along the travel path especially near sensor positions. To delineate structures inside a clouded AE event distribution the collapsing method, which was first reported by Jones and Steward [1997], can be applied. This method describes how the location of an AE event can be moved within its error ellipsoid in order that the distribution of movements for every event of a cloud approximates that of normally distributed location uncertainties. This does not make the location uncertainties in the dataset smaller but it highlights structures already inherent within the unfocussed dataset. 

P. Falsaperla, G. Fonte, J. Z. Chen. Two methods for solving the Dirac equation without variational collapse. Phys. Reo. A, 56(2) (1997) 1240-1248. 

The fuUy relativistic LCAO method for soUds, based on the DKS scheme in the LDA approximation was represented in [541]. The basis set consists of the numerical-type orbitals constructed by solving the DKS equations for atoms. This choice of basis set allows the spurious mixing of negative-energy states known as variational collapse to be overcome. Furthermore, the basis functions transform smoothly to those in the nonrelativistic limit if one increases the speed of light gradually in a hypothetical way. 

What this means for mean-field theory is that the lowest electron eigenvalue of the one-particle matrix that we are diagonalizing can never fall below the lowest eigenvalue of the positive-positive block of the matrix in any one iteration, and therefore there is no problem with variational collapse in a self-consistent field procedure, provided that the set of states in which we have formed the matrix represent the solutions of some one-particle Dirac equation. Failure to ensure a proper representation in a finite basis has been the occasion of problems that appear to exhibit variational collapse. Further discussion of this issue will be postponed to chapter 11, which covers finite basis methods. 

Dirac equation. This method of eliminating the small component is not a procedure that leads to a simplification. It does, however, have some motivation, both physical and practical. First, it projects out the negative-energy states, and leaves a Hamiltonian that may have a variational lower bound, avoiding the potential problem of variational collapse. Second, it removes from explicit consideration the small component, and with the use of the Dirac relation (4.14) it yields a one-component operator that can be used in nonrelativistic computer programs. 

There is far less reported experience for the HF studies of electronic excited states (ESs). Especially, highly, doubly and core hole excited (ionized) states are not often studied. It is clear that existing ground state self-consistent field (SCF) methods cannot be directly applied to excited states of the same symmetry or of the same spin multiplicity as a lower state because of the so-called variational collapse i.e., the optimization procedure will find only the lowest solution of a given symmetry or a given spin multiplicity. Therefore, such calculations for ES cannot be considered as routine. The most powerful scheme for accurate treatment of ESs is based on multireference methods [2-8]. They typically provide an accuracy of about 0.1 eV but require the expense of much computational cost. Thus, it can be quite difficult to carry out the corresponding calculations. Such methods are, however, indispensable to study systems where... 

To avoid variational collapse, it is probably advisable to use an SCF convergence algorithm that is based on direct minimization rather than extrapolation methods such as direct inversion in the iterative subspace (DIIS) " and related methods, which are the default convergence algorithms in most quantum chemistry programs. Direct minimization, while often very slow to reach convergence, is more likely to converge to the desired local minimum in the space of MO coefficients. 

Nevertheless, even direct minimization remains vulnerable to variational collapse, since the newly-occupied MO of the anion is subject to a different potential as compared to virtual MOs that might be nearby in energy. Subsequent SCF iterations can therefore modify the energetic ordering of the MOs, and in such cases, it is unclear which MOs should be the occupied ones at the next SCF iteration. The maximum overlap method (MOM) offers a possible solution to this problem, and a more refined version of the orbital relaxation technique. 

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