Level-shift techniques

If necessary, the convergence of the process may be improved by means of the level shifting technique (Saunders and Hillier, 1973). In our case this means substituting Fk in Eq (32) with the matrix [Fk + [Pg.260]

In the case of an intruder state of intermediate strength, one can in many cases remove them by a level shift technique that will be described below. But also in this case is it better if one can extend the active space such that the intruder states disappear. 

One way to remove the intruder state is to use a level shift technique [47], A level shift, e, is added to the zeroth order Hamiltonian, such that the first order equation becomes ... 

Great difficulties with convergency are often encountered with o-pen shell systems for which even a combination of the techniques noted may be helpless. Such cases are called intrinsically divergent. Among methods which are applicable to these cases we note on the level shifting technique suggested by Guest and Saunders ", The method was... 

In summary, there is agreement between the spectroscopic data computed and available experimental information. The CASSCF/ CASPT2 method has thus shown that it is capable of treating not only the ground state but also a large number of excited states for a molecule that has been a frustrating challenge for quantum chemistry for more than 10 years. With the newly implemented level-shift technique, the intruder state problem is also solved, which makes the calculations less involved and the results more stable. The entire field of spectroscopy of transition metal dimers and their ions is therefore open for an accurate theoretical analysis. 

Unfortunately, CASPT2 calculations on excited states are not a routine task. Intruder states, which lead to a failure of the MBPT method in accounting for correlation energy (as discussed earlier), are much more prevalent in excited states than in ground states. Inclusion of the intruding MOs in the active space is not always possible, especially if several excited states must be modeled simultaneously. Recently, level-shift techniques have been proposed to avoid very small energy denominators, but these must be applied with great care if artifacts are to be avoided. 

There are a number of situations in which additional measures must be taken to speed up or establish convergence. One of the most common problems encountered in peptide systems is the occurrence of asymmetric oscillations in the electron distribution from cycle to cycle. In some cases these oscillations damp out on their own, but more often than not, the energy and density matrix undergoes large fluctuations for an indefinite number of SCF cycles. This particular problem can be dealt with quite effectively in conventional MO calculations using what is commonly known as the level shift technique. ... 

It can be shown that within the CASPT2 approach, the reference state directly interacts only with those states that differ from it through either single or double excitation. It is not uncommon to find that the zeroth order energy of one or more of these excited states can be similar to, or even below, the energy of the reference. Such intmder states cause the perturbational approach to fail, sometimes in dramatic fashion, and must be eliminated through either a redefinition of the active space or via the application of level-shifting techniques [30],... 

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