Perturbation theory shifted denominators

Hamiltonian has been diagonalized, a correction is applied for the appearance of the level shift in the denominators of the expressions for E,2). This level shift method has been applied successfully to a wide variety of problems in the field of spectroscopy and can be considered as a pragmatic solution to the intruder state problem inherent to perturbation theory. All CASSCF / CASPT2 calculations presented here have been done with MOLCAS-4 [27]. 

So what is wrong with it In this form it is still perturbative, implicitly assuming that the correlation perturbation is relatively small. For many problems we want more flexiblity than offered by perturbation theory. This leads to non-perturbative approaches where various categories of terms in MBPT are summed to all orders. One such method would be to make denominator shifts, so a denominator in perturbation theory like Si — Sa could be replaced by Si — — (aiWai). By adding the anti-symmetrized... 

The second factor in (7.60) describes the transition probability for the two-photon transition. It can be derived quantum mechanically by second-order perturbation theory (see, for example, [7.38,7.39]). This factor contains a sum of products of matrix elements Rik kf for the transitions between the initial level i and intermediate molecular levels k or between these levels k and the final state /, see (2.110). The summation extends over all molecular levels k that are accessible by allowed one-photon transitions from the initial state /). The denominator shows, however, that only those levels k that are not too far off resonance with one of the Doppler-shifted laser frequencies (d = (On —V kn (n = 1,2) will mainly contribute. 

On the one hand, the Moller-Plesset partitioning of into and V is not unique and therefore the different orders of perturbation theory are also not uniquely defined. Various other choices of V were proposed " but they all led to different variants of the Epstein-Nesbet perturbation theory with a shifted denominator. This procedure also seems to be feasible for infinite systems, so there is hope that in the future more than 70 to 75% of the correlation energy will be obtained even in the second order. 

The early attempts to develop MR perturbation methods focused on the use of an effective Hamiltonian determined from the Bloch equation [7, 18, 24, 27]. The main drawback of these theories is the intrader state problem, i.e., the appearance of close to zero denominators which give nonphysical contribution to the effective Hamiltonian matrix elements, especially for large CAS spaces where the high lying model functions are energetically not separated from the outer space determinants. To tackle with this problem the application of incomplete model spaces [18, 27], various level shift-based techniques [14, 23, 30, 45] and the concept of intermediate Hamiltonian [26] were intensively studied, but the most common solution is to use a state-specific description, where only a single target state is described [3-5, 8,16,17, 29,44]. 

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