Hilbert space method universal

The former approach is referred to as the valence universal (VU) or Pock space MR CC method [51-54] and the latter one as the state universal (SU) or Hilbert space method [55]. In spite of a great number of papers devoted to both the VU and SU approaches, very few actual applications have been carried out since their inception more than two decades ago. Certainly, no general-purpose codes have been developed. This is not so much due to the increased complexity of the MR formalism relative to the SR one, as it is due to a number of genuine obstacles that have yet to be overcome. 

For systems with more than two open shells, it is in general necessary to resort to multireference methods. This section has dealt only with state-specific coupled-cluster methods, also known as state-universal methods or Hilbert space methods, for which a considerable amount of effort has been expended on nonrelativistic multireference methods. The alternative, which is much more suited to multireference problems, is the valence-universal or Fock space method, which has been developed for relativistic systems by Kaldor and coworkers (Eliav and Kaldor 1996, Eliav et al. 1994, 1998, Visscher et al. 2001). 

The method applied to the calculation of the MWD from GPC- and PDC-measurements is formally the same it is based on the inversion of compact integral operators in the Hilbert space in a numerical way. Like the treatment of the problems connected with the analytical solution of the integrodifferential Eq. (41a b), also the treatment of this inversion method cannot be given here in all details it can be found in Ref. 8). Here, only an orientation in this universal and therefore somewhat abstract theory, stated by Greschner on the basis of a general superposition principle, will be given in a form specified for PDC and GPC, enabling easy application. 

Some comments about nonlinearities in the Hamiltonian may be added here. The case we are considering here is called scalar nonlinearity (in the mathematical literature it is also called nonlocal nonlinearity ) [7] this means that the operators are of the form P(u) = (An, u)Bu where A, B are linear operators and<.,.>is the inner product in a Hilbert space. The literature on scalar nonlinearities applied to chemical problems is quite scarce (we cite here a few papers [2,8]) but the results justified by this approach are of universal use in solvation methods. 

The intermediate Hamiltonian approaches presented here may be applied within any multiroot multireference infinite order method. Recently [43] we implemented the XIH scheme to another all-order relativistic multiroot multireference approach, the Hilbert space or state universal CC, which is the main alternative to and competitor of Fock-space CC. This approach will not be discussed here. We only mention that it allows mixing P-space sectors, which can interact strongly, e.g., 1-particle with 2-particle 1-hole [37]. 

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