Multireference space

L. Meissner and R. J. Bartlett, J. Chem. Phys., 94, 6670 (1991). Transformation of the Hamiltonian in Excitation Energy Calculations Comparison Between Fock-Space Multireference Coupled-Cluster and Equation-of-Motion Coupled-Cluster Methods. 

J. F. Stanton, R. J. Bartlett, and C. M. L. Rittby,/. Chem. Phys., 97,5560 (1992). Fock Space Multireference Coupled-Cluster Theory for General Single Determinant Reference Functions. 

A Fock space multireference coupled cluster method was described by Rittby and Bartlett <91TCA469>, applied to the calculation of ionization potentials and excitation energies of 1,2,4,5-tetrazines, and compared with conventional ab initio calculations and experimental results. 

The Fock space multireference CC methods and the intermediate Hamiltonian techniques (see e.g. Refs. [24-29] and references therein), as well as closely related similarity transformed EOMCC [30-33] are methods particularly suited for calculation of excited/ionized states with a multireference character. Recently, a Brillouin-Wigner formulation of Fock space CC has also been derived [34]. 

The Hilbert space multireference CC (see e.g. Refs. [36-40]), based on the Jeziorski-Monkhorst ansatz for the wave operator [36]. This ansatz can be either combined with the standard (Rayleigh-Schrodinger) Bloch equation, or with the Brillouin-Wigner Bloch equation (cf. Section 18.4), or with a linear combination of both... 

Continuous transition between Brillouin-W ner and Rayleigh-Schrodinger perturbation theory, generalized Bloch equation, and Hilbert space multireference coupled duster Journal of Chemical Physics 118,10676 (2003)... 

The present approach is one of the second-generation multireference perturbation treatments first opened by the CIPSI algorithm 20 years ago. Even if the spirit of these new treatments is different, mainly because the reference space is chosen on its completeness rather than on energetical criteria, it remains that the unavoidable problems of disk storage, bottleneck of variational approaches, can now be conveniently transferred to the problem of CPU time which is less restrictive. 

The methodology presented here expands the recent CASPT2 approach to more flexible zeroth-order variational spaces for a multireference perturbation, either in the Moller-Plesset scheme or in Epsein-Nesbet approach [70-72]. Furthermore, it allows for the use of a wide set of possible correlated orbitals. These two last points were discussed elsewhere [34]. 

By adopting the no-pair approximation, a natural and straightforward extension of the nonrelativistic open-shell CC theory emerges. The multireference valence-universal Fock-space coupled-cluster approach is employed [25], which defines and calculates an effective Hamiltonian in a low-dimensional model (or P) space, with eigenvalues approximating some desirable eigenvalues of the physical Hamiltonian. The effective Hamiltonian has the form [26]... 

Figure 1. Multireference problems involve both dynamical and nondynamical correlation. The nondynamical correlation is accounted for by the CASCI/CASSCF/DMRG wavefunction, which is made of multiple configurations generated in the active space with a fixed number of active electrons. The dynamical correlation is recovered on top of the multiconfigurational reference by correlating the active orbitals with orbitals in the external space (i.e., core and virtual orbitals.)...

Multireference There is no division into occupied and virtual orbitals, all orbitals appear on an equal footing in the ansatz (Equation 8). In particular, the Hartree-Fock reference has no special significance here. For this reason, we expect (and observe) the ansatz to be very well balanced for describing nondynamic correlation in multireference problems (see e.g., refs. 10-12). Conversely, the ansatz is inefficient for describing dynamic correlation, since to treat dynamic correlation one would benefit from the knowledge of which orbitals are in the occupied and virtual spaces. 

So far, we have specified the wave operator H in the BW form (15). If we adopt an exponential ansatz for the wave operator Cl, we can speak about the single-root multireference Brillouin-Wigner coupled-cluster (MR BWCC) theory. The simplest way how to accomplish the idea of an exponential expansion is to exploit the so-called state universal or Hilbert space exponential ansatz of Jeziorski and Monkhorst [23]... 

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