Multi-reference Brillouin-Wigner

Multi-reference Brillouin-Wigner theory overcomes the intruder state problem because the exact energy is contained in the denominator factors. Calculations are therefore state specific , that is they are performed for one state at a time. This is in contrast to multi-reference Rayleigh-Schrddinger perturbation theory which is applied to a manifold of states simultaneously. Multi-reference Brillouin-Wigner perturbation theory is applied to a single state. Wenzel and Steiner [105] write (see also [106]) ... 

When we consider the application of multi-reference Brillouin-Wigner methods to many-body systems, two distinct approaches can be taken which we consider now in turn ... 

We are now ready to define the wave operator for the multi-reference formalism. By analogy with the single-reference wave operator which we defined in Eq. 2.27, the multi-reference Brillouin-Wigner wave operator is defined as... 

Equation 2.50 is the Bloch equation for multi-reference Brillouin-Wigner theory. It should be emphasized that the wave operator (A) has the subscript /z indicating that it is state-specific. 

The multi-reference Brillouin-Wigner reaction operator is defined as... 

Multi-reference Brillouin-Wigner coupled-cluster method with a general model space Molecular Physics 103,2239 (2005)... 

On the generalized multi-reference Brillouin-Wigner coupled duster theory Journal of Physics B Atomic, Molecular Qptical Physics 34 4259 (2001)... 

Whereas the multi-reference Rayleigh-Schrodinger perturbation theory approximates a manifold of states simultaneously, the multi-reference Brillouin-Wigner perturbation theory approach is applied to a single state - it is said to be state-specific . The multi-reference Brillouin-Wigner perturbation theory avoids the intruder state problem. If a particular Brillouin-Wigner-based formulation is not a valid many-body method, then a posteriori correction can be applied. This correction is designed to restore the extensivity of the method. This extensivity may be restored approximately... 

In this section we briefly survey the basic formalism of the multi-reference Brillouin-Wigner perturbation theory. This will serve to introduce our notation. 

Multi-reference Brillouin-Wigner coupled cluster theory... 

Here we shall discuss two versions of the multi-reference Brillouin-Wigner coupled cluster (Bwcc) theory which are based on the use of effective Hamiltonians. 

These two possible formulations of multi-reference Brillouin-Wigner coupled cluster theory are discussed further below. In Section 4.2.2.1, we present a multiroot formulation of Brillouin-Wigner theory. This formalism is employed in Section 4.2.2.2 to develop a multi-root, multi-reference Brillouin-Wigner coupled cluster theory, using a Hilbert space approach. In Section 4.2.2.3, we discuss the basic approximations employed in the multi-reference Brillouin-Wigner coupled cluster method. 

It should be noted that the wave operator 17 no longer depends on the exact energies and therefore represents a much more suitable formulation for practical calculations. Within the multi-reference Brillouin-Wigner perturbation theory, we have been able to construct a multi-root wave operator together with an effective Hamiltonian operator, Jfeff, which formally possess the same properties as those employed in the multi-reference theories based on the Bloch equation. For this reason, the adjective multi-root is clearly not necessary here. 

If we now adopt an exponential ansatz for the wave operator 17, then we are lead to the multi-reference Brillouin-Wigner coupled-cluster (MR Bwcc) theory. 

In the previous section, we have presented three different computational schemes for the determination of the multi-root wave operator within the multi-reference Brillouin-Wigner coupled cluster MR bwcc method. Whilst the first scheme, which is characterized by the set of eqs. (4.84), may be regarded as a pure MR BWCc approach (it must be solved simultaneously with the equation for the effective Hamiltonian), the other two schemes, which are characterized by the set of eqs. (4.85) and (4.87), respectively, represent standard MR BWCC approaches and, in these cases, the adjective Brillouin-Wigner is inappropriate (the wave operators do not depend on exact energies). There is no doubt that of the three schemes the most computationally tractable approach is the second. This scheme is employed, for example. 

Single-root formulation of multi-reference Brillouin-Wigner perturbation theory... 

In Section 4.2.3.1, we have defined the wave operator, 12, in the Brillouin-Wigner form (4.92). If we adopt an exponential ansatz for the wave operator, 12, we can develop the single-root (state-specific) multi-reference Brillouin-Wigner coupled-cluster (MR Bwcc) theory. This is the purpose of the present section. 

We turn now to the calculation of the effective Hamiltonian (4.98) for single-root multi-reference Brillouin-Wigner coupled cluster theory. Using the Hilbert space exponential ansatz of Jeziorski and Monkhorst, expression (4.103), the off-diagonal... 

The application of the Brillouin-Wigner coupled cluster theory to the multireference function electron correlation problem yields two distinct approaches (i) the multi-root formalism which was discussed in Section 4.2.2 and (ii) the single-root formalism described in the previous subsections of this section. Section 4.2.3. The multiroot multi-reference Brillouin-Wigner coupled cluster formalism reveals insights into other formulations of the multi-reference coupled cluster problem which often suffer from the intruder state problem which, and in practice, may lead to spurious... 

In Brillouin-Wigner coupled cluster theory, the simple a posteriori correction described above is exact in the case of the single-reference formalism. In the state-specific multi-reference Brillouin-Wigner coupled cluster theory, the simple a posteriori correction is approximate. An iterative correction for lack of extensivity has been studied by Kttner [38], but this reintroduces the intruder state problem. 

Multi-reference Brillouin-Wigner perturbation theory for limited configuration interaction... 

We turn, in this section, to the multi-reference Brillouin-Wigner perturbation theory. We divide our discussion into two parts. In Section 4.4.2.1, we survey the basic theoretical apparatus of multi-reference second-order Brillouin-Wigner perturbation theory. In Section 4.4.3, we describe an a posteriori correction to multi-reference Brillouin-Wigner perturbation theory. 

If we restrict the order of perturbation admitted in (4.241) then we realize a finite order multi-reference Brillouin-Wigner perturbation theory. Specifically, if we neglect terms of order A are higher, we are led immediately to the second-order theory for which the matrix elements of the effective Hamiltonian (4.239) take the form ... 

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