Brillouin-Wigner Bloch equations

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... 

We emphasize that the wave operator (A) is applied to the /xth state only. It is state specific. Equation 2.29 is termed the Bloch equation [7] in Brillouin-Wigner form. Having introduced the Brillouin-Wigner wave operator, we turn our attention now to the corresponding reaction operator (A). 

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. 

Brillouin- Wigner expansions in quantum chemistry Bloch-like and Lippmanri-Schwinger-like equations... 

BRILLOUIN-WIGNER EXPANSIONS IN QUANTUM CHEMISTRY BLOCH-LIKE AND LIPPMANN-SCHWINGER-LIKE EQUATIONS... 

The Bloch-like equation in Brillouin-Wigner form... 

In the previous section, we have given the Brillouin-Wigner perturbation expansion for the exact wave function for state a developed with respect to some single reference or multireference model function In this section, we define the Brillouin-Wigner wave operator and the corresponding Bloch-like equation [64]. 

By combining equation (41), the Bloch-like equation in Brillouin-Wigner form, with equation (53), the definition of the reaction operator, a Lipp-mann-Schwinger-like equation [122,123] is obtained... 

In the metal, each of the electrons is subject to the influence of all the nuclei and all the other electrons. In view of the periodic arrangement of the atoms, that potential is periodic, becoming infinite at each nucleus and minimal at the points furthest from the nuclei. We cannot hope to solve the Schrodinger equation for so complex a system. Certain calculations have been performed in specific cases - in particular, by Bloch, Brillouin, Wigner, Seitz and Slater, among others. 

Equation (2.133) is the Bloch equation [11]. Together with eq. (2.131), it provides the fundamental equation of the generalized Brillouin-Wigner perturbation theory. 

We do not propose to describe here the details of specific applications of Brillouin-Wigner methods to many-body systems in chemistry and physics. Such details can be found in our article in the Encyclopedia of Computational Chemistry [1] and in our review entitled Brillouin-Wigner expansions in quantum chemistry Bloch-like and Lippmann-Schwinger-like equations [36]. We have established a website at... 

Equation (4.29) can be regarded as the analogue of the Bloch equation [39] in Brillouin-Wigner form. 

Each equation may be regarded as a Bloch equation in the Brillouin-Wigner form for the state a. 

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. 

Second, we derived the Bloch equation starting from the Brillouin-Wigner perturbation expansion, but, in contrast to the approach described by Brandow in his review [74] on Linked-Cluster Expansions for the Nuclear Many-Body Problem, we did not expand the denominator factors in order to remove the exact energy dependence. 

Finally, the two sets of equations given above for the wave operator (4.71) and (4.75), are entirely equivalent. Our first approach represented by the set of eqs. (4.71) may be regarded as a Bloch equation [85] in Brillouin-Wigner form. Similarly, in terms of perturbation theory, the generalized Bloch equation (4.77) may be viewed as a Bloch equation in the Rayleigh-Schrodinger form. 

We recall that the state-specific Brillouin-Wigner analogue of the Bloch equation has the form ... 

Combining the Bloch-/ite equation (4.174) and the definition of the reaction operator, eq. (4.179), gives a Lippmann-Schwinger-ftte equation [160] in Brillouin-Wigner form ... 

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