Generalized Bloch equations

At the same time, Meissner, Kucharski and others [56,57] developed the quadratic MR CCSD method in a spin-orbital form which does not exploit the BCH formula. The unknown cluster amplitudes are calculated from the so-called generalized Bloch equation [45-47,49,64,65] (or in our language the Bloch equation in the Rayleigh-Schrodinger form)... 

Remarkably, when our general ME is applied to either AN or PN in Section 4.4, the resulting dynamically controlled relaxation or decoherence rates obey analogous formulae provided the corresponding density matrix (generalized Bloch) equations are written in the appropriate basis. This underscores the universality of our treatment. It allows us to present a PN treatment that does not describe noise phenomenologically, but rather dynamically, starting from the ubiquitous spin-boson Hamiltonian. 

For generating the RS expansion, it is very suitable to use the so-called generalized Bloch equation... 

Equations (3.23) and (3.24) are valid also for a model space containing several unperturbed energies, e.g. several atomic configurations. These equations will form the basis for our many-body treatment. The generalized Bloch equation is exact and completely equivalent to the Schrodinger equation for the states considered. 

In reality, the cross relaxation rate between a spin pair depends on their distances to other spins surrounding them as well. The evolution of magnetization (peak intensity or peak volume) in the presence of multiple interactions or spin diffusion is governed by the generalized Bloch equation. 

Upon using Eq. (171) in (168), we obtain our generalized Bloch equations for the components of the TLS density matrix (compare with [Cohen-Tannoudji 1992])... 

HeffP = PHQP, VeffP = PVi2P, while the generalized Bloch equation assumes the form... 

The generalized Bloch equation (12) is the basis of the RS perturbation theory. This equation determines the wave operator and, together with Eq. (11), the energy corrections for all states of interest especially, it leads to perturbation expansions which are independent of the energy of the individual states, just referring the unperturbed basis states. Another form, better suitable for computations, is to cast this equation into a recursive form which connects the wave operators of two consecutive orders in the perturbation V. To obtain this form, let us start from the standard representation of the Bloch equation (16) in intermediate normalization and define... 

This is the recursive form of the generalized Bloch equation. In a similar way, we can separate the effective Hamiltonian and effective interaction (15) due to the powers of V. Despite the fact, that the effective Hamiltonian (15) is not hermitian in intermediate normalization, we can diagonalize the corresponding (Hamiltonian) matrix and shall obtain (always) real energies, as they represent the exact energies of fhe system. This property is satisfied for each order independently. 

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)... 

This equation is termed the generalized Bloch equation. Using the resolution of the identity (2.166), the first term on the right-hand side of eq. (2.195) can be rewritten as... 

Introducing (2.196) and (2.197) into the right-hand side of the generalized Bloch equation (2.195) gives... 

The generalized Bloch equation in both (2.195) and (2.199) is completely equivalent to the original Schrbdinger equation for the states in the model space (P. It can be employed to generate a Rayleigh-Schrodinger perturbation expansion as we shall now show. 

The Rayleigh-Schrodinger perturbation expansion is obtained by substituting eq. (2.201) into the generalized Bloch equation, eq. (2.199)... 

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. 

PQP = P. The effective Hamiltonian and wave operator are connected by the generalized Bloch equation, which for a complete model space P may be written in the compact linked form [35]... 

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