Bloch effective Hamiltonian

Joachim C, Launay JP (1986) Bloch effective Hamiltonian for the possibility of molecular switching in the ruthenium-bipyridylbutadiene-ruthenium system. Chem Phys 109 93... 

J.-P. MalrieuandN. Guihdry, Phys. Rev. B 63,5110,2001. These authors formulate a renormalization-group procedure where the renormalized Hamiltonian is defined as a Bloch effective Hamiltonian. This procedure is based on the real-space renormalization-group (RSRG) method (a) K. G. Wilson, Rev. Mod. Phys. 47, 773, 1975. (b) S.R. White and R.M. Noack, Phys. Rev. Lett. 68, 3487, 1992. 

One of the first two-step DGCI treatment was implemented in the RCI (Relativistic Cl) code proposed by Balasubramanian [72]. Although his Hamiltonian cannot be written in a simple Bloch-effective Hamiltonian form, it has neverthe-... 

Multiplying both sides of Eq. (13) on the left by Pq and using the intermediate normalization property leads to the Bloch effective Hamiltonian ... 

The above theory of the Bloch effective Hamiltonian was mainly based on two finite N -dimensional subspaces the model space Sq and the target space 5, with projectors Pq and P respectively. There is no difficulty in choosing the model space on which the reduced quantum information will be projected. The definition of the target space is not so obvious. If the energy levels associated... 

This will be the general expression of the Schmidt-orthogonalized effective Hamiltonian. If P/x a IP, the Bloch effective Hamiltonian will be non-Hermitian. The relative advantages of the Bloch, des Cloizeaux, Schmidt-orthogonalized effective Hamiltonians, or of the intermediate Hamiltonians has never been tested, especially in transfers to large systems. 

The simple projection relation between the right model eigenfunctions of Hg and their true counterparts is an appealing aspect of Bloch s formalism. However, the non-Hermiticity of the resulting effective Hamiltonian represents a strong drawback, as discussed in Section VII. This has led many, beginning with des Cloizeaux [7], to derive Hermitian effective Hamiltonians, des Cloizeaux s method transforms the lag)ol not the... 

An alternative possibility for the construction of an effective Hamiltonian is via partitioning technique [102]. The original formulation of Bloch [103] also differs considerably from the one presented here [89]. ... 

Both VU and SU MR CC methods employ the effective Hamiltonian formalism the relevant cluster amplimdes are obtained by solving Bloch equations and the (in principle exact) energies result as eigenvalues of a non-Hermitian effective Hamiltonian that is defined on a finite-dimensional model space Mq. An essential feature characterizing this formalism is the so-called intermediate or Bloch normalization of the projected target space wave functions I f, ) with respect to the corresponding model space configurations 1, ), namely = 8 (for details, see, e.g. Refs. [172,174]). 

In the remainder of this section, the Bloch formalism (22) of the effective Hamiltonian (23) will be first presented and then applied to the construction of an active space (24) without using arbitrary selection criteria. 

Effective Hamiltonian theory. Following the formalism developed by Bloch, we will define a new operator called the effective Hamiltonian, Hc(t, given by... 

This original approach, first proposed by Bloch (22) in 1958, follows a pedagogical approach to obtain both the wave operator and the effective Hamiltonian. However, from a computational point of view, the perturbative expansion [Eq. (41)] frequently diverges and the first few terms give only an approximation to the exact solution. In vibrational... 

At each iteration step, this procedure involves the evaluation of XN and the effective Hamiltonian HN+l which is then used to calculate XN+1, and so on. The originality of this iteration method comes from the presence of the perturbed diagonal elements (i.e., °(/ // /)0, °(a A]a)0) rather than their zero-order expressions [see Eq. (41)] in Bloch s perturbative expression. The study of simple systems (31,32) has shown that this procedure may accelerate the convergence of the series. 

It would be interesting to establish the possible relationships between the Bloch formalism for constructing effective Hamiltonians and other perturbative approaches, including Van Vleck perturbation theory (161). 

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