Bloch’s effective Hamiltonian

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. 

Recent developments include exact [12-14, 44, 90, 91] and approximate [14, 90, 92-94] iterative schemes to determine Hg, the intermediate Hamiltonian method [21, 24, 95], the use of incomplete model spaces [43, 44] and some multireference open-shell coupled-cluster (CC) formalisms [16-20, 96, 97]. Only some eigenvalues of the intermediate Hamiltonian H, are also eigenvalues of H. The corresponding model eigenvectors of H, are related to their true counterparts as in Bloch s theory. Provided effective operators a are restricted to act solely between these model eigenvectors, the possible a definitions from Bloch s formalism (see Section VI.A) can be used. 

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... 

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. 

The transferability of a valence effective Hamiltonian defined on H2 to H clusters therefore faces a series of basic difficulties, which leaves little hope of success. The situation would be even worse of course if one dealt with boron or carbon atoms since for C2 already one should introduce strongly hybridized (for instance C(p ) + C(p )) or m tiply ionic (for instance C -I- C (s p )) states which are unbound. The choice of the target space is already impossible on the diatom, and the definition of an exact (Bloch, des Cloizeaux,...) effective Hamiltonian from knowledge of the spectrum of the diatom is either impossible or perfectly arbitrary. Even if it were possible, the treatment of B4 or C4 would introduce some multiply ionic valence determinants for which the assessment of an effective energy would be impossible. 

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