Exact diagonalization

Fig. 3. (a) Partially resolved nuclear hyperfine structure in the p.SR spectrum for Mu in GaAs in an applied field of 0.3 T. The structure occurs in the line corresponding to 0 = 90° and Ms = —1/2. (b) Theoretical frequency spectrum obtained by exact diagonalization of the spin Hamiltonian using the nuclear hyperfine and electric quadrupole parameters in Table I for the nearest-neighbor Ga and As on the Mu symmetry axis. Both Ga isotopes, 69Ga and 71Ga, were taken into account. From Kiefl et al. (1987). 

Table 7.2 Comparison between the vibrational frequencies of linear triatomic molecules obtained by exact diagonalization of the Hamiltonian and the l/N (mean field) result."...

Figure 6. Log of the transfer rate vs. absolute temperature as a function of the electronic coupling (e) for A = 3, v. = vc = 450 cm 1. The values of —e are appended to the curves. The present calculations (—) are based on the Weiner method and an exact diagonalization of the model the perturbational results ( — )...

Seniority Truncated Exact Diagonalization (STED) Scheme... 

Hence, the small clusters of anisotropic rectangular lattice with n=2 have a minimal ground state spin at 2 /, =a and intermediate values of So at 21/, >a. The corresponding results of the exact diagonalization study for the small lattice clusters described by the Hamiltonian (1) are in agreement with this conclusion [26]. 

The region J23 > 0, J13 < 0 was studied by different methods in [24, 26], The exact diagonalization of finite chains shows a gap A in the excitation spectrum... 

This way of generating the EDM (-> IDM) collective modes guarantees their one-to-one correspondence to the respective IDM of reactants and, as such, may provide an alternative, convenient framework for describing the CT processes. Like the IDM, such localized EDM preserve the memory of the reactant interaction in M + and should lead to substantial hardness decoupling. This expectation is due to their resemblance to the PNM (IDM) of reactants, for which the diagonal (reactant) blocks of the relevant hardness matrix are exactly diagonal. Thus, with no external hardness interaction between the A and B subsets of EDM, it comes as no surprise that these collective, delocalized charge-displacement modes also bear some similarity to the PNM of M as a whole. 

These are basically analytic methods, but in recent years numerical methods play a major role. Among them, the simplest one is exact diagonalization in which the Hamiltonian matrix obtained with an appropriate expansion basis is numerically diagonalized. This is very elementary, but due to the bosonic character of phonons, the size of the Hamiltonian matrix increases exponentially as N and/or Ne increase. Thus it is not easy to treat the iJ (ge system with more than two sites by this method. 

Fig. 3 Inverse of the mass enhancement factor, mim, as a function of g

Perturbation theory is an extremely useful analytic tool. It is almost always possible to treat a narrow range of. /-values in a multistate interaction problem by exactly diagonalizing a two-level problem after correcting, by nondegenerate perturbation theory or a Van Vleck transformation, for the effects of other nearby perturbers. Such a procedure can enable one to test for the sensitivity of the data set to the value of a specific unknown parameter. 

Even in the simplest cases, to be able to extrapolate properties to the thermodynamic limit, we need to extend the exact diagonal-ization technique to the very limit. This involves exploiting all the... 

The major problem with exact diagonalization methods is the exponential increase in dimensionality of the Hilbert space with the increase in the system size. Thus, the study of larger systems becomes not only CPU intensive but also memory intensive as the number of nonzero elements of the matrix cdso increases with system size. With increasing power of the computers, slightly larger problems have been solved every few years. To illustrate this trend, we consider the case... 

The exact diagonalization method has been widely exploited in the study of polyenes as well as small conjugated molecules. It has also been employed in studying spin systems and systems with interacting fermions and spins such as Kondo lattices. These studies have been mainly confined to low- dimensions. The exact diagonalization techniques also allow bench-marking various approximate many-body techniques for model quantum cell Hamiltonians. 

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