Hamiltonian dimer

Let us first consider tire simplest case of two identical cliromophores (or molecules) tliat are in interaction with each otlier such tliat tlieir dimer Hamiltonian can be written [30, 31] as... 

A unique example of observation of tunneling splitting is given by Oppenlander et al. [1989]. Upon replacing the host benzoic acid dimer by a thioindigo molecule of nearly the same size, the resulting bias accidentally turns out to be small, of order of A. The 4x4 Hamiltonian of the complex of two dimers and the guest molecule is... 

An important property of the electron Hamiltonian (Eq. (3.3)) is that for arbitrary hopping amplitudes the spectrum of the single-electrons slates is symmetric with respect to c=0 if is the electron amplitude on site n of an eigenstate with energy c, then the state with amplitudes —)"< > is also an eigenstate, with energy -c. In particular, in the uniformly dimerized stale, the gap between the empty conduction and the completely filled valence bands ranges from -A, to A(). 

The frustration effects are implicit in many physical systems, as different as spin glass magnets, adsorbed monomolecular films and liquid crystals [32, 54, 55], In the case of polar mesogens the dipolar frustrations may be modelled by a spin system on a triangular lattice (Fig, 5), The corresponding Hamiltonian consists of a two particle dipolar potential that has competing parallel dipole and antiparallel dipole interactions [321, The system is analyzed in terms of dimers and trimers of dipoles. When the dipolar forces between two of them cancel, the third dipole experiences no overall interaction. It is free to permeate out of the layer, thus frustrating smectic order. 

Note that both the delocalized and dimer models can be used to explain temperature dependence of the absorption spectra of R26.Phe-a RCs. This is due to the fact that the interactions [Eq. (2.48b)] among Bchls except for that between B and Ih are small compared with the diagonal element Hamiltonian in Eq. (2.48a). 

Hamiltonian, spin, dimer species, 31 95 Hamiltonian matrix elements, 25 4, 5, 7, 8, 10 Hammett acidity... 

In scheme (9) we could have substituted on the diagonal S(S -i-1)/2 for As a-ASb and BSa-BSb- As it stands, the matrix is a hybrid because it contains matrix elements on the off-diagonal and operators on the diagonal. The present form of (9) is, however, despite its awkward form, a more useful starting point for setting up a spin Hamiltonian for the description of trinuclear and tetranuclear clusters containing delocalized dimers. 

There are many systems of different complexity ranging from diatomics to biomolecules (the sodium dimer, oxazine dye molecules, the reaction center of purple bacteria, the photoactive yellow protein, etc.) for which coherent oscillatory responses have been observed in the time and frequency gated (TFG) spontaneous emission (SE) spectra (see, e.g., [1] and references therein). In most cases, these oscillations are characterized by a single well-defined vibrational frequency, It is therefore logical to anticipate that a single optically active mode is responsible for these features, so that the description in terms of few-electronic-states-single-vibrational-mode system Hamiltonian may be appropriate. 

Ginsberg et al.2917 made a detailed analysis of the average magnetic susceptibility of nickel(II) dimers and fitted the temperature dependence of the magnetic susceptibility of [Ni2(en)4X2]Y2 (X, Y = Cl, Br X = NCS, Y = I). The model Hamiltonian they used is... 

In an ESR study of the interaction of oxovanadium(IV) cupferronate with basic organic solvents, it has been shown that the coordination of basic solvent molecules leads to the destruction of the dimeric nature of the complex molecules.109 Parameters of the spin Hamiltonian of the ESR spectra of the adducts thus formed in 15 cases have been correlated with the basicity of the organic solvents. By contrast, VO(cupferronate)2 retains its diamagnetic dimeric form in the frozen solution of non-coordinating solvents such as chloroform, carbon tetrachloride, cyclohexane or benzene. 

When kT is large with respect to the energy gap, the population of each level is just one over the number of levels or functions. When kT is small with respect to the energy separation, then only the lowest level is occupied. The new energy levels S are linear combinations of the S, Ms > functions of each metal ion. The functions and energies can be calculated by the simple Heisenberg Hamiltonian, that for a dimer is... 

If the nucleus senses another metal ion with a different Am value, then the total shift will be given by the sum of two terms of the type of Eq. (6.9), each containing its own Am and C, values. Ami and Ami can be estimated from the monomeric analogs. The C, i and C,-2 values can be calculated once and for all for each system with (he help of Hamiltonian Eq. (6.3). Their values are reported in Table 6.1 for dimers involving 5, and 52 values from xh to 1k [3], These coefficients can be calculated from the following equations ... 

A combination of the two techniques was shown to be a useful method for the determination of solution structures of weakly coupled dicopper(II) complexes (Fig. 9.4)[119]. The MM-EPR approach involves a conformational analysis of the dimeric structure, the simulation of the EPR spectrum with the geometric parameters resulting from the calculated structures and spin hamiltonian parameters derived from similar complexes, and the refinement of the structure by successive molecular mechanics calculation and EPR simulation cycles. This method was successfully tested with two dinuclear complexes with known X-ray structures and applied to the determination of a copper(II) dimer with unknown structure (Fig. 9.5 and Table 9.9)[119]. 

We choose as model system the simplest conjugated polymers - the trans-polyacetylene. This compound shows a. dimerized structure with an altemance between double bond (1.35A) and single bond (1.45A) the monomer is then simply a double bond (see figure (8)). The tt electrons are assumed to be effectively describe not by the full PPP Hamiltonian but by the Extended Peierls-Hubbardmodel (EPH) [17, 14], a short version of it) for simplicity... 

When both 10ns of a Cr dimer are in the electronic ground state, the exchange interaction can be represented by the well-known Heisenberg Hamiltonian... 

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